# References — Chapter ᚜(Epilogue): The homeward view *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-28, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch28.md) *Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.* ### 28.1 {#r-28.1} The Millennium Prize Problems were announced in Paris, at the Collège de France, on 24 May 2000, a century after Hilbert’s 1900 Paris address, with seven prizes of one million dollars each; Poincare is the solved problem and the other six remain open. *Status: confirmed* - [Clay Mathematics Institute, The Millennium Prize Problems](https://www.claymath.org/millennium-problems/) — Clay states that the problems were announced on 24 May 2000, that the fund was $7 million with $1 million per problem, and identifies Poincare as solved while listing the remaining problems as unsolved; “The prizes were announced at a meeting in Paris, held on May 24, 2000 at the Collège de France,” and the Riemann hypothesis “also appears in the list of twenty-three problems discussed in the address given in Paris by David Hilbert on August 9, 1900.” ### 28.2 {#r-28.2} The Hilbert lecture is dated 8 August 1900 in the congress timetable, while Clay gives 9 August; the chapter epigraph reproduces Hilbert’s opening sentence. Sources differ: 8 August (the congress timetable; Wikipedia, “speaking on August 8 at the Sorbonne”) or 9 August (Clay); corrections welcome. *Status: judgement* - [MacTutor, 1900 ICM Paris](https://mathshistory.st-andrews.ac.uk/ICM/ICM_Paris_1900/) — the timetable places Hilbert’s communication in the Wednesday 8 August session. - [Clay Mathematics Institute, The Millennium Prize Problems](https://www.claymath.org/millennium-problems/) — Clay’s historical note says Hilbert’s address was on August 9, 1900. - [MacTutor, Hilbert’s Mathematical Problems speech](https://mathshistory.st-andrews.ac.uk/Extras/Hilbert_Problems_speech/) — the transcription opens with the sentence used as the epigraph; “The address Mathematische Probleme appeared in Göttinger Nachrichten in 1900, before the Proceedings of the Congress were published.” Editorial note: We follow the congress timetable, which places the address on 8 August; Clay’s 9 August is recorded as the outlier. ### 28.3 {#r-28.3} The Riemann Hypothesis puts the nontrivial zeta zeros on the critical line and controls deviations in the distribution of primes; the de Bruijn–Newman constant is a definite real with RH equivalent to Λ ≤ 0, while Rodgers and Tao proved Λ ≥ 0 in a paper posted to arXiv in January 2018. *Status: confirmed* - [Rodgers and Tao, The De Bruijn–Newman constant is non-negative](https://arxiv.org/abs/1801.05914) — the abstract defines Λ by the heat-flow deformation, states RH is equivalent to Λ ≤ 0, and proves Newman’s conjecture Λ ≥ 0; the page is headed “Submitted on 18 Jan 2018.” - [Clay Mathematics Institute, Riemann Hypothesis](https://www.claymath.org/millennium/Riemann-Hypothesis/) — Clay states that the prime number theorem gives the average distribution of primes and RH controls the deviation from that average. ### 28.4 {#r-28.4} Nicolaas de Bruijn showed in 1950 that the deformed function has only real zeros for parameters at least one half, in The Roots of Trigonometric Integrals, Duke Mathematical Journal 17 (1950), 197–226, and Charles Newman gave the constant its second half in Fourier Transforms with only Real Zeros, Proceedings of the American Mathematical Society 61 (1976), 245–251. Platt and Trudgian rigorously verified RH up to height 3·10^12, yielding the upper bound Lambda <= 0.2 used here; the paper is Bulletin of the London Mathematical Society 53 (2021), pp. 792–797, published online 22 January 2021, doi:10.1112/blms.12460, from the preprint arXiv:2004.09765 of 21 April 2020. Platt and Trudgian’s bound Λ ≤ 0.2 appeared as a preprint in April 2020 and in print in 2021. Newman remarked that his conjecture is “a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so.” *Status: confirmed* - [Platt and Trudgian, The Riemann hypothesis is true up to 3·10^12](https://arxiv.org/abs/2004.09765) — the abstract states that interval arithmetic verifies all zeros with 0 < gamma <= 3·10^12 lie on the critical line. - [de Bruijn-Newman constant, reference summary](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Newman_constant) — the history records the 2020 Platt-Trudgian improvement Lambda <= 0.2. Re-read raw on 17 September 2026 for the two founding dates: “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,” and the two papers are given as de Bruijn, N. G. (1950), The Roots of Trigonometric Integrals, Duke Math. J. 17 (3), 197–226, and Newman, C. M. (1976), Fourier Transforms with only Real Zeros, Proc. Amer. Math. Soc. 61 (2), 245–251. - [J. Andrade, A. Chang, S. J. Miller, Newman’s conjecture in various settings, arXiv:1310.3477](https://arxiv.org/abs/1310.3477) — read from the PDF: “Newman made the conjecture that Λ ≥ 0 along with the remark that “the new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so.”” Editorial note: The verification is dated 2021 throughout the book, which is its publication year. Checked 17 September 2026 and not to be reopened: the arXiv record reads “[Submitted on 21 Apr 2020]” and the Crossref record for doi:10.1112/blms.12460 gives Bulletin of the London Mathematical Society 53 (3), pp. 792–797, published online 22 January 2021 and in print June 2021. 