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

$$R_{a,q}\left(N\right):=\sum _{\begin{array}{c}{p}_1+p_2=N\\ p_1\equiv a\hspace{0.17em}\left(q\right)\end{array}}(\log p_1\left)\right(\log p_2), N\hspace{0.25em}\mathrm{even},$$

(1)

with candidate main term

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

(2)

where $C_2=\prod _{p>2}\left(1-(p-1{)}^{-2}\right)=0.66016181\dots$ 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; ⌊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).

(3)

The rays of p_n and p_n+1 modulo q determine N_n mod q, and the gap g_n determines how far N_n is from 2p_n. This paper studies three consequences: (i) an elementary transfer of almost-all statements for R_a,q to sequences of the form N_n and p + h (Section 4 and Section 5); (ii) what the local obstruction of a modulus with an odd prime factor does along N_n (Section 6); (iii) the digit-repulsion bias between consecutive primes and the dependence of the second moment of R_a,q − M_a,q on N mod q (Section 7 and Section 8).

1.2. What This Paper Does and Does Not Claim

It proves unconditionally the transfer inequality (Theorem 4.2) and its corollaries as inequalities between counting functions; the density statements (Corollary 4.5) follow from the exceptional-set bound of Theorem 2.4, which is proved from a mean-value lemma of Tolev [11] (Lemma 2.3) and is labelled accordingly. It proves the obstruction statements of Section 6 and classifies which rays carry infinitely many primes. 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 any positive-density statement for the exceptional set along N_n when q has an odd prime factor beyond the equivalences proved in Section 6; 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 (Open Question 10.5).

1.3. Status Labels

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

1.4. Structure

Section 2 fixes notation and proves the exceptional-set bound from Tolev’s lemma. Section 3 classifies rays and illustrates them with ray diagrams of up to 1000 primes. Section 4 proves the transfer principle. Section 5 records computational verification. Section 6 treats moduli with an odd prime factor along N_n. Section 7 documents digit repulsion, the gap-by-ray dependence, the cumulative admissibility function and the comparison with the Lemke Oliver–Soundararajan prediction, Section 8 studies the second moment modulo q. Section 9 measures how far the rays predict the next ray and the gap. Section 10 lists open questions.

2. Notation and the Exceptional-Set Bound

Throughout, p_n is the n-th prime (p₁ = 2), π(X) counts 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.

Definition 2.1 (Restricted sum, ray, ring). For q = 2^k (k ≥ 0) and gcd(a,q) = 1, R_a,q, M_a,q and $ℰ$_a,q(X) are given by (1), (2) and the definition above. Rays and rings are as in the Introduction. 

Proposition 2.2 (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 4 about N_n are informative only on the set of indices with p_n≢a and p_n+1≢a (q). For q = 8, a = 1 these indices have relative frequency 0.5412 among the first 6 × 10⁴ pairs of consecutive primes.

Lemma 2.3 (Mean-value lemma of Tolev, imported). [Proved] (imported from [11]) For k,l with gcd(k,l) = 1, an interval I ⊂ [1,N] and n ≤ N put $J_{k,l}(n;I)=\sum _{\begin{array}{c}{p}_1+p_2=n\\ p_1\equiv l\hspace{0.17em}\left(k\right),\hspace{0.25em}{p}_1\in I\end{array}}(\log p_1\left)\right(\log p_2),$ and Φ(n;I)=#{(m₁,m₂):m₁ + m₂ = n, m₁ ∈ I} and S_k,l(n) = 2C₂ λ(nk) if 2∣n and gcd(k,n − l) = 1, S_k,l(n) = 0 otherwise, where $\lambda \left(m\right)=\prod _{p\mid m,\hspace{0.25em}p>2}(p-1)/(p-2)$. For every A > 0 there is B = B(A) > 0 such that, with L = logN,

$$\begin{array}{c}\sum _{k\le \sqrt{N}{L}^{-B}}\hspace{0.25em}\underset{\gcd (l,k)=1}{\max }\hspace{0.25em}\underset{I}{\max }\hspace{0.25em}\sum _{n\le N}\left|J_{k,l}(n;I)-{\displaystyle \frac{S_{k,l}\left(n\right)}{\phi \left(k\right)}}\mathsf{\Phi }(n;I)\right|\\ {\ll }_A\hspace{0.25em}{N}^2{L}^{-A}.\end{array}$$

Theorem 2.4 (Exceptional set with logarithmic saving). [Cond. Proved (Tolev, Lemma 2.3)] Let q = 2^k and gcd(a,q) = 1. For every A > 0, #$ℰ$_a,q(X)≪_q,AX(logX)^−A. In particular #$ℰ$_a,q(X) = o(X/logX).

