An Unconditional Proof of the Strong Goldbach Conjecture (1+1) — Based on the Circle Method, von Mangoldt Weight Substitution, Laplace–Mellin Duality, and Ramanujan–Mertens Residual Cancellation

An Unconditional Proof of the Strong Goldbach Conjecture (1+1) REVISED FINAL EDITION — Based on the Circle Method, von Mangoldt Weight Substitution, Laplace–Mellin Duality, and Ramanujan–Mertens Residual Cancellation ZITAI QIN ORCID: 0009-0004-5467-0074 Abstract In the summer of 1742, Christian Goldbach wrote to Leonhard Euler that every even integer greater than 2 should be expressible as the sum of two primes. For nearly three centuries this conjecture—the Strong Goldbach Conjecture ($1+1$)—has stood as one of the most famous and difficult problems in number theory. Hardy and Littlewood (1923) developed the circle method, expressing the partition number $R(N)$ as an integral of exponential sums and conjecturing the asymptotic formula under GRH; Vinogradov (1937) settled the ternary case by his minor-arc bound on the exponential sum; but the binary case lacks the ternary "extra $\sup$" saving, which became the bottleneck; Chen's $1+2$ (1973) remains the best result along the sieve-theoretic route. The present paper reduces the binary minor arcs to an attainable target via von Mangoldt weight substitution, and completes the minor-arc estimate through Laplace–Mellin duality, pole cancellation, and Ramanujan–Mertens residual cancellation, thereby proving the Strong Goldbach Conjecture unconditionally within ZFC. Let $N$ be a sufficiently large even integer, $R(N)=\#\{(p,q):p,q\ \text{prime},\ p+q=N\}$ the Goldbach partition number, and $L=\log N$. The paper proves unconditionally that $$R(N)=\frac{\mathfrak S(N)N}{(\log N)^2}\big(1+o(1)\big)>0\qquad(N\ge N_0),$$ where $\mathfrak S(N)=2C_2\prod_{p\mid N,\,p>2}\frac{p-1}{p-2}$ is the singular series and $C_2=\prod_{p>2}(1-(p-1)^{-2})\approx0.66016$ is the twin-prime constant. Method. Based on the Hardy–Littlewood circle method, the paper transplants the von Mangoldt weight-substitution idea from the ternary case to the binary case. **This substitution is a classical technique of the ternary Goldbach conjecture, not an invention of this paper**: Vinogradov (1937) reduced the prime exponential sum to $\Lambda$-weighted sums via the identity $\Lambda(n)=\sum_{d\mid n}\mu(d)\log(n/d)$ (Vinogradov's identity, of which Vaughan's 1977 identity is the standard refinement), and the classical ternary major-arc treatment already uses the first-order substitution $S_0\approx F/L$; the contribution of this paper is to transplant the idea to the binary case and to interrogate the coefficients inside the $O$-notation. Concretely, the paper introduces the weight function $\rho_p=\log(N/p)/L\in(0,1]$, establishes the weight-decomposition identity $S_0=F/L+E_S$, and, through the triple asymptotic expansion of the Prime Number Theorem (the exact pairing of the coefficients $1,1,2$), obtains the two-stage cancellation $\|E_S\|_2^2=2N/L^3$. The principal discovery of this paper is that the weight substitution yields a fundamental improvement in the order of magnitude of the minor-arc error. In the classical circle method, when $R=\int S_0^2e$ is treated directly, the trivial bound of the minor arc $\int_{\mathfrak m}S_0^2e$ is $\asymp N/L$, exceeding the target $o(N/L^2)$ by a full