2020 is the preprint and 2021 is the paper; where a source summary here says “the 2020 Platt-Trudgian improvement” it is dating the preprint. ### 28.5 {#r-28.5} For the conductor-37 elliptic curve y²+y=x³−x, the root number is −1, L(E, 1)=0, and LMFDB records analytic rank 1 and algebraic rank 1. The L-function of 37.a1 has sign −1, analytic rank 1 and L(1) = 0. *Status: confirmed* - [LMFDB, elliptic curve 37.a1](https://www.lmfdb.org/EllipticCurve/Q/37/a/1) — the record identifies the curve and gives conductor 37, analytic rank 1, and rank 1 (the sign −1 is on the L-function line below); its BSD formula line gives the leading coefficient as L′(E, 1) = #Ш(E/Q)·Ω_E·Reg(E/Q)·∏c_p / #E(Q)_tor² (as typeset on the page). - [Clay Mathematics Institute, Birch and Swinnerton-Dyer official problem description](https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf) — the official description states the conjectural equality between analytic order of vanishing and algebraic rank and describes the known analytic-rank-zero and analytic-rank-one results. - [LMFDB, L-function of elliptic curve isogeny class 37.a](https://www.lmfdb.org/L/EllipticCurve/Q/37/a/) — “Sign −1,” “Analytic rank 1,” and among the special values L(1) = 0. ### 28.6 {#r-28.6} Poincare posed the three-dimensional conjecture in 1904 and Perelman proved it using Ricci flow in work posted in 2002-2003 and never submitted to a refereed journal; the corresponding bracket is 1. Perelman declined the Fields Medal in 2006 and the Millennium Prize in 2010, having also declined the European Mathematical Society prize in 1996. *Status: confirmed* - [Clay Mathematics Institute, Poincare Conjecture](https://www.claymath.org/millennium-problems/poincare-conjecture/) — Clay describes Poincare’s 1904 question and Perelman’s proof as the solved Millennium problem. - [Clay annual report 2010, The Millennium Prize](https://www.claymath.org/library/annual_report/ar2010/ar2010.pdf) — the report states that Perelman’s three arXiv articles appeared in 2002 and 2003 and solved the Poincare conjecture, and that “Perelman did not take the usual route of submission of his work to a refereed journal.” - [Grigori Perelman, Wikipedia](https://en.wikipedia.org/wiki/Grigori_Perelman) — read as raw HTML: the awards list gives “Saint Petersburg Mathematical Society Prize (1991),” “EMS Prize (1996, declined),” “Fields Medal (2006, declined)” and “Millennium Prize (2010, declined).” Added because #clay’s quick line ends “one has been solved and the prize declined,” which no entry carried, and a quick line is repeated in every popup. - [Poincaré conjecture, Wikipedia](https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture) — read as raw wikitext: “after publication he found his announced theorem to be incorrect. In his fifth and final supplement, published in 1904, he proved this with the counterexample of the Poincaré homology sphere”; “Hamilton’s program was started in his 1982 paper in which he introduced the Ricci flow”; “The Ricci flow usually deforms the manifold towards a rounder shape.” ### 28.7 {#r-28.7} P versus NP asks whether every efficiently checkable problem is efficiently solvable; Cook formulated the problem in 1971, Levin did so independently in work published in 1973, and most complexity theorists believe P ≠ NP. *Status: confirmed* - [Clay Mathematics Institute, P vs NP](https://www.claymath.org/millennium/p-vs-np/) — Clay defines the easy-to-check versus easy-to-find question. - [Stephen Cook, The P versus NP Problem](https://www.claymath.org/wp-content/uploads/2022/06/pvsnp.pdf) — the official problem description gives the Cook, Levin and Karp history and says that most complexity theorists believe P is not equal to NP. ### 28.8 {#r-28.8} The Hodge conjecture concerns rational (p, p)-classes on smooth projective complex varieties, and Atiyah and Hirzebruch’s work published in 1962 shows that the integer-coefficient analogue is false. *Status: confirmed* - [Clay Mathematics Institute, Hodge Conjecture](https://www.claymath.org/millennium/hodge-conjecture/) — Clay’s landing page gives only the informal wording, that Hodge cycles are ‘(rational linear) combinations of geometric pieces called algebraic cycles’. It does not use the phrase type (p, p), and it names the variety class only as “particularly nice types of spaces called projective algebraic varieties,” without saying smooth. The type-(p, p) and smooth-projective halves of this entry’s claim are carried by Deligne below. - [Pierre Deligne, The Hodge Conjecture, Clay Mathematics Institute official problem description](https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf) — printed p. 46: ‘Rational (p, p)-classes are called Hodge classes’, and ‘On a projective non-singular algebraic variety over