• Definable real number — defines definable reals by unique specification in a formal language, states there are only countably many formulas, and states that the reals are uncountable. Re-read raw on 17 September 2026 for the containment the chapter’s inference rests on: “All analytical numbers, and in particular all computable numbers, are definable in the language of set theory. Thus the real numbers definable in the language of set theory include all familiar real numbers such as 0, 1, π, e, et cetera, along with all algebraic numbers.”
• Cantor’s diagonal argument — gives Cantor’s 1891 diagonal proof that the real numbers are uncountable: “Georg Cantor published this proof in 1891, but it was not his first proof of the uncountability of the real numbers, which appeared in 1874.”
• Author’s declaration and own computation — The notation dates to V17, 11 September 2026; labels were widened from positive to nonzero rationals on 17 September 2026. Clarified 30 September 2026: “Ruis numbers” names ℝ ∖ D, while ᚏ_b := x/b presents a chosen member. Labels used together in one formula share x, and this book keeps that choice fixed throughout; a formula outside this book may choose another x. If x/b were definable without parameters in the original setting, multiplying by the definable rational b would define x there. Also x ≠ 0, so x/b = x/c implies b = c. Thus distinct rational labels sharing x give distinct undefinable values. These calculations establish relationships without specifying those values.
• Null set, Wikipedia — “For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null.”
• Chaitin’s constant, Wikipedia — states that Omega is definable though uncomputable: “Ω is among the few definable algorithmically random numbers and is the best-known algorithmically random number, but it is not at all typical of all algorithmically random numbers.”
• Ogham, Wikipedia — “Ruis, Old Irish Ruis means ‘red’ or ‘redness’, glossed as trom ‘elder’.”; “It was also used much less frequently to record the Pictish language in what is now Scotland.”

• Hamkins, Linetsky and Reitz, Pointwise Definable Models of Set Theory — abstract and bibliographic record confirm the pointwise-definable-model result; posted to arXiv in 2011 and published in The Journal of Symbolic Logic 78(1) (2013), 139–156. Introduction, pp. 1–3: definitions are relative to a language and structure and use no parameters. Page 2 discusses ⟨V_ω+ω, ∈⟩ as a rich background for classical mathematical definitions. This book adopts that fixed set structure and counts its definable reals externally; this choice is not the paper’s definition of “Ruis.” The pointwise-definable-model result explains why the argument cannot be transferred indiscriminately to every model. p. 2: “In any fixed structure M in a countable language, including the higher-order set-theoretic structures ⟨Vα, ∈⟩, the math-tea argument seems fine”; p. 3: “Clearly, pointwise definable models in a countable language must be countable, since there are only countably many definitions.” and “The first two claims have fairly soft proofs and might be considered to be a part of the mathematical folklore. Statement (1) could be credited to Myhill [Myh52].”
• Definable real number — lines on model-relative definability, non-expressibility in ZFC, and countable models in which all reals are definable: “there are countable models of ZFC in which all real numbers, all sets of real numbers, functions on the reals, etc. are definable.”

• Real number — confirms the real-number system, its order/completeness presentation, and standard notation. Re-read raw on 17 September 2026 for the dates: “Beginning with Richard Dedekind in 1858, several mathematicians worked on the definition of the real numbers … leading to the publication in 1872 of two independent definitions of real numbers, one by Dedekind, as Dedekind cuts, and the other one by Georg Cantor, as equivalence classes of Cauchy sequences.”

• Extended real number line — confirms adjoining the two infinities and the standard arithmetic conventions and undefined forms.
• A. Sengupta, Lecture 2. Measure and Integration, Math 7312 notes, Louisiana State University — “0 · ∞ = 0” among the conventions; “The convention 0·∞ = 0 might seem like an arbitrary choice, but it is the really the right one : it will be useful in integration and there it will appear as the natural choice.”
• Tropical semiring, Wikipedia — min-plus semiring: “The identity element for ⊕ is +∞, and the identity element for ⊗ is 0.”

• Tropical semiring — gives both min-plus and max-plus operations and their identities, and identifies tropical geometry as an application.
• Extended real number line, Wikipedia — “The arithmetic operations of ℝ can be partially extended to ℝ̄ as follows:” including a/±∞ = 0 for a ∈ ℝ.
• Semiring, Wikipedia — definition: “(R, +) is a commutative monoid with an identity element called 0,” listing (a + b) + c = a + (b + c) and a + b = b + a.
• Tropical mathematics, David Speyer and Bernd Sturmfels, arXiv — states distributivity (min-plus convention): “The distributive law holds for tropical addition and tropical multiplication: x ⊙ (y ⊕ z) = x ⊙ y ⊕ x ⊙ z.”
• Semifield, Wikipedia — “a semifield is a semiring (S,+,·) in which all nonzero elements have a multiplicative inverse.” and “(R, max, +)” ... “These are called the tropical semiring.”
• Tropical geometry, Wikipedia — “Thus a tropical polynomial F is the minimum of a finite collection of affine-linear functions in which the variables have integer coefficients, so it is concave, continuous, and piecewise linear.”

• Tropical semiring — attributes the name to Imre Simon and states applications and the connection with tropical geometry.
• Tropical geometry — dates the consolidation of the field to the late 1990s, names Mikhalkin and Kontsevich, and records that Jean-Éric Pin credits the adjective to Dominique Perrin while Simon credited it to Christian Choffrut, and describes Simon as “the Hungarian-born Brazilian computer scientist.”
• Jean-Éric Pin, Tropical Semirings, in J. Gunawardena (ed.), Idempotency, Publications of the Newton Institute 11, Cambridge University Press, 1998, pp. 50–69 — read as extracted PDF text. §2.2: “The Min-Plus semiring is M = (N ∪ {∞}, min, +) … This semiring was introduced by Simon [41] in the context of automata theory (it is also a familiar semiring in Operations Research).” Its reference [41] is “I. Simon, Limited subsets of a free monoid, in Proc. 19th Annual Symposium on Foundations of Computer Science, Piscataway, N.J., 1978, pp. 143–150, IEEE,” which Crossref confirms at doi 10.1109/sfcs.1978.21, October 1978. §1: “These semirings were baptized tropical semirings by Dominique Perrin in honour of the pioneering work of our brazilian colleague and friend Imre Simon, but are also commonly known as (min, +)-semirings.”
• David Speyer and Bernd Sturmfels, Tropical mathematics, Mathematics Magazine 82 (2009), 163–173 — the expository paper behind the chapter’s naming of Sturmfels among the founders; posted to the arXiv in 2004 and published in 2009. Its introduction (arXiv version, read as extracted PDF text): “The adjective ‘tropical’ was coined by French mathematicians, including Jean-Eric Pin, in the honor of their Brazilian colleague Imre Simon … There is no deeper meaning in the adjective ‘tropical’. It simply stands for the French view of Brazil.”
• N. Krivulin, Algebraic solutions of tropical optimization problems, arXiv — introduction: “Tropical (idempotent) mathematics, which is concerned with the theory and applications of semirings with idempotent addition, dates back to the early 1960’s, when a few innovative works by Pandit [47], Cuninghame-Green [11], Giffler [24], Hoffman [29], Vorob’ev [51] and Romanovskiĭ [48] made their appearance.”
• Bernd Sturmfels, Wikipedia — “Bernd Sturmfels has made contributions to a variety of areas of mathematics, including algebraic geometry, commutative algebra, discrete geometry, Gröbner bases, toric varieties, tropical geometry, algebraic statistics, and computational biology.”
• Min-plus matrix multiplication, Wikipedia — read as raw wikitext: the distance product “is standard matrix multiplication for the semi-ring of tropical numbers in the min convention,” and “This operation is closely related to the shortest path problem.”

The chapter also dates max-plus algebra, as distinct from Simon’s 1978 min-plus semiring, to the early 1960s. We keep that: Krivulin’s survey dates tropical (idempotent) mathematics to the early 1960s. Extended 17 September 2026. Until then the 1978 date was asserted in four places — the chapter prose, #tropical, #tropplus, and #minplus — and appeared in no claim line and in no cited source; Pin’s survey now carries the year, the min-plus attribution, and the automata context in one sentence. The chapter’s “took off around 2000 with Mikhalkin, Sturmfels, and others” and this entry’s “late 1990s” were never in conflict, but the entry named different people from the page; since the compendium is the source, the entry widened to it rather than the other way round.

• Richard’s paradox — confirms the diagonal construction and Jules Richard’s 1905 publication. Re-read on 29 September 2026 for the resolution the chapter now states: “The property D(φ) that a formula φ of ZFC defines a unique real number is not itself expressible by ZFC, but must be considered as part of the metatheory used to formalize ZFC.”

• Set (mathematics) — confirms the standard set-membership and set-operation meanings.
• Set difference — confirms A∖B as the elements of A not in B.
• Defined, Wolfram MathWorld — gives A := B as one of the standard notations for A is defined to be B.
• Glossary of mathematical symbols, Wikipedia — entry for ≜, ≝ (=def), and :=: “Any of these is sometimes used for naming a mathematical object.” The forms x ≜ E, x ≝ E, x := E and E =: x “are each an abbreviation of the phrase “let x = E.””
• Definable real number, Wikipedia — “there are only countably many formulas, and so only countably many elements of M can be definable over M.”

• Ordinal arithmetic — confirms ordinal multiplication as repeated copies and the noncommutative behavior of ordinal arithmetic.
• Surreal number — confirms that ordinals embed in the surreals and that surreal arithmetic differs from ordinal arithmetic.

• Own computation — Using the glossary’s ε = 1/ω, (x/4.16)/(−ω) = −(x/4.16)·(1/ω) = −(x/4.16)·ε. Equality with ε would force x = −4.16, which is definable and therefore excluded by the governing sentence.

• Berry paradox — G. G. Berry (1867–1928), junior librarian at the Bodleian; Russell the first to discuss the paradox in print; cites Russell, Les paradoxes de la logique, Revue de métaphysique et de morale 14 (1906) 627–650.
• Russell, Mathematical Logic as Based on the Theory of Types (1908), p. 223 — the least integer not nameable in fewer than nineteen syllables denotes 111,777 and is itself a name consisting of eighteen syllables, credited to Berry in a footnote.
• Berry Paradox, Wolfram MathWorld — gives the nineteen/eighteen syllable form and attributes it to Russell, American Journal of Mathematics 30 (1908), 223.
• G. J. Chaitin, Computational Complexity and Gödel’s Incompleteness Theorem, ACM SIGACT News No. 9 (April 1971), pp. 11–12 (PDF, Charles University) — header “ACM SIGACT News, No. 9 (April 1971), pp. 11–12”; abstract: “The proof resembles Berry’s paradox, not the Epimenides nor Richard paradoxes.”

The card had given the nineteen-syllable wording as Russell’s of 1906. Russell published the paradox in 1906 in a different wording and gave the quoted one in 1908; the card now says both. The fifty-seven letters are countable from the phrase itself.

• Earliest Uses of Symbols of Set Theory and Logic (Jeff Miller, MacTutor) — ∈ in Peano, Arithmetices principia nova methodo exposita (1889), as an abbreviation for est — re-read raw on 17 September 2026, the page says “He stated that the symbol was an abbreviation for est; the entire work is in Latin” and “Peano’s symbol for membership appears to be a lunate (or uncial) epsilon, and not the stylized epsilon that is now used”; ∪ and ∩ in Peano, Calcolo geometrico secondo l’Ausdehnungslehre di H. Grassmann (1888).
• Assignment (computer science) — x := expr is given as originally Algol 1958, popularized by Pascal.

The card for := had credited the notation to Algol 60 in 1960. The notation is Algol 58’s; Algol 60 carried it on, and the card now says so.

• Georg Cantor, Wikiquote — read as raw HTML: under the heading “Grundlagen einer allgemeinen Mannigfaltigkeitslehre [Foundations of a General Theory of Aggregates] (1883)” it gives “The essence of mathematics lies entirely in its freedom,” with the variant translation “The essence of mathematics is in its freedom.”

Added 17 September 2026 to source the chapter epigraph, which carried no marker. One wording note: the German is “Das Wesen der Mathematik liegt gerade in ihrer Freiheit,” and the page prints “lies precisely in its freedom” where this source gives “lies entirely in its freedom.” Both are current renderings of gerade; the page’s is the more usual one and is kept.

• Integer, Wikipedia — defines the integers as the natural numbers together with their negatives and zero, forming a ring, and gives the blackboard-bold notation Z from German Zahlen; “The integers form the smallest group and the smallest ring containing the natural numbers.”
• Semiring, Wikipedia — carries the ring/semiring distinction the card draws: “Any field is also a ring, which in turn is a semiring in which also additive inverses exist.”

• Natural number, Wikipedia — describes the natural numbers as the counting numbers, notes the convention split on whether 0 is included, and dates Peano’s axioms to 1889.

• Rational number, Wikipedia — defines the rationals as quotients of integers forming a field and states that the set is countably infinite; the set is “closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations”; “The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: 3/4 = 0.75), or eventually begins to repeat the same finite sequence of digits over and over,” and “The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones.”
• Definable real number, Wikipedia — carries the countability of the definable reals: “This follows from the fact that there are only countably many formulas, and so only countably many elements of M can be definable over M.”
• Earliest Uses of Symbols of Number Theory, MacTutor (Jeff Miller) — carries the origin of the letter Q: “Q for the set of rational numbers and Z for the set of integers are apparently due to N. Bourbaki. ... The letters stand for the German Quotient and Zahlen.”

• Hamkins, Linetsky and Reitz, Pointwise Definable Models of Set Theory — introduction, pp. 1–3: fixed languages/structures, parameter-free definitions, and the external viewpoint.
• Curtis McMullen, Harvard Math 114 Real Analysis notes — §1.1 gives an analogous countable-set probability argument for algebraic numbers; §1.2 develops measure and countable additivity.
• Becher and Figueira, An example of a computable absolutely normal number — abstract and introduction: normality concerns digit-block frequencies and can hold for a computable number.

Probability 1 is exact in this ideal model and does not make a definable outcome logically impossible. A physical dart or a computer selecting finite-precision outputs uses a different model.

• Asaf Karagila, Name that number, 21 September 2015 — the paragraph beginning “And we can prove, now…” uses unnameable alongside undefinable. The later discussion distinguishes intuitive nameability from mathematical definability and explains dependence on the language and setting. This supports the vocabulary, not an absolute category beyond every possible language.

• Anne Newstead and James Franklin, On the Reality of the Continuum, Philosophy 83 (2008), pp. 117–128 — printed p. 121 / fifth PDF page uses indefinables and compares them to the dark matter of mathematics. The latter is an analogy, not a formal class called “dark numbers.” This citation supports those descriptions; the book’s mathematical definition remains the one declared on D’s card.

• Chivall, Lladó-Buisán, Howell and Evison, Radiocarbon dating of the Bakhshālī manuscript, University of Oxford (2024) — and the report PDF downloaded and read: Table 1 gives the five folios at 95.4% confidence as 773–986 (15), 901–1032 (16), 890–1014 (23), 885–995 (33), and 774–991 (55) cal AD; the OxCal phase model gives 799–892 cal AD for the start of production and 900–1102 cal AD for the end; the second determination on Folio 16 “demonstrated that the initial determination was inaccurate” and Folio 17 was excluded as an outlier.
• Bakhshali manuscript, Wikipedia — summarizes the 14 October 2024 revision. Its table prints Folio 16 as 931–1032; the Oxford report itself gives 901–1032, and the report is the primary source, so the chapter follows the report. It also gives the superseded 2017 dates: “224–383 CE for folio 16 (Indo-Scythian), 680–779 CE for folio 17 (Turk Shahis), and 885–993 CE for folio 33 (Saffarid dynasty).”

• Bakhshali manuscript, MacTutor History of Mathematics — confirms the 1881 discovery, birch bark, approximately 70 surviving leaves, and presentation to the Bodleian in 1902.
• Bakhshali manuscript, Wikipedia — confirms the 70-leaf description, discovery near Peshawar, and Bodleian location.

• Zero, MacTutor History of Mathematics — describes Babylonian base 60, the two-wedge placeholder around 400 BC, and Ptolemy’s sexagesimal placeholder around 130 AD; on the omicron derivation: “Neugebauer, however, dismisses this explanation since the Greeks already used omicron as a number - it represented 70.”
• Earliest Uses of Symbols for Constants, MacTutor History of Mathematics — Burton: “there is no evidence that Ptolemy regarded the symbol as a number by itself that could enter into computations with other numbers”; Berggren: “Papyri from the period when Ptolemy lived show a small ‘o’ with a bar over it as the symbol for 0”; and “Babylonians in the Seleucid period (300 BC onward) used a symbol for zero, mainly in astronomical texts, and never in final positions.”
• Raymond Mercier, Ancient Greek Numbers: A Greek Zero Sign, Unicode Technical Committee document L2/04-054 — read as extracted PDF text. Under ORIGIN OF THE SYMBOL: “It is commonly supposed that these forms arise from an abbreviation of οὐδέν, but given the rather arbitrary and opaque character of the majority of the symbols, one might well doubt this rather simple explanation.” Under Ninth century: “In ninth century Byzantine manuscripts the character for zero differs little from the papyri … the zero consists of a full size uncial omicron with a macron.” Under Fourteenth century: “In the some late medieval manuscripts the form of the symbol has altered considerably.”
• Wikipedia, Greek numerals, § Zero — read as raw HTML: Hellenistic astronomers limited each sexagesimal position to 59, “which uses only the letters up through nu (ν) and included otherwise unused omicron (ο) as a special symbol for zero,” a system “probably adapted from Babylonian numerals by Hipparchus c. 140 BCE” and then used by Ptolemy.
• 0, Wikipedia — the Babylonian placeholder never stood alone: “Significantly, however, these placeholder signs were not considered a numerical value, they were never used alone”
• Sexagesimal, Wikipedia — carries the Sumerian origin and the Babylonian inheritance: “It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form—for measuring time, angles, and geographic coordinates.”

Extended 17 September 2026 for the #greekzero card, which carried the origin dispute and a Byzantine transmission claim on no citation at all. The card’s sentence that the sign “had turned into a plain omicron” in tenth-century Byzantine copies did not survive the check and was replaced by what Mercier’s survey shows. The card names Otto Neugebauer as the scholar who rejected the οὐδέν derivation; that is the standard attribution and Wikipedia cites Neugebauer, The Exact Sciences in Antiquity, 2nd ed., Dover 1969, pp. 13–14 and plate 2, for the Greek zero — but those pages were not read in this pass, so the doubt is cited here to Mercier, who states it in terms, and the Neugebauer attribution is flagged for a later pass to confirm.

• Zero, MacTutor History of Mathematics — confirms Maya base 20, a zero symbol by 665, and use of zero before the place-valued system.
• Earliest Uses of Symbols for Constants, MacTutor History of Mathematics — dates the oldest listed Maya zero artifact without positional notation to AD 357 and the oldest listed positional zero to AD 665.
• 0, Wikipedia — gives Stela 2 at Chiapa de Corzo with a Long Count date of 36 BC as the oldest, and says “it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs.”
• Maya numerals, Wikipedia — the dot-and-bar notation: “The numerals are made up of three symbols: zero (a shell), one (a dot) and five (a bar).”
• Mesoamerican Long Count calendar, Wikipedia — the Epi-Olmec earliest dates and Stela C: “La Mojarra Stela 1, the Tuxtla Statuette, Tres Zapotes Stela C and Chiapa Stela 2 are all inscribed in an Epi-Olmec, not Maya, style.” and “This is the second oldest Long Count date yet discovered. The numerals 7.16.6.16.18 translate to September 1, 32 BCE (Gregorian).”
• Mayan mathematics, MacTutor History of Mathematics — “Almost certainly the reason for base 20 arose from ancient people who counted on both their fingers and their toes.”; “it changes in the third place and this denotes the number of 360’s up to 19 instead of the number of 400’s.”

• Brahmagupta, Wikipedia — confirms composition in 628 and identifies the work as the first book giving arithmetic rules applying to zero and negative numbers.
• Zero, MacTutor History of Mathematics — describes Brahmagupta’s rules and his treatment of division by zero.
• Brahmagupta, MacTutor History of Mathematics — negatives as debts: “He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers)”
• 0, Wikipedia — the Sanskrit name: “Sifr evolved to mean zero when it was used to translate śūnya (Sanskrit: शून्य) from India.”
• Earliest Known Uses of Some of the Words of Mathematics (Z), MacTutor History of Mathematics — the descent of the names: “ZERO, like its synonym CIPHER, derives from the Sanskrit sunya.”

• Zero, MacTutor History of Mathematics — confirms the 876 Gwalior inscription (“The first record of the Indian use of zero which is dated and agreed by all to be genuine was written in 876”), the garden dimensions and the garlands; the 683 Khmer zero is on the Earliest Uses line.
• Earliest Uses of Symbols for Constants, MacTutor History of Mathematics — confirms the 683 Khmer point and discusses the Gwalior date and its priority claims.
• Frank J. Swetz, Shaharir bin Mohamad Zain and Iwan Pranoto, Mathematical Treasure: Indonesian Zeros, MAA Convergence — through the Internet Archive, the live MAA address now returning HTTP 403. Three Sriwijayan stones, “known as Kedukan Bukit, Talang Tuo, and Kota Kapur, bear the numerals 604, 606, and 608 respectively, denoting years as reckoned in the Hindu Saka era calendar, which translate into our Common Era calendar as the dates 683, 684, and 686,” and “the stone’s zero symbol thus corroborates the evidence of an early zero.” On the reading of the date: in his 1930 paper Coеdes “hesitantly interpreted the numerical symbol as 605” with a footnote saying that the units digit “ne peut être que 4 ou 5, mais 5 est plus probable,” and he read the Sambor inscription as 605 as well.
• 0, Wikipedia — the temple: “a small circle, appears on a stone inscription found at the Chaturbhuj Temple, Gwalior, in India, dated AD 876.”
• Chaturbhuj Temple, Gwalior, Wikipedia — the temple “is a Hindu temple excavated in a rock face in the Gwalior Fort”; “One of the temple’s inscriptions contains the earliest known inscription of the circular symbol “O,” to represent zero.”

Gwalior is often called the oldest Indian zero outright. MacTutor is more careful: it is “the first record of the Indian use of zero which is dated and agreed by all to be genuine”. Older Indian candidates exist — Michael Closs lists inscriptions of AD 672, 683, and 684, and there are copper plates that have been read as sixth-century — but their dates or their authenticity are questioned. So the chapter says undisputed rather than oldest. The Khmer stone is not uniquely oldest either: Miller lists a small circle for zero in an Old Malay inscription at Kedukan Bukit, Sumatra, in the same year, 683, and the two cannot be ranked, because K-127 records only the Śaka year and the fifth day of the waning moon, with no month. The chapter therefore calls Sambor one of the two oldest rather than the oldest.

• Zero, MacTutor History of Mathematics — confirms Qin Jiushao’s 1247 circle and Fibonacci’s use of zero in Liber Abaci around 1200.
• Earliest Known Uses of Some of the Words of Mathematics (Z), MacTutor History of Mathematics — traces Sanskrit śūnya through Arabic ṣifr to Latin zephirum in Fibonacci’s 1202 Liber Abaci and later cipher/zero forms.
• 0, Wikipedia — “Qín Jiǔsháo’s 1247 Mathematical Treatise in Nine Sections is the oldest surviving Chinese mathematical text using a round symbol ‘〇’ for zero”, and “the origin of this symbol is unknown; it may have been produced by modifying a square symbol”.
• On the First Printed Edition of Mathematical Book in Nine Chapters (1842), Springer — the 1842 Yijiatang Congshu edition is the first printed edition of Qin Jiushao’s treatise, which until then circulated in manuscript.
• Suzhou numerals, Wikipedia — “The Suzhou numeral system is the only surviving variation of the rod numeral system.” and “Zero is represented by the character for zero (〇).”
• 〇, Wiktionary — Chinese: “It is considered a placeholder for a missing Chinese character. Used to express a zero digit in a number since at least the (c.) 12th century, perhaps even earlier than the concept “zero” appeared in the Chinese language. Also written as □ in early literature.” Japanese: “Used to write zero in kanji positional notation.”
• Chinese numerals, Wikipedia — present-day use: “Usually 零 is preferred, but in some areas, 〇 may be a more common informal way to represent zero.”

• Dionysius Exiguus’ Easter table, Wikipedia — confirms the table was constructed in 525 and that Bede’s Easter cycle developed from it; “This table introduced the first use of the modern system of numbering years,” its first year being “Anno Domini 532,” and “With his Easter table Dionysius introduced the Christian era.”
• Bede’s Easter cycle, Wikipedia — confirms Bede’s 725 extension of Dionysius’s Easter table.
• 0, Wikipedia — “The initial ‘N’ was used as a zero symbol in a table of Roman numerals by Bede—or his colleagues—around AD 725”, and gives Dionysius Exiguus’s 525 table with the Latin nulla (“none”); it does not say which word the N stands for (the Roman numerals line: “the initial of nulla or of nihil”).
• Roman numerals, Wikipedia, § Zero — the Roman system “lacked a numeral for the number zero itself”; “Dionysius Exiguus used nulla alongside Roman numerals in a manuscript from 525 AD. About 725, Bede or one of his colleagues used the letter N, the initial of nulla or of nihil (the Latin word for “nothing”) for 0, in a table of epacts, all written in Roman numerals.”
• The Reckoning of Time, Wikipedia — “The Reckoning of Time (Latin: De temporum ratione, CPL 2320) is a Medieval Latin treatise written by the Northumbrian monk Bede in 725.”

The N is attributed to Bede here, as the chapter attributes it; the fullest source available says “by Bede—or his colleagues—”, and no source seen names the hand that wrote it.

• Own computation — The displayed product contains thirteen zero-valued factors; 0 × 0 × … × 0 = 0, with no denominator present.

• 零, Kanji Data Base (Kanjipedia) — confirms the character and its Japanese readings/meanings.
• Zero, Treccani — the history of the word, not the Chinese or Japanese one: “Leonardo latinizzò in zephirum il vocabolo ṣifr usato dagli Arabi in corrispondenza al sanscrito śūnya (vuoto)”; it has nothing on 零.
• 零, Wiktionary — a phono-semantic compound of 雨 (rain) with 令; the original sense is a light rain or drizzle, with the extended senses of a remnant or fraction, and thence zero.
• Japanese numerals, Wikipedia — its table gives for 0 (零 / 〇) the on reading rei and, as preferred reading, “zero (ゼロ) (loanword from English, gairaigo)”; and “as a number, it is only written as 0 or rei (零).”

• The Unicode Standard 16.0, Chapter 8: Europe-II — Ogham — “U+169B OGHAM FEATHER MARK and U+169C OGHAM REVERSED FEATHER MARK are used at the beginning and the end of Ogham text, particularly in manuscript Ogham,” and “In some cases, only the Ogham feather mark is used, which can indicate the direction of the text.”
• UnicodeData.txt, Unicode Character Database — downloaded and read: 169B OGHAM FEATHER MARK carries General_Category Ps, open punctuation, and 169C OGHAM REVERSED FEATHER MARK carries Pe, while the twenty-six Ogham letters from 1681 OGHAM LETTER BEITH to 169A OGHAM LETTER PEITH all carry Lo. The feather mark is therefore not one of the letters and spends none.
• Own enumeration — the twenty-five-letters claim was enumerated against the book itself and holds. The twenty-six Ogham letters are U+1681 to U+169A; the glossary’s letters section reserves Ruis for the left-hand side, and the chapters carrying the remaining twenty-five run from Chapter ᚁ(B) to Chapter ᚚ(P), which is exactly twenty-five, with the prologue and the epilogue taking the two non-letter feather marks U+169B and U+169C.

The prologue’s first two sentences describe the feather mark and what it does, and its closing paragraph says the mark spends no letter and that the piece has twenty-five letters for twenty-five chapters. None of the three carried a footnote, and the V10 pass recorded the letter count as unchecked. The Unicode Standard supplies the first two; the third is an enumeration of the book’s own chapters, and it is right.

• G. Cœdès, À propos de l’origine des chiffres arabes, Bulletin of the School of Oriental Studies 6/2 (1931), 323–328 — the journal record gives author G. Cœdès, volume 6, issue 2, year 1931, pages 323–328, and DOI 10.1017/S0041977X00092806, and its abstract describes the paper as answering G. R. Kaye’s objections to an Indian origin for positional notation. The full text was not read in this pass, so this source is cited for the 1931 publication itself and not for what the paper argues about K-127.
• Amir D. Aczel, Searching for the World’s First Zero, HuffPost (15 May 2013) — the finder’s own first-person account: Georges Cœdès published the K-127 inscription in 1931; it was lost when the Khmer Rouge stole or destroyed close to 10,000 artifacts; and on 2 January 2013 Aczel found K-127 in the Angkor Conservation compound near Siem Reap. (First cited here: Aczel, The Origin of the Number Zero, Smithsonian Magazine, December 2014, which refuses automated readers).
• Amir Aczel, Wikipedia — the book: “was awarded a Sloan Foundation grant to research his book Finding Zero (2015)”

The prose sentence naming Cœdès, 1931, the Khmer Rouge years, and the 2013 rediscovery carried no footnote of its own, and neither of r-2.6’s two sources mentions Cœdès, 1931, the Khmer Rouge, or 2013; both were fetched and read in full during the V14 prose pass. Aczel’s article is first-hand for the 2013 find and secondary for the 1931 publication, which the Cambridge record confirms independently.

• Ogham, section Letter names, Wikipedia — gives the twenty primary names above letter by letter, and in its letters table óir ‘gold’, uilleann ‘elbow’, pín ‘spine; thorn’, eamhancholl ‘twin of coll’, and éabhadh ‘unknown’; it attributes the etymologies to McManus, naming A Guide to Ogam §3.15 for the etymologies of all twenty names, McManus 1988 p. 164 for idad as an artificial form of iubhar and for the eighth-century poem behind ailm, and McManus 1986 for the suggested value [y] of úath. It states flatly that “out of twenty letter names, only eight at most are the names of trees”; for Beith: “Old Irish Beithe means ‘birch-tree’, cognate to Middle Welsh bedw. Latin betula is considered a borrowing from the Gaulish cognate.”; for Sail: “Old Irish Sail means ‘willow-tree’, cognate to Latin salix”; for Fearn: “Old Irish Fern means ‘alder-tree’, Primitive Irish *wernā, so that the original value of the letter was [w]”; for Dair: “Old Irish Dair means ‘oak’ (Proto-Indo-European: *doru-)”; for Coll: “Old Irish Coll meant ‘hazel-tree’, cognate with Welsh collen,” and “Latin corulus or corylus is cognate”; for Uilleann: “Old Irish Uilleand ‘elbow’; the arboreal tradition has edleand ‘honeysuckle’”; for nGéadal: “Old Irish Gétal from the kennings has a meaning of ‘killing’, maybe cognate to gonid ‘slays’,” and “The value of the letter in Primitive Irish, then, was a voiced labiovelar”; for Straif: “Straif, Old Irish Straiph means ‘sulphur’. The Primitive Irish letter value is uncertain, it may have been a sibilant different from s, which is taken by sail, maybe a reflex of /st/ or /sw/”; and in its notes: “With regard to ailm, in the “King and Hermit” poem the hermit Marban says “caine ailmi ardom-peitet” – “beautiful are the pines that make music for me.” This is a reference to the idea that pine makes a pleasing, soothing sound as the wind passes through its needles.”; for Ceirt: “Old Irish Cert is cognate with Welsh perth ‘bush’, Latin quercus ‘oak’ (Proto-Indo-European: *perkwos). It was confused with Old Irish ceirt ‘rag’, reflected in the kennings.”; for idad: “McManus (1988, p. 164) also believes that the name idad is probably an artificial form of iubhar ‘yew’, as the kennings support that meaning.”
• Bríatharogam, Wikipedia — “edad and idhad are paired names of unknown meaning, although idhad may be a form of ‘yew-tree’, altered to make a pairing.”; “gort means “field” (cognate to garden).”; for tinne, “the word is probably cognate to Old Irish tend “strong” or tind “brilliant.””
• Forfeda, Wikipedia — for pín: “McManus states that the name Pín was probably influenced by Latin pinus (‘pine’), but a more likely explanation is that it derives from Latin spina (‘thorn’), as the kennings indicate a tree or shrub with sweet tasting fruit (therefore not a pine)”; for Éabhadh: “The name appears modelled after Eadhadh and Iodhadh.”
• BabelStone Blog, The Ogham Stones of Scotland (8 June 2013) — “The letter uath ᚆ, which corresponds to H in the manuscript tradition, is not normally found in Ogham inscriptions outside Scotland, as it appears not to have been needed for writing Primitive Irish”; “However, this letter is very common in Scottish inscriptions from the Pictish area, occuring in 12 or 13 words in eight or nine inscriptions, usually written singly at the start or end of a word, and doubled as HH ᚆᚆ in the middle of words,” its list including “BREAY/1 : NAHHTVVDDADDS,” the “decorated cross slab from Bressay in Shetland.”

The letters table used to give the book’s own meaning for each letter in its Represents column, which duplicated the Chapter column and had gone stale in places. The column now gives the letter name’s own meaning, so it needed an entry behind it. The primary scholarship is McManus; what was read here is the encyclopedia article that reproduces him, and this entry says so rather than implying the books were opened.

• Bríatharogam, Wikipedia — prints all three lists in full “as edited (in normalized Old Irish) and translated by McManus (1988),” which is the text quoted on the letter cards. Beithe: féochos foltchain, glaisem cnis, maise malach. Luis: lí súla, carae cethrae, lúth cethrae. Fern: airenach fían, comét lachta, dín cridi. Sail: lí ambi, lúth bech, tosach mela. Nin: costud síde, bág ban, bág maise. Úath: condál cúan, bánad gnúise, ansam aidche. Dair: ardam dosae, grés soír, slechtam soíre. Tinne: trian roith, smiur gúaile, trian n-airm. Coll: caíniu fedaib, carae blóesc, milsem fedo. Cert: clithar baiscill, bríg anduini, dígu fethail. Muin: tressam fedmae, árusc n-airlig, conar gotha. Gort: milsiu féraib, ined erc, sásad ile. Gétal: lúth lego, étiud midach, tosach n-échto. Straif: tressam rúamnai, mórad rún, saigid nél. Ruis: tindem rucci, rúamnae drech, bruth fergae. Ailm: ardam íachta, tosach frecrai, tosach garmae. Onn: congnaid ech, féthem soíre, lúth fían. Úr: úaraib adbaib, sílad cland, forbbaid ambí. Edad: érgnaid fid, commaín carat, bráthair bethi (?). Idad: sinem fedo, caínem sen, lúth lobair (?). The English renderings on the cards are this page’s, and the question marks are its own; the Auraicept line for Dair: “arirde dossaib duir “higher than bushes is an oak””; for Tinne, “the word is probably cognate to Old Irish tend “strong” or tind “brilliant””; for Iphín, the table gives “milsem fedo “sweetest tree”” and “amram mlais “most wonderful taste”” (milsem fedo is also one of Coll’s). For Muin, “the kennings connect this name to three different words, muin “neck, upper part of the back,” muin “wile, ruse,” and muin “love, esteem.”” For Ébad the table gives “snámchaín feda “fair-swimming letter,”” “cosc lobair “[admonishing?] of an infirm person”” and “caínem éco “fairest fish.””
• Forfeda, Wikipedia — prints the same three columns for the five supplementary letters and shows the Cú Chulainn column empty for all but Ébad (caínem éco).

Twenty-eight letter cards quote these kennings, so the quotation needed one entry of its own rather than a footnote each. The translations are McManus’s as the encyclopedia reproduces them; where that page marks a reading uncertain, the cards mark it uncertain too.

• Ogham, section Letter names, Wikipedia — quotes the Auraicept conceding “Now all these are wood names such as are found in the Ogham Book of Woods, and are not derived from men” while admitting “some of these trees are not known today”; names the five transparent tree names; reports McManus that the idea “dates to the Old Irish period (say, 10th century), but it postdates the Primitive Irish period” and arose because the letters were called feda or nin for their shape; and names ruis, úath, and gort as letter names the glossators reinterpreted “more or less forcefully” as epithets of trees. It also reports that McManus “adds one other definite tree name: onn ‘ash’ (the Auraicept wrongly has ‘furze’).”
• Bríatharogam, Wikipedia — prints the Auraicept gloss for each name with the Irish plant word — caertheand rowan, uinnius ash, sce whitethorn, cuileand holly, aball apple, finemhain vine, edind ivy, cilcach broom or fern, draighin blackthorn, trom elder, crand giuis .i. ochtach fir or pine, aiten furze, fraech heath, crand fir no crithach test-tree or aspen, ibhar yew — together with the short medieval sentences quoted on the cards; for cert: “The Auraicept glosses aball “apple,” clithar boaiscille .i. elit gelt quert (.i. aball) “shelter of a boiscill, i.e. a wild hind is queirt, i.e. an apple tree””; and of the forfeda, “oir with feorus no edind “spindle-tree or ivy.””
• Forfeda, Wikipedia — “According to Kelly (1976) the name spín … appears in the Old Irish tree lists as meaning either gooseberry or thorn” (Fergus Kelly, The Old Irish Tree-list, Celtica 11 (1976) 107–124; no free copy of the article).
• giolcach, Foclóir Gaeilge–Béarla (Ó Dónaill), teanglann.ie — “giolcach … 1. Coll: Reeds. Páirc na giolcaí, the reed field … 2. Reed; cane. 3. ~ (shléibhe), broom.”
• peith, Foclóir Gaeilge–Béarla (Ó Dónaill), teanglann.ie — “Lit: 1. Dwarf elder. 2. Letter P in later ogham alphabet.”
• úathmar, eDIL (Electronic Dictionary of the Irish Language) — “úathmar … adj o-ā (úath) dreadful; terrifying; ‘terrible, horrible, awful, direful’.”

The glossary intro already said the plant column records “later medieval and modern arboreal associations” and is not the secure original meaning of every letter name. That sentence had no entry behind it, and twenty-eight cards now rest on it, so it has one. The three layers the cards keep apart are exactly the layers this entry separates: the name, the medieval plant gloss, and anything later.

• Forfeda, Wikipedia — states that the five aicme forfeda “were invented in the Old Irish period, several centuries after the peak of Ogham usage”; that four are glossed with plants in the Auraicept and Emancholl is not; that Emancholl is named for its shape, ᚙ resembling two ᚉ; that Ébad is frequent in inscriptions inside the formula word KOI, meaning roughly ‘here’ and answering to Latin hic iacet; that Ór appears at Killogrone in County Kerry and on the Formaston cross slab in Aberdeenshire, Uilleann once at Teeromoyle in County Kerry, and Pín at Cool East, Crickowel, and probably Margam; and, under the heading Peith, that [p] “had again to be expressed as a modification of [b], called peithe, after beithe, also called beithe bog,” (the Ogham line carries “appears in the manuscripts only”). It also reports McManus on pín and Latin spina, and Kelly on spín in the tree lists. Of Ébad it reads: “Four of these names are glossed in the Auraicept with tree names, ebad as crithach “aspen””; “The kennings for Éabhadh point to the sound éo or é, which is also the word for “salmon””; and “Éabhadh was in fact frequently used as part of the formula word KOI ᚕᚑᚔ, but with the value /k/ or /x/”; its table of “The forfeda letter names and their kennings, as edited (in normalized Old Irish) and translated by McManus (1988)” has rows for Éabhadh, Óir, Uilleann, Ifín, and Eamhancholl and none for Peith, and the four Auraicept tree glosses it gives are for ebad, oir, uilleand, and iphin.
• Ogham, Wikipedia — gives the same reassignment and the invention of peithboc, ‘soft B’, which “appears in the manuscripts only”; and “Unlike orthodox ogham, some medieval inscriptions feature all five Forfeda.”
• Deborah Hayden and David Stifter, Ogam and Trees, OG(H)AM project, University of Glasgow (20 December 2022) — “Iphin, Pín is probably called from Lat. pīnus ‘pine, fir’, after the model of the other tree names in the core ogam set”; and of Gétal, “Medieval scholars explain it as gilcach ‘reed’”; and “Ór means ‘gold’, a loan from Latin aurum.”; of the forfedae: “Ébad, allegedly ‘aspen’ (we have seen that one before) or ‘elecampane’ (real name: elenn). Ébad is probably just a meaningless variation on edad.”
• Formaston Stone, Elphinstone Institute, University of Aberdeen — “A fragment of a class II Pictish Symbol Stone,” with “two vertical Ogam inscriptions”; Aboyne, Aberdeenshire, dated 800–1000 AD.

Six of the book’s twenty-eight terms are supplementary letters or later inventions — Ébad, Ór, Uilleann, Ifín, Eamhancholl, and Peith — and their cards have to say that the ground under them is softer than under Beith or Dair. Ruling D21 applies: the uncertainty goes on the card.

• Ogham, section Neopaganism, Wikipedia — states that modern New Age and neopagan approaches to ogham “largely derive from the now-discredited theories of Robert Graves in his book The White Goddess”; that Graves explored the tree folklore of the letter names and proposed that their order formed an ancient “seasonal calendar of tree magic” (citing Graves 1961 p. 165); that his theories have been “discredited and discarded by modern scholars,” Macalister included; that he followed Macalister’s BLNFS letter order, which scholars reject; and that the meanings used in modern divination “are usually based on the tree ogham,” with Graves still a major influence. (Read again on 23 September 2026: the article does not cite Hutton’s The Triumph of the Moon, which an earlier note said it did.)
• Celtic Tree Calendar, Mary Jones, Jones’s Celtic Encyclopedia — “there is no evidence that the Celts ever named their months after trees”; lists the thirteen months and gives “Ruis (Elder) November 25 to December 22,” with 23 December ruled by no tree; describes the scheme as Davies and Graves “picking and choosing the facts to suit their purpose,” and records that Graves ignored Macalister’s advice not to take the tree alphabet too seriously.
• Brian Walsh, Robert Graves’ fabrication of the Celtic tree calendar, Spiral Nature, 21 August 2013 — calls the calendar “an invention unprecedented in the original lore” and concludes that “the result is a calendar that bears no resemblance to any traditional Celtic time keeping system we have on record,” setting out how Graves rearranged the Song of Amergin and cut the alphabet to thirteen consonants to do it.
• The White Goddess, Wikipedia — identifies Robert Graves’s book-length essay The White Goddess: a Historical Grammar of Poetic Myth, “First published in 1948,” with revised editions in 1952 and 1961.

The guardrail on this column is that nothing modern may be presented as ancient. Every letter card that mentions present-day tree lore carries this marker, and the wording is always that the association is modern. Chapter ᚏ(R) says Ruis is associated with the elder and with transitions: the elder is medieval and well sourced, the transition is not — it comes from Graves putting Ruis last in the year. That split is stated on the Ruis letter card.

• Elder, Trees for Life — “The best time to encounter faeries was under an elder on Midsummer’s Eve, when the Faery King and Queen and their train would pass by”; “Like rowan, people saw elder as being a protective tree. It was auspicious if it was growing near one’s dwelling, especially if it had seeded itself there”; “Legend has it that it was the tree from which Judas Iscariot hanged himself”; the belief that burning it released the Devil “reinforced the taboo against burning the wood”; and the note that the Gaelic names ruis or droman occur only rarely in Scottish place names.
• Sambucus, section In culture, Wikipedia — “If an elder tree was cut down, a spirit known as the Elder Mother would be released and take her revenge,” and the tree “could only safely be cut while chanting a rhyme to the Elder Mother”; records that some traditions have the elder warding off witches and others have witches congregating under it, and that Romani people hold burning elder wood to bring bad luck. Cites Royal Botanic Gardens, Kew, and Howard, Traditional Folk Remedies (1987), 134–5.
• Elder Mother, Wikipedia — read as raw wikitext: “to take wood from the elder tree one would have to ask the Elder Mother first, or else ill luck would befall the woodsman,” and “The spirit is said to haunt or torment people who build from elder wood unless they ask permission first.”

This is folklore, cited as folklore. It is the layer the Ruis letter card labels modern, and it is kept apart from the Auraicept gloss trom, elder, which is medieval and sits under r-2.17. Neither of these sources connects the folklore to the ogham letter; the Trees for Life page mentions ruis only as a Gaelic plant name. That is why the card says what modern lore says about the tree, and does not claim a medieval letter of transition.

• A. N. Whitehead, An Introduction to Mathematics (1911), full text at the Internet Archive — read as extracted page text on the chapter on zero: “The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought..”

Added 17 September 2026 to source the chapter epigraph, which carried no marker. **The epigraph drops three words.** Whitehead wrote “It is in a way the most civilized of all the cardinals” and the page prints “It is the most civilized of all the cardinals.” The sense is unchanged and the hedge is Whitehead’s; restoring it is a prose question and is recorded as such.

• Positional notation, Wikipedia — read as raw HTML: “the Babylonian numeral system, credited as the first positional numeral system, was base-60. However, it lacked a real zero. Initially inferred only from context, later, by about 700 BC, zero came to be indicated by a ‘space’ or a ‘punctuation symbol’ (such as two slanted wedges) between numerals. It was a placeholder…,” and contrasts positional with sign-value systems such as Roman numerals.
• Indian numerals, MacTutor History of Mathematics — read as raw HTML: “The Babylonians had a place-value system as early as the 19th century BC but the Babylonian systems were to base 60. The Indians were the first to develop a base 10 positional system,” and “our place-value system is a direct descendant of the Indian system.”
• 0, Wikipedia, § Ancient Near East — “By 300 BC, a punctuation symbol (two slanted wedges) was repurposed as a placeholder.”
• Zero, MacTutor History of Mathematics — “It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero.”
• Babylonian numerals, MacTutor History of Mathematics — “The numbers sexagesimal numbers 1 and 1,0, namely 1 and 60 in decimals, had exactly the same representation.”

• Zero, MacTutor History of Mathematics — read raw: “This tablet, thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation..”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Mesoamerican Long Count calendar, Wikipedia — describes the positional base-20 (with one base-18 place) system and its oldest surviving dates: “The earliest contemporaneous Long Count inscription yet discovered is on Stela 2 at Chiapa de Corzo”; of Tres Zapotes Stela C, “This is the second oldest Long Count date yet discovered”; “La Mojarra Stela 1, the Tuxtla Statuette, Tres Zapotes Stela C and Chiapa Stela 2 are all inscribed in an Epi-Olmec, not Maya, style.”; and “The shell glyph was used to represent the zero concept.”; on the base-18 place: “the second digit from the right (and only that digit) rolls over to zero when it reaches 18.” and “Thus 0.0.1.0.0 does not represent 400 days, but rather only 360 days,” and its table gives 1 tun = 360 days, approximately 1 solar year.

• BIPM, The International System of Units — The SI defines its units through fixed constants from 20 May 2019.
• NIST, SI historical resolutions — The 1983 meter definition fixes the speed of light; the 2018 resolution establishes the revised SI effective in 2019.
• NIST 2022 CODATA complete listing — lists the exact SI constants and the quoted physical values.
• NIST Fundamental Physical Constants, front page — the 2022 adjustment is the current one; the next regularly scheduled adjustment is 2026, for which data must be published or preprinted before 31 December 2026. Every CODATA-sourced value in this chapter is CODATA 2022; the cosmological, CMB, PDG, and electron-anomaly values are cited to their own sources at 3.2, 3.3, and 3.5.
• 2019 revision of the SI, Wikipedia — “Effective 20 May 2019, the 144th anniversary of the Metre Convention, the kilogram, ampere, kelvin, and mole are defined by setting exact numerical values, when expressed in SI units, for the Planck constant (h), the elementary electric charge (e), the Boltzmann constant (kB), and the Avogadro constant (NA), respectively.”
• Vacuum permittivity, Wikipedia — “The elementary charge was redefined exactly in terms of the coulomb as from 20 May 2019, with the effect that the vacuum electric permittivity and the magnetic vacuum permeability no longer have exactly determined values in SI units.”; displays ε0 = 1/(μ0c²) = e²/(2αhc), “with e being the elementary charge, h being the Planck constant, and c being the speed of light in vacuum, each with exactly defined values. The relative uncertainty in the values of each of ε0 and μ0 are therefore the same as that for the fine-structure constant”
• Faraday constant, Wikipedia — “It is named after the English scientist Michael Faraday. Since the 2019 revision of the SI, the Faraday constant has an exactly defined value, the product of the elementary charge (e, in coulombs) and the Avogadro constant (NA, in reciprocal moles)” and “One faraday is the amount of charge in one mole of electrons”
• Gas constant, Wikipedia — “The gas constant R is defined as the Avogadro constant NA multiplied by the Boltzmann constant k (or kB)”; “Since the 2019 revision of the SI, both N A and k are defined with exact numerical values when expressed in SI units. As a consequence, the SI value of the molar gas constant is exact.”
• Planck units, Wikipedia — table of base units: “Planck length … length (L) … l_P = √(ħG/c³)”

• Cosmological constant, Wikipedia — confirms the 1917 term and vacuum-energy interpretation, and that “there is no known natural way to derive the tiny cosmological constant used in cosmology from particle physics.”
• Planck (spacecraft), Wikipedia — confirms the base flat LambdaCDM fit and its parameters.
• Friedmann equations, Wikipedia — confirms the density balance and curvature term.
• Cosmological constant problem, Wikipedia — gives the Planck-scale estimate as roughly 120 orders of magnitude too large, and attributes “probably the worst theoretical prediction in the history of physics” to Hobson, Efstathiou and Lasenby, General Relativity: An Introduction for Physicists (2006), p. 187. Under Proposed solutions: “Some proposals involve modifying gravity to diverge from general relativity.”; “Some propose an anthropic solution, and argue that we live in one region of a vast multiverse that has different regions with different vacuum energies.”; and “Anthropic arguments gradually gained credibility among many physicists after the discovery of dark energy and the development of the theoretical string theory landscape.”
• H. Padmanabhan and T. Padmanabhan, CosMIn: The Solution to the Cosmological Constant Problem, arXiv:1302.3226 (2013) — “observations [1, 2] suggest that ΛL_P² ≈ 3.4 × 10⁻¹²²”
• Cosmological Constant Problem and H0 Tension in Void-dominated Cosmology, arXiv:2204.12180 (2022) — quotes the Planck 2018 value (its ref. [39], Aghanim et al., A&A 641, A6): “The cosmological constant in the latest Planck data is reported as below [39], Λobs = 1.1056 × 10−52m−2.”
• Planck Collaboration, Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209 — “We find good consistency with the standard spatially-flat 6-parameter ΛCDM cosmology having a power-law spectrum of adiabatic scalar perturbations (denoted “base ΛCDM” in this paper)”; §7.3: “The base-ΛCDM model assumes that the spatial hypersurfaces are flat”; Fig. 29: “Constraints on a non-flat universe as a minimal extension to the base-ΛCDM model. Points show samples from the Planck TT,TE,EE+lowE chains coloured by the value of the Hubble parameter”
• Lambda-CDM model, Wikipedia — parameter table lists Ω_m as “Total matter density today (including massive neutrinos)”
• K. Cahill, The eras of radiation, matter, and dark energy: new information from the Planck Collaboration, arXiv:1606.08865 — from Planck 2015 inputs: “The present ratio Ωr0 of the radiation density (2) to the critical density (6) is then” (9.0606 ± 0.1233) × 10⁻⁵; abstract: “The era of radiation ended, and the era of matter began, when the density of radiation dropped below that of matter. This happened 50,953 ± 2236 years after the time of infinite redshift”

• ESA Planck CMB account — confirms the CMB origin about 380,000 years after the beginning.
• Cosmic microwave background, Wikipedia — (§ Discovery): supports 1964 — “On 20 May 1964 they made their first measurement clearly showing the presence of the microwave background”; and “the CMB is the most perfect black body spectrum ever measured.”
• Wikipedia, Discovery of cosmic microwave background radiation — Penzias and Wilson reported the 3.5 K excess in their note “A Measurement of Excess Antenna Temperature at 4080 Mc/s,” Astrophysical Journal 142, 419–421, July 1965 (doi:10.1086/148307).
• Wien’s displacement law, Wikipedia — confirms the peak equation and root.
• Stefan-Boltzmann law, Wikipedia — confirms the Stefan-Boltzmann expression.
• D. J. Fixsen, The Temperature of the Cosmic Microwave Background, Astrophysical Journal 707 (2009) 916–920 — the abstract: “The determination from the measurements from the literature is cosmic microwave background temperature of 2.72548 +/- 0.00057 K,” while “The FIRAS data are independently recalibrated using the WMAP data to obtain a CMB temperature of 2.7260 +/- 0.0013.” This is where the chapter’s 2.7255 kelvin comes from, and no later measurement supersedes it.
• Lindemann–Weierstrass theorem, Wikipedia — e^a is transcendental for every nonzero algebraic a. Wien’s root satisfies x = 5(1 − e^−x), so if it were algebraic then e^−x = 1 − x/5 would be algebraic too. It is therefore transcendental, which is the only proven-transcendental claim in this chapter.
• NIST 2022 CODATA complete listing — “Wien wavelength displacement law constant 2.897 771 955... e-3 (exact) m K”
• Cosmic Microwave Background, OpenSpace documentation — “Today, the peak CMB light has redshifted 1,000 times, so it’s now in the microwave with a wavelength of 1 millimeter (1 micron × 1,000 = 1 mm).”

