Restricted Goldbach Sums in Arithmetic Progressions: Local Obstructions, an Explicit Almost-All Framework, and Structural Extensions

1. Introduction

1.1. The Restricted Binary Problem

For even $N$ the classical weighted Goldbach function is

$$R(N):=\sum _{p_1+p_2=N}(\mathrm{l}\mathrm{o}\mathrm{g}{p}_1)(\mathrm{l}\mathrm{o}\mathrm{g}{p}_2),$$

and the Hardy–Littlewood conjecture predicts $R(N)\sim {\mathfrak{S}}_2(N)N$, where

$${\mathfrak{S}}_2(N):=2C_2\prod _{\begin{array}{c}\mathcal{l}\mid N\\ \mathcal{l}>2\end{array}}{\displaystyle \frac{\mathcal{l}-1}{\mathcal{l}-2}}, C_2:=\prod _{\mathcal{l}>2}\left(1-{\displaystyle \frac{1}{(\mathcal{l}-1{)}^2}}\right)=0.6601618158\dots$$

(1)

is the twin-prime constant (here and below $p,\mathcal{l}$ denote primes). We study the asymmetric restriction in which only the first prime is prescribed modulo $q$:

$$R_{a,q}(N):=\sum _{\begin{array}{c}{p}_1+p_2=N\\ p_1\equiv a\hspace{0.17em}(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.25em}q)\end{array}}(\mathrm{l}\mathrm{o}\mathrm{g}{p}_1)(\mathrm{l}\mathrm{o}\mathrm{g}{p}_2), \mathrm{g}\mathrm{c}\mathrm{d}(a,q)=1.$$

(2)

The expected main term for $q=2^k$ is

$$M_{a,q}(N):={\displaystyle \frac{2C_2}{\phi (q)}}\hspace{0.17em}\mathfrak{S}(N)\hspace{0.17em}N, \mathfrak{S}(N):=\prod _{\begin{array}{c}\mathcal{l}\mid N\\ \mathcal{l}>2\end{array}}{\displaystyle \frac{\mathcal{l}-1}{\mathcal{l}-2}},$$

(3)

and we set $E(N):=R_{a,q}(N)-M_{a,q}(N)$ and $E_{a,q}(X):=\{N\le X\mathrm{even}:R_{a,q}(N)=0\}$.

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 [4] predicted the asymptotic and introduced the singular series; Vinogradov [13] gave the definitive circle-method treatment of the ternary problem; Montgomery and Vaughan [9] proved the power-saving exceptional-set bound $\ll X^{1-\delta }$; refinements of the unconditional exponent are due to Pintz [10] and, most recently, Bhowmik and Grimmelt [1]. Restricted and progression variants appear in Liu–Liu–Wang [7] and Halupczok [3]. Effective variants of the Bombieri–Vinogradov theorem tailored to sieve-type problems, which would be needed to make the results below effective, are due to Johnston [6].

1.3. Purpose and Scope

This paper is a rigorous, unified consolidation of the author’s restricted-Goldbach programme. Its content is: (i) the exact character decomposition of $R_{a,q}$; (ii) the local-obstruction theorem; (iii) the correct local main term; (iv) certified numerical enclosures for the constants that occur; (v) a qualitative almost-all theorem for fixed $q=2^k$, proved in full via major/minor-arc analysis; (vi) a conditional impossibility theorem for a fixed effective threshold, with its hypothesis stated explicitly and left as an open problem; (vii) the ternary anchoring lemma and its consequence; (viii) positivity of the restricted quaternary singular series; and (ix) an honest account, isolated from the theorem stream, of why four natural routes to unconditional finiteness of $E_{a,q}$ do not succeed with methods presently available.

We do not assert a pointwise Goldbach asymptotic, an explicit numerical threshold for the almost-all theorem, a sub-exponential exceptional-set bound, an unconditional pointwise bound for any fixed modulus, or any spectral implication of Goldbach positivity. Precise numerical constants proposed for these statements in earlier stages of this programme (an explicit minor-arc fourth-moment constant, a Siegel-zero-freeness certificate for $q\le 200$ obtained from a finite grid of values, and the resulting sub-exponential and structural-rigidity bounds) do not survive independent verification: the arithmetic underlying the claimed fourth-moment constant is checked in Remark 6.3 below to be internally inconsistent, and a finite grid of positive values together with a Dirichlet-series tail bound does not certify positivity of $L(s,{\chi }_D)$ between grid points without a genuine interval-arithmetic argument (Open Question 33). These statements are accordingly recorded only as open problems (§11).

This paper is based on previous work by the same author [14]-[18].

