Uniqueness of the Rank-5 Projection in the Quantum Gauge Theory of Time

This foundational closure note proves that the rank-5 projection used throughout the QGT corpus is unique. The result closes the rank-5 uniqueness gate: r = 5 is the only finite rank compatible with three foundational requirements of the framework — Noetherian conservation, autosimilar second-order closure, and operational metric stability. The proof connects three previously independent strands of the corpus: spectral curvature of the Pincherle operator, holonomy threshold at n = 5, and boundary metric stability. Ranks below 5 fail conservation or autosimilar closure; ranks above 5 introduce sub-threshold redundant modes and violate metric stability. From this point forward, r = 5 is not a modelling choice within QGT. It is the unique admissible finite rank of the non-invertible projection Π : M⁵ → R⁴. The admissible mode set, boundary metric, fine-structure constant, gauge structure, and subsequent QGT results therefore rest on a closed foundational rank theorem.