B20: Phase Geometry, Emergent Gauge Structure, and the Structurally Derived Invariant X = 4π³ + π² + π in BRISM

This paper develops a phase-based structural formulation of the BRISM (Brane Interface Substrate Model) invariant X = 4π³ + π² + π, establishing an additional, independent construction alongside its symmetry-layer framework. Within the BRISM framework, physical observables are understood as projections of an underlying unitary structure onto a real measurement interface. In this context, X has been shown to organize microscopic observables, including an infrared relation for the fine-structure constant with ~10⁻⁹ level agreement without parameter selection. The present work introduces an independent construction based on phase geometry. Starting from phase structure, Born projection, and stability under phase mixing, X is re-derived as the total admissible phase-space capacity of the interface. The contributions π, π², and 4π³ arise from one-, two-, and three-dimensional phase structures combined with interface-specific doubling effects. The central result is that both the symmetry-layer formulation and the phase-geometric construction converge to the same invariant. This indicates that X is not tied to a specific representation, but reflects a structurally overdetermined property of the interface. The work should be understood as a transition step within BRISM: it establishes a dual interpretation of X and points toward a deeper, representation-independent structural description of admissible physical configurations. For a current top-down overview of the BRISM framework and its wider structural context, see: Current BRISM Top‑Down Overview All BRISM papers on Zenodo >> Searchlist Supplementary structural material, dependency maps, and reproducibility notes are available in an accompanying public repository: https://github.com/swencarloheinze/brism-framework