[polyopt] Fox: thermodynamic-limit certificate for the triangular J1-J2 bulk gap

[hungry-foolish-12138]

Team

Team name	Fox
Members	肖元; 冷昊阳

Challenge

Row
Challenge	Can the reported ~0.3 J1 triangular spin-liquid gap survive a boundary-free thermodynamic-limit certificate?
Catalog issue	Addresses #88 — Certified bulk spectral-gap bounds for frustrated spin-1/2 models.
Track	tracks/polyopt/

Result

We turn the reported ~0.3 J1 gap at J2/J1 = 0.10 into a boundary-free thermodynamic-limit SDP certificate search.

• Eight d=1 Rung-A instances have independently verified exact primal witnesses.
• The d=2 Rung-A instance is constructed exactly: 60,467 variables, 507 affine equalities, and PSD block dimensions [1430, 506].
• No certified-infeasible point was obtained; the scientific verdict is explicitly inconclusive, not a claimed gap bound.
• The submission includes an independent exact/interval certificate verifier, full source and tests in the attached archive, machine-readable audit artifacts, and a human-facing mathematical report.

Reproduce

See tracks/polyopt/solutions/Fox/README.md. The adjacent archive contains the full source, tests, and audit artifacts; verify it using SHA256SUMS.

This contribution establishes a reproducible certificate pipeline and sharpens the remaining open problem without turning solver failure into physics.