A Cogenetic Derivation of P versus NP
Morgan Petrik, Militant.AI · Zenodo (CERN European Organization for Nuclear Research) · 2026
This work presents a process-first derivation and Cogenetic resolution of the relation between the classical complexity classes P and NP. It begins with a standard deterministic Turing-machine model and retains computation as a finite temporal configuration process carrying complete program ancestry, transition cost, output, witness or branch registration, validation, discarded alternatives and projection residual. Classical complexity classes are recovered only afterwards as uniform projections over arbitrarily large finite inputs.The derivation shows that generation preceding validation is not itself a separating invariant: the same order occurs in polynomial-time computations, a decision procedure need not reproduce an apparent witness search, and SAT decision and witness search are polynomially interreducible under a polynomial decision hypothesis. A universal-machine carrier and an exact integer polynomial-degree coordinate then recover both classical horns. P = NP corresponds to persistence of one uniform machine–degree pair through every finite input refinement. P ≠ NP corresponds to finite survivor-channel extinction of every bounded machine–degree family, with each rejected candidate retained in the refutation channel.The resulting primitive process relation is written P♥NP. Equality and inequality are recovered as secondary, contextually resolved coordinates: equality records closure of a carried comparison, while inequality records a retained differential. Both classical horn relations remain available, but their Boolean marks become exclusive only after placement in a completed quotient that erases ancestry, differential and residual. The conventional P-versus-NP binary is therefore recovered as a non-injective static-shadow projection of the underlying computation process.The supplementary material includes the complete LaTeX source, bibliography, citation metadata and a deterministic standard-library verifier. The verifier checks 175,452 finite cases across 22 labelled manuscript claims, including configuration refinement, cost accounting, SAT self-reduction, survivor–refutation transport, persistence, quantisation non-injectivity and anti-collapse. These executable checks validate the stated finite constructions; they are not empirical substitutes for the manuscript’s asymptotic or foundational derivations.