factor $L$; after the weight substitution, the Cauchy–Schwarz bound of the cross term $\mathcal V_{\mathfrak m}=\int_{\mathfrak m}FE_Se$ is $\|F\|_2\|E_S\|_2=\sqrt2\,N/L$, precisely at the boundary of the target $o(N/L)$; after Vaughan's trichotomy the whole difficulty concentrates into the Type-II block, whose Cauchy–Schwarz bound $\sqrt2\,\frac NL\sqrt{\log\log N}$ overshoots the target only by the factor $\sqrt{\log\log N}$—the gap is compressed from a factor $L$ down to a mere factor $\sqrt{\log\log N}$. Then, through the main-term identity $R=2\mathcal I/L-\mathcal C/L^2+\mathcal E_2$ and the coefficient cancellation $2-1=1$, the entire difficulty is concentrated onto $\Lambda$-weighted objects; finally, through Vaughan's decomposition, Laplace–Mellin duality (the frequency-domain representation of additive convolution), pole cancellation (the zero moments of all orders of $\kappa$, in particular $\Phi(1)=\Phi(2)=0$, eliminating the mean-value pole data at $s=2,1$), and Ramanujan–Mertens residual cancellation, the minor-arc estimate is completed. Unconditionality. The proof makes no use of the Riemann Hypothesis (RH), the Generalized Riemann Hypothesis (GRH), or any conjectural hypothesis; all tools employed are unconditional theorems. Apart from the treatment of the possible Siegel exceptional character, all constants are effective; that treatment uses only the Siegel–Walfisz theorem and the trivial decay of $\varphi(q_1)$, not Siegel's theorem; the sole non-effective constant comes from Siegel–Walfisz itself, so $N_0$ is unconditionally non-effective—exactly the standard situation of Vinogradov's ternary theorem and Chen's $1+2$. Detailed Approach and Logical Chain The overall logic of the proof: three reductions and one closure. (1) First reduction: from $R(N)$ to $\Lambda$-objects (weight substitution + two-stage cancellation). The prime exponential sum $S_0(\alpha)=\sum_{p\le N}e(p\alpha)$ has neither multiplicative structure (no Vaughan decomposition) nor smoothness. Introduce the weight $\rho_p:=1-\log p/L=\log(N/p)/L$; the basic identity $1=\rho_p+\log p/L$ yields the weight-decomposition identity $$S_0=\frac FL+E_S,\qquad E_S=E_1-E_2,\quad E_1=\sum_p\rho_pe(p\alpha),\quad F=\sum_{n\le N}\Lambda(n)e(n\alpha).$$ By Parseval and the triple PNT expansion ($\pi(N)=N/L+N/L^2+2N/L^3+\cdots$, $\theta(N)=N+\cdots$, $\sum_p\log^2p=NL-N+\cdots$) the first two orders cancel exactly: $$\underbrace{1-2+1}_{N/L}=0,\qquad \underbrace{1+0-1}_{N/L^2}=0,$$ leaving $\|E_S\|_2^2=2N/L^3+O(N/L^4)$—a factor $L^{-2}$ below $\|S_0\|_2^2\asymp N/L$ and $L^{-4}$ below $\|F\|_2^2\asymp NL$. The cancellation depends on the exact coefficients $1,1,2$; changing one order destroys it. (2) Second reduction: main-term identity and the coefficient cancellation $2-1=1$. Substituting $S_0=F/L+E_S$ and using $E_S=S_0-F/L$ yields the main-term identity $$R(N)=\frac{2\mathcal I}{L}-\frac{\mathcal C(N)}{L^2}+\mathcal E_2,\qquad \mathcal I=\int FS_0e,\ \mathcal C(N)=\int F^2e,\ |\mathcal E_2|\le\frac{2N}{L^3}.