Wire–Correlation Tradeoffs at Exponent 7/2 for Depth-Two Threshold Circuits

We prove almost-everywhere correlation lower bounds for unrestricted THR∘THR and SYM∘THR circuits measured by physical wires, raising the wire-exponent frontier for these classes from 3 to 7/2, up to logarithmic factors. Write L = ceil(log_2(n+2)). For every fixed c > 0 and K_T > 5+2c, K_S > 4+2c, there is one language in E^NP, depending on c, K_T, K_S but on nothing else, such that for every fixed pair of multipliers A_T, A_S > 0 and every sufficiently large n: every THR∘THR circuit with at most A_T n^(7/2)/L^(K_T) wires, and every SYM∘THR circuit with at most A_S n^(7/2)/L^(K_S) wires, agrees with the language on less than a 1/2 + L^(-c) fraction of inputs. Further regimes trade the denominator against the advantage: endpoint powers L^5, L^4 at constant advantage with suitably small positive coefficients, a log* n variant, and polynomial advantage n^(-c) at caps n^(7/2-eps) for every fixed 0 < c < eps/2, the range eps > 1/2 following from the author's earlier near-cubic wire theorem. Since a normalized circuit with G bottom-gate occurrences has at most (n+1)G wires, gate-small circuits are wire-small, and the constant-advantage endpoint gives almost-everywhere correlation gate bounds at positive-coefficient caps n^(5/2)/L^5 and n^(5/2)/L^4, improving the author's earlier gate preprint in both the logarithmic denominators and the form of hardness. Two ideas drive the improvement. First, building on the restriction-and-CAPP framework of Chen, Tal, and Wang, coordinate restrictions are replaced by a partition of the cube into products of Hamming-code stars: the partition charges a threshold only for genuinely mixed dependence on the two live halves of a cell — one-sided behavior, however complicated, is absorbed into exact scores — and the mixed difference is nonzero with probability O(r^(3/2) f / n^(3/2)) for fan-in f and r blocks. Second, that structural gain must survive two hostile conversions, deterministic counting and hardness amplification, and most of the new machinery exists to keep each conversion from spending it: a matrix sampler with one fixed trace moment is accurate for all rank-one selectors simultaneously, so the absorbed one-sided behavior never has to be enumerated; repaired count lists keep every reported count state Boolean, with controlled intermediate ranges, even on failed hash seeds; and a sparse masked verifier carries the counting algorithm through the Chen–Lyu–Williams almost-everywhere transfer without multiplying discarded mass by the coefficient mass of an XOR reconstruction. We specify the imported algorithm-to-lower-bound interfaces, prove the new compiler and transfer steps in detail, and give three scoped obstructions to stronger conclusions from these methods.