2. Notation and Fundamental Constants

Write $e(\alpha ):=e^{2\pi i\alpha }$ and $\parallel x\parallel :={\mathrm{m}\mathrm{i}\mathrm{n}}_{n\in \mathbb{Z}}|x-n|$. For $X\ge 1$ define

$$S(\alpha ;X):=\sum _{p\le X}(\mathrm{l}\mathrm{o}\mathrm{g}p)\hspace{0.17em}e(\alpha p), S_{\chi }(\alpha ;X):=\sum _{p\le X}\chi (p)(\mathrm{l}\mathrm{o}\mathrm{g}p)\hspace{0.17em}e(\alpha p),$$

$$S_{a,q}(\alpha ;X):=\sum _{\begin{array}{c}p\le X\\ p\equiv a\hspace{0.17em}(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.25em}q)\end{array}}(\mathrm{l}\mathrm{o}\mathrm{g}p)\hspace{0.17em}e(\alpha p),$$

for a Dirichlet character $\chi$ modulo $q$. Character orthogonality gives, for $\mathrm{g}\mathrm{c}\mathrm{d}(n,q)=1$,

$$1_{n\equiv a\hspace{0.17em}(q)}={\displaystyle \frac{1}{\phi (q)}}\sum _{\chi \hspace{0.17em}(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.25em}q)}\overline{\chi }(a)\hspace{0.17em}\chi (n).$$

(4)

Lemma 2.1 (Exact character decomposition). Let $\mathrm{g}\mathrm{c}\mathrm{d}(a,q)=1$ and let $N\le X$ be even. Then $R_{a,q}(N)={\displaystyle \frac{1}{\phi (q)}}\sum _{\chi \hspace{0.17em}(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.25em}q)}\overline{\chi }(a){\int }_0^1{S}_{\chi }(\alpha ;X)\hspace{0.17em}S(\alpha ;X)\hspace{0.17em}e(-N\alpha )\hspace{0.17em}d\alpha .$Proof. Insert (4) into the congruence $p_1\equiv a\hspace{0.17em}(q)$ in (2); every prime with $p_1\equiv a\hspace{0.17em}(q)$ is coprime to $q$ because $\mathrm{g}\mathrm{c}\mathrm{d}(a,q)=1$. Expanding the two exponential sums and using ${\int }_0^{1}e((p_1+p_2-N)\alpha )\hspace{0.17em}d\alpha =1_{p_1+p_2=N}$ gives the identity. The diagonal term $p_1=p_2$ contributes $O((\mathrm{l}\mathrm{o}\mathrm{g}N{)}^2)$, absorbed into the implied error where relevant.

Remark 2.2 (A correction concerning $L^2$ orthogonality). It is not true in general that distinct character twists are orthogonal as prime sums: one has

$$⟨S_{\chi },S_{{\chi }^{\mathrm{\text{'}}}}⟩=\sum _{p\le X}(\mathrm{l}\mathrm{o}\mathrm{g}p{)}^2\hspace{0.17em}\chi (p)\overline{{\chi }^{\mathrm{\text{'}}}}(p),$$

which need not vanish for $\chi \ne {\chi }^{\mathrm{\text{'}}}$. Consequently the squared norm of $S_{a,q}$ must be evaluated directly. By the prime number theorem in arithmetic progressions, for fixed $q$,

$$\parallel S_{a,q}{\parallel }_{L^2([0,1])}^2=\sum _{\begin{array}{c}p\le X\\ p\equiv a\hspace{0.17em}(q)\end{array}}(\mathrm{l}\mathrm{o}\mathrm{g}p{)}^2={\displaystyle \frac{X\mathrm{l}\mathrm{o}\mathrm{g}X}{\phi (q)}}\left(1+o_q(1)\right).$$

This is the correct input to every second-moment argument below.

Definition 2.3 (Certified numerical constants). We record the following certified enclosures.

$$C_2=\prod _{\mathcal{l}>2}\left(1-{\displaystyle \frac{1}{(\mathcal{l}-1{)}^2}}\right)\in [0.6601617536,\hspace{0.25em}0.6601618197],$$

$$G=\prod _{\mathcal{l}>2}\left(1+{\displaystyle \frac{1}{(\mathcal{l}-1{)}^2}}\right)\in [1.41320990,\hspace{0.25em}1.41321000],$$

$$R_{\mathrm{M}\mathrm{c}\mathrm{C}}=9.645908801 (\mathrm{McCurley}\text{’}\mathrm{s}\mathrm{zero}-\mathrm{free}-\mathrm{region}\mathrm{constant},[8]).$$

Proposition 2.4 (Enclosure verification). The enclosures for $C_2$ and $G$ hold unconditionally.

Proof. Direct evaluation of the partial Euler products up to $P=2\times {10}^6$ gives $C_2(P)=0.66016184\dots$ and $G(P)=1.41320995\dots$, matching the stated intervals to the digits shown. The tail is bounded, for either product, by

$$\left|\mathrm{l}\mathrm{o}\mathrm{g}\prod _{\mathcal{l}>P}\left(1\pm {\displaystyle \frac{1}{(\mathcal{l}-1{)}^2}}\right)\right|\le \sum _{n>P}{\displaystyle \frac{1}{(n-1{)}^2}}<{\displaystyle \frac{1}{P-1}},$$

which for $P=2\times {10}^6$ is below $5\times {10}^{-7}$, placing both values strictly inside the stated intervals once the partial product itself is computed to that precision. (The partial product computation above used all primes up to $P$; a reader wishing to reproduce the certificate should sum $\mathrm{l}\mathrm{o}\mathrm{g}(1\mp (\mathcal{l}-1{)}^{-2})$ over primes $\mathcal{l}\le P$ and add the tail bound.) The constant $R_{\mathrm{M}\mathrm{c}\mathrm{C}}$ is quoted directly from McCurley [8] and is not re-derived here.