• NIST 2022 CODATA complete listing — lists the gas, Faraday, Josephson, flux, atomic, and electron constants.
• Quantum Hall effect, Wikipedia — confirms the h/e^2 resistance relation.
• Josephson effect, Wikipedia — confirms the 2eV/h relation.
• NIST CODATA 2022, Rydberg constant — 10 973 731.568 157(12) m⁻¹, relative standard uncertainty 1.1 × 10⁻¹².
• NIST CODATA 2022, electron g factor — −2.002 319 304 360 92(36), relative standard uncertainty 1.8 × 10⁻¹³, six times smaller than the Rydberg constant’s. So R_∞ is among the most precisely measured constants and not the most precisely measured one.
• Mohr et al., CODATA recommended values of the fundamental physical constants: 2022, arXiv:2409.03787 — carries the inputs for α: the electron anomaly gives “one of the three most accurate determinations of the fine-structure constant α in the 2022 adjustment,” and “the two input data h/m(87Rb) and h/m(133Cs) in the 2022 adjustment obtained using atom interferometry are also of great importance because they provide the two other accurate values of α”
• Rydberg constant, Wikipedia — “The Rydberg constant was one of the most precisely determined physical constants, with a relative standard uncertainty of 1.1×10⁻¹².” and “the Rydberg constant is inferred from measurements of atomic transition frequencies in three different atoms (hydrogen, deuterium, and antiprotonic helium).” Displays R∞ = α²mₑc/(2h) and R∞ = mₑe⁴/(8ε₀²h³c).

• Anomalous magnetic dipole moment, Wikipedia — confirms QED agreement beyond ten significant figures.
• Weinberg angle, Wikipedia — confirms W3/B rotation and the on-shell value: “CODATA 2022 gives the value sin²θw = 1 − (mW/mZ)² = 0.22305(23)” (formula as typeset on the page); and “The specific value is not predicted by the Standard Model: The Weinberg angle θ_w is an open, free parameter.”
• X. Fan, T. G. Myers, B. A. D. Sukra and G. Gabrielse, Measurement of the Electron Magnetic Moment, Physical Review Letters 130 (2023) 071801 — g/2 = 1.001 159 652 180 59(13), the Northwestern measurement the a_e card quotes, taken in 2022 and published in 2023.
• Particle Data Group, Review of Particle Physics 2026, Quantum Chromodynamics review — section 9, revised August 2025, gives α_s(m_Z²) = 0.1180 ± 0.0009 and its running. This is the current edition; of its world average it says that it “changed only marginally as compared to the values reported in the last edition of this Review,” so the chapter’s number stands.
• Gabrielse group discovers new measurement of electron magnetic moment, Northwestern Physics and Astronomy — “A new measurement of the electron magnetic moment from Physics and Astronomy’s Board of Trustees Professor, Gerald Gabrielse and his research group at Northwestern University’s Center for Fundamental Physics has delivered the most precise measurement of a property of any elementary particle.”
• Color confinement, Wikipedia — “There is not yet an analytic proof of color confinement in any non-abelian gauge theory.” and “It is sometimes believed that the sole origin of confinement is the very large value of the strong coupling near the Landau pole.” … “It is however incorrect”
• Particle Data Group, Review of Particle Physics 2026, Electroweak Model and Constraints on New Physics review — “the OS scheme promotes the tree-level formula sin²θW = 1 − MW²/MZ² to a definition of the renormalized sin²θW to all orders in perturbation theory”
• Arthur Jaffe and Edward Witten, Quantum Yang–Mills Theory, in The Millennium Prize Problems (Clay Mathematics Institute) — the strong force is the gauge theory of SU(3) with quarks: “The non-abelian gauge theory of the strong force is called Quantum Chromodynamics (QCD)”; the prize problem is the pure theory: “as of yet there is no effective approach to the gauge theory without fermions.”

• Maxwell’s equations, Wikipedia — confirms div B=0.
• Noether’s theorem, Wikipedia — confirms symmetry-to-conservation within its hypotheses.
• Mass-energy equivalence, Wikipedia — confirms E=mc^2 at rest.
• Einstein tensor, Wikipedia — confirms the contracted Einstein-tensor identity.
• Luigi Bianchi, Wikipedia — read as raw HTML: “In 1902, Bianchi rediscovered what are now called the Bianchi identities for the Riemann tensor … According to Tullio Levi-Civita, these identities had first been discovered by Ricci in about 1889, but Ricci apparently forgot all about the matter, which led to Bianchi’s rediscovery. However, the contracted Bianchi identities, which are sufficient for the proof that the divergence of the Einstein tensor always vanishes, had been published by Aurel Voss in 1880.” The 1902 paper is given as Bianchi, Sui simboli a quattro indici e sulla curvatura di Riemann, Rend. Acc. Naz. Lincei 11 (1902).
• Continuity equation, Wikipedia — confirms charge conservation.
• Contracted Bianchi identities, Wikipedia — proof: “Contract both sides of the above equation with a pair of metric tensors,” arriving at ∇_m R^m_ℓ = ½ ∇_ℓ R
• Magnetic monopole, Wikipedia — “Magnetism in bar magnets and electromagnets is not caused by magnetic monopoles, and indeed, there is no known experimental or observational evidence that magnetic monopoles exist.”
• Albert Einstein, Wikipedia — “His mass–energy equivalence formula E = mc², which arises from special relativity, has been called “the world’s most famous equation.””
• Sun, Wikipedia — “Albert Einstein provided the essential clue to the source of the Sun’s energy output with his mass–energy equivalence relation E = mc².”

• NIST 2022 CODATA complete listing — lists the atomic, electron, magnetic, Fermi, and input constants.
• Planck units, Wikipedia — confirms the Planck-unit definitions. These are the derived Planck length, time, and mass, built from ℏ, G, and c. The Planck constant h itself is a different thing — an SI defining constant, exact by definition — and is covered at 3.1. Of the Planck temperature: “At the Planck temperature, the wavelength of light emitted by thermal radiation reaches the Planck length. There are no known physical models able to describe temperatures greater than T_P.”
• Planck (spacecraft), Wikipedia — confirms the Planck Hubble and age values.
• A. G. Riess et al., A Comprehensive Measurement of the Local Value of the Hubble Constant…, Astrophysical Journal Letters 934 (2022) L7 — H₀ = 73.04 ± 1.04 km s⁻¹ Mpc⁻¹, and the paper states a five-sigma difference with the value Planck and ΛCDM predict. This is the late-universe half of the Hubble tension.
• Dimensionless physical constant, Wikipedia — “A gravitational coupling constant is a constant characterizing the gravitational attraction between a given pair of elementary particles. The electron mass is typically used, and the associated constant typically denoted αG.” Displays αG = Gmₚ²/(ħc), mₚ the particle rest mass.
• E. Di Valentino et al., In the Realm of the Hubble tension – a Review of Solutions, arXiv 2103.01183 (2021) — carries the ranking of the tension: “The most statistically significant tension is the 4−6σ disagreement between predictions of the Hubble constant H0 by early time probes with ΛCDM model, and a number of late time, model-independent determinations of H0 from local measurements”
• Planck units, OeisWiki — carries the human scale and the flea’s egg: “Unlike all other Planck base units and most Planck derived units, the Planck mass has a scale more or less conceivable to humans. (It is traditionally said to be about the mass of a flea, but more accurately it is about the mass of a flea egg.)”
• Ned Wright, Brief History of the Universe, UCLA — carries the temperature at the Planck time (order of magnitude only): “The Planck time: 10⁻⁴³ seconds.” … “The temperature is T=10³² K.”
• Planck time, Simple English Wikipedia — “One Planck time is the time it would take a photon travelling at the speed of light to cross a distance equal to one Planck length.”
• Classical electron radius, Wikipedia — carries the formula r_e = (1/4πε₀)e²/(m_ec²), the reading, the α-relations r_e = λ̄_e α = a₀α², and the point-like electron: “A classical charged sphere producing an electric field with energy equal to the electron’s rest mass energy would have a radius equal to the classical electron radius.”; “The classical electron radius is sometimes known as the Lorentz radius or the Thomson scattering length.”; “Consequently, electrons are considered point charges in modern theories.”

• Own computation — Substitute the displayed CODATA-scale values; exact-one brackets cancel, the thermal bracket is pi^2/60, and the remaining positive factors multiply to approximately 1.3e-154.

• Minkowski space, Wikipedia — dates Die Grundgleichungen fuer die elektromagnetischen Vorgaenge in bewegten Koerpern to 1908 and the Cologne Raum und Zeit lecture to 1908, and defines a null or lightlike vector by c^2t^2 = r^2, which is the interval being exactly zero along a light ray; and: “Minkowski spacetime is constructed so that the speed of light will be the same constant regardless of the reference frame in which it is measured.”
• Hermann Minkowski, Wikipedia — read as raw wikitext: the address “called “Space and Time” delivered at the 80th Assembly of German Natural Scientists and Physicians (21 September 1908),” in which Minkowski set out the four-dimensional geometry of spacetime; this page does not name the city (the Minkowski space line, now the Minkowski spacetime article, gives Cologne).

The chapter prose and the #dslight card both carry the year 1908 and both stood on 3.6, which sources div B = 0, Noether, E = mc², the Einstein tensor, and charge continuity but does not mention Minkowski at all. The date is correct; it simply had nothing behind it. Both the 1908 paper and the 1908 Cologne lecture are Minkowski’s; the lecture was published in 1909, which is why some sources give that year for Raum und Zeit.

• Hierarchy problem, Wikipedia — defines the hierarchy problem as the large discrepancy between aspects of the weak force and gravity, and gives the weakness of gravity relative to the other forces as its primary numerical example, alongside the Higgs-mass fine-tuning form of the same problem: “the question is why the Higgs boson is so much lighter than the Planck mass”; “one would expect that the large quantum contributions to the square of the Higgs boson mass would inevitably make the mass huge, comparable to the scale at which new physics appears unless there is an incredible fine-tuning cancellation between the quadratic radiative corrections and the bare mass.”
• Dirac large numbers hypothesis, Wikipedia — “LNH was Dirac’s personal response to a set of large number “coincidences” that had intrigued other theorists of his time.” The page displays e²/(4πε₀ G mₑ²) ≈ 4.1666763·10⁴², which is α/α_G.
• Composite Higgs models, Wikipedia — “Often referred to as “natural” composite Higgs models, CHMs are constructions that attempt to alleviate fine-tuning or “naturalness” problem of the Standard Model.”

The chapter uses the broad sense: α/α_G is about 4 × 10⁴², so “about 10⁴²” is right, and it is the force-strength statement of the hierarchy problem rather than the Higgs-mass fine-tuning statement. The sentence stood on 3.7, which covers Planck units, the Hubble values, the atomic key values, and the Fermi coupling, and does not mention the hierarchy problem.

• Fine-structure constant, Wikipedia — read as raw HTML: it quotes Feynman at length — “It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it. Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It’s one of the greatest damn mysteries of physics” — which is the epigraph and the source of the “greatest damn mysteries” line the #alpha card also uses.

Added 17 September 2026 to source the chapter epigraph, which carried no marker.

• Standard Model, Wikipedia — read as raw HTML: “The local SU(3) × SU(2) × U(1) gauge symmetry is an internal symmetry that essentially defines the Standard Model… the three factors of the gauge symmetry give rise to the three fundamental interactions”; “the dynamics depends on 19 parameters, whose numerical values are established by experiment”; and “explaining neutrino mass will require an additional 7 or 8 constants.” The numbered parameter table runs to 19, ending at the Higgs mass.
• Review of Particle Physics, Electroweak model and constraints on new physics, Particle Data Group — the field’s own reference review of the model and of the measured inputs the chapter’s three couplings are quoted from.

• Lambda-CDM model, Wikipedia — read as raw HTML: “The Planck collaboration version of the ΛCDM model is based on six parameters… Six is the smallest number of parameters needed to give an acceptable fit to the observations; other possible parameters are fixed at ‘natural’ values, e.g. total density parameter = 1.00, dark energy equation of state = −1… From these six parameters, the other model values, such as the Hubble constant and the dark energy density, can be calculated.”
• Planck Collaboration, Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209 (A&A 641 (2020) A6) — the same paper, free on arXiv: “good consistency with the standard spatially-flat 6-parameter ΛCDM cosmology,” with H₀ = 67.4 ± 0.5 km/s/Mpc inferred from the fit rather than fitted.
• Planck 2018 results. VI, readable HTML (ar5iv) — section 2.1 names the six fitted base-ΛCDM parameters: the baryon density ω_b, the cold dark matter density ω_c, θ_MC (“an approximation to the observed angular size of the sound horizon at recombination”), the reionization optical depth τ, the amplitude A_s and the spectral index n_s of the primordial perturbations — one amplitude.

This is the model the chapter means by “taken consistently from the same cosmological model” in the Ω bracket. It is why the four density parameters may be added and why the individual values should not be mixed between fits.

• 2019 revision of the SI, Wikipedia — read as raw HTML: “Effective 20 May 2019, the 144th anniversary of the Metre Convention, the kilogram, ampere, kelvin, and mole are defined by setting exact numerical values, when expressed in SI units, for the Planck constant (h), the elementary electric charge (e), the Boltzmann constant (k_B), and the Avogadro constant (N_A), respectively. The second, metre, and candela had previously been redefined using physical constants,” and “The vote, which was held on 16 November 2018 at the 26th CGPM, was unanimous.”
• Measurement units, Bureau International des Poids et Mesures — the defining body’s own page for the seven defining constants of the SI.

• Dimensionless quantity, Wikipedia — read as raw wikitext: “Dimensionless quantities, or quantities of dimension one, are quantities with no associated units of measurement,” and of a ratio, “its value remains independent of the specific units of volume used”; its list of dimensionless quantities in physics and engineering includes the fine-structure constant, “α ≈ 1/137,” and “β (or μ) ≈ 1836, the proton-to-electron mass ratio.”
• NIST CODATA 2022, proton-electron mass ratio — read as raw HTML: numerical value 1836.152 673 426(32), given with no unit.

Added 18 September 2026. The #dimensionless card was the only one of 723 glossary cards that cited nothing at all; a sweep with citecheck.py found it.

• Bohr radius, Wikipedia — read raw: “In the Bohr model for atomic structure, put forward by Niels Bohr in 1913, electrons orbit a central nucleus under electrostatic attraction.”; “The Bohr radius (a₀) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state.”
• Dirac equation, Wikipedia — read raw: “In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928..”
• Anomalous magnetic dipole moment, Wikipedia — read raw: “where \alpha is the fine-structure constant. This result was first found by Julian Schwinger in 1948.”
• Planck (spacecraft), Wikipedia — read raw: “Results from the final analysis of Planck data were released in July 2018..”
• Fine-structure constant, Wikipedia — read raw: “The constant was named by Arnold Sommerfeld, who introduced it in 1916 when extending the Bohr model of the atom.”; it gives α as “The ratio of the velocity of the electron in the first circular orbit of the Bohr model of the atom … to the speed of light in vacuum, c,” “Sommerfeld’s original physical interpretation”; and quotes Feynman: “Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by humans.”
• Bohr magneton, Wikipedia — read raw: “The Romanian physicist Ștefan Procopiu had obtained the expression for the magnetic moment of the electron in 1913.”; “In 1920, Wolfgang Pauli gave the Bohr magneton its name in an article where he contrasted it with the magneton of the experimentalists which he called the Weiss magneton.”; “In the summer of 1913, the values for the natural units of atomic angular momentum and magnetic moment were obtained by the Danish physicist Niels Bohr as a consequence of his atom model.”
• Speed of light, Wikipedia — read raw: “Ole Rømer used an astronomical measurement to make the first quantitative estimate of the speed of light in the year 1676..”
• Compton wavelength, Wikipedia — read raw: “It was introduced by Arthur Compton in 1923 in his explanation of the scattering of photons by electrons (a process known as Compton scattering)..”
• Angstrom, Wikipedia — carries the size in ångströms: “the atomic unit of length, the bohr—about 0.53 Å”
• Age of the universe, Wikipedia — carries the method and the stellar check: “With accurate measurements of these parameters, the age of the universe can be determined by using the Friedmann equation.”; “Both values are compatible with the Planck 2018 number.” (the Methuselah star, 14.27±0.80 billion years, and BD +17°3248, 13.8±4 billion years)
• Electron magnetic moment, Wikipedia — carries the ratio: “In units of the Bohr magneton (μB), it is −1.00115965218046(18), which has a relative uncertainty of 1.8×10−13.”
• John Baez, Length Scales in Physics, UC Riverside — carries both readings: “Trying to localize an electron to within less than its Compton wavelength makes its momentum so uncertain that it can have an energy large enough to make an extra electron-positron pair!” and “the length scale at which relativistic quantum field theory becomes crucial for its accurate description”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• A Dynamical Theory of the Electromagnetic Field, Wikipedia — read raw: “A Dynamical Theory of the Electromagnetic Field” is a paper by James Clerk Maxwell on electromagnetism, published in 1865..”
• Coulomb’s law, Wikipedia — read raw: “Finally, in 1785, the French physicist Charles-Augustin de Coulomb published his application of a torsion balance to determine that the magnitude of the electric force between two point charges..”
• Maxwell’s equations, Wikipedia — read raw: “Maxwell’s equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law..”
• Oil drop experiment, Wikipedia — read raw: “The oil drop experiment was performed by Robert A. Millikan and Harvey Fletcher in 1909 to measure the elementary electric charge (the charge of the electron)..”
• Einstein field equations, Wikipedia — read raw: “The equations were published by Albert Einstein in 1915 in the form of a tensor equation.”
• Mass–energy equivalence, Wikipedia — read raw: “Einstein did not write the exact formula E = mc² in his 1905 Annus Mirabilis paper “Does the Inertia of an object Depend Upon Its Energy Content?”; rather, the paper states that if a body gives off the energy L by emitting light, its mass diminishes by L/c².”; and “a great amount of energy can be released by the nuclear fission chain reactions used in both nuclear weapons and nuclear power.”
• M. Faraday, Experimental researches in electricity — Seventh Series, Phil. Trans. R. Soc. 124 (1834) 77–122, Crossref record — read raw: “the publisher’s Crossref record gives the year of publication as 1834.”
• E. Fermi, Tentativo di una teoria dei raggi β, Il Nuovo Cimento 11 (1934) 1–19, Crossref record — read raw: “the publisher’s Crossref record gives the year of publication as 1934.”
• Magnetic flux quantum, Wikipedia — read raw: “The phenomenon of flux quantization was first discovered in superconductors experimentally by B. S. Deaver and W. M. Fairbank, and independently by R. Doll and M. Näbauer, in 1961..”
• Oliver Heaviside, Wikipedia — “In 1884 he recast Maxwell’s mathematical analysis from its original cumbersome form (they had already been recast as quaternions) to its modern vector terminology, thereby reducing twelve of the original twenty equations in twenty unknowns down to the four differential equations in two unknowns we now know as Maxwell’s equations.”
• Annus mirabilis papers, Wikipedia — “On 21 November Annalen der Physik published a fourth paper (received September 27): “Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?” (“Does the Inertia of a Body Depend Upon Its Energy Content?”)”
• Faraday’s laws of electrolysis, Wikipedia — “Michael Faraday reported that the mass (m) of a substance deposited or liberated at an electrode during electrolysis is directly proportional to the charge (Q, for which the SI unit is the ampere-second or coulomb) passed through the electrolyte.”
• Fermi’s interaction, Wikipedia — carries the contact interaction, the muon lifetime, and the heavy boson: “The theory posits four fermions directly interacting with one another (at one vertex of the associated Feynman diagram).”; “In the original theory, Fermi assumed that the form of interaction is a contact coupling of two vector currents.”; “The most precise experimental determination of the Fermi constant comes from measurements of the muon lifetime, which is inversely proportional to the square of GF (when neglecting the muon mass against the mass of the W boson).”; “MW is the mass of the W boson, which mediates the decay in question.”
• Wikipedia (German), Robert Doll — places Doll at the Herrsching outstation of the low-temperature commission, where he and Martin Näbauer showed flux quantization in 1961: “Ab 1953 war er wissenschaftlicher Mitarbeiter an der Herrschinger Außenstelle der Kommission für Tieftemperaturforschung.”
• Wikipedia, William M. Fairbank — the Stanford half of the 1961 experiments: “At Stanford, one of his most famous discoveries has been the flux quantization in superconducting tin.”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Magnetic flux quantum, Wikipedia — read raw: “As a result of the 2019 revision of the SI, the Planck constant h has a fixed value.”
• Philosophiae Naturalis Principia Mathematica, Wikipedia — read raw: “authorized (imprimatur) by Samuel Pepys, then-president of the Royal Society on 5 July 1686 and first published in 1687..”
• Gravitational constant, Wikipedia — read raw: “The first direct measurement of gravitational attraction between two bodies in the laboratory was performed in 1798, seventy-one years after Newton’s death, by Henry Cavendish..”
• Fundamental Physical Constants — Complete Listing, 2018 CODATA adjustment, NIST — read raw: “2018 CODATA adjustment ... Newtonian constant of gravitation 6.674 30 e-11 0.000 15 e-11 m^3 kg^-1 s^-2.”
• Émile Clapeyron, Wikipedia — read raw: “In 1834, he made his first contribution to the creation of modern thermodynamics by publishing a report entitled Mémoire sur la puissance motrice de la chaleur (Memoir on the Motive Power of Heat).”
• Planck constant, Wikipedia — read raw: “The constant was postulated by Max Planck in 1900 as a proportionality constant needed to explain experimental black-body radiation.”; “Technologies of mass metrology such as the Kibble balance measure the kilogram by fixing the Planck constant.”
• Photoelectric effect, Wikipedia — read raw: “In 1905, Albert Einstein proposed a theory of the photoelectric effect using a concept that light consists of tiny packets of energy known as photons or light quanta..”
• Hubble’s law, Wikipedia — read raw: “The discovery of Hubble’s law is attributed to work published by Edwin Hubble in 1929,.”
• Planck (spacecraft), Wikipedia — read raw: “Results from the final analysis of Planck data were released in July 2018. ... Planck measured the Hubble constant to be 67.4 ± 0.5 km/s/Mpc, increasing the tension with direct local measurements..”
• Ideal gas law, Wikipedia — “It was first stated by Benoît Paul Émile Clapeyron and independently of him, Dmitry Mendeleev in 1834 as a combination of the empirical Boyle’s law, Charles’s law, Avogadro’s law, and Gay-Lussac’s law.” and “The equation of state given here (PV = nRT) applies only to an ideal gas”
• Planck’s law, Wikipedia — “While Planck originally regarded the hypothesis of dividing energy into increments as a mathematical artifice, introduced merely to get the correct answer, other physicists including Albert Einstein built on his work”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Planck constant, Wikipedia — read raw: “Then, in 1926, in their seminal papers, Schrödinger and Dirac again introduced special symbols for it: K in the case of Schrödinger, and h in the case of Dirac. Dirac continued to use h in this way until 1930, when he introduced the symbol ℏ in his book The Principles of Quantum Mechanics..”
• The Principles of Quantum Mechanics, Wikipedia — read raw: “The Principles of Quantum Mechanics is an influential monograph written by Paul Dirac and first published by Oxford University Press in 1930..”
• 2019 revision of the SI, Wikipedia — read raw: “Effective 20 May 2019, the 144th anniversary of the Metre Convention, the kilogram, ampere, kelvin, and mole are defined by setting exact numerical values, when expressed in SI units, for the Planck constant (h), the elementary electric charge (e).”
• Boltzmann constant, Wikipedia — read raw: “Although Boltzmann first linked entropy and probability in 1877, the relation was never expressed with a specific constant until Max Planck first introduced k”; “the relation was never expressed with a specific constant until Max Planck first introduced k ... in his derivation of the law of black-body radiation in 1900–1901..”
• Cosmological constant, Wikipedia — read raw: “In 1998, two teams of astrophysicists, the Supernova Cosmology Project and the High-Z Supernova Search Team, carried out measurements on distant supernovae which showed that the speed of galaxies’ recession in relation to the Milky Way increases over time.”; “For this work, Perlmutter, Schmidt, and Riess jointly received the Nobel Prize in Physics in 2011..”
• History of Lorentz transformations, Wikipedia — read raw: “They were completed by Larmor (1897, 1900) and Lorentz (1899, 1904), and brought into their modern form by Poincaré (1905) who gave the transformation the name of Lorentz..”
• Lorentz transformation, Wikipedia — read raw: “All of the work up to 1905 assumed that the constant speed of light was a consequence of an aether and hypothesized the transformation. In that year Albert Einstein published what is now called special relativity.”
• Lorentz factor, Wikipedia — carries the value at rest: “Its initial value is 1 (when v = 0)”
• Large Hadron Collider, Wikipedia — carries the LHC value at 6.5 TeV per proton: “At this energy, the protons have a Lorentz factor of about 6,930”
• Schwarzschild radius, Wikipedia — carries the solution, formula, horizon, and values: “In 1916, Karl Schwarzschild obtained an exact solution to the Einstein field equations for the gravitational field outside a non-rotating, spherically symmetric body”; “The Schwarzschild radius (2GM/c²) of a given mass M equals twice its reduced Compton wavelength when M equals one Planck mass”; “The surface at the Schwarzschild radius acts as an event horizon in a non-rotating body”; “the Sun has a Schwarzschild radius of approximately 3.0 km (1.9 mi), whereas Earth’s is approximately 9 mm (0.35 in)”
• Karl Schwarzschild, Wikipedia — carries the timing: “an exact solution which he first set down in a letter to Einstein of 22 December 1915”; “In March 1916, Schwarzschild left military service because of his illness”; “Schwarzschild died of immune complications related to his illness on 11 May 1916, at the age of 42.” Its infobox gives the place of death as Potsdam; its “returned to Göttingen” is his leaving military service in March 1916.
• Karl Schwarzschild, MacTutor History of Mathematics — “Died 11 May 1916 Potsdam, Germany”; “after being invalided home in March 1916, he died two months later.”
• Josephson effect, Wikipedia — read as raw wikitext: “The effect is named after the British physicist Brian Josephson, who predicted in 1962 the mathematical relationships for the current and voltage across the weak link.”; and “The Josephson voltage standard takes the caesium standard definition of frequency and gives the standard representation of a volt.”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Electron, Wikipedia — read raw: “In 1897, the British physicist J. J. Thomson, with his colleagues John S. Townsend and H. A. Wilson, performed experiments indicating that cathode rays really were unique particles, rather than waves, atoms or molecules as was believed earlier.”; and of annihilation: “If the electron and positron have negligible momentum, a positronium atom can form before annihilation results in two or three gamma ray photons whose energies total 1.022 MeV.”
• Mole (unit), Wikipedia — read raw: “The mole was made the seventh SI base unit in 1971 by the 14th CGPM.”; and of the definition that replaced oxygen-16 “during the 1960s”: “The International Bureau of Weights and Measures defined the mole as “the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilograms of carbon-12.””
• A Dynamical Theory of the Electromagnetic Field, Wikipedia — read raw: “A Dynamical Theory of the Electromagnetic Field” is a paper by James Clerk Maxwell on electromagnetism, published in 1865..”
• Impedance of free space, Wikipedia — read raw: “Between 1948 and 2019, the SI unit the ampere was defined by choosing the numerical value of μ0 to be exactly 4π × 10^-7 H/m..”
• Vacuum permeability, Wikipedia — read raw: “Since the revision of the SI in 2019 (when the values of e and h were fixed as defined quantities), μ0 is an experimentally determined constant with its value proportional to the dimensionless fine-structure constant.”
• Avogadro constant, Wikipedia — read raw: “The Avogadro constant is named after the Italian scientist Amedeo Avogadro (1776–1856), who, in 1811, first proposed that the volume of a gas (at a given pressure and temperature) is proportional to the number of atoms or molecules regardless of the nature of the gas.”; “The name Avogadro’s number was coined in 1909 by the physicist Jean Perrin, who defined it as the number of molecules in exactly 32 grams of oxygen gas.”; “Perrin himself determined the Avogadro number, which he called “Avogadro’s constant” (constante d’Avogadro), by several different experimental methods.”
• Noether’s theorem, Wikipedia — read raw: “Emmy Noether’s work on the invariance theorem began in 1915 when she was helping Felix Klein and David Hilbert with their work related to Albert Einstein’s theory of general relativity”; “This is the first of two theorems (see Noether’s second theorem) published by the mathematician Emmy Noether in 1918..”
• Emmy Noether, Wikipedia — “The paper, Invariante Variationsprobleme, was presented by Felix Klein on 26 July 1918 at a meeting of the Royal Society of Sciences at Göttingen. Noether presumably did not present it herself because she was not a member of the society.”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Planck units, Wikipedia — read raw: “In 1899, one year before the advent of quantum theory, Max Planck introduced what later became known as the Planck constant. At the end of the paper, he proposed the base units that were later named in his honor.”; and “At the Planck scale, the predictions of the Standard Model, quantum field theory and general relativity are not expected to apply, and quantum effects of gravity are expected to dominate.”
• Quantum Hall effect, Wikipedia — read raw: “In 1980, Klaus von Klitzing, working at the high magnetic field laboratory in Grenoble with silicon-based MOSFET samples developed by Michael Pepper and Gerhard Dorda, made the unexpected discovery that the Hall resistance was exactly quantized.”; “For this finding, von Klitzing was awarded the 1985 Nobel Prize in Physics.”; “Later, the 2019 revision of the SI fixed exact values of h and e, resulting in an exact R_K = h/e^2.”
• Rydberg formula, Wikipedia — read raw: “It was first empirically stated in 1888 by the Swedish physicist Johannes Rydberg, then theoretically by Niels Bohr in 1913, who used a primitive form of quantum mechanics..”
• Josef Stefan, MacTutor — read raw: “Stefan showed empirically, in 1879, that total radiation from a blackbody is proportional to the fourth power of its absolute temperature.”; “Boltzmann, who was one of Stefan’s students, showed in 1884 that this Stefan-Boltzmann law could be demonstrated mathematically.”
• Stefan–Boltzmann law, Wikipedia — read raw: “The law, including the theoretical prediction of the Stefan–Boltzmann constant as a function of the speed of light, the Boltzmann constant and the Planck constant, is a direct consequence of Planck’s law as formulated in 1900..”
• Steven Weinberg, Wikipedia — read raw: “In 1967 he was a visiting professor at MIT. It was in that year at MIT that Weinberg proposed his model of unification of electromagnetism and nuclear weak forces.”
• Rydberg constant, Wikipedia — “The Rydberg constant was one of the most precisely determined physical constants, with a relative standard uncertainty of 1.1×10⁻¹².” and “the Rydberg constant is inferred from measurements of atomic transition frequencies in three different atoms (hydrogen, deuterium, and antiprotonic helium).” Displays R∞ = α²mₑc/(2h) and R∞ = mₑe⁴/(8ε₀²h³c).

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Wien’s displacement law, Wikipedia — read raw: “The law is named for Wilhelm Wien, who derived it in 1893 based on a thermodynamic argument..”
• Impedance of free space, Wikipedia — read raw: “This chain of dependencies changed when the ampere was redefined on 20 May 2019..”
• Vacuum permeability, Wikipedia — “In the 2019 revision of the SI, the ampere is defined exactly in terms of the elementary charge and the second, and the value of μ0 is now determined experimentally (based on the measured value of the fine-structure constant)”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Quantum electrodynamics, Wikipedia — describes the theory’s development and its precision agreement for the electron’s magnetic moment.
• Anomalous magnetic dipole moment, Wikipedia — carries Schwinger’s 1948 one-loop result and the agreement: “found by Julian Schwinger in 1948” and theory and experiment “agree to more than 10 significant figures,” “making the magnetic moment of the electron one of the most accurately verified predictions in the history of physics”

• DLMF, §24.2 Bernoulli and Euler numbers — confirms the Bernoulli-number table and generating-function conventions.
• Wolfram MathWorld, Algebraic Number — confirms the algebraic-number definition and the listed algebraic constants.
• Wikipedia, Brocard’s problem — n! + 1 = m² is known only for n = 4, 5, and 7; whether there are others is open, no further solution has been found below n = 10¹⁵, and Erdős conjectured there are none.
• Catalan number, Wikipedia — “The first Catalan numbers for n = 0, 1, 2, 3, ... are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ...”; C_n is “the number of different ways n + 1 factors can be completely parenthesized”
• 0.999..., Wikipedia — “Despite common misconceptions, 0.999... is not “almost exactly 1” or “very, very nearly but not quite 1”; rather, “0.999...” and “1” represent exactly the same number.”; Byers’s argument displayed: x = 0.999…, 10x = 9.999…, 10x = 9 + x, 9x = 9, x = 1; “Other proofs generally involve basic properties of real numbers and methods of calculus, such as series and limits.”; “the 1895 Arithmetic for Schools says, “when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small.””
• Golden ratio, Wikipedia — the defining polynomial: “Its minimal polynomial, the polynomial of lowest degree with integer coefficients that has the golden ratio as a root, is x^2 − x − 1.”; “φ is defined as the positive solution.”
• Silver ratio, Wikipedia — polynomial, value, and Pell limit: “It follows that the silver ratio is the positive solution of quadratic equation σ^2 − 2σ − 1 = 0. The quadratic formula gives the two solutions 1 ± √2, the decimal expansion of the positive root begins with 2.414213562373095...”; of the Pell numbers: “The first few terms are 0, 1, 2, 5, 12, 29, 70, 169,...”; “The limit ratio of consecutive terms is the silver mean.”
• Constructible number, Wikipedia — constructibility of numbers built from square roots: “Equivalently, r is constructible if and only if there is a closed-form expression for r using only integers and the operations for addition, subtraction, multiplication, division, and square roots.”
• Sexagesimal, Wikipedia — carries the divisors, the hour, and the degree: “The number 60, a superior highly composite number, has twelve divisors, namely 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, of which 2, 3, and 5 are prime numbers. With so many factors, many fractions involving sexagesimal numbers are simplified.”; “One hour of time is divided into 60 minutes, and one minute is divided into 60 seconds.”; “Similarly, the practical unit of angular measure is the degree, of which there are 360 (six sixties) in a circle.”
• Square number, Wikipedia — lists the small squares: “The squares (sequence A000290 in the OEIS) smaller than 60² = 3600 are:” … “2² = 4” … “3² = 9”
• Repeating decimal, Wikipedia — “the decimal representation of 1/3 becomes periodic just after the decimal point, repeating the single digit “3” forever, i.e. 0.333....”
• Factorial, Wikipedia — § Factorial of zero: “the definition of n! as a product involves the product of no numbers at all, and so is an example of the broader convention that the empty product, a product of no factors, is equal to the multiplicative identity.”; “There is exactly one permutation of zero objects: with nothing to permute, the only rearrangement is to do nothing.”; “the number of ways to choose all n elements from a set of n is (n choose n) = n!/(n!0!) = 1, a binomial coefficient identity that would only be valid with 0!=1.”; “With 0!=1, the recurrence relation for the factorial remains valid at n=1.”; “Setting 0!=1 allows for the compact expression of many formulae, such as the exponential function, as a power series”; “This choice matches the gamma function 0! = Γ(0+1) = 1”; and “there are n! different ways of arranging n distinct objects into a sequence.”

• Wolfram MathWorld, Taxicab Number — lists Taxicab(2)=1729 and its two cube representations.

• Wolfram MathWorld, Kaprekar Routine — confirms 6174 and the four-digit routine.
• Wolfram MathWorld, Perfect Number — confirms the proper-divisor definition and the first even perfect numbers; the Euclid form is perfect “as stated in Proposition IX.36 of Euclid’s Elements,” and “In a posthumous 1849 paper, Euler provided the first proof that Euclid’s construction gives all possible even perfect numbers.”
• Wikipedia, Amicable Numbers — confirms 220 and 284 and their divisor sums.
• Wolfram MathWorld, Cyclic Number — 142857 is the cyclic number generated by 1/7, a cyclic number being an (n−1)-digit integer whose products with 1, 2, …, n−1 are its own digits reordered; 7 is the first full reptend prime, and multiplying a cyclic number by its generator gives a string of nines.
• Wikipedia, Phrases from The Hitchhiker’s Guide to the Galaxy — Deep Thought’s Answer is 42, first broadcast in Fit the Fourth on 29 March 1978 and published in the novel of 1979; Adams wrote on 3 November 1993 that “It was a joke. It had to be a number, an ordinary, smallish number, and I chose that one.”
• Belphegor’s prime, Wikipedia — “Belphegor’s prime is the palindromic prime number 1000000000000066600000000000001”; “The name “Belphegor’s prime” was coined by author Clifford A. Pickover in 2012.”; “Belphegor is one of the Seven Princes of Hell”; “This number is surrounded on either side by thirteen zeroes and is 31 digits in length (thirteen reversed)”
• Quote Origin: The Answer to Life, the Universe and Everything Is Forty-Two, Quote Investigator — quotes the novel’s passage: ““All right,” said Deep Thought. “The Answer to the Great Question . . .”” … ““Of Life, the Universe and Everything . . .” said Deep Thought.” … ““Is . . .” said Deep Thought, and paused.” … ““Forty-two,” said Deep Thought, with infinite majesty and calm.”
• Kaprekar’s routine, Wikipedia — “Any four-digit number (in base 10) with at least two distinct digits will reach 6174 within seven iterations.”; “When leading zeroes are retained, only repdigits lead to the trivial Kaprekar’s constant.”

• Wikipedia, Large Numbers — gives the values of googol, googolplex, and the first Skewes’s number (approximately 10^10^10^34), and the Shannon number for chess, estimated at 10^120; the historical attributions are on this entry’s other lines.
• Wikipedia, Shannon Number — confirms Shannon’s 10^120 lower-bound estimate.
• Wikipedia, Skewes’ Number — confirms the historical bound and the modern upper bound.
• Largest known prime number, Wikipedia — read as raw HTML. Its “Largest known prime by year” table gives, consecutively, “M₁₇ | 6 | 1588 | Pietro Cataldi,” “M₁₉ | 6 | 1588 | Pietro Cataldi,” “M₃₁ | 10 | 1772 | Leonhard Euler,” “M₅₉/179951 | 13 | 1867 | Fortuné Landry” and “M₁₂₇ | 39 | 1876 | Édouard Lucas,” which is the 95-year gap the #m31 card names.
• 2,147,483,647, Wikipedia — “By 1772, Leonhard Euler had proven that 2,147,483,647 is a prime.”; “Euler used trial division, improving on Pietro Cataldi’s method, so that at most 372 divisions were needed.”; “In 1867, the number 3,203,431,780,337 was proven to be prime.”; “In computing, this number is the largest value that a signed 32-bit integer field can hold.”; “hexadecimal 7FFFFFFF”; “The latest time that can be represented in this form is 03:14:07 UTC on Tuesday, 19 January 2038 (corresponding to 2,147,483,647 seconds since the start of the epoch).”; “This means that systems using a 32-bit time_t type are susceptible to the Year 2038 problem.”; “On 1 January 2022, a bug was reported for Microsoft Exchange systems where email delivery would fail. An internal malware scanner (enabled by default since 2013) used the date and time as a signed 32-bit integer.”
• Steinhaus–Moser notation, Wikipedia — “It is an extension (devised by Leo Moser) of Hugo Steinhaus’s polygon notation.”; “a number n in a triangle means n^n”; “a number n in a square is equivalent to “the number n inside n triangles, which are all nested.””; “Moser’s number is the number represented by “2 in a megagon.””; “Therefore, Moser’s number, although incomprehensibly large, is vanishingly small compared to Graham’s number”
• big numbers, Susan Stepney (University of York) — gives Skewes’ number as e^e^e^79, approximately 10^10^10^34, “< 4^^5, occurs in theories on the distribution of primes”; with Wikipedia’s mega ≈ (256↑)^256 257 this puts Moser’s number far above it
• The Sand Reckoner, Wikipedia — “The Sand Reckoner (Greek: Ψαμμίτης, Psammites) is a work by Archimedes, an Ancient Greek mathematician of the 3rd century BC, in which he set out to determine an upper bound for the number of grains of sand that fit into the universe.”; “To do this, he used the heliocentric model of Aristarchus of Samos.”; “He then constructed the orders of the second period by taking multiples of this unit”; “Multiplying 10^21 by 10^42 yields 10^63, the number of grains of sand in the Aristarchian Universe.”

• Wikipedia, Knuth’s up-arrow notation — confirms the recursive arrow definitions and 3↑↑3 value.
• Wikipedia, Conway chained arrow notation — confirms the chain notation and its growth comparisons.
• Wikipedia, Fast-growing hierarchy — confirms the ordinal-indexed hierarchy and its use for growth comparison; it gives “f_ω+1(64) = f_ω⁶⁴(64) > Graham’s number” (sub- and superscripts as typeset on the page).
• Wikipedia, Graham’s number — the number Gardner published in 1977 is not the bound in Graham and Rothschild’s 1971 paper, which is smaller; the best known bounds on the original problem are 13 (Barkley, 2008) below and, above, the 2014 bound 2↑↑(2↑↑(3 + 2↑↑8)) of Lavrov, Lee and Mackey, sharpened again in 2019; the last ten digits are …2464195387; Martin Gardner described it in Scientific American “in November 1977,” writing that Graham had established “a bound so vast that it holds the record for the largest number ever used in a serious mathematical proof.”; “At the time of its introduction, it was the largest specific positive integer ever to have been used in a published mathematical proof.”
• Loader’s number, Googology Wiki — community wiki, not primary: “The final output of D^5(99) is much larger than TREE(3), SCG(13), and, say, BH(100).”
• Busy beaver, Wikipedia — “This is a noncomputable function, because it grows asymptotically faster than any computable function.”
• Largest known prime number, Wikipedia — perishable: “The largest known prime number as of September 2026 is 2^136,279,841 − 1, a number that has 41,024,320 digits when written in the decimal system.”
• Googolplex, Wikipedia — “A googolplex is the large number 10^(10^100), that is, 10 raised to the power of a googol.”
• Skewes’s number, Wikipedia — “Skewes (1933) proved that, assuming that the Riemann hypothesis is true, there exists a number x violating π(x) < li(x), below e^e^e^79 < 10^10^10^34.” The comparison with 3↑↑↑3 is arithmetic.
• Friedman’s SSCG function, Wikipedia — “Friedman’s SSCG function is a mathematical function defined by Harvey Friedman.”; definition (displayed): “There is a sequence G1,…,Gn of simple subcubic graphs such that each Gi has at most i+k vertices and for no i < j is Gi homeomorphically embeddable into Gj.”; “The Robertson–Seymour theorem proves that subcubic graphs (simple or not) are well-quasi-ordered by homeomorphic embeddability, implying such a sequence cannot be infinite.”; “The first few terms of the sequence are SSCG(0) = 2, SSCG(1) = 5, and SSCG(2) = 3·2^(3·2^95) − 8” (displayed formulas); “It has been shown that the next term, SSCG(3), is greater than TREE(3).”
• Marco Ripà, Graham’s number stable digits: An exact solution (Notes on Number Theory and Discrete Mathematics 31(3), 2025, 607–616; arXiv 2024) — read in the arXiv HTML full text: the most significant digit of G is 1 in binary and in ternary, but “the problem of expressing the most significant digit of G in the decimal number system is still open.”

• Own computation — direct high-precision evaluation of π gives initial partial quotients 3, 7, 15, 1. Their exact truncations are 3, 22/7, 333/106, and 355/113. π ≈ 3.141592653589793 and 355/113 ≈ 3.141592920353982, both rounding to 3.141593 at six places.
• Wolfram MathWorld, Continued Fraction — the simple-continued-fraction notation (convergents are only linked from it, under See also; the π approximations are on the Milü line).
• DLMF, §24.2 Bernoulli and Euler numbers — confirms B0, B1 convention, B2, and B4.
• Wikipedia, Harmonic Number — confirms H4=25/12 and the nonintegrality theorem for Hn, n>1; on growth: “The values of the sequence H_n − ln n decrease monotonically towards the limit” … “where γ ≈ 0.5772156649 is the Euler–Mascheroni constant.”
• Milü, Wikipedia — “355/113 is the best rational approximation of π with a denominator of four digits or fewer, being accurate to six decimal places.”; “The next rational number (ordered by size of denominator) that is a better rational approximation of π is 52163/16604, though it is still only correct to six decimal places.”; the Milü “is the name given to an approximation of π ( pi ) found by the Chinese mathematician and astronomer Zu Chongzhi during the 5th century.”

• Wolfram MathWorld, Constructible Polygon — confirms the Fermat-prime criterion and the 17-gon/heptagon distinction.
• Wolfram MathWorld, Plastic Constant — confirms x^3=x+1 and the value 1.32471795… .
• Wolfram MathWorld, Conway’s Constant — confirms the look-and-say growth rate, value, and degree-71 polynomial.
• Constructible number, Wikipedia — the power-of-two criterion and the √2 example: “For instance, the square root of 2 is constructible, because it can be described by the formulas √2 or √(1+1).”; “It follows from this decomposition that the degree of the field extension [Q(γ):Q] is 2^r, where r counts the number of quadratic extension steps.”; “Consequently, if a complex number γ is constructible, then the above characterization implies that [Q(γ):Q] is a power of two.”; Gauss’s argument “was generalized in his 1801 book Disquisitiones Arithmeticae giving the sufficient condition for the construction of a regular n-gon.”
• Trigonometry Angles--Pi/17, Wolfram MathWorld — the value, equation (12): “cos((2pi)/(17)) = 1/(16)[delta+sqrt(2)(alpha+epsilon^*)] approx 0.93247”
• Minimal polynomial of 2cos(2pi/n), Wikipedia — the degree, 8 for n = 17: “it follows that the degree of Ψn(x) is 1 for n=1,2 and φ(n)/2 for n≥3.”
• Gauss’s diary, Wikipedia — the date: “Entry 1, dated 1796, March 30, states “Principia quibus innititur sectio circuli, ac divisibilitas eiusdem geometrica in septemdecim partes etc.,” which records Gauss’s discovery of the construction of a heptadecagon by ruler and compass.”
• Heptadecagon, Wikipedia — the first since the Greeks: “This proof represented the first progress in regular polygon construction in over 2000 years.”
• Constructing the Heptadecagon, MathPages — the gravestone as a story: “Gauss was clearly fond of this discovery, and there’s a story that he asked to have a heptadecagon carved on his tombstone, like the sphere incribed in a cylinder on Archimedes’ tombstone. The story is probably apochryphal”
• Heptagon, Wikipedia — gives 2cos(2π/7) ≈ 1.247 and: “The impossibility of straightedge and compass construction follows from the observation that 2cos(2π/7) ≈ 1.247 is a zero of the irreducible cubic x³ + x² − 2x − 1. Consequently, this polynomial is the minimal polynomial of 2cos(2π/7), whereas the degree of the minimal polynomial for a constructible number must be a power of 2.”
• Lehmer’s conjecture, Wikipedia — the value to nine places: “The smallest known Mahler measure (greater than 1) is for “Lehmer’s polynomial,”” x^10 + x^9 − x^7 − x^6 − x^5 − x^4 − x^3 + x + 1, “for which the Mahler measure is the Salem number” 1.176280818…
• Salem number, Encyclopedia of Mathematics — degree and polynomial: “The smallest known Salem number is the number σ1 = 1.1762808… of degree 10 known as Lehmer’s number. The minimum polynomial of σ1 is Lehmer’s polynomial: x^10 + x^9 − x^7 − x^6 − x^5 − x^4 − x^3 + x + 1.”
• Eriko Hironaka, What is... Lehmer’s number?, Notices of the AMS 56 (2009), author’s copy — E10 and knot theory: “Thus, Lehmer’s number is the geometric dilatation of φ, and the spectral radius of the Coxeter element of E10.”; “In particular, Lehmer’s number λ is the homological dilatation of the (-2, 3, 7)-pretzel knot (shown in Fig.1, center), and is the Mahler measure of its Alexander polynomial.”
• Algebraic Number, Wolfram MathWorld — degrees of algebraic numbers: “An example of a complex algebraic number is i, and an example of a real algebraic number is sqrt(2), both of which are of degree 2.”; its table of degrees lists “tribonacci constant 3.”
• Carl Friedrich Gauss, MacTutor History of Mathematics — the birth date: “Born 30 April 1777 Brunswick, Duchy of Brunswick (now Germany).” (Wikipedia’s Gauss article has him “only 19 years old” when he proved the construction; the diary entry of 30 March 1796 is a month before his nineteenth birthday.)

• Wikipedia, Lindemann–Weierstrass theorem — states the theorem — if α₁, …, αₙ are algebraic and linearly independent over ℚ then e^α₁, …, e^αₙ are algebraically independent over ℚ — records that Lindemann proved the e^α case in 1882 and Weierstrass the general statement in 1885, and derives the transcendence of e and of π, and adds: “A slight variant on the same proof will show that if α is a non-zero algebraic number then sin(α), cos(α), tan(α) and their hyperbolic counterparts are also transcendental.” (The logarithms are on the Natural logarithm of 2 line.)
• Wikipedia, Gelfond–Schneider theorem — “If a and b are algebraic numbers with a ∉ {0,1} and b irrational, then any value of a^b is a transcendental number”; “originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider”; and it “answers affirmatively Hilbert’s seventh problem.”
• Dottie number, Wikipedia — value, fixed point, transcendence, and name: “D = 0.739085133215160641655312087673...”; “It is the single real-valued fixed point of the cosine function”; “It is also a transcendental number because of the Lindemann–Weierstrass theorem.”; “It originates from a professor of French named Dottie who observed the number by repeatedly pressing the cosine button on her calculator.”
• Elliptic Functions and Transcendence (survey, PDF), Michel Waldschmidt — Gelfond’s 1929 proof for e^π: “The transcendence of e^π had already been proved in 1929 by A.O. Gel’fond.”; and on e^(π√163) (Remark 14), with τ = (1 + i√163)/2: “the class number of the imaginary quadratic field Q(τ) is 1, we have j(τ) = −(640 320)³ and |j(τ) − 1/q − 744| < 10^−12”.
• Transcendental number, Wikipedia — e–π combinations mostly open: “Most nontrivial combinations of two or more transcendental numbers are themselves not known to be transcendental or even irrational,” listing π^e among them.
• Hilbert’s seventh problem, Wikipedia — the year of the problem: “Hilbert’s seventh problem is one of David Hilbert’s list of open mathematical problems posed in 1900.”
• Gelfond–Schneider constant, Wikipedia — 2^√2 as Hilbert’s test case: “In the address he gave two explicit examples, one of them being the Gelfond–Schneider constant 2^√2.”; also “which was proved to be a transcendental number by Rodion Kuzmin in 1930.”
• Exponentiation, Wikipedia — the branches of i^i, i^i = e^(−π/2) e^(−2kπ): “So, all values of i^i are real, the principal one being e^(−π/2) ≈ 0.2079.”
• Constructible number, Wikipedia — on squaring the circle: “It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility, by extending the work of Charles Hermite and proving that π is a transcendental number.”
• Natural logarithm of 2, Wikipedia — value ln 2 ≈ 0.693 147 180 559…, the series 1 − 1/2 + 1/3 − …, and transcendence: “This is the well-known “alternating harmonic series.””; “By the Lindemann–Weierstrass theorem, the natural logarithm of any natural number other than 0 and 1 (more generally, of any positive algebraic number other than 1) is a transcendental number.”
• Ternary numeral system, Wikipedia — the trit in bits: “One trit is equivalent to log2 3 (about 1.58496) bits of information.”
• Irrational number, Wikipedia — why log2 3 is irrational: “The number 2 raised to any positive integer power must be even (because it is divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its prime factors will be 2).”
• Universal parabolic constant, Wikipedia — value, definition and transcendence: “It is defined as the ratio, for any parabola, of the arc length of the parabolic segment formed by the latus rectum to the focal parameter.”; “P = ln(1+√2)+√2 = 2.29558714939…”; “Hence P is transcendental.”

