Constitutive Geometry of Systemic Coherence: Orthogonal Bases, Dimensional Independence, and Faithful Projections in the UCMS/SCFL Framework

Reinsurance internal capital models and Solvency II validations routinely combine risk factors into composite scores and aggregate distributions. When those factors are correlated yet treated as occupying independent dimensions, the resulting models contain irreducible geometric leakage. This leakage is not statistical noise or calibration error — it is a structural consequence of projecting onto non-orthogonal bases. Even sophisticated dependence modeling cannot remove it once the dimensions have been defined as entangled. This paper supplies the missing constitutive geometry. It defines the Coherence Workspace as a finite-dimensional inner-product space whose basis vectors must satisfy four non-negotiable requirements before any operator is applied: orthogonal bases, independent investigative dimensions, distinct coordinate projections, and non-redundant analytic frames. Only under these conditions do drift, acceleration, precursor, and half-life operators produce signals that reflect actual system behavior rather than artifacts of dimensional overlap. The central technical result (Proposition 1) is derived via explicit Gram matrix construction. In a non-orthogonal basis with correlation ρ, the oblique projection estimator of the component along one dimension incurs a residual of ρ² / (1 − ρ²) times the energy in the contaminating dimension. For a typical ρ ≈ 0.5 between composite risk sub-indices, this creates a geometric leakage floor of approximately 33 % of the energy present in the second dimension. This floor cannot be eliminated by recalibration inside the entangled basis; it is removed only by orthogonalizing the investigative dimensions at the point of definition. The framework operationalizes this requirement through the Dimensional Audit — a four-stage protocol that enforces both conceptual and statistical independence before dimensions are admitted as basis vectors. A Projection Fidelity Theorem then shows that only faithful coordinate projections preserve ordering, separation, and threshold integrity; composite indices and variance-driven rotations can make a system appear stable even while coherence is degrading toward rupture. A synthetic case study on the Strait of Hormuz isolates the geometric effect under controlled entanglement and demonstrates a six-week precursor window invisible to conventional composite indices. The geometry is substrate-agnostic and is supported by two independent real-data validations: a retrospective study of 738 rupture-adjacent events in the ORNL EAGLE-I grid archive (2015–2020) with a mean SCFL precursor lead time of 8.9 hours, and a prospective locked-basis, no-look-ahead deployment on ERCOT telemetry in April 2026. The paper positions this constitutive geometry against three adjacent bodies of work: manifold and Riemannian approaches to risk (descriptive rather than prescriptive), statistical orthogonalization techniques (PCA, ICA, Gram-Schmidt), and existing early-warning literature on critical transitions. It supplies the upstream measurement substrate those approaches presuppose but do not define. Primary audiences include reinsurance capital modeling and model validation teams, critical infrastructure operators, hospital system resilience functions, and researchers working on AI constitutional and governance stacks. For reinsurance practitioners the framework offers a rigorous upstream diagnostic layer that can be applied to existing risk factor libraries to quantify structural leakage in current internal models and to improve the reliability of drift and early-warning signals used in capital monitoring. Companion empirical studies Brogdon, R. (2026). Before the Alarm: Detecting Irreversible Coherence Deformation and Loss of Recoverability in the U.S. Power Grid (2015–2020). DOI: 10.5281/zenodo.20514713 Brogdon, R. (2026). The SCFL: A Real-Time Measurement Instrument Validated on ERCOT Grid Stress Events. DOI: 10.5281/zenodo.19488182 Brogdon, R. (2026). Strait of Hormuz Coherence and Drift Index (HCDI 1.2). DOI: 10.5281/zenodo.19105053 Author Ronald Brogdon Team Leader, Upstream Coherence Measurement Stratum (UCMS) ORCID: https://orcid.org/0009-0009-0507-2971 [email protected]