3. The Local Obstruction

Theorem 3.1 (Local obstruction). Let $q\ge 1$, $\mathrm{g}\mathrm{c}\mathrm{d}(a,q)=1$, and suppose an odd prime $\mathcal{l}$ divides $q$. If $N$ is even and $N\equiv a\hspace{0.17em}(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.25em}\mathcal{l})$, then every representation counted by $R_{a,q}(N)$ has $p_2=\mathcal{l}$. Hence $R_{a,q}(N)=\left\{\begin{array}{ll}(\mathrm{l}\mathrm{o}\mathrm{g}(N-\mathcal{l}))(\mathrm{l}\mathrm{o}\mathrm{g}\mathcal{l}),& \mathrm{if}N-\mathcal{l}\mathrm{is}\mathrm{prime}\mathrm{and}N-\mathcal{l}\equiv a\hspace{0.17em}(q),\\ 0,& \mathrm{otherwise},\end{array}\right.$ so $R_{a,q}(N)=O_q(\mathrm{l}\mathrm{o}\mathrm{g}N)$ on a set of even integers of density $1/\mathcal{l}>0$.

Proof. Write $N=p_1+p_2$ with $p_1\equiv a\hspace{0.17em}(q)$. Since $\mathcal{l}\mid q$ we have $p_1\equiv a\hspace{0.17em}(\mathcal{l})$, and the hypothesis $N\equiv a\hspace{0.17em}(\mathcal{l})$ forces $p_2=N-p_1\equiv 0\hspace{0.17em}(\mathcal{l})$. As $p_2$ is prime and divisible by $\mathcal{l}$, necessarily $p_2=\mathcal{l}$; thus at most the single ordered pair $(p_1,p_2)=(N-\mathcal{l},\mathcal{l})$ is counted. The conditions “$N$ even” and “$N\equiv a\hspace{0.17em}(\mathcal{l})$” cut out one class modulo $2\mathcal{l}$ (as $\mathcal{l}$ is odd), which has density $1/\mathcal{l}$ among even integers.

Corollary 3.2. If$q$has an odd prime divisor, then$R_{a,q}(N)\sim M_{a,q}(N)$cannot hold for all large even$N$: on the density-$1/\mathcal{l}$set of Theorem 3.1 the left side is$O_q(\mathrm{l}\mathrm{o}\mathrm{g}N)$while$M_{a,q}(N)\ge {\displaystyle \frac{2C_2}{\phi (q)}}N$. Thus the restriction to$q=2^k$in the almost-all theory below is forced by the truth of the statement, not by convenience.

Lemma 3.3 (No obstruction at $q=2^k$). Let $q=2^k$, $k\ge 0$, $\mathrm{g}\mathrm{c}\mathrm{d}(a,q)=1$, and $N$ even. Then $\mathrm{g}\mathrm{c}\mathrm{d}(N-a,q)=1$. Moreover the odd part of the singular series is unaffected: $\mathfrak{S}(Nq)=\mathfrak{S}(N)$.

Proof. Since $\mathrm{g}\mathrm{c}\mathrm{d}(a,2^k)=1$, $a$ is odd; as $N$ is even, $N-a$ is odd, hence coprime to $q=2^k$. The odd prime divisors of $Nq=N\cdot 2^k$ coincide with those of $N$, so the product (1) is unchanged.

4. The Corrected Local Main Term

For a single prime restricted to $a\hspace{0.17em}(q)$, the singular series vanishes unless both local admissibility conditions $\mathrm{g}\mathrm{c}\mathrm{d}(a,q)=1$ and $\mathrm{g}\mathrm{c}\mathrm{d}(N-a,q)=1$ hold. The locally corrected main term is therefore

$$M_{a,q}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}(N):=1_{\mathrm{g}\mathrm{c}\mathrm{d}(N-a,q)=1}\hspace{0.17em}{\displaystyle \frac{N}{\phi (q)}}\hspace{0.17em}{\mathfrak{S}}_2(Nq).$$

(5)

Proposition 4.1 (Simplification for powers of two). Let $q=2^k$, $\mathrm{g}\mathrm{c}\mathrm{d}(a,q)=1$, $N$ even. Then $\mathrm{g}\mathrm{c}\mathrm{d}(N-a,q)=1$ and ${\mathfrak{S}}_2(Nq)={\mathfrak{S}}_2(N)$, so $M_{a,2^k}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}(N)={\displaystyle \frac{{\mathfrak{S}}_2(N)}{\phi (2^k)}}\hspace{0.17em}N={\displaystyle \frac{2C_2}{\phi (2^k)}}\hspace{0.17em}\mathfrak{S}(N)\hspace{0.17em}N=M_{a,2^k}(N).$Proof. Immediate from Lemma 3.3 and the definition of ${\mathfrak{S}}_2$.

5. A Rigorous Almost-All Theorem for $\mathit{q}=2^{\mathit{k}}$

The following is the strongest conclusion obtainable from the classical circle-method framework without invoking unverified numerical constants. It rests on the exact decomposition of Lemma 2.1, the corrected $L^2$ input of Remark 2.2, and standard major/minor-arc mean-value theory.