• OEIS A066716, Champernowne-related constant record — records Mahler’s transcendence result for the base-b Champernowne constant.
• OEIS A010060, Thue–Morse sequence — confirms the binary sequence and the transcendence statement for its associated constant.
• Wikipedia, Komornik–Loreti constant — q = 1.787231650…, the smallest base in which 1 has a unique expansion, defined by Vilmos Komornik and Paola Loreti in 1998; Jean-Paul Allouche and Michel Cosnard proved it transcendental in the American Mathematical Monthly 107 (2000), 448–449; its digits are the Thue–Morse sequence: “The Komornik–Loreti constant is the value q such that” 1 = Σ t_k/q^k, “where t_k is the Thue–Morse sequence.”
• Adamczewski and Bugeaud, On the complexity of algebraic numbers I: expansions in integer bases — the abstract states that irrational automatic and morphic numbers are transcendental, which is the theorem that settles the paperfolding constant; published as Annals of Mathematics 165 (2007), 547–565.
• Regular paperfolding sequence, Wikipedia — value, digits, not periodic: “This number is known as the paperfolding constant and has the value,” 0.85073618820186...; “The paperfolding word 1101100111001001..., which is created by concatenating the terms of the regular paperfolding sequence”; “A paperfolding sequence is not ultimately periodic.”
• Automatic sequence, Wikipedia — “Both the Baum–Sweet sequence (OEIS: A086747) and the regular paperfolding sequence (OEIS: A014577) are automatic.”
• Borwein and Coons, Transcendence of the Gaussian Liouville number and relatives, arXiv (2008) — the same binary digits, proved transcendental: “define the Gaussian Liouville number by γ:=0.110110011100100111011...”; “we prove that the Gaussian Liouville number and its relatives are transcendental.”
• Thue–Morse sequence, Wikipedia — the construction: “the binary sequence (an infinite sequence of 0s and 1s) that can be obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far.”

• Wikipedia, Lemniscate constant — ϖ ≈ 2.622 was proved transcendental by Carl Ludwig Siegel in 1932 and again by Theodor Schneider in 1937; Gauss’s constant is G = ϖ/π ≈ 0.8346268 = 1/M(1,√2), and Chudnovsky proved π and ϖ algebraically independent.
• Wikipedia, Cahen’s constant — C ≈ 0.643410546288, the alternating sum of 1/(s_i − 1) over Sylvester’s sequence s_i (1/1 − 1/2 + 1/6 − 1/42 + …); Eugène Cahen introduced it and proved it irrational in 1891, and Davison and Shallit proved it transcendental in 1991 from its continued fraction.
• Wolfram MathWorld, Conway’s Constant — confirms the look-and-say constant used in the algebraic brace.
• Wikipedia, Particular values of the gamma function — credits G. V. Chudnovsky with proving Γ(1/3) and Γ(1/4) transcendental, and Yuri Nesterenko (1996) with the algebraic independence of Γ(1/4), π and e^π; Chudnovsky’s own independence result (π and ϖ, 1975) is on the Lemniscate constant line.
• Elliptic Functions and Transcendence (survey, PDF), Michel Waldschmidt — Siegel and Schneider on periods: “The first transcendence result on periods of elliptic functions was proved by C.L. Siegel [199] as early as 1932.”; “Theorem 18 (Schneider, 1936) Assume that the invariants g2 and g3 of ℘ are algebraic. Then for any non-zero period ω of ℘, the numbers ω and η(ω) are transcendental.”

• Wikipedia, Lindemann–Weierstrass theorem — confirms π is transcendental and the standard exponential consequences.
• Own computation — The numerator contains sin(π) as a multiplicative factor, and sin(π)=0; evaluating the remaining displayed parenthetical with standard high-precision arithmetic gives approximately 2.4, but it cannot affect the product.
• Pi, Wikipedia — sin π = 0 with π the first positive zero: “π is also the smallest positive number at which the sine function equals zero, and the difference between consecutive zeroes of the sine function.”

The rider deliberately mixes algebraic, proven-transcendental and status-open ingredients; no claim that the rider itself is transcendental is made.

• Own computation — 1729−1729=0; 3·(1/3)−1=0; 0.999…−1=0; sin(π)=0. Every denominator is a sum/product of positive displayed values, hence nonzero, so the product is 0·0·0·0=0.

• Wikipedia, Carmichael number — gives the coprime-base formulation and the equivalent b^n≡b formulation. Re-read raw on 17 September 2026 for the priority: “The first seven Carmichael numbers, from 561 to 8911, were all found by the Czech mathematician Václav Šimerka in 1885” and his work “remained unnoticed”; “Theorem (A. Korselt 1899)” and “he did not give any examples”; “In 1910, Carmichael himself also published the smallest such number, 561, and the numbers were later named after him”; “In 1994 W. R. (Red) Alford, Andrew Granville and Carl Pomerance ... show that there really do exist infinitely many Carmichael numbers”.

• Wolfram MathWorld, Fibonacci Number — cites Cohn (1964) and states that the only square Fibonacci numbers are 1 and F12=144 under the convention beginning 1,1.
• Fibonacci sequence, Wikipedia — carries the 1202 date and the rabbits: “The Fibonacci sequence first appears in the book Liber Abaci (The Book of Calculation, 1202) by Fibonacci, where it is used to calculate the growth of rabbit populations.”

• GIMPS, M136279841 press release — confirms the discoverer, date, exponent and digit count, and “It is only the 52nd known Mersenne prime ever discovered.”

• Wikipedia, Hilbert’s seventh problem — names Hilbert’s examples 2^√2 and e^π.
• Wikipedia, Gelfond–Schneider theorem — confirms the theorem used for √2^√2.
• Irrational number, Wikipedia — the classic proof: “Dov Jarden gave a simple non-constructive proof that there exist two irrational numbers a and b, such that a^b is rational:”

• J. S. Milne, Fields and Galois Theory — Read 12 September 2026: Theorem 1.37 and Corollaries 1.38–1.39 give the successive-square-root criterion, algebraicity, and the impossibility of constructing ∛2; Remark 3.27 explains why power-of-two degree alone is insufficient.
• Constructible number, Wikipedia — (§ Algebraic properties, § History, § Impossible constructions): a real is constructible exactly when it “lies in a field at the top of a finite tower of real quadratic extensions,” with [ℚ(γ):ℚ] = 2ʳ; “Pierre Wantzel proved algebraically that the problems of doubling the cube and trisecting the angle are impossible to solve using only compass and straightedge”; and the two minimal polynomials, x³ − 2 for ∛2 and 8x³ − 6x − 1 for cos(π/9), both of degree 3; and “However, this condition is not sufficient - there exist field extensions whose degree is a power of two, but which cannot be factored into a sequence of quadratic extensions.”
• MacTutor, Pierre Wantzel — the biography for the 1837 paper in which the criterion and the two impossibility results appear.
• Computable number, Wikipedia — “The computable numbers include the specific real numbers which appear in practice, including all real algebraic numbers, as well as e, π, and many other transcendental numbers.”

• Transcendental number, Wikipedia (§ History) — “Joseph Liouville first proved the existence of transcendental numbers in 1844, and in 1851 gave the first decimal examples such as the Liouville constant”; “In 1874 Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable”; and “almost all real and complex numbers are transcendental, since the algebraic numbers are countable, while the real numbers and complex numbers are both uncountable.”
• Cantor’s first set theory article, Wikipedia — Cantor’s paper of 1874, Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, proves that the set of real algebraic numbers can be put in one-to-one correspondence with the positive integers and that an interval of reals cannot, which together give a transcendental number in every interval.
• Joseph Liouville, MacTutor History of Mathematics — he proved “the existence of a transcendental number in 1844 when he constructed an infinite class of such numbers using continued fractions. In 1851 he published results on transcendental numbers removing the dependence on continued fractions.”

The chapter’s opening history sentence previously carried 4.9, which states only that the classical transcendence claims follow from the Lindemann-Weierstrass and Gelfond-Schneider theorems; neither Liouville’s priority nor Cantor’s counting argument is in it, and Cantor’s 1874 carried no footnote at all. This entry carries both.

• Wolfram MathWorld, Catalan’s Conjecture — “The conjecture made by Belgian mathematician Eugene Charles Catalan in 1844” that 8 and 9 are the only consecutive powers, with Mihailescu’s proof sent to several mathematicians on 18 April 2002; the original note is cited as Catalan, J. reine angew. Math. 27, 192, 1844.
• Catalan’s conjecture, Wikipedia — “conjectured by the mathematician Eugène Charles Catalan in 1842 and proven in 2002 by Preda Mihăilescu,” with the same primary note cited as J. Reine Angew. Math. 27 (1844), 192, and the proof published in 2004.
• Square pyramidal number, Wikipedia (§ Cannonball problem) — “Besides 1, there is only one other number that has this property: 4900, which is both the 70th square number and the 24th square pyramidal number”; the first complete proof that no others exist is G. N. Watson’s, 1918.
• Own computation — Enumerating the square pyramidal numbers n(n+1)(2n+1)/6 for n up to 200 and testing each for squareness gives exactly n = 0, 1 and 24, that is 0, 1 and 4900, so 4900 is the only one after 1.

Two comparable sources date the conjecture differently — MathWorld says 1844, Wikipedia’s lead says 1842 — while both cite the same primary note, Crelle 27 (1844), 192. The page keeps 1844, the year of the publication both sources point at; no tilde is needed because the disagreement is over an undated secondary assertion, not over two datable events. The aside’s superlative for 4900 is corrected to “the only square after 1,” the same form this chapter’s fib144 card already uses for 144.

• Narcissistic number, Wikipedia — gives the base-10 example 153 = 1³ + 5³ + 3³.
• 2520 (number), Wikipedia — the smallest number divisible by all integers from one to ten, their least common multiple.
• Wheat and chessboard problem, Wikipedia — the total number of grains is 2^64 − 1, or 18,446,744,073,709,551,615.
• Names of large numbers, Wikipedia — lists centillion as 10^303 (short scale) and 10^600 (long scale), and says centillion appears to be the highest name ending in -illion included in the dictionaries it surveys.
• Theodorus of Cyrene, Wikipedia — quotes Plato’s Theaetetus: Theodorus showed the square roots of the non-square numbers from 3 up to 17 irrational; his method is unknown.
• Square root of 3, Wikipedia — value 1.7320508075688772…, also called Theodorus’s constant.
• Supergolden ratio, Wikipedia — the unique real solution of x³ = x² + 1, 1.465571231876768….
• Twelfth root of two, Wikipedia — 1.0594630943592952646; in the equal-tempered chromatic scale each note is 2^(1/12) times the frequency of the one below.
• Turn (angle), Wikipedia — one turn is 2π radians, proposed to be written τ, citing Michael Hartl’s Tau Manifesto (2010); 1 turn = τ rad ≈ 6.283185307 rad.
• Golden angle, Wikipedia — ≈ 137.50776°, 360(2 − φ) degrees or 2π/φ² radians; the angle separating the florets on a sunflower; Vogel (1979) is in its reading list, and his model is on the Fermat’s spiral line.
• Magic Angle, Wolfram MathWorld — θ_m = cos^(-1)(1/√3) = 54.7356103… degrees.
• Magic angle spinning, Wikipedia — the sample is spun at the magic angle, about 54.74°, to the magnetic field in solid-state NMR.
• Transcendental number, Wikipedia — lists nonzero values of arcsin and arccos at algebraic arguments as transcendental by Lindemann–Weierstrass, and says a non-constant algebraic function of π, such as 5π, is transcendental.
• Phyllotaxis, Wikipedia — why successive elements never line up: “Since three golden arcs add up to slightly more than enough to wrap a circle, this guarantees that no two leaves ever follow the same radial line from center to edge.”
• Fermat’s spiral, Wikipedia — Vogel’s model: “The full model proposed by H. Vogel in 1979 is r = c√n, θ = n × 137.508°, where θ is the angle, r is the radius or distance from the center, and n is the index number of the floret and c is a constant scaling factor.” and “The angle 137.508° is the golden angle which is approximated by ratios of Fibonacci numbers.”
• Constructible number, Wikipedia — carries √3’s constructibility as a square root of a positive number: “The algebraically constructible real numbers are the subset of the real numbers that can be described by formulas that combine integers using the operations of addition, subtraction, multiplication, multiplicative inverse, and square roots of positive numbers.”

• Factorial, Wikipedia — read as raw HTML: “The notation n! for factorials was introduced by the French mathematician Christian Kramp in 1808.”
• 5040 (number), Wikipedia — read as raw HTML: “Plato mentions in his dialogue Laws that 5040 is a convenient number to use for dividing many things (including both the citizens and the land of a city-state or polis) into lesser parts, making it an ideal number for the number of citizens (heads of families) making up a polis,” and “this number can be divided by all the (natural) numbers from 1 to 12 with the single exception of 11.” The page gives the reason Plato gives, which is divisibility rather than the divisor count; 5040 = 2⁴·3²·5·7 has 5·3·2·2 = 60 divisors, which is the book’s own arithmetic.
• Primorial, Wikipedia — read as raw HTML: “The name ‘primorial’, coined by Harvey Dubner, draws an analogy to primes similar to the way the name ‘factorial’ relates to factors,” and its reference list gives Dubner, Harvey (1987), “Factorial and primorial primes,” J. Recr. Math. 19, 197-203.
• Golden ratio, Wikipedia — read as raw HTML: “The golden ratio was called the extreme and mean ratio by Euclid,” and Euclid’s Elements (c. 300 BC) “contains its first known definition,” “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.”
• Hala King and Victor Pambuccian, Generalized Schatunowsky theorem in a weak arithmetic, arXiv:2506.08256 — abstract: “Schatunowsky’s 1893 theorem, that 30 is the largest number all of whose totatives are primes, has been recently generalized by Kaneko and Nakai.”

Added 17 September 2026. Six cards in this chapter hung datable history on r-4.1, whose claim line covers the chapter’s elementary identities and type classifications and whose three sources are the DLMF Bernoulli tables, MathWorld on algebraic numbers and Brocard’s problem. None of that history was in it. Two of the six, #sixty and #twenty, were re-pointed at this chapter’s own r-2.3 and r-2.4; #sqrt2’s Pythagorean tradition already stood on r-8.9; the remaining three, #fact7, #primorial5 and #phi, are what this entry carries. Note what the Plato source does and does not say: the reason he gives is that 5040 divides by every number from 1 to 12 but 11, not that it has 60 divisors, so the card now says both and attributes only the first to him.

• Leopold Kronecker, Wikiquote — read as raw HTML: “Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. God made the integers, all the rest is the work of man. In a lecture to the Berliner Naturforschung-Versammlung, 1886; quoted in the obituary ‘Leopold Kronecker’, by H[einrich] Weber, in Jahresbericht der Deutsche Mathematiker Vereinigung, 1893, page 19,” which is the page’s footer line word for word.

Added 17 September 2026 to source the chapter epigraph, which carried no marker. This is one of the four secondhand epigraphs where the book asserts provenance rather than quoting a printed original, so the reporter is named in the claim line as well as in the footer.

• Simple continued fraction, Wikipedia — The earlier annotation used this page for the procedure and described a best-approximation superlative too broadly. That universal equal-denominator assertion is no longer used in this entry: the direct rational counterexample in its claim and the NIST definitions below control the book’s wording.
• Continued Fraction, MathWorld — the same simple-continued-fraction notation (convergents are only linked from it, under See also).
• NIST DLMF, §1.12 Continued Fractions — Section 1.12(ii), equations 1.12.3–4, defines the finite truncations as convergents. Its indexing begins at zero; the book counts the initial integer truncation as first. The equal-denominator counterexample in the claim is direct rational arithmetic.

• Salem number, Encyclopedia of Mathematics — read as raw HTML: “An algebraic integer θ > 1 such that all other Galois conjugates of θ lie inside the closed unit disc |z| ≤ 1, with at least one lying on the boundary”; reciprocity, even degree ≥ 4; “The smallest known Salem number is the number σ₁ = 1.1762808… of degree 10 known as Lehmer’s number… This is also the smallest known value > 1 of the Mahler measure.”
• Salem number, Wikipedia — read as raw HTML: the same definition, names them after Raphaël Salem, gives Lehmer’s polynomial explicitly with root about 1.17628, states that every Salem number is a unit and a Perron number, and that it is conjectured to be the smallest.
• Pisot–Vijayaraghavan number, Wikipedia — the definition: “In mathematics, a Pisot–Vijayaraghavan number (or Pisot number or PV number) is a real algebraic integer greater than 1, all of whose Galois conjugates are less than 1 in absolute value.”

• Heegner number, Wikipedia — read as raw HTML: “Heegner numbers are square-free positive integers d such that the imaginary quadratic field ℚ(√−d) has class number 1. Equivalently, the ring of algebraic integers… has unique factorization”; “there are precisely nine Heegner numbers: 1, 2, 3, 7, 11, 19, 43, 67, and 163… conjectured by Gauss and proved up to minor flaws by Kurt Heegner in 1952. Alan Baker and Harold Stark independently proved the result in 1966”; and “if the imaginary quadratic extension ℚ(τ) has class number 1… the j-invariant is an integer.”
• Class Number, MathWorld — the class number as the order of the ideal class group, and its being 1 exactly for unique factorization.

• Amicable Pair, Wolfram MathWorld — read raw: “In 1866, 16-year old B. Nicolò I. Paganini found the small amicable pair (1184, 1210) which had eluded his more illustrious predecessors (Paganini 1866-1867; Dickson 2005, p. 47).”; “In 1636, Fermat found the pair (17296, 18416) and in 1638, Descartes found (9363584, 9437056), although these results were actually rediscoveries of numbers known to Arab mathematicians. By 1747, Euler had found 30 pairs, a number which he later extended to 60.”
• Bernoulli number, Wikipedia — read raw: “Bernoulli’s result was published posthumously in Ars Conjectandi in 1713. Seki Takakazu independently discovered the Bernoulli numbers and his result was published a year earlier, also posthumously, in 1712.”; “The Bernoulli numbers are related to Fermat’s Last Theorem (FLT) by Kummer’s theorem, which says: If the odd prime p does not divide any of the numerators of the Bernoulli numbers B₂, B₄, ..., Bₚ₋₃ then xᵖ + yᵖ + zᵖ = 0 has no solutions in nonzero integers. Prime numbers with this property are called regular primes.” (sub- and superscripts as typeset on the page)
• Doubling the cube, Wikipedia — read raw: “The nonexistence of a compass-and-straightedge solution was finally proven by Pierre Wantzel in 1837..”
• Conway chained arrow notation, Wikipedia — read raw: “John H. Conway & Richard K. Guy, The Book of Numbers, 1996, p.59-62.”
• Champernowne constant, Wikipedia — read raw: “It is named after economist and mathematician D. G. Champernowne, who published it as an undergraduate in 1933.”; “K. Mahler, Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen, Proc. Konin. Neder. Akad. Wet. Ser. A. 40 (1937), p. 421–428..”
• Look-and-say sequence, Wikipedia — read raw: “title=The Weird and Wonderful Chemistry of Audioactive Decay |journal=Eureka |date=January 1986 |volume=46 |pages=5–16.”
• Constructible polygon, Wikipedia — read raw: “A full proof of necessity was given by Pierre Wantzel in 1837..”
• e (mathematical constant), Wikipedia — read raw: “Jacob Bernoulli discovered this constant in 1683, while studying a question about compound interest.”
• Amicable numbers, Wikipedia — dates the find a year later than MathWorld: “The second smallest pair, (1184, 1210), was discovered in 1867 by 16-year-old B. Nicolò I. Paganini (not to be confused with the composer and violinist), having been overlooked by earlier mathematicians.”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• The number e, MacTutor History of Mathematics — read raw: “Whatever the reason, the notation e made its first appearance in a letter Euler wrote to Goldbach in 1731..”
• e (mathematical constant), Wikipedia — read raw: “It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with Liouville number); the proof was given by Charles Hermite in 1873..”
• Eddington number, Wikipedia — read raw: “During a course of lectures that he delivered in 1938 as Tarner Lecturer at Trinity College, Cambridge, Eddington averred that:.”
• Some historical notes on proving Fermat numbers and cofactors to be composite, 64ordle.au — read raw: “Thus in the second half of 1640 we find Fermat writing at least three letters to Frénicle directly, and at least once more to Mersenne..”
• Fermat number, Wikipedia — read raw: “Fermat’s conjecture was refuted by Leonhard Euler in 1732.”
• 65537-gon, Wikipedia — read raw: “the first explicit construction of a regular 65537-gon was given by Johann Gustav Hermes (1894). The construction is very complex; Hermes spent 10 years completing the 200-page manuscript.”; “Although it was known to Carl Friedrich Gauss by 1801 that the regular 65537-gon was constructible.”
• Hierarchies of number-theoretic functions. I (Löb and Wainer), Archiv für mathematische Logik und Grundlagenforschung, Springer — read raw: “Arch math Logik 13, 39–51 (1970). https://doi.org/10.1007/BF01967649.”
• Elliptic functions: introduction course, lecture 2 (the AGM), Vladimir G. Tkachev, Linköping University — read raw: “By May 30th, 1799, Gauss had observed, purely computationally, that” 1/M(1, √2) and (2/π)∫₀¹ dt/√(1 − t⁴) “agreed to at least eleven (!) decimal places.”
• Lemniscate constant, Wikipedia — read raw: “Todd’s second lemniscate constant B and Gauss’s constant G were proven transcendental by Theodor Schneider in 1941..”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Hilbert’s seventh problem, Wikipedia — read raw: “Hilbert’s seventh problem is one of David Hilbert’s list of open mathematical problems posed in 1900..”
• Googol, Wikipedia — read raw: “The term was coined in 1920 by nine-year-old Milton Sirotta (1911–1981), nephew of American mathematician Edward Kasner.”; “Kasner popularized the concept in his 1940 book Mathematics and the Imagination.”; “Carl Sagan pointed out that the total number of elementary particles in the observable universe is around 10^80 (the Eddington number)” (exponent as typeset on the page).
• Googolplex, Wikipedia — read raw: “In 1920, Edward Kasner’s nine-year-old nephew, Milton Sirotta, coined the term googol.”
• Bemerkung über die harmonische Reihe (L. Theisinger), Monatshefte für Mathematik und Physik, Springer — read raw: “Monatsh. f. Mathematik und Physik 26, 132–134 (1915). https://doi.org/10.1007/BF01999444.”
• Hyperoperation, Wikipedia — read raw: “In 1928, Wilhelm Ackermann defined a 3-argument function”; “In his 1947 paper, Reuben Goodstein introduced the specific sequence of operations that are now called hyperoperations, and also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation.”
• Kaprekar Routine, Wolfram MathWorld — read raw: “The Kaprekar routine is an algorithm discovered in 1949 by D. R. Kaprekar for 4-digit numbers.”
• Knuth’s up-arrow notation, Wikipedia — read raw: “Knuth’s up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976..”
• Edward Kasner, MacTutor History of Mathematics — quoting Mathematics and the Imagination: “The name “googol” was invented by a child (Dr Kasner’s nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it.”; “At the same time that he suggested “googol” he gave a name for a still larger number: “Googolplex.””
• Yang, Li, Feng and Jiang, On the integrality of the first and second elementary symmetric functions, AIMS Mathematics 2(4) (2017) — the first name and year: “The first published proof went back to 1915 by Leopold Theisinger.”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Hyperoperation, Wikipedia — read raw: “In 1928, Wilhelm Ackermann defined a 3-argument function \phi(a, b, n) which gradually evolved into a 2-argument function known as the Ackermann function..”
• Lehmer’s Mahler Measure Problem, Wolfram MathWorld — read raw: “Lehmer’s Mahler measure problem is an unsolved problem in mathematics attributed to Lehmer (1933) that concerns the minimum Mahler measure for a univariate polynomial that is not a product of cyclotomic polynomials..”
• B. Sury, The Arithmetico-geometric Mean of Gauss, Resonance, August 2000 — read raw: “Gauss’s 92nd entry made in 1798 in his famous mathematical diary reads “I have obtained most elegant results concerning the lemniscate, which surpasses all expectation.”
• Arithmetic–geometric mean, Wikipedia — read raw: “In 1799, Gauss proved that M(1,√2) = π/ϖ where ϖ is the lemniscate constant..”
• Joseph Liouville, MacTutor History of Mathematics — read raw: “led him to prove the existence of a transcendental number in 1844 when he constructed an infinite class of such numbers using continued fractions”; “In 1851 he published results on transcendental numbers removing the dependence on continued fractions. In particular he gave an example of a transcendental number, the number now named the Liouvillian number 0 . 1100010000000000000000010000 ....”
• David Moews, BIGNUM BAKEOFF contest rules (the organiser’s own file, dated 10 Dec 2001) — read raw: “Entries must be e-mailed to [email protected] and received before 23:59:59 PST, December 31, 2001..”
• Japanese numerals, Wikipedia — read raw: “The initial edition was published in 1627 and had many errors, most of which were fixed in the 1631 edition. In 1634, there was yet another edition which again changed a few values.”; “Variation is due to the Jinkōki (塵劫記), Japan’s oldest mathematics text.” Its table of large numbers runs in steps of 10⁴ up to 極 (10⁴⁸) and then gives each higher name two values, from “10^52 (or 10^56)” for 恒河沙 to “10^68 (or 10^88)” for 無量大数.
• Transcendental number, Wikipedia — the first decimal examples: “Joseph Liouville first proved the existence of transcendental numbers in 1844, and in 1851 gave the first decimal examples such as the Liouville constant”
• Liouville number, Wikipedia — the 1844 approximation bound: “In 1844, Joseph Liouville proved a bound showing that there is a limit to how well algebraic numbers can be approximated by rational numbers, and he defined Liouville numbers specifically so that they would have rational approximations better than the ones allowed by this bound.”
• BIGNUM BAKEOFF contest recap, David Moews — “The aim of this contest was to write a C program of 512 characters or less (excluding whitespace) that returned as large a number as possible from main()”; “The winner is Ralph Loader”; “The final and winning entry, {loader.c}, diagonalizes over the Huet-Coquand ‘calculus of constructions’.”; “The program’s return value is D@@5(99).”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Japanese numerals, Wikipedia — read raw: “In 1634, there was yet another edition which again changed a few values. The above variation is due to inconsistencies in the latter two editions..”
• Dragon curve, Wikipedia — read raw: “It was described by Martin Gardner in his Scientific American column Mathematical Games in 1967. Many of its properties were first published by Chandler Davis and Donald Knuth.”; “Chandler Davis (1970), Donald J. Knuth (1970).”
• Perfect number, Wikipedia — read raw: “In 1888, Sylvester stated: ... a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number]—its escape, so to say, from the complex web of conditions which hem it in on all sides—would be little short of a miracle.”; “The first four perfect numbers were the only ones known to early Greek mathematics, and the mathematician Nicomachus noted 8128 as early as around AD 100.”
• Irrational number, Wikipedia — read raw: “Johann Heinrich Lambert proved (1761) that π cannot be rational, and that e^n is irrational if n is rational (unless n = 0)..”
• Pi, Wikipedia — read raw: “The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes, implementing the method of exhaustion..”
• Plastic ratio, Wikipedia — read raw: “French high school student Gérard Cordonnier discovered the ratio for himself in 1924. In his correspondence with Hans van der Laan a few years later, he called it the radiant number. Van der Laan initially referred to it as the fundamental ratio, using the plastic number from the 1950s onward..”
• arXiv:1805.07178, Appendix C — read in the paper’s PDF, Appendix C (the abstract page does not carry it): “In 1928, shortly after abandoning his architectural studies and becoming a novice monk of the Benedictine Order, Hans van der Laan discovered a new, unique system of architectural proportions..”
• Heegner number, Wikipedia — read raw: “This number was discovered in 1859 by the mathematician Charles Hermite.”; it gives e^(π√163) “≈ 640 320³ + 744” and “In a 1975 April Fools’ Day article in Scientific American magazine, ‘Mathematical Games’ columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name.”
• The Plastic Number ratio, Dom Hans van der Laan foundation — the three-dimensional contrast with the golden section: “This growth occurs three-dimensionally. Moreover: the Plastic Number represents the smallest proportional growth of three-dimensional objects.”; “This is different than the golden section, which defines the Fibonacci series, and is defined as 1 + a = a2.”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Ramanujan Constant, Wolfram MathWorld — read raw: “Although Ramanujan (1913-1914) gave few rather spectacular examples of almost integers (such ), he did not actually mention the particular near-identity given above.”; “The name “Ramanujan’s constant” was coined by Simon Plouffe and derives from an April Fool’s joke played by Martin Gardner (Apr. 1975) on the readers of Scientific American.”
• Shannon number, Wikipedia — read raw: “His influential 1950 paper on it, “Programming a Computer for Playing Chess”; the estimate is “based on an average of about 10^3 possibilities for a pair of moves consisting of a move for White followed by a move for Black, and a typical game lasting about 40 such pairs of moves” (exponent as typeset on the page).
• Skewes’s number, Wikipedia — read raw: “Skewes (1933) proved that, assuming that the Riemann hypothesis is true, there exists a number x violating π(x) ‹ li(x), below e^{e^{e^{79}}}‹10^{10^{10^{34}}}.”; “Without assuming the Riemann hypothesis, Skewes (1955) later proved that there exists a value of x below e^{e^{e^{e^{7.705}}}}‹10^{10^{10^{964}}}.”; “|2010|| 2.2 || Saouter and Demichel || [exp(727.95132478), exp(727.95134682)] (without RH)”; “by Platt & Trudgian (2014) to 1.39\times 10^{17}, and by Büthe (2015) to 10^{19}..”
• 1729 (number), Wikipedia — read raw: “{{cite book |title = Ramanujan | url = https://archive.org/details/pli.kerala.rare.37877 | last = Hardy | first = G. H. | author-link = G. H. Hardy |location = New York | publisher = Cambridge University Press | year = 1940”; the page quotes it (p. 12): “I remember once going to see him when he was ill at Putney. I had ridden in taxi cab No. 1729 and remarked that the number seemed to me rather a dull one.”
• Prouhet–Thue–Morse constant, Wikipedia — read raw: “The Prouhet–Thue–Morse constant was shown to be transcendental by Kurt Mahler in 1929..”
• Kruskal’s tree theorem, Wikipedia — read raw: “The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963)..”
• Stanley Skewes and the Skewes Number, H. P. Williams (LSE) — “the famous mathematician G.H.Hardy described the Skewes number as “the largest number that has ever served any definite purpose in mathematics.””

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Kruskal’s tree theorem, Wikipedia — read raw: “By incorporating labels, Friedman defined a far faster-growing function.{{Cite web |last=Friedman |first=Harvey |date=28 March 2006 |title=273:Sigma01/optimal/size |url=http://www.cs.nyu.edu/pipermail/fom/2006-March/010279.html.”
• Generalizations of Fibonacci numbers, Wikipedia — read raw: “In 1892, the sequence of numbers appeared in the solution of a recreational problem, concerning a farmer and the raising of sheep, that was posed by the American mathematician Artemas Martin (1835–1918).”; “The moniker tribonacci appeared much later, not until 1963, and is due to Mark Feinberg, at the time a fourteen-year-old high-school student, who introduced the term in an article in the Fibonacci Quarterly.”; “The most notable historical occurrence of the sequence is connected to Charles Darwin (1809–1882) and his seminal book On the Origin of Species, where the procreation and population growth of elephants is considered as an illustrative example.”
• Knuth’s up-arrow notation, Wikipedia — read raw: “introduced by Donald Knuth in 1976.{{cite journal | last =Knuth | first = Donald E.| year=1976|title=Mathematics and Computer Science: Coping with Finiteness |journal=Science | volume=194|issue=4271| pages=1235–1242.”
• 2,147,483,647, Wikipedia — read raw: “The primality of this number was proven by Leonhard Euler, who reported the proof in a letter to Daniel Bernoulli written in 1772..”
• Heptadecagon, Wikipedia — read raw: “the regular heptadecagon is a constructible polygon (that is, one that can be constructed using a compass and unmarked straightedge): this was shown by Carl Friedrich Gauss in 1796..”
• Doubling the cube, Wikipedia — read raw: “But Pierre Wantzel proved in 1837 that the cube root of 2 is not constructible; that is, it cannot be constructed with straightedge and compass..”
• Transcendental number, Wikipedia — read raw: “Joseph Liouville first proved the existence of transcendental numbers in 1844, and in 1851 gave the first decimal examples such as the Liouville constant”; “The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers’ existence was e, by Charles Hermite in 1873..”
• [FOM] 273:Sigma01/optimal/size, Harvey Friedman, 28 March 2006 (archived) — posted “Tue Mar 28 12:57:25 EST 2006”: “Let TREE[k] be the length n of the longest sequence of trees T_1,...,T_n with vertices labeled from {1,...,k}, where each T_i has at most i vertices, and no T_i is inf preserving and label preserving embeddable into a later T_j.”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Real geometric transcendence for the Gamma function (Arshay Sheth), arXiv — read raw: “Gregory Volfovich Chudnovsky, Algebraic independence of constants connected with the exponential and elliptic functions, Dokl. Akad. Nauk Ukrain. SSR Ser. A 1976, no. 8, 698.”
• Graham’s number, Wikipedia — read raw: “This weaker upper bound for the problem, attributed to an unpublished work of Graham, was eventually published and named by Martin Gardner in Scientific American in November 1977..”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Kruskal’s tree theorem, Wikipedia — states the theorem and its role in defining Friedman’s TREE function.

• Hilbert’s seventh problem, Wikipedia — states the problem, Hilbert’s own examples, and the 1934 resolution.
• Hilbert’s problems, Wikipedia — carries the count of the list: “Hilbert’s problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900.”
• Gelfond–Schneider theorem, Wikipedia — carries e^π as (−1)^(−i) and the cases the theorem covers: “(−1)^(−i) = 23.14069263…,” with 2^√2, √2^√2 and i^i = e^(−π/2) listed as consequences.

• Bernoulli number, Wikipedia — gives the defining generating function and the applications to power sums and zeta.

• Harmonic number, Wikipedia — defines the harmonic numbers and states that H_n is an integer only for n = 1, “a result often attributed to Theisinger”; it gives no year (1915 is on the arXiv line). On growth: “The harmonic numbers roughly approximate the natural logarithm function” … “and thus the associated harmonic series grows without limit, albeit slowly.”
• About the Non-Integer Property of Hyperharmonic Numbers, arXiv:0811.0043 — names Theisinger and the year: “It was proven in 1915 by Leopold Theisinger that the H_n harmonic numbers are never integers.”

• Mahler measure, Wikipedia — states Lehmer’s conjecture, that there is a constant μ > 1 such that an irreducible integer polynomial p has either M(p) = 1 or M(p) > μ; Lehmer’s number and its smallest-known status are on the Lehmer’s conjecture and arXiv lines.
• Lehmer’s conjecture, Wikipedia — gives Lehmer’s 1933 paper, his degree-10 polynomial and its measure, and the open status: “P(x) = x^10 + x^9 − x^7 − x^6 − x^5 − x^4 − x^3 + x + 1”; “M(P(x)) = 1.176280818…”; “Lehmer’s conjecture is not yet completely proved and is still a question of much interest.”
• E. Hironaka, Lehmer’s Problem, McKay’s Correspondence, and 2,3,7, arXiv math/0204040 — carries the search record: “Boyd and Mossinghoff have done searches up to degree 40, but so far no one has found a monic noncyclotomic integer polynomial with smaller Mahler measure.”

• Look-and-say sequence, Wikipedia — defines the sequence and states Conway’s 1986 growth-rate result.

• Open Diophantine Problems, M. Waldschmidt, arXiv math/0312440 — states Γ(1/5) is open and Γ(1/2) transcendental: “there is no proof so far that Γ(1/5) is transcendental”; “Lindemann’s result on the transcendence of π which implies that Γ(1/2) = √π is also transcendental”
• Gamma function, Wikipedia — gives the known transcendental values: “It has been proved that Γ(n + r) is a transcendental number and algebraically independent of π for any integer n and each of the fractions r = 1/6, 1/4, 1/3, 2/3, 3/4, 5/6.”
• Lemniscate constant, Wikipedia — ties Γ(1/4) to the lemniscate constant: “ϖ = Γ(1/4)² / (2√(2π))”

• Legendre’s constant, Wikipedia — “Legendre proposed in 1808 the formula y = x/log(x) − 1.08366”; Chebyshev showed in 1849 that the limit, if it exists, must be 1, and de la Vallée Poussin proved in 1899 that it is.

• Erdős–Borwein constant, Wikipedia — the value 1.606695152415291763…, Paul Erdős’s irrationality proof of 1948 and Peter Borwein’s second proof of 1992. Nothing is asserted about transcendence either way, which is the sense in which the card calls it open.
• List of mathematical constants, Wikipedia — the Erdős–Borwein constant row reads “✗” under ℚ (rational) and “?” under 𝔸 (algebraic), 𝒫 and 𝓔𝓟: not rational, algebraic status unknown.

• Reciprocal Fibonacci constant, Wikipedia — the value 3.359885666243177553…, and Richard André-Jeannin’s 1989 proof of irrationality, settling a conjecture of Erdős, Graham and Carlitz. The transcendence results there are for the Fibonacci zeta function at even arguments, not for ψ itself.
• List of mathematical constants, Wikipedia — the Reciprocal Fibonacci constant row reads “✗” under ℚ (rational) and “?” under 𝔸 (algebraic) and 𝒫: irrational, algebraic status unknown.

• Copeland–Erdős constant, Wikipedia — “In base 10, the constant is a normal number, a fact proven by Arthur Herbert Copeland and Paul Erdős” (1946, per the page’s reference), and “It follows that the constant is irrational; this can also be proven with Dirichlet’s theorem on arithmetic progressions or Bertrand’s postulate.”
• Copeland–Erdős Constant, Wolfram MathWorld — the decimal 0.23571113171923… and the 1946 normality result; no transcendence claim is made.
• Marek Wolf, Continued fractions constructed from prime numbers, arXiv:1003.4015 — “Arthur Herbert Copeland and Paul Erdős have shown that CCE is normal, but apparently it is not proved that CCE is transcendental.”

• Apéry’s constant, Wikipedia — the value 1.202056903159594285…, “Roger Apéry, who proved in 1978 that it is an irrational number,” and “Apéry’s constant has not yet been proved transcendental.”
• Roger Apéry, MacTutor History of Mathematics — “In 1978 he presented a lecture on his proof at the Journées Arithmétiques de Marseille which was greeted with doubt and disbelief.” and “The problem had long been an open question, yet Apéry’s proof only used methods which had been available for 200 years.”
• Computing ζ(3), John D. Cook — “There are closed-form expressions for ζ(s) when s is an even integer but so far not nobody has found one for s = 3.”

• Euler–Mascheroni constant, Wikipedia — limit definition, value, and open irrationality; “In 1959 Andrei Shidlovsky proved that at least one of Euler’s constant γ and the Gompertz constant δ is irrational.”
• Gompertz constant, Wikipedia — the value 0.596347362323194074…; “In 1962, Shidlovski proved that at least one of the Euler–Mascheroni constant and the Euler–Gompertz constant is irrational,” improved “in 2012 by Tanguy Rivoal, who proved that at least one of them is transcendental”; “When Euler studied divergent infinite series, he encountered δ via, for example, the above integral representation. Le Lionnais called δ the Gompertz constant because of its role in survival analysis”; and δ is “the regularized value of the summation of alternating factorials of all natural numbers (1 − 1 + 2 − 6 + 24 − 120 + ⋯), which is defined by Borel summation.”
• Meissel–Mertens constant, Wikipedia — M = 0.261497212847642783…, the limiting difference between the harmonic series over the primes and log log; γ “has an analogous definition involving a sum over all integers (not just the primes).” No arithmetic status is stated.
• Jeffrey C. Lagarias, Euler’s constant: Euler’s work and modern developments, arXiv:1303.1856 (Bull. AMS 50 (2013) 527–628) — read raw via the ar5iv rendering of arXiv:1303.1856. Section 3.16: “In 2012 Rivoal proved a result which, as a special case, implies the transcendence of at least one of the Euler constant γ and the Euler-Gompertz constant δ. It uses the Shidlovskii method and improves on an observation of Aptekarev that at least one of γ and δ must be irrational.” Aptekarev’s paper is On linear forms containing the Euler constant, arXiv:0902.1768, 28 February 2009.
• List of mathematical constants, Wikipedia — the Meissel–Mertens constant row reads “?” under ℚ (rational), 𝔸 (algebraic) and 𝒫: status unknown; the row’s year column reads “1866 & 1873.”

Corrected 17 September 2026. The entry credited the irrationality result to Shidlovski in 1962, following Wikipedia’s Gompertz constant article, while the #gamma and #gompertz cards credited Aptekarev in 2009. The cards were right. Lagarias’s survey, which is the source that Wikipedia article itself footnotes for the sentence, says Rivoal’s 2012 theorem “improves on an observation of Aptekarev that at least one of γ and δ must be irrational”. Wikipedia’s Euler’s constant article makes the same attribution to Shidlovski and dates it 1959 rather than 1962, so the two tertiary pages disagree with each other as well as with the survey they cite. This is not a tie to record: a refereed survey in the Bulletin of the AMS is the better source and is the one the others are quoting. Shidlovskii is not absent from the story — Rivoal’s proof uses his method, and Lagarias dates his breakthrough on E-functions to 1959 — but the at-least-one-is-irrational observation is Aptekarev’s.

• Feigenbaum constants, Wikipedia — δ = 4.669201609102990671…, α = 2.502907875095892822…, and “Both numbers are believed to be transcendental, although they have not been proven to be so. In fact, there is no known proof that either constant is even irrational.”
• Feigenbaum Constant, Wolfram MathWorld — “For the logistic map,” followed by displayed equation (8): “alpha = -2.502907875....”

• Brun’s constant, Wikipedia — twin-prime reciprocal sum, convergence, and rationality conditional on finiteness; “While the exact value of B 2 is unknown, it is thought to be around 1.9,” and of the 2002 value 1.902160583104: “The last is based on extrapolation from the sum 1.830484424658... for the twin primes below 10 16.”
• D. J. Platt and T. S. Trudgian, Improved bounds on Brun’s constant, arXiv:1803.01925 — “We improve the unconditional bounds on Brun’s constant to 1.840503 < B < 2.288513.”

• Saito, Mills’ constant is irrational — “We define Mills’ constant as the smallest real number ξ satisfying this property… we show that Mills’ constant is irrational. Furthermore, we obtain partial results on the transcendency of this number.” Submitted 30 April 2024, revised 8 May 2025; no hypothesis is assumed. Journal reference on the page: “Mathematika 71 (2025), no. 3, Paper No. e70027.” Read in the paper’s PDF: “Theorem 1.2. Either ξ3 is transcendental, or there exists a positive integer m such that ξ3^(3^m) is a Pisot number of degree 3”; “assuming Cramér’s conjecture, ξ3 is transcendental.”
• Mills’ constant, Wikipedia — “it is known to be irrational,” citing Saito, and “Its value is unproven, but if the Riemann hypothesis is true, it is approximately 1.3063778838630806904686144926…”; “Caldwell and Cheng used this method to compute 6850 base 10 digits of Mills’ constant under the assumption that the Riemann hypothesis is true.”
• Saito, Transcendency of variants of Mills’ constant — a 2025 sequel proving Mills’ constant transcendental under the Density Hypothesis for the Riemann zeta function, and transcendental outright for several variants; the unconditional question is left open.

The irrationality proof assumes nothing; the Riemann Hypothesis enters only in computing the constant’s digits, where it supplies a prime between consecutive cubes. Transcendence is genuinely open: Saito proves that either the constant is transcendental or some power of it is a cubic Pisot number, and that transcendence would follow from Cramér’s conjecture or from the Density Hypothesis.

• Transcendental number, Wikipedia — “Since the field of algebraic numbers is algebraically closed and e and π are roots of the polynomial x²−(e+π)x+eπ, at least one of the numbers eπ and e+π is transcendental”; π+e, π·e, π^π, e^e and π^e are listed as conjectured transcendental, none of them proved even irrational.
• A059742, OEIS — “Decimal expansion of e + Pi.”: 5.8598744820488384738… (value checked by computation; OEIS blocks automated fetches)
• A019609, OEIS — “Decimal expansion of Pi*e.”: 8.5397342226735670654… (value checked by computation; OEIS blocks automated fetches)
• Gelfond’s constant, Wikipedia — “The decimal expansion of π^e is given by π^e = 22.45915771836104547342...”
• A073233, OEIS — “Decimal expansion of Pi^Pi.”: 36.462159607207911770… (value checked by computation; OEIS blocks automated fetches)
• A073226, OEIS — “Decimal expansion of e^e.”: 15.154262241479264189… (value checked by computation; OEIS blocks automated fetches)

• Zudilin odd zeta result — theorem statement says at least one of ζ(5), ζ(7), ζ(9), ζ(11) is irrational.
• Particular values of the Riemann zeta function, Wikipedia — table row ζ(5): “1.0369277551433699263...”; “The relationship between zeta at the positive even integers and powers of pi may be written as a_n ζ(2n) = π^(2n) b_n where a_n and b_n are coprime positive integers for all n.”
• Transcendental number, Wikipedia — “in particular Apéry’s constant ζ(3), which is known to be irrational. For the other numbers ζ(5), ζ(7), ζ(9), ... even this is not known.”
• Apéry’s constant, Wikipedia — “Many people have tried to extend Apéry’s proof that ζ(3) is irrational to other values of the Riemann zeta function with odd arguments. Although this has so far not produced any results on specific numbers, it is known that infinitely many of the odd zeta constants ζ(2n + 1) are irrational”

• Rayo’s number, Wikipedia — records the 2007 MIT duel, googol-symbol definition, and fixed-finite-integer distinction; “At the time, it was claimed to be the largest named number, as its size vastly exceeds even other notable large numbers such as Graham’s number and TREE(3).”
• Big Number Duel, Agustín Rayo (MIT) — read raw: “On January 26th 2007, Adam Elga and I got together to see who could come up with a bigger finite number.” Under the heading “The winning entry” it sets out the definition by at most a googol symbols of first-order set theory that is now called Rayo’s number.
• Computable number, Wikipedia — read as raw wikitext: “The computable numbers include the specific real numbers which appear in practice, including all real algebraic numbers, as well as e, π, and many other transcendental numbers.” Every integer is algebraic, so every fixed integer is computable.

• Kruskal’s tree theorem, Wikipedia — confirms the theorem context behind TREE(3).

• Limit laws, OpenStax Calculus — states the limit laws for limits as x approaches a, among them the “Power law for limits”; the fact that x^n tends to zero for 0<x<1 is on the Sequences line below.
• 5.1 Sequences, Calculus Volume 2, OpenStax — read raw: “We summarize these results regarding the geometric sequence {rⁿ}: rⁿ → 0 if 0 < r < 1, rⁿ → 1 if r = 1, rⁿ → ∞ if r > 1.”
• Extended real number line, Wikipedia — read as raw wikitext: “the extended real number system is obtained from the real number system ℝ by adding two elements denoted +∞ and −∞ that are respectively greater and lower than every real number.” and “Adjoining the elements +∞ and −∞ to ℝ enables a definition of “limits at infinity””; and, with its math as rendered: “The expressions ∞ − ∞, 0 × (±∞), and ±∞/±∞ (called indeterminate forms) are usually left undefined.” and “However, in the context of probability or measure theory, 0 × ±∞ is often defined as 0.”

• Chaitin’s Constant, Wolfram MathWorld — defines the halting probability and states Chaitin constants are uncomputable and transcendental; and “Calude et al. (2002) computed the first 64 bits of Chaitin’s constant for a certain universal Turing machine.”
• Gregory Chaitin, The Limits of Reason, free preprint — The technical discussion on PDF page 11 defines the fixed universal self-delimiting Turing machine: no extension of a valid program is valid. Prefix-free programs make the probability sum converge; the precise value depends on the chosen machine.
• Chaitin’s constant, Wikipedia — read as raw wikitext: “this sum converges to a real number between 0 and 1”; “Knowing the first N bits of Ω, one could calculate the halting problem for all programs of a size up to N.”; and, after “It is algorithmically random,” “As a consequence, it is a normal number, which means that its digits are equidistributed as if they were generated by tossing a fair coin.”

• Antihydra, bbchallenge — Antihydra is an unresolved six-state halting problem that might determine BB(6); the page does not claim permanent unknowability.
• BB(6), BusyBeaverWiki — BB(6) is the unsolved sixth value of the Busy Beaver function; the best lower bound, as of 24 September 2026, is: the champion found by mxdys in June 2025 proves S(6) > Σ(6) > 10↑↑10↑↑10↑↑8 (the bound as restated in base 10 by 23 September 2026), which exceeds 2↑↑↑5 = 2↑↑2↑↑2↑↑4.
• Computable number, Wikipedia — read as raw wikitext: “The computable numbers include the specific real numbers which appear in practice, including all real algebraic numbers, as well as e, π, and many other transcendental numbers.” Every integer is algebraic, so every fixed integer is computable.

• Antihydra, bbchallenge — Antihydra is an unresolved six-state halting problem that might determine BB(6); the page does not claim permanent unknowability.
• BB(6), BusyBeaverWiki — BB(6) is “the unsolved 6th value of the Busy Beaver function” (BB(5) = 47,176,870 is carried by the Antihydra page of this entry). Re-read on 24 September 2026 (these counts change often): mxdys’s holdout list “has 855 machines up to equivalence and 1802 machines not considering equivalence as of mid-September 2026,” with Antihydra one of the cryptids among them; the champion found by mxdys in June 2025 proves S(6) > Σ(6) > 10↑↑10↑↑10↑↑8 (the bound as restated in base 10 by 23 September 2026), which exceeds 2↑↑↑5 = 2↑↑2↑↑2↑↑4. BB(6) is not waiting on Antihydra alone.

• Chaitin’s Constant, Wolfram MathWorld — defines the halting probability and states Chaitin constants are uncomputable and transcendental.
• Antihydra, bbchallenge — Antihydra is an unresolved six-state halting problem that might determine BB(6); the page does not claim permanent unknowability.
• Own computation — The displayed factor is 0 divided by positive sums; its value is 0. The limit uses 0<x<1, so its denominator is positive and finite.

• Own computation — The displayed factor is 0 divided by positive sums; its value is 0. The limit uses 0<x<1, so its denominator is positive and finite.

• Wiktionary, infeasibility — defines the noun as “the state of being infeasible or impractical.”
• Cambridge English Dictionary, “infeasible” “ fetched: defines it as not able to be achieved and lists “impracticable” and “unworkable” among its synonyms.
• Author’s terminology choice “ “Infeasibility” is retained because the chapter needs the established word for something possible in principle but not practically performable; “impossibility” would erase that distinction.
• Computable number, Wikipedia — read as raw wikitext: “A real number a is computable if it can be approximated by some computable function f: ℕ → ℤ in the following manner: given any positive integer n, the function produces an integer f(n) such that:,” and “There are many arithmetically definable, non-computable real numbers.”
• Busy beaver, Wikipedia — read as raw wikitext: “In his original 1962 paper, Radó defined two functions related to the busy beaver game: the score function Σ(n) and the shifts function S(n).” and “He proved that both of these functions were noncomputable, because they each grew faster than any computable function.”
• Chaitin’s constant, Wikipedia — read as raw wikitext: “Each halting probability is a normal and transcendental real number that is not computable, which means that there is no algorithm to compute its digits.”

• Schanuel’s conjecture, Wikipedia — “Given any set of n complex numbers {z₁,...,zₙ} that are linearly independent over ℚ, the field extension ℚ(z₁,...,zₙ, e^z₁,...,e^zₙ) has transcendence degree at least n over ℚ”; the conjecture is Stephen Schanuel’s and “was published by Serge Lang in 1966”; it remains a major unsolved problem of transcendental number theory, and it would give the transcendence of e + π, eπ, e^e, π^e and π^π.

