A characteristic obstruction in higher Koszul homology

Over every field k of positive characteristic p, the free quadratic algebra k<x,y> is Koszul but its higher Koszul homology with coefficients in the algebra and fundamental cocycle e_A has a nonzero degree-one class in total weight p. The explicit class sums the distinct cyclic rotations of x^(p-1)y, split into prefix and final letter. A complete calculation shows that each cyclic word orbit with cardinality divisible by p contributes one higher class in degrees zero and one. This refutes the literal unrestricted-field equivalence stated in Andrea Solotar's contribution to Oberwolfach Report 10/2016, printed pages 479-481, and its version with a degree-zero conjunct. It does not resolve the characteristic-zero vanishing-to-Koszulness converse, the general converse, or uniquely determine the authors' historical intended field scope.

The result is an elementary assembled classical consequence of Loday-Quillen's tensor-algebra Hochschild invariant/coinvariant calculation, Goodwillie's Euler cap formula and the Berger-Lambre-Solotar definition. No new substantial homological mechanism is claimed. The note records an explicit characteristic obstruction and complete elementary specialization. The dated bounded priority audit located no earlier explicitly stated exact counterexample in its inspected corpus; historical firstness, novelty and worldwide current openness are unestablished. Related current literature, including the 2026 published derived-calculus article, is checked with version and final-body access limits recorded.

The deposit contains a three-page research note and a verification archive with editable source, all 65 frozen original files and four evidence archives, independent algebra/source-scope and bounded-priority reviews, exact finite matrix and word-operator controls, integrity checks and negative controls. Fresh diagnostics execute 2110 explicit checks per normal and optimized mode, covering 234 cases, 14 prime witnesses and ten negative controls; deliberately corrupted expectations are rejected. These finite checks supplement the universal algebraic proof and do not certify it or priority. Downloaded primary publications are excluded; source locators and hashes are retained. OpenAI tools were used extensively in development, solving, drafting, coding and adversarial verification. This is an unrefereed preprint; no completed external human peer review or formal proof-assistant certification is claimed.