# Verification and limitations

The analytic proof uses a common H1 form after a determinant-one affine pullback, an analytic eigenpair normalized to mean one, the equilateral symmetry and perimeter Hessian, compactness at bounded perimeter, and an explicit thin-triangle estimate. The cubic formula follows by eliminating the second corrector from two weak equations and applying the divergence theorem.

The originating researcher's detailed self-audit accepts these stated theorems. This is not an independent review or a proof-assistant certificate. The original all-parameter question is not closed, and no numerical value of the common weak-coupling threshold is supplied. The fourth-order remainder in the cubic expansion is fixed-domain only.

The exact standard-library checker verifies 3,477 equalities and 3,456 inequalities, including 1,728 ordered tangency triples, 20 ball moments and the perimeter Hessian. Normal and optimized runs have identical complete reports. The finite checks are supplementary to the analytic proof, not a numerical solution of the PDE or a proof of the universal claim.

The six-page PDF was compiled with the desktop LaTeX compiler and Tectonic. Every rendered page was inspected; no overfull or underfull boxes, undefined references or missing glyphs remain. The source archive is separately extracted, the checker rerun in both modes, and the PDF recompiled before release. Machine-readable receipts retain the results.