Non-vanishing of commutator-character moments on the minimal SU(3) lattice
Year: 2026
Type: preprint
Abstract: Let U0, U1, U2 in SU(3) be independent and Haar-distributed, with group commutators W_ij = U_i U_j U_i^-1 U_j^-1. For a fixed cubic trace polynomial e and irreducible representations X, Y we study the moments E(X,X,Y) = 27 ∫ |e|² χ_X(W01) χ_X(W02) χ_Y(W12). These integrals have signed integrands and reduce to differences of channel sums, so their sign is not evident. We prove that E(1,1,Y) > 0 and E(3,3,Y) > 0 for every irreducible representation Y of SU(3), with explicit rational closed forms in the Dynkin labels of Y. For X = 1 the proof is by hand. A crossing identity, the dual Pieri rule and three Casimir moments give every channel weight in closed form, Z_Y(R_i) = (d_i²/d_Y) ρ_i, with an explicit coefficient ρ_i in [0,1) that depends only on content differences. This yields the two-sided bounds 36|A|/d_Y < E(1,1,Y) < 54|A|/d_Y. For X = 3 the proof combines a Young seminormal path model, a degree bound and exact rational evaluation into a computer-assisted certificate. On every chamber, E(3,3,Y) is an explicit rational function whose shifted numerator has only positive coefficients. Scope: exact finite-dimensional integrals. Nothing is claimed about transfer-operator spectra, mass gaps, infinite volume or the continuum. ### Additional notes (paste as written)Supplement: the X = 3 source bundle (54 files), Parts A–D (the referee carry, computation records, Lean 4 files with build log, and REPRODUCE.txt with a test log). Reproduction commands 1–7 were run by the author on 2026-11-07, and all passed. The two Lean files (Lean 4.35.0 / Mathlib v4.35.0) compile with exit code 0. Disclosure: drafting, computation scripts and refereeing were done with AI assistance. Claude (Anthropic) was the drafter. ChatGPT (OpenAI) and Grok (xAI) were used as independent hostile and verifying referees. The paper has not yet been reviewed by a human. The author is responsible for all content.
Author: CANRIGHT, IAN
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225567737