Unitary Manifold Restoration and the Spectral Topology of the Riemann Zeta Function — A Complete Obstruction Classification for the Riemann Hypothesis

This paper presents the Unitary Manifold Restoration (UMR) framework and a structural analysis of the Riemann Hypothesis obstruction. We prove the following unconditional results: the Viète Convergence Threshold (E (σ) 1/2) ; the Near-Zone Detection Theorem (any failure of monotonicity in the near zone localizes an off-line zero within an explicit disk, with quantitative depth control; monotonicity is therefore unconditional at all heights where RH is verified, and holds under RH in general), with corrected coverage quantification; the Sign Partition Lemma (negativity of a pair-geometry contribution requires vₖ > δ₀ strictly) ; the Cascade Theorem (v1 (σ, γ) > 0 for σ > σ* combined with the Vinogradov–Korobov zero-free region, with no circularity) ; the Pair Geometry Theorems (negative-contribution window, integral identity, single-pair dominance) ; the Explicit-Formula Bridge (ψ (x) − x = −2√x · Re S (x, T) + O (x log²x / T), empirically verified against the first 300 zeros; the coherence bound A 0 for all σ > 1/2) is equivalent to RH — a reformulation, not a simplification.