Selection as Universal Geometry: Darwin, Turing, and the Quantum Bridge

We show that the replicator equation — the mathematical statement of natural selection — is the unique continuous-time limit of a general discrete update kernel, the Darwin kernel, defined on the probability simplex. This kernel is formally identical, up to choice of fitness function, to the Feller kernel of Track U in the Configured Observation Planes framework of Spencer (2026a). The identification implies that selection, in every domain where it operates, is a geometric phenomenon: the flow of a vector field on a probability simplex, governed by the same topological constraints whether the variants are organisms, strategies, memes, or quantum states. We establish, in sequence: the derivation of the Price equation and gauge invariance from the Darwin kernel (Theorems 1–2); the quasispecies as the leading eigenvector of the mutation-selection matrix (Theorem 3); the error threshold as a seam-crossing condition (Theorem 4); chirality lock-in as topological protection of a Z2 holonomy (Theorem 5); Nash equilibria as stationary measures of the Darwin kernel on the strategy simplex (Theorem 6); the correspondence between the blow-up tower and the Turing jump hierarchy (Theorem 7); the Born rule as a consequence of COP gauge invariance (Theorem 8); and the recovery of Track U from the Darwin kernel in the zero-fitness limit (Theorem 9). A sequence of appendices extends these results: the holometabolic seam-crossing as the mechanism of all major evolutionary transitions (Appendix H); the GUClosure theorem establishing that Track G and Track U are the same process seen from two directions (Appendix J); and the application of the framework to cultural and computational systems, including the identification of the internet as a Σ2 system and the characterization of algorithmic recommendation as a self-referential Darwin kernel at Σ3 (Appendix K). The central open question — whether the Darwin kernel and the background geometry of quantum field theory are formally the same object — is consistent with every result proved here and is designated as the primary open program.