Wire-Charged Restrictions: Touching and the Root-Fan-In Frontier

This note develops touching, a wire-sensitive accounting principle for random restrictions of depth-two threshold circuits: if a restriction leaves (k) variables live, the expected number of touched bottom gates is at most (kW/n), where (W) is the wire count, and a good live set can be found deterministically in polynomial time with no bad columns. Combined with a parameter-explicit extension of the Chen–Tal–Wang CAPP pipeline and the Bathie–Williams/Chen–Williams transfer, this yields, for every (ε>0), a deterministic (2^{,n-nδ_ε})-time acceptance-probability algorithm and a worst-case, infinitely-often (n3-ε)-wire lower bound for (THR). The paper also introduces the root-fan-in measure (W1/2=∑_i√f_i) and proves an (n3-ε) frontier for both (THR) and (SYM), subsuming the known (n2.5-ε)-gate and (n3-ε)-wire regimes and yielding mixed gate–wire tradeoffs. It identifies the exponent-3 barrier of the present black-box method and isolates a Live-Literal Lemma whose proof would raise the wire exponent to (3.5). The wire lower-bound consequence is already implied by the stronger almost-everywhere, inverse-polynomial-correlation result in “Almost-everywhere near-cubic wire lower bounds for SYM∘THR and THR∘THR” (https://doi.org/10.5281/zenodo.21984712); the present paper is a complementary account designed to expose the underlying mechanism and its frontier.