Rays and Gaps: Almost-All Goldbach Theorems for Consecutive and Shifted Primes

1. Introduction

1.1. Rays and Gaps in the Restricted Goldbach Programme

Fix q ≥ 1 and gcd(a,q) = 1. The restricted weighted Goldbach sum is

$$R_{a,q}\left(N\right):=\sum _{\begin{array}{c}{p}_1+p_2=N\\ p_1\equiv a\hspace{0.17em}\left(q\right)\end{array}}\left(\log p_1\right)\left(\log p_2\right), N\mathrm{even},$$

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with candidate main term

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

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where $C_2=\prod _{p>2}\left(1-{\left(p-1\right)}^{-2}\right)=0.66016181\dots$ is the twin-prime constant. Put $ℰ$_a,q(X):={N ≤ X, N even:R_a,q(N) = 0} and ${\mathcal{E}}_{a,q}:={\bigcup }_X{\mathcal{E}}_{a,q}\left(X\right)$. For a prime p the residue class of p modulo n is called its ray of modulus n, and ⌊p/n⌋ is its ring. The gap to the next prime is g_n = p_n+1 − p_n, and the even integer built from two consecutive primes is

N_n:=p_n + p_n+1 = 2p_n + g_n  (n ≥ 2).

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The rays of p_n and p_n+1 modulo q determine N_n modulo q, and the gap g_n determines how far N_n is from 2p_n. This paper studies four consequences: (i) the mean-square behaviour of R_a,q − M_a,q for q = 2^k and the resulting exceptional-set bound with a saving of every power of logX (Section 4); (ii) an elementary transfer of almost-all statements for R_a,q to sequences of the form N_n and p + h (Section 6 and Section 7); (iii) what the local obstruction of a modulus with an odd prime factor does along N_n (Section 8); (iv) the digit-repulsion bias between consecutive primes, the dependence of the second moment of R_a,q − M_a,q on N mod q, and the law of the amplitude of the main term along shifted primes (Section 9, Section 10, Section 11 and Section 12).

1.2. What this Paper Does and Does not Claim

It proves unconditionally the mean-square theorem (Theorem 4.10) and the exceptional-set bound (Corollary 4.12), with ineffective implied constants because the Siegel–Walfisz theorem is used; the transfer inequality (Theorem 6.2) and its corollaries as inequalities between counting functions; the obstruction statements of Section 8; the classification of the rays carrying infinitely many primes; and the limit law of Section 12. Numerical statements are labelled [Comp. Verified] with their exact range.

It does not claim that R_a,q(N_n) > 0 for every large n (a pointwise statement); it does not claim a power-saving bound #$ℰ$_a,q(X)≪_qX^1−Δ for q = 2^k (Open Question 13.1); it does not claim any positive-density statement for the exceptional set along N_n when q has an odd prime factor beyond the equivalences proved in Section 8; it does not prove the Lemke Oliver–Soundararajan transition probabilities, which are conditional on the Hardy–Littlewood prime tuples conjecture; and it does not decide whether the second moment of R_a,q(N_n) − M_a,q(N_n) depends on the ray of p_n as n → ∞ (Open Question 13.6).

1.3. Status Labels

Every statement carries one of [Proved], [Cond. Proved (X)], [Comp. Verified], [Conjecture], [Open], [Honest Caveat].

1.4. Structure of the Paper

Section 2 fixes notation and the numerical constants. Section 3 proves the local-obstruction theorem, restricts the analytic core to q = 2^k, and states a conjectural main term for general moduli with a numerical test. Section 4 proves the exact evaluation of the singular series of a progression modulo 2^k, the mean-square theorem and the exceptional-set bound. Section 5 classifies rays. Section 6 proves the transfer principle and the almost-all theorems. Section 7 records computational verification. Section 8 treats moduli with an odd prime factor along N_n. Section 9 documents digit repulsion, the gap-by-ray dependence, the cumulative admissibility function and the comparison with the Lemke Oliver–Soundararajan prediction. Section 10 studies the second moment modulo q. Section 11 measures how far the rays predict the next ray and the gap. Section 12 proves the limit law of the main term along shifted primes. Section 13 lists open questions and Appendix A describes the computational protocol. The relation to the preprint [18] is recorded in Section 1.5.

1.5. Relation to the Preprint [18]

This paper continues [18], which studies the same sum R_a,q(N), the same candidate main term M_a,q(N) and the same exceptional set $ℰ$_a,q(X) for q = 2^k. The points of contact are recorded here so that the two papers can be read together without contradiction. Write E(N):=R_a,q(N) − M_a,q(N). (i) Theorem 3.1 and Proposition 3.3 restate the local-obstruction theorem [18, Theorem 3.1] and the absence of an obstruction at q = 2^k [18, Lemma 3.3]; the proofs are repeated because Section 6 and Section 8 use them along N_n. (ii) Theorem 4.10 proves $\sum _{N\le X}{\left|E\left(N\right)\right|}^2{\ll }_{A,q}{X}^3{\left(\log X\right)}^{-A}$ for every A > 0, which is stronger than the master second-moment bound $\sum _{N\le X}{\left|E\left(N\right)\right|}^2\le {\displaystyle \frac{G}{2\phi \left(q\right)}}{\displaystyle \frac{X^3}{\log X}}\left(1+o\left(1\right)\right)$ of [18, Theorem 6.3], where G is the Gallagher–Goldston constant of Definition 2.1. The bound of [18] therefore remains true but is not sharp, and the discrepancy between the constants discussed in the audit note that follows the proof of [18, Theorem 6.3] concerns only a non-sharp upper bound.

(iii) Taking A = 2 in Theorem 4.10 gives $\sum _{N\le X}{\left|E\left(N\right)\right|}^2=\mathrm{o}\left({\displaystyle \frac{X^3}{\log X}}\right)$. Hence the lower bound $\sum _{N\le X}{\left|E\left(N\right)\right|}^2\gg {\displaystyle \frac{X^3}{\log X}}$, which is Open Question 1 of [18] and the hypothesis of [18, Theorem 7.3], fails, and the conclusion of [18, Theorem 7.3] is not available. Corollary 4.13 gives, ineffectively, a threshold N(logN)^−A that decays, whereas the effective almost-all theorem [18, Theorem 7.2] has a threshold that grows with X. (iv) The sub-exponential bound $\#{\mathcal{E}}_{a,q}\left(X\right){\ll }_{q}X\exp \left(-{\displaystyle \frac{\sqrt{\log X}}{R_{\mathrm{McC}}}}\right)$ stated in [18, Theorem 8.3] is stronger than Corollary 4.12. It is obtained in [18] by a different route, it is not used or reproved here, and nothing in the present paper depends on it. The correspondence between the open questions of the two papers is recorded in Remark 13.10.

2. Notation and Fundamental Constants

Throughout, p,𝓁 denote primes, e(x):=e^2πix, Λ is the von Mangoldt function, φ is Euler’s function, μ the Möbius function, τ = d the divisor function, and ω(n) the number of distinct prime factors of n. We write ∥x∥ for the distance from x to the nearest integer, p_n for the n-th prime (p₁ = 2), π(X) for the number of primes ≤X and π(X;q,c)=#{p ≤ X:p≡c (q)}. We write f≪g for |f| ≤ Cg with an absolute constant, and f≪_qg when C depends on q. For a Dirichlet character χ modulo q no use is made of character sums beyond the Siegel–Walfisz theorem. For X ≥ 3 put

$$S\left(\alpha \right)=\sum _{n\le X}\Lambda \left(n\right)e\left(\mathit{n\alpha }\right), S_{a,q}\left(\alpha \right)=\sum _{\begin{array}{c}n\le X\\ n\equiv a\hspace{0.17em}\left(q\right)\end{array}}\Lambda \left(n\right)e\left(\mathit{n\alpha }\right), v\left(\beta \right)=\sum _{1\le n\le X}e\left(\mathit{n\beta }\right).$$

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Definition 2.1 (Fundamental constants). The twin-prime constant is $C_2=\prod _{\mathcal{l}>2}\left(1-{\left(\mathcal{l}-1\right)}^{-2}\right)$ and the Gallagher–Goldston constant is $G=\prod _{\mathcal{l}>2}\left(1+{\left(\mathcal{l}-1\right)}^{-2}\right)$. For an odd prime 𝓁 let $a_{\mathcal{l}}={\displaystyle \frac{\mathcal{l}-1}{\mathcal{l}-2}}$.

Proposition 2.2 (Certified values). [Comp. Verified] C₂ = 0.66016181… and G = 1.41321…; precisely, the partial products over the odd primes 𝓁 ≤ 10⁷ are C₂(10⁷) = 0.6601618197 and G(10⁷) = 1.4132099903, and over the odd primes 𝓁 ≤ 10⁸ the product for C₂ is 0.66016181616…. The tails satisfy $0\le -\log \prod _{\mathcal{l}>P}\left(1-{\left(\mathcal{l}-1\right)}^{-2}\right)\le \sum _{m>P}{m}^{-2}<1/\left(P-1\right)$ and likewise for G, so each of the two values is determined to the seven decimals shown.

We also record the elementary inequalities, used repeatedly below,

$${\displaystyle \frac{1}{\mathcal{l}}}\le {\displaystyle \frac{1}{\mathcal{l}-2}}-{\displaystyle \frac{1}{2{\left(\mathcal{l}-2\right)}^2}}\le \log a_{\mathcal{l}}\le {\displaystyle \frac{1}{\mathcal{l}-2}}\le {\displaystyle \frac{3}{\mathcal{l}}} \left(\mathcal{l}\ge 3\right).$$

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Proposition 2.3 (Trivial representations). [Proved] For every n ≥ 2 and every q,a: if p_n≡a (q) or p_n+1≡a (q), then R_a,q(N_n) ≥ (logp_n)(logp_n+1) > 0.

Proof. If p_n≡a (q), the pair (p₁,p₂) = (p_n,p_n+1) is counted in (1); if p_n+1≡a (q), so is (p_n+1,p_n). Both terms equal (logp_n)(logp_n+1).

Consequently the statements of Section 6, Section 7 and Section 8 about N_n are informative only on the set of indices with p_n≢a and p_n+1≢a (q). [Comp. Verified] For q = 8, a = 1 these indices have relative frequency 0.5412 among the first 6 ⋅ 10⁴ pairs of consecutive primes.

3. Local Obstructions and the General Main Term

Theorem 3.1 (Local obstruction). [Proved] Let q ≥ 1, gcd(a,q) = 1, let 𝓁 be an odd prime with 𝓁∣q, and let N≡a (𝓁) be even. Then R_a,q(N) ≤ log(N − 𝓁)log𝓁, and R_a,q(N) = 0 unless N − 𝓁 is a prime congruent to a modulo q.

Proof. If p₁≡a (q) then p₁≡a (𝓁), so p₂ = N − p₁≡N − a≡0 (𝓁). Hence p₂ = 𝓁 and p₁ = N − 𝓁. There is at most one such pair. This is the argument of [18, Theorem 3.1].

Corollary 3.2 (Positive lower density of unrepresentable integers). [Proved] Under the hypotheses of Theorem 3.1, the even integers N≡a (𝓁) form a set of relative density 1/𝓁 among the even integers, and $\#\{N\le X\mathrm{even}:R_{a,q}\left(N\right)=0\}\ge {\displaystyle \frac{X}{2\mathcal{l}}}-O\left(\pi \left(X\right)\right)$. In particular no modulus q divisible by an odd prime admits an almost-all theorem of the type proved in Section 4.

Proof. The even N with N≡a (𝓁) form a single residue class modulo 2𝓁. By Theorem 3.1, R_a,q(N) ≠ 0 for such N forces N − 𝓁 to be prime, which happens for at most π(X) values of N ≤ X. Cf. [18, Corollary 3.2].

For q = 2^k and a odd there is no obstruction, since N − a is odd whenever N is even. For the analytic core of the paper, and wherever it is not stated otherwise, we write q = 2^k with k ≥ 1 and take a odd.

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

Proof. Since gcd(a,2^k) = 1, a is odd; as N is even, N − a is odd, hence coprime to q. The odd prime divisors of N/q coincide with those of N, so the product defining 𝔖 is unchanged. Cf. [18, Lemma 3.3].

3.1. The Conjectural Main Term for General Moduli

For a single prime restricted to the class a (mod q) the local singular series vanishes unless both gcd(a,q) = 1 and gcd(N − a,q) = 1 hold. Write e_𝓁 = v_𝓁(q) and define, for 𝓁∣q and for 𝓁∤q with 𝓁 > 2,

$${\Delta }_{\mathcal{l}}\left(N\right)=1_{\mathcal{l}\nmid N-a}\hspace{0.17em}{\mathcal{l}}^{-e_{\mathcal{l}}}{\left(1-{\mathcal{l}}^{-1}\right)}^{-2}, {\sigma }_{\mathcal{l}}\left(N\right)=\left\{\begin{array}{ll}1-{\left(\mathcal{l}-1\right)}^{-2},& \mathcal{l}\nmid N,\\ \mathcal{l}/\left(\mathcal{l}-1\right),& \mathcal{l}\mid N,\end{array}\right.$$

and ε_q = 2 if q is odd, ε_q = 1 if q is even.

Definition 3.4 (Local-density main term). Let q ≥ 1, gcd(a,q) = 1, N even. Define

$$M_{a,q}\left(N\right)={\epsilon }_q\hspace{0.17em}N\prod _{\mathcal{l}\mid q}{\Delta }_{\mathcal{l}}\left(N\right)\prod _{\begin{array}{c}\mathcal{l}\nmid q\\ \mathcal{l}>2\end{array}}{\sigma }_{\mathcal{l}}\left(N\right).$$

For q = 2^k (k ≥ 1) this equals ${\displaystyle \frac{4}{q}}{C}_2\mathfrak{S}\left(N\right)N={\displaystyle \frac{2C_2}{\phi \left(q\right)}}\mathfrak{S}\left(N\right)N$, in agreement with (2), and for q = 1 it is the Hardy–Littlewood main term 2C₂𝔖(N)N [6].

Conjecture 3.5 (General main term). [Conjecture] For every q and a with gcd(a,q) = 1, R_a,q(N) = M_a,q(N)(1 + o(1)) as N → ∞ through even integers with gcd(N − a,q_odd) = 1, where q_odd is the odd part of q.

Proposition 3.6 (Numerical test of Conjecture 3.5). [Comp. Verified] For each pair (q,a) in Table 1 and 120 even integers N were drawn uniformly from [5 ⋅ 10⁶,10⁷] with M_a,q(N) > 0, and R_a,q(N)/M_a,q(N) was computed exactly from the list of primes. The sample means lie in [0.9983,1.0006] and the sample standard deviations in [0.0027,0.0059]. In addition, for 20 integers with M_a,q(N) = 0 per pair, R_a,q(N) ≤ log𝓁logN held in every case in which an obstruction at an odd prime 𝓁 was present, as Theorem 3.1 requires.

4. The Mean-Square Theorem and the Exceptional Set for q = 2^k

Throughout this section q = 2^k with k ≥ 1 and a is odd. The restriction to powers of two enters only through Proposition 4.7: for an odd prime 𝓁∣q the shifts b/r + h/q change the odd part of the denominator and the coefficients no longer factor in this way (and, by Theorem 3.1, the conclusion fails).

