Non-Local Search-to-Decision Reduction over $\mathbb{F}_2$

Non-local search-to-decision asks whether two noncommunicating parties, given the two shares of a bipartite encoding of a uniformly random string $x\in \mathbb{F}_2^n$, can both predict the same random parity $\langle r,x\rangle$ without there also being local measurements with which both parties recover $x$. We prove that if their optimal probability of both recovering $x$ by local measurements is $p$, then their probability of both answering a common parity challenge correctly is at most $\min\{1,\frac12+5p^{1/22}\}$. The result is motivated by applications to unclonable cryptography, including unclonable encryption and quantum copy-protection. The proof is information-theoretic and does not provide an efficient extractor. The proof and the exposition were developed with assistance from ChatGPT using GPT-5.6 Sol Pro and Codex in the Ultra reasoning mode.