Near-Inverse-Linear Barriers for Explicit Affine Witness Isolation

Abstract: We study randomized nonuniform polynomial-size transformations that output explicit binary affine filters for circuit inputs whose nonempty satisfying sets are affine. The filter may depend on the entire input description; no affine basis is supplied. For every fixed $δ&lt;1$, a worst-case singleton success guarantee $Ω(n<sup>{-δ})$, where $n$ is the witness arity, implies $NP\subseteq P/poly$. More generally, any success guarantee $ω(\log n/n)$ yields satisfiability circuits of size $(s+2)<sup>{O(1)}2<sup>{o(v)}$ for length-$s$ descriptions with at most $v$ witness variables, hence subexponential in $v$ when $s=v<sup>{O(1)}$. Conversely, SAT search-to-decision gives deterministic perfect isolation, making polynomial-resource fixed-exponent strong affine isolation equivalent to $NP\subseteq P/poly$. For explicit unions of at most $n<sup>β$ affine components, success $Ω(n<sup>{-δ})$ implies the same collapse whenever $β,δ\ge0$ and $β+δ&lt;1$. The inverse-linear endpoint is not claimed.