The P = W conjecture for fixed-determinant moduli spaces
From OpenAI Aftermath
The P = W conjecture for fixed-determinant moduli spaces concerns the two filtrations that nonabelian Hodge theory places on the same cohomology: the perverse filtration P from the Hitchin map on the moduli space of Higgs bundles and the weight filtration W from the mixed Hodge structure of the character variety. The conjecture asserts that Pk = W2k. A manuscript in OpenAI's public mathematics collection, dated 24 September 2026, claims it for the fixed-determinant, trace-free SLn moduli spaces in every composite rank, on the full cohomology, which with the known prime-rank cases would complete the smooth coprime fixed-determinant case in every rank. The collection lists no Lean formalization of this result.
Background
- As recounted in the manuscript's introduction:
- Origins. The weight symmetries behind the conjecture arose from the arithmetic study of character varieties by Hausel and Rodriguez-Villegas, extended by Hausel, Letellier and Rodriguez-Villegas. De Cataldo, Hausel and Migliorini formulated the filtration comparison and proved it in rank two.
- The general linear group. Generation by tautological classes was established by Hausel and Thaddeus in rank two and by Markman in every rank, and Shende determined their weights. De Cataldo, Maulik and Shen proved the conjecture for GLn in genus two, and Maulik and Shen proved the coprime GLn conjecture in every rank and genus; further proofs are due to Hausel, Mellit, Minets and Schiffmann and to Maulik, Shen and Yin.
- Fixed determinant. The group of n-torsion line bundles acts on the SLn moduli space, and the tautological classes do not generate the part of the cohomology on which it acts nontrivially, the variant cohomology. De Cataldo, Maulik and Shen determined both filtrations on the variant part in prime rank, and Maulik and Shen constructed endoscopic correspondences compatible with the perverse filtrations in every rank, without identifying the filtrations in composite rank.
The original result
- Claim, 24 September 2026. The main theorem of P = W in composite rank for fixed determinant states that for every smooth projective complex curve of genus at least two, every composite rank n ≥ 4, every degree d coprime to n and every line bundle of degree d, the perverse and weight filtrations on the rational cohomology of the twisted SLn character variety satisfy Pk = W2k = W2k+1, including the variant cohomology. The manuscript states that, combined with the prime-rank cases, this resolves the smooth coprime fixed-determinant, trace-free conjecture in every rank, and it derives an answer to a question of Maulik and Shen on the weights of endoscopic correspondences for the canonical-twist operator.
- Scope, as stated: the result concerns the smooth case of degree coprime to the rank.
- Method, as described: one degree-two class satisfies relative hard Lefschetz on the Higgs side and curious hard Lefschetz on the character variety, the latter proved on the full cohomology by lifting Mellit's toric stratifications through a determinant-root cover. A degree −2 operator is built from a logarithmic pole moved over the curve with a full flag, eigenvalue monodromy and a flag correspondence; it lowers the perverse filtration by two and the weights by four, and a lattice Riemann–Roch computation gives the commutator identity that identifies both filtrations as sums of the same eigenspaces. The algebraic pattern follows Hausel, Mellit, Minets and Schiffmann.
- Sources. Commit
adc7f12of the openai/math repository: P = W in composite rank for fixed determinant. - Formalization. The collection lists no Lean scope note or Comparator challenge for this family.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 31 October 2026; AI Proof Watch row 1 November 2026) |
|---|---|---|---|---|
| 2026-10-12 | AI Proved What?, episode 043 P = W, every rank?, remade version of 26 October 2026 | Reading, exposition | Video explainer of the 39-page preprint. It recalls the conjecture of de Cataldo, Hausel and Migliorini, its proofs for GLn and the prime-rank fixed-determinant case of de Cataldo, Maulik and Shen, and states that the result concerns the smooth coprime case and that its answer to Maulik and Shen's question is stated for the canonical-twist operator. | Episode status, last checked 30 October 2026: posted as a preprint, not withdrawn or revised; under expert review, not yet; peer reviewed, not yet; checked in Lean, no; no public assessment by specialists found. It notes that the paper builds on the method of Hausel, Mellit, Minets and Schiffmann and on Mellit's toric stratifications. The site states that it is made with Anthropic's Claude. Read on 31 October 2026. |
| 2026-10-16 | J. Jardine, Lattice24, referee review rank_407 in JJardine919/openai-math-audit | Review, AI-assisted audit | Reports no gap. It states that the reviewer read all nine sections and checked, among other steps, the comparison criterion, the flag correspondence, the dimension counts and the Riemann–Roch computation. Its two minor points concern cited inputs it could not check against the sources: a perversity statement attributed to de Cataldo, Maulik and Shen in arbitrary rank, which it rederives from the endoscopic support statement, and the form of Mellit's stratifications in the relative case, cited from a proof rather than a stated theorem. | Part of the completed Lattice24 audit; the reviews are published under CC BY 4.0. |
| 2026-10-18 | AI Proof Watch, record OpenAI reports P = W for fixed determinant in composite rank | Catalogue record, assessment | Rates the importance as major, as completing the fixed-determinant P = W comparison in every rank coprime to the degree when combined with the earlier prime-rank results, and the AI involvement as AI-led. It lists as limitations that the claim is for smooth curves of genus at least two, coprime degree, fixed determinant, trace-free moduli and rational cohomology. It rates the technical verification as reported and the human understanding as few, the defaults when no higher level is documented, consistent with the Formalization section on this page. | Curated public tracker; record published 18 October 2026, assessment reviewed 31 October 2026. Read on 1 November 2026. |
Notes
- The background summarizes the manuscript's own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
