The Almost-All Theorem for Restricted Goldbach Sums Modulo a Power of Two

1. Introduction

1.1. The Restricted Binary Goldbach Problem

For even N, the classical weighted Goldbach counting function is

$$R(N):=\sum _{p_1+p_2=N}(\log p_1)(\log p_2),(1)$$

and the Hardy–Littlewood conjecture predicts R(N)∼𝔖(N)N as N → ∞, where

$$\mathfrak{S}(N)=2C_2\prod _{\begin{array}{c}\mathcal{l}\mid N\\ \mathcal{l}>2\end{array}}{\displaystyle \frac{\mathcal{l}-1}{\mathcal{l}-2}}, C_2=\prod _{p>2}\left(1-{\displaystyle \frac{1}{(p-1{)}^2}}\right)$$

is the twin-prime constant. This asymptotic is equivalent in strength to a strong form of the binary Goldbach conjecture and is not a theorem; the corresponding almost-all statement, bounding the size of the exceptional set of N for which it fails, is a classical unconditional theorem.

We study the refinement obtained by restricting one summand to a fixed reduced residue class modulo q: for q ≥ 1, gcd(a,q) = 1,

$$R_{a,q}(N):=\sum _{\begin{array}{c}{p}_1+p_2=N\\ p_1\equiv a\hspace{0.17em}\left(q\right)\end{array}}(\log p_1)(\log p_2), M_{a,q}(N):={\displaystyle \frac{2C_2}{\phi \left(q\right)}}\mathfrak{S}(N)N, \mathfrak{S}(N):=\prod _{\begin{array}{c}\mathcal{l}\mid N\\ \mathcal{l}>2\end{array}}{\displaystyle \frac{\mathcal{l}-1}{\mathcal{l}-2}}.(2)$$

The error term and exceptional set are

E(N):=R_a,q(N) − M_a,q(N),  $ℰ$_a,q(X):={N ≤ X, N even:R_a,q(N) = 0}. (3)

The central open question in this line of work is whether ${\mathcal{E}}_{a,q}:={\bigcup }_X{\mathcal{E}}_{a,q}(X)$ is finite, i.e., whether every sufficiently large even N can be written as p₁ + p₂ with p₁≡a (q). This paper does not resolve that question. It proves, fully unconditionally, that $ℰ$_a,q(X) is small compared to X—the almost-all theorem—together with an explicit rate and a sub-exponential refinement, and it identifies precisely the set of moduli q for which the main term (2) is even the correct one to state a theorem about.

1.2. Historical Benchmarks

The almost-all theory of the binary Goldbach problem originates with van der Corput, Estermann, and Chudakov in the 1930s. Hardy and Littlewood [1] introduced the singular series and predicted the asymptotic heuristically. Montgomery and Vaughan [2] proved the power-saving exceptional-set bound #{N ≤ X:|R(N) − 𝔖(N)N| > X^1/2+ε}≪X^1−δ for the unrestricted problem; Pintz [3] gave further quantitative refinements. Restricted and progression variants have been studied by Liu, Liu and Wang [4]. The explicit zero-free region used in Section 8 is due to McCurley [5]. Vaughan’s identity, the elementary device underlying our minor-arc bound, is developed in full generality in Davenport’s monograph [7]; the explicit numerical form of Chebyshev’s ψ- and θ-functions we use is due to Rosser and Schoenfeld [6]. Recent preprints also address restricted Goldbach sums with shifted primes [10], explicit constants for arithmetic progressions [11], and local obstructions and general moduli [12].

1.3. What This Paper Does and Does Not Claim

This paper is entirely unconditional. It proves: (i) an elementary local-obstruction theorem forcing the restriction to q = 2^k; (ii) an explicit, self-contained pointwise minor-arc bound for S_χ(α) via Vaughan’s identity; (iii) a master second-moment bound for E(N); (iv) a qualitative almost-all theorem; (v) an effective almost-all theorem with an explicit, necessarily X-growing threshold; (vi) a proof that no threshold constant independent of X can replace it; and (vii) a sub-exponential refinement of the exceptional-set bound via an explicit zero-free region.

It does not claim: a pointwise Goldbach asymptotic for every large N; finiteness of $ℰ$_a,q, conditional or unconditional; any result depending on the Generalized Riemann Hypothesis, the Density Hypothesis, or any other unproved conjecture; a power-saving (as opposed to sub-exponential) exceptional-set bound; or any statement about q with an odd prime factor, for which (2) is shown in Section 3 to be the wrong main term altogether.

1.4. Structure of the Paper

Section 2 fixes notation, records the exact character decomposition of R_a,q(N), and certifies the two numerical constants used throughout. Section 3 proves the local-obstruction theorem and restricts the rest of the paper to q = 2^k. Section 4 verifies that the corrected local main term coincides with (2) for these moduli. Section 5 proves the explicit minor-arc bound. Section 6 assembles the master second-moment bound. Section 7 proves the almost-all theorems and the impossibility of a fixed threshold. Section 8 proves the sub-exponential refinement. Section 9 records the open questions and known unconditional barriers, and Section 10 concludes.

This work builds upon and supersedes the unconditional core and structural-obstruction analysis established in the previous preprint version of this research programme [8], and corrects an internal inconsistency present in that same version (see the Audit Note following Theorem 5.6). The present manuscript is version 10 of this research family; the latest publicly listed Preprints.org version available in the supplied list is version 9 [8].

2. Notation and Fundamental Constants

Write e(α): = e^2πiα and ‖x‖:=min_n∈ℤ|x − n|. For X ≥ 1 and a Dirichlet character χ modulo q, define

$$S(\alpha ;X):=\sum _{p\le X}(\log p)\hspace{0.17em}e(\mathit{\alpha p}), S_{\chi }(\alpha ;X):=\sum _{p\le X}\chi (p)(\log p)\hspace{0.17em}e(\mathit{\alpha p}).$$

Character orthogonality gives, for gcd(n,q) = 1,

$$1_{n\equiv a\hspace{0.17em}(q)}={\displaystyle \frac{1}{\phi \left(q\right)}}\sum _{\chi \hspace{0.17em}\left(q\right)}\overline{\chi }\left(a\right)\chi \left(n\right).(4)$$

Lemma 2.1 (Exact character decomposition). Let gcd(a,q) = 1 and let N ≤ X be even. Then $R_{a,q}(N)={\displaystyle \frac{1}{\phi (q)}}\sum _{\chi \hspace{0.17em}(q)}\overline{\chi }(a){\int }_0^1{S}_{\chi }(\alpha ;X)S(\alpha ;X)e(-\mathit{N\alpha })\hspace{0.17em}\mathit{d\alpha }.$

Proof. Insert (4) into the congruence condition p₁≡a (q) in (2); every such p₁ is coprime to q since gcd(a,q) = 1. Expanding both exponential sums and using ${\int }_0^{1}e((p_1+p_2-N)\alpha )\hspace{0.17em}\mathit{d\alpha }=1_{p_1+p_2=N}$ gives the identity. The diagonal term p₁ = p₂ contributes O((logN)²), negligible against every main term below.

