We establish an asymptotic bound on the smoothed two-point Liouville correlation S_Φ (M) = Σ λ (Wm + a) λ (Wm + a + h) Φ (m/M) restricted to a residue class modulo a smooth number W. The bound is obtained by direct evaluation of the Mellin inversion integral on the line σ = 1, without contour deformation and without analytic continuation of the associated Dirichlet series Gₐ (s). The argument rests on three inputs, all established as theorems of the present paper: the logarithmically averaged two-point Chowla cancellation of Tao 2016 Forum Math. Pi (unconditional on the Liouville function) ; the Montgomery–Vaughan mean-value theorem applied to Gₐ and its distributional derivatives at σ = 1 (classical) ; and a Sobolev regularity theorem (Target T) for boundary traces of Dirichlet series with uniform Archimedean cancellation, proved as Theorem 3. 3 of the paper with full proof in Appendix A. The main theorem delivers the Gevrey-parametric bound |S_Φ (M) | ≤ A_Φ · M · exp (−c_β (log M) ^1/ (β+1) ) for every M ≥ M₀, with c_β = (β + 1) / (2R) ^1/ (β+1) and β > 1 the Gevrey class of the window. The limit β → 1⁺ delivers the architectural supremum exp (−√ (2 log M / R) ). A structural Carleman-class theorem establishes that no compactly supported window in a Denjoy–Carleman-boundary class beats this limit, locating the architectural ceiling precisely. A rigorous quantitative bound on the Meyer-wavelet rate constant, D ≤ 3/ (2π) ≈ 0. 4775 and hence R ≤ 3√ (log 2) /π ≈ 0. 7950, is established as Proposition 3. 19 via a direct Paley–Wiener estimate using only the Fourier support of the Meyer wavelet. Threshold tables are reported at the working value R = 1, which is conservative: since the threshold is monotonically increasing in R, the entries at R = 1 are upper bounds on the rigorous thresholds at R ≤ 0. 7950. Combined with the bridge lemma of the companion treatise on polynomial correlations of the Liouville function, the bound yields a conditional closing theorem for the binary Goldbach conjecture: every even integer N ≥ M₀ is the sum of two primes. At the nominal constants the threshold is M₀ ≈ 10^202 (Gevrey β → 1⁺) ; under a conjectural Cotlar–Stein reduction of the Besov-assembly constant it falls to M₀ ≈ 10^150 at the working rate constant R = 1, or M₀ ≈ 10^114under the rigorous bound R ≤ 0. 7950 of Proposition 3. 19. The gap to Oliveira e Silva's computationally verified range (N ≤ 4 × 10^18) is at least 95 orders of magnitude: mathematically explicit, physically inaccessible (and robust to quantum computation, the verification being an enumeration over an exponential range rather than a search or period-finding problem). The paper's contribution is therefore an architectural-limit result rather than a resolution of binary Goldbach; the residual gap is quantified, bounded, and localised for future work. The main theorem is conditional in a layered manner. Three layers are distinguished in Section 1. 4. Layer 0 (unconditional inputs): the logarithmic Chowla theorem of Tao 2016, the Mellin–Perron representation, the Montgomery–Vaughan bound, Target T (Theorem 3. 3), and the D-bound (Proposition 3. 19). Layer 1 (conditional on Target T as an analytic engine): the smoothed Perron bound and the threshold tables of Section 7, delivering log₁0 M₀ ≈ 202 at β → 1⁺, R = 1 (and ≈ 1504 at β = 2) unconditional under the standing UAC hypothesis. Layer 2 (conditional on the Cotlar–Stein reduction of C*): the factor-40 reduction, at β = 1. 001 and R = 1, from log₁0 M₀ ≈ 202 to ≈ 150. Layer 3: presentation choices not affecting validity (working R = 1 versus the rigorous R ≤ 0. 7950 yielding ≈ 114; cₛtar confidence band). To the author's knowledge, this is the first explicit-threshold reduction of binary Goldbach via a non-circle-method architecture. The methodological contribution, application of Hardy-spaces-of-Dirichlet- series machinery and Besov regularity of boundary traces to an additive number theory problem, appears to be new in this context. This version (v7. 2) corrects two quantitative inconsistencies in the threshold figures of the previous version: a double-count of the arctan smoothing factor in the Cotlar–Stein column (the consistent value is ρ = 90, not 73, raising the most favourable working-R threshold from 10^147 to 10^150), and a leading-order rather than exact-Γ evaluation of the rate-constant sensitivity (the rigorous-R threshold is ≈ 10^114, not 10^131). The architecture, the main theorems, and the appendix proof of Target T are unchanged. Keywords. Goldbach's conjecture; two-point Chowla cancellation; Liouville function; smoothed Perron formula; Sobolev regularity of Dirichlet series; Montgomery–Vaughan mean-value theorem; Besov spaces; Meyer wavelets; Gevrey classes; Denjoy–Carleman theorem; bridge lemma; Cotlar–Stein almost-orthogonality. Mathematics Subject Classification (2020). Primary: 11N37, 11N64. Secondary: 11M41, 42B35, 46E35.
Theodore Deligiannis (Mon,) studied this question.