Unitary Manifold Restoration and the Spectral Topology of the Riemann Zeta Function — A Structural 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) < ∞ iff 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 v_k > δ_0 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 < 2 implies RH unconditionally); the Supply–Demand Obstruction and Gallagher gap quantification; and the Universal Obstruction Theorem (every classical approach reduces to one of two irreducible obstructions, A or B). We also establish twenty-two Closed Channels, classifying twenty-two independent approaches as impassable (twenty by undershoot, two by overshoot: de Branges positivity via Conrey–Li, and the bounded-supremum/Bohr–Jessen-support channel via Kronecker alignment). Key results include: the Shell Concentration Theorem (pointwise form: at most O(δ_0 log γ) zeros can contribute negatively at any evaluation point, each with damage bounded by the reciprocal shell depth; a single shallow zero realizes the obstruction); the Duality of Obstructions (the two irreducible obstructions are dual faces of the same singularity); the L¹ Neutrality Theorem (each off-line pair is exactly L¹-neutral: integrated methods are blind to displacement, so the obstruction is irreducibly pointwise); and a computational test for the shell concentration. The complete obstruction catalogue is the paper's central contribution. We also establish and verify the On-Line Residual Distributional Law: GUE-conditionally, the on-line residual slope satisfies E[R(γ)] = (1/6)·log²(γ/2π) with a t^(−3/2) upper tail and an e^(−c/r) margin floor; the tail and the variance convergence are measured against Odlyzko's table of the first two million zeros, with all predictions registered in advance of the data (four hits, one directional, one miss in the rigid direction, scorecard preserved). This paper does not contain a proof of the Riemann Hypothesis. The single open problem (v1(σ,γ) > 0 for all σ > 1/2) is equivalent to RH — a reformulation, not a simplification.