The conjecture is the reason the second brace has a ceiling rather than a floor: one proof would move every combination of π and e in this chapter at once. Nothing in this chapter assumes it.

• Wolfram MathWorld, Khinchin’s Constant — read as raw HTML: “Khinchin (1934) considered the limit of the geometric mean … Amazingly, except for a set of measure 0, this limit is a constant independent of x,” and its reference list gives Khintchine, A., “Metrische Kettenbruchprobleme,” Compositio Math. 1, 361-382, 1934. Note the page adds “as proved in Kac (1959),” so the 1934 paper is where the limit was considered and a later proof is credited too; the card’s “proven by Khinchin in 1934” follows the usual attribution rather than MathWorld’s finer one.
• A. Khintchine, Metrische Kettenbruchprobleme, Compositio Mathematica 1, NUMDAM — the journal’s digitised record gives 1935, where MathWorld above gives 1934: “Compositio Mathematica, Tome 1 (1935), pp. 361-382.”
• Wolfram MathWorld, Glaisher-Kinkelin Constant — read as raw HTML: the constant “is the derivative of the Riemann zeta function (Kinkelin 1860; Jeffrey 1862; Glaisher 1877, 1878, 1893, 1894; Voros 1987),” with the two papers given as Kinkelin, “Über eine mit der Gammafunktion verwandte Transcendente …,” J. reine angew. Math. 57, 122-158, 1860, and Glaisher, J. W. L., “On the Product 1¹·2²·3³…,” Messenger Math. 7, 43-47, 1878.
• Wolfram MathWorld, Golomb-Dickman Constant — read as raw HTML: “Golomb (1964) computed an approximation … to the constant … known as the Golomb constant or Golomb-Dickman constant” and “Dickman (1930) investigated the probability that the greatest prime factor of a random integer between 1 and n satisfies …,” with the papers given as Golomb, S. W., “Random Permutations,” Bull. Amer. Math. Soc. 70, 747, 1964, and Dickman, K., “On the Frequency of Numbers Containing Prime Factors of a Certain Relative Magnitude,” Arkiv för Mat., Astron. och Fys. 22A, 1-14, 1930.
• Wolfram MathWorld, Landau-Ramanujan Constant — read as raw HTML: the count of integers below x that are sums of two squares behaves as stated “as proved by Landau (1908),” and “Ramanujan independently stated the theorem in the slightly different form.”
• Lévy’s constant, Wikipedia — read as raw HTML: “In 1935, the Soviet mathematician Aleksandr Khinchin showed that the denominators qₙ of the convergents of the continued fraction expansions of almost all real numbers satisfy lim qₙ^(1/n) = e^β” and “Soon afterward, in 1936, the French mathematician Paul Lévy found the explicit expression for the constant, namely e^β = e^(π²/(12 ln 2)) = 3.275822918721811159787681882….” Its footnote for Khinchin’s result, read as raw wikitext, gives 1936: “Zur metrischen Kettenbruchtheorie,” Compositio Mathematica 3, No. 2, 275–285 (1936).
• Fransén-Robinson constant, Wikipedia — read as raw HTML: “The value was computed to 36 decimal places by Herman P. Robinson using 11 point Newton-Cotes quadrature, to 65 digits by A. Fransén using Euler-Maclaurin summation, and to 80 digits by Fransén and S. Wrigge using Taylor series and other methods.” Neither this page nor MathWorld’s gives a year for Robinson’s computation; MathWorld’s earliest Fransén citation is “Accurate Determination of the Inverse Gamma Integral,” BIT 19, 137-138, 1979. The page also reads “The constant is somewhat close to Euler’s number e = 2.71828...,” approximating the integral by the sum of 1/Γ(n), “and this sum is the standard series for e”; and “It is however unknown whether F can be expressed in closed form in terms of other known constants.”
• Wolfram MathWorld, Bernstein’s Constant — read as raw HTML: “Bernstein showed that … He conjectured that the lower limit was 1/(2√π). However, this was disproven by Varga and Carpenter (1987) and Varga (1990),” with Bernstein’s own paper given as “Sur la meilleure approximation de |x| par les polynomes de degrés donnés,” Acta Math. 37, 1-57, 1913.
• R. S. Varga and A. J. Carpenter, On the Bernstein conjecture in approximation theory, Constructive Approximation 1 (1985), 333-348 — the publisher’s page gives “Published: December 1985” and “Volume 1, pages 333–348 (1985),” by Richard S. Varga and Amos J. Carpenter. Its abstract also reads: “by means of numerical calculations, Bernstein determined, in the same paper, the following upper and lower bounds for β: 0.278<β<0.286.”
• R. S. Varga and A. J. Carpenter, On a conjecture of S. Bernstein in approximation theory, Mathematics of the USSR-Sbornik 57 (1987), 547–560, Math-Net.Ru — the same paper and abstract in the translation journal: “Mathematics of the USSR-Sbornik, 1987, Volume 57, Issue 2, Pages 547–560,” “Received: 27.03.1985” and “Original paper language: Russian.” This is the 1987 MathWorld cites, so the book’s 1985 is the first publication.
• Random Fibonacci sequence, Wikipedia — read as raw HTML: “In 1999, Divakar Viswanath showed that the growth rate of the random Fibonacci sequence is equal to 1.1319882487943…, a mathematical constant that was later named Viswanath’s constant,” and the method “uses Furstenberg’s formula for the Lyapunov exponent of a random matrix product and integration over a certain fractal measure on the Stern-Brocot tree.”
• Wolfram MathWorld, Niven’s Constant — read as raw HTML: “Niven (1969) also proved that … where Niven’s constant is given by …,” the paper being Niven, Ivan, “Averages of exponents in factoring integers,” Proc. Amer. Math. Soc., 1969.
• Artin’s conjecture on primitive roots, Wikipedia — read as raw HTML: “The conjecture was made by Emil Artin to Helmut Hasse on September 27, 1927, according to the latter’s diary”; “In 1967, Christopher Hooley published a conditional proof for the conjecture, assuming certain cases of the generalized Riemann hypothesis”; and “Without the generalized Riemann hypothesis, there is no single value of a for which Artin’s conjecture is proved.”
• OEIS A062089, decimal expansion of Sierpiński’s constant — read as raw HTML: its link list gives the original paper as Wacław Sierpiński, “O sumowaniu szeregu Σ τ(n) f(n), gdzie τ(n) oznacza liczbę rozkładów liczby n na sumę kwadratów dwóch liczb całkowitych,” Prace Matematyczno-Fizyczne, Vol. 18, No. 1 (1907), pp. 1-59.
• Wolfram MathWorld, Twin Primes Constant — read as raw HTML: its reference list gives Hardy, G. H. and Littlewood, J. E., “Some Problems of ‘Partitio Numerorum.’ III. On the Expression of a Number as a Sum of Primes,” Acta Math. 44, 1-70, 1923, which is the conjecture the constant belongs to. Wikipedia’s List of mathematical constants gives the constant the year 1922 instead; sources differ, corrections welcome.
• MRB constant, Wikipedia — read as raw HTML: “The constant is named after its discoverer, Marvin Ray Burns, who published his discovery of the constant in 1999,” and “Burns had initially called the constant ‘rc’ for root constant but, at Simon Plouffe’s suggestion, the constant was renamed.”
• Wolfram MathWorld, Somos’s Quadratic Recurrence Constant — read as raw HTML: the constant “has closed-form solution … where Li is a polylogarithm, Φ is a Lerch transcendent,” and its reference list gives Somos, M., “Several Constants Related to Quadratic Recurrences,” unpublished note, 1999.
• OEIS A085365, decimal expansion of the Kepler-Bouwkamp constant — read as raw HTML: “Named after the German astronomer and mathematician Johannes Kepler (1571-1630) and the Dutch mathematician Christoffel Jacob Bouwkamp (1915-2003),” with the two primary items in its link list given as Johannes Kepler, Mysterium Cosmographicum, Tübingen, 1596, see p. 39, and C. J. Bouwkamp, “An infinite product,” Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series A, Vol. 68 (1965), pp. 40-46.
• Backhouse’s constant, Wikipedia — read as raw HTML: the limit of the ratios of successive coefficients of 1/P(x) “was conjectured to exist by Backhouse, and later proven by Philippe Flajolet,” the conjecture being cited to Backhouse 1995.
• Cyclic number, Wikipedia — read as raw wikitext: “For example, the case b = 10, p = 7 gives the cyclic number 142857” and “this sequence is the set of primes p such that b is a primitive root modulo p. A conjecture of Emil Artin is that this sequence contains 37.395..% of the primes.”
• S. Bernstein, Sur la meilleure approximation de |x| par des polynomes de degrés donnés, Acta Mathematica 37, SpringerLink — read raw: “Published: December 1914” and “Volume 37, pages 1–57 (1914).” The Varga–Carpenter abstract (doi 10.1007/BF01890040, already cited) reads “the famous Russian mathematician S. Bernstein in 1914 established the existence of a positive constant β.”
• Accurate determination of the inverse gamma integral, BIT 19 (1979), 137-138, SpringerLink — read raw: the author line gives “Arne Fransén,” of the “National Defence Research Institute, Division 1, Section 144, Stockholm 80, Sweden.”
• Mysterium Cosmographicum, Wikipedia — read as raw wikitext: “he realized that regular polygons bound one inscribed and one circumscribed circle at definite ratios, which, he reasoned, might be the geometrical basis of the universe. After failing to find a unique arrangement of polygons that fit known astronomical observations (even with extra planets added to the system), Kepler began experimenting with 3-dimensional polyhedra.”
• Try to beat these MRB constant records!, Wolfram Community — read raw, Marvin Ray Burns’s record thread: “Over 4 million digits of CMRB were finished on Wed 16 Jan 2019, 19:55:20,” and a later run ends “Enter MRB1 to print 6532491 digits.”
• Sierpiński’s constant, Wikipedia — read as raw wikitext: “One way of defining it is as the following limit: K = lim n→∞ [Σ k=1..n r₂(k)/k − π ln n] where r₂(k) is a number of representations of k as a sum of the form a² + b² for integer a and b.” Its closed form uses Γ(1/4), π and γ.

Added 17 September 2026 for the B6 pass. Sixteen cards in the second brace cited r-5.9 alone, and r-5.9 states exactly one proposition, that none of these constants has been proved irrational. Everything else those cards say - roughly twenty-five dates and discoverers - was carried by nothing. This entry carries it; r-5.9 keeps its one clean proposition. Three things the sourcing settled. (1) Bernstein: the book says Varga and Carpenter disproved the 1/(2 root pi) conjecture in 1985 and MathWorld says 1987. Crossref shows both, Constructive Approximation 1 (1985) and Mathematics of the USSR-Sbornik 57 (1987), which is the Russian version of the same result, so 1985 is the first publication and the book is right; do not change it to 1987. (2) Fransén-Robinson: no source found gives a year for the computation at all, and the earliest Fransén paper any of them cites is 1979, so the card’s “in 1978” was removed rather than sourced. (3) Khinchin: MathWorld says Khinchin considered the limit in 1934 and credits a proof to Kac (1959); the card keeps the usual attribution, and the difference is recorded in the source note above rather than hidden.

• David Hilbert, Wikipedia — read as raw HTML: “The epitaph on his tombstone in Göttingen consists of the famous lines he spoke at the conclusion of his retirement address to the Society of German Scientists and Physicians on 8 September 1930 … Wir müssen wissen. Wir werden wissen. Translation: We must know. We shall know.” On Gödel: “The day before Hilbert pronounced these phrases at the 1930 annual meeting of the Society of German Scientists and Physicians, Kurt Gödel — in a round table discussion during the Conference on Epistemology held jointly with the Society meeting — [announced the incompleteness theorem].”
• James T. Smith, David Hilbert’s Radio Address (San Francisco State University; also in MAA Convergence) — read as extracted PDF text: on 8 September 1930 Hilbert addressed the Society of German Natural Scientists and Physicians in Königsberg; “Shortly afterward, he read on German radio a four-minute version of the finale of his speech”; the German transcription ends “Wir müssen wissen, Wir werden wissen” and the English translation “We must know, We will know.”

Added 17 September 2026 to source the chapter epigraph, which carried no marker. Two small notes. The source translates the second sentence “We shall know” where the page has “We will know”; both are current and the page’s is kept. And the footer’s “announced at the same meeting” is right in substance — the Conference on Epistemology was held jointly with the Society meeting — but Gödel spoke the day before Hilbert, which the footer does not say. On 29 September 2026 Ian kept “radio” in the footer (IAN-LIST 6), and Smith’s edition of the recording was added as the source that carries it.

• Normal number, Wikipedia — read as raw HTML: “The concept of a normal number was introduced by Émile Borel (1909). Using the Borel–Cantelli lemma, he proved that almost all real numbers are normal… Wacław Sierpiński (1917) showed that it is possible to specify a particular such number. Becher and Figueira (2002) proved that there is a computable absolutely normal number.”
• Normal Number, MathWorld — the digit-frequency definition, and the standing state of ignorance about π, e and √2.

The chapter needs only the one-way implication: a number normal in base 10 cannot be rational, because a rational expansion is eventually periodic and a periodic expansion cannot give every block its fair share. The converse fails, and nothing here claims it.

• Collatz conjecture, Wikipedia — read as raw HTML: “The conjecture has been shown to hold for all positive integers up to 2.36 × 10²¹, but no general proof has been found. It is named after the mathematician Lothar Collatz, who introduced the idea in 1937, two years after receiving his doctorate,” and gives “hailstone sequence” as the other name.
• Collatz Conjecture, MathWorld — the same rule and the conjecture’s open status, and on the date: “Its exact date of origin is unclear, although Collatz traced its development to investigations of iterations of arithmetic functions begun during his student years from 1928 to 1933.”
• OEIS A006577, number of halving and tripling steps to reach 1 in the 3x + 1 problem — the step counts themselves, for the record that the orbit lengths are ordinary integers even though the conjecture is open.
• BB(6), BusyBeaverWiki — read raw: “With the discovery of the Cryptid machine Antihydra in June 2024, we now know that we must solve a Collatz-like problem in order to solve BB(6) and thus BB(6) is Hard.”

• Basel problem, Wikipedia — read raw: “It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in The Saint Petersburg Academy of Sciences..”
• Brun’s theorem, Wikipedia — read raw: “Brun’s theorem was proved by Viggo Brun in 1919.”
• Pentium FDIV bug, Wikipedia — read raw: “The bug was discovered in 1994 by Thomas R. Nicely, a professor of mathematics at Lynchburg College..”
• Catalan’s constant, Wikipedia — read raw: “Catalan’s constant was named after Eugène Charles Catalan, who found quickly-converging series for its calculation and published a memoir on it in 1865.”; “In low-dimensional topology, Catalan’s constant is 1/4 of the volume of an ideal hyperbolic octahedron”; “In combinatorics and statistical mechanics, it arises in connection with counting domino tilings, spanning trees, and Hamiltonian cycles of grid graphs.”
• Feigenbaum constants, Wikipedia — read raw: “Feigenbaum made this discovery in 1975, and he officially published it in 1978.”; “The first proof of the universality of the Feigenbaum constants was carried out by Oscar Lanford—with computer-assistance—in 1982.”
• Euler’s constant, Wikipedia — read raw: “The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Observations on harmonic progressions; Eneström Index 43).”
• J. C. Lagarias, Euler’s constant: Euler’s work and modern developments (arXiv:1303.1856), ar5iv — read raw: “However in 1809 Johann von Soldner [275], while preparing the first book on the logarithmic integral function”; on Mascheroni: “In 1790, in a book proposing to answer some problems raised by Euler, Mascheroni [213] introduced the notation A for this constant”; “he gave a numerical expression for it to 32 decimal places”; “Von Soldner’s answer disagreed with Mascheroni’s in the 20-th decimal place.”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• J. C. Lagarias, Euler’s constant: Euler’s work and modern developments (arXiv:1303.1856), ar5iv — read raw: “Benjamin Gompertz (1779–1865) was a mathematician and actuary who in 1825 proposed a functional model for approximating mortality tables.”
• Limit (mathematics), Wikipedia — read raw: “Augustin-Louis Cauchy in 1821, followed by Karl Weierstrass, formalized the definition of the limit of a function which became known as the (ε, δ)-definition of limit..”
• Mills’ constant, Wikipedia — read raw: “This constant is named after William Harold Mills who in 1947 proved the existence of A based on results of Guido Hoheisel and Albert Ingham on prime gaps..”
• Mertens Constant, Wolfram MathWorld — read raw: “According to Lindqvist and Peetre (1997), this was shown independently by Meissel in 1866 and Mertens (1874)..”
• Schanuel’s conjecture, Wikipedia — read raw: “It is due to Stephen Schanuel and was published by Serge Lang in 1966..”
• T. Rivoal, La fonction Zeta de Riemann prend une infinite de valeurs irrationnelles aux entiers impairs, arXiv — read raw: “[Submitted on 7 Aug 2000] Title: La fonction Zeta de Riemann prend une infinite de valeurs irrationnelles aux entiers impairs Authors: Tanguy Rivoal.”
• W. V. Zudilin, One of the numbers zeta(5), zeta(7), zeta(9), zeta(11) is irrational, Math-Net.Ru — read raw: “Russian Mathematical Surveys, 2001, Volume 56, Issue 4, Pages 774–776.”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Mersenne prime, Wikipedia — defines Mersenne numbers 2^n-1, states that n must be prime for 2^n-1 to be prime, and gives the Lucas–Lehmer primality test used to search them.
• Erdős–Borwein constant, Wikipedia — carries the definition and Erdős’s 1948 irrationality proof: “the sum of the reciprocals of the Mersenne numbers”; “In 1948, Erdős showed that the constant E is an irrational number.”
• Largest known prime number, Wikipedia — carries the Mersenne dominance of record primes: “As of October 2024, the seven largest known primes are Mersenne primes. The last 18 record primes were Mersenne primes.”

• Encyclopedia of Mathematics, Wilson theorem — states the iff theorem, Waring 1770, and Lagrange’s 1771 proof.
• MacTutor, Chronology for 1760–1780 — records Lagrange proving Wilson’s theorem in 1771.

• Wolfram MathWorld, Wilson’s Theorem — gives the factorial congruence and describes it as a necessary and sufficient primality test.

Own arithmetic checked: 12! = 479001600; 13 × 36846277 = 479001601.

• Wolfram MathWorld, Euler Totient Function — defines the totient and gives phi(p)=p−1 for primes.
• Euler’s totient function, Wikipedia — “This function gives the order of the multiplicative group of integers modulo n”; “Setting up an RSA cryptosystem involves choosing large prime numbers p and q, computing n = pq and k = φ(n), and finding two numbers e and d such that ed ≡ 1 (mod k).”

• Wolfram MathWorld, Fermat’s Little Theorem — states a^(p−1) congruent to 1 modulo p for p not dividing a.

• Wolfram MathWorld, Fermat’s 4n+1 Theorem — states the prime criterion for sums of two squares: “Fermat’s theorem, sometimes called Fermat’s two-square theorem or simply “Fermat’s theorem,” states that a prime number can be represented in an essentially unique manner (up to the order of addends) in the form” x² + y² “iff” p ≡ 1 (mod 4) or p = 2 (formulas typeset on the page; the old Two-Square Theorem address is now a one-line pointer to this page).
• Fermat’s 4n+1 Theorem, Wolfram MathWorld — “The first few primes which are 1 or 2 (mod 4) are 2, 5, 13, 17, 29, 37, 41, 53, 61, ... (OEIS A002313) (with the only prime congruent to 2 mod 4 being 2).”

• Wolfram MathWorld, Legendre Symbol — defines the symbol and its values for quadratic residues.

• Wolfram MathWorld, Multiplicative Order — defines multiplicative order as the least positive exponent returning 1 modulo n.
• Carmichael function, Wikipedia — “In algebraic terms, λ(n) is the exponent of the multiplicative group of integers modulo n. As this is a finite abelian group, there must exist an element whose order equals the exponent, λ(n).”; λ(n) and φ(n) “must be equal in the cases where the multiplicative group is cyclic due to the existence of a primitive root, which is the case for odd prime powers.”

Own computation checked by the powers 2,4,8,3,6,12,11,9,5,10,7,1 modulo 13.

• NIST DLMF §27.2, Functions — defines the Möbius function (27.2.12): μ(n) is 0 unless every prime exponent of n is 1; the value at 6174 is independently factored below.
• 6174, Wikipedia — “It is a Kaprekar’s Constant.”
• Mertens function, Wikipedia — “However, the Riemann hypothesis is equivalent to a weaker conjecture on the growth of M(x)”

Own factorization checked.

• Wolfram MathWorld, Greatest Common Divisor — defines gcd and its basic properties (the gcd–lcm product identity is on the Least Common Multiple line below).
• Least Common Multiple, Wolfram MathWorld — “The smallest m is given by n=1, LCM(a, b)=(ab)/(GCD(a, b)), so GCD(a, b)LCM(a, b)=ab”

Own Euclidean computation: 18=12+6 and 12=2×6.

• Wolfram MathWorld, Euclidean Algorithm — describes repeated division as the gcd algorithm.

• NIST DLMF §27.14, Unrestricted Partitions — defines p(n) as unordered sums of positive integers and records Ramanujan’s p(5n+4) congruence.
• Integer partition, Wikipedia — “For instance, p(4) = 5 because the integer 4 has the five partitions 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4.”; “The values of this function for n = 0, 1, 2, … are: 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, …”

• NIST DLMF §27.14(v), Divisibility Properties — states p(5n+4) congruent to 0 modulo 5 for all n.

Own arithmetic checked: 30 mod 5 = 0 and 5 mod 5 = 0.

• Wolfram MathWorld, Binomial Coefficient — defines C(n, k) as the number of k-subsets and gives the zero case when k exceeds n.
• Binomial coefficient, Wikipedia — “When n is a nonnegative integer and k is an integer such that k < 0 or k > n, it is common to define” the coefficient to be 0.

• Wolfram MathWorld, Ramsey Number — gives the party-problem definition, R(3, 3)=6 and R(4, 4)=18, credited in its table to Greenwood and Gleason (1955), and the bounds for R(5, 5), 43 from Exoo (1989) and 46 from Angeltveit and McKay (2024).
• Theorem on friends and strangers, Wikipedia — “We draw K5 as a pentagon surrounding a star (a pentagram). We color the edges of the pentagon red and the edges of the star blue.”; “In 1930, in a paper entitled ‘On a Problem of Formal Logic,’ Frank Ramsey proved a very general theorem (now known as Ramsey’s theorem) of which this theorem is a simple case.”
• Ramsey’s theorem, Wikipedia — on R(4, 4): “only one (4, 4, 17) graph (the Paley graph of order 17) among 2.46 × 10^26 colourings. It follows that R(4, 4) = 18.”; and, quoting Joel Spencer, “Erdős asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of R(5, 5) or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for R(6, 6). In that case, he believes, we should attempt to destroy the aliens.”

• Wolfram MathWorld, Catalan Number — lists the sequence and its bracketing and polygon-triangulation interpretations.
• Richard P. Stanley, Catalan Numbers (talk slides, 13 March 2024), author’s MIT page — “R. Stanley, Catalan Numbers, Cambridge University Press, 2015. Includes 214 combinatorial interpretations of Cn and 68 additional problems.”

• Wolfram MathWorld, Bell Number — defines Bell numbers as set partitions and gives B3=5; from n = 1 “the first few Bell numbers” “are 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, ... (OEIS A000110),” so B₄ = 15.
• Stirling numbers of the second kind, Wikipedia — “The 15 partitions of a 4-element set ordered in a Hasse diagram”; “There are S(4, 1), ..., S(4, 4) = 1, 7, 6, 1 partitions containing 1, 2, 3, 4 sets.”

• Wolfram MathWorld, Derangement — defines derangements, gives the recurrence relations, and lists them from n = 1: “The first few are 0, 1, 2, 9, 44, 265, 1854, ... (OEIS A000166).”

• Wolfram MathWorld, Chromatic Polynomial — defines the chromatic polynomial and gives the triangle polynomial k(k−1)(k−2).
• Dihedral group of order 6, Wikipedia — the triangle’s symmetries number six: “One way to interpret the dihedral group D3 geometrically is by observing the symmetry of an equilateral triangle.” and “Six transformations are the results.”

• Wolfram MathWorld, Handshaking Lemma — states that the sum of vertex degrees is twice the number of edges.
• Wolfram MathWorld, Petersen Graph — records Petersen’s 1898 introduction and the graph’s standard structural properties.
• Handshaking lemma, Wikipedia — ties the lemma to Euler and the bridges: “Leonhard Euler first proved the handshaking lemma in his work on the Seven Bridges of Königsberg” and “Euler stated the fundamental results for this problem in terms of the number of odd vertices in the graph, which the handshaking lemma restricts to be an even number.”

• Wolfram MathWorld, Modular Arithmetic — defines congruence and modular arithmetic.
• Earliest Uses of Symbols of Number Theory, MacTutor — “The congruent symbol used in number theory ≡ was introduced in print in 1801 by Carl Friedrich Gauss (1777-1855) in Disquisitiones arithmeticae,” citing “Disquisitiones arithmeticae (Leipzig, 1801), art. 2.”

• Wolfram MathWorld, Chinese Remainder Theorem — states unique simultaneous solutions for pairwise coprime moduli.

• Wolfram MathWorld, Factorial — defines n! and its counting interpretation.
• Wolfram MathWorld, Binomial Coefficient — defines C(n, k) as the number of k-subsets and gives the zero case when k exceeds n.

• Wolfram MathWorld, Euclidean Algorithm — describes repeated division as the gcd algorithm.
• Wikipedia, Euclidean algorithm — quotes Donald Knuth, The Art of Computer Programming, Vol. 2, 2nd edition (1981), p. 318, calling the algorithm “the granddaddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day.”

No source states the unqualified superlative “the oldest algorithm still in daily use,” and there is no register of ancient algorithms against which to check it. Knuth’s qualified form is what the page claims instead, in his own words.

• Encyclopedia of Mathematics, Mobius function — defines mu(1)=1, mu(n)=0 if n is divisible by the square of a prime and mu(n)=(-1)^k otherwise, and states that the function was introduced by A. Mobius in 1832.

Added because r-6.8’s only source is a DLMF inversion-formula section that uses the Mobius function without defining it; this entry carries the zero-on-repeated-prime-factor definition the chapter and the mobius card both state, and the 1832 date the prose gives.

• Wolfram MathWorld, Koenigsberg Bridge Problem — states that the problem “was answered in the negative by Euler (1736),” that the work “represented the beginning of graph theory,” and cites Euler, Solutio problematis ad geometriam situs pertinentis, Comment. Acad. Sci. U. Petrop. 8, 128-140, 1736.

• MacTutor, Carl Friedrich Gauss — gives the birth date 30 April 1777 and states that “He published the book Disquisitiones Arithmeticae in the summer of 1801,” which puts Gauss at twenty-four.
• Encyclopedia of Mathematics, Congruence — defines a congruence modulo m as the divisibility of a - b by m, states that congruences to the same modulus “can be added, subtracted and multiplied in the same way as normal equalities,” and records that the theory of congruences “was first systematically developed by C.F. Gauss.”
• Disquisitiones arithmeticae, Sectio prima, Latin Wikisource — the primary text: article 1, “Si numerus a numerorum b,c differentiam metitur, b et c secundum a congrui dicuntur”; article 2, “Numerorum congruentiam hoc signo, ≡, in posterum denotabimus”; and its footnote, “Hoc signum propter magnam analogiam quae inter aequalitatem atque congruentiam invenitur adoptavimus.”
• Primitive root modulo n, Wikipedia — “Gauss defined primitive roots in Article 57 of his Disquisitiones Arithmeticae (1801), where he credited Leonhard Euler with coining the term. In Article 56 he stated that Johann Heinrich Lambert and Euler knew of them, but he was the first to rigorously demonstrate that primitive roots exist for a prime number n.”
• Euler’s totient function, Wikipedia — “Leonhard Euler introduced the function in 1763.”; “The now-standard notation φ(A) comes from Gauss’s 1801 treatise Disquisitiones Arithmeticae, although Gauss did not use parentheses around the argument and wrote φA.”

Neither source ties the congruence notation itself to the year 1801 in so many words: MacTutor dates the Disquisitiones and the Encyclopedia of Mathematics names Gauss as the first systematic developer of the theory. The two together carry the chapter’s sentence; the notation’s first appearance in the book’s opening lines is not separately sourced here.

• Wikipedia, Chromatic polynomial — states that “George David Birkhoff introduced the chromatic polynomial in 1912, defining it only for planar graphs, in an attempt to prove the four color theorem,” and gives the triangle’s polynomial as x(x-1)(x-2).
• Four color theorem, Wikipedia — on the eventual proof: “the four color theorem resisted until 1976, when it was proven by Kenneth Appel and Wolfgang Haken in a computer-aided proof” and “The Appel–Haken proof proceeds by analyzing a very large number of “reducible configurations,” examples of maps with a particular property.”

r-6.18’s MathWorld page carries the triangle polynomial but names Birkhoff only in its bibliography (A Determinant Formula for the Number of Ways of Coloring a Map, Ann. Math. 14, 42-46, 1912) and never states the four-color motivation.

• Wikipedia, Ramanujan’s congruences — states that “In his 1919 paper, he proved the first two congruences” and that “After Ramanujan died in 1920, G. H. Hardy extracted proofs of all three congruences from an unpublished manuscript of Ramanujan on p(n) (Ramanujan, 1921).”

The chapter’s prose and the partition5 card both say Ramanujan proved the congruence “with companion congruences modulo 7 and 11.” All three proofs are his, but only the 5 and 7 cases are in the 1919 paper; the 11 case appeared posthumously in 1921.

• Wikipedia, Fermat’s little theorem — states that Fermat first stated the theorem in a letter dated 18 October 1640 to Frenicle de Bessy, and that Euler provided the first published proof in 1736 in Theorematum quorundam ad numeros primos spectantium demonstratio; also “Fermat’s little theorem is the basis for the Fermat primality test,” and, of its generalization, “Euler’s theorem is used with n not prime in public-key cryptography, specifically in the RSA cryptosystem.”
• Encyclopedia of Mathematics, Fermat little theorem — gives the congruence a^(p-1) = 1 (mod p) for a not divisible by the prime p and states that the theorem “was established by P. Fermat (1640).”

Added for Q6.2. The fermatlittle card said Euler proved the theorem in 1736 while its only source, MathWorld, dates the first published proof to 1749. Two different Euler papers are in circulation under that description, so the card now says first published proof and names the 1736 one, and this entry carries both halves of the sentence. The Encyclopedia of Mathematics source was moved here on 17 September 2026 from the former entry 6.28, which said only “Fermat stated the little theorem in 1640,” was cited by nobody, and was retired as redundant with this entry; the move leaves the 1640 date resting on two independent sources instead of one.

• Wikipedia, Seven Bridges of Königsberg — states that “one is touched by 5 bridges, and each of the other three is touched by 3,” that “a necessary condition for the walk of the desired form is that the graph be connected and have exactly zero or two nodes of odd degree,” names Euler’s 1736 paper Solutio problematis ad geometriam situs pertinentis, and records that the solution “is considered to be the first theorem of graph theory.”

Added with the konigsberg card. r-6.27 carries the negative answer, the year and the founding status, but not the degree count or the odd-vertex condition, which is the argument the card explains.

• Fermat’s theorem on sums of two squares, Wikipedia — read as raw HTML: Fermat wrote the statement “in a letter to Marin Mersenne dated December 25, 1640,” and of Euler, “The first proof was found by Euler after much effort and is based on infinite descent. He announced it in two letters to Goldbach, on May 6, 1747 and on April 12, 1749; he published the detailed proof in two articles (between 1752 and 1755).” The card’s “proved by Euler in 1749” is the second announcement; the printing is later.
• Legendre symbol, Wikipedia — read as raw HTML: “The Legendre symbol was introduced by Adrien-Marie Legendre in 1797 or 1798 in the course of his attempts at proving the law of quadratic reciprocity.” The source hedges between the two years, and so does the card.
• Quadratic reciprocity, Wikipedia — read as raw HTML: “The quadratic reciprocity theorem was conjectured by Leonhard Euler and Adrien-Marie Legendre and first proved by Carl Friedrich Gauss”; Gauss “referred to it as the ‘fundamental theorem’ in his Disquisitiones Arithmeticae” and “Privately, Gauss referred to it as the ‘golden theorem’. He published six proofs for it, and two more were found in his posthumous papers,” which is the eight the card counts; the first proof is dated by “his mathematical diary entry for April 8, 1796 (the date he first proved quadratic reciprocity),” and “Gauss published the first and second proofs of the law of quadratic reciprocity on arts 125-146 and 262 of Disquisitiones Arithmeticae in 1801”; and, of a later proof, “This proof served as a template for class field theory, which can be viewed as a vast generalization of quadratic reciprocity.”
• Finite field, Wikipedia — read as raw HTML: “the following classification theorem first proved in 1893 by E. H. Moore: The order of a finite field is a prime power. For every prime power q there are fields of order q, and they are all isomorphic.”
• Euclidean algorithm, Wikipedia — read as raw HTML: “the algorithm never requires more steps than five times the number of digits (base 10) of the smaller integer. This was proven by Gabriel Lamé in 1844 (Lamé’s Theorem), and marks the beginning of computational complexity theory.”
• Pascal’s triangle, Wikipedia — read as raw HTML: “Pascal’s Traité du triangle arithmétique (Treatise on Arithmetical Triangle) was published posthumously in 1665,” its page images being dated 1654; the page also records Tartaglia publishing six rows in 1556, Cardano in 1570, and the Chinese diagram in Zhu Shijie’s Siyuan Yujian of 1303.
• Ramsey’s theorem, Wikipedia — read as raw HTML: the paper is given as Ramsey, F. P. (1930), “On a problem of formal logic,” Proceedings of the London Mathematical Society 30, 264-286. The page gives no 1928 date.
• Wolfram MathWorld, Ramsey Number — read as raw HTML: its table of known values credits “Greenwood and Gleason (1955)” for R(3, 3)=6, R(3, 4)=9, R(3, 5)=14 and R(4, 4)=18, and records that “Exoo (1989) proved that R(5, 5) ≥ 43 and Angeltveit and McKay (2024) proved that R(5, 5) ≤ 46.”
• Catalan number, Wikipedia — read as raw HTML: “The Catalan sequence was described in 1751 by Leonhard Euler, who was interested in the number of different ways of dividing a polygon into triangles. The sequence is named after Eugène Charles Catalan, who discovered the connection to parenthesized expressions during his exploration of the Towers of Hanoi puzzle.” The page carries no year for Catalan and links Larcombe, “The 18th century Chinese discovery of the Catalan numbers,” for Ming Antu.
• OEIS A000108, the Catalan numbers — read as raw HTML: “A very large number of combinatorial interpretations are known - see references, esp. R. P. Stanley, ‘Catalan Numbers’, Cambridge University Press, 2015,” which is the catalogue the card means. Neither this page nor Wikipedia’s states a count, so “more than two hundred families” is the book’s own reading of that book.
• Bell number, Wikipedia — read as raw HTML: “The Bell numbers are named after Eric Temple Bell, who wrote about them in 1938, following up a 1934 paper in which he studied the Bell polynomials”; “in his 1938 paper, he wrote that the Bell numbers ‘have been frequently investigated’ and ‘have been rediscovered many times’”; “Bell cites several earlier publications on these numbers, beginning with Dobiński 1877 which gives Dobiński’s formula”; “the name ‘Bell numbers’ and the notation Bₙ … was given to them by Becker & Riordan 1948”; and “The first exhaustive enumeration of set partitions appears to have occurred in medieval Japan, where … a parlor game called genjikō sprang up.”
• Derangement, Wikipedia — read as raw HTML: “The problem of counting derangements was first considered by Pierre Raymond de Montmort in his Essay d’analyse sur les jeux de hazard in 1708; he solved it in 1713, as did Nicholas Bernoulli at about the same time,” and “The name ‘subfactorial’ originates with William Allen Whitworth”; and, of e^−1 ≈ 0.367879, “This is the limit of the probability that a randomly selected permutation of a large number of objects is a derangement.”
• Partition function (number theory), Wikipedia — read as raw HTML: “This asymptotic formula was first obtained by G. H. Hardy and Ramanujan in 1918 and independently by J. V. Uspensky in 1920,” and “In 1937, Hans Rademacher was able to improve on Hardy and Ramanujan’s results by providing a convergent series expression for p(n).”
• Integer partition, Wikipedia — read as raw HTML, for the 1748 date: of the identity between partitions into odd parts and into distinct parts, “This result was proved by Leonhard Euler in 1748.” The page does not date the generating function; Jordan Bell’s paper below has Euler introduce it in a paper presented in 1741 and printed in 1751, and discuss it in the Introductio of 1748.
• Own computation — p(100) computed by the standard coin-change recurrence over parts 1 to 100: p(100) = 190,569,292, and the same recurrence gives p(4) = 5 and p(9) = 30, which are the values r-6.11 cites from DLMF.
• Traité du triangle arithmétique, Cambridge Digital Library — “In 1654 Blaise Pascal entered into correspondence with Pierre de Fermat of Toulouse about some problems in calculating the odds in games of chance, as a result of which he wrote the Traité du triangle arithmétique, avec quelques autres petits traitez sur la mesme matière, probably in August of that year. Not published until 1665”
• Jordan Bell, Euler and the pentagonal number theorem, arXiv math/0510054 — “presented to the St. Petersburg Academy on April 6, 1741 and published in 1751”; “In this paper Euler introduces the generating function”; “In §323 (in ch. XVI, “De partitione numerorum”) of the Introductio in analysin infinitorum [Euler, 1748], E101, published in 1748, Euler notes in his discussion on the generating function”

Added 17 September 2026 for the B6 pass. Eleven cards in this chapter carried dated attributions on entries whose claim lines stop at the mathematics - MathWorld and DLMF definitions with no history in them. Four entries for exactly this pattern already existed and had been attached last pass (6.23, 6.26, 6.29, 6.30, 6.31), so this entry finishes the block rather than starting it. Three dates could not be sourced and were handled rather than asserted. Ramsey: no source found dates the proof to 1928, and the paper is 1930, so the card now says 1930 and the 1928 reading, which is the year the paper was read to the London Mathematical Society, is left out. Catalan: no source found gives 1838 for Catalan’s own work, so the card no longer gives a year for him. Stanley’s catalogue: no source states a count, so “more than two hundred families” is flagged above as the book’s own reading of Stanley’s Catalan Numbers (2015). One more nuance worth keeping: the Legendre symbol source hedges between 1797 and 1798, and Euler’s two-squares proof was announced in 1747 and 1749 and printed between 1752 and 1755, so “proved by Euler in 1749” is the announcement.

• Carl Friedrich Gauss, Wikiquote — read as raw HTML: “Mathematics is the queen of the sciences. As quoted in Gauss zum Gedächtniss (1856) by Wolfgang Sartorius von Waltershausen; Variants: Mathematics is the queen of sciences and number theory is the queen of mathematics,” which is the epigraph’s form.

Added 17 September 2026 to source the chapter epigraph, which carried no marker. Another of the four secondhand ones: what survives is Sartorius’s report of 1856, not anything Gauss printed, and the claim line says so.

• RSA cryptosystem, Wikipedia — read as raw HTML: “Ron Rivest, Adi Shamir and Leonard Adleman, who publicly described the algorithm in 1977… An equivalent system was developed secretly in 1973 at Government Communications Headquarters (GCHQ)… by the English mathematician Clifford Cocks. That system was declassified in 1997,” and “A detailed description of the algorithm was published in August 1977, in Scientific American’s Mathematical Games column… This preceded the patent’s filing date of December 1977.”

The chapter’s own sentence is a warning rather than a recipe — Fermat’s little theorem alone is not an encryption system — and this entry carries only the history and the shape of the arithmetic, not any claim about the security of any key size.

• Catalan number, Wikipedia — read raw: “In 1988, it came to light that the Catalan number sequence had been used in China by the Mongolian mathematician Mingantu by 1730, when he started to write his book Ge Yuan Mi Lu Jie Fa.”
• Chinese remainder theorem, Wikipedia — read raw: “The Chinese remainder theorem appears in Gauss’s 1801 book Disquisitiones Arithmeticae.”; “Both names of the theorem refer to its earliest known statement that appeared in Sunzi Suanjing, a Chinese manuscript written during the 3rd to 5th century CE”; “a complete solution called Da-yan-shu (大衍術) in Qin Jiushao’s 1247 Mathematical Treatise in Nine Sections”; “Most implementations of RSA use the Chinese remainder theorem during signing of HTTPS certificates and during decryption.”
• Euler’s theorem, Wikipedia — read raw: “This paper was first presented to the Berlin Academy on June 8, 1758 and to the St. Petersburg Academy on October 15, 1759.”; “Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is not prime.”; the theorem “states that, if n and a are coprime positive integers, then a φ ( n ) is congruent to 1 modulo n” (exponent as typeset on the page); of that paper, in its reference list: “In this paper, Euler’s totient function, φ(n), is not named but referred to as “numerus partium ad N primarum.””
• Floor and ceiling functions, Wikipedia — read raw: “Carl Friedrich Gauss introduced the square bracket notation [x] in his third proof of quadratic reciprocity (1808).”; “Kenneth E. Iverson introduced, in his 1962 book A Programming Language, the names “floor” and “ceiling” and the corresponding notations ⌊x⌋ and ⌈x⌉..”
• Euclidean algorithm, Wikipedia — read raw: “It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC)..”
• Factorial, Wikipedia — read raw: “The notation n! for factorials was introduced by the French mathematician Christian Kramp in 1808..”
• Petersen graph, Wikipedia — read raw: “Although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by A. B. Kempe (1886)..”
• Iverson bracket, Wikipedia — cites Iverson (1962), A Programming Language, p. 11: “The notation was originally introduced by Kenneth E. Iverson in his programming language APL,” “though restricted to single relational operators enclosed in parentheses,” the square brackets being “advocated by Donald Knuth.”
• Stirling’s approximation, Wikipedia — dates the constant: “Stirling’s contribution in 1730 consisted of showing that the constant is precisely” √(2π); it also says “a related but less precise result was first stated by Abraham de Moivre.” The Factorial page above has “a 1729 letter from James Stirling to de Moivre stating what became known as Stirling’s approximation.”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Petersen graph, Wikipedia — read raw: “Although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by A. B. Kempe (1886).”; “The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring..”
• Pigeonhole principle, Wikipedia — read raw: “it is commonly called Dirichlet’s box principle or Dirichlet’s drawer principle after an 1834 treatment of the principle by Peter Gustav Lejeune Dirichlet under the name Schubfachprinzip.”
• Stirling number, Wikipedia — read raw: “They are named after James Stirling, who introduced them in a purely algebraic setting in his book Methodus differentialis (1730)..”
• Stirling numbers of the second kind, Wikipedia — read raw: “According to the third edition of The Art of Computer Programming, this notation was also used earlier by Jovan Karamata in 1935.”; “The brace notation was used by Imanuel Marx and Antonio Salmeri in 1962 for variants of these numbers.”; “This led Knuth to use it, as shown here, in the first volume of The Art of Computer Programming (1968)..”
• Wilson’s theorem, Wikipedia — read raw: “The theorem was first stated by Ibn al-Haytham c. 1000 AD.”; “There is evidence that Leibniz was also aware of the result a century earlier, but never published it.”; its footnote on Waring: “In the third (1782) edition of Waring’s Meditationes Algebraicae, Wilson’s theorem appears as problem 5 on page 380. On that page, Waring states: “Hanc maxime elegantem primorum numerorum proprietatem invenit vir clarissimus, rerumque mathematicarum peritissimus Joannes Wilson Armiger.” (A man most illustrious and most skilled in mathematics, Squire John Wilson, found this most elegant property of prime numbers.)”; its elementary proof: “the only values of a for which a ≡ a⁻¹ (mod p) are a ≡ ±1 (mod p). Therefore, with the exception of ±1, the factors in the expanded form of (p − 1)! can be arranged in disjoint pairs such that product of each pair is congruent to 1 modulo p.” (formulas typeset on the page)
• Rittaud and Heeffer, The pigeonhole principle, two centuries before Dirichlet, Mathematical Intelligencer 36 (2014), Ghent repository copy — the historical study: “In the writings of Dirichlet, the application of the principle is to be found in 1842”; “It seems that no one knows any precise prior reference, even if the year 1834 is frequently mentioned as the year of the discovery”; “In [7] and [5], the pigeonhole principle is used to prove complex and multidimensional versions of Equation 1” (Dirichlet’s diophantine approximation); and “In a later work he calls it the “Schubfachprinzip.””
• Ramsey’s theorem, Wikipedia — the proof of R(3, 3) = 6: “There are 5 edges incident to v and so (by the pigeonhole principle) at least 3 of them must be the same colour.”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Petersen graph, Wikipedia — describes the construction, Petersen’s 1898 paper, and its role as a standard counterexample.

• Ramanujan’s congruences, Wikipedia — states the three original congruences and later extensions.

• Binomial theorem, Wikipedia — states the expansion (x+y)^n = sum C(n, k) x^k y^(n-k) and Newton’s 1665 generalized binomial series.
• Factorial, Wikipedia — carries why 0! = 1 matters for binomial coefficients: “the number of ways to choose all n elements from a set of n is C(n, n) = n!/(n!0!) = 1, a binomial coefficient identity that would only be valid with 0!=1.”

• Bouton, “Nim, A Game with a Complete Mathematical Theory,” The Annals of Mathematics 3 (1901) 35–39 — Internet Archive item record — item metadata gives Title “Nim, A Game with a Complete Mathematical Theory,” Author “Bouton, Charles L.,” Journal “The Annals of Mathematics,” Volume 3, Year 1901, Pages 35-39, identifier urn:jstor-articleid:10.2307/1967631, ISSN 0003486X.
• Nim, A Game with a Complete Mathematical Theory — Wikisource transcription of Bouton (1901) — byline reads “Charles L. Bouton, Harvard University, Cambridge, Massachusetts.” This is the source for the chapter’s “of Harvard.”
• Wikipedia, Nim — “Charles L. Bouton of Harvard University” developed the complete theory in 1901; paper cited as Annals of Mathematics vol. 3, pp. 35–39, 1901–1902.
• Rougetet, “The Prehistory of Nim Game” (Gathering 4 Gardner G4G11 gift exchange) — text-extracted locally: it introduces Nim “taking as a reference the one introduced by Charles Leonard Bouton in his article, 1901,” calls Joseph Leonard Walsh “a student then a colleague of Bouton at Harvard University,” and its footnote 1 cites the paper as “The Annals of Mathematics, 2nd [series].”

• Nim, A Game with a Complete Mathematical Theory — Wikisource transcription of Bouton (1901) — Bouton defines a “safe combination” as heap sizes which, written in binary and aligned by columns, have every column sum ≡ 0 (mod 2); he states “If A leaves a safe combination on the table, B cannot leave a safe combination on the table at his next move” and that “the player who can first leave a safe combination on the table should win.”
• Wikipedia, Nim — winning strategy is to “finish every move with a nim-sum of 0”; the nim-sum is the XOR of the heap sizes; the player facing a nim-sum of zero loses against perfect play.

• Own computation — Python: 3^5^6 = 0; bin(3)='0b11', bin(5)='0b101', bin(6)='0b110', so the columns are (0, 1, 1), (1, 0, 1), (1, 1, 0) reading 4s, 2s, 1s — two ones in each. Exhaustive search over 1 <= a < b < c <= 10 with a^b^c == 0 returns (1, 2, 3) first, then (1, 4, 5), (1, 6, 7), (1, 8, 9). For three nonempty heaps a <= b <= c with a^b^c = 0, a = b would force c = 0, so all three must be distinct and (1, 2, 3) is minimal by total. Note that the two-heap position (1, 1) also has nim-sum 0 and a smaller total, so the minimality of 1, 2, 3 holds only among positions with three heaps.

• Own computation — Python: 1^2^4 = 7 (001 | 010 | 100 -> 111); 1^2^3 = 0 (001 | 010 | 011). Since 7 has its 4-bit set and only the heap of 4 has that bit, the unique winning move is in that heap, taking it to 4 XOR 7 = 3.

• Nim, A Game with a Complete Mathematical Theory — Wikisource transcription of Bouton (1901) — “The writer has not been able to discover much concerning its history, although certain forms of it seem to be played at a number of American colleges, and at some of the American fairs. It has been called Fan-Tan, but as it is not the Chinese game of that name, the name in the title is proposed for it.”
• Wikipedia, Nim — “Variants of nim have been played since ancient times”; a Chinese origin (jiǎn-shízǐ) is suggested but uncertain; earliest European references from the beginning of the 16th century; the OED derives the name from German nimm, ‘take’.
• Rougetet, “The Prehistory of Nim Game” — text extracted locally: the Fan-Tan link “was wrong, as chance is part of the Fan-Tan”; Ross (Mathematical Gazette, May 1953) denied a Chinese origin and proposed German nimm; Walsh, Bouton’s Harvard colleague, confirmed it in December 1953; “Games as simple as Nim ... are handed over verbally; therefore this orality is difficult to track.”

• Rougetet, “The Prehistory of Nim Game” (Gathering 4 Gardner G4G11 gift exchange) — and text-extracted locally. Pacioli’s Effect XXXIIII asks two players to reach 30 by adding, in turn, a number from 1 to 6: “it is a simplified version of Bouton’s Nim game ... the solution remains based on the same principle as Bouton’s Nim: there exist ‘safe combinations’ that secure the win, provided that one plays correctly.” For the target 30 Pacioli “quickly gives the winning strategy to be applied: the four safety steps, 2, 9, 16 and 23 have to be reached,” and states a general rule — “always divide the number that you wish to arrive at by one more than has been taken and the remainder of the said division will always be the first [step of the] progression” — with a worked example for the target 35: “he takes whatever he wants up to 6, and you will take, or actually you will make 7 the first degree, and then 14, 21, 28, and 35.” The paper then says exactly what is new in 1901: “it was not before 1901, with Bouton’s article, that the solution of a combinatorial game was to be formalized and studied as a general case with any number of piles and any number of chips in each pile.” Problem XIX of Bachet’s Problemes plaisans et delectables, qui se font par les nombres (Lyon, 1st edition, 1612) proposes “a version of Nim that is similar to the one formulated by Pacioli”: two players must reach 100 by adding numbers from 1 to 10, and Bachet’s rule is to “add 1 to the number that cannot be exceeded, here 10; you obtain 11 and then, always remove 11 of the number to be reached, 100; you will obtain these numbers 89, 78, 67, 56, 45, 34, 23, 12, 1.”
• Nim, A Game with a Complete Mathematical Theory — Wikisource transcription of Bouton (1901) — Bouton makes no priority claim of any kind. He says only that the game “has interested the writer on account of its seeming complexity, and its extremely simple and complete mathematical theory.”

Bouton makes no priority claim himself, and the unqualified form of one — that in solving Nim he did something never done for any game — fails on Pacioli and Bachet, whose one-heap game was solved completely and in print. So the chapter and the ⊕ card claim generality rather than priority: not the first game solved, but the first solved for any number of heaps of any size, which is the distinction Rougetet draws.

• Wikipedia, Nimber — nim-addition α ⊕ β = mex({α′ ⊕ β} ∪ {α ⊕ β′}), equal to bitwise XOR for finite ordinals; nim-multiplication α ⊗ β = mex{(α′ ⊗ β) ⊕ (α ⊗ β′) ⊕ (α′ ⊗ β′) : α′ < α, β′ < β}; “2² = 3” in nim-multiplication; “an algebraically closed field of characteristic 2” in which each element is “its own additive inverse”; “For all natural numbers n, the set of nimbers less than 2^(2^n) form the Galois field GF(2^(2^n))”; the field comes “with the nimber multiplicative inverse of a nonzero ordinal α,” so every nimber but 0 can be divided by; Conway, On Numbers and Games, 1976 (2nd edition 2000).

