A Complete Bottom-Up-to-Top-Down Proof Framework for the Riemann and Generalized Riemann Hypotheses: From the Primon Gas, Self-Dual Gaussian Seed, and Conditional Wall-Crossing Phase Compensator Across the Wall of Uncertainty to Critical-Line Zeros

We construct a complete, unified Bottom-Up-to-Top-Down analytical and operator-theoretic framework for the Riemann Hypothesis (RH) and Generalized Riemann Hypothesis (GRH) governed by the Three-O Principle (One Seed, One Operator, One Law). Starting bottom-up from the Primon Gas (H = Σp (log p) ap†ap ≥ 0) and the Self-Dual Gaussian Seed (g(x) = e−πx2 = ĝ(x)), we bridge across the Heisenberg–Weyl / Paley–Wiener Wall of Uncertainty via two dual channels: (A) The Logarithmic-Cell Weil–Schur Operator Channel (Theorems I–V) Pulling Connes' finite-cutoff Weil quadratic form AL(F) on L2([0, L], du) with L = log M back through the invertible integer-shift cascade EF = Σn ≤ M aF(n) n−1/2 τlog n yields the 100% Prime-Free Operator: −R̆L(F) := EF AL(F) EF* = PolĕL(F) − (DF EF* + EF DF* + EF WL(βF) EF*), where DF = [U, EF] = Σn ≤ M aF(n) (log n / √n) τlog n uses only integer logarithms log n and Dirichlet coefficients aF(n), with zero prime factorization. Partitioning [0, L] into integer logarithmic cells Jk = [log k, log(k+1)] splits L2([0, L]) = Ptot(2) ⊕ H⊥(2,d). By Sylvester's Law of Inertia and the Schur complement theorem, −R̆L(F) > 0 holds if and only if: Pillar I (Intrinsic Cell Induction): θintr(k) := ||Ak,k−1/2 Ck Ak,k−1/2||op < 1 for all k ≥ 1, and Pillar II (Orthogonal Schur Contraction): ||Tk(d,cf)(F)||op < 1. (B) The Conditional Wall-Crossing Phase Compensator & All-t Dominance Channel (Theorems VI–VII) On the critical line Re(s) = 1/2, crossing the Uncertainty Wall at the self-dual saddle horizon N*(t) = √(t / 2π) (and conductor-scaled N*(t; q) = √(qt / 2π) for primitive Dirichlet characters χ mod q in GRH) splits the Hardy function Z(t) = Zbulk(t) + Zwall(t) via the READ → RESIDUE operator into a Pre-Wall finite integer bulk sum Zbulk(t) (READ) and the Conditional Wall-Crossing Phase Compensator (RESIDUE): Zwall(t) = (−1)N−1 (t / 2π)−1/4 [Ψwall(0)(p) + Rwall(τ, p)], Ψwall(0)(p) = cos(2π(p2 − p − 1/16)) / cos(2πp), where N = ⌊√(t / 2π)⌋ and p = {√(t / 2π)} ∈ [0, 1). Key Contributions and Certified Results (Theorems I–VII) Theorem I (Exact Zone-3 Vanishing & 100% Prime-Free Cell-Inversion Identity): Proves that all arithmetic blocks vanish identically in the upper half ⌊k/2⌋ < m ≤ k (Pk,m(F) ≡ 0), and that ΛV1(F) = EV1(F)−1 DV1(F) extracts the generalized von Mangoldt function ΛF(n) to ~10−14 precision with zero prime factorization across ζ(s), Dirichlet L-functions, and the Dedekind zeta function ζℚ(√−5)(s) = ζ(s)L(s, χ−20) to 6.813 × 10−14. Theorem II (Self-Dual Gaussian Seed & Unitary Pole-Prime Balance κ = 1): Proves that the self-duality g = ĝ of the Gaussian seed enforces unitary pole-prime balance (κ = 1) in Zone 1 and induces a strictly positive Gaussian–Weil coercion lower bound μGaussLB(k, d) > 0 on the orthogonal complement. Theorem III & Lemma A (Second-Difference Telescoping, Universal 1/2 Law, Bonnet–Neumann Hierarchy, and Rational Self-Energies): Proves that the constant-mode Cauchy–Legendre Toeplitz block wr(1,1) = (1/2)Δ2(r log r) telescopes in closed form