prime_pair_error: results

All numbers below are in results.json, written by probe.py on 2026-09-06 (run cost in RUNS.md). Notation as in Chou, Haag, Huryn, Ledoan (arXiv:2308.14888): psi_2(N, k) is the von Mangoldt pair count with both members at most N, S(k) the singular series with S(odd) = 0, e(N, k) = psi_2(N, k) - S(k)(N - k) for 1 <= k <= N, and E(N) = 2 sum_{k=1}^{N} e(N, k)^2. R(x) = psi(x) - x throughout. Everything is measured; nothing is a theorem.

1. What replicated

Table 1, all eleven rows. E(N)/(N^2 log^2 N), computed, then truncated to five decimals as the paper does:

The 10^4 row is the one that tells truncation from rounding: 0.1232782 rounds to 0.12328 and truncates to the paper's 0.12327. The Figure 3 curve for N <= 1000 peaks at 0.6158 at N = 16 and ends at 0.09464, as drawn. C_2 from the Euler product to 4 x 10^6 is 0.6601618261 against the pinned 0.6601618158, inside the tail bound.

The implementation against itself. psi_2 by FFT autocorrelation agrees with a pure-Python pair count (no numpy) at N = 2000 to 7 x 10^-12, with a direct numpy pair count at N = 10^5 to 2 x 10^-10, and with direct sums at seven values of k at N = 10^7 to 1.5 x 10^-8. The singular-series sieve agrees with trial-division factorization at every k <= 3 x 10^4 to 9 x 10^-16. The identity sum_{|k| <= N} psi_2(N, k) = psi(N)^2 - sum Lambda(n)^2, which is the paper's Section 3, holds at every N to floating precision, and gives the mean of e over k as psi(N) - N + c(N) with c(N) between 1.21 and 1.32 at all six N.

Beyond the paper's range. The ratio keeps rising:

After the signed part of Section 4 is removed the ratio is 0.1667, 0.2124, 0.2201, 0.2399 at 10^5, 10^6, 3 x 10^6, 10^7. The data through 10^7 do not separate convergence from slow growth; the paper's Corollary 3 allows either up to a power of log N.

2. Where the error sits

Odd separations carry nothing. Their share of E(N) is 4.5 x 10^-2 at N = 10^3, 3 x 10^-4 at 10^5 and 2 x 10^-6 at 10^7; the only pairs at odd k involve a power of 2.

By size of separation, the share of E(N) in each decile of k/N at N = 10^7:

0.172 0.160 0.146 0.132 0.113 0.097 0.077 0.057 0.034 0.012

stable from 10^5 on. A squared error proportional to the prediction S(k)(N - k) would give 0.19, 0.17, 0.15, ..., 0.01; the measured profile is flatter at small k. Small separations dominate, but less than the count of pairs would say.

By prime factors of k, at N = 10^7 (k-share is the fraction of separations in the class, E-share its fraction of E(N), the ratio column is the class mean of e^2 / (S(k)(N - k))):

Two things read off this table. The power of 2 in k is invisible, exactly as it is invisible to S(k): the three 2-adic classes have E-share equal to k-share to three decimals and the same mean. And the odd small primes matter through S(k) but less than proportionally: k divisible by 3 is one sixth of the separations and just under half of E(N), and the ratio e^2/pred falls as the class gets richer in small primes.

The variance is not proportional to the prediction. Binning even k by quintile of S(k) and fifth of k/N, the cell mean of e^2 / (S(k)(N - k)) at N = 10^7 runs from 64.7 (smallest S, smallest k) down to 52.9 across S and up to 79.6 across k. A power fit at small k gives e^2 proportional to S(k)^gamma with gamma = 0.71, 0.75, 0.79, 0.83 at N = 10^5, 10^6, 3 x 10^6, 10^7, drifting toward 1 but not there. A linear fit e^2 = A + B S(k)(N - k) per k-fifth gives an S-independent part A that shrinks with k roughly like N log N (1 - k/N), not a constant floor. Recorded, not explained.

3. The mean of the error is S(k) times the prime number theorem remainder

The paper's identity fixes the k-average of e at psi(N) - N + O(1) and uses it for Theorem 2. The class means above show how that average is distributed: the 3-divides-k class has mean -4470 against -2191 for the rest of the even k, ratio 2.04, and the factor S(k) carries for 3 | k is exactly 2. At N = 10^6, where psi(N) - N = -413 is an unusually large excursion, every class mean is its class mean of S(k) times about -430, within the noise. The offset is not a constant across k; it is S(k) times one number.

4. The one compact relationship, and its test

Write theta_2(N, k) for the pair count over primes only (proper prime powers removed), P(N, k) = psi_2 - theta_2 for the pairs with at least one proper prime power, and

B(N, k) = (psi - theta)(N) + (psi - theta)(N - k) - (psi - theta)(k),

which is about sqrt(N) + sqrt(N - k) - sqrt(k). On even k < N, fit each of e, the prime-only error e_theta = theta_2 - S(k)(N - k), and P by least squares on the three profiles

S(k), S(k) (B(N, k) - mean B), S(k) (R(N - k) - R(k)),

with coefficients alpha, beta, gamma. The first-order heuristic (each factor of the pair carries the one-point fluctuation, and the arcs that build S(k) carry it multiplied by S(k)) predicts alpha = R(N), beta = 0, gamma = 1 for e; alpha = theta(N) - N, beta = -1, gamma = 1 for e_theta; and beta = 1, gamma = 0 for P. Measured:

The relationship was found on N <= 10^6; the two bold rows are the held-out test. At 10^7 the independence-approximation standard errors are 5 on alpha, 0.003 on beta and 0.008 on gamma, so every coefficient is within a few percent of its predicted value and the two that should vanish do. Compactly, for F either psi or theta and R_F(x) = F(x) - x,

F_2(N, k) - S(k)(N - k) = S(k) [ R_F(N) + R_F(N - k) - R_F(k) ] + a residual not modelled here.

The offset alpha_e - (psi(N) - N) is -7, -11, -14, -17, -20, -21 across the six N, about -1.5 log N + 3; it mirrors the positive mean of e on odd k (12 to 32, the pairs with a power of 2), which the exact identity forces the even k to compensate.

How much it explains. For the von Mangoldt count the fitted part carries 1.1, 2.6, 0.4 and 1.9 percent of sum e^2 at 10^5, 10^6, 3 x 10^6, 10^7, tracking |psi(N) - N| / sqrt(N) (0.16, 0.41, 0.00, 0.46). For the prime-only count it carries 20, 22, 12 and 16 percent, because theta(N) - N has the extra -(psi - theta)(N) of size sqrt(N) at every N. After removal the residual ratio is the same for both counts (0.2399 against 0.2455 at 10^7): the residual is shared, and prime powers only move the signed part.

What the beta column says. P(N, k) is S(k) B(N, k) to within 0.2 percent in slope (correlation 0.90; the remaining 12 percent of its variance is the Legendre-symbol modulation of pairs (p^2, p^2 +- k), for instance p^2 + 2 is always divisible by 3). The prime-only count sits below the Hardy-Littlewood count by exactly that amount, and the von Mangoldt count does not see it at all. This is the pair analogue of theta(x) = psi(x) - (psi - theta)(x). It is invisible in a per-k table: at k = 2 and N = 10^7 the deficit S(2) B is about 9 x 10^3 against a per-k residual of about 3 x 10^4, and only pooling over k shows it: the beta_theta entry is hundreds of standard errors from zero under the independence approximation, which overstates the significance since neighbouring k are correlated, but not by a factor that matters.

