# Verification of the Paneitz product theorem

The full analytic argument has been self-audited, including the Laplacian sign convention, completeness of the product basis, all zero-mode coincidences, strict index endpoints, finite spectral cutoffs, the threshold eigenvalue 2/9 and the fixed-surface error term. There are no known gaps in the stated family theorem. The general AIM classification remains unresolved.

The companion checker passes 12,000 exact conditions in normal and optimized Python. Rational mode values are synthetic; the checks do not assert realizability as hyperbolic spectra. The all-parameter proof does not rest on this finite sample.

The built-in document compiler succeeded. All five PDF pages were visually inspected, including the author footnote, formulas, proof endings and references. The final local compiler log has no errors, missing references or overfull/underfull boxes. The sealed package also records extracted-archive execution and recompilation results.

This is originating-researcher self-audit, not independent review or formal verification. Historical priority is undetermined.