Openai 2026 fontaine mazur modularity prime 2

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theorem_1_1: The manuscript's main claim: the odd, regular two-dimensional Fontaine-Mazur conjecture over Q at the prime 2, with no hypothesis on the residual representation, scalar and reducible residues included. Claims checked only.

OpenAI, Fontaine--Mazur modularity at the prime 2, OpenAI Math Release preprint, October 6, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision fd4aeeb2e), folder preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026; the held PDF, paper.pdf in the release, is retained as openai_2026_fontaine_mazur_modularity_prime_2.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:Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026,
  author = {{OpenAI}},
  title = {{Fontaine--Mazur modularity at the prime 2}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026/paper.pdf}{OAI:Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026}},
  year = {2026}
}

The held PDF has 72 pages with a text layer; its title page is dated 6 October 2026 and its metadata records a creation date of 7 October 2026. The TeX source is build/main.tex (91 lines), which inputs eight section files under build/sections/ (introduction.tex, preliminaries.tex, deformation.tex, completed_forms.tex, patching.tex, fields.tex, connectedness.tex and classicality.tex, 5,434 lines together), with build/references.bib (64 entries, all printed, as [1]--[64]). TeX paths below are relative to build/.

Attestation. The release README states that the manuscripts were "produced by an internal OpenAI model", 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, not this family's subject. The manuscript's own README carries the title, the author line "OpenAI", the date, the citation block and a version note saying that this version "updates citations to revised manuscripts in this collection", with a link to the previous version; it adds no sentence about authorship, assistance or verification. The title page and the PDF metadata name OpenAI as the sole author. 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 lists no formalization of this manuscript: its Lean catalog lean/formalization.yaml has no entry for it or its family at revision fd4aeeb2e, and there is no page lean/docs/010.md.

Companions. The release groups this manuscript in its family, "Unrestricted pro-modularity at the prime two", with two others; neither has a card in this library, so neither is linked. The first, Unrestricted pro-modularity at the prime two (October 6, 2026, with an earlier version dated October 4, 2026), is described by the release as a completed-Hecke occurrence result for every continuous, odd, absolutely irreducible two-dimensional $2$-adic representation of $G_{\mathbb Q}$ unramified outside finitely many primes, with no de Rham hypothesis and no claim of classical modularity; its introduction cites Theorem 1.1 here to make a regular de Rham point classical. The Dimension of the Two-Adic Hecke Algebra at Odd Level (October 5, 2026) is described by the release as proving that every irreducible component of the full $2$-adic Hecke algebra of level $\Gamma_1(N)$, $N$ odd, has Krull dimension four, the case $p=2$ of Emerton's dimension conjecture; it and this manuscript do not cite each other. The release also keeps an earlier dated version of this manuscript, Fontaine--Mazur modularity at the prime 2 (September 23, 2026, folder preprints/Fontaine-Mazur-modularity-at-the-prime-2-September-23-2026, 72 pages), which this version's README links as the previous version; its TeX bundle differs from this one only in the date line and in the bibliography link to the release's Hilbert's tenth problem manuscript (the earlier version links that manuscript's September 24 version, this one its October 6 version), so the statements and labels cited on this card are the same in both. Outside the family, Hilbert's tenth problem over the rational numbers (October 6, 2026), filed at the Hilbert's tenth problem card, cites Theorem 1.1 as the modularity input of its height argument (its Section 4), for a two-dimensional $2$-adic representation built from the Tate module of an abelian surface with quaternionic multiplication, and reads weight two off the normalization of Section 2.1 here. The companions' arguments were not read for this card.

Read status: claims checked for Theorem 1.1, together with the statements of Lemma 2.2 (the virtually solvable case), Theorem 5.1 (localized propagation), Theorem 7.1 (potential pro-modularity) and Lemma 8.5 (the dense modular points obtained from Thorne's theorem), read clause by clause in the release's TeX source (sections/introduction.tex label thm:main at lines 20--30, sections/preliminaries.tex label lem:solvable-reduction at lines 146--152, sections/patching.tex label thm:localized-propagation at lines 23--38, sections/connectedness.tex label thm:potential-promodularity at lines 11--24 and sections/classicality.tex label lem:dense-Thorne-points at lines 395--399); the statements of most other numbered results were read, not clause by clause, for the account below; the proofs were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

The numbering below is the PDF's: theorems, propositions, lemmas, corollaries and remarks share one counter within each section.