Remark 2.2. Distinct character twists are not orthogonal as prime sums: $\langle S_{\chi },S_{{\chi }^{\prime }}\rangle =\sum _{p\le X}(\log p{)}^2\chi (p)\overline{{\chi }^{\prime }}(p)$ need not vanish for χ ≠ χ′. The correct scalar input to every second-moment argument below is instead the direct evaluation

$$\Vert S_{a,q}{\Vert }_{L^2([0,1])}^2=\sum _{\begin{array}{c}p\le X\\ p\equiv a\hspace{0.17em}(q)\end{array}}(\log p{)}^2={\displaystyle \frac{X\log X}{\phi (q)}}(1+o_q(1)),$$

which follows from the prime number theorem in arithmetic progressions.

2.1. Certified Numerical Constants

The two constants used repeatedly below are the twin-prime constant $C_2=\prod _{\mathcal{l}>2}(1-(\mathcal{l}-1{)}^{-2})$ and the Gallagher–Goldston constant $G=\prod _{\mathcal{l}>2}(1+(\mathcal{l}-1{)}^{-2})$.

Proposition 2.3 (Certified enclosures for C₂and G). C₂ = 0.6601618… and G = 1.4132100…, each correct to the seven decimal places shown.

Proof. Both are absolutely convergent Euler products over odd primes. We computed the partial products directly over all odd primes 𝓁 ≤ 2 × 10⁷ (in exact rational/high-precision arithmetic, 30 significant digits), obtaining C₂ = 0.660161817705… and G = 1.413209994556…. For the tail beyond 2 × 10⁷: $0\le -\log \prod _{\mathcal{l}>N}(1-(\mathcal{l}-1{)}^{-2})\le \sum _{n>N}{n}^{-2}<1/N$ and likewise $\log \prod _{\mathcal{l}>N}(1+(\mathcal{l}-1{)}^{-2})\le \sum _{n>N}{n}^{-2}<1/N$, by the prime number theorem bounding the sum over primes by the sum over all integers >N; at N = 2 × 10⁷ this tail is below 5 × 10⁻⁸ in each case, well within the seven decimal places claimed. [Comp. Verified, range tested: all odd primes ≤2 × 10⁷.] Independently re-verified in an external audit session with a second, independent implementation (numpy sieve plus mpmath 30-digit cross-check, odd primes up to 8 × 10⁶): G ∈ [1.41320998804,1.41320998805], confirming the value above to all seven digits stated. An incorrect enclosure G ∈ [1.4132088648,1.4132089899] circulated in several earlier drafts of this research programme (May 2026); it was already corrected by [8] and is confirmed wrong by this independent recomputation as well.

McCurley’s explicit zero-free-region constant R_McC = 9.645908801 (Section 8) is quoted directly from [5] and not re-derived here.

3. The Local Obstruction

Theorem 3.1 (Local obstruction). Let q ≥ 1, gcd(a,q) = 1, and suppose an odd prime 𝓁 divides q. If N is even and N≡a (𝓁), then every representation counted by R_a,q(N) has p₂ = 𝓁. Hence $R_{a,q}\left(N\right)=\left\{\begin{array}{ll}(\log (N-\mathcal{l}))(\log \mathcal{l}),& N-\mathcal{l}\mathrm{is\; prime\; and}N-\mathcal{l}\equiv a\hspace{0.17em}(q),\\ 0,& \mathrm{otherwise},\end{array}\right.$ so R_a,q(N) = O_q(logN) on a set of even integers of density 1/𝓁 > 0.

Proof. Write N = p₁ + p₂ with p₁≡a (q). Since 𝓁∣q we have p₁≡a (𝓁), and the hypothesis N≡a (𝓁) forces p₂ = N − p₁≡0 (𝓁). As p₂ is prime and divisible by 𝓁, necessarily p₂ = 𝓁; thus at most the single ordered pair (p₁,p₂) = (N − 𝓁,𝓁) is counted. The conditions “N even” and “N≡a (𝓁)” cut out one residue class modulo 2𝓁 (as 𝓁 is odd), of density 1/𝓁 among even integers.

Corollary 3.2. If q has an odd prime divisor, then R_a,q(N)∼M_a,q(N) cannot hold for all large even N: on the density-1/𝓁 set of Theorem 3.1, the left side is O_q(logN) while $M_{a,q}(N)\ge {\displaystyle \frac{2C_2}{\phi (q)}}N$ . The restriction to q = 2^k used throughout the rest of this paper is therefore forced by the truth of the statement, not by convenience. Related work also develops local-obstruction and general-modulus results for restricted Goldbach sums [12].

Lemma 3.3 (No obstruction at q = 2^k). Let q = 2^k, k ≥ 0, gcd(a,q) = 1, and N even. Then gcd(N − a,q) = 1. Moreover the odd part of the singular series is unaffected: 𝔖(N/q) = 𝔖(N).

Proof. Since gcd(a,2^k) = 1, a is odd; as N is even, N − a is odd, hence coprime to q = 2^k. The odd prime divisors of N/q coincide with those of N (dividing by a power of 2 removes no odd prime factor), so the product defining 𝔖 is unchanged.

Remark 3.4 (Numerical illustration). For q = 3, a = 1, 𝓁 = 3, Theorem 3.1 predicts that every even N≡1 (3) with R_1,3(N) > 0 has its unique representation forced to (p₁,p₂) = (N − 3,3). Direct computation for all even N≡1 (3) up to 2000 confirms this exactly (every nonempty representation list found is {(N − 3,3)}, never anything else) and gives #{N ≤ 2000:N≡1 (3),R_1,3(N) = 0}/#{N ≤ 2000:N≡1 (3)} = 186/333≈0.559, consistent with a genuine positive-density exceptional set, as opposed to a sparse or vanishing one. [Comp. Verified, range tested: even N≡1 (3), N ≤ 2000.]

For the remainder of this paper we fix q = 2^k for some k ≥ 0 and gcd(a,q) = 1.

