Spatial Coherence C_ij as a Substrate-Agnostic Information-Theoretic Primitive — Volume I: Formal Definition, Invariance Claims, and Placement within the Standard Coherence Fidelity Layer (SCFL)

Abstract

This working paper introduces spatial coherence C_ij as a candidate substrate-agnostic, information-theoretic primitive for complex networked dynamical systems under stress, and formally situates it within the Standard Coherence Fidelity Layer (SCFL). While spatial coherence is conventionally treated in electrical engineering and wave physics as a second-order cross-correlation statistic, this volume proposes an alternative reading: when the underlying system is a complex networked dynamical system rather than a linear medium, C_ij can be interpreted as a normalized measure of residual shared information that network topology can still support under systemic stress. The paper makes four contributions. First, it supplies an operational (not group-theoretic) definition of invariance: after canonical reference normalization and conditioning on network topology and stress class, the qualitative decay trajectory of the spatial coherence matrix is hypothesized — not asserted as demonstrated — to be approximately independent of physical or institutional substrate. Second, it derives the formal definition of C_ij via an absolute-deviation functional form against a reference-epoch baseline, and provides an explicit justification for that choice over Pearson correlation, cosine similarity, direct mutual-information estimators, and Wasserstein distance, weighing robustness to non-Gaussianity, responsiveness to sudden deviation, and computational cost for continuous monitoring. Third, it establishes three analytic bridges connecting the operational statistic to classical information theory: a Gaussian bound linking C_ij decay to mutual information decay, a heuristic link between the antisymmetric drift field and directed transfer entropy, and an interpretation of precision-matrix growth as breakdown of local conditional independence (Markov property violation). Fourth, it defines spectral entropy S_eff over the normalized eigenvalue spectrum of C as a dimensionless, substrate-agnostic early-warning candidate observable. A central structural contribution of this volume is the explicit hierarchical separation between the primitive layer and the fidelity layer: C_ij and its derived quantities (spectral entropy, drift, precision-matrix structure) constitute what is measured; SCFL defines the canonical reference conditions, aggregates multiple primitives across spatial, temporal, hierarchical, and cross-layer domains, supplies falsification criteria, and converts raw invariants into decision-grade observables — i.e., how measurement is standardized. This distinction, underspecified in earlier SCFL manuscripts, is made explicit here to remove a persistent source of ambiguity in how primitives relate to the fidelity layer that governs them. All empirical and cross-domain claims in this volume are explicitly framed as testable hypotheses rather than demonstrated results. An illustrative (non-evidentiary) cross-domain mapping table sketches how the same normalized statistic would be interpreted under SCFL conventions across power grid, financial system, supply chain, and healthcare substrates, but the paper is explicit that whether numerical thresholds and collapse trajectories are actually comparable across these domains is the empirical question reserved for Volume II. Limitations are stated directly: sensitivity of the absolute-deviation form to reference-epoch quality and non-stationarity, the local-Gaussian dependency of the mutual-information bound, and the still-heuristic (not rigorously derived) status of the transfer-entropy link. Volume II — Cross-Domain Validation of Spatial Coherence Collapse Signatures — is reserved for controlled empirical testing of the substrate-independence hypothesis (Hypothesis H1) via side-by-side spectral-entropy trajectories and matrix visualizations across a common reference-normalization pipeline, with illustrative candidate stress events including IEEE test systems, ERCOT Winter Storm Uri, SVB-style liquidity freezes, regional healthcare capacity stress, and supply-chain disruptions. This volume is theoretical and definitional in scope. It does not present validated results and should not be cited as empirical confirmation of substrate-agnostic collapse behavior.