Cross-Domain Empirical Evidence for Φ = I·ρ − α·S as an Indicator of Operational Regime

We report the cross-substrate recurrence of a three-coordinate scalar, Φ = I·ρ − α·S, as a bounded empirical indicator of operational regime across tested mechanical, aerospace, electrical, geophysical, quantum, computational, biological, and large-language-model substrates. The parameters α = 0.1 and Φc = 0.25 for engineered-cluster substrates were fixed in production code on 2025-10-21 and reused unchanged in three subsequent repositories through 2025-12-31. A later observational LLM repository reused both values in code on 2026-01-27, with the threshold logged only and not used to gate execution. A USPTO provisional patent application filed on 2025-11-20 provides disclosure evidence independent of git history. The evidence takes three forms. Regime-separation evidence is reported for mechanical bearings, aerospace turbofans, electrical grids, geophysical cases, quantum qubits and two-qubit gates, and computational neural-network architectures: Φ falls below the applicable reference threshold in documented failing or critical cases and remains above it in the tested stable controls, where available, using domain-specific implementations of I, ρ, and S. Scope-boundary evidence is reported for biological cardiac data: under a matched protocol, Φ separates arrhythmia transition-positive windows from normal-symbol windows (AUC = 0.9148) but does not separate atrial fibrillation from non-AFIB rhythm (AUC = 0.5556), establishing a transition-versus-stable-state scope bound. Observational structural-role evidence is reported for large language models: Φ supports the substrate-shared structural role of the three coordinates without threshold-gated regime decisions. Most evidence is retrospective. A preregistered forward test on a mechanical system outside the original scalar validation set produced end-of-life Φ = -0.342, satisfying the preregistered criterion Φ < 0.25 under a commitment chain documented by git history. The pattern is not claimed to generalize beyond the tested substrates, and documented limitations and counterexamples are reported in the manuscript. Every numerical claim traces to a code-output artifact at a pinned commit hash in repositories released with this paper. The methods described in this paper are the subject of U.S. Provisional Patent Application No. 63/921,348 (filed November 20, 2025) and additional U.S. provisional patent applications in the portfolio. No license to implement or commercialize the described methods is granted by this publication. All rights reserved.