Higher Koszul Homology of Square-Zero Algebras in Positive Characteristic

We compute the higher Koszul homology, defined by the Euler cap differential, of A=k plus V with V squared zero and finite-dimensional V. In each total weight, the calculation is the natural map from invariants to coinvariants of a signed cyclic rotation. A word orbit contributes higher classes exactly when its sign is compatible and its length is divisible by the characteristic. An explicit necklace formula gives every bigraded dimension. In every positive characteristic, k[x,y]/(x,y)^2 is Koszul but has nonzero higher Koszul homology; a three-term characteristic-three cycle and boundary witness are displayed. The first nonzero positive homological degree is p-1 in characteristic p. This is a complete counterexample to a characteristic-free extension of Koszul acyclicity and the comparison with left Koszul homology. It does not resolve the characteristic-zero converse associated with UnsolvedMath record OWR-14218-004. Classical cyclic-word decomposition is credited. The English, AI-assisted preprint includes a complete proof, a self-audit and105 exact diagnostic checks. It is unrefereed; no absolute priority or formal verification is claimed.