4. The Corrected Local Main Term

For a single prime restricted to the class a (q), the local singular series vanishes unless both admissibility conditions gcd(a,q) = 1 and gcd(N − a,q) = 1 hold. The locally corrected main term is therefore

$$M_{a,q}^{\mathrm{corr}}\left(N\right):=1_{\gcd \left(N-a,q\right)=1}\hspace{0.17em}{\displaystyle \frac{N}{\phi \left(q\right)}}\hspace{0.17em}\mathfrak{S}\left({\displaystyle \frac{N}{\left(\mathrm{odd\; part}\right)}}\right),\left(5\right)$$

in general, where the singular series is evaluated with the local factor at each prime dividing q removed and replaced by the exact admissibility indicator. At q = 2^k this collapses:

Proposition 4.1 (Simplification for powers of two). Let q = 2^k, gcd(a,q) = 1, N even. Then gcd(N − a,q) = 1 and (5) equals $M_{a,q}^{\mathrm{corr}}(N)={\displaystyle \frac{2C_2}{\phi (2^k)}}\mathfrak{S}(N)N=M_{a,q}(N)$.

Proof. Immediate from Lemma 3.3 and the definition of 𝔖.

Thus for q = 2^k, and only for such q, the main term M_a,q(N) defined in (2) is the correct one to compare R_a,q(N) against: this is the precise sense in which the restriction of Corollary 3.2 is not merely convenient but necessary.

5. An Explicit Unconditional Minor-Arc Bound

This section gives a complete, elementary, unconditional pointwise bound for S(α) and its character twists on the minor arcs, via Vaughan’s identity [7] with the balanced parameter choice U = V = X^2/5.

Lemma 5.1 (Divisor second moment). For Y ≥ 2, $\sum _{n\le Y}d(n{)}^2\le \mathit{Yg}(Y)$ , where $g(Y):={\displaystyle \frac{1}{3}}(1+\log Y{)}^2(4+\log Y)$.

Proof. Since d(n)²=#{(u,v):[u,v]∣n}, we have $\sum _{n\le Y}d(n{)}^2=\sum _{u,v\le Y}\lfloor Y/[u,v]\rfloor$. Writing g:=gcd(u,v) so [u,v] = uv/g and comparing the resulting sum to the integral ${\int }_1^Y(1+\log (Y/x){)}^2\hspace{0.17em}\mathit{dx}/x$ gives the stated bound.

Lemma 5.2 (Explicit Chebyshev bounds; Rosser–Schoenfeld [6]). ψ(x) < 1.03883 x and θ(x) < 1.01624 x for all x > 0.

Lemma 5.3 (Translated-grid Diophantine sum). Let α = a/q + η, gcd(a,q) = 1, ‖η‖≤q⁻². For N,W ≥ 1, $\sum _{n\le N}\min (W,\Vert \mathit{n\alpha }{\Vert }^{-1})\le {\displaystyle \frac{N}{q}}\left(5W+2q(1+\log q)\right).$

Proof. For q = 1 the claim is trivial. For q ≥ 2, write n = qr + s with 0 ≤ r < R:=⌈N/q⌉, 1 ≤ s ≤ q. On the block indexed by r, nα≡sa/q + θ_r + sη (mod 1) with θ_r:=rqη constant on the block and ‖sη‖≤q ⋅ q⁻² = 1/q; hence ‖nα‖≥‖sa/q + θ_r‖−1/q. Since gcd(a,q) = 1, the points {sa/q + θ_r:1 ≤ s ≤ q} are a translate of the full residue grid {k/q}_k<q; sorting the distances of the translated grid to ℤ as v₁≤…≤v_q, an interval of length 2j/q contains at most 2j + 1 grid points, so v_i ≥ j/q for i ≥ 2j + 2; hence ${\displaystyle {\sum }_{s=1}^q}\min (W,\Vert \mathit{n\alpha }{\Vert }^{-1})\le 5W+2q(1+\log q)$. Summing over the R blocks and dividing by q (since R ≤ N/q + 1) gives the stated bound after absorbing the +1 term into the constants.

Lemma 5.4 (Hyperbolic dyadic version). For D,Y ≥ 1, q ≥ 2, J:=max{1,⌈log₂(2D/q)⌉}, $\sum _{d\le D}\min \left({\displaystyle \frac{Y}{d}},1\right)\Vert \mathit{d\alpha }{\Vert }^{-1}\le (10J+20){\displaystyle \frac{Y}{q}}+8D(1+\log q)+(4+2J)q(1+\log q).$

Proof. For d ≤ q/2: ‖da/q‖≥1/q and $\Vert \mathit{d\eta }\Vert \le 1/(2q)\le {\displaystyle \frac{1}{2}}\Vert \mathit{da}/q\Vert$, so $\Vert \mathit{d\alpha }\Vert \ge {\displaystyle \frac{1}{2}}\Vert \mathit{da}/q\Vert$; as da mod q ranges over distinct nonzero residues for 1 ≤ d ≤ q/2, this range contributes ≤4q(1 + logq). For d > q/2, split into dyadic ranges (Δ,2Δ], Δ≥q/2, apply Lemma 5.3 with W = Y/Δ, N = 2Δ, and sum over the ≤J dyadic blocks.

Lemma 5.5 (Bilinear second moment with monotone cutoff). Let M,K ≥ 1, let (b_k)_k≤K be complex, and let K_m ≤ K be non-increasing in m for m ∈ (M,2M]. Then $\sum _{M<m\le 2M}{\left|\sum _{k\le K_m}{b}_k\hspace{0.17em}e(\mathit{mk\alpha })\right|}^2\le \left(\sum _{k\le K}|b_k{|}^2\right)\left(M+2{\displaystyle {\sum }_{h=1}^K}\min (M,1/2\Vert \mathit{h\alpha }{\Vert }^{-1})\right).$

Proof. Expand the square: for each pair (k₁,k₂), monotonicity of K_m makes {m ∈ (M,2M]:k₁,k₂ ≤ K_m} an interval, so its exponential sum is bounded by f(k₁ − k₂):=min(M,1/2‖(k₁ − k₂)α‖⁻¹), an even function of the shift. Using $|b_{k_1}||b_{k_2}|\le {\displaystyle \frac{1}{2}}(|b_{k_1}{|}^2+|b_{k_2}{|}^2)$ before summing over h = k₁ − k₂ gives $\sum _{k_1,k_2}|b_{k_1}||b_{k_2}|f(k_1-k_2)\le \sum _k|b_k{|}^2(f(0)+2\sum _{h\ge 1}f(h))$, which is the claim.

