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 (\varepsilon>0), a deterministic (2^{,n-n^{\delta_\varepsilon}})-time acceptance-probability algorithm and a worst-case, infinitely-often (n^{3-\varepsilon})-wire lower bound for (\mathrm{THR}\circ\mathrm{THR}). The paper also introduces the root-fan-in measure (W_{1/2}=\sum_i\sqrt{f_i}) and proves an (n^{3-\varepsilon}) frontier for both (\mathrm{THR}\circ\mathrm{THR}) and (\mathrm{SYM}\circ\mathrm{THR}), subsuming the known (n^{2.5-\varepsilon})-gate and (n^{3-\varepsilon})-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.