B17: BRISM and Mass–Energy Equivalence: Deriving E = mc² as an Interface Equilibrium Toward a Shared Structural Basis of Quantum Mechanics and Relativity

This work shows that the mass–energy equivalence E = mc² arises within BRISM (Brane Interface Substrate Model) as a necessary equilibrium relation of a two‑sided bulk–brane interface. BRISM is not proposed as a replacement for quantum mechanics (QM) or relativity theory (RT), but as a structural substrate on which both operate as effective descriptions. Starting from interface axioms—phase neutrality, positivity and σ‑additivity, orthogonal additivity, homogeneity, and spectral stability—the interface readout is shown to admit a unique quadratic norm corresponding to the Born density. The bulk–brane architecture then induces a geometric orthogonal decomposition of interface projections into tangential (brane‑parallel) and normal (bulk‑to‑brane) components. Defining momentum as the norm of the tangential component, rest mass as the impedance encoded by the normal component, and energy as the appropriately scaled norm of the total projection flux yields the invariant relation E² = (pc)² + (mc²)², and hence E₀ = mc² in the rest configuration. An explicit non‑circularity audit clarifies that no Lorentz postulates, Minkowski metric, or spacetime kinematics are assumed at any stage; the result follows solely from quadratic norm additivity and interface geometry. In this sense, mass–energy equivalence appears in BRISM as a norm identity of interface equilibrium. The outlook emphasizes that BRISM thereby provides a shared structural basis: quantum mechanics describes the statistical interface readout, while relativity theory captures the effective kinematics and geometry of the interface response. A structured consistency analysis of the underlying BRISM interface framework is provided in the accompanying Supplement (DOI: 10.5281/zenodo.19296467) All BRISM papers on Zenodo >> SearchlistRemark on Versions.Compared to v1.191, version v1.213 provides a logically more robust refinement with clearer status distinctions and an expanded non‑circularity clarification, strengthening its reviewer‑facing presentation