C, any Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles’. This is the official statement, and the source for both halves of the claim above. The same document is cited at r-28.21 for the 1950 date. - [Atiyah and Hirzebruch, Analytic cycles on complex manifolds](https://doi.org/10.1016/0040-9383(62)90094-0) — the Topology article gives the torsion counterexample underlying the failure of the integral version; the journal metadata dates the article to 1962. ### 28.9 {#r-28.9} Navier and Stokes introduced equations for viscous flow in the nineteenth century, Leray proved global weak solutions in three dimensions in 1934, and the Clay problem asks whether three-dimensional solutions remain smooth and unique. For incompressible flow the enstrophy is the integral of the squared vorticity; Leray’s strong solution lasts at least C/‖∇u0‖⁴ (unit viscosity) and ‖∇u‖ must blow up at a finite blow-up time; Leray’s weak solutions exist for all time and satisfy the equations in mean value; smooth solutions are known for small data or for a short time. *Status: confirmed* - [Clay Mathematics Institute, Navier–Stokes Equation](https://www.claymath.org/millennium/Navier-Stokes-Equation/) — Clay describes the fluid equations, their nineteenth-century origin, and the unresolved existence and uniqueness questions. - [Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace](https://doi.org/10.1007/BF02547354) — the 1934 Acta Mathematica paper is Leray’s foundational work on global weak solutions in three-dimensional space. - [Enstrophy, Wikipedia](https://en.wikipedia.org/wiki/Enstrophy) — read as raw wikitext: “In the case that the flow is incompressible, or equivalently that ∇·u = 0, the enstrophy can be described as the integral of the square of the vorticity.” - [W. S. Ożański and B. C. Pooley, Leray’s fundamental work on the Navier–Stokes equations: a modern review, arXiv:1708.09787](https://arxiv.org/abs/1708.09787) — read from the PDF: Corollary 3.11, the maximal existence time of the strong solution satisfies “T0 > C/‖∇u0‖⁴” (ν = 1), and Corollary 3.12 gives blow-up rates, ‖∇u(t)‖ ≥ C/(T0 − t)^(1/4); Leray “constructs a sequence of solutions … converging to a global-in-time weak solution”; the terms “Leray weak solution (or Leray-Hopf weak solution)” recognise the 1934 paper. - [Navier–Stokes existence and smoothness, Wikipedia](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness) — read as raw wikitext: Leray “also proved the existence of so-called weak solutions to the Navier–Stokes equations, which may not satisfy the equations pointwise but do satisfy them in mean value.” - [Clay Mathematics Institute, The Millennium Prize Problems (official problem descriptions, 2006)](https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf) — read from the PDF (Fefferman): “it is known that (A) and (B) hold provided the initial velocity u° satisfies a smallness condition,” and otherwise “if the time interval [0, ∞) is replaced by a small time interval [0, T).” ### 28.10 {#r-28.10} The normalized supremum of reciprocal lifetimes is finite because local existence gives a universal positive lower bound for the normalized data class; it is zero exactly when no smooth solution breaks down. The equivalence of 1/T∗ = 0 with the Clay statement on ℝ³ is this book’s own derivation from two sourced facts: Leray’s lower bound on the existence time in terms of the initial enstrophy, and the scale invariance of the unforced equations. *Status: judgement* - [Clay Mathematics Institute, Navier-Stokes official problem description](https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf) — the official problem is existence and smoothness for three-dimensional Navier-Stokes solutions. - **Own computation** (the author's own work) — For the declared class, T* = inf T(u0) and 1/T* = sup 1/T(u0), with 1/infinity defined as 0. A positive uniform local-existence time makes the supremum finite; if every smooth solution is global, all reciprocals are 0, while any finite-time breakdown makes the supremum positive. - [W. S. Ożański and B. C. Pooley, Leray’s fundamental work on the Navier–Stokes equations: a modern review, arXiv:1708.09787](https://arxiv.org/abs/1708.09787) — read from the PDF: Corollary 3.11, the maximal existence time of the strong solution satisfies “T0 > C/‖∇u0‖⁴” (ν = 1), and Corollary 3.12 gives blow-up rates, ‖∇u(t)‖ ≥ C/(T0 − t)^(1/4); Leray “constructs a sequence of solutions … converging to a global-in-time weak solution”; the terms “Leray weak solution (or Leray-Hopf weak solution)” recognise the 1934 paper. - [Terence Tao, Why global regularity for Navier–Stokes is hard (author’s blog, 2007)](https://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/) — “the Navier-Stokes equation … obeys a scale invariance: specifically, given any scaling parameter λ, and any smooth velocity field u solving the Navier-Stokes equations for some time T, one can form a new velocity field u^(λ) to the Navier-Stokes equation up to time λ²T,” in the absence of an external force. Editorial note: The book’s exact normalized invariant is a construction rather than a standard Clay