Curvature Crystallization on Labeled Metric Charts: Phase Separation Using Discrete Ricci Flux in k-NN Complexes

The central question is geometric: when a discrete Ricci-type flow is driven by a locally defined sign structure on a finite metric space with label-induced boundary conditions, under what conditions do global topological invariants emerge, and what is their structure? The setting is a k-nearest-neighbour complex; the boundary conditions are binary label-induced curvatures κ^S_ij = ±1. Phase separation of the Forman–Ricci spectrum emerges as a consequence of the geometry: intra-class edges converge to a curvature mode κ+ while inter-class edges are expelled in finite time, yielding a bimodal-to-unimodal transition. Classification is a corollary of the geometric fixed point. This paper belongs to discrete differential geometry. Seven results are established. (i) The canonical observable τ* is not a heuristic: it is the unique functional satisfying six independent axioms (A1–A5, A3′) from measure theory, and simultaneously the onlycandidate passing five eliminative filters (F1–F5) from a behavioral perspective; neither system was designed to produce it (Theorems 2.2 and 2.7, Section 2). (ii) A spectral admissibility criterion replaces Hamilton’s pointwise positivity with a mode separation condition on κ_eff (Theorem 4.4). (iii) A Chernoff–Union analysis identifies τ > 0.88 as the critical stability threshold (Theorem 5.2). (iv) The triangle-clustering term of the curvature derivative, γ_3 C e^{κ_F} with γ_3 = 2η, is derived from first principles via discrete BBGKY closure at order 2; the remaining terms constitute an empirical conjecture (R^2 = 0.96, Theorem 5.4 and Conjecture 5.5). (v) A two-phase Lyapunov argument proves bimodal convergence with T_stab < ∞ (Theorem 6.3). (vi) A forbidden-surgery principle establishes that the flow reaches a post-separation fixed point without requiring external intervention (Theorem 7.4). (vii) Curvature crystallization is identified as the geometric engine of Semantic Tolerance compliance: κ_F-crystallization ⇒ τ ↑ ⇒ ρ ↓, where the phase boundary θ(d_eff) = A / d_eff^α follows from the dimensional scaling of the transition tensor, not from empirical fitting (Theorem 8.3, Remark 2.6).