Finite-Rank Selection and the Gauge Structure of the Observable Boundary

The observable world is not fundamental. It is the residue left on R⁴ when a five-dimensional Majorana bulk is projected through a non-invertible rank-5 map. This paper establishes the gauge structure of that boundary. The gauge group U (1) × SU (2) × SU (3), the fine-structure constant α⁻¹QGT = (F₅ + F₄√5) ², and the electroweak mixing angle sin²θW = F₄/ (2F₅ + F₄) = 3/13 are not introduced phenomenologically. They arise as structural residues of the rank-5 Majorana projection. In particular, sin²θW = 3/13 is established as an unconditional theorem. The observable boundary is identified with the Temperley–Lieb algebra TL₄ (φ). Within this structure, the monodromy conditions G1–G3 and the separation Hproj ⊕ Hdyn close the electroweak sector. In the present version, the local route to H3 is also completed independently through the edge-mode boundary condition. The Standard Model is not modified from within. It is recovered as the unique gauge residue of a non-invertible rank-5 Majorana projection. Version 3 (April 2026): clarified distinction between the integer residue σ = 137 and the exact projection pole α⁻¹QGT; completed local H3 route; revised topological boundary section.