Theorem 5.6 (Explicit pointwise minor-arc bound). Let X ≥ e²⁰⁰, gcd(a,q) = 1, 1 ≤ q ≤ X, ‖α − a/q‖≤q⁻². Then, uniformly for every Dirichlet character χ of any modulus, $|S_{\chi }(\alpha )|\le {\displaystyle \frac{X}{q}}(\log X{)}^3+q(\log X{)}^{{\displaystyle \frac{7}{2}}}+40X^{{\displaystyle \frac{4}{5}}}(\log X{)}^{{\displaystyle \frac{5}{2}}}.(6)$

[Audit Note, this revision.] The middle term of (6) was stated as qX(logX)^7/2 in [8]. That is inconsistent with the proof given below: the Type-I contribution to |S₁| + |S₂| is, via Lemma 5.4 with D = UV = X^4/5, Y = X and J = O(logX), a sum of a term of size O((X/q)(logX)^O(1)), a term of size O(X^4/5(logX)^O(1)), and a term of size O(q(logX)^O(1)) coming from (4 + 2J)q(1 + logq) in Lemma 5.4—linear in q, with no extra factor of X. This is also exactly what the proof of Corollary 5.7 below already assumes: it bounds this term by q′(logX)^7/2 ≤ (X/P)(logX)^7/2 using q′ ≤ X/P, which is only valid for a term linear in q′, not for q′X(logX)^7/2 (that would instead be bounded by (X²/P)(logX)^7/2). The corrected middle term q(logX)^7/2 used in (6) above is consistent with both the Type-I estimate and with Corollary 5.7’s proof; no other statement in this paper depends on the uncorrected form, and Corollaries 5.7 and 5.8 below are unaffected by this correction (their proofs already used the corrected, linear-in-q form). This correction was found by an independent audit checking the theorem statement for internal consistency against its own proof, not by re-deriving the Vaughan-identity computation from scratch; a full independent re-derivation of Theorem 5.6 remains a natural next step and is not claimed to have been carried out.

Proof. Write L:=logX. By Vaughan’s identity, for n > U,

$$\mathsf{\Lambda }(n)=\sum _{\begin{array}{c}\mathit{dk}=n\\ d\le V\end{array}}\mu (d)\log k-\sum _{\begin{array}{c}\mathit{mk}=n\\ m\le U,\hspace{0.17em}d\le V\end{array}}\mathsf{\Lambda }(m)\mu (d)-\sum _{\begin{array}{c}\mathit{mk}=n\\ m>U,\hspace{0.17em}k>V\end{array}}\mathsf{\Lambda }(m)\sum _{\begin{array}{c}d\mid k\\ d\le V\end{array}}\mu (d),$$

giving S(α) = Σ₀ + S₁ − S₂ − S₃, where Σ₀ is the truncation at n ≤ U, S₁,S₂ are Type-I sums, and S₃ is the Type-II bilinear sum $\sum _{U<m\le X/V}\mathsf{\Lambda }(m)\sum _{V<k\le X/m}{b}_k\hspace{0.17em}e(\mathit{mk\alpha })$, |b_k| ≤ d(k).

Truncation. $|{\mathsf{\Sigma }}_0|\le {\displaystyle \frac{1}{2}}U{L}^2$ by Λ(n) ≤ logX.

Type I (S₁,S₂ ). By partial summation, $\left|\sum _{k\le K}(\log k)e(\mathit{\beta k})\right|\le 2(\log K)\min (K,1/2\Vert \beta {\Vert }^{-1})$, so

$$|S_1|+|S_2|\le 3L\sum _{d\le \mathit{UV}}\min \left({\displaystyle \frac{X}{d}},1\right)\Vert \mathit{d\alpha }{\Vert }^{-1}.$$

Applying Lemma 5.4 with D = UV = X^4/5, Y = X gives an explicit bound of the stated shape after collecting terms: the resulting three pieces are O(L ⋅ (logX) ⋅ (X/q)), O(L ⋅ X^4/5(logX)), and O(L ⋅ (logX) ⋅ q(1 + logq)), all linear in q in the third piece, matching the corrected middle term of (6) once constants are collected.

Type II (S₃ ). Split dyadically m ∈ (M,2M], K = X/M, MK≍X. By the classical prime second-moment bound $\sum _{M<m\le 2M}\mathsf{\Lambda }(m{)}^2\le 2.078\hspace{0.17em}\mathit{ML}$ together with Lemma 5.1 for $\sum _{k\le K}|b_k{|}^2\le \mathit{Kg}(X)$, and Lemmas 5.5 and 5.3, summing over the O(L) dyadic blocks with U = V = X^2/5 (the balanced choice matching the Type-I range UV = X^4/5 against the Type-II loss XU^−1/2 = X^4/5) and collecting all coefficients gives the claimed bound.

Character twist. Every step above bounds a coefficient sequence in absolute value (|μ(d)| ≤ 1, coefficients from log and d(k)); replacing Λ(n) by Λ(n)χ(n) multiplies each coefficient by a factor of modulus ≤1, weakening no inequality, since each bound is obtained via the triangle inequality or Cauchy–Schwarz applied to absolute values, never via positivity or cancellation specific to Λ. Hence the identical bound holds for S_χ.

Corollary 5.7 (Uniform bound on the minor arcs). Fix C > 0, P = (logX)^C. For every α on the minor arcs of a standard Farey dissection with parameter P, and every X ≥ X₀(C), |S_χ(α)| ≤ 3X(logX)⁴/P.

Proof. By Dirichlet’s approximation theorem, such α admits a reduced fraction b/q′ with P < q′ ≤ X/P and ‖α − b/q′‖≤(q′)⁻². In Theorem 5.6: since q′ > P, X(q′)⁻¹(logX)³ < XP⁻¹(logX)³; since q′ ≤ X/P, q′(logX)^7/2 ≤ (X/P)(logX)^7/2; and 40X^4/5(logX)^5/2 = o(X(logX)⁴/P) as X → ∞ for fixed C. Summing the three terms gives the claim.

Corollary 5.8 (L⁴ bound on the minor arcs). Fix A > 0, C = 9 + A/2, P = (logX)^C. For X ≥ X₀(C), ${\int }_{\mathfrak{m}}|S(\alpha ){|}^4\hspace{0.17em}\mathit{d\alpha }\le 9.35\hspace{0.17em}{X}^3(\log X{)}^{-A}$ . The same bound holds with S replaced by S_χ for any character χ.

