Unified Coherence Framework (UCF) v1.0: A Substrate-Agnostic Derivative-Alignment Architecture for Rupture Geometry Measurement

The Unified Coherence Framework (UCF) is a cross-domain derivative-alignment measurement architecture for detecting and classifying coherence degradation geometries in large-scale dynamical systems prior to system failure. The framework addresses a single governing question: do physically and institutionally distinct systems exhibit separable coherence degradation geometries, and can a single measurement architecture detect them without domain-specific tuning? UCF constructs a scalar coherence functional C (t) = λₘax (M (t) ) from the dominant eigenvalue of a windowed derivative-alignment matrix M (t), where M (t) integrates rank-1 projections of the observable state derivative ẋ (t) over a sliding window. This functional is extended to a seven-field coherence vector G (t) via structural subspace partitioning, producing a multi-dimensional deformation signal that preserves differential coherence loss across physical, operational, and institutional layers. The measurement pipeline is formally substrate-agnostic: the mathematical form is identical regardless of whether the observable state encodes grid frequencies, hospital occupancy rates, or supply chain flows. Four canonical coherence deformation geometry classes are formally defined by necessary and sufficient topological conditions on the deformation trajectory D (t) = G (t) − Gbaseline (t): G1 (Layered Temporal Decay), G2 (Front-Loaded Collapse), G3 (Spatial Gradient Steepening), and G4 (Coordination-Gated Rupture). The classes are discriminated by three independent structural axes — temporal synchrony, spatial concentration, and cross-layer coordination gap — with no two classes sharing the same signature across all three axes. The statistical architecture includes a pre-registered falsification clause, 2-Wasserstein separation methodology, mandatory window sensitivity testing, and a dual decision rule requiring both statistical significance (p θ) simultaneously. A complete calibration envelope specifies validity conditions across five dimensions: sampling rate, observable dimensionality, signal-to-noise ratio, window length, and resolution dependence. This document presents the complete canonical methods spine of UCF v1. 0 in seven parts and one appendix: Measurement Charter (I), Mathematical Core (II), Geometry Taxonomy (III), Statistical Structure (IV), Calibration Envelope (V), Cross-Domain Demonstration Logic (VI), Version and Boundary Statement (VII), and a worked numerical example for a three-node system demonstrating the full pipeline from derivative estimation through Wasserstein separation (Appendix E). An Executive Preview page provides geometry classification reference and falsification condition for rapid orientation. The framework is explicitly versioned and living: v1. 0 establishes the mathematical specification and extension logic. Retrospective validation against historical rupture events, pre-registered separation tests, and prospective deployment constitute the v2. x program. A hard falsification clause — if W₂ (μR, μN) is statistically indistinguishable from W₂ (μN, μN) under pre-registered parameters, the framework is falsified for that domain and observable partition — converts UCF from a measurement proposal into a scientific theory.