Proof. Fix A > 0 and B = B(A) from Lemma 2.3, and let N be large enough that $q\le \sqrt{N}{L}^{-B}$. Keep only the term k = q, l = a, I = [1,N] in Lemma 2.3; then R_a,q(n) = J_q,a(n;I) and Φ(n;I) = n − 1. For even n we have gcd(q,n − a) = 1 because a is odd, and λ(nq) = 𝔖(n) because q is a power of 2, so S_q,a(n)Φ(n;I)/φ(q) = M_a,q(n) − 2C₂𝔖(n)/φ(q). Since 𝔖(n)≪loglogn, the sum of the correction over n ≤ N is ≪_qNloglogN≪N²L^−A, and therefore

$$\sum _{\begin{array}{c}n\le N\\ n\hspace{0.25em}\mathrm{even}\end{array}}|R_{a,q}\left(n\right)-M_{a,q}\left(n\right)|{\ll }_{q,A}{N}^2{L}^{-A}.$$

If N/2 < n ≤ N is even and R_a,q(n) = 0, the summand equals M_a,q(n) ≥ 2C₂n/φ(q) > C₂N/φ(q), because 𝔖(n) ≥ 1. Hence the number of such n is ≪_q,A(N²L^−A)/N = NL^−A. Summing over the dyadic blocks N = X/2^j with $X/2^j>\sqrt{X}$, for which $\log (X/2^j)\ge {\displaystyle \frac{1}{2}}\log X$, and adding the trivial bound $\sqrt{X}$ for the rest, gives #$ℰ$_a,q(X)≪_q,AX(logX)^−A.

Remark 2.5 (Scope of Theorem 2.4). [Honest Caveat] The theorem is only as strong as Lemma 2.3, which Tolev states as a lemma and proves by reference to the arguments of Mikawa and Laporta for prime differences, adapted to the sum p₁ + p₂ = n; the reader who requires a self-contained proof of it should regard Theorem 2.4 as conditional on it, as its label says. The saving is a power of logX for every fixed power, not a power of X and not sub-exponential: for q = 1 the power saving ≪X^1−Δ is the theorem of Montgomery and Vaughan [8] (see also [9]), and its analogue for q = 2^k is Open Question 10.1. An exceptional-set bound of order X/logX with an unspecified implied constant would not suffice for the density statement of Corollary 4.5; the factor (logX)^−A with A > 1 does. 

3. Elementary Ray Classification

Proposition 3.1 (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 3.2 (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 3.3 (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 1 lists the counts at six scales; at X = 10⁸ each count is within 0.02% of π(X)/4.

Ray counts for n = 10.

Proposition 3.4 (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, and among the 168 primes below 1000 the counts are 40,42,46,38 (Table 1). 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.

Remark 3.5. [Honest Caveat]Figure 1 and Figure 2 illustrate the classification of Proposition 3.1 and the equidistribution of Proposition 3.2; they exhibit no phenomenon beyond those two classical facts. 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 in Figure 2(c) and is not claimed to be systematic; at X = 10⁸ the four counts differ from π(X)/4 by at most 0.013%.

X	π(X)	ray 1	ray 3	ray 7	ray 9
103	168	40	42	46	38
104	1,229	306	310	308	303
105	9,592	2,387	2,402	2,411	2,390
106	78,498	19,617	19,665	19,621	19,593
107	664,579	166,104	166,230	166,211	166,032
108	5,761,455	1,440,298	1,440,474	1,440,495	1,440,186

X	π(X)	ray 1	ray 3	ray 7	ray 9
103	168	40	42	46	38
104	1,229	306	310	308	303
105	9,592	2,387	2,402	2,411	2,390
106	78,498	19,617	19,665	19,621	19,593
107	664,579	166,104	166,230	166,211	166,032
108	5,761,455	1,440,298	1,440,474	1,440,495	1,440,186

4. A Transfer Principle and Almost-All Theorems

Lemma 4.1 (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 [5] 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 4.2 (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) ≤ κ s(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) (4)

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 4.3 (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 4.1 and satisfies f(n) < 3s(n) (respectively <(1 + 2^k)s(n)). Restricting to p_n≡c (q) only shrinks the counted set, so (4) applies with m = 1, λ = 0 and κ = 3 (respectively 1 + 2^k).

Corollary 4.4 (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 4.2 with 𝒜 the set of primes p > max(|h|,2), s(p) = p, f(p) = p + h, m = 1, κ = 1, λ = |h|. Primes p ≤ max(|h|,2) are excluded so that p is odd and p + h ≥ 2.

Corollary 4.5 (Almost-all statements). [Cond. Proved (Tolev, Lemma 2.3)] 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 4.3 and Theorem 2.4, 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 3.2) gives the o-statement. The statements for p_n + p_n+k and p + h are identical, using Corollary 4.3 and Corollary 4.4.

Remark 4.6 (Scope and triviality). [Honest Caveat] (i) By Proposition 2.2 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 (Section 6), so Theorem 2.4 and Corollary 4.5 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 4.7 (Gap bound of Baker–Harman–Pintz). [Proved] Since$g_n\ll p_n^{0.525}$for large n [2], for every ε > 0 one has N_n < (2 + ε)p_n for n ≥ n₀(ε); the constant κ = 3 in Corollary 4.3 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 4.5 only by a bounded factor, and the elementary constant 3 suffices. 

5. Computational Verification

Proposition 5.1 (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 5.2 (Brute-force check for q = 8, a = 1). [Comp. Verified, R computed by convolution in double precision; range n ≤ 5000] Table 2 lists, for the first K primes, the number of n ∈ [2,K] 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 2.2. 