to (1/2)[(R+1)log(R+1) − R log R] ~ (1/2) log R against the diagonal self-energy Ak,k(m+1,m+1) = log k + Cdiag(m) + O(1/k), where the Cauchy–Legendre self-energy constant satisfies the exact rational harmonic identity Dm = 1 + Σj=1m 4j/(4j2 − 1) (D0 = 1, D1 = 7/3, D2 = 43/15, …). By the Two-Index Bonnet–Neumann recurrence, all higher blocks (a,b) ≠ (1,1) decay as O(r−(a+b−1)) ≤ O(r−2) with convergent ℓ1 sums (Σr=1∞ wr(1,2) = −√3/4), proving limk→∞ θintr(d)(k) = 1/2 < 1 with non-asymptotic bound max2 ≤ k ≤ 200 θintr(k) = 0.97874097 < 1. Theorem IV & Lemma B (Paley–Wiener Uncertainty Floor, 105-Digit Schur Contraction, & Counterexample Exclusion): Certifies 1 − σ1(Tk(d,cf))2 > 0 in 105-digit Decimal LDLT arithmetic (+2.977134840813 × 10−18 > 0 at k = 10, d = 8) and explicitly excludes 7 classical counterexample families (κ ≠ 1, off-line Za(s), Epstein ζx2+5y2(s), and Davenport–Heilbronn fDH(s)). Theorem V (100% Quadrature-Free & 100% Prime-Free Critical-Line Zero Reconstruction): Reconstructs via subdivided-cell spherical Bessel waves (with Parseval completeness < 10−12 and zero prime factorization or numerical integration) the first 50 critical-line zeros of ζ(s) to 1.884 × 10−9 median error, the first 10 critical-line zeros of L(s, χ−4) to 5.970 × 10−10, the first 10 critical-line zeros of L(s, χ5+) to 1.462 × 10−9, and the complete 12-zero factorization of ζℚ(√−5)(s) = ζ(s)L(s, χ−20) in [0, 22]. Theorem VI (The Conditional Wall-Crossing Phase Compensator & Universal V-Valley Bridge at ξ = 1): Proves that in the normalized Wall coordinate ξwall = 2p ∈ [0, 2] (p = {√(t / 2π)}), the quadratic phase p2 − p of the Mordell–Siegel compensator Ψwall(0)(p) is identically the Universal Bottom-Up V-Valley Parabola fBU(ξwall) = −(1/2)ξwall + (1/4)ξwall2 = −1/4 + (1/4)(ξwall − 1)2 ≡ p2 − p that governs the P ≠ NP complexity valley (ξ = Q/Q* = 1) and the 4D Yang–Mills 40 → 28 mass gap valley (ξ = κ/κ* = 1, κ* = 1/6). Shows that the first critical-line zero γ1 = 14.134725 is born right at the V-valley vertex ξwall(γ1) = 2p1 = 0.99974 ≈ 1 by 100% destructive interference between Zbulk(γ1) = +0.587705 and Zwall(γ1) = −0.587705, and proves exact Stokes threshold jump cancellation ΔZwall(tN) = −ΔZbulk(tN) across every Wall boundary tN = 2πN2 (99.97%–99.99% cancellation). Theorem VII (All-t Wall Dominance, Sector-Parity Selection Theorem, and GRH Conductor Scaling): Proves that Ψwall(0)(1/2 + u) = sin(π/8 + 2πu2) / cos(2πu) (u = p − 1/2) is strictly even, has removable singularities Ψwall(0)(1/4) = Ψwall(0)(3/4) = 1/2, is strictly convex on [0, 1] (min d2Ψwall(0)/dp2 ≈ 3.4979 > 0), and satisfies the exact all-p bounds: 0 < sin(π/8) ≈ 0.38268343 ≤ Ψwall(0)(p) ≤ cos(π/8) ≈ 0.92387953 < 1. Combining this V-valley floor with Gabcke's explicit all-t remainder bound (0.053 τ−3/4) proves that sgn(Zwall(t)) = (−1)N−1 across the entire N-th Wall sector t ∈ [2πN2, 2π(N+1)2) for all t ≥ 12 (N → ∞), enforcing the Sector-Parity Selection Rule across all 1,071 zeros (1,068 arches) in t ∈ [10, 1500] with 0/1,068 parity violations and maximum Dimensionless Defect Ratio ρmax = 0.750515 < 1, and extends the mechanism to GRH via the conductor-scaled Wall horizon N*(t; q) = √(qt / 2π). Author Contact:Oleksandr Khropak (Kyiv, Ukraine) — oleksandr.hropak@gmail.com