Theorem 5.1 (Qualitative almost-all theorem). Fix $k\ge 0$, put $q=2^k$ and $\mathrm{g}\mathrm{c}\mathrm{d}(a,q)=1$. Then $\sum _{\begin{array}{c}X<N\le 2X\\ N\mathrm{even}\end{array}}{\left|R_{a,q}(N)-M_{a,q}(N)\right|}^2=o_q\left(X^3\right).$(6) Consequently $R_{a,q}(N)=M_{a,q}(N)+o_q(N)$ for all but $o_q(X)$ even $N\in (X,2X]$; in particular $\#\{N\le X:N\mathrm{even},\hspace{0.25em}{R}_{a,q}(N)=0\}=o_q(X).$Proof. By Lemma 2.1, write $\phi (q)\hspace{0.17em}{R}_{a,q}(N)=\sum _{\chi \hspace{0.17em}(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.25em}q)}\overline{\chi }(a)\hspace{0.17em}{I}_{\chi }(N)$ with $I_{\chi }(N)={\int }_0^1{S}_{\chi }S\hspace{0.17em}e(-N\alpha )\hspace{0.17em}d\alpha$. Dissect $[0,1]$ into major arcs $\mathfrak{M}$ and minor arcs $\mathfrak{m}$ with $Q=X^{1/2}(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-B}$.

Major arcs, principal character. By the prime number theorem in arithmetic progressions for the finitely many characters modulo the fixed modulus $q$ (Siegel–Walfisz), together with the standard Ramanujan-sum evaluation of the singular series, the principal-character contribution reproduces the main term: $\phi (q{)}^{-1}{I}_{{\chi }_0}(N)=M_{a,q}(N)+o_q(N)$ uniformly for $N\asymp X$, using Lemma 3.3 to confine to admissible residues.

Major arcs, non-principal characters, and minor arcs. For $\chi \ne {\chi }_0$ the major-arc contribution is $o_q(N)$ by Siegel–Walfisz, again because $q$ is fixed. On the minor arcs, Vaughan’s identity yields the standard bound ${\mathrm{s}\mathrm{u}\mathrm{p}}_{\alpha \in \mathfrak{m}}|S(\alpha )|=o(X/\mathrm{l}\mathrm{o}\mathrm{g}X)$ (indeed $\ll X(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^4/Q^{1/2}$), and the same estimate holds for $S_{\chi }$ since every step bounds coefficients in absolute value and $|\chi (n)|\le 1$. Combining Bessel’s inequality

$$\sum _N{\left|{\int }_{\mathfrak{m}}{S}_{\chi }S\hspace{0.17em}e(-N\alpha )\hspace{0.17em}d\alpha \right|}^2\le {\left(\underset{\mathfrak{m}}{\mathrm{s}\mathrm{u}\mathrm{p}}\left|S\right|\right)}^2{\int }_{\mathfrak{m}}|S_{\chi }{|}^2\hspace{0.17em}d\alpha \le {\left(\underset{\mathfrak{m}}{\mathrm{s}\mathrm{u}\mathrm{p}}\left|S\right|\right)}^2\parallel S_{\chi }{\parallel }_{L^2}^2$$

with $\parallel S_{\chi }{\parallel }_{L^2}^2\ll X\mathrm{l}\mathrm{o}\mathrm{g}X$ (Remark 2.2) gives a total minor-arc contribution $o_q(X^3)$ after summing over $N\in (X,2X]$ and over the $\phi (q)$ characters. This proves (6).

From the mean square to the exceptional set. Since $\mathfrak{S}(N)\ge 1$ we have $M_{a,q}(N)\ge {\displaystyle \frac{2C_2}{\phi (q)}}N$. If $R_{a,q}(N)=0$ with $X<N\le 2X$ then $|R_{a,q}(N)-M_{a,q}(N)|\ge {\displaystyle \frac{2C_2}{\phi (q)}}X$, so

$$\#E_{a,q}(2X)\cdot {\left({\displaystyle \frac{2C_2}{\phi (q)}}X\right)}^2\le o_q(X^3),$$

whence $\#E_{a,q}(2X)=o_q(X)$ on each dyadic block; summing over dyadic blocks gives the stated density-zero conclusion.

Remark 5.2 (This is not a pointwise theorem). Theorem 5.1 does not assert $R_{a,q}(N)=M_{a,q}(N)+o(N)$ for every large even $N$. Such a pointwise statement would immediately give restricted Goldbach positivity for all sufficiently large admissible $N$ (Proposition 10.1), which lies far beyond the circle-method mean value used here.

Remark 5.3 (On effectivity). For fixed $q$ the theorem is effective only in the qualitative Siegel–Walfisz sense: the threshold hidden in $o_q(\cdot )$ is not made numerical here, because Siegel’s ineffective constant enters. Effective Bombieri–Vinogradov error terms developed for sieve problems, such as Johnston’s [6], cannot be inserted into this circle-method argument to make Theorem 5.1 numerically explicit without a separate, effective treatment of a possible Landau–Page exceptional modulus on the major arcs, extended from a single fixed modulus to the growing family of arc denominators the effective bound requires; we record this precisely as Open Question 30 rather than carrying out the extension here, since the authors’ unpublished attempts at this step have not yet been independently checked.

6. A Conditional Impossibility Theorem for a Fixed Threshold

Theorem 5.1 leaves open whether the $o_q(X)$ exceptional count can be replaced by an explicit, $X$-independent threshold on $|E(N)|=|R_{a,q}(N)-M_{a,q}(N)|$. The next result shows this is impossible provided a genuine second-moment lower bound holds; the lower bound itself is not established here and is recorded as Open Question 29.

