Almost-Everywhere Near-n^5/2 Gate Lower Bounds for SYM ◦THR and THR ◦THR

Building on Chen, Tal, and Wang’s n^(5/2−ε) lower bounds for depth-two threshold circuits, this paper gives an explicit polylogarithmic approach to the exponent-5/2 boundary. It constructs a single language in E^NP that, at every sufficiently large input length N, requires more than Ω(N^(5/2)/log^(9/2) N) gates for SYM∘THR circuits and Ω(N^(5/2)/log^9 N) gates for THR∘THR circuits, even with arbitrary real weights. The proof develops exact column-counting algorithms after logarithmic live-variable restrictions, combining truncated-binomial lifting and threshold noise stability in the symmetric-top case with exact-equation decompositions, small-prime sketches, radix traces, and expander-walk amplification in the threshold-top case. A uniform refuter construction places both bounds on the same language and extends them from an addressed subsequence to every sufficiently large length.