Phase Transitions in Labeled Metric Spaces: A Scalar Phase Boundary from Geometric-Topological Coherence on k-NN Graphs

Learnability in labeled metric spaces is governed by a scalar phase boundary θ(d_eff) = A / d_eff, emerging from the geometric–topological coherence of k-nearest neighbor graphs. Through a correspondence with the q-state Potts model, the local label coherence τ* maps to an energy density, and its complement Θ = 1 − τ* serves as an effective temperature consistent with the maximum entropy principle [3], with coarse-graining invariance verified at std = 0.003 ≪ 0.05. The functional form θ ∝ 1/d_eff is derived via hyperscaling on the label-coherent subspace, where d_eff denotes its fractal correlation dimension. The form is empirically robust (Spearman ρ = +0.367, p = 0.006 over 54 OpenML datasets) and acts as a strict upper bound in the sub-critical regime d_eff < 1/ν, with zero violations across 362 datasets. The 3D Ising critical exponents (γ = 1.237, ν = 0.630) provide the conceptual scaffolding for the derivation; the proportionality constant A is problem-specific, with empirical invariant I = ρ_c · d_eff = 3.03 ± 0.33 (95% CI [2.45, 3.71]) measured across datasets, reflecting the universality of the geometric form rather than of any fixed numerical prefactor. Beyond the phase boundary, Allen–Cahn nucleation theory establishes an energy barrier growing with disorder, consistent with empirical persistence of dead states under monotone local corruption; full irreversibility (hysteresis) remains untested. Under the associated Topological Relaxation Flow, the effective dimension contracts monotonically—verified on 58,000+ instances—producing a survival gap mechanism G(t) ≥ G(0) > 0 that drives recoverable systems away from the boundary. The contribution is the geometric–topological derivation of a phase boundary whose form is universal across labeled metric spaces: a scalar law of learnability that captures the order–disorder transition without requiring fixed numerical thresholds.