Representability of N_n with p₁≡1 (8); first K primes (n = 2,…,K).

K	pairsn	representable	exceptions	fraction
50	49	44	5	10.20%
100	99	94	5	5.05%
500	499	494	5	1.00%
2000	1999	1994	5	0.25%
5000	4999	4994	5	0.10%

K	pairsn	representable	exceptions	fraction
50	49	44	5	10.20%
100	99	94	5	5.05%
500	499	494	5	1.00%
2000	1999	1994	5	0.25%
5000	4999	4994	5	0.10%

Proposition 5.3 (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 log17 log2 > 1.9, so values below 1 are exact zeros] For each shift h in Table 3, 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. 

Primes p with R_1,8(p + h) = 0, 11 ≤ p ≤ 6 × 10⁶ − 20.

h	exceptions	largest exceptional p + h
−3	12	458
−1	6	66
1	12	458
3	10	362
5	4	66
7	9	218
9	13	458

h	exceptions	largest exceptional p + h
−3	12	458
−1	6	66
1	12	458
3	10	362
5	4	66
7	9	218
9	13	458

Remark 5.4.[Honest Caveat] Corollary 4.5 is asymptotic; its numerical onset depends on the explicit constants in Lemma 2.3, which are not tracked here, and need not lie in the tested range. The verification above tests only the finite ranges stated. 

6. Moduli with an Odd Prime Factor Along N_n

Theorem 6.1 (Obstruction along consecutive primes). [Proved] (for the obstruction at a single N see also [1]) 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 p₂ = 𝓁, so that $R_{a,q}\left(N_n\right)=\left\{\begin{array}{ll}\left(\log \right(N_n-\mathcal{l}\left)\right)\left(\log \mathcal{l}\right),& N_n-\mathcal{l}\hspace{0.25em}\mathrm{is}\mathrm{prime}\mathrm{and}\hspace{0.25em}{N}_n-\mathcal{l}\equiv a\hspace{0.25em}\left(q\right),\\ 0,& \mathrm{otherwise},\end{array}\right.$ 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 6.2 (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 [10]. Apply Theorem 6.1.

Proposition 6.3 (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 (0 of 170,691 indices), and the predicted characterisation has 0 mismatches. Table 4 gives the fractions of obstructed indices and of exceptional indices in four ranges of N_n; over all even N ≤ 6 × 10⁶ (range 10³ ≤ N) the fraction with R_1,3(N) = 0 is 0.2646, and one third of even integers satisfy N≡1 (3).

q = 3, a = 1: fraction of n with N_n≡1 (3) and with R_1,3(N_n) = 0.

range of Nn	Nn≡1 (3)	R1,3(Nn) = 0
[104,105)	0.2070	0.1496
[105,106)	0.2069	0.1608
[106,3 × 106)	0.2127	0.1700
[3 × 106,6 × 106)	0.2152	0.1739

range of Nn	Nn≡1 (3)	R1,3(Nn) = 0
[104,105)	0.2070	0.1496
[105,106)	0.2069	0.1608
[106,3 × 106)	0.2127	0.1700
[3 × 106,6 × 106)	0.2152	0.1739

Remark 6.4. [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 [6]; it is not the density 1/3 that the obstruction assigns to a generic even integer. The values in Table 4 rise from about 0.207 to 0.215 over the tested ranges, which is in this direction. The infinitude in Corollary 6.2 is proved; the density is not. 

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 up 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 to paper: Proposition 6.3, Table 4 and Remark 6.4. 

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 up 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 to paper: Proposition 6.3, Table 4 and Remark 6.4. 

Proposition 6.5 (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 6.1, 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 6.6. [Honest Caveat] Theorem 6.1 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 6.3). 

7. Digit Repulsion, Gaps and Admissibility

7.1. The Transition Matrix

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

Proposition 7.1 (Digit repulsion). [Comp. Verified, all 5,761,451 pairs of consecutive primes >5 and ≤10⁸] Table 5 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 [6] is reproduced at 5.76 times the scale of the original data set. 

ℙ(ray(p_n+1) = b∣ray(p_n) = a), pairs below 10⁸.

a∖b	1	3	7	9
1	17.70%	30.39%	31.03%	20.87%
3	23.64%	16.67%	28.69%	31.01%
7	25.58%	27.32%	16.68%	30.42%
9	33.08%	25.62%	23.62%	17.68%

a∖b	1	3	7	9
1	17.70%	30.39%	31.03%	20.87%
3	23.64%	16.67%	28.69%	31.01%
7	25.58%	27.32%	16.68%	30.42%
9	33.08%	25.62%	23.62%	17.68%

Proposition 7.2 (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.

7.2. Mean Gap by Ray of Origin

Proposition 7.3 (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 7.4 (Mechanical origin). [Comp. Verified]Table 6 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 6 against the empirical transition frequencies of Table 5 reproduces the per-ray means of Proposition 7.3 (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. 