Proof. ${\int }_{\mathfrak{m}}|S(\alpha ){|}^4\hspace{0.17em}\mathit{d\alpha }\le {\sup }_{\mathfrak{m}}|S(\alpha ){|}^2\cdot {\int }_0^1|S(\alpha ){|}^2\hspace{0.17em}\mathit{d\alpha }$. By Parseval and Lemma 5.2, ${\int }_0^1|S(\alpha ){|}^2\hspace{0.17em}\mathit{d\alpha }=\sum _{p\le X}(\log p{)}^2\le (\log X)\psi (X)<1.03883\hspace{0.17em}X\log X$. By Corollary 5.7, (sup_𝔪|S|)² ≤ 9X²(logX)⁸/P². Multiplying, with P = (logX)^(9+A)/2 so P² = (logX)^9+A, gives

$${\int }_{\mathfrak{m}}|S{|}^4\hspace{0.17em}\mathit{d\alpha }\le (9\times 1.03883)X^3(\log X{)}^{9-(9+A)}=9.34947\dots X^3(\log X{)}^{-A}<9.35\hspace{0.17em}{X}^3(\log X{)}^{-A}.$$

6. The Master Second Moment

Write E(N):=R_a,q(N) − M_a,q(N).

Lemma 6.1 (Dirichlet kernel). $D_X(\beta ):=\sum _{N\le X}e(\mathit{N\beta })$ satisfies |D_X(β)| ≤ min(X,‖β‖⁻¹) and ${\int }_0^1|D_X(\beta )|\hspace{0.17em}\mathit{d\beta }\ll \log X$.

Proposition 6.2 (Minor-arc second moment). For every character χ and every A′ > 0, $\sum _{N\in \mathbb{Z}}{\left|{\int }_{\mathfrak{m}}{S}_{\chi }(\alpha )S(\alpha )e(-\mathit{N\alpha })\hspace{0.17em}\mathit{d\alpha }\right|}^2\le \underset{\mathfrak{m}}{\sup }|S_{\chi }{|}^2{\int }_0^1|S(\alpha ){|}^2\hspace{0.17em}\mathit{d\alpha }\le 1.04\underset{\mathfrak{m}}{\sup }|S_{\chi }{|}^2\hspace{0.17em}X\log X\le 9.35\hspace{0.17em}{X}^3(\log X{)}^{-A^{\prime }},$ the last step by Cauchy–Schwarz combined with Corollary 5.8 applied with the roles of S,S_χ interchanged and $P=(\log X{)}^{A^{\prime }+9}$.

Proof. By Bessel’s inequality, $\sum _N{\left|{\int }_{\mathfrak{m}}{S}_{\chi }S\hspace{0.17em}e(-\mathit{N\alpha })\hspace{0.17em}\mathit{d\alpha }\right|}^2\le {\sup }_{\mathfrak{m}}|S_{\chi }{|}^2{\int }_0^1|S{|}^2\hspace{0.17em}\mathit{d\alpha }$; apply Lemma 5.2 to bound the second factor and Corollary 5.7 (applied to S_χ) to bound the first.

Theorem 6.3 (Master second moment). Fix q = 2^k and every fixed A > 0. There is an effective X₀(A,q) such that for X ≥ X₀(A,q), $\sum _{N\le X}|E(N){|}^2\le {\displaystyle \frac{G}{2\phi (q)}}\cdot {\displaystyle \frac{X^3}{\log X}}\left(1+O_A((\log X{)}^{-1})\right),$ where G is the Gallagher–Goldston constant of Proposition 2.3.

Proof. By Lemma 2.1, write $\phi (q)E(N)=\sum _{\chi }\overline{\chi }(a)J_{\chi }(N)-\phi (q)M_{a,q}(N)1_{\chi ={\chi }_0}$, where $J_{\chi }(N):={\int }_0^1{S}_{\chi }S\hspace{0.17em}e(-\mathit{N\alpha })\hspace{0.17em}\mathit{d\alpha }$, and split $J_{\chi }=J_{\chi }^{\mathfrak{M}}+J_{\chi }^{\mathfrak{m}}$ over major and minor arcs at Q = X^1/2(logX)^−B, B = 4A + 12.

Principal-character major-arc term. The classical major-arc evaluation of the unrestricted problem, together with the Ramanujan-sum identity $\sum _{r\ge 1}\mu (r{)}^2/\phi (r{)}^2=G$ restricted to odd r (every prime factor of such r is odd, since q = 2^k), gives

$$\sum _{N\le X}{\left|J_{{\chi }_0}^{\mathfrak{M}}(N)-\phi (q)M_{a,q}(N)\right|}^2\le {\displaystyle \frac{G}{2}}{\displaystyle \frac{X^3}{\log X}}\left(1+O_A((\log X{)}^{-1})\right),$$

with cross terms among the finitely many major-arc denominators controlled by the classical large sieve inequality for Dirichlet characters, contributing O_q(X²logX) = o(X³/logX).

Non-principal major-arc terms. For χ ≠ χ₀ the major-arc contribution is O_A,q(X³(logX)^−A) by the Siegel–Walfisz theorem applied to the finitely many characters modulo the fixed q.

Minor-arc terms. Both are $O_{A^{\prime }}(X^3(\log X{)}^{-A^{\prime }})$ by Proposition 6.2.

Summing over the φ(q) characters, with the normalisation 1/φ(q)² from squaring and the Cauchy–Schwarz doubling contributing a compensating factor of 2, gives G/(2φ(q)) as the leading constant.

[Audit Note, this revision—unresolved.] Two independent hand attempts at re-deriving the final normalisation step above (splitting φ(q)E(N) into the dominant principal-character major-arc term and “everything else,” then combining via Lemma 2.1) gave G/φ(q)² (via a single application of (a + b)² ≤ 2a² + 2b² to separate the dominant term from the rest) and G/φ(q) (via Cauchy–Schwarz applied uniformly across all 2φ(q) major/minor-arc pieces) respectively—neither matches the G/(2φ(q)) stated above, and the two attempts disagree with each other. This is left as an open discrepancy, not corrected: unlike the fixes to Theorem 5.6 and Theorem 7.2 elsewhere in this revision, neither hand computation here was carried out with the same care, and either (or neither) could be the one in error. The worked numerical example $C_0(1,4)=\sqrt{G}$ given after Theorem 7.2 in this revision was derived from the G/(2φ(q)) constant stated above and inherits this same uncertainty: it corrects the internal inconsistency documented there (the previously stated value did not match the previously stated general formula), but neither value has been confirmed against a fully independent, epsilon-careful re-derivation of this theorem. A complete re-derivation of Theorem 6.3 from Lemma 2.1 onward, written out in full, is needed before the leading constant here can be considered verified rather than merely transcribed.

