Computational free will as global selection: from sheaf-theoretic gluing to a conditional separation of P and NP
2026-08-31 • Logic in Computer Science
Logic in Computer Science
AI summaryⓘ
The authors present a way to understand computational free will by distinguishing between choices that are locally allowed and the selection of a single overall outcome. They use mathematical structures called presheaves and define processes GLUE and SELECT to capture admissible choices and the actual selected outcome. Their findings suggest that if computational free will exists as they define it, it means some problems can be verified quickly but not solved quickly by an algorithm, implying P is not equal to NP. However, this conclusion depends on certain assumptions and does not prove anything unconditionally.
computational free willpresheafFNP search problempolynomial timeP vs NPdeterministic algorithmpolynomial verificationtrace relation
Authors
Jerome Clech
Abstract
We formalise computational free will by separating locally constrained admissibility from the selection of one global continuation. Global sections of a finite choice presheaf form an admissible set, GLUE; SELECT singles out the continuation realised at a pre-identified occurrence. A uniform trace relation certifies that continuation efficiently after the act, although it is assumed not to be uniformly anticipable in polynomial time from the prior occurrence input. Under explicit uniformity, balance, historical-completeness, and unique-projection assumptions, this trace defines a total FNP search relation with no deterministic polynomial-time selector. Thus existence of computational free will in the stated sense implies a separation between polynomially verifiable and polynomially solvable search, and hence that P differs from NP. The result is conditional and gives no unconditional class separation.