4.1. Analytic Tools

Theorem 4.1 (Siegel–Walfisz). For every B > 0 there is c₁ = c₁(B) > 0 such that for r ≤ (logt)^B and gcd(c,r) = 1, $\psi \left(t;r,c\right)={\displaystyle \frac{t}{\phi \left(r\right)}}+O\left(t\hspace{0.17em}{e}^{-c_1\sqrt{\log t}}\right)$, where the implied constant depends on B only and is ineffective [4].

Theorem 4.2 (Vaughan). Let gcd(b,r) = 1 and |α − b/r| ≤ r⁻². Then $\sum _{n\le X}\Lambda \left(n\right)e\left(\mathit{n\alpha }\right)\ll \left(X/\sqrt{r}+X^{4/5}+\sqrt{\mathit{rX}}\right){\left(\log X\right)}^4$ with an absolute implied constant [4,17].

Lemma 4.3 (Restricted sum as a sum of shifts). [Proved] For q ≥ 1 and every a, $S_{a,q}\left(\alpha \right)={\displaystyle \frac{1}{q}}\sum _{h\hspace{0.25em}\mathrm{mod}\hspace{0.25em}q}e\left(-{\displaystyle \frac{\mathit{ah}}{q}}\right)S\left(\alpha +{\displaystyle \frac{h}{q}}\right)$. Consequently, for 1 ≤ N ≤ X, $R_{a,q}^{\Lambda }\left(N\right):=\sum _{\begin{array}{c}{n}_1+n_2=N\\ n_1\equiv a\hspace{0.17em}\left(q\right)\end{array}}\Lambda \left(n_1\right)\Lambda \left(n_2\right)={\int }_0^1{S}_{a,q}\left(\alpha \right)S\left(\alpha \right)e\left(-\mathit{N\alpha }\right)\hspace{0.17em}\mathit{d\alpha }.$

Proof. Insert ${\displaystyle \frac{1}{q}}\sum _{h}e\left(h\left(n-a\right)/q\right)=1_{n\equiv a\hspace{0.17em}\left(q\right)}$ into the definition. The integral formula follows from orthogonality, since for N ≤ X the constraint n_i ≤ X is automatic.

Lemma 4.4 (Relation to prime weights). [Proved] For 2 ≤ N ≤ X, $\left|R_{a,q}\left(N\right)-R_{a,q}^{\Lambda }\left(N\right)\right|\ll \sqrt{X}{\left(\log X\right)}^2$.

Proof. The two sums differ only by pairs in which some n_i is a proper prime power. There are $O\left(\sqrt{X}\right)$ proper prime powers up to X, each term is at most (logX)², and each such n_i occurs in at most one pair.

Lemma 4.5 (Major-arc approximation). [Proved] Fix B' > 0. There is c = c(B') > 0 such that for r' ≤ (logX)^B', gcd(b',r') = 1 and real β, $S\left({\displaystyle \frac{b\mathrm{\text{'}}}{r\mathrm{\text{'}}}}+\beta \right)={\displaystyle \frac{\mu \left(r\mathrm{\text{'}}\right)}{\phi \left(r\mathrm{\text{'}}\right)}}v\left(\beta \right)+O\left(X{e}^{-c\sqrt{\log X}}\left(1+X\left|\beta \right|\right)\right).$

Proof. Terms with gcd(n,r') > 1 are prime powers of primes dividing r' and contribute O((logX)²). For a class c coprime to r' put E(t) = ψ(t;r',c) − t/φ(r'). By Theorem 4.1 (applied with B' replaced by 2B' when $t\ge \sqrt{X}$, and trivially for $t<\sqrt{X}$) we have $E\left(t\right)\ll X{e}^{-c_2\sqrt{\log X}}$ uniformly for t ≤ X. Partial summation gives $\sum _{n\le X,\hspace{0.25em}n\equiv c\hspace{0.17em}\left(r\mathrm{\text{'}}\right)}\Lambda \left(n\right)e\left(\mathit{\beta n}\right)={\displaystyle \frac{1}{\phi \left(r\mathrm{\text{'}}\right)}}{\int }_1^{X}e\left(\mathit{\beta t}\right)\hspace{0.17em}\mathit{dt}+O\left(X{e}^{-c_2\sqrt{\log X}}\left(1+X\left|\beta \right|\right)\right)$, and ${\int }_1^{X}e\left(\mathit{\beta t}\right)\hspace{0.17em}\mathit{dt}=v\left(\beta \right)+O\left(1+X\left|\beta \right|\right)$. Multiplying by e(b'c/r') and summing over the φ(r') ≤ (logX)^B' classes, using $\sum _{\left(c,r\mathrm{\text{'}}\right)=1}e\left(b\mathrm{\text{'}}c/r\mathrm{\text{'}}\right)=\mu \left(r\mathrm{\text{'}}\right)$, gives the claim with a slightly smaller c.

Lemma 4.6 (Divisor moments). [Proved] $\sum _{N\le X}\tau {\left(N\right)}^2\ll X{\left(\log X\right)}^3$; moreover $\sum _{m\le y}m/\phi {\left(m\right)}^2\ll \log \left(y+2\right)$ and $\sum _{m>z}\phi {\left(m\right)}^{-2}\ll \min \left(1,1/z\right)$ for z > 0.

Proof. The first estimate is classical [7]. The other two follow by partial summation from $\sum _{m\le t}{\left(m/\phi \left(m\right)\right)}^2\ll t$.

4.2. The Singular Series of a Progression Modulo 2^k

For an integer r ≥ 1, an integer b coprime to r (with b = 0 when r = 1) and h mod q, let r_b,h denote the denominator in lowest terms of b/r + h/q. Define

$$A\left(r,N\right)={\displaystyle \frac{1}{q}}\sum _{h\hspace{0.25em}\mathrm{mod}\hspace{0.25em}q}e\left(-{\displaystyle \frac{\mathit{ah}}{q}}\right)\sum _{\begin{array}{c}b\hspace{0.25em}\mathrm{mod}\hspace{0.25em}r\\ \left(b,r\right)=1\end{array}}{\displaystyle \frac{\mu \left(r\right)\mu \left(r_{b,h}\right)}{\phi \left(r\right)\phi \left(r_{b,h}\right)}}\hspace{0.17em}e\left(-{\displaystyle \frac{\mathit{Nb}}{r}}\right).$$

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Proposition 4.7 (Factorization of A(r,N)). [Proved] Let N be even and write r = r₂r₁ with r₂ a power of 2 and r₁ odd. Then A(r,N) = 0 unless r₂ ∈ {1,2} and r₁ is squarefree, and in that case $A\left(r,N\right)={\displaystyle \frac{2}{q}}\cdot {\displaystyle \frac{\mu {\left(r_1\right)}^2{c}_{r_1}\left(N\right)}{\phi {\left(r_1\right)}^2}}$, where $c_{r_1}\left(N\right)=\sum _{\left(b,r_1\right)=1}e\left(\mathit{bN}/r_1\right)$ is the Ramanujan sum.

Proof. If r is not squarefree then μ(r) = 0 and A(r,N) = 0. Assume r = r₂r₁ squarefree, so r₂ ∈ {1,2}. By the Chinese remainder theorem $b/r\equiv \bar{u}b/r_2+\bar{v}b/r_1\hspace{0.25em}(\mathrm{mod}\hspace{0.25em}1)$ where ur₁ + vr₂ = 1 (so u is odd when r₂ = 2), and (b mod r₂,b mod r₁) runs over the product of the unit groups. The fraction $\bar{v}b/r_1$ has denominator r₁ (odd), hence r_b,h = d r₁ where d is the denominator of $\bar{u}b/r_2+h/q$; therefore μ(r_b,h) = μ(d)μ(r₁) and φ(r_b,h) = φ(d)φ(r₁). The exponential e(−Nb/r) splits as $e\left(-N\bar{u}b/r_2\right)e\left(-N\bar{v}b/r_1\right)$, and the sum over units modulo r₁ produces $c_{r_1}\left(N\right)$. Consequently

$$A\left(r,N\right)={\displaystyle \frac{\mu {\left(r_1\right)}^2{c}_{r_1}\left(N\right)}{\phi {\left(r_1\right)}^2}}\cdot A_2\left(r_2\right), A_2\left(r_2\right)={\displaystyle \frac{1}{q}}\sum _{h\hspace{0.25em}\mathrm{mod}\hspace{0.25em}q}e\left(-{\displaystyle \frac{\mathit{ah}}{q}}\right){\displaystyle \frac{\mu \left(r_2\right)\mu \left(d_h\right)}{\phi \left(r_2\right)\phi \left(d_h\right)}}\hspace{0.17em}e\left(-{\displaystyle \frac{N\bar{u}{b}_2}{r_2}}\right),$$

with the inner factor evaluated at the unique unit class b₂ modulo r₂.

Case r₂ = 1. Here d_h is the denominator of h/q, a power 2^s; μ(d_h) = 0 unless s ≤ 1. The value s = 0 occurs for h = 0 (term 1), and s = 1 for h = q/2 (term e(−a/2)μ(2) = (−1)(−1) = 1, because a is odd). Hence A₂(1) = 2/q.

Case r₂ = 2. Here b₂ = 1 and $\bar{u}{b}_2/2\equiv {\displaystyle \frac{1}{2}}$, so $e\left(-N\bar{u}{b}_2/2\right)=1$ because N is even. The fraction ${\displaystyle \frac{1}{2}}+{\displaystyle \frac{h}{q}}$ has denominator 2 for h = 0 and 1 for h = q/2, and has denominator divisible by 4 otherwise. The two surviving terms are μ(2)μ(2) = 1 for h = 0 and e(−a/2)μ(2)μ(1) = (−1)(−1) = 1 for h = q/2. Hence A₂(2) = 2/q.

Theorem 4.8 (Singular series for the progression). [Proved] For every even N, $\sum _{r\ge 1}A\left(r,N\right)$ converges absolutely and $\sum _{r\ge 1}A\left(r,N\right)={\displaystyle \frac{2}{\phi \left(q\right)}}{C}_2\mathfrak{S}\left(N\right)$, so that $N\sum _{r\ge 1}A\left(r,N\right)=M_{a,q}\left(N\right)$.

Proof. By Proposition 4.7, $\sum _{r}A\left(r,N\right)=\left(A_2\left(1\right)+A_2\left(2\right)\right)\sum _{r_1\mathrm{odd}}\mu {\left(r_1\right)}^2{c}_{r_1}\left(N\right)/\phi {\left(r_1\right)}^2$. Since A₂(1) + A₂(2) = 4/q = 2/φ(q), and the odd sum is absolutely convergent (dominated by $\prod _{\mathcal{l}}\left(1+O\left(1/{\mathcal{l}}^2\right)\right)\prod _{\mathcal{l}\mid N}\left(1+O\left(1/\mathcal{l}\right)\right)$), it equals the Euler product

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

where we used $\mathcal{l}/\left(\mathcal{l}-1\right)=\left(1-{\left(\mathcal{l}-1\right)}^{-2}\right)\cdot {\displaystyle \frac{\mathcal{l}-1}{\mathcal{l}-2}}$.

Lemma 4.9 (Tail and mean-square estimates for A(r,N)). [Proved] For N ≥ 3 even and y ≥ 2: (i) $\sum _{r>y}\left|A\left(r,N\right)\right|{\ll }_q{y}^{-1/2}\tau \left(N\right){\left(\mathrm{loglog}N\right)}^2$; (ii) $\sum _{r\le y}r\left|A\left(r,N\right)\right|{\ll }_q\tau \left(N\right){\left(\mathrm{loglog}N\right)}^2\log y$; and (iii) $\sum _r\left|A\left(r,N\right)\right|{\ll }_q{\left(\mathrm{loglog}N\right)}^3$.

Proof. By Proposition 4.7, $\left|A\left(r,N\right)\right|\le {\displaystyle \frac{2}{q}}\mu {\left(r_1\right)}^2\left(r_1,N\right)/\phi {\left(r_1\right)}^2$, and for squarefree r₁ one has $c_{r_1}\left(N\right)=\mu \left(r_1/d\right)\phi \left(d\right)$ with d = (r₁,N). Write r₁ = dm with d = (r₁,N) and use φ(dm) ≥ φ(d)φ(m). Then

$$\begin{array}{rr}{\displaystyle \sum _{r_1>y}}{\displaystyle \frac{\left(r_1,N\right)}{\phi {\left(r_1\right)}^2}}& \le {\displaystyle \sum _{d\mid N}}{\displaystyle \frac{d}{\phi {\left(d\right)}^2}}{\displaystyle \sum _{m>y/d}}{\displaystyle \frac{1}{\phi {\left(m\right)}^2}}\ll {\displaystyle \sum _{d\mid N}}{\displaystyle \frac{d}{\phi {\left(d\right)}^2}}\min \left(1,{\displaystyle \frac{d}{y}}\right)\\ \le y^{-1/2}{\displaystyle \sum _{d\mid N}}{\displaystyle \frac{d^{3/2}}{\phi {\left(d\right)}^2}}\ll y^{-1/2}\tau \left(N\right){\left(\mathrm{loglog}N\right)}^2\end{array}$$

by Lemma 4.6 and (d/φ(d))²≪(loglogN)². Since r₂ ∈ {1,2}, (i) follows (with y/2 in place of y). For (ii), $\sum _{r_1\le y}{r}_1\left(r_1,N\right)/\phi {\left(r_1\right)}^2\le \sum _{d\mid N}{\left(d/\phi \left(d\right)\right)}^2\sum _{m\le y/d}m/\phi {\left(m\right)}^2\ll \tau \left(N\right){\left(\mathrm{loglog}N\right)}^2\log y$. For (iii), $\sum _{r_1}\mu {\left(r_1\right)}^2\left(r_1,N\right)/\phi {\left(r_1\right)}^2\le \prod _{\mathcal{l}\nmid N}\left(1+{\left(\mathcal{l}-1\right)}^{-2}\right)\prod _{\mathcal{l}\mid N}\left(1+\mathcal{l}{\left(\mathcal{l}-1\right)}^{-2}\right)\ll \exp \left(3\sum _{\mathcal{l}\mid N}1/\mathcal{l}\right)\ll {\left(\mathrm{loglog}N\right)}^3$.

4.3. The Mean-Square Theorem

Theorem 4.10 (Mean-square theorem). [Proved] Let q = 2^k with k ≥ 1, let a be odd, and let A > 0. Then $\sum _{\begin{array}{c}N\le X\\ N\mathrm{even}\end{array}}{\left|R_{a,q}\left(N\right)-M_{a,q}\left(N\right)\right|}^2{\ll }_{A,q}{X}^3{\left(\log X\right)}^{-A}.$ The implied constant is ineffective.

Proof. Let X be large and put P = (logX)^A+10. We prove the bound for $R_{a,q}^{\Lambda }$; by Lemma 4.4 the difference contributes at most X ⋅ X(logX)⁴ to the sum, which is negligible.