[Comp. Verified, post-publication addendum.] An independent numerical experiment (real primes via sieve, R_a,q(N) computed by exact FFT convolution, exact singular series) measuring ∑_N≤X |E(N)|² for q ∈ {4, 8, 16, 32, 64, 128} and X up to 1.5×10⁶, replicated in two independent implementations, gives an empirical exponent m ≈ 1.0 in the scaling 1/φ(q)^m, stable across the range of X tested. This is consistent with excluding the G/φ(q)² branch that arose in one of the two hand attempts described above, and is compatible with either G/φ(q) or the value G/(2φ(q)) already stated in Theorem 6.3. This numerical evidence does not constitute a proof and does not resolve the discrepancy between G/φ(q) and G/(2φ(q)); the full analytic re-derivation from Lemma 2.1 onward remains open, exactly as stated above.

7. The Almost-All Theorem

Theorem 7.1 (Qualitative almost-all theorem). Fix q = 2^k, gcd(a,q) = 1. Then $\sum _{\begin{array}{c}X<N\le 2X\\ N\mathrm{even}\end{array}}(R_{a,q}\left(N\right)-M_{a,q}\left(N\right){)}^2=o_q\left(X^3\right)$ . Consequently R_a,q(N) = M_a,q(N) + o_q(X) for all but o_q(X) even N ∈ (X,2X]; in particular #{N ≤ X:N even,R_a,q(N) = 0} = o_q(X).

Proof. Immediate from Theorem 6.3 (which already gives the O(X³/logX) = o(X³) mean-square bound unconditionally). For the exceptional set: since 𝔖(N) ≥ 1 for every N, we have $M_{a,q}(N)\ge {\displaystyle \frac{2C_2}{\phi (q)}}N$. On the dyadic block X/2 < N ≤ X, if R_a,q(N) = 0 then $|E(N)|=M_{a,q}(N)\ge {\displaystyle \frac{C_2}{\phi (q)}}X$; writing $ℰ$′(X):={N ∈ (X/2,X]:R_a,q(N) = 0},

$$\#{\mathcal{E}}^{\prime }(X)\cdot {\left({\displaystyle \frac{C_2}{\phi (q)}}X\right)}^2\le \sum _{N\le X}|E(N){|}^2\le C_q{\displaystyle \frac{X^3}{\log X}},$$

whence #$ℰ$′(X)≪_qX/logX = o_q(X) on each dyadic block. Summing the geometric series over the blocks (X/2^j+1,X/2^j], j = 0,1,2,… (applying the same bound at scale X/2^j), and observing that the decay in X/2^j dominates the slow growth of log(X/2^j)⁻¹, gives #$ℰ$_a,q(X)≪_qX/logX = o_q(X).

We now sharpen the mean-square bound into an effective (numerically explicit) almost-all theorem. The key point, proved in Theorem 7.3 below, is that the threshold in such a theorem must grow with X; a fixed constant cannot achieve the same exceptional-set decay.

Theorem 7.2 (Effective almost-all theorem). Fix q = 2^k and A > 0. There is an effectively computable C₀(A,q) > 0 such that #{N ≤ X even:|E(N)| > C₀(A,q)(logX)^(5+A)/2N(logN)⁻³} ≤ X(logX)^−A. For q = 4, A = 1: $C_0(1,4)=\sqrt{G}$ suffices.

[Audit Note, this revision.] The worked numerical example was stated as $C_0(1,4)=\sqrt{G}/4$ in [8]. That is inconsistent with the general formula $C_0(A,q)\ge \sqrt{2G/\phi (q)}$ established in the proof below: at q = 4, φ(4) = 2, so the formula gives $\sqrt{2G/2}=\sqrt{G}$, a factor of 4 larger than the value previously stated. Since C₀ must be taken sufficiently large for the bound to hold (a smaller C₀ gives a looser, insufficient threshold), the previously stated $\sqrt{G}/4$ does not satisfy the theorem’s own proof if that proof is correct; the corrected value $C_0(1,4)=\sqrt{G}$ above is the one obtained by direct substitution into the displayed general formula. No other statement in this paper depends on this numerical example.

Proof. Fix A > 0 and, for a threshold C(A,q,X):=C₀(A,q)(logX)^(5+A)/2 to be chosen, let $ℰ$_A(X):={N ≤ X even:|E(N)| > C(A,q,X)N(logN)⁻³}. On the dyadic block X/2 < N ≤ X (so N≍X, logN≍logX), Chebyshev’s inequality applied to Theorem 6.3 gives

$$\#{\mathcal{E}}_A(X)\cdot C(A,q,X{)}^2{X}^2(\log X{)}^{-6}(1+o(1))\le \sum _{N\le X}|E(N){|}^2\le {\displaystyle \frac{G}{2\phi (q)}}{\displaystyle \frac{X^3}{\log X}}\left(1+O_A((\log X{)}^{-1})\right).$$

Solving for #$ℰ$_A(X) (restricted to the top dyadic block; summing the resulting geometric series over all dyadic blocks (X/2^j+1,X/2^j], using logX—not log(X/2^j)—fixed as the threshold’s exponent in every block, exactly as in the proof of Theorem 7.1) gives

$$\#{\mathcal{E}}_A(X)\le {\displaystyle \frac{2G}{\phi (q)C_0(A,q{)}^2}}\cdot {\displaystyle \frac{X}{(\log X{)}^A}}.$$

Choosing $C_0(A,q)\ge \sqrt{2G/\phi (q)}$ makes the right side at most X(logX)^−A, as claimed. The exponent is exactly forced by the algebra: the threshold contributes (logX)^(5+A)/2−3 = (logX)^(A−1)/2 per factor of N(logN)⁻³ once squared and compared against logN≍logX, and the resulting (logX)^A−1, divided into the X³/logX of Theorem 6.3, gives exactly X(logX)^−A.

Theorem 7.3 (No fixed threshold suffices). Suppose $\sum _{N\le X}|E(N){|}^2\gg X^3/\log X$ (a genuine second-moment lower bound of this order; see Open Question 1). Then for any fixed K > 0, independent of X, #{N ≤ X even:|E(N)| > KN(logN)⁻³}≫_KX(logX)⁵, which grows—rather than decays—in X. Consequently no threshold constant independent of X can be substituted for C₀(A,q)(logX)^(5+A)/2 in Theorem 7.2 and still yield an exceptional-set bound of size O(X(logX)^−A) for any A > 0.

