Openai 2026 hilberts tenth problem rational numbers

../

corollary_7_1: The manuscript's consequence of Theorem 1.1: rational solvability of integer polynomials has the degree of the halting problem, and stays undecidable for total degree at most four and for sums of squares of quadratics. Claims checked only.

theorem_1_1: The manuscript's main claim: Hilbert's tenth problem over Q has a negative answer, by a Turing reduction of integer solvability to rational solvability. Claims checked only.

theorem_2_4: The manuscript's five-point height estimate: the Weil height of a rational point of P^1 off five fixed rational points is at most H M^c, where M is the product of the primes outside a fixed finite set at which the point meets one of the five to odd order. Claims checked only.

OpenAI, Hilbert's tenth problem over the rational numbers, OpenAI Math Release preprint, October 6, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision fd4aeeb2e), folder preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026; the held PDF, main.pdf in the release, is retained as openai_2026_hilberts_tenth_problem_rational_numbers.pdf, and the release's TeX bundle in that folder (under build/) is the TeX source cited on this card. The license term was read in the release's root LICENSE file at that revision on 2026-10-12.

@misc{OAI:Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026,
  author = {{OpenAI}},
  title = {{Hilbert's tenth problem over the rational numbers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/main.pdf}{OAI:Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026}},
  year = {2026}
}

The held PDF has 62 A4 pages with a text layer; its title page is dated October 6, 2026. Its metadata records the title and the author "OpenAI" but no creation date, since the TeX source suppresses it (\pdfinfoomitdate=1). The TeX source is build/main.tex (81 lines), which inputs seven section files under build/sections/ (4,184 lines in all: 01-introduction.tex 250, 02-reduction.tex 888, 03-parity.tex 718, 04-height.tex 1,326, 05-elliptic.tex 189, 06-primes.tex 705, 07-consequences.tex 108), the figure file build/figures/reduction-overview.tex (60 lines) and the bibliography build/references/references.bib (86 entries, of which the PDF prints the 74 cited).

Attestation. The release README at this revision says the repository holds manuscripts "produced by an internal OpenAI model", that "Some outputs build upon earlier results produced by the models", that the collection "includes results at different stages of verification", that "Not all have accompanying Lean formalizations" and that "Some of the unformalized results could have issues". Its account of the method says that "The vast majority of results were obtained with the same procedure using an unreleased internal OpenAI model"; the exceptions it names concern a zero-free region for the Riemann zeta function and the Hodge conjecture for CM abelian varieties, and it does not say whether any other manuscript is an exception. The manuscript's own README carries the title, the author line "OpenAI", the date and the citation block, with a version note saying that this version "updates citations to revised manuscripts in this collection"; it adds no sentence about authorship or assistance. The title page names OpenAI as the sole author, and the TeX source carries no statement on how the manuscript was produced. These are the source's own statements, recorded as attestations and not as this corpus's review. No refereed publication, arXiv version or independent review of the manuscript is recorded here, and nothing on this card is independently reviewed.

Formalization. The release's Lean catalog (lean/formalization.yaml) at revision fd4aeeb2e lists no formalization of this manuscript, and its family has no page under lean/docs/; nothing here is a formal proof of any statement.

Companions. The release pairs this manuscript with A pointwise 2-converse for elliptic curves with rational two-torsion (October 7, 2026, with an earlier dated version of September 24, 2026), whose abstract claims that an elliptic curve over $\mathbb Q$ with a nonzero rational two-torsion point and $2^\infty$-Selmer corank $0$ or $1$ has analytic and Mordell--Weil rank equal to that corank and a finite Tate--Shafarevich group; its finiteness statement is the input to Lemma 3.4 here. It is not held in this library, so it is not linked. The release also holds an earlier dated version of this manuscript, Hilbert's tenth problem over the rational numbers (September 24, 2026, folder preprints/Hilberts-tenth-problem-over-the-rational-numbers-September-24-2026, 62 pages); a comparison of the two TeX bundles at the release revision shows that they differ only in the title-page date and in three bibliography links, which the earlier version points at the earlier dated versions of the three cited release manuscripts, so labels, statements and proofs agree. Outside the family, the modularity step of Section 4 cites Theorem 1.1 of the release manuscript Fontaine--Mazur modularity at the prime 2 (October 6, 2026), filed at the Fontaine--Mazur card, and Remark 3.5 names Theorem 1.1 of Goldfeld's analytic density conjecture and the 2-converse for elliptic curves (October 7, 2026) as an alternative input to Lemma 3.4; that manuscript is not held here. The companions' arguments were not read for this card.