notation. We mark the mathematical definition and its equivalence to the smoothness statement as a judgement call, while the underlying Clay question is sourced. ### 28.11 {#r-28.11} The Yang-Mills Millennium problem asks for a rigorous four-dimensional theory with a positive mass gap; no proof of the mass-gap property is known and simulations provide evidence only. *Status: confirmed* - [Clay Mathematics Institute, Yang-Mills & the Mass Gap](https://www.claymath.org/millennium/yang-mills-the-maths-gap/) — Clay: “Experiment and computer simulations suggest the existence of a “mass gap” in the solution to the quantum versions of the Yang-Mills equations. But no proof of this property is known.” The four-dimensional statement is on the Jaffe–Witten line below. - [Arthur Jaffe and Edward Witten, Quantum Yang–Mills Theory, Clay Mathematics Institute official problem description](https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf) — (p. 142): carries the exact wording the bracket quotes — a non-trivial quantum Yang–Mills theory for every compact simple gauge group on ℝ⁴, satisfying the rigorous axioms, with a mass gap Δ > 0. The Clay landing page above carries the mass-gap blurb but not this wording. ### 28.12 {#r-28.12} The Yang-Mills bracket is total: it records both existence of the theory and positivity of the mass gap, so it has value 0 or 1 even before the theory is constructed. *Status: judgement* - [Clay Mathematics Institute, Yang-Mills & the Mass Gap](https://www.claymath.org/millennium/yang-mills-the-maths-gap/) — Clay formulates the problem as the mathematical foundation of quantum Yang-Mills theory together with a positive mass gap. - [Arthur Jaffe and Edward Witten, Quantum Yang–Mills Theory, Clay Mathematics Institute official problem description](https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf) — (p. 142): the official statement the bracket encodes, requiring both that the theory exist and that Δ > 0. - **Own computation** (the author's own work) — An Iverson bracket of the complete proposition ‘the theory exists and Delta > 0’ is defined as 1 when the proposition is true and 0 otherwise; unlike a raw Delta, it is total by construction. Editorial note: This is the chapter’s encoding choice, not a standard physical observable. We leave it as a judgement call and explain why it avoids an undefined mass quotient. ### 28.13 {#r-28.13} The CODATA 2022 proton rest mass is 938.27208943(29) MeV/c^2. *Status: confirmed* - [CODATA 2022 recommended values](https://physics.nist.gov/cuu/Constants/) — the front page of NIST’s CODATA 2022 constants database, whose search and lists lead to the proton mass; the value itself is on the next line. - [NIST, proton mass energy equivalent in MeV (CODATA 2022)](https://physics.nist.gov/cgi-bin/cuu/Value?mpc2mev) — numerical value 938.272 089 43 MeV, standard uncertainty 0.000 000 29 MeV; source: 2022 CODATA recommended values. ### 28.14 {#r-28.14} More than 10^12 nontrivial zeta zeros have been verified on the critical line, with the cited rigorous computation reaching height 3·10^12. *Status: confirmed* - [Platt and Trudgian, The Riemann hypothesis is true up to 3·10^12](https://arxiv.org/abs/2004.09765) — the paper verifies every zero through height 3·10^12 on the critical line. - [Clay Mathematics Institute, Riemann Hypothesis](https://www.claymath.org/millennium/Riemann-Hypothesis/) — “This has been checked for the first 10,000,000,000,000 solutions.” ### 28.15 {#r-28.15} The Mordell-Weil rank counts independent rational points of infinite order, and the conductor-37 curve has rank one. *Status: confirmed* - [LMFDB, elliptic curve 37.a1](https://www.lmfdb.org/EllipticCurve/Q/37/a/1) — the record gives rank 1 for 37.a1; its properties table gives “Conductor 37,” “Discriminant 37,” “Rank 1,” and “Torsion structure” “trivial”; its Mordell-Weil generators table lists the point (0, 0), of infinite order. - [Birch and Swinnerton-Dyer conjecture, Wikipedia](https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture) — read as raw wikitext: “The number of independent basis points with infinite order is called the rank of the curve”; “it is conjectured that the rank of the abelian group E(K) of points of E is the order of the zero of L(E, s) at s=1.” ### 28.16 {#r-28.16} Hamilton introduced Ricci flow in 1982, and Perelman controlled its singularities to prove Poincare and geometrization. *Status: confirmed* - [Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255–306](https://doi.org/10.4310/jdg/1214436922) — Hamilton’s foundational Ricci-flow paper. (The earlier DOI given here resolved to Cannon, Shrinking Cell-Like Decompositions of Manifolds, Annals 110 (1979); corrected.) - [Clay Mathematics Institute, Poincare Conjecture](https://www.claymath.org/millennium-problems/poincare-conjecture/) — Clay describes Perelman’s proof through Ricci flow and its relation to geometrization; its list of Perelman’s articles gives “The entropy formula for the Ricci flow and its geometric applications” and “Ricci flow with surgery on three-manifolds.” - [Ricci