5. The literature

• Chou, Haag, Huryn, Ledoan, Section 3: the k-sum identity and the mean R(N). Our Section 3 is the k-resolved form of that mean.
• Korevaar and te Riele, Average prime-pair counting formula, Math. Comp. 79 (2010), arXiv:0902.4352, Approximation 2.1: for N large and x much larger than N, (1/N) sum_{r <= N} {psi_{2r}(x) - 2 C_{2r} x} = -4 sum_rho x^rho / rho + ... , supported by prime-pair counts to x = 10^12. That is the small-k, k-averaged limit of the relationship above: S(k) averages to 2 over even k, and R(N - k) - R(k) tends to R(N) for k much smaller than N, so the profile averages to 4 R(N), which is their -4 times the zero sum. Their (2.5) splits psi_{2r} - theta_{2r} through the prime-square pair count, the same split as the beta column. So the relationship is a refinement of a published heuristic: the S(k) weighting at each k, the R(N - k) - R(k) term, and the range up to k = N are what is measured here and not stated there, as far as pages 1 to 8 of that paper go. Not claimed new. Keating and Smith (arXiv:1903.07057) derive S(k) from the pair correlation of zeros and do not carry this term.
• Funkhouser, Goldston, Sengupta, Sengupta, Prime difference champions (2020, arXiv:1612.02938), Figure 11: the unweighted analogue, the variance of the prime-difference count over d divided by pi(x)^2, sits near 0.16 for x from 10^4 to 10^7, flat. Our log-weighted ratio rises from 0.12 to 0.24 over the same range. The two normalizations are not the same object and were not reconciled here.
• Friedlander and Goldston (1995), as cited by Korevaar and te Riele: the sum of the singular series to h is h - (1/2) log h + O(log^{2/3} h). Measured sum_{k <= N} S(k) - N + (1/2) log N stays within +-0.55 across 10^3 to 10^7, so the coefficient of log is -1/2 to within 0.1. Korevaar and te Riele's displayed (3.4) reads (1/2) log m for the sum of C_{2r} to m, which would be a full log h here and is inconsistent with this measurement; read it as a normalization slip in the display.

6. What failed, and what was not pursued

• The Poisson-type law e^2 proportional to S(k)(N - k) does not hold (Section 2). No compact variance law was found.
• Whether E(N)/(N^2 log^2 N) converges, and to what, is open at 10^7. A crude model in which the minor-arc part of |S(alpha)|^2 is exponentially distributed about half of N log N gives 1/4; the data are at 0.244 and still rising. That model is a heuristic aside, not a claim.
• The class means at N = 10^5 miss the S(k) R(N) prediction by three to four sigma in the richer classes (3 | k: 94 against 155, sigma 15). The non-principal characters contribute L-function zero sums of the same sqrt(N) order with signs that depend on k modulo small q; those were not modelled. At 10^6 and 10^7 the principal term dominates because |psi(N) - N| is large there.
• A finer profile for P(N, k) with the Legendre-symbol singular series of p^2 +- k would close the last 12 percent of its variance. Not built.

Second pass, 2026-09-06: a fixed-coefficient residue-class correction

All numbers in this part are in results_residue.json, written by residue.py (cost in RUNS.md); tests/test_prime_pair_residue.py pins the identities, the closed form, the frozen coefficients and the recorded numbers. Notation as above. Everything is measured; the identities in Section 8 are exact and are checked as such; nothing is a theorem about the asymptotics.

7. What was checked before anything new was built

The modulo-3 formula. The brief asked for an independent check of a supplied modulo-3 formula. No such formula was found in this repository, in PR #181, in the Fulcrum record, in the private ledger or in any checkout, so the formula checked here is the one the decomposition of Section 8 produces at modulus 3, written in closed form with the character mod 3 (Section 9). It is checked two ways: the closed form, which never forms residue-class sums of the centered function, agrees with the general modulus-q code to rounding (check_chi3_closed_form_vs_general_maxabs, below 2 x 10^-6 at every cutoff, on terms of size 10^3 to 10^4), and the general code is checked against the exact identity with an independent pair count.

The first-pass results. probe.py was re-run to N = 10^6: Table 1 matches at all eleven rows, and the ratio, the three-profile coefficients and the class means at 10^3, 10^4, 10^5, 10^6 are identical to results.json to every printed digit. The three-profile fit was also re-implemented from scratch in residue.py (first_pass_fit_at_fresh_N) and gives alpha = 37.6, gamma = 0.987 at 10^5 and alpha = -430.3, gamma = 0.994 at 10^6, the first pass's numbers.

8. The decomposition at a modulus q

Fix q with phi = phi(q), w = q / phi, and write M_q(n) = w for (n, q) = 1, n >= 1, and 0 otherwise: the periodic model of Lambda with the right mean on the reduced classes. Split Lambda on the reduced classes as Lambda' = M_q + delta', and leave the non-reduced classes, which carry only the powers of the primes dividing q, alone. Then for 1 <= k <= N,

psi_2(N, k) = A_q(N, k) + L_q(N, k) + X_q(N, k) + D_q(N, k), (I1)

A_q = sum_{n <= N-k} M(n) M(n+k) periodic baseline L_q = sum_{n <= N-k} [M(n) delta'(n+k) + delta'(n) M(n+k)] linear endpoint terms X_q = sum over pairs with a member a power of a prime dividing q exceptional D_q = sum_{n <= N-k} delta'(n) delta'(n+k) remaining correlation

(I1) is an exact identity: it is the expansion of (M + delta' + Lambda^x)(n) times the same at n + k, with the three products involving Lambda^x collected into X_q. The pieces have the following exact properties.

(I2) The baseline is the local singular series times the length, plus a bounded periodic term. With nu_q(k) = #{a mod q : (a, q) = 1 = (a + k, q)},

A_q(N, k) = S_q(k) (N - k) + rho_q(N, k), S_q(k) = q nu_q(k) / phi^2, |rho_q| <= w^2 nu_q(k),

and by the Chinese remainder theorem S_q(k) = prod_{p | q} sigma_p(k) with sigma_p(k) = p/(p-1) for p | k and p(p-2)/(p-1)^2 otherwise. For p = 2 this is 2 or 0, for p = 3 it is 3/2 or 3/4, for p = 5 it is 5/4 or 15/16.

(I3) The singular series factors over the modulus. S(k) = S_q(k) S^(q)(k), where S^(q) is the same product over the primes not dividing q, i.e. S^(q)(k) = C_2^(q) prod_{p | k, p > 2, p not | q} (p-1)/(p-2) times sigma_2(k) when q is odd, with C_2^(q) = C_2 / prod_{p | q, p > 2} (1 - 1/(p-1)^2). This holds at every k, odd k included, and the test checks it at every k <= 5000 for q = 1, 2, 3, 5, 30, 210.

