Emergent Origin of the Fine-Structure Constant in the BRISM Interface: A Minimal π-Structure, SU(5) Leakage, and a 10⁻⁹-Level Agreement

This work develops a structural account of the fine‑structure constant within the BRISM (BRane Interface Substrate Model) framework. Under the interface requirements of phase neutrality, positivity, σ‑additivity, norm conservation, and the Naimark–Stinespring dilation embedding, a unique dimensionless interface functional emerges from three invariant geometric contributions associated with U(1), SU(2) and SU(3). This functional lies numerically close to the empirical inverse fine‑structure constant. It is further shown that the small residual between the geometric interface value and the experimental value can be interpreted as transverse leakage into the full 24‑dimensional SU(5) sector that minimally completes the U(1)×SU(2)×SU(3) interface algebra within the dilation space. A first‑order, parameter‑free inverse‑suppressed correction derived from this SU(5) completion reproduces the empirical value at the 10⁻⁹ level in the infrared - in percentage terms, this amounts to approximately 99.99999960% Accuracy. No closed‑form identity is claimed. Instead, the results indicate that the fine‑structure constant in the infrared may be understood as a stability‑fixed emergent quantity governed jointly by the BRISM interface geometry and its minimal SU(5) dilation completion.