Near-Inverse-Linear Barriers for Explicit Affine Witness Isolation

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 $\delta <1$, a worst-case singleton success guarantee $\mathrm{\Omega }(n^{-\delta })$, where $n$ is the witness arity, implies $NP\subseteq P\mathrm{/}poly$. More generally, any success guarantee $\omega (\log n\mathrm{/}n)$ yields satisfiability circuits of size $(s+2{)}^{O(1)}{2}^{o(v)}$ for length-$s$ descriptions with at most $v$ witness variables, hence subexponential in $v$ when $s=v^{O(1)}$. Conversely, SAT search-to-decision gives deterministic perfect isolation, making polynomial-resource fixed-exponent strong affine isolation equivalent to $NP\subseteq P\mathrm{/}poly$. For explicit unions of at most $n^{\beta }$ affine components, success $\mathrm{\Omega }(n^{-\delta })$ implies the same collapse whenever $\beta ,\delta \ge 0$ and $\beta +\delta <1$. The inverse-linear endpoint is not claimed.