The computational complexity class P is traditionally defined as a purely mathematical set of functions computable in polynomial time by a Turing machine. Thisabstract formulation, however, omits the physical constraints under which any real computing device must operate. We argue that the separation P̸ = N P is notmerely a syntactic property of formal systems, but a consequence of the fundamental limits of information processing in a locally causal universe. By restoring thephysical resource axioms originally derived by Alan Turing (1936) and formalized by Robin Gandy (1980), we show that the entropy of the satisfiability backbone atthe critical threshold αc imposes an irreducible information-geometric barrier. This barrier, which we term the Shannon-Lebesgue Gap, prevents any polynomial-timemechanism from resolving the global connectivity of a random k-SAT formula without exponential energetic or temporal cost. Using the framework of harmonic Maassforms and mock modularity, we formally prove that P̸ = N P emerges as an analytical necessity when the channel capacity of the computing agent is bounded by itsown physical locality.Keywords: P vs. N P , Shannon-Lebesgue Gap, satisfiability threshold, backbone entropy, Turing locality, Gandy axioms, Information-Geometric barrier.
Benjamín Felipe Pérez Contreras (Thu,) studied this question.