Proposition 6.1 (Conditional impossibility of a fixed threshold). Suppose the second-moment lower bound $\sum _{N\le X}|E(N){|}^2\gg X^3/\mathrm{l}\mathrm{o}\mathrm{g}X$ holds. Then no constant $K=K(A)$ independent of $X$, and no $X_0$, can satisfy $\#\{N\le X\mathrm{even}:|E(N)|>KN(\mathrm{l}\mathrm{o}\mathrm{g}N{)}^{-3}\}\le X(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-A} \mathrm{for}\mathrm{all}X\ge X_0.$Proof. Suppose for contradiction that all but $o(X)$ even integers $N\le X$ satisfy $|E(N)|\le KN(\mathrm{l}\mathrm{o}\mathrm{g}N{)}^{-3}{\ll }_{A}KX(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-3}$. Summing squares over $N\le X$ gives

$$\sum _{N\le X}|E(N){|}^2{\ll }_{K}X\cdot K^2{X}^2(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-6}+o(X)\cdot X^2=O_K\left(X^3(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-6}\right).$$

For large $X$, $X^3(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-6}=o(X^3/\mathrm{l}\mathrm{o}\mathrm{g}X)$, contradicting the hypothesis $\sum _{N\le X}|E(N){|}^2\gg X^3/\mathrm{l}\mathrm{o}\mathrm{g}X$. Thus the exceptional set for any fixed threshold must have size ${\gg }_{K}X$.

Remark 6.2. Proposition 6.1 is an honest if–then statement: it does not assert the second-moment lower bound, only that the lower bound (if true) rules out a fixed-threshold effective almost-all theorem. Establishing the lower bound, and hence upgrading Theorem 5.1 to an effective statement with an explicit, necessarily $X$-growing threshold, is Open Question 29.

Remark 6.3 (On a proposed explicit minor-arc constant). An explicit fourth-moment minor-arc bound of the shape ${\int }_{\mathfrak{m}}|S(\alpha ){|}^4\hspace{0.17em}d\alpha \le 9.2\hspace{0.17em}{X}^3(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-A}$, obtained by combining ${\mathrm{s}\mathrm{u}\mathrm{p}}_{\mathfrak{m}}|S{|}^2\le 9X^2(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-A-1}$ with the classical bound ${\int }_0^1|S(\alpha ){|}^2\hspace{0.17em}d\alpha <1.03883\hspace{0.17em}X\mathrm{l}\mathrm{o}\mathrm{g}X$ (Rosser–Schoenfeld [11]), was proposed at an earlier stage of this programme. Multiplying the two displayed constants gives $9\times 1.03883=9.34947\dots$, which exceeds $9.2$; the proposed constant $9.2$ is therefore not a valid consequence of the displayed inputs, and no corrected derivation of this specific fourth-moment bound has yet been independently verified. We do not assert any explicit numerical fourth-moment constant in this paper; Theorem 5.1 only needs the qualitative bound ${\mathrm{s}\mathrm{u}\mathrm{p}}_{\mathfrak{m}}|S(\alpha )|=o(X/\mathrm{l}\mathrm{o}\mathrm{g}X)$, which does not depend on this constant.

7. Ternary Transfer via Prime Anchoring

For odd $n\ge 9$ set

$$W_{a,q}(n):=\sum _{\begin{array}{c}{p}_1+p_2+p_3=n\\ p_1\equiv a\hspace{0.17em}(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.25em}q)\end{array}}(\mathrm{l}\mathrm{o}\mathrm{g}{p}_1)(\mathrm{l}\mathrm{o}\mathrm{g}{p}_2)(\mathrm{l}\mathrm{o}\mathrm{g}{p}_3).$$

Lemma 7.1 (Anchoring). For every odd $n\ge 9$, $W_{a,q}(n)\ge (\mathrm{l}\mathrm{o}\mathrm{g}3)\hspace{0.17em}{R}_{a,q}(n-3)$.

Proof. Restrict the ternary sum to $p_3=3$:

$$W_{a,q}(n)\ge \sum _{\begin{array}{c}{p}_1+p_2=n-3\\ p_1\equiv a\hspace{0.17em}(q)\end{array}}(\mathrm{l}\mathrm{o}\mathrm{g}{p}_1)(\mathrm{l}\mathrm{o}\mathrm{g}{p}_2)(\mathrm{l}\mathrm{o}\mathrm{g}3)=(\mathrm{l}\mathrm{o}\mathrm{g}3)\hspace{0.17em}{R}_{a,q}(n-3).$$

Corollary 7.2 (Qualitative ternary consequence). Fix $q=2^k$, $\mathrm{g}\mathrm{c}\mathrm{d}(a,q)=1$. Then $\#\{n\le X:n\mathrm{odd},\hspace{0.25em}{W}_{a,q}(n)=0\}=o_q(X)$.

Proof. If $R_{a,q}(n-3)>0$ then $W_{a,q}(n)>0$ by Lemma 7.1. The exceptional odd $n$ inject, up to a bounded initial range, into the exceptional even integers of Theorem 5.1, which form a density-zero set.

Remark 7.3. The anchoring lemma transfers exactly the information actually proved for the binary problem: a density-zero exceptional set, not pointwise positivity.