Arcs. Let 𝔐(r,b) = {α:|α − b/r| ≤ P/(rX)} for 1 ≤ r ≤ P, (b,r) = 1, and let 𝔐 be their union, taken modulo 1; for large X the arcs are pairwise disjoint because 2P/X < 1/P². Let 𝔪 = [0,1)\𝔐. By Lemma 4.3, for even N ≤ X, $R_{a,q}^{\Lambda }\left(N\right)=J^{\mathfrak{M}}\left(N\right)+J^{\mathfrak{m}}\left(N\right)$ with $J^{\Omega }\left(N\right)={\int }_{\Omega }{S}_{a,q}\left(\alpha \right)S\left(\alpha \right)e\left(-\mathit{N\alpha }\right)\hspace{0.17em}\mathit{d\alpha }$.

Minor arcs. Let α ∈ 𝔪. By Dirichlet’s theorem with parameter X/P there are coprime b,r with r ≤ X/P and |α − b/r| ≤ P/(rX) ≤ r⁻²; since α ∉ 𝔐 we must have r > P. Theorem 4.2 gives, using P < r ≤ X/P,

$$\left|S\left(\alpha \right)\right|\ll \left({\displaystyle \frac{X}{\sqrt{P}}}+X^{4/5}+{\displaystyle \frac{X}{\sqrt{P}}}\right){\left(\log X\right)}^4\ll {\displaystyle \frac{X{\left(\log X\right)}^4}{\sqrt{P}}}.$$

The numbers J^𝔪(N) are the Fourier coefficients of 1_𝔪S_a,qS ∈ L², so by Parseval and ${\int }_0^1{\left|S_{a,q}\right|}^2=\sum _{n\le X,\hspace{0.25em}n\equiv a}\Lambda {\left(n\right)}^2\ll X\log X$,

$$\sum _N{\left|J^{\mathfrak{m}}\left(N\right)\right|}^2={\int }_{\mathfrak{m}}{\left|S_{a,q}\right|}^2{\left|S\right|}^2\le \underset{\mathfrak{m}}{\sup }{\left|S\right|}^2{\int }_0^1{\left|S_{a,q}\right|}^2\ll {\displaystyle \frac{X^3{\left(\log X\right)}^9}{P}}.$$

(7)

Only the untwisted sum S enters the supremum.

Major arcs. Let α = b/r + β ∈ 𝔐(r,b), so |β| ≤ P/X. For each h mod q write b/r + h/q = b'/r' in lowest terms, so that α + h/q = b'/r' + β, with r' = r_b,h ≤ rq ≤ qP ≤ (logX)^A+11 for large X. Lemma 4.5 applied to S(α) and to S(α + h/q) gives

$S\left(\alpha \right)={\displaystyle \frac{\mu \left(r\right)}{\phi \left(r\right)}}v\left(\beta \right)+O\left(X{e}^{-c\sqrt{\log X}}P\right), S\left(\alpha +{\displaystyle \frac{h}{q}}\right)={\displaystyle \frac{\mu \left(r\mathrm{\text{'}}\right)}{\phi \left(r\mathrm{\text{'}}\right)}}v\left(\beta \right)+O\left(X{e}^{-c\sqrt{\log X}}P\right).$Since |S| ≤ ψ(X)≪X and |v| ≤ X, and the total measure of 𝔐 is at most 2P²/X, Lemma 4.3 yields

$$J^{\mathfrak{M}}\left(N\right)=\sum _{r\le P}\sum _{\begin{array}{c}b\hspace{0.25em}\mathrm{mod}\hspace{0.25em}r\\ \left(b,r\right)=1\end{array}}{\displaystyle \frac{1}{q}}\sum _{h}e\left(-{\displaystyle \frac{\mathit{ah}}{q}}\right){\displaystyle \frac{\mu \left(r\right)\mu \left(r\mathrm{\text{'}}\right)}{\phi \left(r\right)\phi \left(r\mathrm{\text{'}}\right)}}e\left(-{\displaystyle \frac{\mathit{Nb}}{r}}\right)I_r\left(N\right)+O\left(X{P}^3{e}^{-c\sqrt{\log X}}\right),$$

where $I_r\left(N\right)={\int }_{\left|\beta \right|\le P/\left(\mathit{rX}\right)}v{\left(\beta \right)}^{2}e\left(-\mathit{N\beta }\right)\hspace{0.17em}\mathit{d\beta }$. Over the full period ${\int }_{-1/2}^{1/2}{v}^{2}e\left(-\mathit{N\beta }\right)\hspace{0.17em}\mathit{d\beta }=N-1$, and since |v(β)| ≤ (2∥β∥)⁻¹, ${\int }_{P/\left(\mathit{rX}\right)<\left|\beta \right|\le 1/2}{\left|v\left(\beta \right)\right|}^2\hspace{0.17em}\mathit{d\beta }\le {\displaystyle \frac{\mathit{rX}}{2P}}$, so $I_r\left(N\right)=N-1+{\theta }_r{\displaystyle \frac{\mathit{rX}}{2P}}$ with |θ_r| ≤ 1.

The double sum over b and h is exactly A(r,N) from (6). Therefore, with ${\Sigma }_P\left(N\right)=\sum _{r\le P}A\left(r,N\right)$,

$$J^{\mathfrak{M}}\left(N\right)=\left(N-1\right){\Sigma }_P\left(N\right)+O\left({\displaystyle \frac{X}{P}}\sum _{r\le P}r\left|A\left(r,N\right)\right|\right)+O\left(X{P}^3{e}^{-c\sqrt{\log X}}\right).$$

By Theorem 4.8, $N\sum _{r}A\left(r,N\right)=M_{a,q}\left(N\right)$, hence

$$J^{\mathfrak{M}}\left(N\right)-M_{a,q}\left(N\right)\ll X\sum _{r>P}\left|A\left(r,N\right)\right|+\sum _r\left|A\left(r,N\right)\right|+{\displaystyle \frac{X}{P}}\sum _{r\le P}r\left|A\left(r,N\right)\right|+X{P}^3{e}^{-c\sqrt{\log X}}.$$

(8)

Mean square. By Lemma 4.9, the first three terms of (8) are ${\ll }_{q}X{P}^{-1/2}\tau \left(N\right){\left(\mathrm{loglog}X\right)}^2+{\left(\mathrm{loglog}X\right)}^3+{\displaystyle \frac{X}{P}}\tau \left(N\right){\left(\mathrm{loglog}X\right)}^2\log P$. Squaring, summing over N ≤ X and using Lemma 4.6 gives ≪X³(logX)³(loglogX)⁴/P. Adding (7) we obtain a total of ≪X³(logX)⁹/P = X³(logX)^−A−1, which proves the theorem.

Remark 4.11. For q = 1 the same argument, with A(r,N) = μ(r)²c_r(N)/φ(r)², is the classical circle-method proof that almost all even integers are sums of two primes [3,5,16,17]; the estimate $\sum _{N\le X}{\left|R\left(N\right)-2C_2\mathfrak{S}\left(N\right)N\right|}^2\ll X^3{\left(\log X\right)}^{-A}$ is also recorded in [15].

Corollary 4.12 (Exceptional set). [Proved] For q = 2^k, k ≥ 1, a odd and every A > 0, #$ℰ$_a,q(X)≪_A,qX(logX)^−A. In particular #$ℰ$_a,q(X) = o(X/logX).

Proof. By Lemma 4.4 and Theorem 4.10 (applied with A + 1), $\sum _{N\le X}{\left|R_{a,q}\left(N\right)-M_{a,q}\left(N\right)\right|}^2\ll X^3{\left(\log X\right)}^{-A-1}$. For even N ∈ (X/2,X] with R_a,q(N) = 0 we have |R_a,q(N) − M_a,q(N)| = M_a,q(N) ≥ 2C₂N/φ(q) > C₂X/φ(q), because 𝔖(N) ≥ 1 and N > X/2. By Chebyshev’s inequality the number of such N is ≪_qX³(logX)^−A−1/X² = X(logX)^−A−1. Summing over the dyadic blocks (X/2^j+1,X/2^j] with $X/2^j>\sqrt{X}$, for which $\log \left(X/2^j\right)\ge {\displaystyle \frac{1}{2}}\log X$, and adding the trivial bound $\sqrt{X}$ for the rest, gives the claim.

Corollary 4.13 (Relative accuracy). [Proved] For q = 2^k, a odd and A,A' > 0, #{N ≤ X even:|R_a,q(N) − M_a,q(N)| > N(logN)^−A}≪_A,A',qX(logX)^−A'.

Proof. Apply Theorem 4.10 with A₀ = 2A + A' and Chebyshev’s inequality on dyadic blocks, as in Corollary 4.12.

Remark 4.14 (Scope). [Honest Caveat] The saving is a power of logX for every fixed power; it is neither a power of X nor sub-exponential. For q = 1 a power saving ≪X^1−Δ is the theorem of Montgomery and Vaughan [9], with explicit Δ due to Pintz [11]; its analogue for q = 2^k is Open Question 13.1. An exceptional-set bound of order X/logX with an unspecified implied constant would not suffice for the density statement of Corollary 6.5; the factor (logX)^−A with A > 1 does. A stronger, sub-exponential bound is stated in [18, Theorem 8.3]; it comes from a different method and is not used here (see Remark 4.16).

Proposition 4.15 (Numerical comparison for q = 2^k). [Comp. Verified] The ratio R_a,q(N)/M_a,q(N), computed from the primes below 10⁷ for 200 random even N ∈ [5 ⋅ 10⁶,10⁷], has sample mean and standard deviation 0.9996, 0.0041 for (q,a) = (4,3); 0.9989, 0.0052 for (8,3); and 1.0004, 0.0077 for (16,7).

Remark 4.16 (Relation to [18]). [Honest Caveat] Theorem 4.10 and Corollaries 4.12 and 4.13 are proved above from Theorem 4.1 (major arcs) and Theorem 4.2 (minor arcs) alone and use no result of [18]. They strengthen the bounds of [18, Theorems 6.3 and 7.2] as explained in Section 1.5. The sub-exponential bound of [18, Theorem 8.3] is obtained in [18] by a different method; no statement of the present paper depends on it, and the sentence in Remark 4.14 that the saving is not sub-exponential refers only to the saving proved here.

5. Elementary Ray Classification

Proposition 5.1 (Surviving rays). [Proved] Fix n ≥ 2. A residue class c (mod n) contains infinitely many primes if and only if gcd(c,n) = 1. Hence the number of infinite rays is φ(n).

Proof. If d = gcd(c,n) > 1, every m≡c (n) is divisible by d, so m is composite once m > d. Conversely Dirichlet’s theorem gives infinitely many primes in every class coprime to n [4].

Proposition 5.2 (Equidistribution among rays). [Proved] For fixed n and gcd(c,n) = 1, π(X;n,c)∼φ(n)⁻¹π(X) (the prime number theorem for arithmetic progressions [4]).

Proposition 5.3 (Ray counts for n = 10). [Comp. Verified, sieve of Eratosthenes, all p ≤ 10⁸; zero exceptions] π(10⁸) = 5,761,455; exactly 5,761,453 of these primes lie in a ray {1,3,7,9} and the remaining two are 2 and 5. Table 2 lists the counts at six scales; at X = 10⁸ each count is within 0.02% of π(X)/4 (the largest relative deviation is 0.0123%).

Proposition 5.4 (Rays among the first primes). [Comp. Verified, first 100, 500 and 1000 primes] Among the first 100, 500 and 1000 primes the numbers lying in the rays 1,3,7,9 are 24,26,24,24; 123,123,127,125; and 245,253,254,246 respectively; the remaining primes are 2 and 5. Among the first 1000 primes the largest deviation of a ray count from 250 is 5. For the first 1000 primes (largest prime 7919, ring 791) the mean number of rings between two consecutive primes of the same ray is 3.23, 3.13, 3.12, 3.22 for the rays 1,3,7,9.

Figure 1. Ray portrait modulo 16 and the class of N_n. What is plotted: (a) the first 2500 primes at angle 2π(p mod 16)/16 and radius ⌊p/16⌋; the eight occupied spokes are the odd classes, and the class a = 1 of the restricted summand is drawn in red; (b) the conditional probability that N_n = p_n + p_n+1 lies in the class b modulo 8 given that p_n lies in the ray c modulo 8, over all pairs of consecutive primes >7 with N_n ≤ 6 ⋅ 10⁶. What it shows: only the φ(16) = 8 rays are occupied (Proposition 5.1), and the class b = 2 is reached with probability 0.166 from the rays 1,5 but 0.268 from the rays 3,7. Connection: Propositions 5.1 and 10.6.

Figure 1. Ray portrait modulo 16 and the class of N_n. What is plotted: (a) the first 2500 primes at angle 2π(p mod 16)/16 and radius ⌊p/16⌋; the eight occupied spokes are the odd classes, and the class a = 1 of the restricted summand is drawn in red; (b) the conditional probability that N_n = p_n + p_n+1 lies in the class b modulo 8 given that p_n lies in the ray c modulo 8, over all pairs of consecutive primes >7 with N_n ≤ 6 ⋅ 10⁶. What it shows: only the φ(16) = 8 rays are occupied (Proposition 5.1), and the class b = 2 is reached with probability 0.166 from the rays 1,5 but 0.268 from the rays 3,7. Connection: Propositions 5.1 and 10.6.

Remark 5.5. [Honest Caveat] The slight excess of the rays 3 and 7 among the first 1000 primes (about 3 to 4 primes out of 250) is of the size of the fluctuations of the counting function and is not claimed to be systematic; at X = 10⁸ the four counts differ from π(X)/4 by at most 0.013%.

6. A Transfer Principle and Almost-All Theorems

Lemma 6.1 (Anchoring). [Proved] For every n ≥ 2: (i) N_n is even; (ii) N_n+1 − N_n = p_n+2 − p_n > 0, so n ↦ N_n is strictly increasing and injective; (iii) N_n < 3p_n. More generally, for fixed k ≥ 1 and N_n,k:=p_n + p_n+k: N_n,k is even for n ≥ 2, strictly increasing in n, and N_n,k < (1 + 2^k)p_n.

Proof. For n ≥ 2 both primes are odd, giving (i). For (ii), N_n+1 − N_n = p_n+2 − p_n > 0. For (iii), Bertrand’s postulate gives p_n+1 < 2p_n, so g_n < p_n and N_n = 2p_n + g_n < 3p_n. Iterating, p_n+k < 2^kp_n and hence N_n,k < (1 + 2^k)p_n; parity and monotonicity are as before.

Theorem 6.2 (Transfer principle). [Proved] Let q,a be as in the definition. Let 𝒜 be a set, s:𝒜 → [1,∞) a “size” function and f:𝒜 → 2ℕ a map such that, for some constants m ≥ 1, κ ≥ 1 and λ ≥ 0, #f⁻¹(N) ≤ m for every N and f(n) ≤ κs(n) + λ for every n ∈ 𝒜. Then for all X ≥ 1 #{n ∈ 𝒜:s(n) ≤ X, R_a,q(f(n)) = 0} ≤ m⋅#$ℰ$_a,q(κX + λ). (9) The inequality holds for every q ≥ 1 and every a with gcd(a,q) = 1, and the constants do not depend on any residue class of the elements of 𝒜.