(I4) The endpoint terms are linear in the prime counts of residue classes. Writing Psi'b(x) = sum{n <= x, n = b (q)} delta'(n) = psi(x; q, b) - w #{n <= x : n = b (q)} for reduced b,

L_q(N, k) = w sum_{b : (b - k, q) = 1} [Psi'_b(N) - Psi'b(k)] + w sum{a : (a + k, q) = 1} Psi'_a(N - k),

both sums over reduced residues. So L_q reads psi(x; q, b) at x = N, k and N - k only, plus an exactly known sawtooth. In characters, psi(x; q, b) = (1/phi) sum_chi chi-bar(b) psi(x, chi), so L_q carries the principal-character remainder with weight S_q(k) and each non-principal psi(x, chi) with a weight that is a twisted Ramanujan-type sum over the reduced pairs (a, a + k); for q = 30 that is eight characters, of conductors 1, 3, 5 and 15.

(I5) The exceptional term is small and exact. X_q(N, k) is a sum over the O(log N) powers of 2, 3, 5 up to N, each paired with Lambda at distance k; it is O(log^2 N) at every k and is computed by enumeration. For q = 1 it is empty. Its k-mean on even k is what the first pass saw as the offset alpha_e - (psi(N) - N) of about -1.5 log N: the pairs (2^j, 2^j +- k) that the exact identity forces the even k to compensate for.

(I6) The q = 1 member. For q = 1, M = 1, rho = 0, X = 0 and L_1 = R(N) + R(N - k) - R(k), so the first pass's profile is the endpoint term of the trivial modulus.

9. The one heuristic, and the three corrections

Everything above is an identity. The heuristic is one sentence, applied once:

(H_q) the primes not dividing q act on the reduced classes mod q as all primes act on the integers,

i.e. the centered correlation carries the factor (S^(q)(k) - 1) on everything the model already carries: D_q(N, k) = (S^(q)(k) - 1) [A_q(N, k) + L_q(N, k)] + a remainder that (H_q) takes to have mean zero. Under (H_q) the predicted signed part of e(N, k) = psi_2 - S(k)(N - k) is, using (I2) and (I3),

c_q(N, k) = S^(q)(k) [ rho_q(N, k) + L_q(N, k) ] + X_q(N, k),

and the residual Z_q = e - c_q = D_q - (S^(q) - 1)(A_q + L_q) is what (H_q) says is centered. Every coefficient is fixed at 1 by the derivation; the FROZEN table in residue.py records the six numbers a fit could have moved, all at their derived values, and the test refuses any other value. (H_q) is the same first-order heuristic the first pass used, restricted to a residue class pair; the un-linearised form, S^(q)(k) (q/(N-k)) sum_a psi(N-k; q, a) [psi(N; q, a+k) - psi(k; q, a+k)], differs from c_q by at most 73 at any k < N at every cutoff, against endpoint terms of size 10^3 to 10^4, and changes no reduction by more than 0.002 points.

The three corrections compared, and the one diagnostic:

• Original (q = 1): c_1 = S(k) [R(N) + R(N - k) - R(k)], the first pass's profile with its predicted coefficients (R(N), 0, 1), not the fitted ones.
• Modulo 3 (q = 3): with chi the character mod 3, R'(x) = psi(x) - x - (log 3) floor(log_3 x),

c_3 = S(k) [R'(N) + R'(N-k) - R'(k)]

• S(k) chi(k) [psi(N-k, chi) - psi(N, chi) + psi(k, chi)]
• S^(3)(k) [rho_3(N, k) + sawtooth] + X_3(N, k).

The character term has opposite signs on k = 1 and k = 2 mod 3 and vanishes on 3 | k; the sign comes from the twisted sums sum_{a : (a, 3) = 1 = (a + k, 3)} chi(a) = chi(k) and sum chi(a + k) = -chi(k), which for prime q are the values of a Ramanujan sum twisted by chi.

• Period 30 (q = 30): c_30 from (I2) to (I5) with the eight reduced residues mod 30 and the eight characters; not the sum of the q = 2, 3, 5 corrections, because the residue constraint (a, 30) = 1 = (a + k, 30) couples the three primes and the baseline factor S_30 = sigma_2 sigma_3 sigma_5 is a product, not a sum.
• Diagnostic (q = 210): c_210 is computed only to ask whether what c_30 leaves behind is what the next prime predicts. It is not one of the three candidates.

10. The comparison at fresh cutoffs

The first pass evaluated at 10^3, 10^4, 10^5, 10^6, 3 x 10^6 and 10^7. The five cutoffs below were never used before; the coefficients were frozen (Section 9) before the run; nothing was looked at in between. The reduction is 1 - sum_k (e - c_q)^2 / sum_k e^2 over all 1 <= k <= N, in percent (restricting to even k changes the third decimal); the slope is the post-hoc least-squares coefficient of e on c_q over even k, a diagnostic whose derived value is 1.

Three things to read off it.

The order is the same at every cutoff: each refinement removes more than the last, and the modulus-30 correction removes between 1.2 and 16 times what the original does. The size of the reduction at a given N is set by how large the one-point excursions happen to be there: at 2 x 10^5, psi(N) - N is 27 and the original carries 0.6 percent, while the character mod 3 has a large excursion and carries 7 percent on its own. That is the first pass's observation (the q = 1 share tracks |psi(N) - N| / sqrt N) with more characters to track.

The coefficient is 1. The slope of e on c_30 is within 0.2 percent of 1 at all five cutoffs, on c_3 within 3.5 percent, on c_1 within 7 percent; the modulus-1 slope is the noisiest because c_1 is the smallest of the three. The share sum c_q^2 / sum e^2 equals the reduction to two decimals at every row, which is the same statement.

What each piece weighs, at 7 x 10^6 on even k: the endpoint term S^(q) L_q has rms 1838, 2334 and 2984 for q = 1, 3, 30; the exceptional term X_q has rms 43 and 70 for q = 3, 30 (maximum 102 at q = 210); the bounded term rho_q is at most 0.8 and 16.9. The per-k residual has rms 1.4 x 10^4, which is why a correction that is right to one percent in slope removes only a few percent of the squared error: most of E(N) remains unexplained by the tested corrections, and the signed part they capture is a tenth of the residual in amplitude. The residual ratio (sum (e - c_30)^2 in units of N^2 log^2 N, doubled for +-k) is 0.1707, 0.1922, 0.2124, 0.2260, 0.2320, still rising.

The offset the first pass could not place. The mean of e - c_1 on even k is -22 to -28 at the five cutoffs, the first pass's alpha_e - (psi(N) - N) of about -1.5 log N. The mean of e - c_3 is the same; the mean of e - c_30 is 2.3 to 2.5 at all five. The offset is the exceptional term of the prime 2, the pairs (2^j, 2^j +- k), which the modulus-30 decomposition carries exactly and the modulus-3 one does not (X_3 holds only the powers of 3). What is left, +2.4, is constant across N and is not explained.

11. Where the remaining discrepancy sits

For the residual Z_30 = e - c_30 on even k < N, and for e, Z_1, Z_3 as contrast; z is the class mean divided by its standard error under independence, which overstates significance for neighbouring k but not by a factor that matters here.