• Section 1, Introduction (pp. 1--5). Calls a continuous $r:G_{\mathbb Q}\to\mathrm{GL}_2(\overline{\mathbb Q}_p)$ odd if $\det r(c)=-1$ for a complex conjugation $c$, and regular de Rham at $p$ if its restriction to $G_{\mathbb Q_p}$ is de Rham with two distinct Hodge--Tate weights, and states Theorem 1.1 (p. 1), the case $p=2$ of the conjecture, with every semisimple residual representation included. Together with the results of Pan (2022, Theorems 1.0.2 and 1.0.4) and Zhang (2025, Theorem 1.0.2) at odd primes, it says, the conclusion holds at every prime. It names the release's Hilbert's tenth problem manuscript as a user of the theorem. Section 1.1 (pp. 2--3) reviews Wiles and Taylor--Wiles, Serre's conjecture (Khare--Wintenberger, Kisin), Skinner--Wiles, Allen's nearly ordinary dyadic theorem with solvable residual image, Kisin and Emerton, and Tung's dyadic lifting theorem, whose global residual image is nonsolvable. It credits Thorne's Theorem D (a 2026 arXiv preprint) with an "essential set of classical points": ordinary, potentially crystalline representations with weights $\{0,1\}$ are modular at every prime with no residual irreducibility hypothesis. Section 1.2 (pp. 3--5) defines a pro-modular locus, a closed trace locus cut out by a completed Hecke algebra of definite quaternionic forms with no local condition at $2$, and lays out the three stages (potential occurrence for a whole family, classical vectors at the target, solvable descent) with the roadmap Figure 1 (p. 4). Section 1.3 (p. 5) lists the constructions: the Laurent-series deformation theory of Böckle--Iyengar--Paškūnas, patching after Taylor--Wiles, Kisin, Khare--Wintenberger, Pan and Zhang with finite jets, and marked local lines. A table of contents fills p. 6.
• Section 2, Conventions and initial reductions (pp. 7--10). Table 1 (p. 7) fixes geometric Frobenius, $\epsilon=x|x|$ with Hodge--Tate weight $-1$, weights counted as cyclotomic exponents, the central character $\psi=\chi\epsilon$ and the block parameter $r\otimes\epsilon$. Section 2.1 writes $\det r=\chi_{\mathrm{fin}}\epsilon^w$ and fixes the arithmetic normalization $V_f$ of an eigenform's representation, with cyclotomic exponents $\{0,k-1\}$ in weight $k$, and the conversion of a dual realization. Lemma 2.1 (p. 8) shows that any continuous representation of a profinite group on $\overline{\mathbb Q}_2^n$ is defined over a finite extension of $\mathbb Q_2$ and has a stable lattice there. Lemma 2.2 (p. 8) proves Theorem 1.1 for virtually solvable $r$ (scalar identity component excluded by distinct weights, induction from a real quadratic field excluded by regularity, induction from an imaginary quadratic field given by a theta series) and shows that otherwise $r$ stays absolutely irreducible over every finite extension. Section 2.3 (pp. 9--10) fixes the base fields $F$ (totally real, solvable over $\mathbb Q$, of even degree, $2$ split completely), $d=[F:\mathbb Q]$, $s=2d+u$ with $u$ the number of allowed odd finite places, two-dimensional determinants in Chenevier's sense, the pseudodeformation ring $R_F^{\mathrm{ps},\chi}$, the quaternion algebra $D_F$ ramified exactly at the real places, and characteristic-two curves.
• Section 3, Deformations and local geometry (pp. 10--24). Proposition 3.1 (p. 10) on representation spaces under Mazur's condition $\Phi_2$, with recorded invariant lines; Lemma 3.2 bounds the dimension lost in passing to traces by three plus the undistinguished lines; Lemma 3.3 gives dense curves and coefficient points; Lemma 3.4 (p. 13) identifies deformations at a curve with residue field $k((t))$ through an enhanced coefficient ring, following Böckle--Iyengar--Paškūnas, and Lemma 3.5 computes curve-field cohomology. Proposition 3.6 (p. 15) bounds generators minus relations over the local framed base below by $3-6m-h^0(D^\vee(1))$; Corollary 3.7 gives a local line chart of dimension at least five at $\mathbb Q_2$. Lemma 3.8 (p. 17) shows dihedral and reducible loci are small and Lemma 3.9 kills inertia by solvable base change. Proposition 3.10 (p. 19) describes the real--dyadic local base (a normal complete intersection domain of dimension $1+8d$) and two tame models, unipotent and with inertia eigenvalues $\pm i$, sharing a special fiber; Lemma 3.12 (p. 22) shows that marked scalar dyadic factors give an integral base of dimension $2d+5m$.
• Section 4, Completed forms and their local parameters (pp. 24--32). Proposition 4.1 (p. 24): completed definite forms over $F$ are finite free over an Iwasawa algebra, their locally algebraic vectors are classical, the regular non-character eigenpackets transfer to Hilbert cusp forms with Galois representations, regular classical vectors are dense, and each Hecke factor is a reduced torsion-free quotient of $R_F^{\mathrm{ps},\chi}$. Proposition 4.2 (p. 26), from Paškūnas--Tung's block finiteness: the local block action on completed forms is the restriction of the global Galois action, the Hecke algebra is finite over the local pseudodeformation ring, and absolutely irreducible regular de Rham local parameters give locally algebraic vectors. Proposition 4.4 (p. 29) bounds every Hecke component's dimension below by $1+2d$; Lemma 4.5 (p. 30) gives exact auxiliary-level identities and Lemma 4.6 (p. 32) forward solvable base change.