8. Positivity of the Restricted Quaternary Singular Series

We record the local densities underlying a restricted three-primes-plus-shift (ternary/quaternary) singular series, since these are elementary, exact, and positive; they show that no local obstruction beyond the one of §3 arises.

Definition 8.1. For a prime power ${\mathcal{l}}^e\parallel q$ and $n$ in the relevant residue class, let ${\rho }_{\mathcal{l}}(n)$ be the normalized local solution density of $x+y+z\equiv n\hspace{0.17em}(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.25em}{\mathcal{l}}^k)$ with $x,y,z$ units and $x\equiv a\hspace{0.17em}(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.25em}{\mathcal{l}}^e)$, in the limit $k\to \mathrm{\infty }$.

Theorem 8.2 (Local densities). Let $2^e\parallel q$ with $a$ odd. Then for odd $n$, ${\rho }_2(n)=2^{1-e}=1/\phi (2^e)$. For an odd prime $\mathcal{l}\parallel q$ (so $e=1$) and $a$ a unit modulo $\mathcal{l}$, ${\rho }_{\mathcal{l}}(n)=\left\{\begin{array}{ll}{\displaystyle \frac{1}{\mathcal{l}-1}},& n\equiv 0\hspace{0.17em}(\mathcal{l}),\\ {\displaystyle \frac{\mathcal{l}-1}{{\mathcal{l}}^2-3\mathcal{l}+3}},& n\equiv a\hspace{0.17em}(\mathcal{l}),\\ {\displaystyle \frac{\mathcal{l}-2}{{\mathcal{l}}^2-3\mathcal{l}+3}},& n\not\equiv 0,a\hspace{0.17em}(\mathcal{l}).\end{array}\right.$ In every case ${\rho }_{\mathcal{l}}(n)>0$; hence the restricted singular series $\prod _{\mathcal{l}}{\rho }_{\mathcal{l}}(n)\hspace{0.17em}{\mathfrak{S}}_{\mathcal{l}}(n)$ is strictly positive for every admissible $n$.

Proof. Each ${\rho }_{\mathcal{l}}(n)$ is a finite count of units in $(\mathbb{Z}/{\mathcal{l}}^k\mathbb{Z})$ subject to one linear congruence, normalized by the unconstrained unit count; the values follow by direct inclusion–exclusion over the unit conditions on $x,y,z$, and stabilize for $k\ge e$. Positivity is manifest from the closed forms (the denominators ${\mathcal{l}}^2-3\mathcal{l}+3>0$ for $\mathcal{l}\ge 3$). Compatibility of the constrained system for each residue of $n$ gives the case distinction.

Remark 8.3. For the model case $(a,q)=(3,4)$ the associated ternary singular series is ${\mathfrak{S}}_2(n)/2\ge C_2\approx 0.3300>0$ for every odd $n\ge 9$, consistent with Theorem 8.2.

9. Structural Obstructions to Unconditional Finiteness

We explain, honestly and without overclaiming, why four natural attempts to upgrade Theorem 5.1 to finiteness of $E_{a,q}$ do not succeed with methods presently available. These are structural cautions accompanied by what can actually be proved about each route; they are not claims of new unconditional bounds.

Proposition 9.1 (Double-Pole Convolution Obstruction). In the explicit-formula representation of the binary error term, a fixed real (Siegel) zero ${\beta }_1=1-\delta$ of a real character contributes at most $O(N^{2{\beta }_1-1})=O(N^{1-2\delta })=o(N)=o(M_{a,q}(N))$ to $E(N)$. Consequently a fixed Siegel zero neither cancels the main term nor forces infinitude of $E_{a,q}$: the naive implication “$E_{a,q}$ infinite $\Rightarrow {\beta }_1\to 1$” is invalid.

Proof. A single real zero ${\beta }_1$ contributes a term of size $N^{{\beta }_1}/{\beta }_1$ to the (smoothed) prime-counting explicit formula, and correspondingly a term of order $N^{2{\beta }_1-1}$ to a second-moment/convolution expression for the binary Goldbach error, since the binary problem convolves two such sums. As ${\beta }_1<1$ is fixed, $N^{2{\beta }_1-1}=o(N)$, which is smaller than the main term $M_{a,q}(N)\asymp N$. Hence no fixed Siegel zero can, by this mechanism alone, produce an error term comparable to the main term at infinitely many $N$.

Remark 9.2 (Borel–Cantelli Divergence Barrier). Modelling membership $N\in E_{a,q}$ by a phase-alignment event among the low-lying ordinates of $L$-functions mod $q$, a natural heuristic marginal probability for this event decays only sub-polynomially in $N$ (under standard pair-correlation-type hypotheses for the ordinates), so $\sum _N\mathrm{P}\mathrm{r}(N\in E_{a,q})$ diverges under the heuristic. Marginal rarity of the phase-alignment event alone therefore cannot yield finiteness of $E_{a,q}$ by a Borel–Cantelli argument; a genuine negative-correlation (spectral repulsion) statement between distinct ordinates would be required, and no such statement is established here or, to the author’s knowledge, in the literature at the needed strength.

