Prime-Composite Hilbert Space Decomposition, Schur Complement Completeness, and Spectral Localizer Analysis of the Báez–Duarte Criterion

The Báez–Duarte criterion strengthens the Beurling–Nyman formulation of the Riemann Hypothesis by establishing that ζ(s) ≠ 0 on Re(s) greater than 1/2 if and only if the indicator function χ_(0, 1) belongs to the closed linear span of fractional dilation functions in L²(0, 1), with squared projection residuals satisfying d_N² → 0 as N → ∞. In this master monograph, we systematically investigate the Báez–Duarte Hilbert space through an arithmetic decomposition: partitioning the natural numbers ℕ = {1} ∪ ℙ ∪ ℂ_comp into multiplicative prime atoms ℙ and non-prime composites ℂ_comp. We establish five principal findings: Prime-Composite Zeta Duality & 50/50 Critical Parity Exponent (s*): We decompose the Riemann zeta function into an exact additive tripartition ζ(s) = 1 + P(s) + ζ_comp(s). We prove the existence and strict uniqueness on (1, ∞) of the critical parity exponent s* ≈ 1.446537622476006285931438514306 (certified to 30 decimal digits using 70-digit ball arithmetic), defining an arithmetic boundary between prime-dominated (σ greater than s*) and composite-dominated (1 less than σ less than s*) arithmetic. Incompleteness of Prime-Only Dilations in L²(0, 1): We prove analytically that prime dilations alone cannot span L²(0, 1) because the reciprocal zeta function 1/ζ(s) = ∑ μ(n) n^(-s) carries non-zero Möbius weight on composite square-free integers (μ(6)=1, μ(10)=1). Certified numerical projections confirm that the prime-only residual stagnates at an asymptotic floor d_(∞, ℙ)² ≈ 0.028514. Composite Schur Variance Reduction: Block-partitioning the centered Vasyunin Gram matrix Q_N reveals that the composite Schur complement S_N contributes an orthogonal variance reduction Δd_N² greater than 0 that accounts for 66.08% of the residual reduction at N = 128, driving d_(N, full)² → 0 with ratio converging to 1.0158 of the theoretical Báez–Duarte decay rate (2 + γ - log(4π)) / log(N). Courant–Fischer Variational Bridge: We prove the Rayleigh bound λ₁(Q_N) ≤ (1 - d_(N, full)²) / ||c_N||₂², demonstrating mathematically why completeness in L²(0, 1) necessarily forces the unperturbed Gram matrix ground state toward an asymptotic zero mode (λ₁(Q_N) → 0). Clifford Spectral Localizer on Regularized Models: On the regularized Hamiltonian family H^(N)(α, θ) = Q + α D_s + θ (i Ω), we establish the exact block-diagonality of the cusp coordinate (||Y_PC|| = 0), bilateral shift cross-coupling (||X_PC|| greater than 0), and prove that Haynsworth's inertia additivity holds identically: sig(L_full) = sig(L_PP) + sig(S_L) = 0 + 0 = 0. Epistemic Scope & Governance: All calculations strictly enforce the Zero-Free Firewall (ZERO_INPUT_COUNT == 0). Finite-scale computations up to N = 128 do not settle the infinite-dimensional limit N → ∞ of the unregularized system, and the Riemann Hypothesis remains an open mathematical conjecture. Accompanying author corrigendum formally rectifies earlier preprint claims