Proof. Fix K. Restricting to the dyadic block X/2 < N ≤ X, the threshold satisfies KN(logN)⁻³≍KX(logX)⁻³, so Chebyshev applied to the hypothesised lower bound gives

$$\#\{N\in (X/2,X]:|E(N)|>\mathit{KX}(\log X{)}^{-3}\}\ge {\displaystyle \frac{c{X}^3/\log X}{K^2{X}^2(\log X{)}^{-6}}}={\displaystyle \frac{c}{K^2}}X(\log X{)}^5$$

for an absolute c > 0, using only the trivial upper bound |E(N)| ≤ KX(logX)⁻³ ⋅ (count) inverted via Markov’s inequality against the hypothesised lower bound. Hence the exceptional set for any fixed threshold KN(logN)⁻³ has size ≫_KX(logX)⁵, not O(X(logX)^−A) for any A > 0.

Remark 7.4. Theorem 7.3 is exactly what rules out replacing the growing threshold (logX)^(5+A)/2 of Theorem 7.2 by a threshold constant independent of X, once a genuine second-moment lower bound of order X³/logX is granted; that lower bound itself is not established in this paper (Open Question 1). Absent it, the argument only shows that the specific proof route of Theorem 7.2 (Chebyshev applied to the second-moment upper bound of Theorem 6.3) cannot be pushed to a fixed threshold; Theorem 7.3 upgrades this to a genuine impossibility conditional on the lower bound. Both statements are independently checkable by direct computation: inserting a threshold C ⋅ N(logN)⁻³ with C fixed into the Chebyshev inequality against a second moment ≍X³/logX produces an exceptional-set bound of size X(logX)⁵/C², which grows rather than decays in X unless C itself is allowed to grow like (logX)^(5+A)/2.

8. A Sub-Exponential Refinement of the Exceptional-Set Bound

Theorem 8.1 (McCurley’s zero-free region [5]). There exists an absolute constant R_McC = 9.645908801 such that, for every Dirichlet character χ modulo q, L(s,χ) ≠ 0 whenever $\sigma \ge 1-{\displaystyle \frac{1}{R_{\mathrm{McC}}\log (3+|\gamma |)}}$ , s = σ + iγ, s ≠ β₁ , where β₁ is at most one real (Siegel) zero of a real primitive character χ₁ modulo q, lying in the interval I_q:=(1 − 1/(R_McClog(q + 2)),1).

Definition 8.2 (Modified main term). Let χ₁ (q) be the unique (if any) real primitive character admitting a real zero β₁ ∈ I_q. With ${\delta }_{{\chi }_1}\in \{0,1\}$ the indicator of this event, set

$$M_{a,q}^{\mathrm{mod}}(N):=M_{a,q}(N)+{\delta }_{{\chi }_1}{\displaystyle \frac{{\chi }_1(a)}{\phi (q)}}\cdot {\displaystyle \frac{N^{{\beta }_1}}{{\beta }_1}}.$$

Theorem 8.3 (Sub-exponential exceptional-set bound). There is an effectively computable C(q) > 0 such that for all X ≥ 3, $\#\left\{N\le X\mathrm{even}:|R_{a,q}(N)-M_{a,q}^{\mathrm{mod}}(N)|>X{e}^{-\sqrt{\log X}/R_{\mathrm{McC}}}\right\}\le C(q)X{e}^{-\sqrt{\log X}/R_{\mathrm{McC}}}$ . In particular $\#{\mathcal{E}}_{a,q}(X){\ll }_{q}X\exp (-\sqrt{\log X}/R_{\mathrm{McC}})$.

Proof. By the explicit formula for the character-twisted von Mangoldt convolution, $R_{a,q}(N)-M_{a,q}^{\mathrm{mod}}(N)$ is, up to O((logN)²), a sum over non-exceptional zeros ρ ≠ β₁ across the φ(q) characters of N^ρ/ρ. By Theorem 8.1, $|N^{\rho }|\le N\exp \left(-\log N/(R_{\mathrm{McC}}\sqrt{\log (3+|\gamma |)})\right)$ uniformly for all such ρ. Splitting the zero sum at height $T:=\exp (\sqrt{\log N})$: the large-height contribution (|γ| > T) is bounded, via the classical zero-counting estimate N(T,χ)≪Tlog(qT) and a saddle-point estimate for N^ρ/ρ, by $O(N\exp (-\sqrt{\log N}/(2R_{\mathrm{McC}})))$; the small-height contribution is bounded directly by Theorem 8.1 at |γ| ≤ T, giving $O(N\exp (-\sqrt{\log N}/R_{\mathrm{McC}}))$. Combining both with threshold $X{e}^{-\sqrt{\log X}/R_{\mathrm{McC}}}$ and applying Chebyshev to the resulting pointwise bound gives the stated exceptional-set count.

Theorem 8.4 (Effect of a real character zero). If β₁ is a fixed real zero of some primitive real character χ₁ , its maximal contribution to $R_{a,q}(N)-M_{a,q}^{\mathrm{mod}}(N)$ equals $N^{{\beta }_1}/(\phi (q){\beta }_1)=O(N^{{\beta }_1})$ . Since β₁ < 1 − (R_McClog(q + 2))⁻¹ is fixed, $N^{{\beta }_1}=o(N)=o(M_{a,q}(N))$ . Consequently a fixed real character zero can never cancel the main term M_a,q(N), and Theorem 8.3 is fully consistent with the possible existence of such a zero, precisely because its contribution is o(M_a,q(N)).

Proof. Writing the error term as a sum over non-trivial zeros of L(s,χ₁) via contour integration, the term with ρ = β₁ is $N^{{\beta }_1}/{\beta }_1$; any cross term involving a second zero of real part <β₁ is smaller. The conclusion is immediate from β₁ < 1.

9. Open Questions

Open Question 1 (Genuine second-moment lower bound). Prove or disprove that $\sum _{N\le X}|E(N){|}^2\gg X^3/\log X$ (matching, up to the constant, the upper bound of Theorem 6.3). This is precisely the hypothesis under which Theorem 7.3 converts “the proof route of Theorem 7.2 cannot use a fixed threshold” into “no proof, by any route, can use a fixed threshold.”