Proof. Let F(X) be the set on the left. For n ∈ F(X), f(n) is even, satisfies f(n) ≤ κX + λ and R_a,q(f(n)) = 0, so f(n) ∈ $ℰ$_a,q(κX + λ). Each element of $ℰ$_a,q(κX + λ) has at most m preimages, whence #F(X) ≤ m #$ℰ$_a,q(κX + λ).

Corollary 6.3 (Consecutive primes, uniform in the ray). [Proved] For every c and every X ≥ 3, #{p_n ≤ X:p_n≡c (q), R_a,q(N_n) = 0}≤#$ℰ$_a,q(3X), and for k ≥ 1, #{p_n ≤ X:p_n≡c (q), R_a,q(p_n + p_n+k) = 0}≤#$ℰ$_a,q((1 + 2^k)X).

Proof. Take 𝒜 = {n ≥ 2}, s(n) = p_n and f(n) = N_n (respectively N_n,k), which is injective by Lemma 6.1 and satisfies f(n) < 3s(n) (respectively <(1 + 2^k)s(n)). Restricting to p_n≡c (q) only shrinks the counted set, so (9) applies with m = 1, λ = 0 and κ = 3 (respectively 1 + 2^k).

Corollary 6.4 (Shifted primes). [Proved] For every odd integer h (positive or negative) and X ≥ 1, #{p ≤ X:p > max(|h|,2), R_a,q(p + h) = 0}≤#$ℰ$_a,q(X + |h|), uniformly over the rays of p modulo q.

Proof. For an odd prime p and odd h, p + h is even and p ↦ p + h is injective; apply Theorem 6.2 with 𝒜 the set of primes p > max(|h|,2), s(p) = p, f(p) = p + h, m = 1, κ = 1, λ = |h|.

Corollary 6.5 (Almost-all statements). [Proved] Let q = 2^k and gcd(a,q) = 1. For every ray c with gcd(c,q) = 1 and every A > 0, #{p_n ≤ X:p_n≡c (q), R_a,q(N_n) = 0}≪_q,AX(logX)^−A = o(π(X;q,c)), so for almost all primes p_n in every ray the even integer N_n = p_n + p_n+1 has a restricted Goldbach representation with first summand in the class a (q). The same holds for p_n + p_n+k with k fixed, and for p + h with h odd fixed.

Proof. By Corollary 6.3 and Corollary 4.12, the left side is at most #$ℰ$_a,q(3X)≪_q,A3X(log3X)^−A≪X(logX)^−A. Taking A = 2 and using π(X;q,c)∼X/(φ(q)logX) (Proposition 5.2) gives the o-statement. The statements for p_n + p_n+k and p + h are identical, using Corollary 6.3 and Corollary 6.4.

Remark 6.6 (Scope and triviality). [Honest Caveat] (i) By Proposition 2.3 the conclusion for N_n is automatic when p_n or p_n+1 lies in the class a; only the pairs of rays (c,c') with c≢a≢c' carry content. (ii) The transfer inequality holds for every modulus q, but for a modulus with an odd prime factor the exceptional set has positive density (Corollary 3.2), so Corollaries 4.12 and 6.5 are statements about q = 2^k. (iii) None of this bears on the pointwise question whether R_a,q(N_n) > 0 for all large n; an infinite subsequence of exceptional n of density zero is not excluded.

Remark 6.7 (Gap bound of Baker–Harman–Pintz). [Proved] Since $g_n\ll p_n^{0.525}$ for large n [1], for every ε > 0 one has N_n < (2 + ε)p_n for n ≥ n₀(ε); the constant κ = 3 in Corollary 6.3 may then be replaced by 2 + ε for large X, with n₀(ε) not asserted to be effective. Since log((2 + ε)X) = logX + O(1), this changes the conclusion of Corollary 6.5 only by a bounded factor, and the elementary constant 3 suffices

7. Computational Verification

Proposition 7.1 (Anchoring inequalities). [Comp. Verified, sieve; zero exceptions] Over all 216,814 pairs of consecutive primes with p_n ≥ 3 and p_n+1 ≤ 3 ⋅ 10⁶, and over all 5,761,453 such pairs with p_n+1 ≤ 10⁸: N_n+1 − N_n > 0 (minimum increment 4), N_n/p_n < 3 (maximum ratio 8/3, attained at the smallest admissible n), and N_n is even.

Proposition 7.2 (Brute-force check for q = 8, a = 1). [Comp. Verified; R computed by convolution in double precision; range n ≤ 5000] Table 3 lists, for the first K primes, the number of n ∈ [2,K − 1] for which N_n has no representation N_n = p₁ + p₂ with p₁≡1 (8). The exceptional values are the same at every K: N_n ∈ {8,12,18,42,68}; no new exception occurs as K grows by a factor 100. None of the exceptions has p_n≡1 or p_n+1≡1 (8), in accordance with Proposition 2.3.

Proposition 7.3 (The full exceptional sets for q = 4,8,16,32). [Comp. Verified, even N ≤ 6 ⋅ 10⁶, a = 1; R_a,q by double-precision FFT, every nonzero value at least log2log5 > 1.1] The sets $ℰ$_1,q(6 ⋅ 10⁶) have 5, 25, 45, 115 elements for q = 4,8,16,32 and largest elements 62, 458, 1058, 3074; for (q,a) = (16,9) there are 53 elements, the largest being 1298. For q = 8 the elements are 4,6,8,10,12,14,16,18,26,32,38,42,50,56,62,66,68,74,98,122,188,218,242,362,458. No exceptional element occurs above the stated maximum up to 6 ⋅ 10⁶ (Figure 2).

Proposition 7.4 (Shifted primes, q = 8, a = 1). [Comp. Verified, all primes 11 ≤ p ≤ 6 ⋅ 10⁶ − 20 (412,844 primes); R_1,8(N) by double-precision FFT for N ≤ 6 ⋅ 10⁶; every nonzero value is at least log17log2 > 1.9, so values below 1 are exact zeros] For each shift h in Table 4, the number of primes p with R_1,8(p + h) = 0 is small and the largest exceptional value of p + h is at most 458; no exception occurs above 458 in the range.

Remark 7.5. [Honest Caveat] Corollary 6.5 is asymptotic; its numerical onset depends on explicit constants in Theorem 4.10, which are ineffective, and need not lie in the tested range. The verification above tests only the finite ranges stated.

8. Moduli with an Odd Prime Factor along N_n

Theorem 8.1 (Obstruction along consecutive primes). [Proved] Let q have an odd prime factor 𝓁, gcd(a,q) = 1, and let n ≥ 2 with p_n,p_n+1 ≠ 𝓁. Write c≡p_n and c'≡p_n+1 modulo 𝓁. If c + c'≡a (𝓁), then every representation counted by R_a,q(N_n) has

, so that $R_{a,q}\left(N_n\right)=\left\{\begin{array}{ll}\left(\log \left(N_n-\mathcal{l}\right)\right)\left(\log \mathcal{l}\right),& N_n-\mathcal{l}\mathrm{is\; prime\; and}{N}_n-\mathcal{l}\equiv a\hspace{0.17em}\left(q\right),\\ 0,& \mathrm{otherwise},\end{array}\right.$and in particular R_a,q(N_n) = O_q(logN_n).

Proof. Suppose N_n = p₁ + p₂ with p₁≡a (q). Then p₁≡a (𝓁) and N_n≡c + c'≡a (𝓁), hence p₂ = N_n − p₁≡0 (𝓁). As p₂ is prime, p₂ = 𝓁 and p₁ = N_n − 𝓁.

Corollary 8.2 (The obstructed set is infinite for 𝓁 = 3). [Proved] Let q = 3, a = 1. Then N_n≡1 (3) if and only if p_n≡p_n+1≡2 (3), and there are infinitely many such n. Consequently R_a,q(N_n) = O(logN_n) for infinitely many n, and no asymptotic R_a,q(N_n)≍N_n/φ(q) can hold along N_n.

Proof. For p_n,p_n+1 > 3 the residues are 1 or 2 modulo 3, and c + c'≡1 (3) only for c = c' = 2. Infinitely many pairs of consecutive primes lie in the class 2 modulo 3 by Shiu’s theorem [13]. Apply Theorem 8.1.

Proposition 8.3 (Exact description for q = 3, a = 1). [Comp. Verified, all 216,813 pairs p_n > 3, N_n ≤ 6 ⋅ 10⁶; zero exceptions] R_a,q(N_n) = 0 holds if and only if N_n≡1 (3) and N_n − 3 is composite. In particular no n outside the class p_n≡p_n+1≡2 (3) has R_a,q(N_n) = 0, and the predicted characterisation has 0 mismatches (there are 36,809 zeros). Table 5 gives the fractions of obstructed indices and of exceptional indices in four ranges of N_n; over all even N ≤ 6 ⋅ 10⁶ with N ≥ 10³ the fraction with R_1,3(N) = 0 is 0.2646, and one third of even integers satisfy N≡1 (3).

Figure 3. Obstruction along consecutive primes for q = 3, a = 1. What is plotted: the fraction of n with N_n≡1 (3) and with R_1,3(N_n) = 0, in logarithmic windows of N_n from 10³ to 6 ⋅ 10⁶, against the values 1/3 (generic even integers) and 1/4 (independent rays). What it shows: the obstructed fraction stays well below 1/3 and rises slowly towards 1/4. Connection: Proposition 8.3 and Remark 8.4.

Figure 3. Obstruction along consecutive primes for q = 3, a = 1. What is plotted: the fraction of n with N_n≡1 (3) and with R_1,3(N_n) = 0, in logarithmic windows of N_n from 10³ to 6 ⋅ 10⁶, against the values 1/3 (generic even integers) and 1/4 (independent rays). What it shows: the obstructed fraction stays well below 1/3 and rises slowly towards 1/4. Connection: Proposition 8.3 and Remark 8.4.

Remark 8.4. [Honest Caveat] Along N_n the obstructed fraction is the joint frequency of the diagonal pair of rays (2,2) modulo 3, which by the Lemke Oliver–Soundararajan bias is suppressed below 1/4 and, conditionally on the Hardy–Littlewood conjecture, tends to 1/4 slowly [8]; it is not the density 1/3 that the obstruction assigns to a generic even integer. The infinitude in Corollary 8.2 is proved; the density is not.

Proposition 8.5 (The case 𝓁 = 5). [Comp. Verified, all 216,812 pairs of consecutive primes p_n > 5 with N_n ≤ 6 ⋅ 10⁶] For q = 5 and a = 1,2,3,4, the fraction of n with N_n≡a (5) is 0.1735, 0.1885, 0.1865, 0.1733 and the fraction with R_a,5(N_n) = 0 is 0.1358, 0.1507, 0.1504, 0.1361. The prediction of Theorem 8.1, namely R_a,5(N_n) = 0 if and only if N_n≡a (5) and N_n − 5 is not prime, has exactly three failures in the range, all with R_a,5(N_n) = 0 although N_n≢a (5): N_n = 152 = 73 + 79 for a = 1, and N_n = 18 = 7 + 11 and N_n = 68 = 31 + 37 for a = 4. The fraction of obstructed indices is below the independent-rays value 1/4 for every a.

Remark 8.6. [Honest Caveat] Theorem 8.1 proves only that N_n≡a (𝓁) forces R_a,q(N_n) ∈ {0,(log(N_n − 𝓁))log𝓁}; the converse (non-obstructed N_n have R_a,q(N_n) > 0) is not proved, and the three exceptions above are small values at which it fails for 𝓁 = 5. For 𝓁 = 3 no failure was found in the tested range (Proposition 8.3)

Remark 8.7 (Comparison with [18]). [Comp. Verified, even N ≤ 2000, N≡1(3)] In [18, Remark 3.4] it is reported that for q = 3,a = 1 every nonempty representation list of such N is {(N − 3,3)}, and that R_1,3(N) = 0 for 186 of the 333 even N ≤ 2000 with N≡1(3), a proportion 186/333≈0.559. Both statements were recomputed here by direct summation over the primes below 2000 and agree with [18]; they are the case 𝓁 = 3,a = 1 of Theorem 3.1.

9. Digit Repulsion, Gaps and Admissibility

9.1. The Transition Matrix

For n = 10 and a prime p > 5 write ray(p) = p mod 10 ∈ {1,3,7,9}.

Proposition 9.1 (Digit repulsion). [Comp. Verified, all 5,761,451 pairs of consecutive primes >5 and ≤10⁸] Table 6 gives the conditional transition matrix. The diagonal is suppressed to between 16.7% and 17.7% against 25% under independence, and every off-diagonal entry exceeds 20.8%; the qualitative bias reported in [8] is reproduced at 5.76 times the scale of the original data set.

Proposition 9.2 (Decay with lag). [Comp. Verified, same 5,761,451 primes] The probability that ray(p_n+𝓁) = ray(p_n) equals 17.18, 23.28, 24.50, 24.38, 24.40, 24.48, 24.58, 24.63 (%) for 𝓁 = 1,…,8.

Proposition 9.3 (Ray–gap identity). [Proved] For n with p_n > 5: g_n≡ρ_n+1 − ρ_n (mod 10), where ρ_n = ray(p_n). In particular ρ_n+1 = ρ_n if and only if 10∣g_n, and then g_n ≥ 10; the ray of p_n+1 is determined by ρ_n and g_n mod 10, and conversely.

Proof. g_n = p_n+1 − p_n reduces modulo 10 to ρ_n+1 − ρ_n. If ρ_n+1 = ρ_n then 10∣g_n > 0.

Figure 4. Digit repulsion as a flow on the four rays. What is plotted: (a) the four occupied rays modulo 10 as nodes, with an arrow from the ray a to the ray b of width proportional to ℙ(ray(p_n+1) = b∣ray(p_n) = a) (labels in %) and a red loop of width proportional to the probability of repeating the ray, for all consecutive primes 5 < p_n < p_n+1 ≤ 10⁸; (b) for six classes of gap size, the share of all gaps (blue) and the proportion of those gaps that return to the same ray (red), against the independent value 20%. What it shows: the loops are the thinnest arrows (between 16.7% and 17.7%); the gaps 2,4,6,8 make up 34.4% of all gaps and cannot return. Connection: Propositions 9.1, 9.3 and 9.6.

Figure 4. Digit repulsion as a flow on the four rays. What is plotted: (a) the four occupied rays modulo 10 as nodes, with an arrow from the ray a to the ray b of width proportional to ℙ(ray(p_n+1) = b∣ray(p_n) = a) (labels in %) and a red loop of width proportional to the probability of repeating the ray, for all consecutive primes 5 < p_n < p_n+1 ≤ 10⁸; (b) for six classes of gap size, the share of all gaps (blue) and the proportion of those gaps that return to the same ray (red), against the independent value 20%. What it shows: the loops are the thinnest arrows (between 16.7% and 17.7%); the gaps 2,4,6,8 make up 34.4% of all gaps and cannot return. Connection: Propositions 9.1, 9.3 and 9.6.