Read status: claims checked for Theorem 1.1 (build/sections/01-introduction.tex, label thm:main, lines 7--11), Theorem 2.4 with Corollary 2.5 (build/sections/02-reduction.tex, labels hei:height, lines 232--248, and hei:enlarge-exceptions, lines 270--279) and Corollary 7.1 (build/sections/07-consequences.tex, label thm:degree-normal-form, lines 8--23), together with the statements of Propositions 2.1 and 2.2, Lemma 2.3, Corollary 2.6, Lemma 2.7, Proposition 2.8, Lemmas 2.9 and 2.10 and Propositions 2.11 and 2.12 (02-reduction.tex lines 27--34, 109--127, 143--175, 294--322, 435--438, 477--486, 545--553, 643--654, 722--733 and 755--759, in that order), Lemmas 3.3--3.4 (03-parity.tex lines 300--315 and 464--468) and Lemma 6.1 (06-primes.tex lines 21--40), read clause by clause. The proofs were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

Within each section all numbered statements, remarks included, share one counter; the numbers below are the PDF's, and its printed page numbers agree with the PDF page numbers.

• Section 1, Introduction (pp. 3--6; 01-introduction.tex). States Theorem 1.1 (p. 3). The historical setting (pp. 3--5) recalls Hilbert's problem, the Davis--Putnam--Robinson--Matiyasevich theorem, and two sufficient routes to undecidability over $\mathbb Q$ that the manuscript says it does not need: an existential definition of $\mathbb Z$ in $\mathbb Q$, and a Diophantine model of integer arithmetic (Cornelissen and Zahidi 2000). It reviews Julia Robinson's first-order definition (1949), Poonen's definition with two universal and then seven existential quantifiers (2009) and Koenigsmann's universal one (2016); Mazur's conjecture on the topology of rational points (1995), which would rule out such definitions and models but, the text says, not undecidability itself; Poonen's large subrings $\mathbb Z[S^{-1}]$ (2003) and the Turing equivalences of Eisenträger, Miller, Park and Shlapentokh (2017); Poonen's rank-one transfer between rings of integers (2002), the route of Mazur and Rubin conditional on the conjectured evenness of $\dim_{\mathbb F_2}\mathrm{Sha}[2]$ (2010), and the unconditional results for rings of integers of number fields of Koymans and Pagano (2026) and of Alpöge, Bhargava, Ho and Shnidman (2026), which the text notes concern finitely generated $\mathbb Z$-algebras and not $\mathbb Q$; the elliptic divisibility approach of Cornelissen and Zahidi (2007), and the undecidability of Garcia-Fritz, Pasten and Vidaux (2025) in a language enlarged by height comparisons. The proof strategy (pp. 5--6) sketches the argument outlined on the Theorem 1.1 page and names the two release manuscripts used: the pointwise 2-converse in Section 3 and the Fontaine--Mazur theorem at $2$ in Section 4.
• Section 2, Finite rational tests for integer solvability (pp. 6--17; 02-reduction.tex). Fixes $F=\mathbb Q(\sqrt2)$, $d=75-53\sqrt2$ and $E:y^2=x^3-d^2x$. Proposition 2.1 (p. 7): $E(F)$ has rank one, so $mE(F)=\mathbb ZP$ for some $m\ge1$ and a nontorsion $P$. For a characteristic-zero domain $R$ with $\sqrt2\notin\operatorname{Frac}R$ it states the point axioms ($T_0=O$, $T_1=P$, $T_{a+a'}=T_a+T_{a'}$, $T_a\in mE(F_R)$, and $T_a$ finite with $xy\ne0$ for $a\ne0$) and, inside an elementary extension of the many-sorted structure carrying $\mathbb Q$, $\mathbb Z$, $\mathbb R$, $F$, valuations, heights and elliptic multiples, defines the index map $\eta$ by $\widetilde T_a=\eta(a)P$, which is additive and injective but not assumed multiplicative. With $z=-x/y$ and the truncated formal logarithm $\ell_K$, Proposition 2.2 (p. 8) gives a finite set $S_E$ outside which $z$-valuations read off index ratios and truncated-logarithm ratios approximate integral index ratios to order $K$, on $E$ and on its conjugate $E^\sigma$. Lemma 2.3 (local comparison, p. 8): at an ambient prime $q\notin S_E$, $q>K+1$, a primary test on $E$ and an alternate test on $E^\sigma$, with auxiliary $h,j_0$ fixed for all $a$, force every embedded $\widetilde a$ to be $q$-integral with $\widetilde a\equiv\eta(a)\pmod{q^K}$. The section then states Theorem 2.4 (the five-point height estimate, p. 9) with Corollary 2.5 (p. 10), and Corollary 2.6 (ordinary contact representations, p. 10): a fixed finite multiplier set $\mathcal K$ such that every integer $A$ is $u_1/v+u_2/v+u_3/v$ with each of the thirteen contact values $v$ and $L_b(u_i,v)$ ($b\ne\infty$) equal to $\pm k r$, $k\in\mathcal K$, $r$ a large prime split in $F$ and good for both curves, with test witnesses at both places above $r$. Section 2.3 (pp. 11--12) writes the recursive theory $\mathcal T$: for every ordinary precision $K$, a representation axiom asking for such sums with ring-language versions of the two tests at the two branch ideals cut out by a square root of $2$ modulo $r$. Lemma 2.7 (p. 12): $\mathcal T$ is recursive and has the model $\mathbb Z$ with $T_a=aP$; the text adds that the fixed constants are hardcoded and need no effective search. Proposition 2.8 (p. 12): a positive existential formula $\Phi(b)$ with rational coefficients and a finite prime set $S_1$ such that $\Phi$ holds on every integer and any $b\in\mathbb Q^\times$ satisfying $\Phi$ has even valuation at each $q\notin S_1$ where $v_q(b)<0$. The finite tests Skolemize $\mathcal T$ plus $f(\mathbf a)=0$ and impose ring values, functionality, ground instances of the universal axioms and $\Phi$ on every ground-term value; each finite initial part reduces to finitely many rational-solvability queries. Lemma 2.9 (p. 13) is the compactness step; Lemma 2.10 (p. 14) coarsens the valuation at a contact prime by the convex subgroup generated by the poles of ring elements, so that pole parity selects a tested branch; Proposition 2.11 (p. 15) gives the index congruence at every prime counted by a contact radical; Proposition 2.12 (p. 16) says that for nonconstant $f$, if every finite test is solvable then $f$ has an integer zero. Figure 1 (p. 16) diagrams that proof, and the proof of Theorem 1.1 closes the section (p. 17).
• Section 3, An existential condition excluding odd poles (pp. 18--26; 03-parity.tex). Proves Proposition 2.8, comparing it with the existential divisibility predicates of Demeyer and Van Geel (2006). The formula uses the curves $E_l:y^2=x(x-l)(x+3l)$: for $b\ne0$, $\Phi(b)$ asks, for listed multipliers $m,m_d$ and rational $t,e$, that two twists $E_e$ and $E_{eH'}$, with $H'$ a rational function of $b$ and $t$, both have a rational point whose $x$-coordinate lies in the square class of $mb$. Lemma 3.1 (p. 18) proves the pole restriction by a valuation count. Lemma 3.2 (p. 20) prepares square classes for an integer input. Lemma 3.3 (p. 21): for squarefree $l>0$ prime to $6$ with $l\equiv3\pmod4$, $l\equiv2\pmod3$, a condition on its prime support and a kernel and image condition on a Monsky-type matrix of Legendre symbols over that support, $\dim_{\mathbb F_2}\mathrm{Sel}_2(E_l/\mathbb Q)=3$ with a prescribed Selmer class outside the two-torsion classes. Lemma 3.4 (p. 23): such an $E_l$ has a rational point with $x$-coordinate in the prescribed square class; the proof shows rank one and $\mathrm{Sha}(E_l/\mathbb Q)[2]=0$, applying the finiteness of the Tate--Shafarevich group from the pointwise 2-converse and the alternating Cassels--Tate pairing (Cassels 1962; Poonen and Stoll 1999). Remark 3.5 (p. 24) names the unrestricted 2-converse as an alternative. Lemma 3.6 (p. 24) builds one parameter serving both twists by prescribing Legendre symbols and applying Dirichlet's theorem. Monsky's appendix to Heath-Brown (1994), Mokrani (2020) and Wei and Guo (2022) are cited for the matrix viewpoint on these $\pi/3$-congruent number curves.