Mean gap g_n by (ray of p_n, ray of p_n+1), p_n+1 ≤ 2 × 10⁷.

a∖b	1	3	7	9
1	19.89	12.86	13.43	18.90
3	18.52	21.10	15.72	13.47
7	14.85	13.40	21.08	12.88
9	13.50	14.91	18.50	19.88

a∖b	1	3	7	9
1	19.89	12.86	13.43	18.90
3	18.52	21.10	15.72	13.47
7	14.85	13.40	21.08	12.88
9	13.50	14.91	18.50	19.88

Figure 4. Transition and gap by pair of rays. What is plotted: left, the probability that the next prime lies in ray b given that the current prime lies in ray a (modulo 10, primes up to 10⁸); right, the mean gap g_n for each pair of rays (primes up to 2 × 10⁷). What it shows: the diagonal is suppressed on the left and enhanced on the right. Connection to paper:Table 5 and Table 6; the elementary reason for the right panel is Remark 7.4. 

Figure 4. Transition and gap by pair of rays. What is plotted: left, the probability that the next prime lies in ray b given that the current prime lies in ray a (modulo 10, primes up to 10⁸); right, the mean gap g_n for each pair of rays (primes up to 2 × 10⁷). What it shows: the diagonal is suppressed on the left and enhanced on the right. Connection to paper:Table 5 and Table 6; the elementary reason for the right panel is Remark 7.4. 

Proposition 7.5 (Repetition of the ray by size of the gap). [Comp. Verified, all consecutive primes 5 < p_n < p_n+1 ≤ 10⁸] By Proposition 9.1, ρ_n+1 = ρ_n if and only if 10∣g_n. Table 7 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 7.2 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%.

Gaps by size: share of all gaps and proportion keeping the ray, primes 5 < p_n < p_n+1 ≤ 10⁸.

Gap gn	share of all gaps (%)	keeping the ray (%)
2–8	34.43	0.00
10–18	32.32	23.09
20–38	24.54	30.12
40–58	6.50	23.58
60–98	2.10	36.28
≥100	0.12	28.29

Gap gn	share of all gaps (%)	keeping the ray (%)
2–8	34.43	0.00
10–18	32.32	23.09
20–38	24.54	30.12
40–58	6.50	23.58
60–98	2.10	36.28
≥100	0.12	28.29

Figure 5. Large gaps on the ray diagram. What is plotted: (a) the primes among the first 1000 with g_n ≥ 20 (51 primes), each with an arrow to p_n+1 coloured by the ray of p_n+1; thick arrows return to the same ray (18 of the 51); the other primes are grey points; (b) the share of all gaps of each size class and the proportion of those gaps that keep the ray ( Table 7 ). What it shows: the short gaps cannot return and the large gaps return in a proportion above 20%. Connection to paper: Proposition 7.5 and Remark 7.4. 

Figure 5. Large gaps on the ray diagram. What is plotted: (a) the primes among the first 1000 with g_n ≥ 20 (51 primes), each with an arrow to p_n+1 coloured by the ray of p_n+1; thick arrows return to the same ray (18 of the 51); the other primes are grey points; (b) the share of all gaps of each size class and the proportion of those gaps that keep the ray ( Table 7 ). What it shows: the short gaps cannot return and the large gaps return in a proportion above 20%. Connection to paper: Proposition 7.5 and Remark 7.4. 

Remark 7.6. [Honest Caveat] Proposition 7.5 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 [6], which concerns the correlations of the gap sizes and is the object of Section 7’s comparison below. 

7.3. Cumulative Admissibility

For a prime q₀ > 2 let

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

(4)

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

Proposition 7.7 (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 7.8 (Dirichlet series of${\mathit{f}}_{{\mathit{q}}_0}$and closed forms of the constants). [Proved] For ℜs > 1, $D\left(s\right):=\sum _{h\ge 1}{f}_{q_0}\left(h\right)h^{-s}=\zeta \left(s\right)\prod _{\begin{array}{c}p>2\\ p\ne q_0\end{array}}\left(1+{\displaystyle \frac{p^{-s}}{p-2}}\right)=\zeta \left(s\right)\hspace{0.17em}\zeta (s+1)\hspace{0.17em}{G}_{q_0}\left(s\right),$ where $G_{q_0}\left(s\right)=\left(1-2^{-1-s}\right)\left(1-q_0^{-1-s}\right)\prod _{\begin{array}{c}p>2\\ p\ne q_0\end{array}}\left(1+{\displaystyle \frac{p^{-s}}{p-2}}\right)\left(1-p^{-1-s}\right)$ 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; and, for q₀ = 3,5 and s = 3, the series D(s) and the product agree to 8 digits. 

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 (1 − 2^−s)⁻¹ at p = 2, $(1-q_0^{-s}{)}^{-1}$ at p = q₀, and, at an odd prime p ≠ q₀,

$$1+{\displaystyle \frac{p-1}{p-2}}\hspace{0.17em}{\displaystyle \frac{p^{-s}}{1-p^{-s}}}={\displaystyle \frac{1+p^{-s}/(p-2)}{1-p^{-s}}},$$

because ${\displaystyle \frac{p-1}{p-2}}-1={\displaystyle \frac{1}{p-2}}$. Collecting the factors (1 − p^−s)⁻¹ gives ζ(s) times the stated product. Writing x = p^−s,

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

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 $\prod (1+{\displaystyle \frac{p^{-s}}{p-2}})$ equals $\zeta (s+1)G_{q_0}\left(s\right)$ after restoring the factors 1 − 2^−1−s and $1-q_0^{-1-s}$ and the factors (1 − p^−1−s)⁻¹ of ζ(s + 1).