flow, Wikipedia](https://en.wikipedia.org/wiki/Ricci_flow) — read as raw wikitext: a Ricci flow assigns to each t a metric gₜ “such that ∂gₜ/∂t = −2 Ric^(gₜ)”; “It is often said to be analogous to the diffusion of heat and the heat equation”; “For a 3-dimensional manifold, Perelman showed how to continue past the singularities using surgery on the manifold.” - [Poincaré conjecture, Wikipedia (Hamilton’s program)](https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture) — read as raw wikitext: “Hamilton’s program was started in his 1982 paper in which he introduced the Ricci flow on a manifold and showed how to use it to prove some special cases of the Poincaré conjecture”; “The Ricci flow usually deforms the manifold towards a rounder shape.” ### 28.17 {#r-28.17} NP-complete problems are in NP and every NP problem reduces to them in polynomial time; Cook established satisfiability as NP-complete and Karp used that result to show twenty other natural problems NP-complete in 1972. *Status: confirmed* - [Stephen Cook, The P versus NP Problem](https://www.claymath.org/wp-content/uploads/2022/06/pvsnp.pdf) — the official description gives the P/NP framework and states that Karp used Cook’s result to show twenty other natural problems NP-complete. It does not name them. - [Karp’s 21 NP-complete problems, Wikipedia](https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problems) — lists all twenty-one. Chromatic number and knapsack are on it; the travelling salesman problem is not — Karp has Directed Hamilton Circuit and Undirected Hamilton Circuit. The card named traveling salesman until this was checked. ### 28.18 {#r-28.18} A Hodge class is a rational cohomology class of type (p, p), and algebraic subvarieties supply such classes. *Status: confirmed* - [Clay Mathematics Institute, Hodge Conjecture](https://www.claymath.org/millennium/hodge-conjecture/) — states that Hodge cycles are rational linear combinations of algebraic cycles. It does not use the phrase “type (p, p).” - [Pierre Deligne, The Hodge Conjecture, Clay Mathematics Institute official problem description](https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf) — the official description, which is where the rational type-(p, p) formulation is stated. ### 28.19 {#r-28.19} A glueball is a bound state of gluons; lattice calculations place the lightest scalar glueball near 1.7 GeV, while experiment has not isolated an unambiguous glueball. *Status: confirmed* - [Morningstar and Peardon, The glueball spectrum from an anisotropic lattice study](https://doi.org/10.1103/PhysRevD.60.034509) — the paper reports the lattice glueball spectrum, including the scalar ground state near 1.7 GeV. The APS page refuses automated readers; the arXiv full text (hep-lat/9901004) gives “m(0++) = 1730(50)(80) MeV.” - [Chen et al., Glueball spectrum and matrix elements on anisotropic lattices](https://doi.org/10.1103/PhysRevD.73.014516) — the paper reports the scalar glueball mass estimate and lattice uncertainties. - [Morningstar, Update on Glueballs](https://arxiv.org/abs/2502.02547) — the 2025 review describes the continuing difficulty of identifying glueballs experimentally and the mixing of glueball candidates with ordinary hadrons. - [Glueball, Wikipedia](https://en.wikipedia.org/wiki/Glueball) — read as raw wikitext: “It consists solely of gluons, without valence quarks.” ### 28.20 {#r-28.20} Hilbert’s list contained 23 problems; the first was the continuum hypothesis, the eighth included the Riemann Hypothesis, the seventh concerned 2^(√2), and the tenth was answered no in 1970 by Yuri Matiyasevich, completing twenty-one years of work with Martin Davis, Hilary Putnam and Julia Robinson — the result is also called the MRDP theorem for that reason. *Status: confirmed* - [MacTutor, 1900 ICM Paris](https://mathshistory.st-andrews.ac.uk/ICM/ICM_Paris_1900/) — the congress record identifies Hilbert’s mathematical-problems address and the 23-problem context. - [Wikipedia, Hilbert’s problems](https://en.wikipedia.org/wiki/Hilbert%27s_problems) — the problem list identifies the eighth, seventh and tenth problems and their standard historical resolutions; “Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the Sorbonne. The complete list of 23 problems was published later”; its table gives the 1st as “The continuum hypothesis: there is no set whose cardinality is strictly between that of the integers and that of the real numbers.” (spacing as normalized here). - [Gelfond–Schneider constant, Wikipedia](https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant) — read as raw wikitext: on the seventh problem, “In the address he gave two explicit examples, one of them being the Gelfond–Schneider constant 2^√2.” - **Own reading** (the author's own work) — The seven Millennium Prize Problems (28.1) are the Birch and Swinnerton-Dyer conjecture, the Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP, the Poincaré conjecture, the Riemann Hypothesis, and Yang–Mills existence and the mass gap. Set beside Hilbert’s twenty-three (the Hilbert’s problems line above), only the Riemann Hypothesis is on both