The Chimerical Foundations of Computing: How Computational Complexity Repeated Hilbert's Program

The standard formulation of P vs. N P treats computation as a purely mathematical process — an abstract Turing machine stepping through states withoutphysical constraints. We argue this is a category error. Turing (1936) derived his machine from the physical limitations of a bounded agent; Gandy (1980) proved thatthese limitations are necessary and sufficient: any discrete deterministic device satisfying four physical principles computes exactly the Turing-computable functions. The class P is therefore defined by physics. We show that the Cobham–Cook program (1964–1971), which axiomatized P as “intrinsic” and launched the P vs. N P question, rests on six implicit axioms — each of which is refuted by its own authors’ words. The framework is not incorrect but incomplete: it captured the syntax of polynomial time while discarding the physical semantics that explain why the class exists. Restoring those semantics, we prove that the disorder ratio ρϕ > 0 for every polynomial-time representation ϕ ∈ FP of random k-SAT at the satisfiability threshold αc (k ≥ k0, Ding–Sly–Sun 2022). The proof proceeds via a sensitivity dichotomy: low-sensitivity representations suffer geometric collisions between SAT and UNSAT formulas (unconditionally), while high-sensitivity representations require simultaneously satisfying seven mutually incompatible properties — the impossibility profile established in the companion paper [23]. Combined, these yield ρϕ > 0 for all ϕ ∈ FP, hence P̸ = N P .