Nothing is left at the modulus. Over the fifteen even classes k mod 30, the largest |z| is 53, 29, 67, 104, 79 for e; 25, 25, 56, 70, 52 for Z_3; and 0.6, 0.4, 0.3, 0.2, 0.3 for Z_30. By k mod 3 at 7 x 10^6: e has class means 1639, 1704, 24 (se 22, 17, 17); Z_1 has -83, +840, -839, the antisymmetric pair the character term predicts and the original correction cannot see; Z_3 has -35, -23, -17, the offset of Section 10; Z_30 has 6, -1, 2. The class k = 0 mod 3 of Z_30 sits at +6 with se 15 to 22 at every cutoff, consistent with zero and consistently positive; recorded.

Everything is left at the next primes. The largest |z| of Z_30 by k mod 7, 11, 13 is 15, 15, 10 at 2 x 10^5 and 97, 66, 36 at 7 x 10^6. The modulus-210 member of the same family, with the same frozen coefficients, predicts the mod-7 class means of Z_30 to within one standard error at every cutoff; at 7 x 10^6 the seven class means of Z_30 are -34, +688, +1566, -2733, +913, +912, -1295 and the increment c_210 - c_30 averages -42, +687, +1564, -2737, +912, +912, -1296 on the same classes, with standard errors of 17 to 30. After c_210 the largest |z| by k mod 7 is 0.35 at every cutoff. The correlation of Z_30 with the increment is 0.13 to 0.20 and the slope 0.99 to 1.00. The mod-11 and mod-13 structure remains and would be the next two rungs; the cost of a rung is linear in phi(q).

Not in the size of k, not in S(k). The share of sum Z_30^2 by decile of k/N at 7 x 10^6 is 0.174, 0.162, 0.148, 0.133, 0.115, 0.096, 0.076, 0.054, 0.032, 0.010, the first pass's profile for e itself to the third decimal; the correction does not move where the squared error lives. The mean of Z_30 by quintile of S(k) is within one standard error of zero in every cell at every cutoff (at 7 x 10^6: +20, -8, -7, +13, -7 with se 20 to 29).

12. One statement to prove

Let chi be the non-principal character mod 3 and

T(N) = sum_{1 <= k <= N} chi(k) [psi_2(N, k) - S(k)(N - k)] = sum_{n < n' <= N} Lambda(n) Lambda(n') chi(n' - n) - sum_k chi(k) S(k)(N - k).

Under (H_3), with S(k) replaced by its class mean 3/2 on even k not divisible by 3 and the class sum by a third of the full sum, T(N) is I(N) = sum_{x < N} psi(x, chi)

• ((N - 1)/2) psi(N, chi). Measured, against the original correction's version:

The statement, in the form the data support:

(T) For the non-principal character chi mod 3, as N -> infinity,

sum_{1 <= k <= N} chi(k) [psi_2(N, k) - S(k)(N - k)] = integral_0^N psi(x, chi) dx - (N/2) psi(N, chi) + O(N^{1 + epsilon}),

equivalently, by the explicit formula for psi(x, chi) with rho over the non-trivial zeros of L(s, chi), = sum_rho N^{rho + 1} (rho - 1) / (2 rho (rho + 1)) + O(N^{1 + epsilon}).

Why this one. First, it is the exact character analogue of the mechanism behind the paper's Theorem 2, which lower-bounds E(N) by Omega(N^{1 + 2 Theta - epsilon}) from the variability of psi(N) - N through the k-mean of e: by Cauchy-Schwarz, E(N) >= T(N)^2 / N, so (T) with Landau's theorem would give E(N) = Omega(N^{1 + 2 Theta_chi - epsilon}) with Theta_chi the supremum of the real parts of the zeros of L(s, chi_3); the pair error would then be bounded below by the zeros of L(s, chi_3) as well as by those of zeta, and by the same route by every L(s, chi). Second, unlike the k-mean the paper uses, T(N) does not reduce to one-point counts: the sum over ordered pairs n != n' of Lambda(n) Lambda(n') chi(n' - n) is sum_{a, b} psi(N; 3, a) psi(N; 3, b) chi(b - a), which vanishes identically because chi is odd, so T(N) is the antisymmetric half n < n' and carries genuine pair information. It is the character-weighted analogue of Korevaar and te Riele's Approximation 2.1 at the level of the secondary term. Third, it is sharp enough to be wrong: the main term is of order N^{3/2} and the measured discrepancy is at most 0.035 N log N in absolute value at the five cutoffs, so an error term of order N^{3/2 - delta} for any delta > 0 is what the data can distinguish; they cannot tell O(N log N) from O(N^{1 + epsilon}) and they say nothing about the constant. The zero-sum form is derived from the integral form and was not measured: no zero of L(s, chi_3) was computed here; psi(x, chi) was summed directly from Lambda. Part 3 below (Sections 14 to 19) takes (T) apart: it holds under RH for zeta and for L(s, chi) with O(N log^4 N) in place of O(N^{1+eps}), and the lower bound on E(N) it was named for holds unconditionally.

13. The literature, and what was not done

• Chou, Haag, Huryn, Ledoan, Theorem 2: E(N) = Omega(N^2 (log log log N)^2) and Omega(N^{1 + 2 Theta - epsilon}), proved, in their words, from "the variability in the number of primes up to N", i.e. through the k-mean of e, which is the q = 1 endpoint term of Section 8. Sections 9 to 12 are that mechanism run through the characters.
• Korevaar and te Riele (Math. Comp. 2010): the k-averaged small-k form with zeta zeros only; no characters, no residue classes of k.
• Bhowmik, Halupczok, Matsumoto, Suzuki, Goldbach representations in arithmetic progressions and zeros of Dirichlet L-functions, Mathematika 65 (2019) 57-97, arXiv:1704.06103: the average number of representations of an integer as a sum of two primes in arithmetic progressions, with the secondary term as a sum over zeros of Dirichlet L-functions, conditionally. That is the sum side (n + n' = m, averaged over m) of what Section 8 does on the difference side, unaveraged in k, with the sign rule chi(k). The difference-side statement in this form was not found; the search covered abstract pages and search listings, not paper bodies. Not claimed new.
• Fujii (Acta Arith. 1991) and Languasco and Zaccagnini (2012): the average Goldbach count sum_{n <= N} G(n) = N^2/2 - 2 sum_rho N^{rho+1}/(rho(rho+1)) + error, proved under RH. The zero-sum form of (T) has the same shape with L(s, chi_3) in place of zeta and (rho - 1)/(2 rho (rho + 1)) in place of -2/(rho(rho+1)); whether their method reaches the difference side is exactly what proving (T) would settle.
• Bogomolny and Keating (arXiv:1307.6010): the Hardy-Littlewood conjecture for primes in arithmetic progressions used in the other direction, from prime pairs to the pair correlation of zeros of L(s, chi); their finite products over the primes dividing the modulus are the factorisation (I3).

Not done: the modulus-210 member is a diagnostic here, not a candidate, and the mod-11 and mod-13 structure it leaves is unmeasured against the modulus-2310 member; the second-order (product-form) term was bounded, not modelled; the prime-only count theta_2 was not redone at modulus q (the first pass's prime-power profile is orthogonal to everything here); no zeros of any L-function were computed; and the question of whether E(N)/(N^2 log^2 N) converges is unchanged, 0.2372 at 7 x 10^6 and rising. The constant +2.4 in the residual mean is recorded and not explained.

