THE SPECTRAL SIEVE Filtration Theory, Arithmetic Architecture, Projector-Based Computing, and Hidden Geometric Structure

We present a definitive unified treatment of the Spectral Sieve — a canonical subspace filtration framework executing the dimensional cascade V₁₀ → V₅ → M³ → H₂ → span (P̂₂) through four architectural axioms. This paper amalgamates five prior documents into a single work that is strictly superior to any individual source: incorporating corrected benchmarks, the most rigorous proofs, all empirical discoveries, the complete open-problems programme with proof strategies, and an original hidden-structure analysis. Two outer transitions are rigorously proved: the 10→5 linear Z₂ reflection sieve and the 2→1 U (2) projector-sector invariance. The 5→3 scale contraction (Hard Lemma) is characterised precisely as a rank-2/nullity-3 dissipation problem, verified numerically, and given a new proof strategy. Corrected benchmarks on level ground confirm the sieve's advantage is variance reduction (~40% lower standard deviation than PCA at moderate noise), not mean fidelity — precisely the theoretical prediction. An arithmetic pillar demonstrates that approximately ten dominant Riemann zero channels account for the recoverable large-scale structure of the von Mangoldt fluctuation field, with out-of-sample prediction achieving ~4% relative error across two orders of magnitude. Phase-weighted carriers improve reconstruction dramatically (r₃₀: 0. 665 → 0. 964), identified here as Berry phase compensation — coherent arithmetic phase alignment. A systematic hidden-structure scan across the full mathematical landscape of the framework yields seven new structures not present in any source paper: the doublet ladder is identified as the arithmetic quantum harmonic oscillator (Eₙ = n + ½) ; the projector P̂₂ is the maximum-entropy (infinite-temperature) density matrix of H₂; the reflection operator is identified as the operator form of the functional equation ξ (s) =ξ (1−s) ; and the degradation law rN ~ 1 − C (log N) ²/ (2N) is derived analytically from the explicit formula via Parseval. Three mathematical frameworks — Heisenberg uncertainty, Poincaré recurrence, and the Renormalisation Group — then yield nine further structural results: the Hard Lemma is a Heisenberg necessity; the O (ε²) fidelity advantage is the Casimir theorem, not empirical; εcrit is a Poincaré-Hopf bifurcation on S²; the 5→3 step is Wilsonian RG integration with γ₄, γ₅ relevant and γ₁, γ₂, γ₃ marginal; and P̂₂ is a topological RG fixed point.