lists, as part of the eighth problem; Hilbert’s sixth (the axioms of physics) and his nineteenth and twentieth (on variational and boundary-value problems) lie near Yang–Mills and Navier–Stokes but state neither. Clay says only that the Riemann hypothesis “also appears” in Hilbert’s list, so “only” rests on this comparison. ### 28.21 {#r-28.21} Hodge posed the conjecture in his address to the 1950 International Congress of Mathematicians, and the official Clay problem description states it for rational cohomology classes of type (p, p) on a projective non-singular algebraic variety over the complex numbers. *Status: confirmed* - [Pierre Deligne, The Hodge Conjecture, Clay Mathematics Institute official problem description](https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf) — read at printed pp. 45-53. Page 46: “Rational (p, p)-classes are called Hodge classes,” and “In [6], Hodge posed the Hodge Conjecture. On a projective non-singular algebraic variety over C, any Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles.” Bibliography entry [6], page 53, is W. V. D. Hodge, The topological invariants of algebraic varieties, in Proceedings ICM 1950, AMS, Providence, RI, 1952, 181-192, which is what dates the conjecture to 1950. The Clay landing page cited at r-28.8 and r-28.18 gives only the informal “(rational linear) combinations of algebraic cycles” wording, with no year and no type (p, p). ### 28.22 {#r-28.22} In September 2026 a finite-time singularity was announced for a Navier–Stokes flow driven by an external force. The Clay Mathematics Institute removed the problem from its list of unsolved problems, marked its page Active, and by 17 September 2026 had given it a heading of its own: the problem list now reads Active problems, holding Navier–Stokes alone, then Unsolved problems, holding the other five, then Solved problems, holding Poincaré. The official problem is stated by Fefferman as four separate statements; a forced blow-up bears on the forced statements (C) and (D), while the number this book builds is defined over unforced flows, which is statement (A). Clay has published no adjudication, and the book continues to treat the problem as open. *Status: judgement* - [Clay Mathematics Institute, The Millennium Prize Problems](https://www.claymath.org/millennium-problems/) — Read as raw HTML rather than through a summarising fetcher. The page now carries three group headings.

Active problems

is followed by
holding one tile,

Navier-Stokes Equation

.

Unsolved problems

is followed by
holding five — Birch and Swinnerton-Dyer, Hodge, P vs NP, Riemann, and Yang-Mills & the Mass Gap.

Solved problems

is followed by
holding one, the Poincaré conjecture. On 11 September 2026 there was no Active group at all and Navier-Stokes appeared in the page’s navigation but in neither list. - [Clay Mathematics Institute, Navier-Stokes Equation](https://www.claymath.org/millennium/navier-stokes-equation/) — The page carries

Active

, the status label that now heads a group of its own on the problem list, and its metadata gives dateModified 2026-09-11T07:43:31+00:00. It states no result, names no claimant, and awards nothing. Re-read raw on 18 September 2026: unchanged, as is the problem list. - [Navier–Stokes existence and smoothness, Wikipedia](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness) — read as raw wikitext: records the 8 September 2026 announcement, the priority dispute that followed it, and that as of 18 September 2026 the construction “has not yet been independently verified”. - [C. Fefferman, Existence and smoothness of the Navier–Stokes equation, in The Millennium Prize Problems (Clay Mathematics Institute)](https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf) — “we ask for a proof of one of the following four statements”; statement (C), “Breakdown of Navier–Stokes solutions on R3,” is a breakdown for some smooth force f(x, t). Editorial note: Added 11 September 2026 and filed as a judgement rather than a confirmation, because what is confirmed is only what Clay’s own pages show: a status label and a removal from one list. The announced result is unrefereed, publicly disputed, and priority-contested, the claimants decline the prize, and Clay has adjudicated nothing. Under the book’s recency rule a newer source supersedes when it is a retraction, a correction, or a re-measurement; an unrefereed announcement is none of those. The author ruled on 11 September 2026 that the problem stays open in the book and that the chapter should say the news. The forced/unforced distinction is the reason the chapter’s own number is untouched: 1/T∗ is a supremum over flows with no force term. Re-read raw on 17 September 2026: the problem page is still Active with no result, no claimant, and no award, and the only change is that the problem list has grown an explicit Active problems group holding Navier-Stokes alone. Nothing supersedes the entry’s claim; only its description of the list page had gone stale, and that is what was rewritten. On 18 September 2026 the chapter’s Navier–Stokes paragraph was given the