Third pass, 2026-09-06: Section 12's sum is a Wronskian, and what is proved about it

Independent review, 2026-09-06. REFEREE.md audits the complete proofs from commit 37c2a53a, supplies the uniform Perron calculation and the actual double-zero kernel, and records the corrections incorporated below. A retains its exponent and four logarithms; B's case split is valid. The unproved RH equivalence, error-term optimality and noncoincidence assertions are withdrawn. These are reviewed written proofs, not formal verification. The numerical JSON files are unchanged.

All numbers in this part are in results_wronskian.json, written by wronskian.py (cost in RUNS.md); section 5 of tests/test_prime_pair_residue.py pins the identities on arbitrary weights as well as on Lambda, and the recorded numbers. Notation as above, and in addition P(x) = psi(x, chi) = sum_{n <= x} Lambda(n) chi(n) with chi the non-principal character mod 3, R(x) = psi(x) - x, and

T(N) = sum_{k=1}^{N} chi(k) [psi_2(N, k) - S(k)(N - k)], I(N) = integral_0^N P(x) dx - (N/2) P(N) = sum_{j=0}^{N-1} P(j) - (N/2) P(N),

so that (T) of Section 12 reads T(N) = I(N) + O(N^{1+eps}). Section 12 measured T/I within 0.5 percent at five cutoffs and stated (T) as a conjecture. This part asks what is true of T(N) - I(N). In one line: it is a Wronskian of the two prime-counting remainders R and P plus an explicit term of order N; the Wronskian is bounded by N^{Theta + Theta_chi} (log N)^4 unconditionally, Theta and Theta_chi the suprema of the real parts of the zeros of zeta(s) and of L(s, chi); so (T) holds under RH for zeta and for L(s, chi) with O(N log^4 N) in place of O(N^{1+eps}), and, unconditionally, E(N) = Omega(N^{1 + 2 Theta_chi - eps}), the lower bound by the zeros of L(s, chi) that Section 12 named. Not proved: (T) without the two hypotheses, and anything sharper than log^4 N.

The sections separate four questions: what is an identity (14, 16), the auxiliary asymptotic (15), what is a theorem and under which hypothesis (17, 18), what is known elsewhere (19).

14. The reduction, verified

Write A(n), B(n) for the sums of Lambda over m <= n in the classes 1 and 2 mod 3, so P = A - B and, with r(n) = A(n) + B(n) - n, A = (n + r + P)/2 and B = (n + r - P)/2. Lambda' = Lambda on 3 not | n is the reduced part; the rest is the powers of 3.

(E0) The character is antisymmetric and linear on reduced pairs. For 3 dividing neither n nor m,

chi(m - n) = (chi(n) - chi(m)) / 2.

Four cases: (n, m) = (1, 2) mod 3 gives chi(1) = 1 = (1 - (-1))/2; (2, 1) gives chi(2) = -1 = (-1 - 1)/2; (1, 1) and (2, 2) give chi(0) = 0. Everything below rests on this line, and it is special to the modulus 3, the one modulus whose two reduced classes are exchanged by negation with a non-zero character (mod 4 and mod 6 have chi(m - n) = 0 on every reduced pair).

(E1) The reduced pairs. By (E0) and the definition of the prefix sums, for any weight f supported on 3 not | n,

sum_{n < m <= N} f(n) f(m) chi(m - n) = sum_{n=1}^{N} [A(n-1) dB(n) - B(n-1) dA(n)] = sum_{j=0}^{N-1} P(j) - ((N-1)/2) P(N) + W(N),

W(N) = (1/2) sum_{n=1}^{N} [P(n-1) dr(n) - r(n-1) dP(n)], df(n) = f(n) - f(n-1),

all cumulative functions vanishing at 0. The second line is the expansion of A and B in n + r and P: A(n-1) dB(n) - B(n-1) dA(n) = (1/2)[P(n-1)(1 + dr(n)) - (n - 1 + r(n-1)) dP(n)], and sum_n (n-1) dP(n) = (N-1) P(N) - sum_{j=0}^{N-1} P(j) by partial summation. So the coefficient of P(N) is (N-1)/2 and not N/2; the two differ by P(N)/2, and the test checks that the N/2 form misses by exactly that.

(E2) The pairs with a power of 3. chi(3^j + k) = chi(k) and chi(3^j - k) = -chi(k), so

X_chi(N) = sum_{n < m <= N, 3 | nm} Lambda(n) Lambda(m) chi(m - n) = (log 3) sum_{3^j <= N} [P(N) - 2 P(3^j)],

and for a general weight f the same with f(3^j) for log 3 and the sum over all 3 | n.

(E3) The identity. With H_chi(N) = sum_{k <= N} chi(k) S(k) (N - k), (E1) and (E2) applied to Lambda give

T(N) - I(N) = W(N) + P(N)/2 + X_chi(N) - H_chi(N). (E3)

This is the identity the brief proposed, confirmed, including its (N-1)/2; the P(N)/2 is what separates that from the N/2 of (T).

Checks. (E1) and (E2) are checked against an O(N^2) pair enumeration on Gaussian random weights at N = 7, 50, 301, 1000 (agreement to 5 x 10^-12 absolute) and on Lambda at N = 3000 against the pure-Python pair count (3 x 10^-11). (E3) is checked at the five fresh cutoffs against T(N) from the FFT pair count: the residual is at most 1.2 x 10^-3 on T of size 10^8 to 2 x 10^9, rounding. The T(N) used is the one residue.py recorded, to 10^-9 relative.

15. The auxiliary term is of order N exactly

The brief proposed P(N)/2 + X_chi(N) - H_chi(N) = O(N log^2 N) without proof. It holds, and it is not sharp: the term is -c_H N + o(N), with -c_H positive and approximately 0.31305. The asymptotic below uses the fixed-character PNT and a contour shift.

• P(N)/2 is O(N exp(-c sqrt(log N))) by the prime number theorem for the character; |P(N)| <= psi(N) < 2N by Chebyshev in any case.
• X_chi(N) is exact by (E2): floor(log_3 N) terms, each at most 3 max_{x <= N} |P(x)|, so O(N log N) by Chebyshev and O(N log N exp(-c sqrt(log N))) = o(N) by the same theorem.
• H_chi(N) = sum_k chi(k) S(k) (N - k) = integral_0^N M(x) dx, M(x) = sum_{k <= x} chi(k) S(k). With S(k) = 2 C_2 1[2 | k] sum_{d | k, d odd squarefree} g(d), g(p) = 1/(p - 2), and chi(2dj) = -chi(d) chi(j), sum_{j <= y} chi(j) in {0, 1}:

M(x) = -2 C_2 sum_{d odd squarefree, 3 not | d, d <= x/2} g(d) chi(d) 1[floor(x/2d) = 1 mod 3],

so |M(x)| <= 2 C_2 sum_{d <= x/2} g(d) << log x by Mertens, and H_chi(N) << N log N with nothing beyond Chebyshev. Sharper: the Dirichlet series of chi(k) S(k) is D(s) = -2^{1-s} C_2 L(s, chi) F(s), F(s) = sum_d g(d) chi(d) d^{-s} = L(1 + s, chi) K(s) with K(s) an Euler product absolutely convergent for Re s > -1/2; D is analytic there, on Re s = -1/4 the two factors satisfy L(s, chi) << (1+|t|)^{3/4} and L(1+s, chi) <<_nu (1+|t|)^{1/8+nu} for any fixed 0 < nu < 1/8, by the functional equation and convexity. Thus D(s)/(s(s+1)) is integrable on this line, and the Riesz-mean form H_chi(N) = (1/2 pi i) integral D(s) N^{s+1} / (s (s+1)) ds, shifted to Re s = -1/4, leaves the residue at s = 0 alone:

