Configured Observation Planes: Topological Emergence, Discrete Holonomy, and Wave Dynamics from Probabilistic Foundations

The relationship between geometry and probability is usually imposed from outside: one assumes a space, then studies processes on it. This paper develops the Configured Observation Planes (COP) framework, in which geometric and wavetheoretic structure emerges from a probabilistic foundation without being assumed. The starting point is a Markov model on the probability simplex Δ2, whose boundary—the seam—plays the role of a singularity. The central geometric construction is the projective blow-up, which replaces the seam with the real projective plane RP2,encoding the direction of approach as irreducible data. Four main results are established. (i) The seam-resolved triadic Markov kernel is Feller on a compact state space, admits at least one stationary probability measure, and any stationary measuresatisfies an explicit seam-mass upper bound; for the triadic map the seam-entry probability admits a closed-form estimate via Beta distribution tails (Theorem 29). (ii) Klein–Gordon and Schrödinger wave equation dynamics are derived from the overlap-mismatch structure of adjacent probe states under internal assumptions that are discharged from framework-first principles (Discharges A–D). (iii) From probe transition statistics alone, without prior geometric assumptions, we construct a discrete connection, define loop holonomies as curvature proxies, prove stability under graph refinement with explicit error budgets, and establish that persistent nontrivial holonomy constitutes an operational obstruction to flatness; the resulting curvature density proxy is gauge-invariant under arbitrary relabeling of probe vertices (Sections 14–22). (iv) Every stationary measure of the Track U chain satisfies an explicit lower bound on expected Shannon entropy, with the degree of entropic preference controlled quantitatively by the seam parameters of Theorem 29 (Corollary 30). As a structural consequence of the blow-up construction, the oriented lift q : S2 → RP2 carries Z2 holonomy identical in structure to the spinor phase acquired under a 2π rotation. The coupling of the geometry lane (Track G) and the Markov lane (Track U) into a single unified construction, and the full B-strong refinement robustness program, are designated as open problems. The COP framework additionally contains, as a provable special case, the core mathematical structure of Darwinian selection dynamics; this connection—including chirality lock-in theorems, a quantitative diversity floor, and the Z2 parity correspondence—is developed in Appendix G.