Discrete Crystallization on Labeled Metric Spaces Pruning Capacityand Finite-Time Reachability under a Label-Conditioned RicciFlow

This paper studies the finite geometric interval between an already encoded labeled-locality tensor and a preglobal discrete crystallized state. The primitive locality kernel, its categorical representation T^Y_{cc'j}, and the calibrated diagonal contraction

                                                                                 τ_i^*(t) = (1/k) ∑_{j ∈ N_k(i,t)} 1[Y_j = Y_i]

are inherited from the static locality theory. A label-conditioned Ricci-type flow is then used as a geometric driver that moves the points, changes the metric rankings, and thereby changes the realization of τ_i^*.

For a class containing n_{Y_i} observations, the unique local crystallization capacity is

                                                                                          τ_i^max = min{1, (n_{Y_i} - 1)/k}.

Hence complete observed crystallization is possible exactly when n_{Y_i} ≥ k + 1; smaller classes generate unavoidable capacity defects. We define the canonical fixed-budget pruning projection, derive the exact flow pair-distance identity with its directed reciprocity factor, and characterize uniform active inter-label expansion by a trajectory parameter α^* < 1.

A signed geometric velocity cone derives the coherent-radius, inter-label barrier, and active expansion conditions directly from the SRF field. A finite branchwise certificate pulls these inequalities back to the initial configuration, while a terminal capacity region supplies absorption. An integer potential then proves finite entry into the moving pruning projection for this explicit certified class. The result is an entry-and-invariance theorem, not a metric fixed-point theorem.

The graph-edge diagnostic is defined by one weighted Forman functional with unit vertex weights and is typed separately before and after crystallization. Its phasewise ranges depend on the actual degrees and edge-weight ratios; no universal k-only range is assumed, and neither diagnostic drives the flow.

Finally, observed purity is separated from externally validated fidelity. A finite hypergeometric baseline and a null false-crystal lemma distinguish aligned, null, and anti-aligned crystallization. The only global statement retained is spectral. In the successfully crystallized regime, the order-2 BBGKY closure cannot resolve the Fiedler mode.

Under the critical P5 spectral model and target-mode equipartition, the missing Dirichlet-energy fraction is

                                                                                             1 - R_E^2 = (3 - √5)/16 ≈ 0.04775,

so the transition preserves most, but not all, of the label-geometric signal under those hypotheses. A separate finite-sample experiment reported R^2 = 0.96; that numerical observation is not used in the theorem. No metric fixed point or global topological classification is claimed.