H_chi(N) = c_H N + O(N^{3/4}), c_H = D(0) = -(2 C_2 / 3) prod_{p > 3} (1 + chi(p)/(p - 2)) < 0,

using L(0, chi) = 1/3 and F(0) = prod_{p > 3} (1 + chi(p)/(p-2)), the product taken in the order of the primes (it is L(1, chi) K(0), and both converge). The JSON's G_infinity = 0.71133109 and c_H = -0.31306242 are products truncated at 2000000, not exact limiting constants. The independent calculation and convergent product in REFEREE.md Section 5 place the limit near -0.31305. Measured H_chi(N)/N is -0.3131, -0.3131, -0.3130, -0.3130, -0.3130 at the five cutoffs. H_chi_by_divisor_expansion evaluates the divisor form with no code in common with the singular-series sieve; the two agree to 10^-10 relative.

So P(N)/2 + X_chi(N) - H_chi(N) = -c_H N + O(N log N exp(-c sqrt(log N))): established, unconditionally, as O(N), and it is Theta(N). At N = 200000 the independent check gives H_chi(N)/N = -0.3131254443 and the auxiliary term divided by N = +0.3431617351; the difference is P(N)/(2N) + X_chi(N)/N = 0.0300362908. Finite agreement does not establish the asymptotic. An o(N) error in (T) would require W(N) = c_H N + o(N), which has not been excluded. No optimality of O(N) is claimed.

16. What W is: the Wronskian of the two remainders

(E4) Partial summation of sum_n P(n-1) dR(n) against sum_n R(n) dP(n), with dR(n) = Lambda(n) - 1 and dP(n) = Lambda(n) chi(n), gives the exact split

W(N) = R(N) P(N) / 2 - J(N) + Q(N), J(N) = sum_{m <= N} Lambda(m) chi(m) R(m),

Q(N) = (1/2) [ sum_{m <= N} Lambda(m)^2 chi(m) - P(N) - (log 3) sum_{3^j <= N} P(3^j - 1)

• sum_{m <= N} Lambda(m) chi(m) psi_0(m - 1) ],

psi_0(x) the sum of Lambda over the powers of 3 up to x, so |Q(N)| <= 2 psi(N) log_3 N + 2N = O(N log N) by Chebyshev. split_W evaluates the three pieces; the test checks W - (RP/2 - J + Q) = 0 to rounding at N = 3000 and at the five cutoffs.

So W is, up to O(N log N), (1/2) integral (P dR - R dP): the Wronskian of R = psi - x and P = psi(., chi). The symmetric left-endpoint combination sum [P(n-1) dR(n) + R(n-1) dP(n)] equals R(N) P(N) - sum Lambda^2 chi + P(N), a one-point quantity; the antisymmetric one is the pair content of T(N), and it does not reduce to one-point counts. In (E4) it is J(N), the zeta remainder sampled at the prime powers and weighted by the character:

J(N) = sum_{p^a <= N} (log p) chi(p^a) [psi(p^a) - p^a].

The reduction, in one line. By (E3), (E4) and Section 15,

T(N) - I(N) = R(N) P(N) / 2 - J(N) + O(N log N), (E5)

so (T) is equivalent to J(N) - R(N) P(N) / 2 = O(N^{1+eps}) for every eps > 0. Under RH for zeta and for L(s, chi) the product term is O(N log^4 N) on its own and (T) is the one statement J(N) = O(N^{1+eps}): the values psi(p) - p at the primes p = 1 mod 3 and at the primes p = 2 mod 3 agree, in the log-weighted mean, to relative accuracy N^{-1/2+eps}.

Why a pointwise bound does not do it. |J(N)| <= sum_m Lambda(m) |R(m)| <= psi(N) max_{x <= N} |R(x)|, which is N^{3/2} log^2 N under RH: N^{1/2} short of the target. The target needs the sign of chi(m) to cancel R(m) across the two classes, and nothing about the size of R alone says that it does. Section 17 gets the cancellation from the zeros: J is a double sum over pairs (rho, rho') of a zero of zeta and a zero of L(s, chi), each term of modulus N^{Re rho + Re rho'}, and the count of pairs, weighted by the coefficients, is a power of log N.

Measured. At the five cutoffs J/N = 0.61, 0.10, 0.33, 0.55, 0.83 and J/(N log N) = 0.050, 0.007, 0.022, 0.036, 0.053; R(N) P(N)/(2N) = 0.03, -0.01, 0.00, 0.06, -0.02; |Q|/N is at most 0.022. J is positive at all five cutoffs, recorded and not explained. The data are consistent with W = O(N) and cannot distinguish that from O(N log N); the bound of Section 17 is a power of log N above either.

17. Theorem A: the Wronskian is bounded by the zeros

Let Theta = sup Re rho over the non-trivial zeros of zeta(s) and Theta_chi = sup Re rho' over those of L(s, chi), so 1/2 <= Theta, Theta_chi <= 1, equality to 1/2 being the two Riemann hypotheses.

Theorem A. W(N) << N^{Theta + Theta_chi} (log N)^4. Hence T(N) = I(N) + O(N^{Theta + Theta_chi} log^4 N), and under RH for zeta(s) and for L(s, chi), T(N) = I(N) + O(N log^4 N).

The unconditional form is empty when Theta + Theta_chi = 2 and is (T) with log^4 N for N^eps when both suprema are 1/2. It is the difference-side analogue of Bhowmik, Halupczok, Matsumoto and Suzuki's Theorem 1(1) (Section 19), whose error term is x^{1 + B_q} for the same reason.

Standard facts used (Davenport, Multiplicative Number Theory, chapters 15 to 19; Montgomery and Vaughan, Multiplicative Number Theory I, chapters 5, 10 and 12), all unconditional:

• (F1) For x >= 2 and T >= 2, psi(x) = x - sum_{|gamma| <= T} x^rho / rho
• O(x T^{-1} log^2(xT) + log x), and psi(x, chi) = -sum_{|gamma'| <= T} x^{rho'} / rho'
• O(x T^{-1} log^2(xT) + log x). These are the right-continuous forms of Montgomery and Vaughan, Theorems 12.5 and 12.10: their midpoint formulas differ by Lambda(x)/2 and Lambda(x)chi(x)/2 at prime powers. The bounded constant and trivial-zero terms are absorbed for x >= 2. Both formulas allow arbitrary T; the proofs choose a nearby good height and bound the intervening zeros.
• (F2) The number of zeros of zeta, or of L(s, chi), with ordinate in [t, t + 1] is O(log(|t| + 2)); hence sum_{0 < |gamma| <= T} 1/|gamma| << log^2 T for either function.
• (F3) For -1 <= sigma <= 2 and |t| >= 2, L'/L(s, chi) = sum_{|t - gamma'| <= 1} 1/(s - rho') + O(log(|t| + 2)). At bounded heights one must include the trivial-zero poles or stay a fixed distance from them. For sigma <= -1/4 at distance >= 1/40 from the trivial zeros, L'/L(s, chi) << log(|s| + 2) by the functional equation. The corresponding zeta statements also retain its pole at 1 when working at bounded heights.
• (F4) Perron's formula with remainder (Montgomery and Vaughan, Corollary 5.3).
• (F5) The classical zero-free region: no zero of zeta with Re rho > 1 - c/log(|gamma| + 2), hence, by rho -> 1 - conj(rho), none with Re rho < c/log(|gamma| + 2).