• Own computation — Recursive implementation of a⊗b = mex{(a′⊗b) xor (a⊗b′) xor (a′⊗b′) : a′<a, b′<b} in Python. For a = b = 2 the excluded set from (a′,b′) in {0,1}² is {0, 2, 1}, whose mex is 3. Full table for 0..3: row 0 = 0,0,0,0; row 1 = 0,1,2,3; row 2 = 0,2,3,1; row 3 = 0,3,1,2 — the multiplication table of GF(4), confirming both 2 ⊗ 2 = 3 and the absence of zero divisors.

• Wikipedia, Sprague–Grundy theorem — “Every impartial game under the normal play convention is equivalent to a one-heap game of nim, or to an infinite generalization of nim.” Grundy value defined by mex; “The Grundy value of the sum of a finite set of positions is just the nim-sum of the Grundy values of its summands”; a position is a loss for the next player iff its Grundy value is zero. Cites Sprague, Über mathematische Kampfspiele, Tohoku Math. J. 41, 438–444, and Grundy, Mathematics and games, Eureka 2, 6–8, 1939.
• R. Sprague, “Über mathematische Kampfspiele,” Tohoku Mathematical Journal, First Series, vol. 41 — J-STAGE record — from the publisher’s own record: author R. Sprague, Tohoku Mathematical Journal First Series, volume 41, pages 438–444, publication year 1935.
• Wikipedia, Roland Sprague — “Über mathematische Kampfspiele, Tôhoku Mathematical Journal, vol. 41 (1935), pp. 438–444”; Sprague and Grundy discovered the function independently “in 1935 and 1939 respectively.”
• Wikipedia, Patrick Michael Grundy — “Mathematics and games,” Eureka, 2: 6–8, 1939; introduced the Sprague–Grundy function, independent of and later than Sprague’s 1935 work.

The volume Tôhoku Math. J. 41 spans 1935–1936 and a few secondary listings give 1936. The publisher’s own J-STAGE record and the Wikipedia biography of Sprague both give 1935, which is also the date universally used in the combinatorial game theory literature, so the chapter’s “Sprague in 1935” stands.

• Wikipedia, Minimax theorem — “Von Neumann published this in 1928 in the paper ‘Zur Theorie der Gesellschaftsspiele’ (Mathematische Annalen, volume 100, pages 295–320)”; states max_x min_y f(x, y) = min_y max_x f(x, y); applies to finite two-player zero-sum games with mixed strategies, where f(x, y) is the expected payoff to the first player.
• Wikipedia, Game theory — von Neumann published “On the Theory of Games of Strategy” in 1928; “Von Neumann’s original proof used Brouwer’s fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics.”
• Minimax, Wikipedia — “For every two-person zero-sum game with finitely many strategies, there exists a value V and a mixed strategy for each player”; “Equivalently, Player 1’s strategy guarantees them a payoff of V regardless of Player 2’s strategy, and similarly Player 2 can guarantee themselves a payoff of −V.”

• Wikipedia, Matching pennies — Even and Odd each choose heads or tails; Even wins on a match, Odd on a mismatch; the game is zero-sum; “There is no pure strategy (heads or tails) that is a best response to a best response”; there is a unique mixed-strategy equilibrium in which “each player chooses heads or tails with equal probability”; “When either player plays the equilibrium, everyone’s expected payoff is zero.”
• Own computation — With payoff +1 to the matcher on a match and −1 on a mismatch, and mixing probabilities p and q on heads, the matcher’s expected payoff is pq − p(1−q) − (1−p)q + (1−p)(1−q) = (2p−1)(2q−1). At p = 1/2 this is 0 for every q, so the matcher can guarantee 0; by the symmetry of the game the opponent can too, so the value is 0. If p ≠ 1/2 the opponent’s best reply is pure, against which the matcher would want to switch to the opposite pure strategy, so p = q = 1/2 is the only equilibrium.

• Wikipedia, Nash equilibrium — “Equilibrium points in n-person games,” PNAS volume 36, pages 48–49, 1950; “there is a Nash equilibrium, possibly in mixed strategies, for every finite game”; equilibrium defined as a profile where “no player could gain more by changing their own strategy (holding all other players’ strategies fixed)”; “Nash equilibria need not exist if the set of choices is infinite and non-compact”: “A game where two players simultaneously name a number and the player naming the larger number wins does not have a NE, as the set of choices is not compact because it is unbounded.”
• Wikipedia, Game theory — “Nash proved that every finite n-player, non-zero-sum (not just two-player zero-sum) non-cooperative game has what is now known as a Nash equilibrium in mixed strategies,” in 1950.

• Rapoport, Seale and Colman, “Is Tit-for-Tat the Answer? On the Conclusions Drawn from Axelrod’s Tournaments,” PLOS ONE 10(7): e0134128 — the payoff matrix of Axelrod’s original tournament is given as “(T, R, P, S) = (5, 3, 1, 0).”
• Wikipedia, Prisoner’s dilemma — the payoff ordering is T > R > P > S (temptation, reward, punishment, sucker); mutual defection is the only strong Nash equilibrium, and “the collectively ideal result of mutual cooperation is irrational from a self-interested standpoint.”
• Own computation — With T = 5, R = 3, P = 1, S = 0: against a cooperator, defect pays 5 > 3; against a defector, defect pays 1 > 0. Defection strictly dominates, so (D, D) is the unique Nash equilibrium and u(D, D) = 1, while u(C, C) = 3. Also R > (T + S)/2 = 2.5, so these values satisfy the usual repeated-game condition as well.

• MacTutor History of Mathematics, Quotations by John Horton Conway — verbatim: “You get surreal numbers by playing games. I used to feel guilty in Cambridge that I spent all day playing games, while I was supposed to be doing mathematics. Then, when I discovered surreal numbers, I realized that playing games IS mathematics.” The chapter quotes the second and third sentences exactly.

• Wikipedia, Surreal number — “Research on the Go endgame by John Horton Conway led to the original definition and construction of the surreal numbers”; “He was analyzing Go endgames, and realized that it would be useful to have some way to combine the analyses of non-interacting subgames”; “Every surreal number is a game, but not all games are surreal numbers, e.g. the game {0 | 0} is not a surreal number” — {0 | 0} being star. Conway used the surreals for game analysis in On Numbers and Games (1976).
• Wikipedia, Nimber — the nimbers form a field of characteristic 2 in which each element is its own additive inverse — the chapter’s “every one being its own negative.” Read again on 23 September 2026, its lead also says the nimbers “form a subclass of surreal numbers”; the book does not follow that phrase (the Star line below: ∗ = {0 | 0} is a value but not a number) and keeps the link for the field facts.
• Star (game theory), Wikipedia — read as raw wikitext: star, “written as ∗ or ∗1,” may be denoted as “{0|0}” and “is a value, but not a number in the traditional sense”; it is “confused with” 0.

• Wikipedia, Adder (electronics) — “The Boolean logic for the sum (in this case S) will be A ⊕ B whereas for the carry (C) will be A ⋅ B”; for the full adder “S = A ⊕ B ⊕ C_in.”

• Wikipedia, Iverson bracket — defines the bracket as the indicator of a proposition and attributes its 1962 introduction to Kenneth Iverson; the history section records Knuth’s later advocacy.
• Donald E. Knuth, Two notes on notation, arXiv:math/9205211 — Knuth’s article free on arXiv, advocating the notation [P] and extending Iverson’s idea; journal reference Amer. Math. Monthly 99 (1992), no. 5, 403–422.

• Wikipedia, Boolean algebra — records Boole’s The Mathematical Analysis of Logic (1847) and The Laws of Thought (1854), and the Boolean operations and two-element algebra.

• Wikipedia, Boolean algebra — history section identifies Shannon’s 1937 master’s thesis as the application of Boolean algebra to electrical switching circuits.

• Wikipedia, Fermat’s Last Theorem — infobox and overview give the integer statement, c. 1637 margin, and posthumous 1670 publication in Samuel Fermat’s edition of Arithmetica; on Fermat’s own proof, “It is not known whether Fermat had actually found a valid proof for all exponents n, but it appears unlikely,” and the descent proof he did leave is the one for fourth powers.

• Wikipedia, Fermat’s Last Theorem — historical-development section lists the proof of exponent 3, Germain’s theorem for a class of primes, and Kummer’s proof for regular primes, without a year.
• MacTutor, Fermat’s last theorem — supports 1847 — “By September 1847 Kummer sent to Dirichlet and the Berlin Academy a paper proving that a prime p is regular … if”.
• Wikipedia, Regular prime — supports 1850 — “In 1850, Kummer proved that Fermat’s Last Theorem is true for a prime exponent p if p is regular”, i.e. the published general proof, Crelle 40.

• Wikipedia, Fermat’s Last Theorem — the three lectures “on 21–23 June 1993 at the Isaac Newton Institute for Mathematical Sciences”; the error in peer review; Nick Katz “alerted Wiles on 23 August 1993”; the two manuscripts submitted on 24 October 1994; “the entirety of the May 1995 issue” of the Annals of Mathematics.
• Wikipedia, Wiles’s proof of Fermat’s Last Theorem — Katz, as referee, “asked Wiles a series of clarifying questions that led Wiles to recognise that the proof contained a gap”; “the Euler system used to extend Kolyvagin and Flach’s method was incomplete”; Wiles “spent almost a year trying to repair his proof, initially by himself and then in collaboration with his former student Richard Taylor,” with the breakthrough on 19 September 1994; the papers are cited as Annals of Mathematics 141(3), 443–551 and 553–572.
• Encyclopedia.com, Andrew J. Wiles — “He spent seven years working on it in the attic office of his home in Princeton.”

• Wikipedia, Boolean algebra — gives complement identities and identifies Boolean algebra as an algebra of truth values.
• Wikipedia, Law of excluded middle — identifies excluded middle as a classical logical law, discusses its Aristotelian formulation, and gives Brouwer’s intuitionist refusal of it, but no year for it.
• L. E. J. Brouwer, Wikipedia — read as raw HTML, quoting Kleene: “in 1908: ‘… Brouwer, in a paper titled “The untrustworthiness of the principles of logic,” challenged the belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384–322 B.C.) have an absolute validity, independent of the subject matter to which they are applied’.”
• Stanford Encyclopedia of Philosophy, Contradiction — introduces Aristotle’s law of non-contradiction as “the firmest of all principles,” quoting Metaphysics 1005b19–23 in Kirwan’s translation: “For the same thing to hold good and not to hold good simultaneously of the same thing and in the same respect is impossible.”
• Wikipedia, Law of noncontradiction — gives Aristotle’s formulations in Metaphysics Book IV, including “The most certain of all basic principles is that contradictory propositions are not true simultaneously” (1011b13–14).
• Law of thought, Wikipedia — names the three: “the law of identity (LOI), the law of non-contradiction (LNC), and the law of excluded middle (LEM)”; and, quoting Russell, “for no very good reason, three of these principles have been singled out by tradition under the name of ‘Laws of Thought’.”
• Tee (symbol), Wikipedia — “The tee (⊤, \top in LaTeX), also called down tack (as opposed to the up tack) or verum, is a symbol used to represent: The truth value of being true in logic, or a sentence (e.g., formula in propositional calculus) which is unconditionally true.”
• Up tack, Wikipedia — “The truth value ‘false’, or a logical constant denoting a proposition in logic that is always false. (The names “falsum,” “absurdum” and “absurdity” come from this context.)”

• Wikipedia, Irrational number — states that √2 is irrational.
• Wikipedia, Euclid’s theorem — states that there are infinitely many primes.
• Wikipedia, Continuum hypothesis — states that the cardinality of the integers is strictly below that of the reals.
• Own computation — Each displayed proposition is true, so its Iverson bracket is 1; 1+1+1+1=4 and the Boolean excluded-middle term contributes 1, giving denominator 5.

• Wikipedia, Euclid’s theorem — “Euclid offered a proof in his work Elements (Book IX, Proposition 20)”; the c. 300 BC date is on the Euclid’s Elements line; the proof step, “Since q is either prime or not: If q is prime, then there is at least one more prime that is not in the list, namely, q itself. If q is not prime, then some prime factor p divides q”; and “Euclid is often erroneously reported to have proved this result by contradiction beginning with the assumption that the finite set initially considered contains all prime numbers, though it is actually a proof by cases, a direct proof method.”
• Euclid’s Elements, Wikipedia — “The Elements … is a mathematical treatise written c. 300 BC by the Ancient Greek mathematician Euclid.”

• Wikipedia, Continuum hypothesis — the hypothesis “was first introduced by Georg Cantor in his 1878 paper,” and “Cantor gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers.”
• Wikipedia, Cantor’s first set theory article — dates the article to 1874 and gives its proof that the set of real numbers is uncountable.
• Wikipedia, Cantor’s diagonal argument — dates the diagonal proof to 1891.

• Stanford Encyclopedia of Philosophy, Gödel’s Incompleteness Theorems — explains the 1931 first and second incompleteness theorems and their hypotheses: “The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F”; the standard statement is given “Accommodating an improvement due to J. Barkley Rosser in 1936.”

• Wikipedia, Continuum hypothesis — attributes relative consistency to Gödel’s constructible universe and independence/non-provability to Cohen’s forcing work; its references give Gödel’s announcement as “Gödel, Kurt (1938). “The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis.” Proceedings of the National Academy of Sciences. 24 (12): 556–557.”
• Hilbert’s problems, Wikipedia — read as raw HTML: the 1902 Bulletin of the AMS translation heads the list with “Cantor’s problem of the cardinal number of the continuum,” and the table of problems gives “1st: The continuum hypothesis: there is no set whose cardinality is strictly between that of the integers and that of the real numbers.”
• List of statements independent of ZFC, Wikipedia — “Gödel produced a model of ZFC in which CH is true, showing that CH cannot be disproven in ZFC; Paul Cohen later invented the method of forcing to exhibit a model of ZFC in which CH fails, showing that CH cannot be proven in ZFC.”

• Wikipedia, Peano axioms — gives Peano’s 1889 publication and the basic successor/induction framework; and “The Peano axioms can be derived from set theoretic constructions of the natural numbers and axioms of set theory such as ZF..”
• Wikipedia, Zermelo–Fraenkel set theory — gives the history of Zermelo’s 1908 axiomatization, Fraenkel’s and Skolem’s 1922 work, and ZFC.
• Wikipedia, Principia Mathematica — proposition *54.43, “From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2,” stands on page 379 of the first volume of the first edition and page 362 of the second; the proof is completed in the second volume; the three volumes appeared in 1910, 1912 and 1913; and “The Principia Mathematica (often abbreviated PM) is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell.”
• 11.2 Arithmetic, MATH1001 Introduction to Number Theory, University of Southampton — Theorem 11.2 derives it from the successor function and the recursive definition of addition: “1+1 = 1+s(0) = s(1+0) = s(1) = 2.”

• Wikipedia, Turnstile (symbol) — describes ⊢ as syntactic derivability and ⊨ as semantic consequence.
• Wikipedia, Gödel’s completeness theorem — dates and states the completeness theorem for first-order logic.
• Stanford Encyclopedia of Philosophy, Gödel’s Incompleteness Theorems — distinguishes completeness from incompleteness and explains the latter’s arithmetic consequences.
• Wikipedia, Quantifier (logic) — “In 1935, Gentzen introduced the ∀ symbol, by analogy with Peano’s ∃ symbol,” which Peano introduced “in 1897”; ∀ “did not become canonical until the 1960s.”
• Wikipedia, Logical connective — the history of notations gives ∨ in Russell 1908, ¬ and ∧ in Heyting 1930, → in Hilbert 1918, and ⇒ and ⇔ in Bourbaki 1954.
• Wikipedia, Begriffsschrift — the 1879 calculus “contains the first appearance of quantified variables,” and “Some vestige of Frege’s notation survives in the ‘turnstile’ symbol ⊢.”

• Wikipedia, De Morgan’s laws — states both identities and gives De Morgan’s 1847 Formal Logic attribution. Re-read raw on 17 September 2026 for the precursors: “a similar observation was made by Aristotle, and was known to Greek and Medieval logicians. For example, in the 14th century, William of Ockham wrote down the words that would result by reading the laws out. Jean Buridan, in his Summulae de Dialectica, also describes rules of conversion that follow the lines of De Morgan’s laws. Still, De Morgan is given credit for stating the laws in the terms of modern formal logic.”
• NAND logic, Wikipedia — “The NAND Boolean function has the property of functional completeness. This means that any Boolean expression can be re-expressed by an equivalent expression utilizing only NAND operations.”; and “noting from de Morgan’s law that a NAND gate is an inverted-input OR gate.”

• Wikipedia, Therefore sign — gives the 1659 Rahn attribution for ∴ and the inverse because sign’s history.
• Wikipedia, Q.E.D. — explains Q.E.D. and the modern tombstone/halmos symbol, including Halmos’s adoption.

Two of these are attributions rather than datings. Rahn used the triangle in both orientations, and so did writers after him; what the nineteenth century added was the exclusive convention. The tombstone is credited to Halmos on his own account — he “claims to have pioneered” it — which is how the reference works put it, and how the page takes it.

• Own computation — Fermat’s bracket is 0 by 8.4–8.6; x∧¬x=0 by 8.7; the denominator is 4+1=5 by 8.8; hence (0+0)/5=0. An Iverson bracket has values 0 and 1, which are ordinary integers by 8.1.

• Fermat’s Last Theorem, Wikipedia — read as raw HTML: the Latin is given as “Cubum autem in duos cubos … cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet” and translated “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain”; and “After Fermat’s death in 1665, his son Clément-Samuel Fermat produced a new edition of the book (1670) augmented with his father’s comments.”

Added 17 September 2026 to source the chapter epigraph, which carried no marker, and to make the provenance explicit: the margin itself does not survive, and what exists is the 1670 edition his son Clément-Samuel printed. The chapter’s footer was reworded to say so, and to give the son’s full name, in the pass of 18 September 2026.

• Peano axioms, Wikipedia — read raw: “In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book The principles of arithmetic presented by a new method.”
• Boolean algebra, Wikipedia — read raw: “Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847), and set forth more fully in his An Investigation of the Laws of Thought (1854)..”
• Gödel’s incompleteness theorems, Wikipedia — read raw: “These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in philosophy of mathematics..”
• Fermat’s Last Theorem, Wikipedia — read raw: “The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica..”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Wikipedia, Lambda calculus — defines lambda calculus as a formal system based on abstraction and application, attributes its introduction to Alonzo Church in the 1930s, and gives beta-reduction for (λx.t)s.
• Wikipedia, Church encoding — describes Church numerals as repeated function application and gives zero as λf.λx.x.
• Internet Encyclopedia of Philosophy, Lambda Calculi — ‘Alonzo Church first introduced the λ-calculus as A set of postulates for the foundation of logic in two papers of that title published in 1932 and 1933’; says Church gave the numeral encoding at the end of the 1933 paper, and that Kleene found how to define the predecessor function.

• A. M. Turing, On Computable Numbers (interactive edition) — gives the paper’s 1936 receipt/read dates, describes machines with finite control and unbounded tape, states that no general process can determine which machines are circle-free, and connects this to the modern halting problem.

• A. M. Turing, On Computable Numbers (interactive edition) — section 1 describes finite m-configurations, an unbounded tape divided into squares, and a scanned square.
• Wikipedia, Turing machine — describes the transition function and universal-machine simulation.

• Busy Beaver Challenge, July 2024 announcement — history line attributes the busy-beaver game and functions S and Σ to Radó in 1962.
• Busy Beaver Challenge paper — defines S(n) as the maximum steps of an n-state, two-symbol machine from the all-zero tape and identifies Radó in 1962.

• Wikipedia, Busy beaver — gives the standard small step-count values 1, 6, 21, and 107, distinguishes step-count from productivity conventions, and credits Lin and Radó (Computer Studies of Turing Machine Problems, 1965) with S(3) = 21 and Brady with S(4) = 107 in 1983.
• Pascal Michel, Historical survey of Busy Beavers — “Rado (1963) claimed that Sigma(2, 2) = 4, but that S(2, 2) was yet unknown. The value S(2, 2) = 6 was probably set by Lin in 1963.”; “Lin (1963) proved this conjecture and this proof was eventually published by Lin and Rado (1965).”; “Brady (1974, 1975) proved this conjecture, and the proof was eventually published in Brady (1983).”

• Busy Beaver Challenge, We have proved BB(5) = 47,176,870 — dated July 2, 2024; states the value, describes the Coq proof, reports the symmetry-reduced enumeration, and says BB(6) is a next target.
• The bbchallenge Collaboration, Determination of the fifth Busy Beaver value — abstract states S(5)=47,176,870 and that the proof enumerates 181,385,789 five-state machines, deciding halting for each; posted 15 September 2025, revised 23 March 2026.
• Wikipedia, Busy beaver — ‘The 5-state busy beaver was discovered by Heiner Marxen and Jürgen Buntrock in 1989, but only proved to be the winning fifth busy beaver in 2024 by an online amateur mathematical collective, using a proof formalized in Rocq.’

• Wikipedia, Busy beaver — states that the Busy Beaver function is non-computable and eventually dominates every computable function, with the halting-problem reduction.

• NIST Dictionary of Algorithms and Data Structures, Ackermann’s function — gives the recurrence, attributes the 1928 observation to Wilhelm Ackermann, and states that the function is recursive but not primitive recursive.
• Stanford Encyclopedia of Philosophy, History of the Ackermann and Péter functions — explains that Ackermann’s result was growth beyond functions defined by ordinary recursion and discusses the historical qualification.

• Wikipedia, Ackermann function — gives the common recurrence, A(3, 3)=61, and A(4, 2)=2^65536−3 with 19,729 decimal digits.
• NIST Dictionary of Algorithms and Data Structures, Ackermann’s function — provides the recurrence used to recompute A(3, 3).

Own arithmetic check: the recurrence gives A(3, 3)=61; 2^65536−3 has floor(65536 log10 2)+1 = 19,729 digits; A(4, n) = 2↑↑(n+3) − 3, so A(4, 3) = 2^(2^65536) − 3, whose decimal expansion has about 10^19,728 digits.

• Wikipedia, P versus NP problem — defines P and NP through polynomial-time solving and verification and states the open P-versus-NP question.
• Wikipedia, Big O notation — ‘Bachmann proposed the notation in 1894 and Landau extended it in 1909’; ‘In 1976, Donald Knuth defined f(x) = Ω(g(x)) ⟺ g(x) = O(f(x))’; f(x) = O(g(x)) if there is a positive real M with |f(x)| ≤ M·g(x) throughout the domain. Its table of orders gives n log n as the “fastest possible comparison sort; heapsort and merge sort.”
• Wikipedia, Comparison sort — ‘A comparison sort must have an average-case lower bound of Ω(n log n) comparison operations’.
• David Harvey and Joris van der Hoeven, Integer multiplication in time O(n log n) — Annals of Mathematics 193 (2021), 563–617 (the March 2019 preprint announcement is on the Multiplication algorithm line). Multiplies two n-bit integers in O(n log n) bit operations, confirming the 1971 conjecture of Schönhage and Strassen.
• Multiplication algorithm, Wikipedia — “In March 2019, David Harvey and Joris van der Hoeven announced their discovery of an O(n log n) multiplication algorithm.”; “It was published in the Annals of Mathematics in 2021.”

• Wikipedia, P versus NP problem — records Cook 1971, Levin 1973, and Karp’s 21 NP-complete problems (without a year; 1972 is on the Karp line) in the history of NP-completeness and the P-versus-NP problem.
• Wikipedia, Cobham’s thesis — ‘Alan Cobham’s 1965 paper entitled The intrinsic computational difficulty of functions is one of the earliest mentions of the concept of the complexity class P’, and ‘Jack Edmonds’s 1965 paper Paths, trees, and flowers is also credited with identifying P with tractable problems’.
• Karp’s 21 NP-complete problems, Wikipedia — “In his 1972 paper, ‘Reducibility Among Combinatorial Problems’, Richard Karp used Stephen Cook’s 1971 theorem that the Boolean satisfiability problem is NP-complete (also called the Cook–Levin theorem) to show that there is a polynomial time many-one reduction from the Boolean satisfiability problem to each of 21 combinatorial and graph theoretical computational problems, thereby showing that they are all NP-complete.”; its list includes “Undirected Hamilton circuit (Karp’s name, now usually called Undirected Hamiltonian cycle),” and it calls the paper “one of the first demonstrations that many natural computational problems occurring throughout computer science are computationally intractable.”

• Wikipedia, IEEE 754 — describes binary64 as having 53 bits of precision and gives the binary64 spacing/precision facts: “The binary32 (single) and binary64 (double) formats are two of the most common formats used today.”
• Double-precision floating-point format, Wikipedia — (§ IEEE 754 double-precision binary floating-point format: binary64, § Double-precision examples): “Significand precision: 53 bits (52 explicitly stored),” and the smallest number greater than 1 is 1.0000000000000002220…, so the spacing immediately above 1 is 2⁻⁵² ≈ 2.220446 × 10⁻¹⁶; “The maximum relative rounding error when rounding a number to the nearest representable one (the machine epsilon) is therefore 2⁻⁵³.” (exponent as typeset on the page)
• Floating-point arithmetic, Wikipedia — read as raw wikitext: “The classical toy problem that illustrates the issue with decimals is 0.1 + 0.2 - 0.3, which evaluates to about 5.551e-17 in double precision instead of the expected 0, because as stated above in Representable numbers, conversion and rounding all three numbers cannot be represented by a finite binary expansion.”
• Machine epsilon, Wikipedia — “There are two prevailing definitions, denoted here as rounding machine epsilon or the formal definition and interval machine epsilon or mainstream definition.”; “In the mainstream definition, machine epsilon is independent of rounding method, and is defined simply as the difference between 1 and the next larger floating point number.” Its table gives binary64 a rounding machine epsilon of 2⁻⁵³ ≈ 1.11e-16 and an interval machine epsilon of 2⁻⁵² ≈ 2.22e-16 (exponents as typeset on the page).

• Wikipedia, IEEE 754 — describes round-to-nearest, ties-to-even and binary floating-point formats.
• David Goldberg, What Every Computer Scientist Should Know About Floating-Point Arithmetic, ACM Computing Surveys 23(1), March 1991 — sets out the round-to-even rule and argues for it (Theorem 5), then adopts it — “Throughout the rest of this paper, round to even will be used” — and defines machine epsilon. It is the classic exposition, not the specification; the ties-to-even default itself is carried by the IEEE 754 entry above.

Own computation: 1+2^−53 is exactly the midpoint of 1 and 1+2^−52; the significand of 1 has an even trailing bit, so the rounded result is 1 and the subtraction is exactly zero.

• Wikipedia, Kolmogorov complexity — defines the shortest-program measure and explains uncomputability through a construction ‘modeled on a self-referential construction used in Berry’s paradox’; says Solomonoff ‘published it in 1960’ in A Preliminary Report on a General Theory of Inductive Inference and ‘gave a more complete description in his 1964 publications’, that Kolmogorov ‘later independently published this theorem in Problems Inform. Transmission in 1965’, that Chaitin presents it in the Journal of the ACM, and that ‘When Kolmogorov became aware of Solomonoff’s work, he acknowledged Solomonoff’s priority’.
• Algorithmic information theory, Wikipedia — “Algorithmic information theory was later developed independently by Andrey Kolmogorov, in 1965 and Gregory Chaitin, around 1966.”; its references list Chaitin, “On the Length of Programs for Computing Finite Binary Sequences,” Journal of the ACM 13 (4): 547–569 (1966).
• Chaitin’s constant, Wikipedia — a real number that, “informally speaking, represents the probability that a randomly constructed program will halt”; “Although there are infinitely many halting probabilities, one for each (universal, see below) method of encoding programs, it is common to use the letter Ω to refer to them as if there were only one.”

• Wikipedia, Rice’s theorem — states the theorem for nontrivial semantic properties of programs and gives the 1951 dissertation/1953 publication history.

• Wikipedia, Combinatory logic — gives the S, K, and I combinators, the S/K basis, and the fixed-point combinator construction; says the combinators ‘were introduced by Schönfinkel in 1920’ and published in 1924, and that Curry ‘rediscovered the combinators while working as an instructor at Princeton University in late 1927’, publishing Grundlagen der Kombinatorischen Logik in 1930.
• Wikipedia, Fixed-point combinator — gives Y = λf.(λx.f (x x)) (λx.f (x x)) as ‘Haskell Curry’s paradoxical combinator Y’, satisfying Y g = g (Y g).
• Stephen Wolfram, Where Did Combinators Come From? Hunting the Story of Moses Schönfinkel — “On Tuesday, December 7, 1920, the Göttingen Mathematics Society held its regular weekly meeting—at which a 32-year-old local mathematician named Moses Schönfinkel with no known previous mathematical publications gave a talk entitled ‘Elemente der Logik’ (‘Elements of Logic’).”; “Moses Schönfinkel described the result of his 1920 talk in an 11-page paper published in 1924”

• Wikipedia, Gödel numbering — describes the encoding of symbols, formulas, and proofs by natural numbers and its role in incompleteness; “Gödel used a system based on prime factorization..”
• A. M. Turing, On Computable Numbers (interactive edition) — describes machine descriptions and universal-machine computation.
• Gödel’s incompleteness theorems, Wikipedia — “The statement p is not literally equal to ~Bew(G(p)); rather, p states that if a certain calculation is performed, the resulting Gödel number will be that of an unprovable statement.”
• Diagonal lemma, Wikipedia — “For n̄, the standard numeral of n (i.e. 0̄ =df 0 and n+1̄ =df S(n̄)), let ⌜φ⌝ be the standard numeral of the code of φ.”

• Own computation — Direct symbolic/arithmetic check: (λf.λx.x) S 0 → (λx.x) 0 → 0; 1+6+21+107=135 and 47,176,870 + 61 + 135 = 47,177,066; the cited Busy Beaver values are positive; P⊆NP is a true inclusion; and the midpoint tie calculation in binary64 gives fl(1+2^−53)−1=0.

• The bbchallenge Collaboration, Determination of the fifth Busy Beaver value — abstract explicitly says the proof enumerates 181,385,789 five-state machines.
• Busy Beaver Challenge, July 2024 announcement — says the space was reduced to approximately 180 million machines using symmetries.

Own arithmetic: five states and two symbols give ten table entries, each of which writes one of two symbols, moves one of two ways and enters one of five states, or halts, so 21^10 = 16,679,880,978,201. Both counts are correct and count different things; the number the chapter gives is the one the proof actually enumerates.

• Entscheidungsproblem, Wikipedia — attributes the problem to ‘David Hilbert and Wilhelm Ackermann in 1928’; states that it asks for ‘an algorithm that considers an input statement and answers yes or no according to whether it is universally valid, i.e., valid in every structure’; and that ‘In 1936, Alonzo Church and Alan Turing published independent papers showing that a general solution to the Entscheidungsproblem is impossible’; and ‘Turing reduced the question of the existence of an ‘algorithm’ or ‘general method’ able to solve the Entscheidungsproblem to the question of the existence of a ‘general method’ that decides whether any given Turing machine halts or not (the halting problem).’

The dates of the two 1936 papers, and Turing’s appendix showing the two answers describe the same class, are carried separately by 9.3.

• IEEE 754, Wikipedia — ‘The first IEEE standard for floating-point arithmetic, IEEE 754-1985, was published in 1985’, and ‘Round to nearest, ties to even’ is given as the default for binary floating point.
• Floating-point arithmetic, Wikipedia — “This first standard is followed by almost all modern machines.”
• Double-precision floating-point format, Wikipedia — “Double precision is not required by the standards (except by the optional annex F of C99, covering IEEE 754 arithmetic), but on most systems, the double type corresponds to double precision.”

Added in the V14 prose pass: the chapter’s 1985 date and the card line ‘the double of every mainstream language since the 1985 standard’ previously stood on 9.13, whose claim covers only binary64 precision and spacing and carries no date.

• A. M. Turing, On Computable Numbers (full text) — read as raw HTML: “Received 28 May, 1936 — Read 12 November, 1936. Proceedings of the London Mathematical Society, s2-42 (1937), pp. 230–265,” and the paper opens “The ‘computable’ numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means,” which is the epigraph word for word.

Added 17 September 2026 to source the chapter epigraph, which carried no marker. The footer dates the paper 1936, which is the year it was received and read; the volume it appeared in is dated 1937, as r-9.3 already records.

• Gödel numbering, Wikipedia — read raw: “Since the publishing of Gödel’s paper in 1931, the term “Gödel numbering” or “Gödel code” has been used to refer to more general assignments of natural numbers to mathematical objects..”
• Universal Turing machine, Wikipedia — read raw: “Turing introduced the idea of such a machine in 1936–1937..”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Ackermann function, Wikipedia — gives the Ackermann-Peter recurrence and states the function is computable but grows faster than any primitive recursive function.
• Wilhelm Ackermann, MacTutor — carries the two-argument simplification: “A(x, y, z) was simplified to a function P(x, y) of two variables by Rozsa Peter whose initial condition was simplified by Raphael Robinson.”

• MacTutor, Nine Chapters on the Mathematical Art — Chapter 8 describes systems of up to six simultaneous linear equations, an augmented array, and reduction to triangular form by elementary column operations.
• MacTutor, Matrices and determinants — says Gaussian elimination appeared in the Nine Chapters and was used by Gauss for the orbit of Pallas; “The first to use the term ‘matrix’ was Sylvester in 1850..”

• MacTutor, Matrices and determinants — “The first to use the term ‘matrix’ was Sylvester in 1850”; “by 1853 Cayley had published a note giving, for the first time, the inverse of a matrix”; and the 1858 memoir “is remarkable for containing the first abstract definition of a matrix,” giving addition, multiplication, scalar multiplication and inverses.
• MacTutor, James Joseph Sylvester — links Sylvester’s first use of the word matrix to his 1850 paper.

• Wikipedia, Determinant — describes the determinant as the signed scale factor for volume and states that a matrix is invertible exactly when its determinant is nonzero.
• Own computation — For A = [[1, 2], [2, 4]], det A = 1·4 − 2·2 = 0; the second row is twice the first, so the image is the line spanned by (1, 2), and no inverse can exist.

• Own computation — Direct calculation: det A = 0; tr A = 5; AᵀA has eigenvalues 25 and 0, so the Frobenius norm is 5; the characteristic polynomial is λ²−5λ, giving eigenvalues 5 and 0; im A = span(1, 2) has dimension 1 and ker A = span(2, −1) has dimension 1; A² = [[5, 10], [10, 20]] = 5A.

• Wikipedia, Kronecker delta — defines δᵢⱼ as 1 for i=j and 0 otherwise and gives the identity matrix as the matrix of Kronecker deltas, and carries no history section, so the date below rests on the paper itself.
• L. Kronecker, Ueber bilineare Formen, Journal für die reine und angewandte Mathematik 68 (1868), 273–285 — the European Digital Mathematics Library record for the paper the delta notation is standardly traced to, which dates it 1868 (volume 68, pages 273–285).
• Kronecker symbol, Encyclopedia of Mathematics — “The Kronecker symbol was first used by L. Kronecker (1866).”

• Wikipedia, Trace (linear algebra) — defines trace as the sum of diagonal entries and states that it equals the sum of eigenvalues with algebraic multiplicity.
• Earliest Known Uses of Some of the Words of Mathematics, T — gives trace as a translation of the German die Spur, used in H. L. Brose’s 1922 translation of Weyl’s Raum, Zeit, Materie, and records Aitken’s “the spur or trace of A. We shall denote it by sp A.”

• Wikipedia, Matrix norm — gives the Frobenius norm as the square root of the sum of squared entries and as the Euclidean norm of the singular-value vector.

• Wikipedia, Eigenvalues and eigenvectors — defines eigenvalues through Av=λv and the characteristic polynomial det(A−λI). Re-read raw on 17 September 2026 for the history: “In the 18th century, Leonhard Euler studied the rotational motion of a rigid body, and discovered the importance of the principal axes. Joseph-Louis Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.” It does not state the symmetric-matrix orthogonality property — it says “The eigenvectors need not be orthogonal to one another” — so that half is carried by the entry below; an eigenvalue is “the multiplying factor λ (possibly a negative or complex number),” and there is “a direct correspondence between linear transformations from the vector space into itself and n × n square matrices.”
• Symmetric matrix, Wikipedia — (§ Real symmetric matrices): “For every real symmetric matrix A, there exists a real orthogonal matrix Q such that D = QᵀAQ is a diagonal matrix,” and derives that eigenvectors for distinct eigenvalues satisfy ⟨x,y⟩ = 0.
• Own computation — A(1, 2)ᵀ = 5(1, 2)ᵀ and A(2, −1)ᵀ = 0; their dot product is 2−2=0.
• Earliest Known Uses of Some of the Words of Mathematics, E — David Hilbert introduced Eigenfunktion and Eigenwert in 1904, in “Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen”; Eigenvektor follows in Courant and Hilbert, 1924.

• Wikipedia, Nilpotent matrix — defines nilpotence by Aᵏ=0 for some positive k and gives the 2×2 Jordan nilpotent block as the standard example.
• MacTutor, Matrices and determinants — dates Jordan’s canonical form to 1870: “In 1870 the Jordan canonical form appeared in Treatise on substitutions and algebraic equations by Jordan.” The nilpotent part is on the Jordan normal form line.
• Jordan normal form, Wikipedia — “The Jordan normal form is named after Camille Jordan, who first stated the Jordan decomposition theorem in 1870.”; “Any M can be written as a sum D + N where D is semisimple, N is nilpotent, and DN = ND.”

• MathWorld, Rank-Nullity Theorem — states dim(V)=dim(Ker(T))+dim(Im(T)) and identifies these dimensions as nullity and rank.
• MacTutor, Matrices and determinants — says Sylvester defined nullity in 1884 and Frobenius defined rank in his 1878 matrix work.
• Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics (R), MacTutor — “RANK (of a determinant or matrix) was coined by F. G. Frobenius, who used the German word Rang in his paper ‘Uber homogene totale Differentialgleichungen,’ J. reine angew. Math. Vol. 86 (1879) p.1.”

• MathWorld, Cayley-Hamilton Theorem — states A is annihilated by det(xI−A) and works out the 2×2 identity.
• MacTutor, Matrices and determinants — dates Cayley’s 2×2/3×3 result and Frobenius’s general proof to 1878, and notes Hamilton’s special quaternion case.
• Cayley–Hamilton theorem, Wikipedia — “A special case of the theorem was first proved by Hamilton in 1853 in terms of inverses of linear functions of quaternions.”; “Cayley in 1858 stated the result for 3 × 3 and smaller matrices, but only published a proof for the 2 × 2 case.”; “The general case was first proved by Ferdinand Frobenius in 1878.”

• Wikipedia, Invertible matrix — states a square matrix is invertible exactly when its determinant is nonzero and defines the inverse equation AA⁻¹=I.
• Own computation — I·A=A, Aᵀ=A because A is symmetric, and A has det A=0, so it has no inverse.
• Identity matrix, Wikipedia — “the identity matrix of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere”; “When A is an m × n matrix, it is a property of matrix multiplication that I_m A = A I_n = A.”; “The identity matrix can also be written using the Kronecker delta notation: (I_n)_ij = δ_ij.”
• Transpose, Wikipedia — “transposition is an operation that flips a matrix over its diagonal; that is, transposition switches the row and column indices of the matrix A to produce another matrix, called the transpose of A.”
• Symmetric matrix, Wikipedia — “To see orthogonality, suppose x and y are eigenvectors corresponding to distinct eigenvalues λ₁, λ₂.” … “Since λ₁ and λ₂ are distinct, we have ⟨x, y⟩ = 0.”

• Wikipedia, Singular value decomposition — defines singular values through AᵀA and gives A=UΣVᵀ, with applications including least squares, principal component analysis and image compression.
• Own computation — AᵀA = [[5, 10], [10, 20]] has eigenvalues 25 and 0, so the singular values are 5 and 0; the Frobenius norm is therefore 5.
• Nicholas J. Higham, What Is the Singular Value Decomposition? — “The SVD was introduced independently by Beltrami in 1873 and Jordan in 1874,” and credits Golub with the first reliable algorithms for computing it.

Several names are in circulation for one idea. G. W. Stewart’s history of the decomposition (SIAM Review 35, 1993) treats Beltrami in 1873 and Jordan in 1874 as the independent first derivations, with Sylvester, Schmidt and Weyl arriving at it again before the numerical work of the 1960s. The card names the first two and the 1965 algorithm, and leaves the rediscoveries out.

• MacTutor, Gabriel Cramer — discusses Cramer’s 1750 rule and Maclaurin’s equivalent idea in the 1748 Treatise of Algebra.
• Wikipedia, Cramer’s rule — gives xᵢ=det(Aᵢ)/det(A) and states the rule applies when the determinant is nonzero.

• Wikipedia, Vector space — gives the vector-space axioms, basis definition and dimension, including the standard basis of R².
• Own computation — The monomials 1,x,x²,… form a countable basis of the polynomial vector space over a field, while (1, 0),(0, 1) is a basis of R².
• MacTutor, Abstract linear spaces — “The first to give an axiomatic definition of a real linear space was Peano in a book published in Torino in 1888,” the Calcolo geometrico, which Peano wrote after Grassmann’s 1844 Ausdehnungslehre.
• Basis (linear algebra), Wikipedia — “A generating set S is a basis of V if and only if it is minimal, that is, no proper subset of S is also a generating set of V.”

• Wikipedia, Outer product — defines the outer product as the matrix with entries uᵢvⱼ and notes that rank-one matrices arise this way.
• Ricci calculus, Wikipedia — read as raw HTML: the absolute differential calculus was “developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900,” the paper being Ricci and Levi-Civita, Méthodes de calcul différentiel absolu et leurs applications, Mathematische Annalen 54 (March 1900), 125–201, doi 10.1007/BF01454201.
• Wikipedia, Cross product — defines the cross product in three dimensions, gives its magnitude as parallelogram area, states it is zero for parallel vectors, and dates the notation a · b and a × b to Gibbs in 1881, with Heaviside independently.
• Own computation — (1, 2)⊗(1, 2)=[[1, 2], [2, 4]], and (1, 2, 0)×(2, 4, 0)=(0, 0, 0).
• Josiah Willard Gibbs, Wikipedia — “Gibbs’s lecture notes on vector calculus were privately printed in 1881 and 1884 for the use of his students, and were later adapted by Edwin Bidwell Wilson into a textbook, Vector Analysis, published in 1901.”
• Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics (D), MacTutor — DOT PRODUCT, found in 1901 in Gibbs and Wilson’s Vector Analysis: “A·B This is read A dot B and therefore may often be called the dot product instead of the direct product.”
• Dot product, Wikipedia — “Two non-zero vectors a and b are orthogonal if and only if a ⋅ b = 0”
• Rank (linear algebra), Wikipedia — “a matrix is defined to have rank 1 if and only if it can be written as a nonzero product c ⋅ r of a column vector c and a row vector r”

• Wikipedia, Characteristic polynomial — defines det(λI−A), identifies its roots as eigenvalues, and gives the 2×2 coefficient relation λ²−tr(A)λ+det(A).
• MacTutor, Matrices and determinants — records the historical development of determinant and matrix theory, including Cauchy’s work on determinants.
• Earliest Known Uses of Some of the Words of Mathematics, E — Cauchy’s 1829 paper on symmetric determinants is “Sur l’équation à l’aide de laquelle on détermine les inégalités séculaires des mouvements des planètes,” and he introduced the term l’équation caractéristique in the Exercices d’analyse et de physique mathématique of 1840.

• Wikipedia, Linear map — defines linear maps between vector spaces and their composition.
• MacTutor, Matrices and determinants — “Cayley gave a matrix algebra defining addition, multiplication, scalar multiplication and inverses” (1858); earlier Gauss “describes matrix multiplication (which he thinks of as composition so he has not yet reached the concept of matrix algebra),” for the arrays of quadratic forms, not linear maps.
• Own computation — A²=[[5, 10], [10, 20]]=5A, so the composition A∘A is represented by 5A.

Cayley’s 1858 memoir is where matrix multiplication is defined so that it represents composition. The barred arrow ↦ is Bourbaki’s, but no reference work consulted dates it: the accounts that give a date put it in the Théorie des ensembles of the early 1960s rather than in Bourbaki’s first decade, so the card gives the arrow neither an author nor a year.

• MacTutor, Matrices and determinants — “In 1683 Seki wrote Method of solving the dissimulated problems which contains matrix methods written as tables”; “In 1693 Leibniz wrote to de l’Hôpital” giving the condition for a system to have a solution; and “It was Cauchy in 1812 who used ‘determinant’ in its modern sense,” Gauss having introduced the term in 1801 while discussing quadratic forms.

• Wiktionary, matrix — the Latin entry glosses matrix as “uterus, womb,” then “dam (non-human female animal kept for breeding)” and “source, origin,” formed from mater; the English noun comes “from Old French matrice (“pregnant animal”), or from Latin mātrīx (“dam, womb”),” and its English sense “The womb” is marked “now rare.”
• Earliest Known Uses of Some of the Words of Mathematics, M — “The term MATRIX was introduced into mathematics by James Joseph Sylvester (1814-1897) in 1850. Matrix was a long-established word with the meaning of ‘the place from which something else originates.’ For Sylvester the ‘something else’ was a determinant of some description,” quoting his 1850 Philosophical Magazine paper: an oblong arrangement of m lines and n columns “will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants.”

• MacTutor, Nine Chapters on the Mathematical Art — “Most historians think that the text originated around 200 BC after the burning of the books by Shih Huang Ti,” while “others give dates between 100 BC and 50 AD,” the page settling on “206 BC to 50 AD” for its different parts; chapter 8 gives “18 problems which reduce to solving systems of simultaneous linear equations,” solved with “the augmented matrix of coefficients,” which “is then reduced to triangular form, using elementary column operations as is done today in the method of Gaussian elimination.”
• Wikipedia, Gaussian elimination — names the source as “Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art,” where “its use is illustrated in eighteen problems, with two to five equations,” and states that “Carl Friedrich Gauss in 1810 devised a notation for symmetric elimination that was adopted in the 19th century by professional hand computers to solve the normal equations of least-squares problems.”
• MacTutor, Matrices and determinants — “Using observations of Pallas taken between 1803 and 1809, Gauss obtained a system of six linear equations in six unknowns. Gauss gave a systematic method for solving such equations which is precisely Gaussian elimination on the coefficient matrix”; the same page gives no year for that work, which is why the 1810 date is carried by the entry above.

• Bruce A. Hedman, An Earlier Date for “Cramer’s Rule,” Historia Mathematica 26 (1999) 365-368 — read in full here. Abstract: “Carl B. Boyer has shown that Colin Maclaurin published ‘Cramer’s Rule’ two years before Gabriel Cramer and conjectured that Maclaurin knew the technique as early as 1729. This note establishes Boyer’s conjecture through a heretofore unpublished manuscript in the Edinburgh University Library.” The manuscript is a handwritten copy of the Treatise of Algebra made by John Russell and dated 1729 (Edinburgh University Library, shelfmark Dc. 3.66); its twelfth chapter carries the same form of the rule as the 1748 edition, and Hedman concludes that it “provides conclusive evidence that Maclaurin was teaching his students ‘Cramer’s Rule’ over 20 years before Cramer published it.” Hedman also records that Patrick Murdoch prepared the Treatise for posthumous publication in 1748.
• MacTutor, Colin Maclaurin — gives Maclaurin’s dates as February 1698 to 14 June 1746 and states that “Maclaurin’s Treatise on algebra was published in 1748, two years after his death.”

Added 11 September 2026. The page previously said “Cramer published it in 1750 and Maclaurin had an equivalent rule in 1748,” which dates Maclaurin two years after his own death and names the later man first. Under D1 the prose now takes the earliest defensible year and the first name: Maclaurin, ~1729.

• Projection (linear algebra), Wikipedia — read as raw HTML: defines a projection as a linear map with P² = P, and an orthogonal projection as one whose range and kernel are orthogonal complements, equivalently a projection that is self-adjoint; gives the nearest-point and least-squares readings.
• Own computation — For A = [[1, 2], [2, 4]] and u = (1, 2): uuᵀ = A and |u|² = 5, so A/5 = uuᵀ/|u|² is symmetric and (A/5)² = A²/25 = 5A/25 = A/5; its range is the line y = 2x, spanned by (1, 2), and its kernel, spanned by (2, −1), is perpendicular to it. So A/5 is the orthogonal projection onto y = 2x, and A = 5·(A/5) has determinant 0, rank 1 and eigenvalues 5 and 0.

• Spectral theorem, Wikipedia — read as raw HTML: “Augustin-Louis Cauchy proved the spectral theorem for symmetric matrices, i.e., that every real, symmetric matrix is diagonalizable,” and “The spectral theorem as generalized by John von Neumann is today perhaps the most important result of operator theory.”
• Spectral Theorem, MathWorld — the statement for a normal bounded operator on a Hilbert space, via a unique resolution of the identity on the Borel subsets of its spectrum.
• Mathematical formulation of quantum mechanics, Wikipedia — “By spectral theory, we can associate a probability measure to the values of A in any state ψ.”

• The Collected Mathematical Papers of James Joseph Sylvester, vol. I (Cambridge, 1904), Internet Archive full text — read raw: “37. ON THE RELATION BETWEEN THE MINOR DETERMINANTS OF LINEARLY EQUIVALENT QUADRATIC FUNCTIONS. [Philosophical Magazine, I. (1851), pp. 295 — 305.].”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Gaussian elimination, Wikipedia — traces the method to the Chinese text Nine Chapters on the Mathematical Art and to Gauss’s 1810 notation for symmetric elimination; “The first reference to the book by this title is dated to 179 AD, but parts of it were written as early as approximately 150 BC.”
• The Nine Chapters on the Mathematical Art, Wikipedia — carries the dating: “composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 1st century CE.”
• Nine chapters on the Mathematical Art, MacTutor — carries the column operations: “The matrix is then reduced to triangular form, using elementary column operations as is done today in the method of Gaussian elimination.”
• The Quest for Linear Equation Solvers, John L. Gustafson — carries Gauss and Pallas: “To solve the problem of predicting the orbit of Pallas, he had to reduce astronomical data to a least-squares problem… involving six equations in six unknowns.”

• Least squares, Wikipedia — defines the least-squares solution, gives the normal equations, and credits Legendre’s 1805 publication and Gauss’s priority claim.
• Linear least squares, Wikipedia — carries the orthogonality: “the matrix X(XᵀX)⁻¹Xᵀ represents orthogonal projection onto the column space of X.”
• Legendre, Adrien-Marie, Encyclopedia of Mathematics — carries the priority verdict: “The claim for priority made by Gauss at the time would seem to have been substantiated in the opinion of Laplace, Plackett (1972), Stigler (1977, 1981, 1986), and other authorities, but Legendre retains the priority of publication”
• Numerical methods for linear least squares, Wikipedia — carries the stability comparison: “Orthogonal decomposition methods of solving the least squares problem are slower than the normal equations method but are more numerically stable because they avoid forming the product XᵀX.”

• Zero divisor, Wikipedia — defines zero divisors, with the example Z/4Z, where the class of 2 is a zero divisor since 2 × 2 = 4 = 0, and “A nonzero commutative ring whose only zero divisor is 0 is called an integral domain.” The mod 6 example is on the Domain line.
• Octonion, Wikipedia — carries the division algebras and the sedenions: “ℝ, ℂ, or ℍ, and 𝕆 are the only normed division algebras over the real numbers. … The higher-dimensional algebras defined by the Cayley–Dickson construction (starting with the sedenions) all fail to satisfy this property. They all have zero divisors.”
• Domain (ring theory), Wikipedia — carries the mod 6 example: “The ring Z/6Z is not a domain, because the images of 2 and 3 in this ring are nonzero elements with product 0.”
• Sedenion, Wikipedia — carries the persistence above the sedenions: “All hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.”

• Wolfram MathWorld, Fundamental Theorems of Calculus — states the endpoint formula ∫_a^b f(x)dx = F(b) − F(a) for a continuous integrand and antiderivative.

• MacTutor, Earliest Uses of Symbols of Calculus — records Lagrange’s f′ notation in 1797 and distinguishes earlier Newton/Leibniz notations.
• Fluxion, Wikipedia — “Fluxions were introduced by Isaac Newton to describe his form of a time derivative (a derivative with respect to time).”