sentence the ruling asked for, and the entry gained the announcement’s own page and an independent record of it, since until then it had cited only Clay’s pages, which describe a status and never the announcement. ### 28.23 {#r-28.23} A 3-manifold is a topological space in which every point has a neighborhood homeomorphic to Euclidean 3-space; closed means compact and without boundary. The Poincaré conjecture, the Clay problem solved by Grigori Perelman, asks whether every simply connected closed 3-manifold is homeomorphic to the 3-sphere. *Status: confirmed* - [3-manifold, Wikipedia](https://en.wikipedia.org/wiki/3-manifold) — read as raw HTML: defines a 3-manifold by local homeomorphism with Euclidean 3-space, and closed as compact without boundary, and states the Poincaré conjecture in the simply-connected form. - [Poincaré Conjecture, Clay Mathematics Institute](https://www.claymath.org/millennium/poincare-conjecture/) — the prize body’s own statement of the problem Perelman settled. ### 28.24 {#r-28.24} A projective variety is the zero set in projective space of a family of homogeneous polynomials; over the complex numbers a smooth projective variety is a nonsingular such set, and it is compact, which is what makes its cohomology well behaved. The Hodge conjecture is stated for smooth complex projective varieties. *Status: confirmed* - [Projective variety, Wikipedia](https://en.wikipedia.org/wiki/Projective_variety) — read as raw HTML: a projective variety as the zero locus in projective space of a family of homogeneous polynomials, and the complex-analytic consequences of projectivity, in particular compactness. - [Hodge Conjecture, Clay Mathematics Institute](https://www.claymath.org/millennium/hodge-conjecture/) — the prize page; read again on 23 September 2026 it says only “projective algebraic varieties,” and the non-singular complex setting is carried by the official description (the MPPc line below). - [Clay Mathematics Institute, The Millennium Prize Problems (official problem descriptions, 2006)](https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf) — read from the PDF (Deligne): “Hodge Conjecture. On a projective non-singular algebraic variety over C, any Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles.” ### 28.25 {#r-28.25} A gauge theory is a field theory whose Lagrangian is invariant under a group of local transformations. The concept and the name come from Hermann Weyl’s work of 1918, where Eichinvarianz meant invariance under a local change of scale; after quantum mechanics, Weyl with Vladimir Fock and Fritz London replaced the scale factor by a phase, making it a U(1) gauge symmetry. The Clay problem asks, for every compact simple gauge group, for a nontrivial quantum Yang–Mills theory on ℝ⁴ with a mass gap. *Status: confirmed* - [Gauge theory, Wikipedia](https://en.wikipedia.org/wiki/Gauge_theory) — read as raw HTML: “The concept and the name of gauge theory derives from the work of Hermann Weyl in 1918… conjectured that Eichinvarianz or invariance under the change of scale might also be a local symmetry… Weyl used the word ‘gauge’ to describe this concept, after a Latin word referring to standard cask sizes… Weyl, Vladimir Fock and Fritz London replaced the simple scale factor with a complex quantity and turned the scale transformation into a change of phase, which is a U(1) gauge symmetry”; “For each group generator there necessarily arises a corresponding field (usually a vector field) called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called gauge invariance). When such a theory is quantized, the quanta of the gauge fields are called gauge bosons.” - [Yang–Mills and the Mass Gap, Clay Mathematics Institute](https://www.claymath.org/millennium/yang-mills-the-maths-gap/) — the prize body’s own page for the problem: “Progress in establishing the existence of the Yang-Mills theory and a mass gap will require the introduction of fundamental new ideas both in physics and in mathematics.” The wording for any compact simple gauge group is in the Jaffe–Witten official description, on the line below. - [Hermann Weyl, Wikipedia](https://en.wikipedia.org/wiki/Hermann_Weyl) — read as raw wikitext: “In 1918, he introduced the notion of gauge, and gave the first example of what is now known as a gauge theory. Weyl’s gauge theory was an unsuccessful attempt to model the electromagnetic field and the gravitational field as geometrical properties of spacetime.” - [Clay Mathematics Institute, The Millennium Prize Problems (official problem descriptions, 2006)](https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf) — read from the PDF (Jaffe–Witten, p. 142): “Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on ℝ⁴ and has a mass gap Δ > 0.” ### 28.26 {#r-28.26} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Platt and Trudgian rigorously verified in 2021 that every zero up to height 3*10^12 lies on the critical line; Morningstar and Peardon’s lattice glueball spectrum, 1999; Chen et al.’s lattice glueball masses, 2006; Cook’s 1971 result established Boolean satisfiability as NP-complete; Claude-Louis