Full explainer:
P = W compares two filtrations on one cohomology: the perverse filtration from the Hitchin map and the weight filtration of the twisted character variety.
• An OpenAI preprint, “P=W in composite rank for fixed determinant” (September 24, 2026; not yet peer reviewed), proves P = W on the full cohomology of the twisted SL_n character variety for every composite n ≥ 4, including the variant part.
• Our rung ladder (Claude, Codex, Gemini and Grok, with Muse in round 1; each step audited by another model) counted matrices over finite fields: for SL_4 at genus 2 the count splits by character order into 1 + 15 + 240 pieces, computed exactly in Tungsten, and by hand each variant piece equals a shifted endoscopic expression, what a Prym torus times a smaller twisted character variety on a covering curve contributes to one deck character (an equality of E-polynomials; one step of reading it as that variety stays open).
• With older tools (Maulik and Shen, Hausel and Thaddeus, Shende) those counts prove P = W through degree 18 for SL_4 at genus 2, and on every full-order piece in every rank. At degree 19 the order-2 classes form Lefschetz strings of lengths 7 and 6, and that route stops; the preprint’s lowering operator settles it. By hand, refereed by AI models, not kernel-checked; the counting method is Mereb’s. The partial results were not found in what we could read.
Code, certificates and an interactive explorer:
The preprint, in OpenAI’s openai/math repository:
Tungsten, the language every computation and frame is written in:
Satori, our proof checker:
Animation drawn in Tungsten; proofs checked in Satori; narration by ElevenLabs.
Independent channel, not affiliated with @OpenAI.
#Shorts #math #openai #geometry
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