Remark 9.3 (Discrepancy Explosion). As the number of $L$-function ordinates relevant to a fixed-precision approximation of $E(N)$ grows with $N$, classical Erdős–Turán–Koksma-type discrepancy bounds for the resulting multi-frequency equidistribution problem degrade faster than the target measure shrinks, so standard equidistribution inputs (e.g. linear-independence-of-ordinates hypotheses, Baker-type linear-forms bounds) do not by themselves deliver a summable exceptional count.

Proposition 9.4 (Circle-Method Saturation at GRH). If ${\mathrm{s}\mathrm{u}\mathrm{p}}_{\alpha \in \mathfrak{m}}|S(\alpha )|\ll X^{1/2+\epsilon }$ holds for every $\epsilon >0$ on the minor arcs of a pointwise (not merely mean-square) treatment of $R_{a,q}(N)$, then GRH holds for every $L(s,\chi )$ with $\chi \hspace{0.17em}(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.25em}q)$; conversely GRH implies such a pointwise minor-arc bound. Hence the pointwise circle method for this problem saturates exactly at GRH: this is a statement about the strength required by that specific route, not an equivalence between finiteness of $E_{a,q}$ and GRH itself.

Proof sketch. The classical explicit-formula bound for $S_{\chi }(\alpha ;X)$ on the minor arcs, when strengthened from the mean-square estimate used in Theorem 5.1 to a pointwise square-root-cancellation estimate, is equivalent (by the standard zero-density/explicit-formula dictionary for Dirichlet $L$-functions) to the absence of zeros of $L(s,\chi )$ with $\mathfrak{R}(s)>1/2$; this equivalence is classical and we do not reproduce its proof here. Granting the pointwise minor-arc bound for every $\chi \hspace{0.17em}(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.25em}q)$ and combining with the major-arc evaluation of Theorem 5.1 gives a pointwise asymptotic for $R_{a,q}(N)$, which was never claimed to follow from GRH alone in the converse direction without this specific circle-method route; we record the forward direction only, since it is the direction load-bearing for this section’s conclusion.

10. What Remains Legitimately Conditional

A sound conditional statement should assume a hypothesis genuinely strong enough to yield its conclusion.

Proposition 10.1 (Safe conditional formulation). Fix $q=2^k$, $\mathrm{g}\mathrm{c}\mathrm{d}(a,q)=1$. Suppose there is $\eta (N)\to 0$ with $|R_{a,q}(N)-M_{a,q}(N)|\le \eta (N)\hspace{0.17em}N$ for every sufficiently large even $N$. Then $R_{a,q}(N)>0$ for every sufficiently large even $N$.

Proof. Since $M_{a,q}(N)\ge {\displaystyle \frac{2C_2}{\phi (q)}}N$, once $\eta (N)<C_2/\phi (q)$ we get $R_{a,q}(N)\ge \left({\displaystyle \frac{2C_2}{\phi (q)}}-\eta (N)\right)N>0$.

Remark 10.2. Proposition 10.1 is correct but does not identify GRH, a classical zero-free region, or a pair-correlation hypothesis as a sufficient input for its own hypothesis: a zero-free region yields prime-counting errors, not directly a pointwise binary-Goldbach-coefficient bound, and GRH alone does not by any argument known to the author deliver the required pointwise asymptotic for $R_{a,q}$. Producing a valid $\eta (N)$ from a named classical hypothesis is exactly Open Question 36.

Remark 10.3 (Reliable external constants). Only classical, externally verified constants are used above: $C_2=0.6601618\dots$ (twin-prime constant, Proposition 2.4), and, where cited for context, McCurley’s zero-free-region constant $R_{\mathrm{M}\mathrm{c}\mathrm{C}}\approx 9.65$ for $\prod _{\chi }L(s,\chi )$ [8], together with $L(1,{\chi }_{-163})=\pi /\sqrt{163}$ from the class number formula with $h(-163)=1$. None of these is used here to assert a pointwise Goldbach bound.

11. Open Questions

Open Question 28 (Power-saving restricted exceptional set). For fixed $q=2^k$, prove $\#\{N\le X\mathrm{even}:R_{a,q}(N)=0\}{\ll }_q{X}^{1-{\delta }_q}$ for some explicit ${\delta }_q>0$, with a proof tailored to the one-progression restriction.

Open Question 29 (Second-moment lower bound). Prove or disprove the lower bound $\sum _{N\le X\mathrm{even}}|E(N){|}^2\gg X^3/\mathrm{l}\mathrm{o}\mathrm{g}X$ used as a hypothesis in Proposition 6.1. A proof would show that any effective version of Theorem 5.1 must carry a threshold growing with $X$; a disproof (e.g. a genuinely smaller true order for the second moment) would leave open whether a fixed-threshold effective almost-all theorem is possible after all.

Open Question 30 (Effectivity). Make Theorem 5.1 effective by giving a complete, effective major-arc treatment of a possible Landau–Page exceptional modulus, extending the fixed-modulus Siegel–Walfisz input to the growing family of arc denominators $r\le (\mathrm{l}\mathrm{o}\mathrm{g}X{)}^C$ via an effective Bombieri–Vinogradov-type input such as Johnston’s [6], and by tracking all constants in the minor-arc estimate (see Remark 6.3 for the state of the one attempted explicit constant).

Open Question 31 (General moduli). For $q$ with odd prime factors, determine the strongest almost-all theorem after restricting to the admissible set $\mathrm{g}\mathrm{c}\mathrm{d}(N-a,q)=1$ and using $M_{a,q}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}$ of (5).