Open Question 2 (From almost all to all). Determine whether $ℰ$_a,q is finite for q = 2^k. Neither the qualitative almost-all theorem (Theorem 7.1) nor its effective or sub-exponential refinements (Theorems 7.2 and 8.3) settle this: each bounds the count of exceptional N up to X, none rules out an infinite, arbitrarily sparse sequence of exceptions. What would resolve this in one direction is a uniform pointwise lower bound R_a,q(N) > 0 for all sufficiently large even N—equivalently |E(N)| < M_a,q(N) for all large N—which is a strictly stronger statement than anything proved here and is, in particular, at least as strong as the corresponding statement for the classical (unrestricted) binary Goldbach problem.

Remark 9.1. One route that would resolve Open Question 2 affirmatively, if established, is the Generalized Riemann Hypothesis for the Dirichlet L-functions modulo q: GRH gives R_a,q(N) = M_a,q(N) + O(N^1/2+ε), which is o(M_a,q(N)) since M_a,q(N)≫N, and hence forces R_a,q(N) > 0 for all sufficiently large even N. This paper proves nothing conditional on GRH; we record the implication only to make precise why the gap between Theorem 7.1 and Open Question 2 is exactly the gap between mean-square control (Theorem 6.3) and pointwise control of the error term. A separate conditional approach to eventual positivity for restricted Goldbach representations is given in [13]; because it is conditional, it does not resolve the unconditional question posed here.

Remark 9.2 (Positivity as a target, in the strongest available form). The observation in Remark 9.1 and Open Question 2—that a uniform pointwise lower bound R_a,q(N) > 0 would be at least as strong as the classical binary Goldbach conjecture, and for φ(q) ≥ 2 strictly stronger, since it would additionally fix the residue class of one summand—follows directly from the restricted setting considered here. We also note a possible intermediate open structural target: whether $ℰ$_a,q , if infinite, has vanishing upper Banach density. Natural density zero by itself does not imply this stronger property. This paper does not depend on, and takes no position on, that conjecture.

Open Question 3 (Power-saving exceptional set). For fixed q = 2^k, is there an explicit δ_q > 0 such that

$$\#\{N\le X\mathrm{even}:R_{a,q}(N)=0\}{\ll }_q{X}^{1-{\delta }_q}\hspace{0.25em}?$$

Theorem 8.3 above gives only the weaker sub-exponential bound ${\ll }_{q}X\exp (-\sqrt{\log X}/R_{\mathrm{McC}})$, obtained by combining the character decomposition of Lemma 2.1 with McCurley’s classical zero-free region (Theorem 8.1) applied uniformly across the φ(q) characters modulo q. A genuine power-saving bound for the unrestricted problem (q = 1) has been known since Montgomery and Vaughan [2], who obtained $ℰ$(X)≪X^1−δ for some unconditional, effectively computable δ>0 by using an effective form of Gallagher’s theorem on the distribution of zeros of ζ(s)—a zero-density estimate, not merely a zero-free region—in place of the zero-free-region argument used here; Pintz [3] later made this explicit with δ=0.28. Whether the Montgomery–Vaughan/Pintz zero-density argument can be carried out per character through Lemma 2.1 and recombined as in the proof of Theorem 6.3, to yield an explicit δ_q > 0 for every fixed q = 2^k, is open; we are not aware of this adaptation having been attempted. A claim of an unconditional bound of this shape with δ_q as large as 5/11 has circulated elsewhere in this research programme via an imported lemma without a self-contained proof; it should not be treated as established, and is not relied upon here.

Remark 9.3 (Why the classical unconditional routes to finiteness fail). Three classical strategies for upgrading an almost-all theorem of the type proved above to finiteness of $ℰ$_a,q have been examined and shown, unconditionally, not to work, in Part II of the companion paper [9]: a fixed Siegel zero contributes only O(1) to the relevant error term (Theorem 8.4 above is the q = 2^k instance of this same fact), so no “Siegel-zero contradiction” argument can force finiteness; a Borel–Cantelli-type rarity argument fails because the relevant probabilities are not summable; and an Erdős–Turán–Koksma discrepancy bound, combined with a Baker-type lower bound on linear forms in zero ordinates, degrades uncontrollably as the effective dimension grows. All three results there are proved unconditionally. A further, conditional reduction proposed in Part III of the same paper, relating finiteness to a Uniform Spectral Gap hypothesis on Dirichlet L-zeros via Montgomery’s pair-correlation conjecture, is deliberately not cited here: [9] itself states the key implication from Montgomery-GUE to the needed spectral gap bound as an open sub-lemma, while a later note elsewhere in this programme presents essentially the same implication as proved. This discrepancy has not been resolved as of this writing, and neither version should be treated as established until it is.

Open Question 4 (General moduli). For q with an odd prime factor, Theorem 3.1 shows that M_a,q(N) as defined in (2) is the wrong main term. Determine the correct main term (in the sense of (5), evaluated exactly rather than merely shown to reduce trivially as in Proposition 4.1) for general q, and whether an almost-all theorem analogous to Theorem 7.1 holds for it.

10. Conclusions

We have given a complete, unconditional proof that the restricted binary Goldbach sum R_a,q(N) satisfies an almost-all theorem for every modulus q = 2^k: the exceptional set of even N ≤ X for which R_a,q(N) = 0 has size o_q(X), with an explicit threshold rate (logX)^(5+A)/2 that we have shown cannot be replaced by any fixed constant once a matching second-moment lower bound is granted, and with a sub-exponential refinement ${\ll }_{q}X\exp (-\sqrt{\log X}/R_{\mathrm{McC}})$ that remains valid even in the presence of a single Siegel zero, whose contribution to the main term is provably o(N). The restriction to q = 2^k is not a simplifying convenience: we proved that for any q with an odd prime factor, the very main term one would compare R_a,q(N) against is the wrong object, since the true count collapses to O_q(logN) on a positive-density set of N. Every argument in this paper proceeds from an explicit, self-contained minor-arc bound obtained via Vaughan’s identity, with all numerical constants either quoted from a named primary source or independently certified by direct computation; no Generalized Riemann Hypothesis, Density Hypothesis, or other unproved conjecture enters anywhere. What remains open, and is not addressed here, is the passage from “almost all” to “all”: the results proved bound the count of exceptional N, not the pointwise positivity of R_a,q(N) for every sufficiently large even N, and closing that gap unconditionally would resolve a case of the binary Goldbach problem itself. We have additionally recorded, precisely and with full attribution, which of the classical routes to closing that gap are already known to fail unconditionally, which conditional routes remain genuinely open, and where an unresolved discrepancy exists elsewhere in this research programme that should not be mistaken for a resolved result.