9.2. Mean Gap by Ray of Origin

Proposition 9.4 (Gap by ray). [Comp. Verified, all pairs with p_n > 5 and p_n+1 ≤ 2 ⋅ 10⁷ (1,270,603 pairs)] The global mean gap is 15.741. Conditioned on ray(p_n) = 1,3,7,9 the mean gap is 15.474, 16.523, 14.836, 16.129, with deviations z = −11.5, +33.2, −38.4, +16.6 standard errors of the per-ray sample mean; the counts per ray are 317,587, 317,743, 317,690 and 317,583.

Remark 9.5 (Mechanical origin). [Comp. Verified] Table 7 conditions the mean gap on the pair (ray of p_n, ray of p_n+1). The diagonal entries (19.88 to 21.10) exceed every off-diagonal entry (12.86 to 18.90). The reason is elementary: a return to the same ray forces g_n≡0 (10), and the smallest such gap is 10, whereas a change of ray admits gaps 2,4,6,8. Averaging the rows of Table 7 against the empirical transition frequencies of Table 6 reproduces the per-ray means of Proposition 9.4 (law of total expectation). The dependence of the mean gap on the ray of p_n is therefore a corollary of the transition bias and the divisibility constraint, and is not an independent phenomenon.

Proposition 9.6 (Repetition of the ray by size of the gap). [Comp. Verified, all consecutive primes 5 < p_n < p_n+1 ≤ 10⁸] By Proposition 9.3, ρ_n+1 = ρ_n if and only if 10∣g_n. Table 8 gives, for six classes of gap size, the share of all gaps in the class and the proportion of those gaps that return to the same ray. The gaps 2,4,6,8 make up 34.4% of all gaps and none of them can return; overall 17.18% of the gaps are multiples of 10 (the value 17.18% of Proposition 9.2 at lag 1), against 20% if the five even residues of g_n modulo 10 were equally likely. Among the gaps of size at least 20 the proportion returning to the same ray is between 23% and 37% in every class, above 20%.

Remark 9.7. [Honest Caveat] Proposition 9.6 displays the arithmetic part of the repulsion: more than a third of all gaps are too short to return to the ray. It does not prove that this accounts for the whole bias, nor does it replace the mechanism of Lemke Oliver and Soundararajan [8], which concerns the correlations of the gap sizes.

9.3. Cumulative Admissibility

For a prime q₀ > 2 let

$$f_{q_0}\left(h\right):=\prod _{\begin{array}{c}p\mid h,\hspace{0.25em}p>2\\ p\ne q_0\end{array}}{\displaystyle \frac{p-1}{p-2}}$$

(10)

be the Hardy–Littlewood admissibility function of h with the local factor at q₀ removed.

Proposition 9.8 (Cumulative sum). [Comp. Verified, least squares over every integer 100 ≤ H ≤ 4000] Fitting $\sum _{h\le H}{f}_{q_0}\left(h\right)=c_{1}H-c_2\log H+c_0$ gives c₁ = 1.136085, c₂ = 0.188854, c₀ = −0.127389 with R² = 0.99999999 for q₀ = 3, and c₁ = 1.420105, c₂ = 0.282050, c₀ = −0.298717 with R² = 0.99999996 for q₀ = 5; in both cases c₂ > 0, i.e. the secondary term has the sign opposite to that extrapolated from the main term.

Proposition 9.9 (Dirichlet series of $f_{q_0}$ and closed forms of the constants). [Proved] For ℜs > 1, $D\left(s\right):=\sum _{h\ge 1}{f}_{q_0}\left(h\right)h^{-s}=\zeta \left(s\right)\prod _{\begin{array}{c}p>2\\ p\ne q_0\end{array}}\left(1+{\displaystyle \frac{p^{-s}}{p-2}}\right)=\zeta \left(s\right)\zeta \left(s+1\right)G_{q_0}\left(s\right),$ where $G_{q_0}\left(s\right)=\left(1-2^{-1-s}\right)\left(1-q_0^{-1-s}\right)\prod _{p>2,\hspace{0.17em}p\ne q_0}\left(1+{\displaystyle \frac{p^{-s}}{p-2}}\right)\left(1-p^{-1-s}\right)$ converges absolutely, is holomorphic in ℜs > −1/2 and positive for real s > −1/2. Hence D extends meromorphically to ℜs > −1/2 with exactly two poles, both simple, at s = 1 and at s = 0. [Comp. Verified] With $c_1^{*}:=\zeta \left(2\right)G_{q_0}\left(1\right)$ and $c_2^{*}:=1/2G_{q_0}\left(0\right)$, evaluated with the products taken over all primes ≤10⁸, one finds $c_1^{*}=1.136085$, $c_2^{*}=0.189348$ for q₀ = 3 and $c_1^{*}=1.420106$, $c_2^{*}=0.284021$ for q₀ = 5.

Proof. The function $f_{q_0}\left(h\right)$ depends only on the set of odd primes ≠q₀ dividing h, so D(s) is an Euler product whose local factor is ${\left(1-2^{-s}\right)}^{-1}$ at p = 2, ${\left(1-q_0^{-s}\right)}^{-1}$ at p = q₀ and, at an odd prime p ≠ q₀,

$$1+{\displaystyle \frac{p-1}{p-2}}\cdot {\displaystyle \frac{p^{-s}}{1-p^{-s}}}={\displaystyle \frac{1+p^{-s}/\left(p-2\right)}{1-p^{-s}}},$$

because ${\displaystyle \frac{p-1}{p-2}}-1={\displaystyle \frac{1}{p-2}}$. Collecting the factors ${\left(1-p^{-s}\right)}^{-1}$ gives ζ(s) times the stated product. Writing x = p^−s, $\left(1+{\displaystyle \frac{x}{p-2}}\right)\left(1-x/p\right)=1+{\displaystyle \frac{2x}{p\left(p-2\right)}}-{\displaystyle \frac{x^2}{p\left(p-2\right)}}$, so the product defining $G_{q_0}$ converges absolutely for ℜs > −1/2, since $\sum _p{p}^{-2-\sigma }$ and $\sum _p{p}^{-2-2\sigma }$ converge for σ > −1/2. Each factor is positive for real s > −1, and the product $\prod \left(1+{\displaystyle \frac{p^{-s}}{p-2}}\right)$ equals $\zeta \left(s+1\right)G_{q_0}\left(s\right)$ after restoring the factors 1 − 2^−1−s and $1-q_0^{-1-s}$ and the factors ${\left(1-p^{-1-s}\right)}^{-1}$ of ζ(s + 1).

Remark 9.10. [Honest Caveat] Proposition 9.9 identifies the poles of D at s = 1 and s = 0 and shows that the residue of D(s)H^s/s at the double pole s = 0 contributes $\zeta \left(0\right)G_{q_0}\left(0\right)\log H=-c_2^{*}\log H$, with the sign of ζ(0) = −1/2 opposite to that of the main term $c_1^{*}H$. It does not prove the asymptotic formula $\sum _{h\le H}{f}_{q_0}\left(h\right)=c_{1}H-c_2\log H+O\left(1\right)$ for the sharp cutoff h ≤ H: that requires control of the contour integral, which [8,14] obtain for a smooth weight. Numerically $c_1^{*}$ agrees with the fitted c₁ in all printed digits and $c_2^{*}$ is within 0.3% (q₀ = 3) and 0.7% (q₀ = 5) of the fitted c₂ of Proposition 9.8, a difference compatible with the O(1) remainder.

Figure 5. Cumulative admissibility sum for q₀ = 3. What is plotted: (a) $S_3\left(H\right)=\sum _{h\le H}{f}_3\left(h\right)$ computed directly for H ≤ 4000 and the least-squares fit c₁H − c₂logH + c₀ over 100 ≤ H ≤ 4000; (b) the same data after subtracting c₁H, on a logarithmic H-axis, against −c₂logH + c₀. What it shows: the curves in (a) are indistinguishable at this scale, and (b) isolates the secondary term, with c₂ > 0. Connection: Propositions 9.8 and 9.9.

Figure 5. Cumulative admissibility sum for q₀ = 3. What is plotted: (a) $S_3\left(H\right)=\sum _{h\le H}{f}_3\left(h\right)$ computed directly for H ≤ 4000 and the least-squares fit c₁H − c₂logH + c₀ over 100 ≤ H ≤ 4000; (b) the same data after subtracting c₁H, on a logarithmic H-axis, against −c₂logH + c₀. What it shows: the curves in (a) are indistinguishable at this scale, and (b) isolates the secondary term, with c₂ > 0. Connection: Propositions 9.8 and 9.9.

Remark 9.11 (Role in the bias). [Honest Caveat] Proposition 9.8 is a numerical fit. The mechanism of Lemke Oliver and Soundararajan [8] derives the transition probabilities from such secondary terms under the Hardy–Littlewood prime tuples conjecture, so the transition formulas are conditional and are not asserted here as theorems. For q₀ = 3 the two-ray transition matrix (joint probabilities, diagonal/off-diagonal) was recomputed at four scales (Table 9) and moves monotonically towards 25% in every cell as N grows.

Proposition 9.12 (Comparison with the Lemke Oliver–Soundararajan prediction, q₀ = 3). [Comp. Verified, primes 3 < p_n < p_n+1 ≤ N for N = 3 ⋅ 10⁶,10⁷,10⁸,10⁹; the scale 10⁹ uses a segmented sieve] Let d(N) = 1/8loglogN/logN and let the prediction be the joint frequency 1/4 − d(N) for the cells (1,1) and (2,2) and 1/4 + d(N) for the cells (1,2) and (2,1). Table 10 gives the mean absolute error of the four observed joint frequencies against this prediction and against the value 1/4. The error against the prediction is smaller than the error against 1/4 at every N, by the factors 0.397, 0.381, 0.353, 0.334, and both errors decrease with N.

Figure 6. Mod-3 joint frequencies of consecutive primes at four scales. What is plotted: (a) the observed joint frequencies of (p_n mod 3,p_n+1 mod 3) for the cells (1,1), (2,2) and (1,2) = (2,1) at N = 3 ⋅ 10⁶,10⁷,10⁸,10⁹, with the prediction 1/4∓d(N) (dotted); (b) the mean absolute error against the prediction and against 25%. What it shows: the diagonal cells are suppressed, the off-diagonal cells inflated, all moving towards 25%; the prediction reduces the error by a factor 0.33 to 0.40. Connection: Proposition 9.12 and Remark 9.13.

Figure 6. Mod-3 joint frequencies of consecutive primes at four scales. What is plotted: (a) the observed joint frequencies of (p_n mod 3,p_n+1 mod 3) for the cells (1,1), (2,2) and (1,2) = (2,1) at N = 3 ⋅ 10⁶,10⁷,10⁸,10⁹, with the prediction 1/4∓d(N) (dotted); (b) the mean absolute error against the prediction and against 25%. What it shows: the diagonal cells are suppressed, the off-diagonal cells inflated, all moving towards 25%; the prediction reduces the error by a factor 0.33 to 0.40. Connection: Proposition 9.12 and Remark 9.13.

Remark 9.13. [Honest Caveat] The prediction is the leading secondary term of the argument of Lemke Oliver and Soundararajan as presented by Tao [14], and it is derived under the Hardy–Littlewood prime tuples conjecture; it is not a theorem. The agreement in Proposition 9.12 is with the leading term only, and a residual of about 1 percentage point remains at N = 10⁸, which the formula does not describe.

10. The Second Moment Modulo q

Let q = 2^k, D_a,q(N):=(R_a,q(N) − M_a,q(N))/M_a,q(N) and write b≡N (mod q).

10.1. A Symmetry

Proposition 10.1 (Symmetry of the restricted sum). [Proved] For classes a,a' modulo q let $R_{a,a\mathrm{\text{'}}}\left(N\right)=\sum _{p_1+p_2=N,\hspace{0.25em}{p}_1\equiv a,\hspace{0.25em}{p}_2\equiv a\mathrm{\text{'}}}\left(\log p_1\right)\left(\log p_2\right)$. Then R_a,a'(N) = R_a',a(N). In particular, for N≡2a (q) the representations counted by R_a,q(N) are closed under (p₁,p₂) ↦ (p₂,p₁), and $R_{a,q}\left(N\right)=2\sum _{p_1<p_2,\hspace{0.25em}{p}_1+p_2=N,\hspace{0.25em}{p}_1\equiv a}\left(\log p_1\right)\left(\log p_2\right)+1_{N/2\mathrm{prime},\hspace{0.25em}N/2\equiv a\hspace{0.17em}\left(q\right)}{\left(\log N/2\right)}^2$, whereas for N≢2a (q) each unordered pair {p,p'} with p≡a, p'≡N − a≢a is counted exactly once.

Proof. Exchange p₁,p₂ in the defining sum. If N≡2a then p₁≡a implies p₂ = N − p₁≡a, so both orders are counted; the term p₁ = p₂ = N/2 is counted once. If N≢2a, the class of p₂ differs from that of p₁, and only one order of an unordered pair is counted.

Remark 10.2 (Heuristic consequence). [Honest Caveat] If the unordered pair counts fluctuated like independent Poisson variables with the same mean μ, the doubling in Proposition 10.1 would multiply the variance of R_a,q(N) by 4 ⋅ (μ/2)/μ = 2 when N≡2a (q). This is a heuristic explanation of the numerical ratio below, not a proof.

Proposition 10.3 (Second moment by class of N). [Comp. Verified; R_a,q(N) computed by FFT convolution of the weighted prime indicators for all even N ≤ 6 ⋅ 10⁶, M_a,q from an exact factorisation sieve] (i) For q = 8, a = 1 and even N ∈ [10⁵,6 ⋅ 10⁶], the mean of $D_{1,8}^2$ (in units of 10⁻⁵) is 9.70, 24.21, 9.76, 9.29 for N≡0,2,4,6 (8), and the mean of D_1,8 (in units of 10⁻³) is −2.991, −5.715, −2.920, −2.999. (ii) For even N ∈ [10⁶,6 ⋅ 10⁶], let ρ_a,q be the mean of $D_{a,q}^2$ on N≡2a (q) divided by its mean on the other classes. For q = 8, a = 1,3,5,7: ρ = 2.44, 2.01, 2.04, 2.03; for q = 16 and all eight a: 2.06, 2.07, 2.04, 2.00, 2.18, 2.08, 2.08, 2.06 (mean 2.07); for q = 32 and all sixteen a the mean is 2.04 and the range 1.94 to 2.21. Delete-one jackknife standard errors over 20 interleaved blocks are 0.17 to 0.29 for q = 8 and 0.12 to 0.19 for the values of a tested at q = 16,32; every value is within 1.6 standard errors of 2, including ρ_1,8 = 2.44 ± 0.29.

Figure 7. Variance ratio on the class N≡2a (q) for every admissible class. What is plotted: the ratio ρ_a,q of the mean of $D_{a,q}^2$ on N≡2a (q) to its mean on the other classes, for N ∈ [10⁶,6 ⋅ 10⁶] and for every odd a modulo q = 8 (blue), 16 (red), 32 (green), with delete-one jackknife error bars over 20 interleaved blocks, against the symmetry value 2 (dashed). What it shows: every one of the 28 values lies within 1.6 standard errors of 2. Connection: Proposition 10.3 and Remark 10.2.