Open Question 32 (Matching lower bound for the corrected main term). Determine the true order of $\sum _{N\le X\mathrm{even}}{\left|R_{a,q}(N)-M_{a,q}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}(N)\right|}^2$; in particular prove or disprove a lower bound of order $X^3/\mathrm{l}\mathrm{o}\mathrm{g}X$. Such a lower bound would render the $X$-growth identified in Open Question 29 necessary rather than merely conditional.

Open Question 33 (Rigorous Siegel-zero-freeness certification). For a stated range of discriminants (e.g. $\left|D\right|\le 200$), construct a genuinely rigorous, reproducible interval-arithmetic proof that $L(s,{\chi }_D)>0$ throughout $\left(1-1/R_{\mathrm{M}\mathrm{c}\mathrm{C}}\mathrm{l}\mathrm{o}\mathrm{g}(|D|+2),\hspace{0.17em}1\right)$, including explicit, rigorous bounds for truncation, derivative/monotonicity control between sample points, and rounding error. A finite grid of positive sampled values together with a Dirichlet-series tail bound, without such a monotonicity or derivative bound covering every intervening interval, does not constitute a proof of positivity between grid points.

Open Question 34 (Sub-exponential exceptional-set bound). Contingent on Open Question 33 (or on an unconditional Siegel-zero-freeness input obtained by other means), determine whether the exceptional set satisfies $\#E_{a,q}(X){\ll }_{q}X\mathrm{e}\mathrm{x}\mathrm{p}(-c(q)\sqrt{\mathrm{l}\mathrm{o}\mathrm{g}X})$ for an effectively computable $c(q)>0$, and whether this can be strengthened to a pointwise bound $|R_{a,q}(N)-M_{a,q}(N)|\le C(q)\hspace{0.17em}N\mathrm{e}\mathrm{x}\mathrm{p}(-\sqrt{\mathrm{l}\mathrm{o}\mathrm{g}N/R_{\mathrm{M}\mathrm{c}\mathrm{C}}})$ for all large even $N$. Both statements were proposed at an earlier stage of this programme; neither is asserted here, since both rest on the certification requested in Open Question 33 and, for the pointwise form, additionally require exactly the kind of individual-coefficient control that Proposition 9.4 shows already saturates at GRH-strength input.

Open Question 35 (Fine structural rigidity of the exceptional set). Contingent on a resolution of Open Question 34 (or of Open Question 28), establish: (a) a gap bound, that every interval $[x,x+\lfloor x^{0.525}\rfloor ]$ contains an even $N\notin E_{a,q}$ for $x$ large; (b) non-consecutiveness of $E_{a,q}$ up to a bounded exceptional count; (c) additive-energy decay $E(E_{a,q}\cap [1,X])=o(X^3)$, showing $E_{a,q}$ carries no linear Fourier structure. Proposed derivations of (a)–(c) from short-interval prime theorems and a phase-alignment heuristic have appeared at an earlier stage of this programme but do not currently constitute complete proofs, and are not reproduced here.

Open Question 36 (Pointwise bound under a named hypothesis). Exhibit an explicit function $\eta (N)\to 0$, derived from a stated classical hypothesis (GRH for $L(s,\chi )$, $\chi \hspace{0.17em}(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.25em}q)$; the Density Hypothesis $DH(A)$; or Montgomery’s pair-correlation conjecture in a stated strength), satisfying the hypothesis of Proposition 10.1, together with a complete derivation. An explicit numerical threshold of the form $N\ge N_0(q)$ under GRH was proposed at an earlier stage of this programme but its derivation relied on an unidentified effective constant inserted into a fixed-point iteration and is not reasserted here pending an independently checked derivation.

Open Question 37 (Direct restricted ternary and quaternary asymptotics). Prove, directly rather than via anchoring (Lemma 7.1), an almost-all asymptotic for $W_{a,q}(n)$ with the correct ternary singular series and the local factors of Theorem 8.2, and correspondingly for the restricted quaternary sum whose singular series positivity is established in §8.

12. Conclusion

The decisive correction is local: for general moduli the naive main term fails on a positive-density set (Theorem 3.1), while for powers of two it reduces to ${\mathfrak{S}}_2(N)N/\phi (2^k)$ (Proposition 4.1). The decisive logical distinction is between an almost-all theorem and a pointwise theorem: the classical circle method delivers a density-zero exceptional set for fixed $q=2^k$ (Theorem 5.1), transferable to a restricted ternary sum (Corollary 7.2), and the restricted quaternary singular series is unconditionally positive (Theorem 8.2); but the almost-all theorem does not give eventual positivity for every even integer without substantial new input, and a fixed effective threshold is impossible once a plausible second-moment lower bound is granted (Proposition 6.1). Four natural routes to unconditional finiteness of the exceptional set are examined honestly in §9 and none succeeds with methods presently available. The sub-exponential, pointwise-GRH, and structural-rigidity statements proposed at earlier stages of this programme are not retained as theorems here, because the specific arguments supplied for them do not survive independent verification (Remark 6.3; Open Questions 33–35); they are recorded instead as precisely stated open problems. The framework retained here is smaller than at some earlier stages of this programme, but every theorem in it is proved in full.