Proof. By (E4) and Section 15 it is enough to bound J(N) and R(N) P(N).

Step 0, the product. By (F1) with T = N and (F2), R(N) << N^Theta log^2 N and P(N) << N^{Theta_chi} log^2 N, so R(N) P(N) << N^{Theta + Theta_chi} log^4 N.

Step 1, R inside J through its zeros. Insert (F1) with T = N into J(N) = sum_m Lambda(m) chi(m) R(m); at a prime power m the formula is off by at most Lambda(m), absorbed in the O(log m). Then

J(N) = - sum_{|gamma| <= N} (1/rho) A_rho(N) + O(N log^2 N), A_rho(N) = sum_{m <= N} Lambda(m) chi(m) m^rho,

the error being sum_m Lambda(m) [(m/N) log^2 N + log m] << N log^2 N.

Step 2, the twisted prime sum through the zeros of L. Fix rho = beta + i gamma with |gamma| <= N. The Dirichlet series of a_m = Lambda(m) chi(m) m^rho is -L'/L(w - rho, chi), absolutely convergent for Re w > 1 + beta. Apply (F4) with c = 1 + beta + 1/log N and a height T_rho in [2N, 2N + 1] chosen so that the two lines Im(w - rho) = +-T_rho - gamma are at distance >= 1/(C log N) from every ordinate gamma' of L(s, chi); by (F2) the excluded set has measure below 1/2 for C large, so such T_rho exists. The near-endpoint part of (F4) runs over N/2 < m < 2N, not just m <= N. There |a_m| <= (2N)^beta log(2N), so its sum is << N^beta log N sum_{1 <= h <= N} min(1, N/(T_rho h)) << N^beta log^2 N. The other part is bounded by (1+N^c)/T_rho times sum_m Lambda(m) m^{-1 - 1/log N} << N^beta log N. Restoring the full Perron endpoint costs at most Lambda(N) N^beta/2. Thus the remainder is << N^beta log^2 N uniformly in rho. Move the contour to Re w = -delta, delta in {1/8, 3/16} chosen so that |beta - 1 + delta| >= 1/40, which keeps the vertical line 1/40 away from the trivial zero of L(w - rho, chi) at w = rho - 1. The poles crossed are w = rho + rho' for every zero rho' of L(s, chi) with |gamma + gamma'| < T_rho, with residue -N^{rho + rho'} / (rho + rho') counted with multiplicity; w = 0, with residue -L'/L(-rho, chi); and, when beta > 1 - delta, the trivial zero at w = rho - 1, with residue -N^{rho - 1} / (rho - 1) = O(N^{beta - 1} / |gamma|). On the horizontal segments Im(w - rho) is at least N in absolute value, so where -1 <= Re(w - rho) <= 2 (F3) gives O(log N) terms each << C log N by the choice of T_rho, and where Re(w - rho) < -1 the point is at height >= N and so at distance >= 1/40 from every trivial zero, where the functional-equation bound gives << log N; either way << log^2 N, and the two segments contribute << log^2 N integral_{-delta}^{c} N^sigma d sigma / T_rho << N^beta log N. On the vertical segment Re(w - rho) = -delta - beta < 0 while every non-trivial zero has Re rho' > 0, so |w - rho - rho'| >= delta for all of them and (F3) gives << log N where it applies, the functional-equation bound << log N where Re(w - rho) < -1; so it contributes << N^{-delta} log^2 N. The residue at 0 is << log^2 N by (F3) and (F5), since its terms 1/(-rho - rho') have |rho + rho'| >= beta + beta' >= beta >> 1/log N. Altogether

A_rho(N) = - sum_{rho' : |gamma + gamma'| < T_rho} N^{rho + rho'} / (rho + rho') + O(N^beta log^2 N).

Step 3, the double sum. For fixed rho split the zeros rho' by j = floor(|gamma + gamma'|). For j >= 1, |rho + rho'| >= j and there are O(log N) zeros in the interval (all have |gamma'| <= 3N + 1), so these contribute << sum_{j <= 2N + 1} log N / j << log^2 N to sum_{rho'} 1/|rho + rho'|. For j = 0, |rho + rho'| >= beta + beta' >= beta >> 1/log N by (F5) and there are O(log N) such rho': again << log^2 N. With |N^{rho + rho'}| <= N^{Theta + Theta_chi} and (F2),

sum_{|gamma| <= N} (1/|rho|) |sum_{rho'} N^{rho + rho'} / (rho + rho')| << N^{Theta + Theta_chi} log^2 N sum_{|gamma| <= N} 1/|gamma| << N^{Theta + Theta_chi} log^4 N.

For the precise J - RP/2 kernel with compatible truncations, REFEREE.md Section 3 derives K(rho,rho') = (rho'-rho)/(2 rho rho' (rho+rho')) on these same slanted cutoff sets and bounds its absolute sum by O(log^4 N). No infinite double sum or interchange of its limits is used.

The remainders of Step 2 add sum_rho N^beta log^2 N / |rho| << N^Theta log^4 N. With Step 1, J(N) << N^{Theta + Theta_chi} log^4 N + N log^2 N, and Theta + Theta_chi >= 1 absorbs the second term. With Step 0 and (E4), W(N) << N^{Theta + Theta_chi} log^4 N; with (E3) and Section 15, so is T(N) - I(N). QED.

Where each hypothesis enters, and where the cancellation comes from. No Riemann hypothesis is used in the argument; the two suprema enter only through |N^{rho + rho'}| = N^{beta + beta'} in Step 3, and the zero-free region (F5) only to keep the near-diagonal denominators |rho + rho'| away from 0 (under RH they are at least 1). The cancellation that a pointwise bound on R cannot see is Step 2: the character-twisted prime sum A_rho(N), which is N^{1 + beta} pointwise, is N^{beta + beta'} per zero of L(s, chi) once its own explicit formula is used, and Step 3 counts the pairs. The N^{1/2} that Section 16 said was missing is N^{Theta_chi} with Theta_chi = 1/2. Nothing is imported: under the two hypotheses every term has modulus N and their weighted number is log^4 N.

What the bound is not. It is a bound and not an asymptotic. An opposite-ordinate pair would give a non-oscillating term. Its absence has not been proved and is not assumed in the bound. The exponent 4 is what counting gives. On the sum side the analogous exponent went from Fujii's (x log x)^{4/3} to x log^5 x (Bhowmik and Schlage-Puchta, Gallagher's lemma with Cramer's and Saffari-Vaughan's mean squares under RH) and to x log^3 x (Languasco and Zaccagnini, the exponential sum with weight e^{-n/N} and their Lemma 1), and that route should transfer: the pair part of T(N) is integral_0^1 |S(alpha)|^2 conj(X(alpha)) d alpha with X(alpha) = sum_{k <= N} chi(k) e(k alpha), peaked at alpha = 1/3 and 2/3 rather than at 0, and the cross term Im(S' conj(S_chi)) there is the Wronskian in Fourier form. Not carried out; the exponent is not what (T) is about.