Figure 7. Variance ratio on the class N≡2a (q) for every admissible class. What is plotted: the ratio ρ_a,q of the mean of $D_{a,q}^2$ on N≡2a (q) to its mean on the other classes, for N ∈ [10⁶,6 ⋅ 10⁶] and for every odd a modulo q = 8 (blue), 16 (red), 32 (green), with delete-one jackknife error bars over 20 interleaved blocks, against the symmetry value 2 (dashed). What it shows: every one of the 28 values lies within 1.6 standard errors of 2. Connection: Proposition 10.3 and Remark 10.2.

Proposition 10.4 (Size of the second moment). [Comp. Verified; q = 8, a = 1, all even N ≤ X, X ≤ 6 ⋅ 10⁶] Table 11 gives $\sum _{N\le X}{\left(R_{a,q}\left(N\right)-M_{a,q}\left(N\right)\right)}^2$ relative to X³ and to X³/logX. The second ratio decreases monotonically in X, from 4.68 × 10⁻⁴ at X = 2 × 10⁵ to 3.19 × 10⁻⁵ at X = 6 × 10⁶.

Figure 8. Size and class dependence of the second moment, q = 8, a = 1. What is plotted: (a) $\sum _{N\le X}{\left(R-M\right)}^2/X^3$ for 2 ⋅ 10⁵ ≤ X ≤ 6 ⋅ 10⁶ on logarithmic axes with the line ∝X⁻¹; (b) the mean of $D_{1,8}^2$ in units of 10⁻⁵ on the four classes of N modulo 8 (N ∈ [10⁵,6 ⋅ 10⁶]), the class N≡2a in red; (c) the variance ratio ρ_1,q for q = 8,16,32 against the value 2. What it shows: the local exponent in (a) is about −0.85; the class N≡2 carries more than twice the variance of the others. Connection: Propositions 10.4 and 10.3.

Figure 8. Size and class dependence of the second moment, q = 8, a = 1. What is plotted: (a) $\sum _{N\le X}{\left(R-M\right)}^2/X^3$ for 2 ⋅ 10⁵ ≤ X ≤ 6 ⋅ 10⁶ on logarithmic axes with the line ∝X⁻¹; (b) the mean of $D_{1,8}^2$ in units of 10⁻⁵ on the four classes of N modulo 8 (N ∈ [10⁵,6 ⋅ 10⁶]), the class N≡2a in red; (c) the variance ratio ρ_1,q for q = 8,16,32 against the value 2. What it shows: the local exponent in (a) is about −0.85; the class N≡2 carries more than twice the variance of the others. Connection: Propositions 10.4 and 10.3.

Remark 10.5. [Honest Caveat] Proposition 10.4 is a finite-range measurement and proves nothing about the limit. By Theorem 4.10, for every fixed A > 0 one has $\sum _{N\le X}{\left|R_{a,q}\left(N\right)-M_{a,q}\left(N\right)\right|}^2{\ll }_{A,q}{X}^3{\left(\log X\right)}^{-A}$ for q = 2^k (the case q = 1 being classical, Remark 4.11); this is compatible with the decrease seen in Table 11, and it excludes any lower bound of order X³/logX for the second moment. The finite range does not by itself decide the true order between X³(logX)^−A and a power saving in X; the measured local exponent −0.85 in X³ ⋅ X^−0.85 is a statement about X ≤ 6 ⋅ 10⁶ only.

10.2. From the Ray of p_n to the Class of N_n

Since N_n≡c + c' (q) with c,c' the rays of p_n,p_n+1, the distribution of N_n mod q given c is read off the transition matrix of consecutive primes modulo q.

Proposition 10.6 (Class of N_n given the ray, q = 8). [Comp. Verified, all 216,811 pairs of consecutive primes >7 with N_n ≤ 6 ⋅ 10⁶] Table 12 gives ℙ(N_n≡b (8)∣p_n≡c (8)). For rays c = 1,5 the class b = 2 has probability 0.166 and 0.163, and for rays c = 3,7 it has probability 0.268 and 0.267: the classes b≡2 (8) arise from the pairs (1,1), (5,5), (3,7), (7,3), and the two diagonal pairs are suppressed by the digit-repulsion bias.

Proposition 10.7 (Second moment along N_n by ray, q = 8, a = 1). [Comp. Verified; 211,683 pairs of consecutive primes >7 with 10⁵ ≤ N_n ≤ 6 ⋅ 10⁶; jackknife over 20 interleaved blocks n mod 20] Put D = D_1,8(N_n) and let $\overline{D_c^2}$ be the mean of D² over p_n≡c (8), in units of 10⁻⁵. Then $\overline{D_c^2}=10.04,\hspace{0.17em}11.43,\hspace{0.17em}9.94,\hspace{0.17em}11.74$ for c = 1,3,5,7, and the mean over all n is 10.79. The pairwise differences are: rays 1 − 3: −1.39 ± 0.18 (z = −7.8); 1 − 7: −1.70 ± 0.25 (z = −6.9); 5 − 3: −1.49 ± 0.19 (z = −8.0); 5 − 7: −1.80 ± 0.19 (z = −9.3); 1 − 5: +0.10 ± 0.16; 3 − 7: −0.31 ± 0.24. The second moment along N_n therefore depends on the ray of p_n: rays 3 and 7 exceed rays 1 and 5.

Proposition 10.8 (Prediction by the class of N_n). [Comp. Verified; same sample] Let $\widehat{D^2}\left(N_n\right)$ be the mean of $D_{1,8}^2$ over all even N ∈ [10⁵,6 ⋅ 10⁶] in the same cell, where a cell is a logarithmic window of N (15 windows with equally spaced logarithms) together with the class of N modulo 8 (respectively modulo 8,3 and 5). Then, in units of 10⁻⁵: (i) windows alone predict the overall mean as 14.22; (ii) adding the class modulo 8 gives 13.66, and predicts the ray differences 1 − 3, 1 − 7, 5 − 3, 5 − 7, 1 − 5, 3 − 7 as −1.68, −1.66, −1.75, −1.73, +0.07, +0.02; (iii) adding the classes modulo 3 and 5 gives 11.05 for the overall mean (observed 10.79), the per-ray means 10.36, 11.72, 10.34, 11.78 (observed 10.04, 11.43, 9.94, 11.74), and the ray differences −1.36, −1.41, −1.38, −1.43, +0.02, −0.05, against the observed values of Proposition 10.7. The differences observed minus predicted in (iii) for rays 1,3,5,7 are −0.32 ± 0.12, −0.29 ± 0.14, −0.41 ± 0.07 and −0.04 ± 0.15.

Remark 10.9 (Interpretation). [Honest Caveat] The ray of p_n acts on the second moment through the class of N_n: the transition matrix of Proposition 10.6 sends more mass to N_n≡2 (8) from rays 3,7, and that class has twice the variance (Proposition 10.3). The ordering and size of the differences are predicted by the class modulo 8 alone to about 20% (−1.7 against observed −1.4 to −1.8). The inclusion of the classes modulo 3 and 5 was decided after seeing that windows and the class modulo 8 over-predict the level; at q = 8 it is therefore a fit choice and not a pre-specified prediction; Proposition 10.10 applies the same recipe, unchanged, at q = 16. The remaining discrepancy (−0.3 × 10⁻⁵ for rays 1,3,5, two to four standard errors) is about 3% of the level and is not explained. The sample is restricted to N_n ≥ 10⁵ and is compared with a reference population in the same range of N, because D² is larger by an order of magnitude at small N.

Proposition 10.10 (Prediction fixed in advance, q = 16, a = 1). [Comp. Verified; 211,683 pairs of consecutive primes >7 with 10⁵ ≤ N_n ≤ 6 ⋅ 10⁶; jackknife over 20 interleaved blocks] The recipe of Proposition 10.8(iii) was applied unchanged with the class modulo 8 replaced by the class of N modulo 16: cells are 15 logarithmic windows of N together with the classes of N modulo 16, 3 and 5, and the prediction for N_n is the mean of $D_{1,16}^2$ over all even N ∈ [10⁵,6 ⋅ 10⁶] in the same cell. Then, in units of 10⁻⁵: the overall mean is observed 19.77 and predicted 20.19; the means over the eight rays c = 1,3,5,7,9,11,13,15 of p_n modulo 16 are observed 17.79, 21.07, 19.75, 19.90, 18.34, 20.59, 20.03, 20.64 and predicted 18.52, 21.64, 20.37, 20.29, 18.42, 21.31, 20.50, 20.48. The spread between the largest and smallest ray mean is 3.28 observed and 3.22 predicted, and the correlation between observed and predicted ray means is 0.962. The differences of the ray means against ray 1 (rays 3,5,7,9,11,13,15) are observed +3.28, +1.96, +2.10, +0.55, +2.80, +2.24, +2.85 (z = +6.1, +4.6, +3.6, +1.0, +4.6, +3.5, +4.2) and predicted +3.12, +1.84, +1.77, −0.10, +2.79, +1.98, +1.95. The differences observed minus predicted for the eight ray means range from −0.73 to +0.16, with standard errors 0.20 to 0.39; the largest in absolute z is ray 5 (−0.61 ± 0.20, z = −3.1).

Proposition 10.11 (The same recipe at other pairs (q,a)). [Comp. Verified; same sample of 211,683 pairs and same reference population; recipe of Proposition 10.10 unchanged, with the class modulo q in place of the class modulo 16] Table 13 lists, for four pairs (q,a), the overall means of $D_{a,q}^2$ along N_n (observed and predicted, in units of 10⁻⁵), the relative difference observed minus predicted with its delete-one jackknife z-value, the spread between the largest and smallest ray mean (observed and predicted), and the correlation between observed and predicted ray means over the reduced rays modulo q. In every case the observed overall mean is below the predicted one, by 2.1% to 3.0%, and the number of rays whose observed mean is below the prediction is 4/4, 7/8, 8/8 and 15/16 respectively.

Figure 9. Observed against predicted second moment along N_n, by ray of p_n. What is plotted: (a) for (q,a) = (8,1),(16,1),(16,9),(32,1) the ray means observed and predicted, each divided by its own overall mean (points on the dotted diagonal are perfectly predicted); (b) for q = 16, a = 1, the observed and predicted mean of $D_{1,16}^2$ (units of 10⁻⁵) on the eight rays of p_n, the recipe being fixed in advance. What it shows: the recipe reproduces the pattern among rays at every pair, with a small systematic over-prediction of the level. Connection: Propositions 10.8, 10.10, 10.11 and Remark 10.13.

Figure 9. Observed against predicted second moment along N_n, by ray of p_n. What is plotted: (a) for (q,a) = (8,1),(16,1),(16,9),(32,1) the ray means observed and predicted, each divided by its own overall mean (points on the dotted diagonal are perfectly predicted); (b) for q = 16, a = 1, the observed and predicted mean of $D_{1,16}^2$ (units of 10⁻⁵) on the eight rays of p_n, the recipe being fixed in advance. What it shows: the recipe reproduces the pattern among rays at every pair, with a small systematic over-prediction of the level. Connection: Propositions 10.8, 10.10, 10.11 and Remark 10.13.

Proposition 10.12 (Effect of further small primes, exploratory). [Comp. Verified; exploratory, the additional moduli were chosen after seeing Table 13] Adding the class of N modulo 7, and then modulo 7 and 11, to the cells of the prediction changes the relative difference observed minus predicted (with z in parentheses) from −2.4% (−4.0) to −1.6% (−2.8) and −1.3% (−2.3) at (q,a) = (8,1); from −2.1% (−3.5) to −1.0% (−1.7) and −0.6% (−1.6) at (16,1); and from −3.0% (−5.1) to −2.0% (−3.6) and −1.4% (−2.9) at (16,9).

Remark 10.13. [Honest Caveat] The prediction at q = 16 uses the recipe of q = 8 without adjustment, so it is a test and not a fit. It reproduces the ordering, the spread and the level of the second moment by ray, but it over-predicts by about 3% for most rays, and the eight differences are not independent (they share a common shift), so the z-values above must not be combined as independent evidence. The over-prediction recurs at every pair tested (Table 13) and decreases monotonically when the classes modulo 7 and 11 are added (Proposition 10.12), which is consistent with N_n = 2p_n + g_n differing from a generic even integer through its residues modulo small primes; this reading is exploratory, and a residual of about 1% to 1.4% remains at the pairs tested. At q = 8 the spread among rays is under-predicted (1.43 against 1.80), while at q = 32 it is over-predicted (6.81 against 6.50). All results are for one range of N, 10⁵ ≤ N ≤ 6 ⋅ 10⁶.

11. Predicting the Next Ray and the Gap

Let ray(p) = p mod 10 for p > 5, and write ρ_n = ray(p_n) ∈ {1,3,7,9}.

Proposition 11.1 (Prediction out of sample). [Comp. Verified; recipe fixed before the run; all pairs (p_n,p_n+1) with 5 < p_n < p_n+1 ≤ 10⁸ (5,761,451 pairs); temporal split: training p_n ≤ 5 ⋅ 10⁷ (3,001,131 pairs), test p_n > 5 ⋅ 10⁷ (2,760,320 pairs)] For the context (ρ_n,ρ_n−1,…,ρ_n−k+1) the predictor of ρ_n+1 is the most frequent next ray in the training set, and the predictor of g_n/logp_n is the training mean in the same context. Table 14 gives, on the test set, the accuracy for ρ_n+1 (baseline 0.2500, standard error 0.0003), the information gain in bits over the marginal distribution of ρ_n+1 (uniform: 2 bits), and the coefficient of determination R² for g_n/logp_n against the training global mean. Accuracy is 0.311 with one ray and 0.316 with four, an excess of about 6.5 points over 0.25; the information gain is 0.036 bits out of 2; and R² for the normalised gap is at most 0.0019. The most likely next ray given ρ_n = 1,3,7,9 is 7,9,9,1, with test accuracy 0.3075, 0.3072, 0.3020, 0.3277.

Proposition 11.2 (Reference level for the gap). [Comp. Verified; same split] Using the exact value of g_n mod 10 (which is not available before p_n+1 is known) as the context, the training means of g_n/logp_n by class give R² = 0.0355 on the test set.

Remark 11.3 (Interpretation). [Honest Caveat] The rays of the past predict the ray of the next prime better than chance, and essentially all of this comes from the last ray: the gain from k = 1 to k = 4 is 0.003 bits, in line with the decay with lag in Proposition 9.2. The size of the gap is not predicted: the class of g_n mod 10, even if known, would explain 3.6% of the variance of g_n/logp_n, and the rays of the past explain at most 0.19%, which is about a twentieth of that reference. The decrease at k = 6 reflects the 4,096 contexts estimated from the training set. The accuracy of about 31.6% is a measurement of the Lemke Oliver–Soundararajan bias as a predictor and is not a new mechanism; it is not asserted to persist at larger scales, where the bias decays. The context of one ray is fixed by the transition matrix of Table 6, whose row maxima (31.0%, 31.0%, 30.4%, 33.1% in sample) are reproduced out of sample.

