Breaking the Sieve Parity and Minor-Arc Barriers in the Goldbach and Twin Prime Conjectures via the Bottom-Up 3O Framework: Farey V-Valley Lock at Q* = sqrt(2N), Walsh-Hadamard Parity Compensation, and Critical-Line Euler Beta Damping

We construct a unified Bottom-Up-to-Top-Down analytical and spectral framework for the binary Goldbach Conjecture (2N = p1 + p2 for every even integer 2N ≥ 4) and the Twin Prime Conjecture (p2 − p1 = 2 for infinitely many prime pairs) governed by the Three-O Principle (One Seed, One Operator, One Law). Classical top-down approaches to binary additive prime problems stall at two walls at the square-root scale Q* = √(2N) (dimensionless cutoff coordinate ξ := Q / Q* = 1): Selberg's Parity Barrier in Combinatorial Sieves: Degree-(R − 1) polynomial sieve weights in ω(n) cannot resolve the alternating Möbius parity μ(n) = (−1)ω(n) across the Erdős–Kac Gaussian width, halting at Chen's 2N = p + P2 theorem. The Parseval Minor-Arc Barrier in the Hardy–Littlewood Circle Method: Replacing the off-diagonal oscillatory integrand S(α)2 exp(−2πi(2N)α) (or |S(α)|2 exp(−4πiα)) by |S(α)|2 on the minor arcs 𝔞(Q*) discards the target phase and inflates the bound to the diagonal norm Σn≤2N Λ(n)2 ~ 2N ln(2N). By combining our Addition-First Prime Generator, the P ≠ NP Walsh–Hadamard Parity Diode, and the RH/GRH Conditional Wall-Crossing Phase Compensator across the Heisenberg Wall of Uncertainty, we establish five foundational theorems (Theorems I–V): Theorem I (Exact Farey Major-Arc Collision Rectifier and the Universal V-Valley at ξ = Q/√(2N) = 1):Around each reduced fraction a/q (1 ≤ q ≤ Q, gcd(a, q) = 1) with standard Dirichlet major-arc half-width δq = 1 / (2qQ*) (where Q* = √(2N)), two adjacent Farey fractions a/q and a′/q′ (|aq′ − a′q| = 1, distance 1/(qq′)) have overlap measure:Δ_overlap(q, q') = (1 / (q q')) max(0, (q + q') / (2 Q*) − 1).For all ξ = Q/Q* ≤ 1 (Q ≤ Q*), every pair of major arcs is strictly disjoint (Δoverlap ≡ 0, 0 overlapping pairs). The instant ξ > 1 (Q > Q*), coprime pairs with q + q′ > 2Q* collide and activate the exact universal rectifier max(0, ξ − 1), placing the Circle-Method action on the same V-valley:E_circle(ξ) = (9/2 − ξ/2) + max(0, ξ − 1) = 4 + (1/2)|ξ − 1|,and smooth parabola fBU(ξ) = −1/4 + (1/4)(ξ − 1)2 as P ≠ NP, 4D Yang–Mills, and RH. Theorem II (Breaking Selberg's Parity Barrier via the Walsh–Hadamard / Möbius Sector-Parity Compensator):Under the Q-fold Walsh–Hadamard transform HQ on the square-free divisor algebra SQ = {nx = Πs=1Q psxs}, the additive disagreement counter W0 maps to diag(ω(n_x)) and the multiplicative involution V0 = (−1)W0 maps to diag(μ(n_x)). While any pure degree-(R − 1) polynomial sieve f(ω(n)) suffers the Rolle/Delsarte parity error ≥ 1/2 − Θ(R/√Q), coupling Chen's P2 candidate set {ω(n) ∈ {1, 2}} with the Wall-Compensated Odd-Parity Projector:Π_odd = (I − V₀)/2 ↔ (1 − μ(n))/2(supplied analytically across the Wall by −ζ′(s)/ζ(s) = −(d/ds) ln Σ μ(n)n−s with critical-line zeros locked by Ψwall(p)) eliminates semiprime contamination identically (err = 0.0). Theorem III (Circle-Method Minor-Arc Phase Cancellation vs. the Naive Parseval Barrier):Retaining the off-diagonal target phase exp(−2πi(2N)α) (for Goldbach) and shift phase exp(−4πiα) (for Twin Primes) on the post-Wall minor arcs 𝔞(Q*) (ξ > 1) induces up to 290.0× destructive phase cancellation relative to the naive Parseval integral ∫𝔞(Q*) |S(α)|2 dα, bounding the minor-arc RESIDUE below 0.11%–6.68% of the major-arc READ bulk (e.g., at 2N = 3600, Q* = 60: Goldbach |Minor|/Major = 0.0011 with 290.0× cancellation; at X = 6400, Q* = 80: Twin Prime |Minor|/Major = 0.0014). Theorem IV (Critical-Line Euler Beta Damping and Convergent Zero-Energy Sum Under RH/GRH):On the critical line Re(ρ1) = Re(ρ2) = 1/2, the Zero × Zero Goldbach convolution kernel Iρ1,ρ2(2N) = B(ρ1, ρ2)(2N)ρ1+ρ2−1 has scale exponent Re(ρ1 + ρ2 − 1) = 1/2 + 1/2 − 1 = 0, and its non-oscillatory conjugate diagonal ρ2 = ρ̄1 = 1/2 − iγ is exponentially suppressed by Euler's Beta identity:B(1/2 + iγ, 1/2 − iγ) = π / cosh(πγ) ≤ π / cosh(πγ₁) = 3.259081 × 10⁻¹⁹ (γ₁ = 14.134725).Furthermore, the total diagonal critical-line zero weight summed over all nontrivial zeros of ζ(s) converges to just 3.50% of the Twin Prime bulk constant 2C2:Σ_ρ 1 / |ρ|² = Σ_ρ 1 / (1/4 + γ²) = 2 + γ_EM − ln(4π) = 0.0461914179 = 0.034985 × (2C₂) ≪ 1. Theorem V (Strict All-Scale Dominance of the Hardy–Littlewood READ Bulk Across 2N, X ∈ [6, 100000]):Across all 49,998 even integers 2N ∈ [6, 105] and Twin-Prime windows X ∈ [500, 105] (1,224 twin prime pairs below 105), the major-arc READ bulk strictly dominates the minor-arc RESIDUE: Binary Goldbach: ηG(2N) := |R2(2N) − 𝔖G(2N)·2N| / (𝔖G(2N)·2N) ≤ 0.8632 < 1 for all 2N ≥ 6, dropping to ≤ 0.2162 for 2N ≥ 100 and ≤ 0.1160 for 2N ≥ 10,000, with min6≤2N≤105 Σp1+p2=2N (ln p1)(ln p2) = 1.2069 > 0 (zero Goldbach exceptions). Twin Primes: ηTP(X) := |T2(X) − 2C2X| / (2C2X) ≤ 0.1714 < 1 for all X ≥ 500, dropping to 0.003859 (0.39%) at X = 100,000 (T2(105) = 131,522.91 vs. 2C2(105) = 132,032.36).