The P = W conjecture for fixed-determinant (SL_n) Higgs moduli in composite rank

Theorem 1.1: $P_k=W_{2k}=W_{2k+1}$ on all of $H^*(X^B,\mathbb Q)$ for the smooth coprime fixed-determinant, trace-free moduli in every composite rank $n\ge4$ , genus $g\ge2$ , including the variant (non- $\Gamma$ -invariant) cohomology; with the earlier prime-rank theorems this gives the fixed-determinant conjecture in every coprime rank. A corollary answers Maulik-Shen's Question 5.5 on Betti weights of endoscopic images. Non-coprime (singular) cases and other groups are not treated.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The P = W conjecture for fixed-determinant (SL_n) Higgs moduli in composite rank · P = W for SL_n, composite rank

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Registry verification: unreviewed · announcement · partial

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VibeMathed record: The P = W conjecture for fixed-determinant (SL_n) Higgs moduli in composite rank

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as the paper states it; it claims $P_kH^m=W_{2k}H^m=W_{2k+1}H^m$ on the full rational cohomology for every composite $n\ge4$ , coprime $d$ , every $L$ and every genus $g\ge2$ curve. The proof (an $\mathfrak{sl}_2$ operator built from logarithmic moving-pole moduli, following Hausel-Mellit-Minets-Schiffmann) was not refereed and is not formalized. It uses Mellit's toric stratifications and published logarithmic Hodge moduli constructions as inputs.

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The P = W conjecture for fixed-determinant (SL_n) Higgs moduli in composite rank

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