Development and Validation of Compatibility Metrics for Structural Organisation

This paper presents the complete development and validation of the Compatibility Metrics framework as a unified operational methodology for quantifying structural organisation within constrained physical systems. The framework is built around three primary observables—the Compatibility Index (C), Transition Index (D), and Cumulative Reorganisation (ΣD)—supported by complementary measures of modal concentration, participation ratio, spectral entropy, accessibility, and observer-limited reconstruction.

Validation is performed through six sequential investigations examining compatibility evolution, spectral organisation, trend decomposition, observer-limited accessibility, intrinsic stabilisation, and collective compatibility under competing organisational basins. Across all studies, the metrics consistently identify progressive structural organisation, cumulative reorganisation, localized transition dynamics within globally stable organisation, and deterministic behaviour under finite observer accessibility. The results establish the Compatibility Metrics as a transferable operational toolkit for analysing structural organisation across diverse constrained systems and provide the methodological foundation for subsequent Phase II applications to particle physics, gravitational accessibility, spin dynamics, and other complex physical systems.