12. The Main Term Along Shifted Primes

By Corollary 6.4 the integers p + h (h odd fixed) are covered by the almost-all theorem, but the shifted primes have density zero and Corollary 4.12 is silent on how the size of the main term behaves along them. Since $M_{a,q}\left(p+h\right)={\displaystyle \frac{2C_2}{\phi \left(q\right)}}\left(p+h\right)\hspace{0.17em}\mathfrak{S}\left(p+h\right)$ for q = 2^k, the whole dependence on the shift sits in the amplitude 𝔖(p + h), which we now study. Throughout, h ≥ 1 is odd and $\mathfrak{S}\left(n\right)=\prod _{\mathcal{l}\mid n,\hspace{0.25em}\mathcal{l}>2}{a}_{\mathcal{l}}$ as in (2).

Lemma 12.1 (Brun–Titchmarsh). [Proved] For 1 ≤ d < x and gcd(c,d) = 1, $\pi \left(x;d,c\right)\le {\displaystyle \frac{2x}{\phi \left(d\right)\log \left(x/d\right)}}$ [4].

Let ${\left(B_{\mathcal{l}}\right)}_{\mathcal{l}>2}$ be independent Bernoulli variables with ℙ(B_𝓁 = 1) = 1/(𝓁 − 1), and define

$$Y_h=\prod _{\begin{array}{c}\mathcal{l}>2\\ \mathcal{l}\nmid h\end{array}}{a}_{\mathcal{l}}^{B_{\mathcal{l}}}, M_h\left(z\right)=\mathbb{E}\hspace{0.17em}{Y}_h^z, A_m\left(h\right)=M_h\left(m\right).$$

(11)

The choice ℙ(B_𝓁 = 1) = 1/(𝓁 − 1) is the density of primes p with 𝓁∣p + h, and the primes 𝓁∣h are omitted because 𝓁∣p + h forces p = 𝓁.

Proposition 12.2 (The model). [Proved] (i) Y_h is a.s. finite and Y_h > 1 a.s. (ii) For every complex z the product $M_h\left(z\right)=\prod _{\mathcal{l}>2,\hspace{0.25em}\mathcal{l}\nmid h}\left(1+{\displaystyle \frac{a_{\mathcal{l}}^z-1}{\mathcal{l}-1}}\right)$ converges, and M_h is entire of order at most 1. (iii) $M_h\left(z\right)=M_1\left(z\right)\prod _{\mathcal{l}\mid h,\hspace{0.25em}\mathcal{l}>2}{\left(1+{\displaystyle \frac{a_{\mathcal{l}}^z-1}{\mathcal{l}-1}}\right)}^{-1}$.

Proof. (i) $\mathbb{E}\log Y_h=\sum _{\mathcal{l}}{\displaystyle \frac{\log a_{\mathcal{l}}}{\mathcal{l}-1}}<\mathrm{\infty }$ by (5), so logY_h < ∞ a.s. Also $\mathbb{P}\left(Y_h=1\right)=\mathbb{P}\left(B_{\mathcal{l}}=0\hspace{0.25em}\forall \mathcal{l}\right)=\prod \left(1-{\displaystyle \frac{1}{\mathcal{l}-1}}\right)=0$ since $\sum 1/\mathcal{l}=\mathcal{\infty }$. (ii) Theorem 12.8(a) below gives $\mathbb{E}{Y}_h^s<\mathrm{\infty }$ for every real s, hence $Y_h^{\mathfrak{R}z}\in L^1$; dominated convergence applied to the partial products $\prod _{\mathcal{l}\le L}{a}_{\mathcal{l}}^{z{B}_{\mathcal{l}}}$ (dominated by $Y_h^{\mathfrak{R}z}$) and independence give the product formula and analyticity. The bound |M_h(z)| ≤ M_h(ℜz) ≤ exp(|z|log|z| + O(|z|)) of Theorem 12.8(a) shows order at most 1. (iii) is immediate.

Theorem 12.3 (Limit law). [Proved] Let h ≥ 1 be odd. As x → ∞, the random variable 𝔖(p + h), with p uniform among the primes ≤x, converges in distribution to Y_h.

Proof. Fix y ≥ 3 and put ${\mathfrak{S}}_y\left(n\right)=\prod _{\mathcal{l}\mid n,\hspace{0.25em}2<\mathcal{l}\le y}{a}_{\mathcal{l}}$, $Y_{h,y}=\prod _{2<\mathcal{l}\le y,\hspace{0.25em}\mathcal{l}\nmid h}{a}_{\mathcal{l}}^{B_{\mathcal{l}}}$.

Step 1 (bounded range). For p > h no prime 𝓁∣h divides p + h. The indicators ${\left(1_{\mathcal{l}\mid p+h}\right)}_{2<\mathcal{l}\le y,\hspace{0.25em}\mathcal{l}\nmid h}$ are determined by p modulo $m_y=\prod _{2<\mathcal{l}\le y}\mathcal{l}$, and 𝓁∣p + h is one of the 𝓁 − 1 unit classes modulo 𝓁. By the prime number theorem in arithmetic progressions the primes are asymptotically equidistributed among the φ(m_y) unit classes modulo m_y, so by the Chinese remainder theorem the vector of indicators converges in distribution to independent Bernoulli variables with parameters 1/(𝓁 − 1). Hence 𝔖_y(p + h) ⇒ Y_h,y.

Step 2 (tail). Take y ≥ 5. Let $D_y\left(n\right)=\log \mathfrak{S}\left(n\right)-\log {\mathfrak{S}}_y\left(n\right)=\sum _{\mathcal{l}\mid n,\hspace{0.25em}\mathcal{l}>y}\log a_{\mathcal{l}}\le \sum _{\mathcal{l}\mid n,\hspace{0.25em}\mathcal{l}>y}2/\mathcal{l}$, since ${\displaystyle \frac{1}{\mathcal{l}-2}}\le {\displaystyle \frac{2}{\mathcal{l}}}$ for 𝓁 ≥ 4. Put x' = x + h. For n = p + h ≤ x' there is at most one prime divisor $\mathcal{l}>\sqrt{x\mathrm{\text{'}}}$, contributing at most $2/\sqrt{x\mathrm{\text{'}}}$. For $y<\mathcal{l}\le \sqrt{x\mathrm{\text{'}}}$ with 𝓁∤h, Lemma 12.1 gives #{p ≤ x:𝓁∣p + h} ≤ 2x/(φ(𝓁)log(x/𝓁)) ≤ 4x/(φ(𝓁)logx). Hence ${\mathbb{E}}_x{D}_y\le {\displaystyle \frac{2}{\sqrt{x\mathrm{\text{'}}}}}+{\displaystyle \frac{2}{\pi \left(x\right)}}\sum _{y<\mathcal{l}\le \sqrt{x\mathrm{\text{'}}}}{\displaystyle \frac{1}{\mathcal{l}}}\cdot {\displaystyle \frac{4x}{\left(\mathcal{l}-1\right)\log x}}+O\left({\displaystyle \frac{1}{\pi \left(x\right)}}\right)\le {\displaystyle \frac{9}{y}}+o\left(1\right)$ as x → ∞. By Markov’s inequality, limsup_xℙ_x(D_y > ε) ≤ 9/(εy) → 0 as y → ∞.

Step 3. Y_h,y ↑ Y_h almost surely as y → ∞, and hence in distribution. The approximation theorem for weak convergence [2], applied to log𝔖_y(p + h), log𝔖(p + h) and logY_h,y, followed by the continuous map exp, gives the theorem.

Lemma 12.4 (Uniform integrability). [Proved] Let h ≥ 1 be odd and λ > 0. Then ${\sup }_{x\ge x_0}{\displaystyle \frac{1}{\pi \left(x\right)}}\sum _{p\le x}\exp \left(\lambda \sum _{\mathcal{l}\mid p+h,\hspace{0.25em}\mathcal{l}>2}{\displaystyle \frac{1}{\mathcal{l}}}\right)<\mathrm{\infty }$. Consequently, for every m > 0, ${\sup }_x{\displaystyle \frac{1}{\pi \left(x\right)}}\sum _{p\le x}\mathfrak{S}{\left(p+h\right)}^m<\mathrm{\infty }$.

Proof. Only p > h matter. Choose y ≥ 2λ and put x' = x + h. Split the primes 𝓁∣n = p + h into 𝓁 ≤ y, $y<\mathcal{l}\le \sqrt{x\mathrm{\text{'}}}$, and $\mathcal{l}>\sqrt{x\mathrm{\text{'}}}$. The first range contributes a factor at most $K_y=\exp \left(\lambda \sum _{\mathcal{l}\le y}1/\mathcal{l}\right)$, the last at most $e^{\lambda /\sqrt{x\mathrm{\text{'}}}}\le 2$. For the middle range, write $T\left(n\right)=\sum _{y<\mathcal{l}\le \sqrt{x\mathrm{\text{'}}},\hspace{0.25em}\mathcal{l}\mid n}1/\mathcal{l}$. Then $e^{\mathit{\lambda T}\left(n\right)}=\prod _{y<\mathcal{l}\le \sqrt{x\mathrm{\text{'}}},\hspace{0.25em}\mathcal{l}\mid n}\left(1+\left(e^{\lambda /\mathcal{l}}-1\right)\right)=\sum _{d\mid n}g\left(d\right)$, $g\left(d\right)=\prod _{\mathcal{l}\mid d}\left(e^{\lambda /\mathcal{l}}-1\right)\le {\left(2\lambda \right)}^{\omega \left(d\right)}/d$, the sum running over squarefree d with all prime factors in $(y,\sqrt{x\mathrm{\text{'}}}]$; we used e^t − 1 ≤ 2t for $0\le t\le {\displaystyle \frac{1}{2}}$. Terms with $d>\sqrt{x\mathrm{\text{'}}}$ contribute at most τ(n)(2λ)^ω(n)x'^−1/2≪x'^−1/2+ε. For $1<d\le \sqrt{x\mathrm{\text{'}}}$, Lemma 12.1 gives ℙ_x(d∣p + h) ≤ 5/φ(d) for large x. Therefore ${\mathbb{E}}_x{e}^{\mathit{\lambda T}}\le 1+5\sum _{d>1}g\left(d\right)/\phi \left(d\right)+o\left(1\right)\le 1+5\left(\prod _{\mathcal{l}>y}\left(1+{\displaystyle \frac{2\lambda }{\mathcal{l}\left(\mathcal{l}-1\right)}}\right)-1\right)+o\left(1\right)\le 2$ for y = y(λ) large. The last statement follows from $\mathfrak{S}{\left(n\right)}^m\le \exp \left(3m\sum _{\mathcal{l}\mid n}1/\mathcal{l}\right)$, by (5).

Theorem 12.5 (Convergence of all moments). [Proved] For odd h ≥ 1 and every m > 0, $\underset{x\to \mathrm{\infty }}{\lim }{\displaystyle \frac{1}{\pi \left(x\right)}}\sum _{p\le x}\mathfrak{S}{\left(p+h\right)}^m=A_m\left(h\right)=\prod _{\begin{array}{c}\mathcal{l}>2\\ \mathcal{l}\nmid h\end{array}}\left(1+{\displaystyle \frac{a_{\mathcal{l}}^m-1}{\mathcal{l}-1}}\right).$Proof. By Theorem 12.3, $\mathfrak{S}{\left(p+h\right)}^m\Rightarrow Y_h^m$. By Lemma 12.4 applied with m replaced by m + 1, the family {𝔖(p + h)^m} is uniformly integrable, so the expectations converge [2] to $\mathbb{E}{Y}_h^m=M_h\left(m\right)$. The product formula is Proposition 12.2(ii).

Proposition 12.6 (Certified enclosures of A_m(h)). [Comp. Verified] Let Q ≥ 2m + 3 and A_m(h;Q) be the product at z = m truncated to 𝓁 ≤ Q. Then A_m(h;Q) ≤ A_m(h) ≤ A_m(h;Q)exp(2m/(Q − 1)). With Q = 10⁷ (double precision, fsum over the 664,578 odd primes ≤Q), Table 15 gives enclosures of A_m(1) and A_m(3).

Proof. For 𝓁 > Q we have $a_{\mathcal{l}}^m-1\le e^{m/\left(\mathcal{l}-2\right)}-1\le 2m/\left(\mathcal{l}-2\right)$ (as $m/\left(\mathcal{l}-2\right)\le {\displaystyle \frac{1}{2}}$), so $\log \left(1+{\displaystyle \frac{a_{\mathcal{l}}^m-1}{\mathcal{l}-1}}\right)\le {\displaystyle \frac{2m}{\left(\mathcal{l}-2\right)\left(\mathcal{l}-1\right)}}$, and $\sum _{n>Q}{\displaystyle \frac{2m}{\left(n-2\right)\left(n-1\right)}}={\displaystyle \frac{2m}{Q-1}}$. The tabulated values are the truncated products and their inflation by this factor.

Proposition 12.7 (Dependence on the shift and amplification). [Proved; numerical value Comp. Verified] (i) If h,h' are odd and have the same set of odd prime divisors, then Y_h and Y_h' have the same law, and $A_m\left(h\right)=A_m\left(1\right)\prod _{\mathcal{l}\mid h}{\left(1+{\displaystyle \frac{a_{\mathcal{l}}^m-1}{\mathcal{l}-1}}\right)}^{-1}$; in particular A₁(3) = A₁(1)/3/2. (ii) Let $U=\prod _{\mathcal{l}>2}{a}_{\mathcal{l}}^{C_{\mathcal{l}}}$ with independent ℙ(C_𝓁 = 1) = 1/𝓁 (the model for 𝔖(n) over all even n). Then 𝔼U = 1/C₂ and 𝔼U^m < A_m(1) for every m > 0; in particular C₂A₁(1) ∈ [1.15048,1.15049] ⊂ (1,∞).

Proof. (i) is immediate from the product formula. (ii) $\mathbb{E}U=\prod _{\mathcal{l}>2}\left(1+{\displaystyle \frac{a_{\mathcal{l}}-1}{\mathcal{l}}}\right)=\prod {\displaystyle \frac{{\left(\mathcal{l}-1\right)}^2}{\mathcal{l}\left(\mathcal{l}-2\right)}}=1/C_2$. Termwise, $1+{\displaystyle \frac{a_{\mathcal{l}}^m-1}{\mathcal{l}}}<1+{\displaystyle \frac{a_{\mathcal{l}}^m-1}{\mathcal{l}-1}}$. The numerical range follows from Propositions 2.2 and 12.6.

Theorem 12.8 (Moment generating function and tails). [Proved] There is an absolute constant C₀ such that:

• for all odd h and real s ≥ 3, logM_h(s) ≤ sloglogs + C₀s; in particular $\mathbb{E}{Y}_h^s<\mathrm{\infty }$ for every real s;
• with$c_0=e^{-C_0-1}$, ℙ(Y_h > y) ≤ exp(−exp(c₀y)) for all y with exp(c₀y) ≥ 3;
• for h = 1 and y ≥ 1, ℙ(Y₁ > y) ≥ exp(−2exp(ey)). Hence $c_{0}y\le \mathrm{loglog}{\displaystyle \frac{1}{\mathbb{P}\left(Y_1>y\right)}}\le \mathit{ey}+1$ for large y.