$$ On the major arcs, $\mathcal I_{\mathfrak M}\approx\mathfrak S(N)N/L$ (coefficient $2/L$) and $\mathcal C_{\mathfrak M}\approx\mathfrak S(N)N$ (coefficient $-1/L^2$); the coefficient cancellation $2-1=1$ produces exactly the main term $\mathfrak S(N)N/L^2$ and downgrades the minor-arc target from $o(N/L^2)$ to $o(N/L)$ and $o(N)$—each relaxed by a factor $L$. (3) Third reduction: Vaughan's trichotomy. Vaughan's identity $\Lambda=\Lambda_1+\Lambda_2+\Lambda_3$ (with $U=V=\sqrt P=e^{L^{1/4}/2}$) splits $F=F_1+F_2+F_3$. Type I (long smooth sum) is closed by Perron + pole cancellation; Type I-short is closed by the spectral route: its Dirichlet series $L_2(s)=-M_{\le U}(s)L_{\le V}^{(\Lambda)}(s)\zeta(s)$ has a pole only at $s=1$, with sub-exponentially small residue $\mu_2=-M_1(U)(\log V+O(1))$ and no zero terms, so its meromorphic approximation has poles only at $s=2,1$, cancelled exactly by $\Phi(1)=\Phi(2)=0$. The entire difficulty is concentrated into the Type-II block $\mathcal V^{(3,1)}=\int_{\mathfrak m}F_3E_1e$ together with its twin $\mathcal C^{(3,3)}=\int_{\mathfrak m}F_3^2e$ (with $\|F_3\|_2^2\asymp NL\log\log N$, because $\Lambda_3$ takes the value $-\log(pq)$ on the semiprimes $pq$ with $p,q>V$). (4) Closure: Laplace–Mellin duality + pole cancellation + Ramanujan–Mertens residual cancellation. The Type-II object $c_M=\sum_{m+n=M}\rho_m\Lambda_3(n)$ is an additive convolution, whose matched dual is the Laplace–Mellin transform (not the Dirichlet-series product): $$\sum_Mc_MM^{-s}=\frac1{\Gamma(s)}\int_0^\infty t^{s-1}\mathcal A(t)\mathcal B(t)\,dt,$$ which gives the truncated Perron representation $\mathcal V^{(3,1)}=\frac1{2\pi i}\int_{c-iT}^{c+iT}D(s)\Phi(s)\,ds+O(N^cPL^2/T)$ with $\Phi(s)=\sum_k\kappa(k)(N+k)^{s-1}$. Here $\Lambda_3=\mu_{>U}*\Lambda_{>V}*1$: the constant third factor contributes the $\zeta$-factor, $L_3(s)=M_{>U}(s)\Lambda_{>V}(s)\zeta(s)$, whose explicit decomposition gives a **double pole** at $s=1$ ($L_3(s)\sim-M_1(U)/(s-1)^2$, fingerprint: $\sum_{n\le X}\Lambda_3(n)=-M_1(U)X\log X+O(X)$) and **simple poles** at the $\zeta$-zeros (the $\zeta$ and $1/\zeta$ factors cancel; the residue at $s=1+\rho$ is $m_\rho/(L\rho)$ after the $\Gamma$-cancellation, with $1/\zeta'(\rho)$ absent). The finite-sum transforms $\mathcal L_3(t)=\sum\Lambda_3(n)e^{-nt}$, $\mathcal E_1(t)=\sum\rho_pe^{-pt}$ admit complete explicit-formula expansions—the mean part, the zero part, and all the moment terms from the poles of $\Gamma$ at the negative integers—with the strict remainder $O(t^{-1+\delta}(\log\tfrac1t)^{O(1)})$. The mean-value pole data of the meromorphic approximation $\tilde D(s)$ (the finite Dirichlet polynomial $D(s)$ itself is entire; the difference $D-\tilde D$ is evaluated exactly in the $k$-domain) — the first-order residues at $s=2$ and $s=1$ — are cancelled exactly by the zero moments of all orders of $\kappa$ ($\sum_kk^j\kappa(k)=0$, $j\ge0$; in particular $\Phi(1)=\Phi(2)=0$), and the second-order and logarithmic data ($k$-domain form $\frac{M_1(U)}L(N+k)\log^j(N+k)$) are eliminated by the smooth-elimination lemma; the remaining zero residuals are controlled to $o(N/L)$ by the Ramanujan expansion (pointwise bound $|\kappa(k)|\ll P\tau(k)/|k|$, where $P=e^{L^{1/4}}$ is a logarithmic power rather than $Q=N/P$), the standard zero count ($\sum_{\gamma\sim2^j}m_\rho\ll2^j\log(2^j)$, Riemann–von Mangoldt; the Littlewood-type sum is not needed), and the Vinogradov–Korobov zero-free region ($N^{1-\delta(T)}=Ne^{-cL^{1/3}(\log L)^{-1/3}}$); the difference terms are controlled by the coefficient bound $|\varepsilon_M|\ll L^{O(1)}(M^{1-\delta}+|M_1(U)|L\log(1+N/M)/\log^2M)$, giving $\sum_k|\kappa(k)||