• Section 5, Propagation from a characteristic-two curve (pp. 32--43). Theorem 5.1 (p. 32): a potentially pro-modular, non-virtually-solvable characteristic-two curve with finite local images at the allowed odd places makes every irreducible closed locus containing it potentially pro-modular, with no local condition at $2$. Lemmas 5.2--5.3 choose auxiliary primes and twist characters; Lemma 5.4 (p. 38) patches finite jets over an ultrapower with a uniform Artin--Rees bound; Proposition 5.5 (p. 40) gives the patched regular sequence and support dimension at least $2d+4s+h$; Proposition 5.6 (p. 41) gives full support through character twists, the irreducible $\pm i$ comparison model and sign intertwiners. The proof of Theorem 5.1 closes on p. 43.
• Section 6, Fields with controlled Selmer groups (pp. 43--53). Proposition 6.1 (p. 43) builds abelian totally real $2$-extensions $\mathbb Q\subset A\subset F$, split completely at $2$, of arbitrarily large degrees with a bounded number of allowed odd places, preserving the residual data, with marked dyadic places when the residual character ratio is trivial at $2$, a dual Selmer vanishing and a strict Selmer bound. The construction uses cyclic subfields of $\mathbb Q(\zeta_r)$ and a one-prime genus argument (Lemma 6.2), switching of local conditions in the manner of Ramakrishna, with the needed variant proved in the text (Lemma 6.3), Chebotarev evaluations (Lemma 6.4), the distinct-character construction (Section 6.4, with Lemma 6.5) and an explicit conductor count (Section 6.5).
• Section 7, Connectedness and potential pro-modularity (pp. 53--58). Theorem 7.1 (p. 53): for $r$ as in Theorem 1.1 and not virtually solvable, $r$ is pro-modular over a solvable totally real Galois extension of even degree split completely at $2$, and so is the closed trace image of any complete integral framed deformation family over $\mathbb Q$ through $r$, with determinant $\det r$ and finite ramification, over one field and at one tame level. Lemma 7.2 (p. 53) gives a classical cuspidal lift of the residue (Serre's conjecture when absolutely irreducible; when reducible, existence is credited to Billerey--Menares and an Eisenstein-difference argument is given). Lemmas 7.3--7.5 (pp. 55--56) show that charts with transverse marked lines have large components joined by large intersections, that potential pro-modularity holds on all or none of a chart's components, and that the transverse residual fiber is geometrically connected. The proof (pp. 57--58) joins a seed curve and a target curve through that fiber and propagates.
• Section 8, Classicality and descent (pp. 58--67). For reducible $r|_{G_{\mathbb Q_2}}$: Lemma 8.1 (p. 59) picks the invariant line of larger cyclotomic exponent; Lemma 8.2 builds a family $\mathscr D$ of dimension at least two recording that line's character; Proposition 8.3 (p. 61) proves the ordinary part finite over the torus Iwasawa algebra; Lemma 8.4 (p. 62) makes $\mathscr D$ finite over a one-dimensional weight space; Lemma 8.5 (p. 63) shows that at a dense set of its points the representation, after a global twist and dualizing, is ordinary and potentially crystalline with Hodge--Tate weights $\{0,1\}$, hence modular by Thorne's Theorem D; Section 8.5 (p. 64) carries the resulting inducing order to the target; Lemmas 8.6--8.7 (pp. 65--66) give a genuine ordinary eigenvector and prove it locally algebraic by a polynomial-orbit argument. Section 8.8 (p. 67) treats absolutely irreducible $r|_{G_{\mathbb Q_2}}$ by the $p$-adic local Langlands correspondence (Colmez; Colmez--Dospinescu--Paškūnas), passes by Jacquet--Langlands to a Hilbert cusp form over $F$, and descends along prime-degree cyclic steps with finite-character corrections to conclude Theorem 1.1.
• References (pp. 68--72), 64 items, among them the 2026 preprints of Thorne (arXiv 2608.07186v1) and Emerton--Gee--Pan--Zhu and the release's Hilbert's tenth problem manuscript.

The manuscript flags nothing as numerical, computer-assisted or conditional. Its reducible local branch rests on Thorne's Theorem D, a 2026 preprint cited at statement level; Emerton--Gee--Pan--Zhu and Pan's 2026 classicality theorem are discussed for comparison and are not used as inputs. The release folder for this manuscript holds only the PDF, the TeX build directory and the README; there is no verification/ folder.

Bears on

The manuscript names no Erdős problem, and no problem page is identified as using it. Where a problem page or one of its claim pages records an argument through elliptic curves or modularity, as for Problems 364, 368, 937, 939 and 960, the curves are over $\mathbb Q$; for Problem 368 the argument is the modular approach through Shimura curves that its Pasten claim page records. Elliptic curves over $\mathbb Q$ are already known to be modular by Breuil, Conrad, Diamond and Taylor, so Theorem 1.1 adds no step there.