THE SPECTRAL SIEVE Filtration Theory, Arithmetic Architecture, Projector-Based Computing, and Hidden Geometric Structure with Complete Proof of the Hard Lemma

We present the complete and unified treatment of the Spectral Sieve — a canonical subspace filtration framework that compresses high-dimensional sensor, arithmetic, and operator data through the dimensional cascade 10 → 5 → 3 → 2 → 1. This edition contains the first complete proof of the Hard Lemma governing the 5 → 3 renormalisation contraction, integrates ten new hidden structures identified via Heisenberg, Poincaré, and Wilsonian RG analysis, and establishes the global necessity of the contraction semigroup structure through six independent architectural constraints. The cascade rests on four architectural axioms. The outer transitions are rigorously proved: Z₂ reflection (10→5), Axiom 4 doublet isolation (3→2), and U(2) Casimir / Poincaré recurrence (2→1). The central hard lemma — the 5→3 Wilsonian RG contraction — is now fully proved via two independent routes: (i) Axiom 3 [globally forced by six independent architectural constraints] + Hille-Yosida theorem; (ii) the arithmetic Bell inequality argument ruling out Re(λᵢ) = 0 from three independent measurements of attractor dimension 3. Twenty-six total results are established: thirteen proved theorems, the Hard Lemma now closed, the SCC now a corollary, ten new hidden structures, five named hidden variables, and five new open problems. The cascade is not chosen — it is forced, at every level, by the global self-consistency of the arithmetic architecture.