A smoothed Perron bound for two-point Liouville correlations via Sobolev regularity at σ = 1

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 thearchitectural 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 saddle-point coefficient c_β is monotonically decreasing 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, with M₀ ≈ 10^140 in the most favourable parameter combination considered (working R = 1; after the arctan sieve refinement applied to the baseline 10^147 at β = 1. 001, ρ = 73). Under the rigorous R ≤ 0. 7950, the corresponding threshold is log₁0 M₀ ≤ 131. The gap to Oliveira e Silva's computationally verified range (N ≤ 4 × 10^18) is approximately 120 orders of magnitude: mathematically explicit, physically inaccessible. 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 ≈ 1500 at β = 2, R = 1 unconditional under the standing UAC hypothesis. Layer 2 (conditional on the Cotlar–Stein reduction of C*): the factor-40 reduction from log₁0 M₀ ≈ 1500 to ≈ 147 at the same parameters. Layer 3: presentation choices not affecting validity (working R = 1, 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. 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; Gevrey classes; Denjoy–Carleman theorem; bridge lemma. Mathematics Subject Classification (2020). Primary: 11N37, 11N64. Secondary: 11M41, 42B35, 46E35.