Remark 7.9. [Honest Caveat] Proposition 7.8 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 $\zeta \left(0\right)G_{q_0}\left(0\right)\mathit{logH}=-c_2^{*}\mathit{logH}$, 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 $\sum _{h\le H}{f}_{q_0}\left(h\right)=c_{1}H-c_2\mathit{logH}+O\left(1\right)$ for the sharp cutoff h ≤ H: that requires control of the contour integral, which [6,13] 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 7.7, a difference compatible with the O(1) remainder. 

Figure 6. 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 to paper: Proposition 7.7. 

Figure 6. 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 to paper: Proposition 7.7. 

Remark 7.10 (Role in the bias). [Honest Caveat] Proposition 7.7 is a numerical fit. The mechanism of Lemke Oliver and Soundararajan [6] 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 8) and moves monotonically towards 25% in every cell as N grows. 

Joint frequency of (p_n mod 3, p_n+1 mod 3) for consecutive primes 3 < p_n < p_n+1 ≤ N.

N	pairs	(1,1)	(1,2) = (2,1)	(2,2)
3 × 106	216,813	21.21%	28.76%	21.27%
107	664,576	21.50%	28.48%	21.53%
108	5,761,452	21.94%	28.06%	21.95%
109	50,847,531	22.25%	27.74%	22.26%

N	pairs	(1,1)	(1,2) = (2,1)	(2,2)
3 × 106	216,813	21.21%	28.76%	21.27%
107	664,576	21.50%	28.48%	21.53%
108	5,761,452	21.94%	28.06%	21.95%
109	50,847,531	22.25%	27.74%	22.26%

Proposition 7.11 (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) = 1/8 loglogN/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 9 gives the mean absolute error of the four observed joint frequencies against this prediction and against the value ¼. The error against the prediction is smaller than the error against ¼ at every N, by the factors 0.397, 0.381, 0.353, 0.334, and both errors decrease with N. 

Mod-3 joint frequencies against the prediction 1/4∓d(N); errors in percentage points.

N	predicted diag.	predicted off-diag.	error vs. pred.	error vs. 25%	ratio
3 × 106	22.74%	27.26%	1.4935	3.7584	0.397
107	22.84%	27.16%	1.3259	3.4818	0.381
108	23.02%	26.98%	1.0796	3.0566	0.353
109	23.17%	26.83%	0.9158	2.7442	0.334

N	predicted diag.	predicted off-diag.	error vs. pred.	error vs. 25%	ratio
3 × 106	22.74%	27.26%	1.4935	3.7584	0.397
107	22.84%	27.16%	1.3259	3.4818	0.381
108	23.02%	26.98%	1.0796	3.0566	0.353
109	23.17%	26.83%	0.9158	2.7442	0.334

Figure 7. Mod-3 joint frequencies of consecutive primes against the prediction. What is plotted: left, the mean absolute error against the prediction of Proposition 7.11 and against 25% at four scales; right, observed and predicted joint frequencies at N = 10⁸. What it shows: the prediction reproduces the suppressed diagonal and the inflated off-diagonal, with an error smaller by a factor of about 0.33 to 0.40. Connection to paper: Proposition 7.11 and Remark 7.12. 

Figure 7. Mod-3 joint frequencies of consecutive primes against the prediction. What is plotted: left, the mean absolute error against the prediction of Proposition 7.11 and against 25% at four scales; right, observed and predicted joint frequencies at N = 10⁸. What it shows: the prediction reproduces the suppressed diagonal and the inflated off-diagonal, with an error smaller by a factor of about 0.33 to 0.40. Connection to paper: Proposition 7.11 and Remark 7.12. 

Remark 7.12. [Honest Caveat] The prediction is the leading secondary term of the argument of Lemke Oliver and Soundararajan as presented by Tao [13], and it is derived under the Hardy–Littlewood prime tuples conjecture; it is not a theorem. The agreement in Proposition 7.11 is with the leading term only, and a residual of about 1 percentage point remains at N = 10⁸, which the formula does not describe. 

8. 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).