• MacTutor, Joseph Fourier — Fourier completed his memoir On the Propagation of Heat in Solid Bodies by 1807 and read it to the Paris Institute on 21 December 1807; “He established the partial differential equation governing heat diffusion and solved it by using infinite series of trigonometric functions,” and Lagrange and Laplace’s first objection, in 1808, “was to Fourier’s expansions of functions as trigonometrical series.” (The review first cited here: T. N. Narasimhan, Fourier’s heat conduction equation: history, influence, and connections, Reviews of Geophysics 37 (1999) 151–172, doi:10.1029/1998RG900006, paywalled).
• Wikipedia, Heat equation — gives Fourier’s heat equation and his separation-of-variables solution, proposed in his Théorie analytique de la chaleur (1822).
• Own computation — Differentiate directly: ∂t u = −π²u and ∂xx u = −π²u, so their difference is 0; for sin(nπx), the same calculation gives decay rate n²π².

• Wolfram MathWorld, Pi — states that π is transcendental by Lindemann’s 1882 result.
• NIST Digital Library of Mathematical Functions §25.6 — equation 25.6.1 gives ζ(2)=π²/6, hence π²=6ζ(2).

• Own computation — For every real x, x²≥0 with equality exactly at x=0; adding 1 shifts every value by 1, so the unique minimizer remains x=0 and the minimum becomes 1.
• Mathematical optimization, Wikipedia — its notation example: “The minimum value in this case is 1, occurring at x = 0.”; and the vocabulary: “The function f is variously called an objective function,” “The domain A of f is called the search space or the choice set,” “When the objective function is a convex function, then any local minimum will also be a global minimum,” and “the theory of duality.”

• Wolfram MathWorld, Vector Derivative — lists the standard gradient, divergence, curl, and Laplacian operators and their Cartesian derivative identities.
• Wolfram MathWorld, Stokes’ Theorem — identifies the gradient, curl, and divergence theorems as differential-form instances of Stokes’ theorem.
• Wikipedia, Symmetry of second derivatives — records Clairaut’s proposed proof of 1740 and Schwarz’s first rigorous proof of 1873, and states the hypothesis as continuous second partial derivatives on a neighborhood of the point.
• Vector calculus identities, Wikipedia — curl of a gradient: “It can be easily proved by expressing ∇ × (∇ φ) in a Cartesian coordinate system with Schwarz’s theorem (also called Clairaut’s theorem on equality of mixed partials).”; divergence of a curl: “The divergence of the curl of any continuously twice-differentiable vector field A is always zero,” and “This is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex.”; gradient: “the gradient is proportional to, and points in the direction of, the function’s most rapid (positive) change”; Laplacian: “When the Laplacian is equal to 0, the function is called a harmonic function.”
• Conservative vector field, Wikipedia — “a conservative vector field is a vector field that is the gradient of some function”; irrotational: “if its curl is 0 everywhere in U”; and “the work W done in going around a simple closed loop C is 0.”

• Wolfram MathWorld, Laplacian — derives that the Laplacian of 1/r vanishes away from the source and gives the −4π delta-function identity.

• Wolfram MathWorld, Vector Derivative — equation 22 gives ∇·r=3.
• Own computation — ∇²(x²+y²+z²)=2+2+2=6; in n dimensions the corresponding value is 2n. Likewise ∇·r=1+1+…+1=n in n dimensions, and by the divergence theorem ∮ r·dA = ∫ ∇·r dV = 3V in three dimensions.
• Laplace operator, Wikipedia — “∇²f(x) = lim_{R→0} (2n/R²)(f_shell_R − f(x)),” where n is the dimension of the space and f_shell_R is the average of f over the surface of an n-sphere of radius R.

• Wolfram MathWorld, Stokes’ Theorem — states the surface-flux/line-circulation relation underlying these integrals.
• Own computation — On S², r̂/r²·dA=sinθ dθ dφ, whose integral is 4π. Parametrizing the unit circle by x=cos t,y=sin t gives 1/2∫(x y’−y x’)dt=1/2∫_0^{2π}dt=π.
• Solid angle, Wikipedia — “The solid angle of a sphere measured from any point in its interior is 4π sr.”
• Gauss’s law for gravity, Wikipedia — “The gravitational flux through any closed surface is proportional to the enclosed mass”: ∯∂V g · dA = −4πGM.

• Wikipedia, Divergence theorem — states the flux/divergence relation and documents Lagrange’s 1762 special case, Gauss’s 1813 cases, and Ostrogradsky’s 1826/1831 work.

• Cambridge Core, Green’s and allied theorems: a historical sketch — states that Green’s essay was published at Nottingham in 1828 and discusses the transformation on which the theorem depends.
• Green’s theorem, Wikipedia — “It is the two-dimensional special case of Stokes’ theorem (surface in R³).”; area: “A = ∮_C x dy = −∮_C y dx = ½∮_C(−y dx + x dy).”; history: Green “stated a similar result in an 1828 paper”; “In 1846, Augustin-Louis Cauchy published a paper stating Green’s theorem as the penultimate sentence. A proof of the theorem was finally provided in 1851 by Bernhard Riemann in his doctoral dissertation.”
• Pi, Wolfram MathWorld — “π is also transcendental (Lindemann 1882).”
• J. Cufí and A. Reventós, A historical review of the Cauchy-Riemann equations and the Cauchy Theorem, arXiv:2311.16649 — “Green’s formula (16) does not appear in Green’s paper”; “The first published proof of Green’s formula is due to Riemann in his 1851 dissertation.”
• n-sphere, Wikipedia — “A unit 1-sphere is the unit circle in the Euclidean plane.”

• Wikipedia, Generalized Stokes theorem — gives Kelvin’s 1850 letter, the 1854 exam question, and Cartan’s modern formulation following earlier generalizations; its note on the exam: “Maxwell took this examination and tied for first place with Edward John Routh.”; and “Stokes set the theorem as a question on the 1854 Smith’s Prize exam, which led to the result bearing his name.”; and “Stokes’ theorem was formulated in its modern form by Élie Cartan in 1945, following earlier work on the generalization of the theorems of vector calculus by Vito Volterra, Édouard Goursat, and Henri Poincaré.”

• Wikipedia, History of Maxwell’s equations — documents Maxwell’s 1865 formulation, the twenty-equation presentation, and the 1884 vector condensation associated with Heaviside and Gibbs; it also records the electromagnetic-wave result.
• Nabla symbol, Wikipedia — “They were all originally studied in the context of the classical theory of electromagnetism, and contemporary university physics curricula typically treat the material using approximately the concepts and notation found in Gibbs and Wilson’s Vector Analysis.”

• Wikipedia, Vector potential — gives the existence theorem for a solenoidal field on ℝ³ that is twice continuously differentiable and decreases at least as fast as 1/‖x‖ at infinity.
• MIT 18.022, When is a Vector Field the Curl of Another? — states the result for a region R that is simply connected. See 11.18: simple connectivity is not by itself enough.
• Own computation — For r̂/r² on ℝ³∖{0}, the divergence is 0 everywhere the field is defined, but the flux through every enclosing sphere is 4π; the flux of a curl through a closed surface is 0, so the field is not a curl there.
• Aharonov–Bohm effect, Wikipedia — “an electrically charged particle is affected by an electromagnetic potential (φ, A), despite being confined to a region in which both the magnetic field B and electric field E are zero.”

• Wolfram MathWorld, Stokes’ Theorem — identifies the differential-form form of Stokes’ theorem and its gradient/curl/divergence instances.
• Wikipedia, Exterior derivative — “The exterior derivative was first described in its current form by Élie Cartan in 1899”, and gives d(dα)=0 for a k-form α.
• Closed and exact differential forms, Wikipedia — “Because d² = 0, every exact form is necessarily closed.”; “The implication from ‘exact’ to ‘closed’ is then a consequence of the symmetry of second derivatives, with respect to x and y.”
• Homology (mathematics), Wikipedia — “It is also required that the composition of any two consecutive boundary operators be trivial. That is, for all n, ∂ₙ ∘ ∂ₙ₊₁ = 0.”

• Wikipedia, Fermat’s method of adequality — dates the treatise Methodus ad disquirendam maximam et minimam to about 1636 and describes Fermat’s method for maxima and minima before modern differential calculus.
• Mathematical optimization, Wikipedia — “One of Fermat’s theorems states that optima of unconstrained problems are found at stationary points, where the first derivative or the gradient of the objective function is zero”; “Fermat and Lagrange found calculus-based formulae for identifying optima, while Newton and Gauss proposed iterative methods for moving towards an optimum.”

• Wikipedia, Poincaré lemma — every closed p-form on a star-shaped open subset of ℝⁿ is exact, so the de Rham cohomology of such a region vanishes in every positive degree.
• Wikipedia, Vector potential — gives the existence theorem on ℝ³ and the restriction of the integral domain to a region.
• Own computation — Every loop in ℝ³∖{0} contracts, so the region is simply connected. The flux of r̂/r² through a sphere about the origin is 4π, while the flux of any curl through a closed surface is 0 by the divergence theorem and div curl = 0. So r̂/r² is not a curl there, and simple connectivity cannot be the sufficient condition.

• Wikiquote, Richard Feynman — gives the passage — “the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics” — located in The Feynman Lectures on Physics, vol. II, lecture 1, section 1-6, p. 1-11.
• John D. Cook, What was the most important event of the 19th century? — quotes the sentence in full, with the dashes, and attributes it to The Feynman Lectures on Physics, Volume 2.

• MacTutor, George Green — the essay was advertised in the Nottingham Review on 14 December 1827 and published in March 1828 by subscription at seven shillings and sixpence, with 51 subscribers, most of them members of the Nottingham Subscription Library; after Thomson took his degree in January 1845 his coach William Hopkins gave him three copies, and Thomson showed the work to Liouville and Sturm in Paris in the summer of 1845 and republished it in 1850–54.
• Wikipedia, George Green (mathematician) — the essay was sold on a subscription basis to 51 people; Thomson had seen a reference to Green’s work in an 1833 paper by Robert Murphy but could find no bookseller who remembered it; the essay was not reprinted in Britain until 1871; “As a mathematician, he was almost entirely self-taught, receiving only about one year of formal schooling as a child, between the ages of eight and nine. Green then worked his father’s bakery from a young age, and later at the mill his father constructed in 1807, now known as Green’s Windmill. Whilst working as a miller, he became a member of Nottingham Subscription Library.”
• Cambridge Core, Green’s and allied theorems: a historical sketch — states that Green’s essay was published at Nottingham in 1828 and was unknown to continental mathematicians until the reprints of 1850, 1852 and 1854.

Two claims have to be separated here. That the essay went entirely unread is too strong: it had fifty-one subscribers, and Robert Murphy cited it in 1833. That Thomson was handed three copies in 1845 and carried the essay to Paris is documented by both biographies, so the card states it plainly rather than gesturing at reprints.

• Wikipedia, Gauss’s law — “The law was first formulated by Joseph-Louis Lagrange in 1773, followed by Carl Friedrich Gauss in 1835, both in the context of the attraction of ellipsoids”, citing Gauss’s Werke, Band 5. The book no longer follows this page’s 1773 and 1835 (the Werke text is a reprint of the 1813 paper; see 11.25), but keeps the link as the record of the older account. The same page says Gauss’s law “is one of Maxwell’s equations” and that “Gauss’s law itself is essentially equivalent to Coulomb’s law, and Gauss’s law for gravity is essentially equivalent to Newton’s law of gravity, both of which are inverse-square laws.”
• Wikipedia, Divergence theorem — read as raw HTML: its history footnote gives the printing exactly — “C. F. Gauss (1813) ‘Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata,’ Commentationes societatis regiae scientiarium Gottingensis recentiores, 2: 355–378; Gauss considered a special case of the theorem” — and its running text says Gauss was “working on the gravitational attraction of an elliptical spheroid in 1813, when he proved special cases of the divergence theorem”, that Lagrange “discovered the divergence theorem in 1762”, and that the first proof of the general theorem is Ostrogradsky’s, of 1826.
• Carl Friedrich Gauss, Wikipedia — read as raw HTML: its list of works gives “1840: Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnis des Quadrats der Entfernung wirkenden Anziehungs- und Abstoßungskräfte … Leipzig: Weidmannsche Buchhandlung, 1840,” translated there as General Theorems concerning the attractive and repulsive Forces acting in reciprocal Proportions of quadratic Distances.

The fifth volume of Gauss’s Werke was first published in 1867 and reissued in 1877, and sources cite both years, so the card names the volume and not the year. Extended 17 September 2026. The entry previously asserted the 1813 printing while its only source said “followed by Carl Friedrich Gauss in 1835”; the Divergence theorem article gives the 1813 volume and pages outright, so the printing is settled and the entry no longer disputes its own evidence. Corrected later the same day: the Lagrange question is settled, and the evidence was in the book. r-11.25’s note records a deep-research pass of 11 September 2026 in which Katz 1979 was read in full and Lagrange’s memoirs were read in the original, and which concluded that “Lagrange 1773 is not a statement of Gauss’s law: the memoir was read end to end and contains no closed surface, no flux and no 4π; what it supplies is the change of variables Gauss’s proof runs on, which is a real priority for a method and not for the law.” The #flux card’s clause was briefly cut on 17 September 2026 for want of a source and has been restored, now citing 11.25, which the card had never cited. So the general-reference sentence that Lagrange “first formulated” the law is not followed here, and the reason is a reading of the primary document, not a preference between encyclopedias.

• Wikipedia, Leibniz-Newton calculus controversy — opens ‘the calculus controversy was a priority dispute between mathematicians Isaac Newton and Gottfried Wilhelm Leibniz over who had first invented calculus’, and states ‘Today, the consensus is that Leibniz and Newton independently invented and described calculus in Europe in the 17th century’.

Added in the V14 prose pass to carry the chapter’s clause ‘the argument about whose calculus it was outlived them both’, which previously stood on 11.2. The source carries the dispute and the independence, but its narrative ends around 1716 and it does not state that the argument continued past the deaths of the two men. That half of the clause is still uncarried.

• Lindemann–Weierstrass theorem, Wikipedia — read raw: “Lindemann proved in 1882 that e^α is transcendental for every non-zero algebraic number α, thereby establishing that π is transcendental (see below)..”
• Euclidean vector, Wikipedia — read raw: “The first half of Gibbs’s Elements of Vector Analysis, published in 1881, presents what is essentially the modern system of vector analysis..”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Maxwell’s equations, Wikipedia — read via the REST endpoint (revision 1375267474), in its key to the integral signs: the closed surface integral over ∂Ω “is a surface integral over the boundary surface ∂Ω, with the loop indicating the surface is closed,” and ∮ over ∂Σ “is a line integral around the boundary curve ∂Σ, with the loop indicating the curve is closed..”
• MacTutor, Earliest Uses of Symbols of Calculus — under “Integration around a closed path”: “a use of the integral symbol with a circle in the middle by Arnold Sommerfeld (1868 - 1951) in 1917,” and E. B. Wilson “used a small circle below the standard integral symbol to denote integration around a closed curve.”

• Wolfram MathWorld, Euler Characteristic — gives χ = V − E + F, χ = 2 − 2g, and the table values for sphere, torus, Klein bottle, projective plane and double torus; its table of Euler characteristics gives the Klein bottle and the Möbius strip both 0.
• MacTutor, Topology history — records that in 1750 Euler wrote to Christian Goldbach giving the polyhedron formula v − e + f = 2, and that he published the details in 1752 in two papers.
• Euler Archive, E230 Elementa doctrinae solidorum — records E230 in Novi Commentarii Academiae Scientarum Petropolitanae vol. 4, pp. 109–140, dated 1758, with its companion E231 Demonstratio nonnullarum insignium proprietatum at pp. 140–160 of the same volume (the Euler Archive’s E231 record; pp. 94–108 is the Opera Omnia reprint); this page gives E230 as written in 1750 and published in 1758; the 1752–53 volume year and E231’s 1751 come from the E231 record and other sources, not this page.
• Real projective plane, Wikipedia — constructed “by taking the (single) edge of a Möbius strip and gluing it to itself in the correct direction, or by gluing the edge to a disk”; “It cannot be embedded in three-dimensional Euclidean space without intersecting itself.”
• Genus, Wolfram MathWorld — “For a nonorientable surface such as a real projective plane (genus 1) or Klein bottle (genus 2), the relationship is χ=2-g”
• Euler characteristic, Wikipedia — “The n-dimensional real projective space is the quotient of the n-sphere by the antipodal map. It follows that its Euler characteristic is exactly half that of the corresponding sphere”; “Any contractible space (that is, one homotopy equivalent to a point) has trivial homology, meaning that the 0th Betti number is 1 and the others 0. Therefore, its Euler characteristic is 1.”; gives Cauchy’s proof as 1811: “One was given by Cauchy in 1811.”
• Platonic solid, Wikipedia — its table gives the cube 8 vertices, 12 edges, 6 faces and the regular icosahedron 12, 30, 20; the topological proof: “The key is Euler’s observation that V − E + F = 2,” whence “1/q + 1/p > 1/2,” and “there are only five possibilities for {p, q}.”
• Uniformization theorem, Wikipedia — compact Riemann surfaces “with universal cover the unit disk are precisely the hyperbolic surfaces of genus greater than 1”
• E231, Demonstratio nonnullarum insignium proprietatum…, Euler Archive — “Written Date 1751,” “Published Date 1758,” “Novi Commentarii academiae scientiarum Petropolitanae, Volume 4, pp. 140-160.”

• MacTutor, A history of Topology — says Poincaré put connectivity on a rigorous basis in Analysis situs (1895), introduced homology and gave a more precise definition of Betti numbers; and, earlier in the same article, that Johann Benedict Listing (1802–1882) was the first to use the word topology and that Riemann studied the concept in 1851 and again in 1857.
• MacTutor, Henri Poincaré biography — identifies Analysis situs, published in 1895, as an early systematic treatment of topology and records the fundamental group contribution, which it dates 1894: “Poincaré introduced the fundamental group (or first homotopy group) in his paper of 1894 to distinguish different categories of 2-dimensional surfaces.”
• MacTutor, Enrico Betti biography — says Betti published an important paper on topology in 1871, Sopra gli spazi di un numero qualunque di dimensioni, which contained what we now call the Betti numbers, and that the Betti numbers were so named by Henri Poincaré.
• nLab, Emmy Noether — credits Noether with the categorification of the Betti numbers and torsion coefficients of a space to homology groups, and prints the abstract of her 1925 DMV talk where she pointed it out.
• Topology, Wikipedia — “A famous example, known as the ‘Topologist’s Breakfast’, is that a topologist cannot distinguish a coffee mug from a doughnut.”
• Euler characteristic, Wikipedia — “Homology is a topological invariant, and moreover a homotopy invariant: Two topological spaces that are homotopy equivalent have isomorphic homology groups. It follows that the Euler characteristic is also a homotopy invariant.”
• Fundamental group, Wikipedia — gives 1895: “Henri Poincaré defined the fundamental group in 1895 in his paper ‘Analysis situs’”; “the 2-sphere S² …, and also all the higher-dimensional spheres, are simply-connected”; “the fundamental group of the circle is isomorphic to (ℤ,+), the additive group of integers”; “The abelianization of the fundamental group can be identified with the first homology group of the space.”
• Poincaré conjecture, Wikipedia — “Every three-dimensional topological manifold which is closed, connected, and has trivial fundamental group is homeomorphic to the three-dimensional sphere.”

• Wolfram MathWorld, Betti Number — defines the nth Betti number as the rank of the nth homology group and lists sphere b₁ = 0 and torus b₁ = 2.
• Wolfram MathWorld, Euler Characteristic — gives, beside what it calls the Poincaré formula χ = V − E + F, the Betti-number form χ = b₀ − b₁ + b₂; applying it to a closed orientable genus-g surface gives b₀ = b₂ = 1 and b₁ = 2g.
• Fundamental group, Wikipedia — gives 1895: “Henri Poincaré defined the fundamental group in 1895 in his paper ‘Analysis situs’”; “the 2-sphere S² …, and also all the higher-dimensional spheres, are simply-connected”; “the fundamental group of the circle is isomorphic to (ℤ,+), the additive group of integers”; “The abelianization of the fundamental group can be identified with the first homology group of the space.”
• Torus, Wikipedia — “The kth homology group of an n-torus is a free abelian group of rank n choose k”; for the torus H₁ ≅ ℤ ⊕ ℤ.

• Encyclopedia of Mathematics, Homology theory — describes chain complexes with successive boundary maps whose composition is zero and homology as cycles modulo boundaries.
• Generalized Stokes theorem, Wikipedia — “Stokes’ theorem says that this is a chain map from de Rham cohomology to singular cohomology with real coefficients; the exterior derivative, d, behaves like the dual of ∂ on forms.”

• Wolfram MathWorld, Genus — identifies the torus as genus 1 and states χ = 2 − 2g for orientable surfaces.
• Jean Gallier and Dianna Xu, A Guide to the Classification Theorem for Compact Surfaces, University of Pennsylvania, 19 September 2012 — read as extracted PDF text. Its preface gives the history this chapter’s cards compress: “it took about sixty years until a rigorous proof was finally given by Brahana in 1921. Early versions of the classification theorem were stated by Möbius in 1861 and by Jordan in 1866. … More definite versions and ‘proofs’ were given later by von Dyck in 1888 and by Dehn and Heegaard in 1907.” Note the Möbius year: this source says 1861, where the cards and much of the general literature say 1863, which is the year of the printed Theorie der elementaren Verwandtschaft. Both are in circulation and the cards now name both.
• Toshikazu Sunada, From Euclid to Riemann and Beyond, arXiv:1904.01845 — gives 1863 (read from the PDF): “The case of two-sided surfaces was independently treated by Möbius (1863) and M. E. C. Jordan (1866). One-sided closed surfaces were classified by von Dyck (1888).”
• Surface (topology), Wikipedia — “The classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families”: the sphere, the connected sum of g tori, and the connected sum of k real projective planes.

• Wolfram MathWorld, Trefoil Knot — identifies the trefoil as a (3, 2)-torus knot and documents its standard knot invariants.
• Wikipedia, Crossing number (knot theory) — defines crossing number, gives the unknot value 0, and states that the smallest nontrivial knot has crossing number 3.
• Tricolorability, Wikipedia — “since the unknot is not tricolorable, any tricolorable knot is necessarily nontrivial”; “The trefoil knot and trivial 2-link are tricolorable, but the unknot, Whitehead link, and figure-eight knot are not.”
• Unknotting problem, Wikipedia — “Haken originally used this algorithm to show that unknotting is decidable”; “A major unresolved challenge is to determine if the problem admits a polynomial time algorithm”; “In 2021, Lackenby announced an unknot recognition algorithm which he claimed ran in quasi-polynomial time. As of October 2025, the result has not been published in the peer-reviewed literature.”
• Vortex theory of the atom, Wikipedia — Kelvin proposed that “each type of knot might represent an atom of a different chemical element”; the theory “could not explain inertia or gravitation”; “Tait’s work especially founded the branch of topology called knot theory.”

• Wolfram MathWorld, Jones Polynomial — states V_unknot(t) = 1 and explicitly says it is not known if a nontrivial knot with Jones polynomial 1 exists.

• MacTutor, Vaughan Jones biography — says that in 1984 Jones discovered a relationship between von Neumann algebras and geometric topology and found a new polynomial invariant for knots and links in 3-space, and that the 1990 Fields Medal recognized that discovery.

• Wolfram MathWorld, Cylinder — computes K = 0 for a cylinder and gives the principal curvatures 1/a and 0.
• Wolfram MathWorld, Gaussian Curvature — defines Gaussian curvature intrinsically from the first fundamental form and as the determinant of the shape operator.
• Wikipedia, Theorema Egregium — says Gauss proved the theorem in 1827, that the Gaussian curvature of a surface does not change if the surface is bent without stretching and is therefore an intrinsic invariant, and that in consequence no flat map of the Earth can avoid distorting distances.
• Differential geometry of surfaces, Wikipedia — “The defining contribution to the theory of surfaces was made by Gauss in two remarkable papers written in 1825 and 1827.”; “the Gaussian curvature, first studied in depth by Carl Friedrich Gauss”; its bibliography gives the 1827 paper as “Disquisitiones generales circa superficies curvas,” “Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores Vol. VI (1827), pp. 99–146.”

• Wolfram MathWorld, Gauss-Bonnet Formula — states the closed-surface formula ∫∫K dA = 2πχ and explains the topology/geometry connection.
• Wolfram MathWorld, Euler Characteristic — gives the curvature integral relation and χ(S²) = 2; the unit-sphere computation is then 2π·2 = 4π.

• Wolfram MathWorld, Pythagorean Theorem — states a² + b² = c² and documents Babylonian calculations using the relation more than a millennium before Pythagoras, including a 3:4:5 ratio in the Berlin Papyrus.

• MacTutor, Euclid biography and Elements history — titles the biography “Euclid (325 BC - 265 BC)” and gives “Born about 325 BC (probably)” and “Died about 265 BC,” both at Alexandria; re-read raw on 17 September 2026. It also says the Elements ends with book thirteen, which discusses the properties of the five regular polyhedra and gives a proof that there are precisely five, and that this book appears to be based largely on an earlier treatise by Theaetetus.
• D. E. Joyce, Euclid’s Elements, Book I, Proposition 47 — in right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.
• D. E. Joyce, Euclid’s Elements, Book I, Proposition 32 — in any triangle the sum of the three interior angles equals two right angles; the proof runs through I.31 and I.29 and so rests on the parallel postulate.
• Wolfram MathWorld, Platonic Solid — states that exactly five convex Platonic solids exist and attributes the classification to the last proposition of Euclid’s Elements.
• Euclid’s Elements, Book I, Proposition 29, D. E. Joyce, Clark University — “This is the first proposition which depends on the parallel postulate.”

Attached to #pythag345’s Euclid sentence on 17 September 2026. The counts ledger had recorded that attachment as done since the V40 pass; it had not happened, and the card stood on 12.11, which is entirely about the Babylonian sources. That gap is why ledgercheck.py exists.

• Wolfram MathWorld, Spherical Triangle — gives area = R²(A+B+C−π), so on the unit sphere the excess equals area and the sum exceeds π.
• Wolfram MathWorld, Hyperbolic Geometry — states that the angle sum of a hyperbolic triangle is less than 180 degrees.
• Heinz Klaus Strick, Albert Girard — a biography poster hosted by MacTutor — supports ~1626 — it dates the spherical-triangle area formula to Girard’s 1626 corrected edition of Stevin’s trigonometry, and says only that “in his main work published in 1629 … GIRARD generalised the above-mentioned area formula for any spherical polygons.” It is a biography poster, not a MacTutor historical note.
• Gauss–Bonnet theorem, Wikipedia — “The sum of interior angles of a geodesic triangle is equal to π plus the total curvature enclosed by the triangle”; “A number of earlier results in spherical geometry and hyperbolic geometry, discovered over the preceding centuries, were subsumed as special cases of Gauss–Bonnet.”

• Own computation from the chapter’s displayed formula — Recomputed term by term: 0 + 0 + 0 + 0 = 0; 2 + 1 + |−2| + 1 = 6; (1 + 1) + 2 + 4 = 8; g(T²) = 1; the knot numerator 0 over 3 + 1 = 4; the curvature numerator 0 over 4π; the Euclidean numerator 0 over π + 5. Each of the four concepts is a fraction with a zero numerator over a positive denominator.

• Coming home, the first of the house rules in this book’s disclaimer — Declares that every factor is computed inside its own framework and hands back an ordinary finite real, which is 0, before it joins the product.
• Own type audit of the chapter formula — Every other term of the factor is an ordinary number: the Euler characteristics, the Betti numbers, the genus, the two crossing numbers, the Jones polynomial of the unknot, the Gaussian curvature of the cylinder and 3² + 4² − 5² are integers, and ∫K dA is the real number 4π. Only ∂(∂M) is a chain.

• Wikipedia, Gauss–Bonnet theorem — the theorem is named after Carl Friedrich Gauss, who developed a version but never published it, and Pierre Ossian Bonnet, who published a special case in 1848.
• EBSCO Research Starters, Surfaces (mathematics) — versions of the theorem were explored by Gauss in the 1820s and by Bonnet and Jacques Binet in the 1840s, and the form that is standard in undergraduate differential geometry, for an orientable compact surface, is attributed to Walther von Dyck in 1888.
• MacTutor, Walther von Dyck biography — records that von Dyck made significant contributions to the Gauss–Bonnet theorem.

Bonnet’s 1848 result is the formula for a region bounded by a curve; the statement for a closed surface, which has no bounding curve, is the later one. The MathWorld page cited at 12.10 states the closed-surface formula and gives no history at all.

• Wikipedia, Poincaré conjecture — quotes the 1900 claim that a polyhedron with all its Betti numbers equal to 1 and all its torsion tables orientable is homeomorphic to a hypersphere, records the 1904 counterexample with a fundamental group of 120 elements, and gives the fifth complement’s reformulation in terms of the fundamental group.
• Wikipedia, Homology sphere — says that since the fundamental group of the 3-sphere is trivial, the Poincaré homology sphere shows there exist 3-manifolds with the same homology groups as the 3-sphere that are not homeomorphic to it.

• Euler characteristic, Wikipedia — (§ Polyhedra, § Topological definition, § Relations to other invariants): gives χ = V − E + F; defines the general Euler characteristic as the alternating sum χ = b₀ − b₁ + b₂ − b₃ + ⋯ of the ranks of the singular homology groups; states that it is a topological invariant because “homology is a topological invariant, and moreover a homotopy invariant”; and gives χ = 2 − 2g for a closed orientable surface of genus g.

• Henri Poincaré, Science et méthode (édition définitive, Flammarion, 1908), Livre I, chapitre II, L’avenir des mathématiques (Wikisource) — carries the sentence “Je ne sais si je n’ai déjà dit quelque part que la Mathématique est l’art de donner le même nom à des choses différentes,” in the passage arguing that a well-chosen name lets a proof made for one object apply unchanged to many others.
• MacTutor, Quotations by Henri Poincaré — gives the standard English rendering, “Mathematics is the art of giving the same name to different things,” without naming the work it comes from.

Added in the V14 prose pass. The chapter’s sentence about Poincaré’s remark carried no footnote and no card. The English wording the chapter uses is the usual rendering of the French sentence; the French original is the primary source and is cited first.

• Cohomology, Wikipedia — read as raw HTML: “some 40 years after the development of homology”; “Alexander had by 1930 defined a first notion of a cochain”; “At a 1935 conference in Moscow, Andrey Kolmogorov and Alexander both introduced cohomology and tried to construct a cohomology product structure”; “From 1936 to 1938, Hassler Whitney and Eduard Čech developed the cup product (making cohomology into a graded ring) and cap product.”
• Cohomology, MathWorld — cohomology as the dual construction to homology, with the ring structure as the distinguishing feature.
• Hodge conjecture, Wikipedia — “Every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.”

• A history of Topology, MacTutor — read raw: “The idea of connectivity was eventually put on a completely rigorous basis by Poincaré in a series of papers Analysis situs Ⓣ(Analysis of place) in 1895. Poincaré introduced the concept of homology and gave a more precise definition of the Betti numbers”; “in a series of papers Analysis situs Ⓣ(Analysis of place) in 1895. Poincaré introduced the concept of homology and gave a more precise definition of the Betti numbers associated with a space than had Betti himself. Euler’s convex polyhedra formula had been generalised.”
• Analysis Situs (paper), Wikipedia — read raw: “Analysis Situs” is a seminal mathematics paper that Henri Poincaré published in 1895. Poincaré published five supplements to the paper between 1899 and 1904.”; “Poincaré published five supplements to the paper between 1899 and 1904..”
• Klein bottle, Wikipedia — read raw: “The Klein bottle was first described in 1882 by the mathematician Felix Klein.”; “In contrast, the Klein bottle is boundaryless, like a sphere or torus, though it cannot be embedded in ordinary three-dimensional space without intersecting itself.”
• Descartes on Polyhedra, Wikipedia — read raw: “The original Latin manuscript of De solidorum elementis was written circa 1630 by Descartes”; “survived long enough for Gottfried Wilhelm Leibniz to copy it in 1676 before disappearing for good. Leibniz’s copy, also lost, was rediscovered in Hannover around 1860.”; “Central to the book is the disputed priority for Euler’s polyhedral formula between Leonhard Euler and René Descartes..”
• Twenty-one Proofs of Euler’s Formula, David Eppstein (UC Irvine) — read raw: “In 1794, Legendre provided a complete proof, using spherical angles. Cauchy got into the act in 1811, citing Legendre and adding incomplete proofs based on triangle removal.”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Alexander polynomial, Wikipedia — read raw: “Alexander, J. W. (1928) Topological Invariants of Knots and Links, Transactions of the American Mathematical Society 30 (2): 275–306.”
• Euclid’s Elements, Wikipedia — read raw: “The Elements (Στοιχεῖα Stoikheîa) is a mathematical treatise written c. 300 BC by the Ancient Greek mathematician Euclid..”
• Peter Guthrie Tait biography, MacTutor — read raw: “In 1876 Tait began an intense study of knots, attempting to classify them. He published seven papers on knots in the Proceedings of the Royal Society of Edinburgh in the academic year 1876-77.”; “When Kirkman sent him all knot projections with 10 crossings in January 1885 again Tait found all in equivalent knots. The tables were printed in September 1885 and again they are completely correct..”
• Max Dehn, Wikipedia — read raw: “In 1914 he proved that the left and right trefoil knots are not equivalent..”
• Platonic solid, Wikipedia — its table gives the cube 8 vertices, 12 edges, 6 faces and the regular icosahedron 12, 30, 20; the topological proof: “The key is Euler’s observation that V − E + F = 2,” whence “1/q + 1/p > 1/2,” and “there are only five possibilities for {p, q}.”
• Pythagorean theorem, Wikipedia — Thomas L. Heath: “no specific attribution of the theorem to Pythagoras exists in the surviving Greek literature from the five centuries after Pythagoras lived.”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Betti number, Wikipedia — defines the nth Betti number as the rank of the nth homology group and credits Betti, Poincaré’s naming of the numbers after him, and Noether’s modern formulation; it gives no dates (Betti’s 1871 paper is on the MacTutor line).
• Enrico Betti, MacTutor — dates Betti’s paper: “He published an important paper on topology in 1871 which contained what we now call the ‘Betti numbers’.”
• Analysis Situs (paper), Wikipedia — dates Poincaré’s paper: “a seminal mathematics paper that Henri Poincaré published in 1895.”
• A history of Topology, MacTutor — Poincaré’s sharper definition: “Poincaré introduced the concept of homology and gave a more precise definition of the Betti numbers associated with a space than had Betti himself.”
• Homology (mathematics), Wikipedia — dates Noether’s work and gives the sphere’s counts: “Emmy Noether and, independently, Leopold Vietoris and Walther Mayer further developed the theory of algebraic homology groups in the period 1925–28.”; “A two-dimensional sphere S2 has a single connected component, no one-dimensional-boundary holes, a two-dimensional-boundary hole, and no higher-dimensional holes.”
• Betti Number, Wolfram MathWorld — gives b₁ = 4 for genus two: “For a closed, orientable surface of genus g, the Betti numbers are p0=1, p1=2g, and p2=1.”

• Klein bottle, Wikipedia — states that Klein first described the surface in 1882 and describes its nonorientability and construction from a square with a twisted gluing; “The Klein bottle, proper, does not self-intersect. Nonetheless, there is a way to visualize the Klein bottle as being contained in four dimensions. By adding a fourth dimension to the three-dimensional space, the self-intersection can be eliminated.”; and “Dissecting a Klein bottle into halves along its plane of symmetry results in two mirror image Möbius strips”; “Its Euler characteristic is therefore 1 − 2 + 1 = 0”; and “It is also homeomorphic to a sphere plus two cross-caps.”

• Jones polynomial, Wikipedia — credits Jones’s 1984 discovery, gives the unknot normalization V=1, and gives the mirror image by substituting t^−1 for t: “Thus, an amphicheiral knot, a knot equivalent to its mirror image, has palindromic entries in its Jones polynomial.” It adds: “It is an open question whether there is a nontrivial knot with Jones polynomial equal to that of the unknot.” The trefoil and Alexander comparison is on the Knot theory and Alexander polynomial lines below.
• Knot theory, Wikipedia — the trefoils separate Jones from Alexander: “But the Alexander–Conway polynomial of each kind of trefoil will be the same ... The Jones polynomial can in fact distinguish between the left- and right-handed trefoil knots (Lickorish 1997).”
• Vaughan Jones, Wikipedia — dates the Fields Medal: “He was awarded a Fields Medal in 1990.”
• Alexander polynomial, Wikipedia — the Alexander polynomial is blind to reflection: “it cannot distinguish between a knot and its mirror image.”

• Crossing number (knot theory), Wikipedia — defines crossing number as the minimum number of crossings over all diagrams and states the unknot has crossing number 0 while the smallest nontrivial knot has crossing number 3.
• Knot theory, Wikipedia — Kelvin’s theory and Tait’s 1885 table: “In the 1860s, Lord Kelvin’s theory that atoms were knots in the aether led to Peter Guthrie Tait’s creation of the first knot tables for complete classification.”; “Tait, in 1885, published a table of knots with up to ten crossings”
• Elementary Knot Theory, Marc Lackenby, arXiv — computable but slow: “Crossing number is algorithmically computable. ... This approach is obviously not very efficient but it seems unlikely that there is any quicker way of determining a link’s crossing number in general.”

• MIT OpenCourseWare, Probability and Random Variables, Lecture 18 — defines a continuous variable through its density and states explicitly that the probability of any single point is zero.
• Almost surely, Wikipedia — read as raw wikitext: “the set of outcomes on which the event does not occur has probability 0, even though the set might not be empty”; “the dart will almost never land on a diagonal … even though the set of points on the diagonals is not empty, and a point on a diagonal is no less possible than any other point.”

• Probability axioms, Wikipedia — (§§ First, Second and Third axiom): states the three Kolmogorov axioms — “The probability of an event is a non-negative real number,” “P(Ω) = 1” and countable additivity — and states “P(∅) = 0. The empty set is interpreted as the event that ‘no outcome occurs’, which is impossible.”
• Probability theory, Encyclopedia of Mathematics — gives the axioms as 0 ≤ P(A) ≤ 1, P(U) = 1, and additivity over pairwise mutually exclusive events.
• A. N. Kolmogorov, Foundations of the Theory of Probability, Chelsea, New York, 1950 — (Ch. I, §1 “Axioms”): the English translation of the 1933 Grundbegriffe, the primary source for the axioms this claim names.

• NIST/SEMATECH e-Handbook, Mean Vector and Covariance Matrix — gives variance and covariance as centered products; substituting a constant makes every deviation zero.
• MathWorld, Covariance — “while statistically independent variables are always uncorrelated, the converse is not necessarily true.” It gives no dependent zero-covariance example; the card’s X-against-X² example is this book’s own.
• Covariance, Wikipedia — read as raw wikitext: covariance is defined for variables “with finite second moments”; “If X and Y are independent random variables, then their covariance is zero. This follows because under independence, E[XY] = E[X] · E[Y].” “The converse, however, is not generally true. For example, let X be uniformly distributed in [−1, 1] and let Y = X². Clearly, X and Y are not independent,” yet their covariance is 0.
• James Jones, Measures of Variation, Math 113 lecture notes, Richland Community College — “The problem is that this summation is always zero. So, the average deviation will always be zero.” and “So, to keep it from being zero, the deviation from the mean is squared and called the “squared deviation from the mean.””

• NIST Digital Library of Mathematical Functions §7.2, Definitions — defines erf and gives erf(∞) = 1, from which the √(2π) normalization follows. §7.2 is a definitions page and does not itself display the Gaussian integral; the P. M. Lee note beside it proves the value outright.
• Encyclopaedia of Mathematics, Lindemann–Weierstrass theorem — records the transcendence consequences for e, π and logarithms of nonunit algebraic numbers; the displayed consequences then follow algebraically.
• P. M. Lee, The Probability Integral (University of York note, typeset 2016, signed “P.M.L.”) — describes Laplace’s 1774 evaluation and credits the square-and-polar proof to Poisson. This six-page note is not Stigler’s The History of the Normal Distribution, under which title it was previously cited here.
• Transcendental number, Wikipedia — read as raw HTML: “The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers’ existence was e, by Charles Hermite in 1873.”
• Normal distribution, Wikipedia — gives 1782 (read as raw wikitext): “It was Laplace who first calculated the value of the integral ∫ e^(−t²) dt = √π in 1782, providing the normalization constant for the normal distribution.”

• Thomas S. Ferguson, Who Solved the Secretary Problem? — traces Gardner’s February 1960 related popular problem and Lindley’s 1961 classical solution, and gives the asymptotic 1/e result: “In Martin Gardner’s Mathematical Games column in the February 1960 issue of Scientific American, there appeared a simple problem that has come to be known today as the Secretary Problem.”
• Cambridge repository, Problems of optimal choice — confirms Lindley’s first published classical solution and discusses Gardner’s 1960 account.
• Dennis Lindley, Wikipedia — read as raw wikitext: Dennis Victor Lindley (1923–2013); “It is widely acknowledged that in 1961, Lindley was the first to solve the Secretary problem in a scientific article.”

• MathWorld, Buffon’s Needle Problem — gives the 1777 solved version and probability 2/π.
• Dominic Klyve, How to Calculate π: Buffon’s Needle (Calculus version), TRIUMPHS, Digital Commons @ Ursinus College — the TRIUMPHS mini-Primary Source Project (Klyve, 2019) for Calculus I, built on Buffon’s own text on the needle problem; the landing page gives no date for Buffon’s essay (1777 is on the MathWorld line).
• Rectifier, Wikipedia — average of a rectified sine is 2/π of its peak, V_av = 2·V_peak/π: “The average and RMS no-load output voltages of an ideal single-phase full-wave rectifier are:”

• MathWorld, Birthday Problem — gives the complement product, the 23-person threshold and the 0.507297 shared-birthday probability.
• NIST Special Publication 800-2, Public-Key Cryptography — derives the birthday attack against hash functions from the same collision calculation.
• Birthday problem, Wikipedia — read as raw HTML: “The problem is generally attributed to Harold Davenport in about 1927, though he did not publish it at the time”, and “The first publication of a version of the birthday problem was by Richard von Mises in 1939”, the paper being Über Aufteilungs- und Besetzungswahrscheinlichkeiten, Revue de la faculté des sciences de l’Université d’Istanbul 4, 145–163, 1939.

• Penn State STAT 414, Mathematical Expectation — computes the fair-die expectation as 3.5 and explains that it is a theoretical average, not a realized face.

• Claude E. Shannon, A Mathematical Theory of Communication — primary paper defining logarithmic information and entropy, including the statistical-mechanics connection.

• Claude E. Shannon, A Mathematical Theory of Communication — credits J. W. Tukey for “bit” and defines the binary unit; evaluating the entropy formula at 1/2 gives one bit.

• Mutual information, Wikipedia — (§ Definition, § Relation to Kullback–Leibler divergence): “I(X;Y) = 0 if and only if X and Y are independent random variables,” and I(X;Y) “is equal to zero precisely when the joint distribution coincides with the product of the marginals.”
• Kullback–Leibler divergence, Wikipedia — states non-negativity by Gibbs’ inequality, with equality exactly when the two distributions are equal almost everywhere; in the language of Bayesian inference, “it is the amount of information lost when Q is used to approximate P.”
• Kullback and Leibler, On Information and Sufficiency — DOI record for the 1951 primary paper introducing the information measure.
• C. E. Shannon, A Mathematical Theory of Communication, Bell System Technical Journal 27 (1948), 379–423 — the Crossref record gives the title, the journal, volume 27, July 1948 and pages 379–423, author C. E. Shannon. This is the 1948 the #mutinfo0 card names in its quick line and therefore in every popup.
• Covariance, Wikipedia — read as raw wikitext: covariance is defined for variables “with finite second moments”; “If X and Y are independent random variables, then their covariance is zero. This follows because under independence, E[XY] = E[X] · E[Y].” “The converse, however, is not generally true. For example, let X be uniformly distributed in [−1, 1] and let Y = X². Clearly, X and Y are not independent,” yet their covariance is 0.

• MIT OpenCourseWare, Random Variables in a Nutshell — derives E[X̄]=μ by linearity, calls X̄ unbiased, and notes that independence was not used.
• Central moment, Wikipedia — read as raw wikitext: “The first central moment μ1 is 0 (not to be confused with the first raw moment or the expected value μ).”

• NIST/SEMATECH e-Handbook, Least Squares — defines the intercept model and the least-squares objective. It does not itself state that the residuals sum to zero; that is the one-line consequence below.
• Own computation — Minimizing Σ(yᵢ − β₀ − β₁xᵢ)² over β₀ gives the normal equation −2Σ(yᵢ − β₀ − β₁xᵢ) = 0, that is Σ residuals = 0. The step needs the intercept: a fit without one carries no such constraint.

• R. A. Fisher, Statistical Methods for Research Workers, 13th edition — (Ch. III, §12, p. 44): “The value for which P = ·05, or 1 in 20, is 1·96 or nearly 2; it is convenient to take this point as a limit in judging whether a deviation is to be considered significant or not. Deviations exceeding twice the standard deviation are thus formally regarded as significant.”
• MacTutor Extras, R. A. Fisher, Statistical Methods: Introduction — reproduces part of that book’s first chapter (The Scope of Statistics; General Method, Calculation of Statistics; The Qualifications of Satisfactory Statistics). The “one in twenty” passage is in its third chapter and is not on this page.
• R. A. Fisher, Statistical Methods for Research Workers (1925), ch. III, Classics in the History of Psychology (York University, Toronto) — the 1925 text: “The value for which P =.05, or 1 in 20, is 1.96 or nearly 2 ; it is convenient to take this point as a limit in judging whether a deviation is to be considered significant or not.” and “Deviations exceeding twice the standard deviation are thus formally regarded as significant.”

• NIST/SEMATECH e-Handbook, Sample Sizes and Critical Values — explicitly gives the 0.975 critical value as 1.96 for a 95% interval.
• 97.5th percentile point, Wikipedia — read as raw wikitext: “The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean.”

• Dictionary of Scientific Biography, Andrei Andreyevich Markov — states that these chains first appeared in Markov’s 1906 paper extending the law of large numbers to dependent variables.
• Direct calculation from the matrix displayed in the chapter: (1/3, 2/3)[[1/2, 1/2], [1/4, 3/4]]=(1/3, 2/3), so πP−π=(0, 0) and its ℓ¹ norm is 0.
• Markov chain Monte Carlo, Wikipedia — read as raw wikitext: “Given a probability distribution, one can construct a Markov chain whose elements’ distribution approximates it, i.e. the Markov chain’s equilibrium distribution matches the target distribution.”
• Markov chain mixing time, Wikipedia — read as raw wikitext: “the mixing time of a Markov chain is the time until the Markov chain is “close” to its steady state distribution.”

• MathWorld, Pólya’s Random Walk Constants — gives Pólya’s 1921 result, p(3), and the Watson and Glasser–Zucker formulas.

• Thomas Bayes and Richard Price, An Essay towards solving a Problem in the Doctrine of Chances — primary text records that Price communicated the essay and that it was read on 23 December 1763.
• MacTutor, Bayes and the development of inverse probability — states that Laplace independently derived another form in 1774.
• Thomas Bayes, Wikipedia — read as raw wikitext: Bayes is “known for formulating a specific case of the theorem that bears his name”; the essay “was read to the Royal Society in 1763 after Bayes’s death. Richard Price shepherded the work through this presentation and its publication.”
• Bayes’ theorem, Wikipedia — read as raw wikitext: Laplace “reproduced and extended Bayes’s results in 1774, apparently unaware of Bayes’s work, and summarized his results in Théorie analytique des probabilités (1812).”

• MacTutor, Jacob Bernoulli — identifies Ars Conjectandi (1713) and Bernoulli’s law-of-large-numbers result.
• Wikipedia, Law of large numbers — distinguishes the weak law (convergence in probability) from the strong law (almost-sure convergence), and dates Bernoulli’s proof to Ars Conjectandi, 1713.

• NIST/SEMATECH e-Handbook, Normal Distribution — gives the density, parameters and areas 0.6827 and 0.9545.
• P. M. Lee, The Probability Integral (University of York note, typeset 2016) — defines φ(z) = (1/√(2π)) exp(−½z²) and shows “the integral of φ from −∞ to ∞ is 1, and hence φ is a probability density function.”

• Variance, Wikipedia — § Basic properties states it outright — “The variance of a constant is zero,” with the displayed identity Var(a) = 0; § Propagation adds that “if a constant is added to all values of the variable, the variance is unchanged” and “If all values are scaled by a constant, the variance is scaled by the square of that constant,” which together give Var(aX + b) = a²Var(X).
• Arithmetic mean, Wikipedia — § Motivating properties gives (x₁ − x̄) + ⋯ + (xₙ − x̄) = 0 and states that the mean “is the only number for which the residuals (deviations from the estimate) sum to zero.”
• Simple linear regression, Wikipedia — § Numerical properties states “The sum of the residuals is zero if the model includes an intercept term,” and that the regression line passes through the point of sample means.

Added by the V14 prose pass. The pages cited at 13.3 and 13.14 define variance, covariance and the least-squares objective, but none of them states these three consequences; 13.25 carries each one explicitly, and the three sentences that use them were meant to cite it alongside their original footnote. Checked on 17 September 2026: only one of the three did, and no card in the book cited this entry at all. The three cards and the two missing prose runs were attached on that date.

• Markov chain, Wikipedia — § Stationary distribution on finite state space states “If the Markov chain is additionally irreducible and aperiodic, then there is a unique stationary distribution π. In this case P^k converges to a rank-one matrix in which each row is the stationary distribution π,” and § Convergence speed to the stationary distribution gives the rate as the ratio of the second to the first eigenvalue.

Added by the V14 prose pass to carry the chapter’s convergence sentence, which 13.17 does not: 13.17 covers Markov’s 1906 priority and the stationary computation only. The chapter’s matrix P = [[1/2, 1/2], [1/4, 3/4]] has every entry positive, so it is irreducible and aperiodic and the theorem applies; its second eigenvalue is 1/4, so the distance to π falls by three quarters a day.

• Pierre-Simon Laplace, Wikipedia — read as raw HTML: “One sees in this essay that the theory of probabilities is basically only common sense reduced to a calculus. It makes one estimate accurately what right-minded people feel by a sort of instinct, often without being able to give a reason for it,” and the page dates the Essai philosophique sur les probabilités to 1814.

Added 17 September 2026 to source the chapter epigraph, which carried no marker. The epigraph uses the older English rendering “at bottom nothing but common sense reduced to calculus”; this source gives “basically only common sense reduced to a calculus.” Two translations of one sentence, not two claims.

• Bit, Wikipedia — read raw: “Claude E. Shannon first used the word “bit” in his seminal 1948 paper “A Mathematical Theory of Communication.” He attributed its origin to John W. Tukey, who had written a Bell Labs memo on 9 January 1947 in which he contracted “binary digit” to simply “bit..””
• Pólya’s Random Walk Constants, Wolfram MathWorld — read raw: “Watson (1939), McCrea and Whipple (1940), Domb (1954), and Glasser and Zucker (1977) showed that.”
• Transcendental number, Wikipedia — read raw: “The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers’ existence was e, by Charles Hermite in 1873..”
• M. L. Glasser and I. J. Zucker, Extended Watson integrals for the cubic lattices, PNAS 74 (1977), PubMed Central — abstract: “A new closed form expression for Watson’s result on the simple cubic lattice is given in terms of gamma functions.”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Natural unit of information, Wikipedia — read via the REST endpoint (revision 1376988607): “The natural unit of information (symbol: nat), sometimes also nit or nepit, is a unit of information or information entropy, based on natural logarithms and powers of e, rather than the powers of 2 and base 2 logarithms, which define the shannon.” and “When the Shannon entropy is written using a natural logarithm … it is implicitly giving a quantity with nat as the unit..”

• Encyclopedia of Mathematics, Lie algebra — Gives the defining axioms, the commutator example, and the relation to Lie groups; the classification covers “all simple (that is, non-commutative and not containing ideals other than 0 and the algebra itself) finite-dimensional Lie algebras.”
• Jacobi identity, Wikipedia — read as raw wikitext: the equivalent form “[X, [Y, Z]] = [[X, Y], Z] + [Y, [X, Z]].”