Navier wrote the equations of viscous flow in 1822; George Stokes wrote the equations of viscous flow in 1845; Perelman’s three papers of 2002-2003 completed Hamilton’s Ricci-flow program; Chen Ning Yang and Robert Mills wrote the non-abelian gauge theory in 1954. The lightest glueball’s lattice mass is about 1.8 times the proton’s, a ratio of the book’s own. *Status: confirmed* - [Platt and Trudgian, The Riemann hypothesis is true up to 3*10^12, Bulletin of the London Mathematical Society 53(3), 792-797 - Crossref record](https://api.crossref.org/works/10.1112/blms.12460) — read raw: “the publisher’s Crossref record gives the year of publication as 2021.” - [Morningstar and Peardon, The glueball spectrum from an anisotropic lattice study, arXiv abstract page](https://arxiv.org/abs/hep-lat/9901004) — read raw: “Phys.Rev.D60:034509,1999”; the full text gives “m(0++) = 1730(50)(80) MeV and m(2++) = 2400(25)(120) MeV.” - [Chen et al., Glueball spectrum and matrix elements on anisotropic lattices, arXiv abstract page](https://arxiv.org/abs/hep-lat/0510074) — read raw: “Phys.Rev.D73:014516,2006”; the full text’s table of masses gives 0++ at 1710(50)(80) MeV. - [Stephen Cook, The P versus NP Problem, Clay Mathematics Institute official problem description](https://www.claymath.org/wp-content/uploads/2022/06/pvsnp.pdf) — read raw: “in 1971 the present author [9] introduced a notion of NP-completeness as a polynomial-time analog of c.e.-completeness.” - [Claude Louis Marie Henri Navier, MacTutor](https://mathshistory.st-andrews.ac.uk/Biographies/Navier/) — read raw: “He gave the well known Navier-Stokes equations for an incompressible fluid in 1821 while in 1822 he gave equations for viscous fluids..” - [George Gabriel Stokes, MacTutor](https://mathshistory.st-andrews.ac.uk/Biographies/Stokes/) — read raw: “he published On the theories of the internal friction of fluids in motion in 1845..” - [Poincare Conjecture, Clay Mathematics Institute](https://www.claymath.org/millennium/poincare-conjecture/) — read raw: “Nearly a century passed between its formulation in 1904 by Henri Poincaré and its solution by Grigoriy Perelman, announced in preprints posted on ArXiv.org in 2002 and 2003..” - [Robert Mills (physicist), Wikipedia](https://en.wikipedia.org/w/index.php?title=Robert_Mills_(physicist)) — read raw: “C.N. Yang and Robert Mills formulated in 1954 a theory now known as the Yang–Mills theory.” - **Own computation** (the author's own work) — with the CODATA 2022 proton rest mass 938.27208943 MeV/c² (28.13), 1730/938.27 ≈ 1.84 and 1710/938.27 ≈ 1.82 for the two lattice masses above, so the lightest glueball is about 1.8 times the proton. Editorial note: 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. ### 28.27 {#r-28.27} States the Birch and Swinnerton-Dyer conjecture, that an elliptic curve’s rank equals the order of vanishing of its L-function at s=1, formulated in the early 1960s. It was formulated by Bryan Birch and Peter Swinnerton-Dyer; the conductor-37 curve 37.a1 has rank 1 and analytic rank 1. It is one of the seven Millennium Prize Problems, and no proof yet covers curves of rank greater than 1. *Status: confirmed* - [Birch and Swinnerton-Dyer conjecture, Wikipedia](https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture) — states the conjecture and its formulation by Birch and Swinnerton-Dyer from numerical computations in the early 1960s; “The conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay Mathematics Institute” and “There are currently no proofs involving curves with a rank greater than 1.” Also: “An L-function L(E, s) can be defined for an elliptic curve E by constructing an Euler product from the number of points on the curve modulo each prime p,” and “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. From these numerical results, he and his colleague Bryan John Birch conjectured ….” - [Elliptic curve 37.a1, LMFDB](https://www.lmfdb.org/EllipticCurve/Q/37.a1/) — lists Mordell–Weil rank 1 and analytic rank 1 for the curve: “This is the elliptic curve of minimal conductor with positive rank.” ### 28.30 {#r-28.30} States Goldbach’s conjecture, that every even integer greater than 2 is the sum of two primes, in Goldbach’s 1742 letter to Euler, and its verification by computer up to large bounds. Euler’s reply recalled Goldbach’s own even-number form. It is verified up to 4×10¹⁸, sits within Hilbert’s eighth problem of 1900 with the Riemann Hypothesis, and Chen’s 1973 theorem is the closest proven result. *Status: confirmed* - [Goldbach’s conjecture, Wikipedia](https://en.wikipedia.org/wiki/Goldbach%27s_conjecture) — states the conjecture, dates Goldbach’s 1742 letter to Euler, and gives the scale of computer verification. - [Hilbert’s eighth problem, Wikipedia](https://en.wikipedia.org/wiki/Hilbert%27s_eighth_problem) — carries Goldbach and the Riemann hypothesis within the eighth problem: “Hilbert’s eighth problem is one of David Hilbert’s list of open mathematical problems posed in 1900.”; “(where the twin prime conjecture and Goldbach conjecture are special cases of this equation)”