# Verification of the cohomology annihilator formulas

Version 1.1 gives complete written proofs for the two named base families in every dimension and characteristic, as well as for every nonfield PID. The proof has received an originating-researcher self-audit, not independent peer review or proof-assistant certification.

The transfer lemma identifies both directions of factorization through finite projectives. The polynomial matrix factorization supplies an exact two-periodic resolution, so eventual annihilation forces the scalar endomorphism to factor through a projective. Scaled Jordan modules bound the entire eventual ideal. A kernel projection over a PID and scalar averaging when n is invertible give the nonzero lower bounds. Parameter-power Koszul resolutions prove the least indices.

The modular extension uses a classical symmetric-power nilpotent operator in characteristic p. Its top power is congruent to a scalar matrix modulo (a^(mp),b^(mp)). A coefficient comparison in the single-block transfer identity, not the trace of the original transfer equation, yields the necessary condition c in (a^m,b^m). Intersecting over all positive m forces c=0. Restriction along the finite-free inclusion S[y]/(y^p) into S[x]/(x^(hp)), y=x^h, treats all n=hp. No division by h occurs, including when p divides h. The argument does not claim that the colon condition is sufficient for p>2.

The two standard-library programs perform 55,889 and 6,148 exact finite diagnostics. The extension tests primes 2, 3, 5, 7 and 11, power substitutions, matrix-unit transfers and induction multipliers, with negative controls for the characteristic, full-block and two-parameter hypotheses. Arithmetic is shared with the hash-pinned original checker. Normal and optimized executions agree. These diagnostics support the written argument but do not establish its universal quantifiers by enumeration.

The package preserves both accepted proof records and their evidence hashes, and includes no third-party source PDF. The new manuscript has been compiled and all seven pages visually inspected. A separate extracted-source build and checker runs verify portability. The general source record remains PARTIALLY_SOLVED. Neither a new full source closure nor a new manuscript is counted, and novelty and absolute priority remain unestablished.