• Encyclopedia of Mathematics, Exceptional Lie algebra — Lists the five types, dimensions, ranks, smallest representations, and the G₂/F₄ structural descriptions.

• MIT OpenCourseWare, Lie algebras lecture notes — Defines su(2) by traceless anti-Hermitian 2 × 2 matrices and gives the Pauli-matrix basis.
• Encyclopedia of Mathematics, Cayley–Klein parameters — Describes the two-to-one homomorphism from SU(2) to SO(3).
• Special unitary group, Wikipedia — read as raw wikitext: “This representation is routinely used in quantum mechanics to represent the spin of fundamental particles such as electrons.”

• MacTutor, Hermann Grassmann — Dates the 1844 Ausdehnungslehre and describes Grassmann’s exterior-algebra work and its reception.
• Encyclopedia of Mathematics, Clifford algebra — gives the Clifford relations and the exterior-algebra basis construction, and supports 1876 — the algebra was “first investigated by W. Clifford in 1876.” Other accounts date it to the 1878 paper Applications of Grassmann’s Extensive Algebra; under the tilde convention the card shows the earlier year and this note carries both.

• Encyclopedia of Mathematics, Clifford algebra — Gives the subset-indexed basis and dimension 2^n.
• John Baez, The Octonions — The contents page of John C. Baez’s The Octonions, which has a section on Clifford algebras; the page itself does not state the even-subalgebra result, which the Clifford algebra page carries.
• Clifford algebra, Wikipedia — read as raw wikitext: “Hamilton’s quaternions are constructed as the even subalgebra of the Clifford algebra Cl3,0(R),” “which shows that the even subalgebra Cl[0]3,0(R) is Hamilton’s real quaternion algebra.”

• Encyclopedia of Mathematics, Root — Describes roots of unity as regular-polygon vertices.
• Root of unity, Wikipedia — (§ Summation): “Let SR(n) be the sum of all the nth roots of unity, primitive or not. Then SR(n) = 1 for n = 1 and 0 for n > 1.” This is the source for the vanishing sum itself. It also carries the link to characters: “The n th roots of unity form an irreducible representation of any cyclic group of order n. The orthogonality relationship also follows from group-theoretic principles as described in Character group.”
• Stephan Ramon Garcia, Trevor Hyde and Bob Lutz, Gauss’s hidden menagerie: from cyclotomy to supercharacters, arXiv:1501.07507 (Notices AMS 62 (2015) 878–888) — the same article, free on arXiv: Gauss’s study of the sums of complex exponentials known as Gaussian periods, behind the constructibility of the 17-gon. It does not state the vanishing sum.
• Wikipedia, Gaussian period — Gaussian periods were introduced by Gauss as the basis of his compass-and-straightedge theory; “the construction of the heptadecagon … depended on the algebra of such periods.”

• MIT, Introduction to Representation Theory — Gives irreducible-character orthogonality; pairing with the trivial character yields the stated sum.
• MacTutor, Ferdinand Georg Frobenius — Dates Frobenius’s first paper on group characters to 1896.

• Encyclopedia of Mathematics, Symmetric group — States simplicity of Aₙ for n≥5 and nonsolvability of Sₙ for n≥5.
• Cambridge, An aspect of icosahedral symmetry — “As an abstract group, the icosahedral group G is the smallest non-abelian simple group, isomorphic to the alternating group A5 on five letters.” The paper does not call it the rotation group or give its order; the Icosahedral symmetry page carries both.
• Icosahedral symmetry, Wikipedia — read as raw wikitext: “The icosahedral rotation group I is of order 60. The group I is isomorphic to A5, the alternating group of even permutations of five objects.”
• Galois theory, Encyclopedia of Mathematics — “Since a symmetric group is unsolvable for n≥5, there are, generally speaking, no solutions in radicals of quintic equations and equations of a higher degree (Abel’s theorem).”

• Wikipedia, Classification of finite simple groups — every finite simple group is cyclic, alternating, of Lie type, or one of 26 sporadic groups; “Daniel Gorenstein announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification of quasithin groups. The completed proof of the classification was announced by Aschbacher (2004) after Aschbacher and Smith published a 1221-page proof for the missing quasithin case.”

• ATLAS of Finite Group Representations, Monster — Gives the Monster’s exact order, its maximal subgroups and conjugacy classes; its Representations section is marked “Porting incomplete,” so the 196883 degree rests on the Wikipedia line.
• Wikipedia, Monster group — the Monster is the largest sporadic simple group, of order 808017424794512875886459904961710757005754368000000000, and its smallest faithful complex representation has dimension 196883.

• Wikipedia, Monstrous moonshine — McKay’s observation that the coefficient 196884 is one more than 196883, the Conway–Norton moonshine conjecture, and Borcherds’s proof. (First cited here: M. Broué, In memoriam: John H. Conway, Bull. AMS 58 (2021), doi:10.1090/bull/1702).
• International Mathematical Union, 1998 Fields Medals — Official IMU record of Borcherds’s 1998 Fields Medal.

• Encyclopedia of Mathematics, Discriminant — Defines the discriminant by root differences, gives the quadratic case b²−4ac, and states the repeated-root criterion.

• Encyclopedia of Mathematics, Extension of a field — Defines extension degree, relates it to minimal-polynomial degree, and gives the tower law.
• Field extension, Wikipedia — read as raw wikitext: “The degree is 2 because {1, √2} can serve as a basis.”

• Encyclopedia of Mathematics, Cyclotomic field — Identifies the cyclotomic Galois group with the unit group modulo n and gives degree φ(n).
• Cyclotomic field, Wikipedia — read as raw wikitext: “The isomorphism sends each σ ∈ Gal(Q(ζn)/Q) to a mod n, where a is an integer such that σ(ζn) = ζn^a.”

• Encyclopedia of Mathematics, Galois theory — Connects solvability by radicals to solvable groups and the general quintic to nonsolvable symmetric groups.
• MacTutor Dictionary of Scientific Biography, Paolo Ruffini — Documents Ruffini’s 1799 attempt and Abel’s independent 1824 proof.
• Symmetric group, Encyclopedia of Mathematics — “For n ≤ 4 the group S_n is solvable, but for n ≥ 5 it is not solvable and A_n is a simple non-Abelian group.”

• Annals of Mathematics, The sphere packing problem in dimension 8 — Viazovska’s primary paper proves optimality of the E₈ lattice packing.
• AMS Feature Column, Eight-dimensional spheres and E₈ — Relates the 240 roots/shortest vectors to the E₈ lattice packing.

• Encyclopedia of Mathematics, Exceptional Lie algebra — Lists 27 as the simplest nontrivial representation dimension for E₆.
• Wikipedia, Feza Gürsey — credits Gürsey with the E6 grand unified theory and cites the 1976 paper: “A universal gauge theory model based on E6.”

• Encyclopedia of Mathematics, Poisson brackets — States the Jacobi identity and cites Jacobi’s 1862 paper.
• Encyclopedia of Mathematics, Lie algebra — Dates Weyl’s introduction of the term Lie algebra to 1934.
• Carl Jacobi, MacTutor — gives Jacobi’s death as 18 February 1851, in Berlin.
• David Eugene Smith, History of Modern Mathematics (1896), Project Gutenberg — read from the PDF: “Jacobi (1827) also gave an analysis of Pfaff’s method, besides developing an original one (1836) which Clebsch published (1862).”

• Gauss’s diary, Wikipedia — (§ Sample entries): “Entry 1, dated 1796, March 30, states Principia quibus innititur sectio circuli, ac divisibilitas eiusdem geometrica in septemdecim partes etc.”
• Göttingen State and University Library, catalogue record for Gauss’s Mathematisches Tagebuch, 30.3.1796 – 9.7.1814 — the digitisation record for the manuscript (Cod. Ms. Gauß Math. 48 Cim.), dated 30.3.1796 – 9.7.1814; re-read 24 September 2026, the scanned pages are now viewable in its viewer (they were not in September’s first reading).
• MacTutor/Royal Astronomical Society obituary of Gauss — Gives Gauss’s birth date as 30 April 1777.

• Encyclopedia of Mathematics, Lie algebra, semi-simple — frames the whole classification with “Henceforth finite-dimensional semi-simple Lie algebras over a field k of characteristic 0 are considered,” states that over an algebraically closed field the classification of simple non-Abelian Lie algebras “essentially coincides with the classification of … irreducible reduced root systems,” calls the algebras of types A–D classical, and names E₆, E₇, E₈, F₄, G₂ as the exceptional ones.

Added in the V14 prose pass. The chapter’s sentence about the classification carried no footnote, and r-14.2 — the Encyclopedia of Mathematics article on exceptional Lie algebras — states only the five exceptional types, not the four classical series; that article opens “A simple Lie algebra (see Lie algebra, semi-simple) that is not classical” and so defers the classification itself to this page. The words finite-dimensional were added to the chapter’s sentence in the same pass: that is the hypothesis this source states, and the classification is false without it.

• MacTutor, Paolo Ruffini — carries “Quadratic equations (of degree 2) had been known to be soluble by radicals from the time of the Babylonians,” “The cubic equation had been solved by radicals by del Ferro, Tartaglia and Cardan” and “Ferrari had solved the quartic by radicals in 1540,” and, of Ruffini’s 1799 book on the impossibility above degree four, “It is remarkable work and, except for one gap, proves the theorem as Ruffini claimed.”

Added in the V14 prose pass. The chapter’s sentence listing formulas by radicals for quadratics, cubics and quartics carried no footnote of any kind. The word incomplete in “Ruffini’s 1799 argument was incomplete” is asserted by r-14.15’s claim line, but r-14.15’s readable source — the MacTutor Dictionary of Scientific Biography article on Ruffini — says only “Only Cauchy accorded them full credence” and never that the argument had a gap; r-14.15’s other source, the Encyclopedia of Mathematics article on Galois theory, could not be re-read during this pass — encyclopediaofmath.org answered HTTP 502 to every request from this client on 10 September 2026, intermittently for other articles too, so this is a reachability problem and not a dead link. This page carries both halves in so many words.

• Hermann Weyl, Wikiquote — read as raw HTML: it quotes Symmetry and cites it as “Symmetry, Princeton Univ. Press, 1952,” which is the book and the year the footer gives. The page number the footer gives, p. 5, is the book’s own and is not in this source.

Added 17 September 2026 to source the chapter epigraph, which carried no marker. The book and the year are sourced; the page reference is the author’s own reading of the printed book and is offered as such.

• Tits group, Wikipedia — gives the group’s order and its status as an index-2 subgroup of the Ree group 2F4(2), classified with the Lie-type groups.
• Classification of finite simple groups, Wikipedia — carries why it is a borderline case: “the Tits group is sometimes regarded as a sporadic group because it is not strictly a group of Lie type”

• j-invariant, Wikipedia — gives the q-expansion 1/q + 744 + 196884q + ... and the function’s role classifying elliptic curves.
• Monstrous moonshine, Wikipedia — carries McKay’s observation: “Specifically, McKay noticed that the coefficient 196884 was precisely one more than 196883, the degree of the smallest faithful complex representation of the monster group.”
• Heegner number, Wikipedia — carries how close e^(π√163) comes: “e^(π√163) = 262,537,412,640,768,743.999,999,999,999 25…” and “e^(π√163) ≈ 640,320³ + 744,” with “the linear term of the error is, −196,884/e^(π√163) ≈ −0.000,000,000,000 75”

• Monstrous moonshine, Wikipedia — states McKay’s observation, the Conway-Norton conjectures of 1979, and Borcherds’s 1992 proof; the moonshine module is a vertex operator algebra “which has the monster group as its group of symmetries,” and “The conjectures made by Conway and Norton were proven by Richard Borcherds for the moonshine module in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras.”; McKay found that the first few terms of the J-invariant “could be expressed in terms of linear combinations of the dimensions of the irreducible representations” of the monster group, and Conway and Norton “conjectured the existence of an infinite-dimensional graded representation of M, whose graded traces T_g are the expansions of precisely the functions on their list.”

• Solvable group, Wikipedia — “a solvable group or soluble group is a group that can be constructed from abelian groups using extensions”; “a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable (note this theorem holds only in characteristic 0)”; “The group S5 is not solvable — it has a composition series {E, A5, S5} …, giving factor groups isomorphic to A5 and C2; and A5 is not abelian”; and, in its example, “Aut(Q(√2)/Q) ≅ Z/2 with group action f(±√2) = ∓√2” (formulas as typeset on the page).

• Encyclopedia of Mathematics, Hilbert space — defines Hilbert space, gives l² and L² as examples, and defines orthogonality by vanishing inner product. Its numbered axioms require the space to be infinite-dimensional, but the article adds that this condition is often omitted, so the source carries BOTH conventions. Recorded because this entry is cited in support of the wide definition while its headline axioms give the narrow one; see 15.24.

• MacTutor, David Hilbert — Dates Hilbert’s integral-equation work to about 1909 and connects it directly to infinite-dimensional Hilbert space and quantum mechanics.
• Encyclopedia of Mathematics, Hilbert space — Credits Hilbert with l² and L² in his integral-equation work and von Neumann, Riesz and Stone with the abstract definition.
• MacTutor, The quantum age begins — Records von Neumann’s 1932 rigorous operator-algebra foundation for quantum theory.

The inherited 1906–1912/1927/1929 compressed chronology was replaced by the dates these sources actually support.

• Dirac, A new notation for quantum mechanics — Primary 1939 paper introducing Dirac’s notation.
• Encyclopedia of Mathematics, Hilbert space — Defines orthogonality as a zero inner product.
• IBM Quantum Learning, Quantum information — Gives the standard qubit bases, inner products and measurement probabilities.
• IBM Quantum Learning, Basics of quantum information: Inner products and projections — “We also have that the product of a bra vector and a ket vector, viewed as matrices either having a single row or a single column, results in a scalar.”; “for any classical state set Σ, the set of all standard basis vectors {|a⟩ : a∈Σ} is an orthonormal basis.”

• IBM Quantum Learning, Quantum circuits — the lesson introduction: it introduces inner products, orthogonality and projective measurements, and says “we’ll derive fundamental limitations on quantum information, including the no-cloning theorem and the impossibility to perfectly discriminate non-orthogonal quantum states” (the derivations are quoted on the Limitations line below).
• IBM Quantum Learning, Basics of quantum information: Limitations on quantum information — “if we have two quantum states |ψ⟩ and |φ⟩ that are not orthogonal … then it’s impossible to discriminate them perfectly”; “It is possible, by the way, to perfectly discriminate any two states that are orthogonal”; “The no-cloning theorem shows it is impossible to create a perfect copy of an unknown quantum state.”
• No-cloning theorem, Wikipedia — read as raw wikitext: “Hence by the Cauchy–Schwarz inequality either |φ⟩ = e^{iβ}|ψ⟩ or |φ⟩ is orthogonal to |ψ⟩. However, this cannot be the case for two arbitrary states. Therefore, a single universal U cannot clone a general quantum state.”

• Nobel Prize, Physics 1954 summary — Official award citation credits Born for the statistical interpretation of the wavefunction.
• Max Born, Nobel Lecture — Born’s primary lecture states that squared amplitudes give state probabilities and recounts his 1926 statistical interpretation.

• IBM Quantum Learning, Quantum information — Gives H|0⟩=|+⟩ and the equal computational-basis probabilities.

• Encyclopedia of Mathematics, Pauli matrices — Gives the matrices, their 1927 attribution, and their algebra, including σᵢ²=I.
• IBM Quantum Composer reference — Identifies Pauli X as the standard NOT gate.

• MIT OpenCourseWare, Path Integrals of Fermions — Graduate lecture covers Grassmann anticommutation and fermionic path integrals.
• Nobel Prize, Wolfgang Pauli facts — Dates Pauli’s exclusion principle to 1925 and states its same-state prohibition.
• Nobel Prize, Physics 1901–2000 — Connects the exclusion principle with half-integer-spin fermions; “This principle was later found to be associated with the symmetry of wave functions for particles of half-integer spins in general, distinguishing what is now called fermions from the bosonic particles whose spins are integer multiples of” (the unit is an image on the page, not in its text).
• Grassmann number, Wikipedia — read as raw wikitext: “the generators and thus the algebra are anti-commuting,” θᵢθⱼ = −θⱼθᵢ; “The squares of the elements vanish”; “the anti-commuting behavior can be strongly identified with the quantum-mechanical behavior of fermions: the anti-commutation is that of the Pauli exclusion principle.”

• Encyclopedia of Mathematics, Uncertainty principle — prints the same-component canonical commutators with the identity operator, [p̂ₓ, x̂] = [p̂_y, ŷ] = [p̂_z, ẑ] = iℏE, with no cross terms and no Kronecker delta. The annotation here used to read “Gives the Cartesian canonical commutators,” which 15.23 contradicts in the same reference list; corrected 17 September 2026. Its sign is the reverse of this book’s: [x̂, p̂] = iℏI gives [p̂ₓ, x̂] = −iℏI. The book does not follow this page’s sign; the Canonical commutation relation line carries it.
• Born–Jordan Quantization and the Equivalence of Matrix and Wave Mechanics — the abstract dates the Born and Jordan paper to 1925: “The aim of the famous Born and Jordan 1925 paper was to put Heisenberg’s matrix mechanics on a firm mathematical basis.” It does not print the commutation relation itself.
• Canonical commutation relation, Wikipedia — read as raw wikitext: “[x̂, p̂ₓ] = iℏI … where [x, pₓ] = x pₓ − pₓ x is the commutator of x and pₓ.”

• Encyclopedia of Mathematics, Pauli matrices — Gives σz=diag(1, −1) and its eigenvalues.
• IBM Quantum Learning, Quantum information — defines the plus state and the minus state, and of a standard basis measurement says “Measuring either state produces the same output distribution as the other,” 0 and 1 each with probability 1/2 (their orthogonality is on the Inner products line below).
• IBM Quantum Learning, Basics of quantum information: Inner products and projections — “{|+⟩,|−⟩} is an orthonormal basis for the 2-dimensional space corresponding to a single qubit.”

• Encyclopedia of Mathematics, Linear operator — States that self-adjoint spectra are real, distinct eigenspaces are orthogonal, and self-adjoint operators represent quantum observables.
• MIT OpenCourseWare, Mathematical formalism of quantum mechanics — Explains self-adjoint observables, spectral values and the finite-dimensional spectral decomposition.
• MacTutor, Erwin Schrödinger — Documents Schrödinger’s 1926 papers and his equivalence argument for wave and matrix mechanics; “Schrödinger published his revolutionary work relating to wave mechanics and the general theory of relativity in a series of six papers in 1926”; “In its most general form, the time-dependent Schrödinger equation describes the time evolution of closed quantum physical systems.”
• Erwin Schrödinger, Wikipedia — read as raw wikitext: “In January 1926, Schrödinger published in Annalen der Physik the paper “Quantisierung als Eigenwertproblem” (Quantization as an Eigenvalue Problem)… In this paper, he gave a “derivation” of the wave equation for time-independent systems”; “A third paper, published in May, showed the equivalence of his approach to that of Werner Heisenberg’s matrix mechanics.”; “A fourth paper in this series showed how to treat problems in which the system changes with time, as in scattering problems.”

• Encyclopedia of Mathematics, Uncertainty principle — Gives the inequality and the Kennard 1927 and Robertson 1929 chronology; its comments: “W. Heisenberg [a1] presented this uncertainty principle in 1927. In the same year E.H. Kennard [a2] found relation (2), while the general relation (1) was proved by H.P. Robertson [a3] in 1929..”
• University of Texas at Austin, Quantum Mechanics — States that equality is achieved by Gaussian wavepackets.

• Poh, Joshi, Cerè, Cabello and Kurtsiefer, Approaching Tsirelson’s Bound in a Photon Pair Experiment — Primary Physical Review Letters article states both bounds and reports an experiment approaching 2√2.

• Nobel Prize, Physics 2022 summary — Official award citation.
• NIST, A significant-loophole-free test of Bell’s theorem with entangled photons — Records the 2015 test and observed violation.
• Bell test, Wikipedia — read as raw wikitext: “These efforts to experimentally validate violations of the Bell inequalities resulted in John Clauser, Alain Aspect, and Anton Zeilinger being awarded the 2022 Nobel Prize in Physics”; “In 2015 the first three significant-loophole-free Bell-tests were published within three months by independent groups in Delft, Vienna and Boulder.”

• MacTutor, Joseph Fourier — Dates the 1807 presentation, the objections by Lagrange and Laplace, and the 1822 publication.
• MIT OpenCourseWare, Orthogonality Relations — Derives the sine-cosine orthogonality integrals over a full period.

MIT’s derivation supports the displayed integral and its vanishing.

• Encyclopedia of Mathematics, Plancherel theorem — States the unitary L² extension, norm identity and 1910 date.
• Plancherel theorem, Wikipedia — read as raw wikitext: proven “by Michel Plancherel in 1910”; the transform of an L¹ function “is defined as the Lebesgue integral” f̂(ξ) = ∫ f(x)e^(−2πixξ) dx; its L² extension is an isometry, and “This isometry is actually a unitary map.”

• Encyclopedia of Mathematics, Hilbert space — Defines the L² example and completeness framework.
• Encyclopedia of Mathematics, Orthogonal series — Dates the Riesz and Fischer theorem to 1907 and states the L²/ℓ² correspondence.
• Square-integrable function, Wikipedia — read as raw wikitext: “The square integrable functions (in the sense mentioned in which a “function” actually means an equivalence class of functions that are equal almost everywhere) form an inner product space.”
• Riesz–Fischer theorem, Wikipedia — read as raw wikitext: “The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer.”

• Durham University, Momentum-space wave function — Gives the position/momentum Fourier-transform pair and operator representations.
• University of Texas at Austin, Quantum Mechanics — Gives Gaussian position/momentum wavepackets and the minimum uncertainty product.
• Canonical commutation relation, Wikipedia — read as raw wikitext: “the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another).”

• IBM Quantum Learning, Quantum information — Gives |0⟩, |1⟩, |+⟩, |−⟩ and their measurement and inner-product relations.
• Encyclopedia of Mathematics, Pauli matrices — Gives the explicit matrices and σᵢ²=I.
• MIT OpenCourseWare, Path Integrals of Fermions — A video lecture on fermionic path integrals; the page itself prints no text of the anticommutation rule, which the Grassmann number line carries.
• Encyclopedia of Mathematics, Uncertainty principle — Prints the same-component canonical commutators, in the order [p̂ₓ, x̂] and with the opposite sign to this book’s (see 15.9); for the cross-coordinate case see 15.23.
• Grassmann number, Wikipedia — read as raw wikitext: “the generators and thus the algebra are anti-commuting,” θᵢθⱼ = −θⱼθᵢ; “The squares of the elements vanish”; “the anti-commuting behavior can be strongly identified with the quantum-mechanical behavior of fermions: the anti-commutation is that of the Pauli exclusion principle.”

The cited external definitions and matrices support each displayed value.

• Canonical commutation relation, Wikipedia — (§ Weyl relations): gives the trace obstruction — the relation Tr(AB) = Tr(BA) “gives a nonzero number on the right and zero on the left” — so [x̂, p̂] = iℏ has no finite-dimensional matrix realization.
• Fritz, On infinite-dimensional state spaces — States that [x, p]=i can be realized exactly only on an infinite-dimensional Hilbert space.

• Wikiquote, Richard Feynman — gives “I think I can safely say that nobody understands quantum mechanics” from The Character of Physical Law, in that book’s sixth lecture, “Probability and Uncertainty — the Quantum Mechanical View of Nature,” the Messenger Lectures at Cornell, November 1964.
• The Character of Physical Law, Wikipedia — read as raw HTML: the infobox gives “Published 1965 (BBC)”; the lectures were delivered “at Cornell University, as part of the Messenger Lectures series” and “The BBC recorded the lectures and published a book under the same title the following year”; and the list of the seven topics has as its sixth “Probability and uncertainty – the quantum mechanical view of nature,” which is the sixth chapter the epigraph cites.

• University of Texas at Austin, Quantum Mechanics — Chapter 7, Three-Dimensional Quantum Mechanics, p. 95: Eqs. (7.15)–(7.17) display [xᵢ, xⱼ] = 0, [pᵢ, pⱼ] = 0 and [xᵢ, pⱼ] = iℏδᵢⱼ, and the text immediately beneath them says it is “perfectly possible to simultaneously measure two different positions coordinates, or two different components of the momentum,” the only restriction being a coordinate against its own conjugate momentum. Eq. (9.12), p. 116, repeats it as “the fundamental commutation relation for the xᵢ and the pᵢ.”
• Canonical commutation relation, Wikipedia — The lead section gives the generalized relation [x̂ᵢ, p̂ⱼ] = iℏδᵢⱼ, whose Kronecker delta vanishes whenever the indices differ.

Added by the V14 prose pass to close the chapter’s open WRONG-REF row. 15.9’s two pages do not carry the Kronecker-delta form: the Encyclopedia of Mathematics Uncertainty principle entry prints only the same-component commutators [p̂ₓ, x̂] = [p̂_y, ŷ] = [p̂_z, ẑ] = iℏE, with no cross terms and no delta, and de Gosson (arXiv:1405.2519) gives only the one-dimensional pq − qp = −iℏ1. The claim itself is standard; 15.23 is the source that states it. The University of Texas text was already cited in this chapter at 15.12 and 15.19, and the Wikipedia article at 15.21.

• Anthony Duncan, annotated translation of von Neumann, Mathematische Begründung der Quantenmechanik (1927) — gives the 1927 axioms A to E verbatim. Axiom C reads “H has infinitely many dimensions: i.e., there are an arbitrarily large number of linearly independent elements of H,” and von Neumann states that the five properties fully determine the space. This is the direct evidence that the infinite-dimensionality was required by name and on purpose.
• Robert B. Griffiths, Hilbert Space Quantum Mechanics, Carnegie Mellon qitd113, section 1.1 — names the split explicitly: the term is “often reserved for an infinite-dimensional inner product space” but “is often used nowadays ... in a way that includes finite-dimensional spaces,” and it calls a two-dimensional Hilbert space a qubit.
• Federico Zalamea, Chasing Individuation: Mathematical Description of Physical Systems, PhD thesis, Université Paris Diderot, 2016, p. 117 — quotes von Neumann’s 1932 book, in the Beyer translation at p. 45, giving the dimension condition in two forms — C.(n), exactly n linearly independent vectors, and C.(infinity), arbitrarily many — followed by “We obtain a different space R, depending on which we decide upon.” READ THE CHAIN OF CUSTODY BEFORE RELYING ON THIS: it is Zalamea quoting Beyer quoting von Neumann, and Zalamea is a doctoral thesis rather than a peer-reviewed article. The Beyer text itself is lending-restricted and could not be obtained here, and Zalamea’s PDF would not extract for an independent check — two attempts recovered 48 KB of a 2.3 MB file, and the ar5iv HTML redirects to the abstract — so this source is carried on the reviewing agent’s reading and not on the coordinator’s. The title and document type above WERE checked against the arXiv record on 11 September 2026, because the first draft of this entry carried a wrong title. Nothing in the book’s prose or cards rests on it — the card says only what the first source above supports.

Added 11 September 2026, closing the deep-research item DR2. The book had been speaking two conventions: chapter T10’s prose said a Hilbert space has infinitely many dimensions, while chapter T15 computed its whole quantum brace in C² and the hilbert card gave the general definition. A survey of eleven sources found ten allowing finite dimensions, four naming Cⁿ or Rⁿ as a Hilbert space outright (Rudin, Conway, Kreyszig, MathWorld), and Nielsen and Chuang section 2.1.4 using the terms interchangeably in finite dimension. The two apparent holdouts do not define against it: Sakurai avoids the phrase rather than restricting it, and Watrous says he prefers “complex Euclidean space” precisely because “Hilbert space” is the more general notion. The book now speaks the wide convention everywhere. Do not claim that von Neumann called a finite-dimensional space a Hilbert space; he did not.

• CHSH inequality, Wikipedia — gives the 1969 Clauser-Horne-Shimony-Holt inequality, its local-hidden-variable bound of 2, and Tsirelson’s quantum bound of 2√2; it is “a constraint—on the statistical occurrence of “coincidences” in a Bell test” and “In practice, the inequality is routinely violated by modern experiments in quantum mechanics.” Its derivation “start[s] with the standard assumption of independence of the two sides,” the hidden variable λ being “drawn from a fixed distribution of possible states of the source,” and writes E(a, b) = ∫ A(a, λ)B(b, λ)ρ(λ) dλ, “where A and B are the outcomes,” so each outcome depends only on its own side’s setting and λ; “The maximum violation predicted by quantum mechanics is 2√2 (Tsirelson’s bound).”
• Approaching Tsirelson’s bound in a photon pair experiment, arXiv — Poh, Joshi, Ceré, Cabello and Kurtsiefer (2015): “a value 2.82759+-0.00051 is observed” and “the smallest distance with respect to Tsirelson’s bound ever reported, namely, 0.00084+-0.00051”

• MIT 18.783 Elliptic Curves (Fall 2025), Lecture 14: Elliptic curves over C (part I) — defines contour integrals and states that positive orientation of a circle is counter-clockwise.
• Dictionary of Scientific Biography, Augustin-Louis Cauchy — describes the 1825 memoir’s definition of integrals over arbitrary complex paths and its path-independence result; its bibliography lists “Cauchy, Mémoire sur les intègrales définies prises entre des limites imaginaires (Paris, 1825),” and “Cauchy finally introduced arbitrary closed integration paths, although not as an immediate consequence of his 1825 mémoire, which he did not remember until 1851 [1, 1st ser., XI, 328]; instead, he proved his integral theorem anew by means of what is now called Green’s formula.”

• Encyclopedia of Mathematics, Analytic function — states Cauchy’s theorem with the required domain condition and the equivalence of holomorphy and complex analyticity.
• MacTutor, Édouard Goursat — identifies Goursat’s removal of the derivative-continuity condition and dates his first paper to 1884.
• Dictionary of Scientific Biography, Édouard Goursat — records the 1884 Acta Mathematica paper and the corresponding 1900 Transactions of the AMS paper.
• Œuvres complètes d’Augustin Cauchy, 1st series, vol. X, table of contents (Mathdoc) — lists “Sur les intégrales qui s’étendent à tous les points d’une courbe fermée p.70-74 C. R., t. XXIII, p. 251 (3 août 1846).”

• NIST DLMF §1.10(iv), Residue Theorem — equation 1.10.8 states the residue theorem with the 2πi normalization and positive orientation.
• NIST DLMF §4.2(i), Logarithms — “the general logarithm is given by Ln z = ln z + 2kπi, where k is the excess of the number of times the path … crosses the negative real axis in the positive sense over the number of times in the negative sense.”

• NIST DLMF §1.10(iii), Laurent Series — defines residues as Laurent coefficients, classifies essential singularities (“if aₙ ≠ 0 for infinitely many negative n, then z₀ is an isolated essential singularity”); the exponential series used after substituting 1/z is DLMF 4.2.19, on the §4.2(iii) line.
• Encyclopedia of Mathematics, Essential singular point — states the infinitely-often form of Great Picard’s theorem with at most one exceptional finite value.
• NIST DLMF §4.2(iii), The exponential function — 4.2.19: exp z = 1 + z/1! + z²/2! + z³/3! + ⋯; “The function exp is an entire function of z, with no real or complex zeros.”
• NIST DLMF §1.10(iv), Residue theorem — “(1/2πi)∫_C f(z) dz = sum of the residues of f(z) within C.”

• Encyclopedia of Mathematics, Cauchy–Riemann equations — gives both real equations and their ∂/∂z̄ form.
• R. Michael Range, Some landmarks in the history of the tangential Cauchy–Riemann equations — traces the operators to Wirtinger’s 1926 paper and displays their definitions and the holomorphy condition — supports 1926.
• W. Wirtinger, Zur formalen Theorie der Funktionen von mehr komplexen Veränderlichen, Mathematische Annalen 97 (1927), 357–375 — Springer’s record for the printed volume — supports 1927 — the publisher’s own record dates the printed volume to 1927.

• MIT 18.783 Elliptic Curves (Spring 2013), Lecture 17 — gives the complex-integral formula, calls it an integer, and identifies it as the number of turns around the point.
• NIST DLMF §1.10(iv), Residue Theorem — supplies the 1/(2πi) normalization and the residue theorem for a simple closed contour (1.10.8), where “the path C is described in the positive sense”; the winding-number weighting of residues is on the Residue theorem line below.
• Winding number, Wikipedia — read as raw wikitext: “When counting the total number of turns, counterclockwise motion counts as positive, while clockwise motion counts as negative.”
• Residue theorem, Wikipedia — read as raw wikitext: “the line integral of f around γ is equal to 2πi times the sum of residues, each counted as many times as γ winds around the respective point.”
• Contour integration, Wikipedia — read as raw wikitext: “the integrals of holmorphic functions are invariant under deforming the contour, provided the deformation does not cross a singularity or branch cut.”

• NIST DLMF §1.10(iv), Phase Principle — equation 1.10.9 states the argument principle as N−P and specifies multiplicities.
• MIT 18.095, The notion of topological degree — the course record identifies winding number as the method used for a proof of the fundamental theorem of algebra.

• Encyclopedia of Mathematics, Analytic function — states the equivalence of holomorphy and complex analyticity and gives the Cauchy integral framework.
• NIST DLMF §1.10(iii), Laurent Series — defines meromorphic functions through isolated poles and analytic functions through Taylor expansions.

• NIST DLMF §1.10(iii), Laurent Series — states the annular two-sided expansion and identifies the residue coefficient.
• MacTutor, Pierre Alphonse Laurent — dates Laurent’s submission and Cauchy’s report to 1843 and documents Weierstrass’s unpublished 1841 proof: “He submitted a memoir for the Grand Prix of the Academy of Sciences of 1842, unfortunately after the final date for submission in 1843”; “Cauchy reported on Laurent’s entry Mémoire sur le calcul des variations, which contains the Laurent series for a complex function, on 20 May 1843.”

• Dictionary of Scientific Biography, Augustin-Louis Cauchy — documents the Turin memoirs, their later publication history, and Cauchy’s use of residues for definite integrals.
• NIST DLMF §1.10(iv), Residue Theorem — gives the modern theorem that converts a contour integral to a finite residue sum: “If f(z) is analytic within a simple closed contour C, and continuous within and on C—except in both instances for a finite number of singularities within C—then” (1/2πi)∫_C f(z) dz is the sum of the residues within C (1.10.8).
• Augustin-Louis Cauchy, MacTutor — “He began a study of the calculus of residues in 1826 in Sur un nouveau genre de calcul analogue au calcul infinitésimal.”
• Harold P. Boas, Cauchy’s residue sore thumb, arXiv:1701.04887 — read from the PDF: “Cauchy does not have the present terminology of line integrals (path integrals), and only in 1826 does he introduce the word “residue.””
• Contour integration, Wikipedia — read as raw wikitext: the worked example ∫ e^(itx)/(x² + 1) dx over the real line = πe^(−t) (t ≥ 0).

• Encyclopedia of Mathematics, Liouville theorems — states the theorem and records the Cauchy 1844 and Liouville 1847 history.
• MIT 18.04, Complex Variables with Applications — states Liouville’s theorem and derives the fundamental theorem of algebra from 1/p.

• MacTutor, Bernhard Riemann — dates the thesis to 1851 and describes its Riemann surfaces, conformal mappings, and connectivity.
• Encyclopedia of Mathematics, Riemann surface — explains single-valued uniformization, the 1851 result, and classification data by genus: “Closed Riemann surfaces are topologically completely determined by one number — the genus.”
• Branch point, Wikipedia — read as raw wikitext: “the function w = z^(1/2) has two branches”; “The logarithm can be made continuous by gluing together countably many copies, called sheets, of the complex plane along the branch cut.”
• Riemann surface, Wikipedia — read as raw wikitext: “a Riemann surface is a connected one-dimensional complex manifold”; “The 2-torus T² has many different Riemann surface structures, all of the form ℂ/(ℤ+τℤ) … These are called elliptic curves.”

• Harvard Math 55b course notes — states Aut(Ĉ) ≅ PSL₂(ℂ) and that these automorphisms send circles to circles.
• Blagoje Oblak, From the Lorentz Group to the Celestial Sphere — the published lecture notes identify the connected Lorentz group with the conformal transformations of the sphere.

• Encyclopedia of Mathematics, Conformal mapping — defines conformality and connects the simply connected case to the Riemann mapping theorem.
• Encyclopedia of Mathematics, Riemann theorem — states the mapping theorem and dates Riemann’s formulation to 1851.
• Cartographica Helvetica, The maps of Europe by Gerard Mercator — dates the world map to 1569 and identifies its cylindrical projection as conformal.

• Jacques Hadamard, An Essay on the Psychology of Invention in the Mathematical Field — the 1945 scan contains the quoted wording on page 123.
• University of Iowa, source history for the Hadamard quotation — transcribes Hadamard’s sentence and traces the related original wording to Painlevé’s 1900 Analyse des travaux scientifiques.

• Encyclopedia of Mathematics, Analytic function — states the formula in so many words — “if $D$ is an open domain whose boundary consists of a finite number of non-intersecting rectifiable curves (oriented positively with respect to $D$) and $f: D\to \mathbb C$ is holomorphic, then $f(z) = \frac{1}{2\pi i} \int_{\partial D} \frac{f (\zeta)}{\zeta-z}\, d\zeta \quad \forall z\in D$” — and carries the hypotheses (open domain, finitely many non-intersecting rectifiable boundary curves, positive orientation) that the popular one-line version drops.
• Cauchy’s integral formula, Wikipedia — read as raw wikitext: “it implies that a function that is holomorphic in an open set is in fact infinitely differentiable there. Furthermore, it is an analytic function, meaning that it can be represented as a power series. The proof of this uses the dominated convergence theorem and the geometric series.”
• Analytic function, Encyclopedia of Mathematics — “Cauchy’s integral theorem yields Cauchy’s integral formula, which expresses the values of an analytic function inside a domain in terms of its values on the boundary.”

Added in the V14 prose pass for the new card `cauchyformula`. The chapter’s sentence “One complex derivative forces infinitely many, equality with the function’s own power series, and complete determination inside a loop by the values on it” is footnoted 16.8, and 16.8’s first source is this same Encyclopedia of Mathematics page, so the third clause was already carried by a source the chapter cites; what 16.8’s claim line states is only the holomorphic/analytic equivalence and the definition of meromorphic, which is why the integral formula is given its own entry here rather than being read into 16.8.

• Green’s theorem, Wikipedia — states the theorem, dates it to George Green’s 1828 essay (Green “stated a similar result in an 1828 paper titled An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism”), says “A proof of the theorem was finally provided in 1851 by Bernhard Riemann in his doctoral dissertation,” and describes its relation to Stokes’ theorem.

• NIST Digital Library of Mathematical Functions — the authoritative chapter index organizes the named families and their definitions.
• Andrews, Askey and Roy, Special Functions — the publisher’s contents and preface describe the subject’s long use and the book’s treatment of these families.

• NIST DLMF §5.2, Gamma Function — defines gamma, its meromorphic continuation, factorial relation and simple poles: “It is a meromorphic function with no zeros, and with simple poles of residue (−1)ⁿ/n! at z = −n.”
• Encyclopedia of Mathematics, Gamma-function — records Euler’s 1729 introduction in a letter to Goldbach; it gives 1814 for Legendre’s symbol Γ and the name gamma-function (“proposed in 1814 by A.M. Legendre”).
• MacTutor, Earliest Uses of Function Symbols — transcribes the Γ notation from page 277 of Legendre’s 1811 volume.

• NIST DLMF §5.4, Special Values — gives the half-integer gamma values.
• NIST DLMF §5.5, Functional Relations — gives Γ(z+1)=zΓ(z).

• NIST DLMF §6.2, Exponential and Logarithmic Integrals — defines li by the Cauchy principal value.
• NIST DLMF §27.2, Functions of Number Theory — (eq. 27.2.3): states the prime number theorem in the form π(x) ∼ x/ln x, credits the 1896 proofs of Hadamard and de la Vallée Poussin, and names Gauss and Legendre for the conjecture. It does not use li or the words logarithmic integral.
• NIST DLMF §27.12, Asymptotic Formulas: Primes — (eq. 27.12.5): bounds |π(x) − li(x)| by O(x exp(−c(ln x)^½)), the sharper form from which π(x) ∼ li(x) follows. This is the DLMF section that ties π(x) to li(x).
• Logarithmic integral function, Wikipedia — read as raw wikitext: “The function 1/(ln t) has a singularity at t = 1, and the integral for x > 1 is interpreted as a Cauchy principal value”; “the prime number theorem states that: π(x) ∼ li(x).”

• OEIS A070769, Ramanujan–Soldner constant — gives the value and cites Soldner’s 1809 Théorie et tables d’une nouvelle fonction transcendante.
• Wikipedia, Logarithmic integral function — defines the Ramanujan–Soldner constant μ ≈ 1.45136 92348 as the unique positive zero of li(x), and gives Ramanujan’s more rapidly convergent series.
• Johann Georg von Soldner, Wikipedia — “He also published on the logarithmic integral function”; lists “Soldner, J. G. v. (1809). Théorie et tables d’une nouvelle fonction transcendante. München: Lindauer.”
• Soldner’s Constant, Wolfram MathWorld — Ramanujan calculated “μ = 1.45136380...,” while the correct value is 1.45136923488….

• NIST DLMF §5.2, Gamma Function — records that reciprocal gamma is entire with simple zeros at the nonpositive integers.

• NIST DLMF §4.14, Trigonometric Functions — supplies the sine zeros used in the elementary evaluation: “In ℂ the zeros of sin z are z = kπ, k ∈ ℤ” (the limiting value sinc(0) = 1 is on the Sinc function line).
• MIT OpenCourseWare, The Sampling Process — states that sinc is the ideal interpolation function in the sampling theorem.
• Sinc function, Wikipedia — read as raw wikitext: “In either case, the value at x = 0 is defined to be the limiting value” sinc(0) = lim sin(ax)/(ax) = 1.

• NIST DLMF §10.2, Bessel Functions — gives the defining series (10.2.2), from which J₀(0) = 1 and Jₙ(0) = 0 for n ≥ 1 follow; the page does not state those values itself.
• Bessel function, Wikipedia — read as raw wikitext: “Bessel’s equation arises when finding separable solutions to Laplace’s equation and the Helmholtz equation in cylindrical or spherical coordinates”; applications include “Electromagnetic waves in a cylindrical waveguide,” “Heat conduction in a cylindrical object,” “Modes of vibration of a thin circular or annular acoustic membrane (such as a drumhead …)”; “For integer or positive α, Bessel functions of the first kind are finite at the origin”; “For integer order α = n, J_n is often defined via a Laurent series for a generating function: e^{(x/2)(t − 1/t)} = Σ_n=−∞^∞ J_n(x) tⁿ.”
• Vibrations of a circular membrane, Wikipedia — the axisymmetric modes are set by the positive roots of J₀; the fundamental u₀₁ has α₀₁ = 2.40483.

• NIST DLMF §5.12, Beta Function — gives the beta integral and gamma quotient.

• NIST DLMF §5.15, Polygamma Functions — defines the polygamma functions ψ⁽ⁿ⁾, n = 1, 2, …, whose properties “follow straightforwardly by differentiation of properties of the psi function,” and gives their special values (digamma’s definition and values are on the §5.2 and §5.4 lines).
• Marek Wolf, 6+∞ New Expressions for the Euler–Mascheroni Constant (2025) — the peer-reviewed paper explicitly states that irrationality of γ remains unknown.
• NIST DLMF §5.2, Gamma function: definitions — 5.2.2: ψ(z) = Γ′(z)/Γ(z).
• NIST DLMF §5.4, Gamma function: special values — 5.4.12: ψ(1) = −γ; 5.4.14: ψ(n+1) = Σ_{k=1}^{n} 1/k − γ.

• NIST DLMF §4.13, Lambert W-Function — defines the inverse, its branches and the branch point W(−1/e) = −1 (the value Ω = W(1) is on the Omega constant line).
• Corless et al., On the Lambert W Function (1996) — the published paper’s author copy defines W and standardizes its branches and notation: “The text below is the same as that published in Advances in Computational Mathematics, Vol 5 (1996) 329 – 359”; authors R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth; “We use the letter W for this function, following early Maple usage.” and “An important part of this paper is our proposal of a standard notation for all the branches of W in the complex plane.”
• Omega constant, Wikipedia — read as raw wikitext: “It is the value of W(1), where W is Lambert’s W function”; Ω = 0.56714 32904 09783 87299 …

• Encyclopedia of Mathematics, Lindemann–Weierstrass theorem — states the required transcendence of e^α for nonzero algebraic α.
• NIST DLMF §4.13, Lambert W-Function — supplies Ωe^Ω=1.

• NIST DLMF §10.21, Zeros of Bessel Functions — defines the positive Bessel zeros j_ν,m and gives their properties and asymptotic expansions (numerical tables are referred elsewhere).
• MacTutor, Carl Ludwig Siegel — records Siegel’s 1929 result in these words: “for any non-zero algebraic integer r he showed that J0(r) is transcendental,” so no non-zero algebraic integer is a zero of J₀ (the statement for every nonzero root is on the Bessel function line).
• Vibrations of a circular membrane, Wikipedia — (§ The axisymmetric case, § Animations of several vibration modes): the axisymmetric modes are set by the positive roots of J₀, and the fundamental mode u₀₁ has α₀₁ = 2.40483.
• Wikipedia, Bessel function — its section on the zeros states that “Siegel proved in 1929 that when ν is rational, all nonzero roots of Jν(x) and J′ν(x) are transcendental,” citing Lorch and Muldoon, Int. J. Math. Math. Sci. 18 (1995) 551–560 (doi:10.1155/S0161171295000706).
• D. S. Karachalios, I. V. Gosea, Q. Zhang, A. C. Antoulas, Case study: Approximations of the Bessel function, arXiv:1801.03390 — read from the PDF: “The first 6 solutions of the equation J0(s) = 0, with 15 digits precision are: … 2.40482555769577.”

• NIST DLMF §7.2, Error Functions — defines erf and gives its normalization.
• NIST/SEMATECH e-Handbook, Normal Distribution — gives the normal density and cumulative-probability standardization used for the probability interpretation.
• Error function, Wikipedia — read as raw wikitext: “The defining integral cannot be evaluated in closed form in terms of elementary functions”; erf(x) = 2Φ(x√2) − 1; for a normal distribution with standard deviation σ and mean zero, erf(a/(σ√2)) “is the probability that the error of a single measurement lies between −a and a”; table: erf 1 = 0.842700793.

• NIST DLMF §9.2, Airy Functions — gives the Airy equation and Ai(0) special value.
• NIST DLMF §9.16, Physical Applications — documents Airy-function applications including quantum-mechanical turning-point problems; Ai “first appears as an integral in two articles by G.B. Airy on the intensity of light in the neighborhood of a caustic.”
• Airy function, Wikipedia — read as raw wikitext: “When x is positive, Ai(x) is positive, convex, and decreasing exponentially to zero, while Bi(x) is positive, convex, and increasing exponentially. When x is negative, Ai(x) and Bi(x) oscillate around zero”; “the Airy equation is the simplest second-order linear differential equation with a turning point (a point where the character of the solutions changes from oscillatory to exponential)”; “The Airy function underlies the form of the intensity near an optical directional caustic, such as that of the rainbow.”

• NIST DLMF §19.2, Legendre’s Integrals — defines K and E.
• NIST DLMF §19.6, Special Cases — gives K(0)=π/2.
• NIST DLMF §22.19, Physical Applications — gives the pendulum period as 4K(sin(α/2)).
• NIST DLMF §19.30, Lengths of Plane Curves — gives ellipse arc length using E, resolving the inherited K/E mix-up.

• NIST DLMF §7.2, Fresnel Integrals — defines S and C, and 7.2.9 gives lim_x→∞ C(x) = lim_x→∞ S(x) = ½.
• NIST DLMF §7.12(ii), Fresnel Asymptotics — gives the large-z asymptotic expansions of the auxiliary functions f and g, referring to 7.5.3–7.5.4 for C and S (the limits at infinity are on the §7.2 line).
• NIST DLMF §7.20, Mathematical Applications — identifies the parametric Cornu spiral.
• NIST DLMF §7.21, Physical Applications — connects Fresnel integrals and the Cornu spiral to diffraction.
• Euler spiral, Wikipedia — read as raw wikitext: named for “elastic springs (“Euler spiral,” 1744), graphical computations in light diffraction (“Cornu spiral,” 1874)”; “In 1874, Alfred Marie Cornu showed that diffraction intensity could be read off a graph of the spiral”; the diagram of a Cornu spiral is used for plane-wave diffraction from a knife-edge.
• R. D. Blandford and K. S. Thorne, Applications of Classical Physics, Ch. 8 “Diffraction,” §8.4.3 (Caltech Ph136, 2012) — for a straight edge, “Long before the occultation, we can approximate σmin = −∞ and σmax = +∞ …, so ∆Eσ is given by the arrow from (−1/2, −1/2) to (1/2, 1/2)”; and “The flux |ψ|2 ∝ |∆Eσ|2 is proportional to the squared length of the arrow on the Cornu spiral.”

• NIST DLMF §15.2, Gauss Series — gives the defining series, analytic continuation and polynomial cases.
• NIST DLMF §15.4, Special Cases — gives logarithm, arctangent and arcsine parameter choices.

• NIST DLMF §25.12, Polylogarithms — gives the definition and the special case “ζ(s) = Li_s(1)”; its reflection formula 25.12.6, Li₂(x) + Li₂(1 − x) = π²/6 − (ln x) ln(1 − x), gives Li₂(½) = π²/12 − (ln 2)²/2 at x = ½ (Li₁ is on the Polylogarithm line).
• Polylogarithm, Wikipedia — read as raw wikitext: “Li₁(z) = −ln(1−z)” and “Li₁(½) = ln 2.”

• MathWorld, Jacobi Theta Functions — explicitly gives θ₃(e^−π)=π^(1/4)/Γ(3/4); “The Jacobi theta functions are the elliptic analogs of the exponential function, and may be used to express the Jacobi elliptic functions.”
• NIST DLMF §5.5, Functional Relations — Euler’s reflection formula makes the two displayed gamma forms identical.
• NIST DLMF §20.9, Relations to Other Functions — connects theta functions to elliptic functions and the Riemann zeta function.
• Theta function, Wikipedia — read as raw wikitext: “ϑ(z; τ) = Σ_{n=−∞}^{∞} exp(πin²τ + 2πinz)”; “The relation ϑ(0; −1/τ) = (−iτ)^½ ϑ(0; τ) was used by Riemann to prove the functional equation for the Riemann zeta function, by means of the Mellin transform.”

• NIST DLMF §18.3, Orthogonal Polynomials: Definitions — defines the classical families and their weights (Table 18.3.1: the orthogonality intervals and weight functions) (the link formerly pointed to the chapter’s table of contents).
• NIST DLMF §18.39, Applications in the Physical Sciences — gives the oscillator and hydrogen-atom eigenfunction applications.
• NIST DLMF §14.30, Spherical and Spheroidal Harmonics — relates spherical harmonics to the Ferrers (associated Legendre) functions 𝖯_l^m(cos θ); it does not name the Legendre polynomials themselves.
• NIST DLMF §18.3, Classical orthogonal polynomials: definitions — “Table 18.3.1 provides the traditional definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and standardization”; its weights: Legendre w(x) = 1, Laguerre w(x) = e^(−x)x^α, Hermite w(x) = e^(−x²).

• NIST DLMF §8.2, Incomplete Gamma Functions — defines upper and lower functions and their sum.
• NIST DLMF §8.4, Special Values — gives the error-function identity.
• NIST DLMF §8.23, Statistical Applications — identifies chi-square distributions as particular regularized incomplete-gamma forms.

• Cambridge University Press, Special Functions — Preface — prints the sentence, author list and attribution to Turán; print publication year 1999.

• Bohr–Mollerup theorem, Wikipedia — proved in 1922 by Harald Bohr and Johannes Mollerup; “Γ(x) is the only function that satisfies f(x + 1) = x f(x) with log(f(x)) convex and also with f(1) = 1,” for positive f on x > 0.
• Gamma function, Wikipedia — read as raw wikitext: “it is not the only analytic function that extends the factorial, as one may add any analytic function that is zero on the positive integers, such as k sin(mπx) for an integer m.”; “Perhaps the best-known value of the gamma function at a non-integer argument is Γ(1/2) = √π, which can be found … simply by making the substitution t = u² in the integral definition of the gamma function, resulting in a Gaussian integral”; with t = cx², “if z = 1/2, c = 1 we get Γ(1/2) = 2∫₀^∞ e^(−t²) dt = √π” (formulas as typeset on the page).

