Translation-invariant Fourier covariance and path-correlated codeword search

Translation-invariant second moments on a finite abelian group have diagonal Fourier spectrum. We observe that this standard diagonalization equates the expected coherent bad mass after subgroup addition with the ordinary bad mass, for any fixed subgroup and bad-error set. The identity applies to translation-equivariant ensembles generated by local constraints, independently of the constraint graph. We apply it to path-correlated codeword search for folded Reed-Solomon codes under an independent-row, exactly half-balanced random bit oracle. For every integer λ ≥ λ0 := 3604, a uniform quantum circuit uses exactly 2(n − 1) queries to the same bit oracle available classically, poly(λ) gates and qubits, and exact finite-gate synthesis. Its average success is at least 1 − 4bλ, while every adaptive classical strategy making at most ⌊2λ1/4⌋ queries has success at most e−n/2000; here n = 2⌊log2 λ⌋ − 1 and bλ ≤ 3e−n/800. Conditioning on an explicit event of probability at least 1 − 2√bλ yields a worst-case bounded-error promise separation. The algorithmic framework and general decoding comparison are those of Yamakawa-Zhandry. The path application supplies the correlated error law, adaptive pair-query bound, and resource assembly. Complexity is measured in unary λ; complete algorithms and lower bounds for general constraint graphs remain open. MSC 2020: 81P68; 68Q12; 94B35. The package contains the paper PDF, LaTeX source, vector figures, figure data, resource ledger, and checksums.