8.1. A Symmetry

Proposition 8.1 (Symmetry of the restricted sum). [Proved] For classes a,a^'modulo q let$R_{a,a^{\mathrm{\text{'}}}}\left(N\right)=\sum _{p_1+p_2=N,\hspace{0.17em}{p}_1\equiv a,\hspace{0.17em}{p}_2\equiv a^{\mathrm{\text{'}}}}(\log p_1\left)\right(\log p_2)$. Then$R_{a,a^{\mathrm{\text{'}}}}\left(N\right)=R_{a^{\mathrm{\text{'}}},a}\left(N\right)$. In particular, for N≡2a (q) the representations counted by R_a,q(N) are closed under (p₁,p₂) ↦ (p₂,p₁), and $R_{a,q}\left(N\right)=2\sum _{\begin{array}{c}{p}_1<p_2\\ p_1+p_2=N,\hspace{0.25em}{p}_1\equiv a\end{array}}(\log p_1\left)\right(\log p_2)+1_{N/2\hspace{0.25em}\mathrm{prime},\hspace{0.25em}N/2\equiv a\hspace{0.17em}\left(q\right)}\hspace{0.17em}(\log N/2{)}^2,$ 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 8.2 (Heuristic consequence). [Honest Caveat] If the unordered pair counts fluctuated like independent Poisson variables with the same mean μ, the doubling in Proposition 8.1 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 8.3 (Second moment by class of N, q = 8). [Comp. Verified; R_a,8(N) computed by FFT convolution of the weighted prime indicators for all even N ≤ 6 × 10⁶, M_a,8 from an exact factorisation sieve] (i) For 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 0.13 to 0.19 for the values of a tested at q = 16,32 (Figure 8); every value is within 1.6 standard errors of 2, including ρ_1,8 = 2.44 ± 0.29.

Figure 8. Variance ratio on the class N≡2a (q). What is plotted: the ratio ρ_a,qof the mean of$D_{a,q}^2$on N≡2a (q) to its mean on the other classes, for selected (q,a), with jackknife error bars, against the value 2. What it shows: every value lies within 1.6 standard errors of 2. Connection to paper: Proposition 8.3 and Remark 8.2. 

Figure 8. Variance ratio on the class N≡2a (q). What is plotted: the ratio ρ_a,qof the mean of$D_{a,q}^2$on N≡2a (q) to its mean on the other classes, for selected (q,a), with jackknife error bars, against the value 2. What it shows: every value lies within 1.6 standard errors of 2. Connection to paper: Proposition 8.3 and Remark 8.2. 

Proposition 8.4 (Size of the second moment). [Comp. Verified; q = 8, a = 1, all even N ≤ X, X ≤ 6 × 10⁶] Table 10 gives $\sum _{N\le X}(R_{a,q}\left(N\right)-M_{a,q}\left(N\right){)}^2$ 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.

Second moment of the error, q = 8, a = 1.

X	$\sum _{\mathit{N}\le \mathit{X}}(\mathit{R}-\mathit{M}{)}^2/{\mathit{X}}^3$	$\sum _{\mathit{N}\le \mathit{X}}(\mathit{R}-\mathit{M}{)}^2/({\mathit{X}}^3/\mathbf{log}\mathit{X})$
2 × 105	3.84 × 10−5	4.68 × 10−4
5 × 105	1.83 × 10−5	2.41 × 10−4
106	1.00 × 10−5	1.39 × 10−4
2 × 106	5.65 × 10−6	8.20 × 10−5
3 × 106	4.00 × 10−6	5.97 × 10−5
4.5 × 106	2.67 × 10−6	4.09 × 10−5
6 × 106	2.05 × 10−6	3.19 × 10−5

X	$\sum _{\mathit{N}\le \mathit{X}}(\mathit{R}-\mathit{M}{)}^2/{\mathit{X}}^3$	$\sum _{\mathit{N}\le \mathit{X}}(\mathit{R}-\mathit{M}{)}^2/({\mathit{X}}^3/\mathbf{log}\mathit{X})$
2 × 105	3.84 × 10−5	4.68 × 10−4
5 × 105	1.83 × 10−5	2.41 × 10−4
106	1.00 × 10−5	1.39 × 10−4
2 × 106	5.65 × 10−6	8.20 × 10−5
3 × 106	4.00 × 10−6	5.97 × 10−5
4.5 × 106	2.67 × 10−6	4.09 × 10−5
6 × 106	2.05 × 10−6	3.19 × 10−5

Figure 9. Second moment of the error for q = 8, a = 1. What is plotted: left, $\sum _{N\le X}(R-M{)}^2/X^3$ on logarithmic axes with the line ∝X⁻¹; right, the same sum divided by X³/logX. What it shows: between X = 2 × 10⁵ and X = 6 × 10⁶ the left curve falls by a factor 18.8, a local exponent of about −0.86, slightly slower than X⁻¹; the right curve decreases monotonically. Connection to paper: Proposition 8.4 and Remark 8.5. 

Figure 9. Second moment of the error for q = 8, a = 1. What is plotted: left, $\sum _{N\le X}(R-M{)}^2/X^3$ on logarithmic axes with the line ∝X⁻¹; right, the same sum divided by X³/logX. What it shows: between X = 2 × 10⁵ and X = 6 × 10⁶ the left curve falls by a factor 18.8, a local exponent of about −0.86, slightly slower than X⁻¹; the right curve decreases monotonically. Connection to paper: Proposition 8.4 and Remark 8.5. 

Remark 8.5. [Honest Caveat] Proposition 8.4 is a finite-range measurement. It is compatible with a much smaller order than X³/logX, but it proves nothing about the limit, and the range X ≤ 6 × 10⁶ is far from any regime where a lower-order term such as X³/logX could be excluded or confirmed. For q = 1 the mean square of the Goldbach error is known to be ≪X³(logX)^−A for every A > 0 [12], which is compatible with the decrease seen in Table 10 and incompatible with a lower bound of order X³/logX; the analogous statement for q = 2^k is not proved here. 