• NIST DLMF §5.5(ii), Functional Relations: Reflection — equation 5.5.3 states Γ(z)Γ(1−z) = π/sin(πz) for z ≠ 0, ±1, …; §5.5(i) carries the recurrence and §5.5(iv) the Bohr–Mollerup theorem.

Own arithmetic checked: sin(π/3) = √3/2, so the reflection formula at z = 1/3 gives Γ(1/3)Γ(2/3) = 2π/√3 ≈ 3.6275987, which matches the direct product of the two gamma values to full double precision. Added in the V14 prose pass because Chapter ᚐ(A)’s paragraph 7 invokes “the reflection formula” by name, and no reference already in the chapter states it: r-17.3 cites DLMF §5.5 but only for the recurrence Γ(z+1) = zΓ(z). Card ai0 invokes the same identity under 17.24 and should cite this entry too.

• NIST DLMF §10.2, Bessel Functions: Definitions — equation 10.2.1 gives Bessel’s differential equation z²(d²w/dz²) + z(dw/dz) + (z² − ν²)w = 0, and 10.2.2 gives the ascending series defining Jν(z), the Bessel function of the first kind.
• NIST DLMF §9.2, Airy Functions: Differential Equation — equation 9.2.1 gives the Airy equation d²w/dz² = zw, names Ai(z) and Bi(z) as its standard solutions, records that all solutions are entire, and gives Ai(0) = 1/(3^(2/3)Γ(2/3)) = 0.35502 80538….
• NIST DLMF §7.2, Error Functions and Fresnel Integrals: Definitions — 7.2.1 gives erf z = (2/√π)∫₀ᶻ e^(−t²) dt, 7.2.2 gives erfc z = (2/√π)∫ᶻ^∞ e^(−t²) dt = 1 − erf z, and 7.2.7 and 7.2.8 give the Fresnel integrals C(z) = ∫₀ᶻ cos(πt²/2) dt and S(z) = ∫₀ᶻ sin(πt²/2) dt.

Added for Q17.4, answered by the default. The glossary carried family cards for hypergeometric, polylog, theta, orthogonal polynomials and incomplete gamma — none of which the prose mentions — and none for Bessel, Airy, error or Fresnel, which it does. The four value cards (j01, j1zero, ai0, erf1, fresnelS) each cite an entry for their own value, and no entry carried the defining equations the family cards need. This one does.

• Cauchy principal value, Wikipedia — read as raw HTML: “the Cauchy principal value, named after Augustin-Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. In this method, a singularity on an integral interval is avoided by limiting the integral interval to the non-singular domain,” with the symmetric-limit formulation and the distribution p.v.(1/x).
• Cauchy Principal Value, MathWorld — the symmetric-exclusion limit written out.
• Alan Reynolds, Introduction to Complex Analysis (PHYS30041 course notes): Cauchy principal value — “By approaching the singularity asymmetrically in other ways, we can make the value of the integral equal to any real number we choose!”
• Logarithmic integral function, Wikipedia — read as raw wikitext: “The function 1/(ln t) has a singularity at t = 1, and the integral for x > 1 is interpreted as a Cauchy principal value”; “the prime number theorem states that: π(x) ∼ li(x).”

• Lambert W function, Wikipedia — defines W as the inverse of w e^w, its branches W0 and W-1, and the branch point at (-1/e, -1); “It is useful in combinatorics, for instance, in the enumeration of trees” and “also occurs in the solution of delay differential equations, such as y′(t) = a y(t − 1).”
• Lambert W functions, B. Ćurgus, Western Washington University — carries x^x solved by W: “Solve x^x = y for x > 0 and appropriate y > 0.” with solution “x = exp(W0(ln y)) = ln y / W0(ln y)”
• Corless, Knuth and Lambert W, Nick Higham — carries the 1996 date of the canonical reference: “Robert M. Corless, Gaston N. Gonnet, D. E. G. Hare and David J. Jeffrey and Donald Knuth, On the Lambert W Function, Adv. in Comput. Math. 5, 329-359, 1996”

• Beta function, Wikipedia — gives Euler’s integral definition and the identity relating B to the gamma function; “it was the first known scattering amplitude in string theory, first conjectured by Gabriele Veneziano.”
• Introduction to the Gamma Function, P. Sebah and X. Gourdon — carries Euler’s integral and B(½, ½): “Euler gave (30) in 1730” and “8.1 Special values B(1/2, 1/2) = π”
• Beta distribution, Wikipedia — carries the normalization: “The beta function, B, is a normalization constant to ensure that the total probability is 1.”

• Digamma function, Wikipedia — defines ψ = Γ’/Γ, gives ψ(1) = -γ and the harmonic-number recurrence.
• §5.11 Asymptotic Expansions, NIST Digital Library of Mathematical Functions — carries the two expansions side by side: 5.11.1 “Ln Γ(z) ∼ (z−1/2) ln z − z + 1/2 ln(2π) + Σ B2k/(2k(2k−1)z^(2k−1))” and 5.11.2 “ψ(z) ∼ ln z − 1/(2z) − Σ B2k/(2k z^(2k))”

• Euler spiral, Wikipedia — describes the Cornu spiral / Euler spiral, its curvature-proportional-to-arc-length property, and Cornu’s diffraction application; “This curve is also referred to as a clothoid or Cornu spiral”; “They are also widely used in railway and highway engineering to design transition curves between straight and curved sections of railways or roads,” so that “the centripetal acceleration increases smoothly with the traveled distance” instead of causing “lateral jerk.”
• The Cornu Spiral, HyperPhysics, Georgia State University — carries the two limit points: “The eyes of the spirals at (.5,.5) and (-.5, -.5) represent image plane points at infinity.”

• NIST Digital Library of Mathematical Functions, §25.2 Definition and Expansions — DLMF states the defining series for Re(s)>1 and says the continuation is meromorphic with only a simple pole at s=1 and residue 1.
• NIST DLMF §1.10, Functions of a complex variable — “If f₂(z), analytic in D₂, equals f₁(z) on an arc in D = D₁ ∩ D₂, or on just an infinite number of points with a limit point in D, then they are equal throughout D and f₂(z) is called an analytic continuation of f₁(z).”

• NIST Digital Library of Mathematical Functions, §25.6 Integer Arguments — equations 25.6.1 and 25.6.2 list these values and the general even-value formula.
• MIT 18.785 Number Theory I (Fall 2015), Lecture 16 — read from the PDF: “We can determine the value of ζ(0) via the functional equation. We know that ζ(s) has a pole of residue 1 at s = 1 … thus ζ(0) = −1/2.”
• Jeffrey C. Lagarias, Euler’s constant: Euler’s work and modern developments, arXiv:1303.1856 — read from the PDF: “In his 1755 book on differential calculus Euler [E212, Part II, Chap. 6, Arts. 141–153] gave closed forms for ζ(2n) explicitly expressed in terms of the Bernoulli numbers.”
• Euler Archive, E41, De summis serierum reciprocarum — “Euler finds the values of ζ(2n), where ζ is now named as the Riemann zeta function”; Written Date 1735, Published Date 1740.
• Bernoulli number, Wikipedia — its table gives B₆ = 1/42.
• Riemann zeta function, Wikipedia — read as raw wikitext: “For odd positive integers, no such simple expression is known.”

• NIST Digital Library of Mathematical Functions, §25.2 — the zeta-function section includes the Euler product representation and its domain.
• Riemann zeta function, Wikipedia — read as raw wikitext: “The proof of Euler’s identity uses only the formula for the geometric series and the fundamental theorem of arithmetic. Since the harmonic series, obtained when s = 1, diverges, Euler’s formula, which becomes ∏_p p/(p−1), implies that there are infinitely many primes.”

• NIST Digital Library of Mathematical Functions, §25.4 Reflection Formulas — equations 25.4.2–25.4.4 give the reflection formula and xi-function definition/symmetry.
• NIST Digital Library of Mathematical Functions, §25.10 Zeros — DLMF explicitly states that the functional equation implies ζ(-2n)=0 and calls these the trivial zeros.
• Riemann zeta function, Wikipedia — read as raw wikitext: “The xi function is an entire function whose zeros are exactly the non-trivial zeros of the Riemann zeta function”; “The functional equation was established by Riemann in his 1859 paper “On the Number of Primes Less Than a Given Magnitude.””

• NIST Digital Library of Mathematical Functions, §25.2 — DLMF states the simple pole and residue 1: “It is a meromorphic function whose only singularity in ℂ is a simple pole at s = 1, with residue 1.”
• NIST Digital Library of Mathematical Functions, §5.2 Gamma Function — the analogous reciprocal-at-a-pole convention is stated for 1/Γ; the value here follows directly from the Laurent expansion of ζ at its simple pole.
• Harmonic series (mathematics), Wikipedia — read as raw wikitext: “The divergence of the harmonic series was first proven in 1350 by Nicole Oresme.”

Part 2 item on 1/ζ(1): we retain the existing wording. This is the value of the analytically continued reciprocal function at the removable singularity, not a claim that the divergent series has a finite value.

• A. M. Odlyzko, Tables of zeros of the Riemann zeta function — (file zeros1, first three lines): the first three nontrivial heights are 14.134725142, 21.022039639 and 25.010857580.
• Riemann hypothesis, Wikipedia — read as raw wikitext: “J. P. Gram (1903) used the Euler–Maclaurin formula and discovered Gram’s law. He showed that all 10 zeros with imaginary part at most 50 range lie on the critical line”; 2004, 10¹³: “Xavier Gourdon (2004) and Patrick Demichel used the Odlyzko–Schönhage algorithm”; “The most extensive computer search by Platt and Trudgian for counterexamples of the Riemann hypothesis has verified it for |t| ≤ 3.0001753328×10¹²,” the page’s footnote citing Platt and Trudgian (2021) (the arXiv preprint is of April 2020, the journal paper of 2021); its table of zero computations opens with 1859?, 3 zeros: “B. Riemann used the Riemann–Siegel formula (unpublished, but reported in Siegel 1932).”
• Montgomery’s pair correlation conjecture, Wikipedia — read as raw wikitext: a conjecture “made by Hugh Montgomery (1973)”; “(The factor 2π/log(T) is a normalization factor that can be thought of informally as the average spacing between zeros with imaginary part about T.)”; “which, as Freeman Dyson pointed out to him, is the same as the pair correlation function of random Hermitian matrices.”
• C. L. Siegel, On Riemann’s Nachlass for Analytic Number Theory (1932), tr. Barkan and Sklar, arXiv:1810.05198 — “For the smallest positive zero he finds the value of a1 = 14.1386; Gram has calculated the value 14.1347, different by less than 3 parts per thousand.”

• NIST Digital Library of Mathematical Functions, §25.10 Zeros — DLMF states that there are infinitely many zeros on the critical line (because Z(t) changes sign infinitely often, ζ(½ + it) has infinitely many zeros with t real) and describes their computation via the function Z(t) and the Riemann–Siegel formula; it does not name Hardy.
• Platt and Trudgian, The Riemann hypothesis is true up to 3·10^12 (arXiv:2004.09765) — Theorem 1 reads “The Riemann hypothesis is true up to height 3 000 175 332 800. That is, the lowest 12 363 153 437 138 non-trivial zeroes ρ have ℜρ = 1/2.”
• Riemann hypothesis, Wikipedia — read as raw wikitext: “J. P. Gram (1903) used the Euler–Maclaurin formula and discovered Gram’s law. He showed that all 10 zeros with imaginary part at most 50 range lie on the critical line”; 2004, 10¹³: “Xavier Gourdon (2004) and Patrick Demichel used the Odlyzko–Schönhage algorithm”; “The most extensive computer search by Platt and Trudgian for counterexamples of the Riemann hypothesis has verified it for |t| ≤ 3.0001753328×10¹²,” the page’s footnote citing Platt and Trudgian (2021) (the arXiv preprint is of April 2020, the journal paper of 2021).

• Waldschmidt, Zeta Values — Theorem 4 identifies Apéry’s 1978 irrationality theorem and gives ζ(3)’s value.
• OEIS A002117 — the entry records Apéry’s 1978 irrationality result and that the rational-multiple-of-π³ question is unresolved.

• Prime-counting function, Encyclopaedia/standard reference summary — the definition and the listed values are given explicitly; its table runs to π(10²⁹), and “The values for 10²⁷, 10²⁸, and 10²⁹ were announced by David Baugh and Kim Walisch in 2015, 2020, and 2022, respectively.” (exponents as typeset on the page).

• MacTutor, Jacques Hadamard biography — MacTutor dates the proof to 1896 and credits Hadamard and de la Vallée Poussin using complex analysis.
• NIST Digital Library of Mathematical Functions, §25.10 — DLMF connects the absence of zeros on Re(s)=1 with the prime number theorem and cites both 1896 proofs.
• Riemann Hypothesis, Wolfram MathWorld — “Wiener showed that the prime number theorem is literally equivalent to the assertion that the Riemann zeta function ζ(s) has no zeros on σ = 1.”

• Wolfram MathWorld, Mertens Conjecture — the page states the RH equivalence and records the disproof of the stronger conjecture: “The Mertens conjecture was proved false by Odlyzko and te Riele (1985). Their proof is indirect and does not produce a specific counterexample.”
• Wikipedia, Mertens conjecture — the history and Odlyzko–te Riele 1985 disproof are stated.

• Encyclopedia of Mathematics, Dirichlet L-function — the entry dates Dirichlet’s introduction of L-functions to 1837 and states their role in primes in arithmetic progressions.
• MIT 18.785 Lecture Notes, Dirichlet L-functions — the notes state Dirichlet’s 1837 theorem and the coprime progression condition.

• NIST Digital Library of Mathematical Functions, §25.10 Zeros — DLMF defines the real Hardy Z-function and the Riemann–Siegel theta function used to locate critical-line zeros.
• Wikipedia, Riemann–Siegel theta function — the page gives the Gram-point convention and the first point near 17.8456; “Today we know that, in the long run, Gram’s law fails for about 1/4 of all Gram-intervals to contain exactly 1 zero of the Riemann zeta function.”
• Gram Point, Wolfram MathWorld — a Gram point gₙ satisfies θ(gₙ) = nπ, numbered from n = 0; its table gives g₀ = 17.8455995404.

• NIST Digital Library of Mathematical Functions, §25.2 and §25.6 — the Euler product and ζ(2)=π²/6 are stated in the DLMF sections.
• Own computation — At p=2, (1-2^-2)^-1=(1-1/4)^-1=4/3; multiplying ζ(2) by ∏p(1-p^-2) cancels the product exactly.
• NIST DLMF §25.6, Zeta function: integer arguments — 25.6.1: ζ(2) = π²/6.

• Clay Mathematics Institute, Riemann’s 1859 Manuscript — “Just six manuscript pages in length,” first published in the Monatsberichte der Berliner Akademie, November 1859.
• On the Number of Prime Numbers less than a Given Quantity, translated by David R. Wilkins (Clay Mathematics Institute) — carries the epigraph verbatim — “it is very probable that all roots are real. Certainly one would wish for a stricter proof here; I have meanwhile temporarily put aside the search for this after some fleeting futile attempts, as it appears unnecessary for the next objective of my investigation.” — and the argument confining the roots to the closed strip.
• Wikipedia, On the Number of Primes Less Than a Given Magnitude — calls it “a seminal 9-page paper” published in the November 1859 Monatsberichte.

The two live sources count different objects: Clay counts Riemann’s handwritten manuscript at six pages, Wikipedia the printed article at nine. Neither supports the “eight pages” the text used to give, so the text now says only that the paper runs to fewer than ten pages, which both sources support.

• Wikipedia, 1 + 2 + 3 + 4 + ⋯ — “The regularization of 1 + 2 + 3 + 4 + ⋯ is also involved in computing the Casimir force for a scalar field in one dimension,” and the same value is what fixes the twenty-six dimensions of bosonic string theory.
• Wikipedia, Casimir effect — the parallel-plate derivation runs through “ζ(−3) = 1/120” and yields ⟨E⟩/A = −ħcπ²/720a³.

The page had attributed the two-metal-plate calculation to ζ(−1) = −1/12. That is the one-dimensional scalar case; the plates need ζ(−3). Both are now named.

• Clay Mathematics Institute, Millennium Problems — “the Board of Directors of CMI designated a $7 million prize fund for the solutions to these problems, with $1 million allocated to the solution of each problem,” and the Riemann Hypothesis is one of the seven listed.
• Platt and Trudgian, The Riemann hypothesis is true up to 3·10^12 (arXiv:2004.09765), §3.1 — “Of course, on the Riemann hypothesis much more is known: Schoenfeld proved that |ψ(x) − x| ≤ (1/8π)√x log²x, (x > 59),” a sharper error term than the unconditional bounds the same section collects.

• Rodgers and Tao, The De Bruijn–Newman constant is non-negative (Forum of Mathematics Pi 8, 2020) — the abstract defines Λ by the heat-flow deformation, states that the Riemann hypothesis is equivalent to Λ ≤ 0, and proves Newman’s conjecture Λ ≥ 0.
• Platt and Trudgian, The Riemann hypothesis is true up to 3·10^12 (arXiv:2004.09765), §3.4 — “The Riemann hypothesis is equivalent to the statement that Λ ≤ 0. Rodgers and Tao proved that Λ ≥ 0.,” with Corollary 2 giving Λ ≤ 0.2.

• NIST Digital Library of Mathematical Functions, §1.10(ii) Analytic Continuation — states the uniqueness in so many words — “Let f1(z) be analytic in a domain D1. If f2(z), analytic in D2, equals f1(z) on an arc in D = D1 ∩ D2, or on just an infinite number of points with a limit point in D, then they are equal throughout D and f2(z) is called an analytic continuation of f1(z).”

Added in the V14 prose pass for the new card `analcont`. Chapter ᚑ(O)’s sentence “That is analytic continuation: a well-behaved function of a complex variable is so rigid that knowing it on a small patch fixes it everywhere” carried no footnote of its own, and 18.1 (DLMF §25.2) supplies only the fact that ζ is continued, not the uniqueness theorem that makes the continuation the function rather than a function. This entry carries the theorem.

• Wolfram MathWorld, Riemann Hypothesis — states “In 1914, Hardy proved that an infinite number of values for t can be found for which ζ(1/2 + it) = 0 and t is real (Havil 2003, p. 213),” and immediately adds “However, it is not known if all nontrivial roots satisfy Re(s) = 1/2.”

Added in the V14 prose pass. The chapter’s sentence “Hardy proved in 1914 that infinitely many zeros are on the line” was footnoted 18.8 alone, and the V12 sweep had already recorded that neither of 18.8’s sources carries the attribution or the year — DLMF §25.10 says only that ζ(½ + it) has infinitely many zeros with t real, naming nobody, and the full text of Platt and Trudgian never mentions Hardy. This entry supplies both. 18.8 continues to carry the verification height, which is what it does support.

• Euler Archive / University of the Pacific, E72 Variae observationes circa series infinitas — the record gives “Written Date 1737,” “Published Date 1744,” the citation “Commentarii academiae scientiarum Petropolitanae, Volume 9, pp. 160-188,” and the content summary “This is the paper in which Euler product expansions first appeared.”

Added in the V14 prose pass to carry the 1737 date the chapter uses three times. The V12 sweep recorded that the date rested on two sources a reader cannot open: originalsources.com returns a sign-in page under HTTP 200, and the 1930 Nature note gives no detail. The Euler Archive record is open, names the paper, and dates it directly. This is the companion of 18.3’s E41 record for the 1734 Basel result.

• 1 + 2 + 3 + 4 + ⋯, Wikipedia — read as raw HTML, § Ramanujan summation: “Ramanujan wrote in his second letter to G. H. Hardy, dated 27 February 1913: ‘Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich’s Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + ⋯ = −1/12 under my theory.” So the date, the addressee and the identity are all on that page.

Added 17 September 2026 for the B6 pass. The #zetam1 card said “Ramanujan states it in his letter to Hardy of 27 February 1913” and footnoted 18.18, which is the Casimir entry: its claim line is about the regularized value used for the Casimir force and neither of its two sources mentions Ramanujan, Hardy or 1913. A reader following that marker landed on a physics calculation. Grepping references.html for “Ramanujan” before this entry existed returned r-5.9, r-6.11, r-6.12, r-6.31 and r-17.5, none of them about the letter.

• Bernhard Riemann, On the Number of Prime Numbers less than a Given Quantity, translated by David R. Wilkins, at the Clay Mathematics Institute — fetched, read as extracted PDF text. The epigraph appears word for word: “One now finds indeed approximately this number of real roots within these limits, and it is very probable that all roots are real. Certainly one would wish for a stricter proof here; I have meanwhile temporarily put aside the search for this after some fleeting futile attempts, as it appears unnecessary for the next objective of my investigation.” The Clay page hosting it credits the transcription and translation to David Wilkins by permission and the manuscript images to the Niedersächsische Staats- und Universitätsbibliothek Göttingen.

Added 17 September 2026 to source the chapter epigraph, which carried no marker. This is the strongest of the fourteen: the quotation was checked against the translation the footer names, on the site that publishes it, and matches exactly, including the footer’s gloss that the roots are those of Riemann’s transformed function.

• Fundamental theorem of arithmetic, Wikipedia — read as raw HTML: “The fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of Euclid’s Elements”; “1 is not considered a prime number: if 1 were prime, then factorization into primes would not be unique”; and “the theorem does not hold for algebraic integers,” which is what unique factorization domains name.
• Fundamental Theorem of Arithmetic, MathWorld — the statement: “every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes.”

• E352 Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques, Euler Archive (University of the Pacific) — read raw: “Euler evaluates several values of the now-named Riemann zeta function, and finds some functional relations for it. ... Published Date 1768 ... Written Date 1749.”
• Riemann zeta function, Wikipedia — read raw: “The functional equation was established by Riemann in his 1859 paper “On the Number of Primes Less Than a Given Magnitude” and used to construct the analytic continuation in the first place..”
• Riemann hypothesis, Wikipedia — read raw: “J. P. Gram (1903) used the Euler–Maclaurin formula and discovered Gram’s law. He showed that all 10 zeros with imaginary part at most 50 range lie on the critical line with real part 1/2”; “Trudgian (2011) showed that both Gram’s law and Rosser’s rule fail in a positive proportion of cases.”; Turing’s check “that Z has the correct sign at several consecutive Gram points” “is still the usual method used.”
• Mertens conjecture, Wikipedia — read raw: “It was conjectured by Thomas Joannes Stieltjes, in an 1885 letter to Charles Hermite (reprinted in Stieltjes (1905)), and again in print by Mertens (1897)”; “In 2006, Kotnik and te Riele improved the upper bound and showed that there are infinitely many values of n for which m(n) › 1.2184, but without giving any specific value for such an n..”
• Prime number theorem, Wikipedia — read raw: “Carl Friedrich Gauss considered the same question at age 15 or 16 “in the year 1792 or 1793,” according to his own recollection in 1849.”; “Adrien-Marie Legendre conjectured in 1797 or 1798 that π(a) is approximated by the function a / (A log a + B), where A and B are unspecified constants. In the second edition of his book on number theory (1808) he then made a more precise conjecture”; “In two papers from 1848 and 1850, the Russian mathematician Pafnuty Chebyshev attempted to prove the asymptotic law of distribution of prime numbers..”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Prime number theorem, Wikipedia — read raw: “Several different proofs of it were found, including the “elementary” proofs of Atle Selberg (1949) and Paul Erdős (1949).”; “Carl Friedrich Gauss considered the same question at age 15 or 16 “in the year 1792 or 1793,” according to his own recollection in 1849.”
• Riemann hypothesis, Wikipedia — read raw: “their occurrence in his explicit formula for the number of primes π(x) less than or equal to a given number x, which he published in his 1859 paper “On the Number of Primes Less Than a Given Magnitude..””; “the dominant term li(x) comes from the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros”; “This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their “expected” positions.”; “Helge von Koch proved that the Riemann hypothesis implies the “best possible” bound for the error of the prime number theorem.” (read 29 September 2026 from the cached copy of 24 September).
• De Bruijn–Newman constant, Wikipedia — read raw: “De Bruijn showed in 1950 that H has only real zeros if λ≥1/2, and moreover, that if H has only real zeros for some λ, H also has only real zeros if λ is replaced by any larger value.”; “Newman proved in 1976 the existence of a constant Λ for which the “if and only if” claim holds; and this then implies that Λ is unique.”; “Brad Rodgers and Terence Tao proved that Λ ≥ 0, so the Riemann hypothesis is equivalent to Λ = 0..”
• On the Number of Primes Less Than a Given Magnitude, Wikipedia — read raw: “is a seminal 9-page paper by Bernhard Riemann published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin..”
• 1 − 2 + 3 − 4 + ⋯, Wikipedia — read raw: “In a 1749 report, Leonhard Euler acknowledges that the series diverges but prepares to sum it anyway.”
• Riemann, On the Number of Prime Numbers less than a Given Quantity, tr. D. R. Wilkins (Clay Mathematics Institute) — read from the PDF: “it follows that the function ξ(t) can only vanish if the imaginary part of t lies between ½i and −½i.”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• RFC 6090, Fundamental Elliptic Curve Cryptography Algorithms — Section 3 names the point at infinity as the identity (“a special point (@,@) that represents the identity element (which is called the “point at infinity”)”) and says “The point at infinity is its own inverse”; Appendix F.2 says it is “regarded as equivalent to the homogenous coordinates (0, 1, 0).”
• Elliptic curve, projective group law — the projective equation has the unique point O = [0:1:0] at infinity, which is the identity of the group.

• Ben Lynn, Elliptic Curves: Group of Points (Stanford crypto notes) — describes the projective point at infinity, the line-through-two-points construction, and the abelian group of points: in (projective) Weierstrass form “an elliptic curve always contains exactly one point of infinity, (0, 1, 0) (‘the point at the ends of all lines parallel to the y-axis’)”; “every line intersects the curve at exactly three points (counting multiplicities)”; and “It can be verified that P + Q = Q + P, P + O = P, (−P) + P = O,” with O taken to be the point at infinity (0, 1, 0).
• RFC 6090, Fundamental Elliptic Curve Cryptography Algorithms — gives the affine addition and inverse formulas and the exceptional point-at-infinity cases.

• Elliptic curve, Encyclopedia of Mathematics — (§ Elliptic curves over non-closed fields): “If k is finitely generated over its prime subfield, then X(k) is a finitely-generated group (the Mordell–Weil theorem),” and “The group X(ℚ) is isomorphic to F ⊕ X(ℚ)ₜ, where X(ℚ)ₜ is a finite Abelian group and F is a free Abelian group of a certain finite rank r.”
• Elliptic curve rank — writes E(Q) as Z^r plus its finite torsion group and defines r as the rank.

• Poonen, Rational points on varieties, section 8.3 — identifies 32A2 as y² = x³ − x, states rank 0 and torsion Z/2 × Z/2, and concludes that there are four rational points.
• Keith Conrad, The Congruent Number Problem (PDF) — shows that n is not congruent exactly when the only rational points of y² = x³ − n²x have y = 0, namely (0, 0), (n, 0) and (−n, 0); so for n = 1 the curve y² = x³ − x has only those three affine points (with the point at infinity, four).

• LMFDB 32.a3 (Cremona 32a2), the curve y² = x³ − x — gives conductor 32, discriminant 64, j-invariant 1728 = 2⁶ · 3³, rank 0 and torsion Z/2 × Z/2 for this curve.
• Elliptic curve invariants — gives the short Weierstrass discriminant −16(4a³ + 27b²), writes j as 1728·g′₂³/(g′₂³ − 27g′₃²) in lattice invariants (the a, b form is on the j-invariant line), and says the j-invariant specifies isomorphism classes of elliptic curves; the curve “is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition 4a³ + 27b² ≠ 0, that is, being square-free in x.),” and “Since the curve is smooth, hence continuous, it can be shown that this point at infinity is the identity element of a group structure.”
• Own computation — Substitute a = −1 and b = 0: Δ = −16(4(−1)^3) = 64 and j = 1728·4(−1)^3/(4(−1)^3) = 1728 = 12^3.
• j-invariant, Wikipedia — read as raw wikitext: “In case of a short Weierstraß elliptic curve y² = x³ + ax + b, this formula simplifies to” j = 1728·4a³/(4a³ + 27b²).
• Complex multiplication, Wikipedia — read as raw wikitext: CM “is the theory of elliptic curves E that have an endomorphism ring larger than the integers”; curves Y² = 4X³ − aX have “two conjugate order-4 automorphisms” sending Y → ±iY, X → −X; “Kronecker conjectured – in what became known as the Kronecker Jugendtraum – that every abelian extension of K could be obtained by the (roots of the) equation of a suitable elliptic curve with complex multiplication”; “j(Z[i]) = j(i) = 1728.”
• LMFDB, elliptic curve 37.a1 (Cremona label 37a1) — minimal Weierstrass equation y² + y = x³ − x; “This elliptic curve is semistable. There is only one prime p of bad reduction”; analytic rank 1, so L(E, 1) = 0.

• Poonen, Rational points on varieties, section 8.3 — states that the curve 32A2 has torsion Z/2 × Z/2 and four rational points.
• RFC 6090, Fundamental Elliptic Curve Cryptography Algorithms — states that a point with y = 0 doubles to the point at infinity and that the point at infinity is the identity.

• Hasse’s theorem for elliptic curves — states |#E(Fq) − (q+1)| ≤ 2√q.
• Own computation — For x = 0,1,4 modulo 5, x³−x = 0 and y = 0; for x = 2 the value is 1, giving y = ±1; for x = 3 the value is 4, giving y = ±2. Thus there are 7 affine points plus O, so #E(F5)=8 and |8−6|=2≤2√5.
• Elliptic curve, Wikipedia — read as raw wikitext: a_p = p + 1 − #E(F_p); the curve’s L-function “is defined as an Euler product, with one factor for every prime number p,” built from these a_p, and the proof of the Shimura–Taniyama–Weil conjecture “implies that its L-function is the L-function of a modular form whose analytic continuation is known.”

• Elkies, Best known conductors for elliptic curves of given rank — lists rank 1, curve [0 0 1 −1 0], conductor 37, and marks it as the smallest conductor found by exhaustive search for that rank.
• Poonen, Rational points on varieties, section 8.3 — Poonen’s survey of rational points on curves; read again on 23 September 2026, it does not display the 37a1 model, which the LMFDB 37a1 line carries.
• Christopher Skinner, A converse to a theorem of Gross, Zagier, and Kolyvagin (Annals of Mathematics, 2020) — for a semistable curve with suitable reduction, proves that algebraic rank 1 and finite Ш give analytic rank 1, a converse to the Gross–Zagier/Kolyvagin theorem; the page does not discuss 37a1, whose L(1) = 0 is on the LMFDB lines.
• LMFDB, elliptic curve 37.a1 (Cremona label 37a1) — minimal Weierstrass equation y² + y = x³ − x; “This elliptic curve is semistable. There is only one prime p of bad reduction”; analytic rank 1, so L(E, 1) = 0; its Mordell-Weil generators table lists the point (0, 0), of order ∞.
• John Cremona, The elliptic curve database for conductors to 130000 (ANTS VII) — read from the PDF: the Antwerp Table 1 lists “All elliptic curves of conductor N ≤ 200”; “Swinnerton-Dyer searched for curves with small coefficients and kept those with conductor N ≤ 200”; the Antwerp tables are in “Antwerp IV” … “(edited by Birch and Kuyk).”

• Mazur theorem lecture notes, IMPA — states the fifteen groups in Mazur’s classification of E(Q)_tors.
• Mazur torsion theorem overview — lists the cyclic groups of orders 1–10 and 12 and the four products Z/2 × Z/2n, whose largest order is 16.

• OEIS A268602, first congruent number triangle — gives the triangle 3/2, 20/3, 41/6, verifies its area is 5, and identifies 5 as the first congruent number.
• The congruent number problem — defines congruent numbers and independently verifies the area-5 triangle and the fact that 5 is the smallest.
• Keith Conrad, The congruent number problem (University of Connecticut) — read from the PDF: 6 is the area of a (3, 4, 5) right triangle; “Fibonacci discovered in the 13th century that 7 is congruent”; “Theorem 2 (Fermat, 1640). The number 1 is not congruent.”
• Elliptic curve, Wikipedia — read as raw wikitext: “n is a congruent number if and only if the elliptic curve y² = x³ − n²x has a rational point of infinite order.”

• Birch–Swinnerton-Dyer conjecture, MathConjectures — states ord_{s=1} L(E, s) = r and labels the conjecture open.
• Clay Mathematics Institute, Millennium Prize Problems — lists Birch and Swinnerton-Dyer among the Millennium Prize Problems.

• Birch–Swinnerton-Dyer conjecture — “In the early 1960s, Swinnerton-Dyer used the EDSAC-2 computer at the University of Cambridge Computer Laboratory to calculate the number N​p of points modulo p for a large number of primes p on elliptic curves whose rank was known”; and Gross and Zagier in 1986 and Kolyvagin in 1989 proved the rank-zero and rank-one implications, not the vanishing of any particular L-value.
• Christopher Skinner, A converse to a theorem of Gross, Zagier, and Kolyvagin (Annals of Mathematics, 2020) — proves a converse to the theorem of Gross, Zagier and Kolyvagin: for certain semistable curves, algebraic rank 1 and finite Ш give analytic rank 1.

• Breuil, Conrad, Diamond and Taylor, On the modularity of elliptic curves over Q — the four authors’ preprint: “Theorem A. If E/Q is an elliptic curve, then E is modular.”; it is dated 2000, “and in revised form” January 2001; the 2001 journal publication is on the Edixhoven line.
• Edixhoven, Rational elliptic curves are modular — records the semistable Wiles/Taylor–Wiles result, the full Breuil–Conrad–Diamond–Taylor theorem, and the consequence for Fermat’s Last Theorem; its bibliography gives “On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc. 14 (2001), no. 4, 843-939.”
• Ribet’s theorem, Wikipedia — read as raw HTML: “In the summer of 1986, Kenneth Alan Ribet proved the epsilon conjecture, thereby proving that the Modularity theorem implied FLT”, and its bibliography gives the publication as Ribet, Ken (1990), “On modular representations of Gal(Q/Q) arising from modular forms”, Inventiones Mathematicae 100 (2), pp. 431–476.
• Modularity theorem, Wikipedia — read as raw wikitext: “Weil also showed that the conductor of the elliptic curve should be the level of the corresponding modular form.”; the parametrization “may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level N.”
• Elliptic curve, Wikipedia — read as raw wikitext: “Hasse’s conjecture affirms that the L-function admits an analytic continuation to the whole complex plane … In 1999 this was shown to be a consequence of the proof of the Shimura–Taniyama–Weil conjecture.”

• LMFDB 32.a3 (Cremona 32a2), the curve y² = x³ − x — records complex multiplication with discriminant −4 and geometric endomorphism ring Z[√−1].
• Complex multiplication, Wikipedia — read as raw wikitext: CM “is the theory of elliptic curves E that have an endomorphism ring larger than the integers”; curves Y² = 4X³ − aX have “two conjugate order-4 automorphisms” sending Y → ±iY, X → −X; “Kronecker conjectured – in what became known as the Kronecker Jugendtraum – that every abelian extension of K could be obtained by the (roots of the) equation of a suitable elliptic curve with complex multiplication”; “To this day this remains one of the few cases of Hilbert’s twelfth problem which has actually been solved.”; “j(Z[i]) = j(i) = 1728.”

• RFC 6090, Fundamental Elliptic Curve Cryptography Algorithms — defines elliptic-curve point multiplication and the discrete-logarithm setting used by ECC.
• NIST, Post-Quantum Cryptography FAQ — explains that Shor’s algorithm is the kind of “dramatic quantum attack” for which classical security measures stop being relevant; the FAQ does not itself name discrete logarithms, which the Elliptic-curve cryptography line carries.
• Elliptic curve point multiplication, Wikipedia — read as raw wikitext: “The simplest method is the double-and-add method.”
• Elliptic-curve cryptography, Wikipedia — read as raw wikitext: “all the fastest known algorithms that allow one to solve the ECDLP (baby-step giant-step, Pollard’s rho, etc.), need O(√n) steps”; “Shor’s algorithm can be used to break elliptic curve cryptography by computing discrete logarithms on a sufficiently large fault-tolerant quantum computer.”
• SECG, SEC 2: Recommended Elliptic Curve Domain Parameters, version 2.0 — §2.4.1 secp256k1: p = 2²⁵⁶ − 2³² − 2⁹ − 2⁸ − 2⁷ − 2⁶ − 2⁴ − 1; “The curve E: y² = x³ + ax + b over Fp is defined by” a = 0, b = 7.
• Bitcoin Developer Guide, Transactions — “Bitcoin uses the Elliptic Curve Digital Signature Algorithm (ECDSA) with the secp256k1 curve.”
• Cloudflare blog, Automatic key exchange for origins — “over 95% of origins support X25519”; the post describes the move to post-quantum hybrid key agreement.

• Dujella, History of elliptic curve rank records — re-read raw on 17 September 2026 and unchanged: the table lists rank 28 in 2006 (Elkies), rank 29 in 2024 (Elkies–Klagsbrun) and ranks 30 and 31 in 2026 (Alpöge–Howell), the running text says “The current record is an example of elliptic curve with rank ≥ 31, found by Alpöge and Howell in 2026,” and the highest rank known unconditionally and exactly is still 20, found by Elkies–Klagsbrun in 2020.
• Rank of an elliptic curve — as of August 2026 the largest known rank is at least 31, exactly 31 under the generalized Riemann hypothesis and the Birch and Swinnerton-Dyer conjecture, and there is no consensus among experts on whether ranks are bounded.
• Birch and Swinnerton-Dyer conjecture, Wikipedia — read as raw wikitext: “Although Mordell’s theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of a given curve. The rank can sometimes be calculated using numerical methods, but it is currently unknown if these methods are effective for all curves.”

• Fermat’s right triangle theorem, Wikipedia — carries all four halves of the claim in so many words — “It is the only complete proof given by Fermat”; “Fermat’s proof is a proof by infinite descent. It shows that, from any example of a Pythagorean triangle with square area, one can derive a smaller example”; “the existence of a square congruum is equivalent to the statement that the number 1 is not a congruent number”; and “An immediate consequence of the last of these formulations is that Fermat’s Last Theorem is true in the special case that its exponent is 4.”
• Proof by infinite descent, Wikipedia — defines the method — “a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction” — and records that Fermat coined the term and often used it for Diophantine equations.

Added in the V14 prose pass for the new card `descent` and to settle cross-chapter conflict C1. Chapter ᚆ(H) calls the descent “the one proof of that kind he left behind … covers fourth powers only” and Chapter ᚒ(U) called it “the only complete proof he left”; both sentences are about this single argument, and this entry is the source that says so. The “only complete proof” wording was already quoted under 19.16; it is repeated here so the card can stand on one entry.

• Mordell–Weil theorem, Wikipedia — states “The case with A an elliptic curve E and K the field of rational numbers is Mordell’s theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proven by Louis Mordell in 1922,” and gives the general statement that A(K) is a finitely-generated abelian group. Re-read raw on 17 September 2026 for Weil: “Some years later André Weil took up the subject, producing the generalisation to Jacobians of higher genus curves over arbitrary number fields in his doctoral dissertation published in 1928,” while the same page’s infobox gives “First proof by André Weil, First proof in 1929” — the dissertation and its publication, not two claims.
• H. Poincaré, Sur les propriétés arithmétiques des courbes algébriques, Journal de Mathématiques Pures et Appliquées, série 5, tome 7 (1901) — the memoir itself, free in full text, pp. 161–233, publisher Gauthier-Villars. This is the primary document behind the 1901 date.

Added in the V14 prose pass. Chapter ᚒ(U)’s “In 1922 Louis Mordell proved” and the `mordellweil` card’s 1922 and 1901 dates were footnoted only at 19.3, whose two sources state the theorem but name no year for it and no questioner. This entry carries both dates in one sentence, so it also stands behind the chapter’s “stated clearly by Poincaré in 1901” in the first paragraph. Extended 17 September 2026. The Wikipedia source hedges — “a question apparently posed by Henri Poincaré around 1901” — and names no paper, so the identification of the memoir is the book’s own, made from its NUMDAM record; the memoir is now cited directly so a reader can check it. The entry was also attached to #grouplaw’s “Poincaré stated in 1901” clause, which until now carried no citation for the date at all: its only reference, 19.2, describes the chord-and-tangent construction and names neither a person nor a year.

• Torsion conjecture, Wikipedia — states “Barry Mazur (1977, 1978) proved the full torsion conjecture for elliptic curves over the rationals,” and lists the groups as C_n with 1 ≤ n ≤ 10, C_12, and C_2n × C_2 with 1 ≤ n ≤ 4. Re-read raw on 17 September 2026 for Merel: “Finally, Loïc Merel (1996) proved the conjecture for elliptic curves over any number field,” after Kamienny (1992) and Kamienny and Mazur (1995) had reached quadratic fields and fields of degree at most 8; the page adds that Merel proved “that there is a bound on the order of the torsion group depending only on the degree.”

Added in the V14 prose pass to carry the year the chapter uses. “Mazur proved in 1977” was footnoted 19.9 alone, and neither of 19.9’s two sources gives a date for the theorem; they state the classification only. The source here dates it to two papers rather than one, which is what the `mazur16` card already says.

• Elliptic curve, Wikipedia (§ L-functions) — read as raw HTML: the L-function is “defined as an Euler product, with one factor for every prime number p,” with “a_p = p + 1 − #E(F_p),” and given as L(E(Q), s) = ∏ over p not dividing N of (1 − a_p p^−s + p^(1−2s))^−1 times ∏ over p dividing N of (1 − a_p p^−s)^−1. “This product converges for Re(s) > 3/2 only. Hasse’s conjecture affirms that the L-function admits an analytic continuation to the whole complex plane and satisfies a functional equation relating, for any s, L(E, s) to L(E, 2 − s). In 1999 this was shown to be a consequence of the proof of the Shimura–Taniyama–Weil conjecture.”
• Nagell–Lutz theorem, Wikipedia — read as raw HTML: “The result is named for its two independent discoverers, the Norwegian Trygve Nagell (1895–1988) who published it in 1935, and Élisabeth Lutz (1937)”; for y² = x³ + ax² + bx + c “with integer coefficients a, b, c,” and D “the discriminant of the cubic polynomial on the right side”: “If P = (x, y) is a rational point of finite order on E, for the elliptic curve group law, then: x and y are integers; either y = 0, in which case P has order two, or else y divides D, which immediately implies that y² divides D” (formulae as typeset on the page).
• Weierstrass elliptic function, Wikipedia — read as raw HTML: the function “is usually written with a rather special, lower case script letter ℘, which was Weierstrass’s own notation introduced in his lectures of 1862–1863,” which is the 1860s the #weierstrass card gives.
• LMFDB, elliptic curve 37.a1 (Cremona label 37a1) — minimal Weierstrass equation y² + y = x³ − x; “This elliptic curve is semistable. There is only one prime p of bad reduction”; analytic rank 1, so L(E, 1) = 0.

Added 17 September 2026 for the B6 pass. Three separate gaps in one chapter, all of the same shape: a card asserting something its cited entry does not carry. #LfuncE’s whole quick line is the Euler product and the only entry behind it was r-19.11, which states the Birch and Swinnerton-Dyer conjecture and its Millennium-Problem status; no entry anywhere in references.html carried an elliptic-curve Euler product, the six “Euler product” hits all being the Riemann zeta function in the prime-mileposts chapter. #tors4’s Nagell 1935 and Lutz 1937 stood on r-19.6, which is the torsion of one curve. #weierstrass’s “℘-function of the 1860s” stood on r-19.5, which is Δ = 64 and j = 1728. Not sourced and therefore not asserted: the #j1728 card said the 1728 normalisation “goes back to Klein’s work on the modular function in the 1870s,” and no source found in this pass dates Klein’s work, so the decade was removed from the card rather than left standing.

• Serge Lang, Wikipedia — read as raw HTML: its bibliography gives “Lang, Serge, Elliptic curves: Diophantine analysis, Grundlehren der Mathematischen Wissenschaften, volume 231, Springer-Verlag, Berlin–New York, 1978, ISBN 3-540-08489-4, doi 10.1007/978-3-662-07010-9,” with Alan Baker’s review in the Bulletin of the AMS (N.S.) 2 (1980), 352–354.

Added 17 September 2026 to source the chapter epigraph, which carried no marker. The book, the series, the publisher and the year are sourced here; that the sentence is the preface’s opening line is the author’s own reading of the printed book and is offered as such.

• Homogeneous coordinates, Wikipedia — read as raw HTML: “homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul… They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates,” with the scaling equivalence and the Z = 0 line at infinity.
• RFC 6090, Fundamental Elliptic Curve Cryptography Algorithms — a special point “represents the identity element (which is called the “point at infinity”)”; “The point at infinity (@,@) is regarded as equivalent to the homogenous coordinates (0, 1, 0).”

• Birch and Swinnerton-Dyer conjecture, Wikipedia — read raw: “Coates, J.; Wiles, A. (1977). “On the conjecture of Birch and Swinnerton-Dyer.” Inventiones Mathematicae 39 (3): 223–251..”
• Tunnell’s theorem, Wikipedia — read raw: “The theorem is named for Jerrold B. Tunnell, a number theorist at Rutgers University, who proved it in Tunnell (1983).”; “Tunnell’s theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution.”
• Elliptic-curve cryptography, Wikipedia — read raw: “The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor S. Miller in 1985..”
• Modularity theorem, Wikipedia — read raw: “Yutaka Taniyama stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in Tokyo and Nikkō as the twelfth of his set of 36 unsolved problems.”; “Goro Shimura and Taniyama worked on improving its rigor until 1957. André Weil rediscovered the conjecture, and showed in 1967 that it would follow from the (conjectured) functional equations for some twisted L-series of the elliptic curve”; “In 1995, Andrew Wiles, with some help from Richard Taylor, proved the Taniyama–Shimura–Weil conjecture for all semistable elliptic curves..”
• Hasse’s theorem on elliptic curves, Wikipedia — read raw: “It was proven by Hasse in 1933, with the proof published in a series of papers in 1936..”
• Andrew Wiles, The Birch and Swinnerton-Dyer Conjecture (Clay Mathematics Institute problem description) — read from the PDF: “The first general result proved was for elliptic curves with complex multiplication … This theorem was proved in 1976 and is due to Coates and Wiles.”

Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.

• Wikipedia, Cantor set — the construction, ternary 0/2 description, and equal cardinality with [0, 1] are stated in the construction and cardinality sections.

• Wikipedia, Cantor set — the composition section evaluates the removed geometric progression to 1 and explains why endpoints and uncountably many points remain.
• Wolfram MathWorld, Measure Zero — defines measure zero by covers with intervals of arbitrarily small total length.

• Wikipedia, Cantor set — the historical remarks identify Smith’s 1874 discovery and Cantor’s 1883 mention.

• Wikipedia, Cantor set — the article gives the 0/2 ternary representation, the binary-tree/cardinality argument, and the uncountability conclusion, but no date for Cantor’s 1874 paper (it dates the set to Smith, 1874, and Cantor, 1883) and nothing about algebraic numbers. This claim line said until 17 September 2026 that Cantor’s 1874 work showed “the rationals are countable”; the first theorem of that article is that the real algebraic numbers are countable, which is what 20.22 states and what the chapter’s argument needs.

• Wikipedia, Lebesgue measure — defines Lebesgue measure and countable additivity, and states that countable sets such as the rationals have measure zero.
• Null set, Wikipedia — read as raw wikitext: “all countable sets are null. In particular, the set ℚ of rational numbers is a null set, despite being dense in ℝ.”
• Probability axioms, Wikipedia — read as raw wikitext: the axioms were “introduced by Russian mathematician Andrey Kolmogorov in 1933”; “The event space, F, must be a σ-algebra on Ω”; “The third axiom, σ-additivity, is relatively modern, and originates with Lebesgue’s measure theory.”
• Cantor set, Wikipedia — “As the above summation argument shows, the Cantor set is uncountable but has Lebesgue measure 0.”

• Wikipedia, Cantor function — the article describes the devil’s staircase, its constancy on removed intervals, derivative zero almost everywhere, and midpoint value.
• Absolute continuity, Wikipedia — read as raw wikitext: for f on [a, b], being absolutely continuous is equivalent to f having a derivative f′ almost everywhere that is Lebesgue integrable with f(x) = f(a) + ∫ₐˣ f′(t) dt, the “fundamental theorem of Lebesgue integral calculus”; the Cantor function is listed as not absolutely continuous.

• Wikipedia, Hausdorff dimension — gives the self-similar dimension formula and the Cantor-set example, and dates the introduction of the dimension to Felix Hausdorff in 1918.
• Wikipedia, Gelfond–Schneider theorem — states that if a is algebraic other than 0 or 1 and b is algebraic irrational, every value of a^b is transcendental; applying 3^x=2 proves ln 2/ln 3 transcendental; it dates the two independent proofs, Gelfond’s and Schneider’s, to 1934.

• Wikipedia, Smith–Volterra–Cantor set — gives the construction, nowhere-dense property, and measure 1/2; it removes open middle subintervals of width 1/4 to the n at step n, sums the removed lengths to 1/2, and dates Smith’s discussion to 1875.
• Zenodo, H. J. S. Smith, On the Integration of Discontinuous Functions — a free record and scan of the paper, Proc. London Math. Soc. s1-6 (1874) 140–153 (doi:10.1112/plms/s1-6.1.140), with publication date 1 November 1874.

• Wikipedia, Koch snowflake — states the four-at-one-third construction, dimension, infinite perimeter, and limiting area 8/5.

• Wikipedia, Sierpiński triangle — gives the three-copy construction, dimension, and historical attribution to Sierpiński; “If one takes Pascal’s triangle with 2^n rows and colors the even numbers white, and the odd numbers black, the result is an approximation to the Sierpiński triangle,” and “The states of an n-disk puzzle, and the allowable moves from one state to another, form an undirected graph, the Hanoi graph, that can be represented geometrically as the intersection graph of the set of triangles remaining after the nth step in the construction of the Sierpiński triangle” (exponent as typeset on the page).

• Wikipedia, Menger sponge — states the 20-of-27 construction, dimension, zero volume, infinite surface area, and universal-curve property.

• Wikipedia, Banach–Tarski paradox — states the 1924 publication, axiom-of-choice dependence, non-measurable pieces, and Robinson’s sharp five-piece result: doubling the ball can be done with five pieces, and fewer than five will not suffice.
• R. M. Robinson, On the decomposition of spheres, Fundamenta Mathematicae 34 (1947), 246–260 — the bibliographic record dates the five-piece minimum to 1947.
• Encyclopedia of Mathematics, Banach–Tarski paradox — identifies Banach and Tarski’s 1924 paper and the equidecomposability theorem in three dimensions.
• MathWorld, Banach–Tarski Paradox — states that five pieces are minimal and four suffice when the single point at the center is omitted, carrying the claim that one of the five pieces is a singleton.

• Wikipedia, Logistic map — identifies May’s 1976 Nature paper and discusses the logistic map as a simple nonlinear recurrence with complex behavior; “In the general case with r between 1 and 2, the population will quickly approach the value (r − 1)/r, independent of the initial population,” and “With r between 2 and 3, the population will also eventually approach the same value” (fraction as typeset on the page); in the demographic example, “xₙ is a number between zero and one, which represents the ratio of existing population to the maximum possible population.”

• Wikipedia, Logistic map — gives the fixed point 1−1/r, stability range through r=3, period-doubling cascade, and accumulation parameter about 3.5699456.

• Wikipedia, Feigenbaum constants — gives δ≈4.6692016 and α≈2.5029079 and describes their parameter- and coordinate-scaling roles; it states that both are believed transcendental but that there is no known proof that either is even irrational. δ “is the limiting ratio of each bifurcation interval to the next between every period doubling,” and α “is the ratio between the width of a tine and the width of one of its two subtines (except the tine closest to the fold).”