Proof. (a) For real s > 0, $Y_h^s=\prod a_{\mathcal{l}}^{s{B}_{\mathcal{l}}}$ is a product of independent nonnegative factors, so by monotone convergence $\mathbb{E}{Y}_h^s$ equals the product in [1,∞]; each factor is ≥1, whence M_h(s) ≤ M₁(s). For 3 ≤ 𝓁 ≤ s + 2, $a_{\mathcal{l}}^s\le e^{s/\left(\mathcal{l}-2\right)}$ and log(1 + e^t) ≤ t + log2 give a contribution $\le {\displaystyle \frac{s}{\mathcal{l}-2}}+\log 2$; summing over these 𝓁 and using Mertens’ theorem, this is ≤sloglogs + C₁s. For 𝓁 > s + 2, t = s/(𝓁 − 2) < 1, $a_{\mathcal{l}}^s-1\le 2t$, so the contribution of these primes is at most $2s\sum _{n>s+2}{\displaystyle \frac{1}{\left(n-2\right)\left(n-1\right)}}\le 2$. This proves (a) with an absolute C₀.

(b) By Markov’s inequality, ℙ(Y_h > y) ≤ M_h(s)e^−slogy ≤ exp(s(loglogs + C₀ − logy)). Choose s = exp(c₀y), so that loglogs = logy − C₀ − 1; the exponent is −s.

(c) If B_𝓁 = 1 for all 3 ≤ 𝓁 ≤ r then, by (5) and Mertens’ theorem, $\log Y_1\ge \sum _{3\le \mathcal{l}\le r}{\displaystyle \frac{1}{\mathcal{l}}}\ge \mathrm{loglog}r-1$. Taking r=⌈exp(ey)⌉ this exceeds logy. The probability of this event is $\prod _{3\le \mathcal{l}\le r}{\displaystyle \frac{1}{\mathcal{l}-1}}\ge \prod _{\mathcal{l}\le r}{\displaystyle \frac{1}{\mathcal{l}}}=e^{-\theta \left(r\right)}\ge e^{-1.01624\hspace{0.17em}r}$ by Rosser–Schoenfeld [12], which is ≥exp(−2exp(ey)). The two-sided statement follows from (b) and (c).

Theorem 12.9 (Continuity). [Proved] Let Q(X;λ) = sup_tℙ(t ≤ X ≤ t + λ). There is an absolute constant 𝒜 such that for every odd h and $0<\lambda <{\displaystyle \frac{1}{2}}$, $Q\left(\log Y_h;\lambda \right)\le \mathcal{A}{\left(\sum _{\mathcal{l}>2,\hspace{0.25em}\mathcal{l}\nmid h,\hspace{0.25em}\log a_{\mathcal{l}}>\lambda }{\displaystyle \frac{1}{\mathcal{l}-1}}\right)}^{-1/2}{\ll }_h{\left(\mathrm{loglog}\left(1/\lambda \right)\right)}^{-1/2}$. In particular logY_h, and hence Y_h, has no atoms.

Proof. By independence, Q(S + T;λ) ≤ Q(S;λ) for a partial sum S and independent remainder T. Apply the Kolmogorov–Rogozin inequality [10], $Q\left(\sum X_i;\lambda \right)\le \mathcal{A}\lambda {\left(\sum {\lambda }_i^2\left(1-Q\left(X_i;{\lambda }_i\right)\right)\right)}^{-1/2}$ for 0 < λ_i ≤ λ, with λ_i = λ to the variables X_𝓁 = B_𝓁loga_𝓁, 𝓁∤h. For λ < loga_𝓁 the variable X_𝓁 takes two values with probabilities $1-{\displaystyle \frac{1}{\mathcal{l}-1}}$ and ${\displaystyle \frac{1}{\mathcal{l}-1}}$, so $1-Q\left(X_{\mathcal{l}};\lambda \right)={\displaystyle \frac{1}{\mathcal{l}-1}}$ (recall 𝓁 ≥ 3). By (5), every prime 𝓁 < 1/λ satisfies loga_𝓁 > λ, and $\sum _{\mathcal{l}<1/\lambda }{\displaystyle \frac{1}{\mathcal{l}-1}}\ge \mathrm{loglog}\left(1/\lambda \right)-c_h$ by Mertens.

Corollary 12.10 (Law of the main term along p + h). [Proved] Let q = 2^k (k ≥ 1), a odd, h ≥ 1 odd. For N = p + h, ${\displaystyle \frac{M_{a,q}\left(N\right)}{2C_{2}N/\phi \left(q\right)}}=\mathfrak{S}\left(N\right)$, so, as p → ∞, this ratio converges in distribution to Y_h, its moments converge to A_m(h), and for every y the limiting probability that the ratio exceeds y equals ℙ(Y_h > y) ≤ exp(−exp(c₀y)) (the law of Y_h has no atoms by Theorem 12.9, so every y is a continuity point).

Proof. The identity is (2); the rest is Theorems 12.3, 12.5 and 12.8.

Remark 12.11. [Honest Caveat] The shifted primes have density zero, so Corollary 4.12 is silent on N = p + h. Corollary 12.10 says instead that the expected number of restricted representations of p + 1, normalised by its size, is on average C₂A₁(1)≈1.1505 times the value for a generic even integer (Proposition 12.7), and that its fluctuations are governed by the explicit law Y₁, which is continuous and has a doubly exponentially thin upper tail. This is a statement about the main term; no claim is made about R_a,q(p + h) itself beyond Corollary 6.4.

Proposition 12.12 (Comparison with the primes up to 10⁷). [Comp. Verified] Over the 664,578 odd primes p ≤ 10⁷, the empirical moments of 𝔖(p + h) are 1.742811, 3.460872, 7.62999, 18.1785 for m = 1,2,3,4 at h = 1, and 1.161863, 1.384401, 1.69582, 2.1394 at h = 3, each inside the enclosures of Table 15 up to the finite-x effect. For h = 1, the Kolmogorov distance between the empirical distribution function and that of Y₁ (estimated from 2 × 10⁵ independent samples of Y₁ with all primes 𝓁 ≤ 2 × 10⁶, seed 12345), restricted to the interval [1.001,6], is 0.0100, against a Monte Carlo fluctuation of about 0.003. The interval starts at 1.001 because the truncated model places mass 0.1055 below 1.001 against 0.0980 empirically; the discrepancy reflects the slow convergence of the atom-like mass near 1 (integers with all odd prime factors large) and is consistent with a finite-x effect.

Figure 10. The amplitude 𝔖(p + 1) and its limit law Y₁. What is plotted: (a) the empirical density of 𝔖(p + 1) over the odd primes p ≤ 10⁷ (filled) and of 2 × 10⁵ simulated samples of Y₁ (outline) on [1,6]; (b) the upper tails ℙ(⋅>y) on a logarithmic scale. What it shows: the two laws agree to about 0.01 in Kolmogorov distance, and the tail falls faster than any exponential. Connection: Theorems 12.3 and 12.8 and Proposition 12.12.

Figure 10. The amplitude 𝔖(p + 1) and its limit law Y₁. What is plotted: (a) the empirical density of 𝔖(p + 1) over the odd primes p ≤ 10⁷ (filled) and of 2 × 10⁵ simulated samples of Y₁ (outline) on [1,6]; (b) the upper tails ℙ(⋅>y) on a logarithmic scale. What it shows: the two laws agree to about 0.01 in Kolmogorov distance, and the tail falls faster than any exponential. Connection: Theorems 12.3 and 12.8 and Proposition 12.12.

13. Open Questions

Open Question 13.1 (Power saving for q = 2^k). [Open] Prove $\#{\mathcal{E}}_{a,q}\left(X\right){\ll }_q{X}^{1-{\Delta }_q}$ for some Δ_q > 0, q = 2^k, k ≥ 1 and every a, with full details for the character sums modulo q; the case q = 1 is the theorem of Montgomery and Vaughan [9], and the difficulty is to adapt the zero-density input to the character sums arising from Lemma 4.3. A power saving would give, via Corollary 6.3, $\#\{p_n\le X:R_{a,q}\left(N_n\right)=0\}{\ll }_q{X}^{1-{\Delta }_q}$.

Open Question 13.2 (Effective version). [Open] Theorem 4.10 is ineffective because it uses the Siegel–Walfisz theorem. Give an explicit X₁ = X₁(q,a,c) beyond which #$ℰ$_a,q(3X) < π(X;q,c), i.e. not merely asymptotically; this requires an effective replacement of the Siegel–Walfisz input on the major arcs.

Open Question 13.3 (From almost all to all). [Open] Determine whether $ℰ$_a,q is finite for q = 2^k. Neither Theorem 4.10 nor Corollary 4.12 settles this: each bounds the count of exceptional N up to X and none rules out an infinite, arbitrarily sparse sequence of exceptions. A uniform pointwise lower bound R_a,q(N) > 0 for all sufficiently large even N would be at least as strong as the corresponding statement for the classical binary Goldbach problem. The data of Proposition 7.3 (the largest exceptional integers 62,458,1058,3074 for q = 4,8,16,32 in N ≤ 6 ⋅ 10⁶) are compatible with finiteness and do not prove it.

Open Question 13.4 (Density of the obstruction). [Open] For q = 3, a = 1 decide whether the natural density of {n:p_n≡p_n+1≡2 (3)} exists; conditionally on the Hardy–Littlewood conjecture it is 1/4 [8]. Determine the density of {n:R_a,q(N_n) = 0}, which by Proposition 8.3 equals that of {n:N_n≡1 (3)} once the primes among N_n − 3 have density zero along this set.

Open Question 13.5 (Class-dependent second moment). [Open] For q = 2^k, determine whether $\sum _{N\equiv 2a\hspace{0.17em}\left(q\right)}{D}_{a,q}{\left(N\right)}^2/\sum _{N\not\equiv 2a\hspace{0.17em}\left(q\right)}{D}_{a,q}{\left(N\right)}^2$ (suitably normalised by the number of terms and the range) has a limit as X → ∞, and its value; Proposition 10.3 gives values between 1.94 and 2.44 at q = 8,16,32, all within 1.6 standard errors of 2.

Open Question 13.6 (Ray-dependent second moment). [Open] Fix q = 2^k and gcd(a,q) = 1. For rays c coprime to q define $W_c\left(X\right)=\sum _{p_n\le X,\hspace{0.25em}{p}_n\equiv c\hspace{0.17em}\left(q\right)}{D}_{a,q}{\left(N_n\right)}^2$ and π(X;q,c)⁻¹W_c(X). Decide whether there exist rays c₀ ≠ c₁ for which the limit inferior and superior of the ratio of these normalised quantities are not both 1. Propositions 10.7, 10.8 and 10.10 show a significant dependence at q = 8 and q = 16 in the range 10⁵ ≤ N_n ≤ 6 ⋅ 10⁶, which the class of N_n predicts to about 20% at q = 8 and to about 2% in level and spread at q = 16; whether it persists as X → ∞ depends on Open Question 13.5 and on the limiting transition probabilities.

Open Question 13.7 (General modulus, closed form). [Open] Prove Conjecture 3.5, at least in the mean-square sense, for moduli divisible by odd primes and N with gcd(N − a,q_odd) = 1; Table 1 gives numerical support. Derive, for q with φ(q) > 2 (in particular q = 10 and q = 2^k, k ≥ 3), the analogue of the two-ray transition formula for consecutive primes, and compare it with Table 6 at the same rigour as for q = 3.

Open Question 13.8 (Shift dependence of the rate). [Open] For fixed odd h, Corollary 6.4 bounds the primes p with R_a,q(p + h) = 0. Determine whether the rate depends on h through 𝔖(p + h), and whether the dependence is ray-independent at leading order. Corollary 12.10 describes the main term along p + h but not the error term.

Open Question 13.9 (Type of the law of Y_h and rate). [Open] By Theorem 12.9, Y_h is continuous. Is its distribution absolutely continuous, or singular continuous? Determine also the rate of convergence, in the Kolmogorov distance, in Theorem 12.3, as a function of x; Proposition 12.12 shows a distance of about 0.01 at x = 10⁷.

Remark 13.10 (Correspondence with [18]). [Honest Caveat] Open Question 13.1 is Open Question 3 of [18]; between Corollary 4.12 and a power saving lies the sub-exponential bound of [18, Theorem 8.3], which is not used here. Open Question 13.2 asks for an effective form of Theorem 4.10; an effective but weaker statement, with a threshold that grows with X, is [18, Theorem 7.2]. Open Question 13.3 is Open Question 2 of [18] (see also [18, Remark 9.2]), and Open Question 13.7 corresponds to Open Question 4 of [18]. Open Question 1 of [18], the second-moment lower bound, is answered in the negative by Theorem 4.10 (Section 1.5).

14. Conclusion

For the restricted Goldbach sum modulo a power of two, the additive-shift decomposition of Lemma 4.3 reduces every step of the classical circle-method argument to statements about the unrestricted exponential sum S; combined with the Siegel–Walfisz theorem on the major arcs and the classical Vaughan bound on the minor arcs it gives an ineffective mean-square theorem with saving (logX)^−A for every A > 0 and the exceptional-set bound #$ℰ$_a,q(X)≪_A,qX(logX)^−A. An injective map into the even integers with linear growth transfers every such bound, so that for consecutive primes, for the k-th next prime and for shifted primes one obtains statements uniform in the ray of p_n, which are informative only where the obvious representation by p_n and p_n+1 themselves is not available. For a modulus with an odd prime factor the same sequence exposes the local obstruction in a concrete form: the condition N_n≡a (𝓁) is a condition on the pair of rays of two consecutive primes, it holds infinitely often by Shiu’s theorem, and for q = 3 it accounts exactly for every zero of R_1,3(N_n) in the tested range. The frequency of the obstructed set is governed by the bias between consecutive primes that Lemke Oliver and Soundararajan discovered, which the tables and figures above reproduce at scales up to 10⁹.

The symmetry p₁ ↔ p₂ doubles the relative variance of R_a,q(N) on the class N≡2a (q), a value fixed in advance and confirmed for every class tested at q = 8,16,32. Since the ray of p_n controls the class of N_n through the transition matrix modulo q, the second moment along N_n depends on the ray, significantly (z = 7 to 9) in the tested range, and the class of N_n predicts the size of this dependence to about 20% (modulo 8 alone) and to within about two standard errors when the classes modulo 3 and 5 are added, with the caveat of Remark 10.9; the same recipe applied unchanged at q = 16 reproduces the level and the pattern among rays, with a small systematic over-prediction. The rays of the past predict the next ray with accuracy about 0.31 to 0.32 out of sample but not the size of the gap. Along shifted primes, the amplitude of the main term has an explicit limit law with all moments given by absolutely convergent Euler products, a doubly exponentially thin upper tail and a continuous distribution. Unconditionally proved are the mean-square theorem, the exceptional-set bound, the transfer principle and its corollaries, the obstruction theorems and the shifted-prime law; conditional or conjectural is only Conjecture 3.5, and the statements labelled [Comp. Verified] are finite computations. The precise limiting statements are recorded as open questions. The paper continues [18]: Theorem 3.1 restates its local-obstruction theorem and Theorem 4.10 strengthens its second-moment estimate (Section 1.5).