Almost-Everywhere Near-Cubic Wire Lower Bounds for SYM ◦THR and THR ◦THR

This paper proves almost-everywhere near-cubic wire lower bounds against depth-two threshold circuit classes. For every fixed (c>0), it constructs a language in (ENP) that, at every sufficiently large input length, cannot be approximated with agreement (1/2+n⁻ᶜ) by (SYM) circuits having (O(n^3/log¹⁰n)) wires or by (THR) circuits having (O(n^3/log¹²n)) wires; at any fixed positive advantage, the denominators improve to (log^5 n) and (log^9 n). The proof develops a deterministic circuit-acceptance-probability algorithm whose cost depends on the wires touching a restricted set of live variables, combining exact residualization, multiscale counting polynomials, signed rectangular multiplication, hardness amplification, and an algorithm-to-lower-bound transfer.