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 " 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\ prime,\ p+q=N\ the Goldbach partition number, and L=log N. The paper proves unconditionally that R(N)= S(N)N/(log N)²(1+o(1))>0(N≥ N₀), where S(N)=2C₂∏p N,\,p>2p-1/p-2 is the singular series and C₂=∏p>2(1-(p-1)⁻²)≈0.66016 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 Λ-weighted sums via the identity Λ(n)=∑d nμ(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 S₀≈ 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 ρₚ=log(N/p)/L∈(0,1], establishes the weight-decomposition identity S₀=F/L+ES, 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 \|ES\|₂²=2N/L³. 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=∫ S₀²e is treated directly, the trivial bound of the minor arc ∫ₘS₀²e is N/L, exceeding the target o(N/L²) by a full factor L; after the weight substitution, the Cauchy–Schwarz bound of the cross term Vₘ=∫ₘFESe is \|F\|₂\|ES\|₂=√2\,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 √2\, NL√loglog N overshoots the target only by the factor √loglog N—the gap is compressed from a factor L down to a mere factor √loglog N. Then, through the main-term identity R=2 I/L- C/L²+ E₂ and the coefficient cancellation $2-1=1$, the entire difficulty is concentrated onto Λ-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 κ, in particular Φ(1)=Φ(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 φ(q₁), not Siegel's theorem; the sole non-effective constant comes from Siegel–Walfisz itself, so N₀ 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 Λ-objects (weight substitution + two-stage cancellation). The prime exponential sum S₀(α)=∑p≤ Ne(pα) has neither multiplicative structure (no Vaughan decomposition) nor smoothness. Introduce the weight ρₚ:=1-log p/L=log(N/p)/L; the basic identity 1=ρₚ+log p/L yields the weight-decomposition identity S₀= FL+ES, ES=E₁-E₂, E₁=∑ₚρₚe(pα), F=∑n≤ NΛ(n)e(nα). By Parseval and the triple PNT expansion (π(N)=N/L+N/L²+2N/L³+⋯, θ(N)=N+⋯, ∑ₚlog²p=NL-N+⋯) the first two orders cancel exactly: 1-2+1N/L=0, 1+0-1N/L²=0, leaving \|ES\|₂²=2N/L³+O(N/L⁴)—a factor L⁻² below \|S₀\|₂² N/L and L⁻⁴ below \|F\|₂² 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 S₀=F/L+ES and using ES=S₀-F/L yields the main-term identity R(N)=2 I/L- C(N)/L²+ E₂, I=∫ FS₀e,\ C(N)=∫ F²e,\ | E₂|≤2N/L³. On the major arcs, IM≈ S(N)N/L (coefficient $2/L$) and CM≈ S(N)N (coefficient -1/L²); the coefficient cancellation $2-1=1$ produces exactly the main term S(N)N/L² and downgrades the minor-arc target from o(N/L²) to $o(N/L)$ and $o(N)$—each relaxed by a factor L. (3) Third reduction: Vaughan's trichotomy. Vaughan's identity Λ=Λ₁+Λ₂+Λ₃ (with U=V=√ P=e^L1/4/2) splits F=F₁+F₂+F₃. Type I (long smooth sum) is closed by Perron + pole cancellation; Type I-short is closed by the spectral route: its Dirichlet series L₂(s)=-M≤ U(s)L≤ V(Λ)(s)ζ(s) has a pole only at $s=1$, with sub-exponentially small residue μ₂=-M₁(U)(log V+O(1)) and no zero terms, so its meromorphic approximation has poles only at $s=2,1$, cancelled exactly by Φ(1)=Φ(2)=0. The entire difficulty is concentrated into the Type-II block V(3,1)=∫ₘF₃E₁e together with its twin C(3,3)=∫ₘF₃²e (with \|F₃\|₂² NLloglog N, because Λ₃ 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 cM=∑m+n=MρₘΛ₃(n) is an additive convolution, whose matched dual is the Laplace–Mellin transform (not the Dirichlet-series product): ∑McMM⁻ˢ=1Γ(s)∫₀^∞ tˢ⁻¹ A(t) B(t)\,dt, which gives the truncated Perron representation V(3,1)=12π i∫c-iTc+iTD(s)Φ(s)\,ds+O(NᶜPL²/T) with Φ(s)=∑ₖκ(k)(N+k)ˢ⁻¹. Here Λ₃=μ>U*Λ>V*1: the constant third factor contributes the ζ-factor, L₃(s)=M>U(s)Λ>V(s)ζ(s), whose explicit decomposition gives a **double pole** at $s=1$ (L₃(s)~-M₁(U)/(s-1)², fingerprint: ∑n≤ XΛ₃(n)=-M₁(U)Xlog X+O(X)) and **simple poles** at the ζ-zeros (the ζ and 1/ζ factors cancel; the residue at s=1+ρ is m_ρ/(Lρ) after the Γ-cancellation, with 1/ζ'(ρ) absent). The finite-sum transforms L₃(t)=∑Λ₃(n)e⁻ⁿᵗ, E₁(t)=∑ρₚe⁻ᵖᵗ admit complete explicit-formula expansions—the mean part, the zero part, and all the moment terms from the poles of Γ at the negative integers—with the strict remainder O(t-1+δ(log1t)O(1)). The mean-value pole data of the meromorphic approximation D(s) (the finite Dirichlet polynomial $D(s)$ itself is entire; the difference D- 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 κ (∑ₖkʲκ(k)=0, j≥0; in particular Φ(1)=Φ(2)=0), and the second-order and logarithmic data (k-domain form M₁(U)L(N+k)logʲ(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 |κ(k)| Pτ(k)/|k|, where P=e^L1/4 is a logarithmic power rather than $Q=N/P$), the standard zero count (∑γ~2ʲm_ρ2ʲlog(2ʲ), Riemann–von Mangoldt; the Littlewood-type sum is not needed), and the Vinogradov–Korobov zero-free region (N1-δ(T)=Ne^-cL1/3(log L)-1/3); the difference terms are controlled by the coefficient bound |εM| LO(1)(M1-δ+|M₁(U)|Llog(1+N/M)/log²M), giving $∑_k|κ(k)||