• Section 4, A height bound from odd contact with five points (pp. 26--44; 04-height.tex). Proves Theorem 2.4 through Lemmas 4.1--4.8, with Remark 4.3 giving a second route to Lemma 4.2; the result page outlines the argument: the discriminant-$210$ Shimura curve and its Atkin--Lehner quotient $\mathbf P^1$ with five rational branch values (Section 4.1, citing Long, Maclachlan and Reid and Nualart Riera), an abelian surface family with quaternionic multiplication, ramification of fiber fields only at odd-contact primes (Lemma 4.1, p. 31), a base-height comparison through theta heights (Lemma 4.2, p. 32), the split of the sign obstruction (Lemma 4.5, p. 37), modularity and the level bound $N\le CM^a$ (Lemma 4.7, p. 40, where the proof allows $a=2$), the modular factor (Lemma 4.8, p. 42) and the final isogeny estimate (p. 43), with the proof closing on p. 44. Remark 4.9 (p. 44) compares related Faltings-height bounds of von Känel and of von Känel and Kret.
• Section 5, The rank-one elliptic group and its local parameters (pp. 44--46; 05-elliptic.tex). Proves Proposition 2.1 by a two-isogeny descent over $\mathbb Z[\sqrt2]$ (class number one, $d$ of norm $7$, the point $(-dw,2\sqrt2\,d^2)$ with $w=-5+4\sqrt2$), and Proposition 2.2 from the valuation properties of the formal logarithm, with a recursion for its coefficients; it compares Lemmas 9 and 11 of Poonen's rank-one paper.
• Section 6, Prime patterns for the two conjugate local tests (pp. 46--56; 06-primes.tex). Lemma 6.1 (prime patterns, p. 47): a finite set $\mathcal K\subset\mathbb Z_{>0}$ such that for every $A\in\mathbb Z$ and $B>0$ the thirteen values above can all be made $\pm k r$ with $k\in\mathcal K$ and $r>B$ prime, split in $F$, good for $E$ and $E^\sigma$, and with the two reduction orders $d_1,d_2$ satisfying $\gcd(d_1,d_2)\mid4$ and $r\nmid d_1d_2$. Corollary 6.2 (p. 47) builds the test witnesses $h,j_0$ by the Chinese remainder theorem, and Corollary 2.6 follows (p. 48). The proof of Lemma 6.1 restricts to primes $r\equiv1\pmod8$, $r\equiv3\pmod7$, where $E$ and $E^\sigma$ reduce to opposite twists of $y^2=x^3-x$ with orders $r+1\pm2a$ ($r=a^2+b^2$); fixes a finite multiplier catalog; obtains $\gg X^3/(\log X)^{13}$ patterns from the Green--Tao--Ziegler theorem on linear equations in primes (Green and Tao 2010 and 2012, Green, Tao and Ziegler 2012 with its 2024 erratum); and removes patterns with a large common odd factor of the orders by a Selberg upper sieve (Lemma 6.3, p. 52) and a fiber sieve, closing on p. 56.
• Section 7, Turing degree and a quartic normal form (pp. 56--57; 07-consequences.tex). States and proves Corollary 7.1 (p. 56).
• References (pp. 57--62), 74 entries, three of them release manuscripts: the pointwise 2-converse [48], Fontaine--Mazur modularity at $2$ [49] and the Goldfeld and 2-converse manuscript [50].

The manuscript flags nothing as numerical, computer-assisted or conditional. Two load-bearing inputs are other release manuscripts that are themselves unreviewed: the pointwise 2-converse (Lemma 3.4, hence Proposition 2.8) and Fontaine--Mazur modularity at $2$ (Section 4.7, hence Theorem 2.4). The fixed data the theory and the tests use (the five points and their contact forms, the multiplier sets and $m,P$) are asserted to exist and are hardcoded rather than computed, and the text says their existence suffices for its argument (p. 12); the exceptional set and constants $S,H,c$ of Theorem 2.4 enter only the proof, which gives no values for them, and its bound for the fibers with complex multiplication in Section 4 rests on a finiteness that "need not be effective" (p. 36). The release folder for this manuscript holds only the README, main.pdf and the TeX build directory; there is no verification/ folder.

Bears on

The manuscript names no Erdős problem, and no problem page is identified as using it. Problem pages were searched for decision problems, rational points on curves, height bounds in terms of squarefree parts or radicals, and the elliptic-curve families of Sections 3 and 5; none records an argument, a conditional route or an open question about the objects these results control.