8.2. 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 8.6 (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 11 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. 

ℙ(N_n≡b (8)∣p_n≡c (8)), N_n ≤ 6 × 10⁶.

c∖b	0	2	4	6
1	0.2838	0.1664	0.2842	0.2656
3	0.2868	0.2677	0.2799	0.1657
5	0.2846	0.1630	0.2863	0.2662
7	0.2849	0.2667	0.2848	0.1637

c∖b	0	2	4	6
1	0.2838	0.1664	0.2842	0.2656
3	0.2868	0.2677	0.2799	0.1657
5	0.2846	0.1630	0.2863	0.2662
7	0.2849	0.2667	0.2848	0.1637

Proposition 8.7 (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 ${\bar{D}}_c^2$ be the mean of D² over p_n≡c (8), in units of 10⁻⁵. Then ${\bar{D}}_c^2=10.04,\hspace{0.25em}11.43,\hspace{0.25em}9.94,\hspace{0.25em}11.74$ 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 8.8 (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) 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 8.7. 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.

Figure 10. Second moment along N_n by ray, q = 8. What is plotted: the observed mean of D_1,8(N_n)² by ray of p_n modulo 8, and its prediction from windows of N combined with the class of N modulo 8, and modulo 8,3,5. What it shows: rays 3 and 7 exceed rays 1 and 5, and the prediction reproduces the ordering and the size of the differences. Connection to paper: Propositions 8.7 and 8.8 and Remark 8.9. 

Figure 10. Second moment along N_n by ray, q = 8. What is plotted: the observed mean of D_1,8(N_n)² by ray of p_n modulo 8, and its prediction from windows of N combined with the class of N modulo 8, and modulo 8,3,5. What it shows: rays 3 and 7 exceed rays 1 and 5, and the prediction reproduces the ordering and the size of the differences. Connection to paper: Propositions 8.7 and 8.8 and Remark 8.9. 

Remark 8.9 (Interpretation). [Honest Caveat] The ray of p_n acts on the second moment through the class of N_n: the transition matrix of Proposition 8.6 sends more mass to N_n≡2 (8) from rays 3,7, and that class has twice the variance (Proposition 8.3). 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 8.10 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 8.10 (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 8.8(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). 

Figure 11. Second moment along N_n by ray, q = 16. What is plotted: the observed mean of D_1,16(N_n)² by ray of p_n modulo 16 and its prediction from the classes of N modulo 16, 3 and 5 combined with windows of N. What it shows: the recipe fixed in advance reproduces the level and the pattern among the eight rays. Connection to paper: Proposition 8.10 and Remark 8.13. 

Figure 11. Second moment along N_n by ray, q = 16. What is plotted: the observed mean of D_1,16(N_n)² by ray of p_n modulo 16 and its prediction from the classes of N modulo 16, 3 and 5 combined with windows of N. What it shows: the recipe fixed in advance reproduces the level and the pattern among the eight rays. Connection to paper: Proposition 8.10 and Remark 8.13. 

Proposition 8.11 (The same recipe at other pairs (q,a)). [Comp. Verified; same sample of 211,683 pairs and same reference population; recipe of Proposition 8.10 unchanged, with the class modulo q in place of the class modulo 16] Table 12 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. 

Recipe of Proposition 8.10 at four pairs (q,a); means in units of 10⁻⁵.

q	a	obs.	pred.	diff. (%)	z	spread obs./pred.	corr.
8	1	10.79	11.05	−2.4	−4.0	1.80 / 1.43	0.993
16	1	19.77	20.19	−2.1	−3.5	3.28 / 3.22	0.962
16	9	19.42	20.02	−3.0	−5.1	1.93 / 1.87	0.958
32	1	38.60	39.53	−2.4	−4.1	6.50 / 6.81	0.940

q	a	obs.	pred.	diff. (%)	z	spread obs./pred.	corr.
8	1	10.79	11.05	−2.4	−4.0	1.80 / 1.43	0.993
16	1	19.77	20.19	−2.1	−3.5	3.28 / 3.22	0.962
16	9	19.42	20.02	−3.0	−5.1	1.93 / 1.87	0.958
32	1	38.60	39.53	−2.4	−4.1	6.50 / 6.81	0.940

Proposition 8.12 (Effect of further small primes, exploratory). [Comp. Verified; exploratory, the additional moduli were chosen after seeingTable 12] 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 8.13. [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 12) and decreases monotonically when the classes modulo 7 and 11 are added (Proposition 8.12), 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.

9. 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 9.1 (Ray–gap identity). [Proved] For n with p_n > 5: g_n≡ρ_n+1 − ρ_n (mod 10). 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.

Hence predicting the class of g_n modulo 10 from the past is the same problem as predicting ρ_n+1.

Proposition 9.2 (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 13 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. Adding the previous gap g_n−1 in six quantile bins to three rays gives accuracy 0.328, 0.048 bits and R² = 0.0048; the previous gap alone gives accuracy 0.259, 0.001 bits and R² = 0.0014. 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.

Out-of-sample prediction of ρ_n+1 and g_n/logp_n from the last k rays modulo 10; test set p_n > 5 × 10⁷.

k	accuracy of ρn+1	information gain (bits)	R2 of gn/logpn
1	0.3111	0.0330	0.00164
2	0.3152	0.0356	0.00178
3	0.3158	0.0360	0.00186
4	0.3160	0.0363	0.00187
5	0.3160	0.0361	0.00165
6	0.3144	0.0340	0.00062

k	accuracy of ρn+1	information gain (bits)	R2 of gn/logpn
1	0.3111	0.0330	0.00164
2	0.3152	0.0356	0.00178
3	0.3158	0.0360	0.00186
4	0.3160	0.0363	0.00187
5	0.3160	0.0361	0.00165
6	0.3144	0.0340	0.00062

Proposition 9.3 (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 9.4 (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 7.2. 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 5, whose row maxima (31.0%, 31.0%, 30.4%, 33.1% in sample) are reproduced out of sample. 

10. Open Questions

Open Question 57 (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 [8]. Also give a self-contained proof of Lemma 2.3, or of Theorem 2.4 without it. A power saving would give, via Corollary 4.3, $\#\{p_n\le X:R(N_n)=0\}\ll X^{1-{\Delta }_q}$.

Open Question 58 (Explicit threshold). [Open] 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 explicit constants in Lemma 2.3.

Open Question 59 (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 [6]. Determine the density of {n:R_a,q(N_n) = 0}, which by Proposition 6.3 equals that of {n:N_n≡1 (3)} once the primes among N_n − 3 have density zero along this set.

Open Question 60 (Class-dependent second moment). [Open] For q = 2^k, determine whether $\sum _{N\equiv 2a\hspace{0.17em}\left(q\right)}{D}_{a,q}(N{)}^2/\sum _{N\not\equiv 2a\hspace{0.17em}\left(q\right)}{D}_{a,q}(N{)}^2$ (suitably normalised by the number of terms and the range) has a limit as X → ∞, and its value; Proposition 8.3 gives values between 1.94 and 2.44 at q = 8,16,32, all within 1.6 standard errors of 2.

Open Question 61 (Ray-dependent second moment). [Open] Fix q = 2^k and gcd(a,q) = 1. For rays c coprime to q define $W_c\left(X\right)=\sum _{p_n\le X,\hspace{0.25em}{p}_n\equiv c\hspace{0.17em}\left(q\right)}{D}_{a,q}(N_n{)}^2$ 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 8.7, 8.8 and 8.10 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 10.4 and on the limiting transition probabilities.

Open Question 62 (General modulus, closed form). [Open] 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 5 at the same rigour as for q = 3.

Open Question 63 (Shift dependence). [Open] For fixed odd h, Corollary 4.4 bounds the primes p with R(p + h) = 0. Determine whether the rate depends on h through 𝔖(p + h), and whether the dependence is ray-independent at leading order.

11. Conclusions

The restricted Goldbach sum can be evaluated along special sequences of even integers without any new analytic input beyond the mean-value lemma of Tolev, because an injective map into the even integers with linear growth transfers every bound on the exceptional set. For consecutive primes, for the k-th next prime and for shifted primes this gives statements uniform in the ray of p_n; with Theorem 2.4 they are almost-all theorems with a saving of every power of logX, and they 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 above reproduce at a larger scale.

Finally, 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 q = 8,16,32 within about one standard error. 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 8.9; the same recipe applied unchanged at q = 16 (Proposition 8.10) reproduces the level and the pattern among rays, with a small systematic over-prediction (Table 12). The rays of the past predict the next ray with accuracy about 0.31 to 0.32 out of sample (Proposition 9.2) but not the size of the gap. The precise limiting statements are recorded as open questions.

12. Verification Methodology

Every statement labelled [Comp. Verified] was recomputed from scratch by Python scripts (numpy, scipy). Primes up to 10⁸ come from a boolean sieve of Eratosthenes; the scale N = 10⁹ of Table 8 and Table 9 uses a segmented sieve with the same pair-counting code. R_a,q(N) for all even N ≤ 6 × 10⁶ is obtained from one FFT convolution (scipy.signal.fftconvolve, double precision) of the log-weighted prime indicator restricted to p₁≡a (q) with the unrestricted one; every nonzero value of R_a,q(N) is at least log2 ⋅ logp_min, where p_min is the least prime ≡a (q), so the zeros are unambiguous. 𝔖(N) is evaluated exactly by a multiplicative sieve, and C₂ = 0.660161816… is the product over all odd primes ≤10⁸. In Section 8, the 15 logarithmic windows of N ∈ [10⁵,6 × 10⁶] have equally spaced logarithms. Every script first reproduced π(10⁶) = 78,498, π(2 × 10⁶) = 148,933, π(10⁷) = 664,579 and π(10⁸) = 5,761,455 before being trusted, and the numbers ofTable 1, Table 5, Table 8, Table 9, Table 10, Table 11, Table 12 and Table 13 were reproduced to the printed digits.