18. Theorem B: the lower bound by the zeros of L(s, chi)

This is the payoff Section 12 named: a lower bound on E(N), not the upper bound the original RH strategy would need.

Theorem B. For every eps > 0, E(N) = Omega(N^{1 + 2 Theta_chi - eps}). With Chou, Haag, Huryn and Ledoan's Theorem 2, E(N) = Omega(N^{1 + 2 max(Theta, Theta_chi) - eps}).

Proof. E(N) = 2 sum_{k <= N} e(N, k)^2 >= 2 T(N)^2 / N by Cauchy-Schwarz, since sum_{k <= N} chi(k)^2 <= N. If Theta_chi <= Theta the claim is their Theorem 2, E(N) = Omega(N^{1 + 2 Theta - eps}), cited and not re-derived here; it is the same Cauchy-Schwarz step applied to the unweighted k-sum, whose main term is N (psi(N) - N) by their Section 3. Otherwise Theta < Theta_chi <= 1. The Mellin transform of I(x) = sum_{m <= x} Lambda(m) chi(m) (x/2 - m) is, for Re s > 1,

integral_1^infinity I(x) x^{-s-2} dx = -(L'/L)(s, chi) (1 - s) / (2 s (s + 1)),

which has a pole at every zero rho' of L(s, chi), the factor (1 - rho')/(2 rho' (rho' + 1)) being non-zero. For every eps' > 0 the definition of supremum supplies a zero with Re rho' > Theta_chi - eps'; no attained supremum is required. If I(x) were O(x^{1 + Theta_chi - eps'}) the left side would converge absolutely and be analytic in Re s > Theta_chi - eps', where the right side has a pole; so I(x) = Omega(x^{1 + Theta_chi - eps'}) for every eps' > 0. Only absolute convergence implies analyticity is used, so no sign condition on the coefficients is needed; this is the direction of Landau's oscillation theorem that does not require one (Ingham, ch. V; Montgomery-Vaughan, section 15.1). Put eps' = min(eps, 1 - Theta)/2. The preceding contradiction in fact proves I is not O of that power. For N <= x < N+1, I(x)-I(N) = (x-N)P(N)/2 = O(N), negligible since 1+Theta_chi-eps' > 1. Thus |I(N)|/N^{1+Theta_chi-eps'} is unbounded on the integers. Along an unbounded sequence with |I(N)| >= N^{1 + Theta_chi - eps'}, (E3), Section 15 and Theorem A give |T(N)| >= N^{1 + Theta_chi - eps'} - O(N^{Theta + Theta_chi} log^4 N + N log N)

= (1/2) N^{1 + Theta_chi - eps'}, because Theta + Theta_chi < 1 + Theta_chi - eps'.

Hence E(N) >= (1/2) N^{1 + 2 Theta_chi - 2 eps'} >= (1/2) N^{1 + 2 Theta_chi - eps}. QED.

The conclusion is along that sequence, not an eventual lower bound for all N.

What it says and does not say. Theorem B is unconditional, and it is informative only if RH fails for L(s, chi): with Theta_chi = 1/2 it gives Omega(N^{2 - eps}), below the Omega(N^2 (log log log N)^2) their Theorem 2 gets unconditionally from Littlewood's oscillation, and below the N^2 log^2 N their conjecture predicts. That is the shape of their Theorem 2, informative only if RH fails for zeta. The statement is: the pair error is bounded below by the zeros of L(s, chi_3) as it is by those of zeta, through the same mechanism, the k-mean of e weighted by a character in place of 1. It is not evidence about RH (docs/08) and it says nothing about the upper bound E(N) = O(N^2 log^2 N), which is the direction the original objective needed. Whether the same holds for the characters of every modulus is a thread, not pursued: (E0) is special to modulus 3; for a general q the pair part of sum_k chi(k) e(N, k) is a bilinear antisymmetric form in the psi(x, psi_j) over the characters psi_j mod q, to which Step 2 applies pairwise, but its main term would have to be derived afresh.

19. Placement: known, consequence, or additional

• Is (T) a known estimate? Not in what was read. The sum-side average sum_{n <= N} r_2(n), r_2(n) = sum_{a + b = n} Lambda(a) Lambda(b), has this structure: main term N^2/2, zero sum -2 sum_rho N^{rho + 1} / (rho (rho + 1)), and an error term O((N log N)^{4/3}) under RH (Fujii, Acta Arith. 58 (1991) 173-179; Proc. Japan Acad. 67 (1991) 248-252 and 278-283), O(N log^5 N) under RH with Omega(N log log N) unconditionally (Bhowmik and Schlage-Puchta, Nagoya Math. J. 200 (2010) 27-33), O(N log^3 N) under RH (Languasco and Zaccagnini, Proc. AMS 140 (2012) 795-804, Theorem 1, from Theorem 2 there, the e^{-n/N}-weighted sum, and Languasco-Perelli's integral_{-xi}^{xi} |S(alpha) - 1/z|^2 << N xi log^2 N under RH; reproved by Goldston and Yang, arXiv:1601.06902, Theorem 1, by Bhowmik and Schlage-Puchta's method with an average over the length), and O(N) for the Cesaro average of order 1 under RH (Goldston and Yang, Theorem 2). Granville (Funct. Approx. 37 (2007) 159-173; corrigendum 38 (2008) 235-237) showed that sum_{n <= x} (G(n) - J(n)) << x^{3/2 + o(1)} is equivalent to RH. All of these are the sum side a + b = n averaged over n; none is the difference side, and none carries a character on the difference. With normalized weight (1-n/N)^k/Gamma(k+1), the sum-side quadratic term is sum_{rho, rho'} N^{rho + rho'} Gamma(rho) Gamma(rho') / Gamma(rho + rho' + k + 1) (Languasco and Zaccagnini, Forum Math. 27 (2015) 1945-1960, Theorem 1, unconditional with a Cesaro weight of order k > 1) carries Gamma factors. Their double series converges absolutely for k > 1/2, but the proved formula requires k > 1. The difference-side term of Step 3 carries 1/(rho (rho + rho')) and no Gamma factor, which is why the near-coincidences gamma' = -gamma are the terms to control and why (F5) is needed unconditionally.
• Primes in progressions. Bhowmik, Halupczok, Matsumoto and Suzuki (Mathematika 65 (2019) 57-97, arXiv:1704.06103) treat S(x; q, a, b) = sum_{n <= x} sum_{l + m = n, l = a, m = b (q)} Lambda(l) Lambda(m): Theorem 1(1), S = x^2 / (2 phi(q)^2) + O(x^{1 + B_q}) unconditionally with B_q the supremum of Re over the zeros of every L(s, chi) mod q; Theorem 2, the explicit formula with those zeros and error O(x^{2 B_q^*} (log qx)^5); Theorem 1(2), a converse under DZC and nonvanishing character coefficients, with an additional B_q=1 possibility unless a=b; Theorem 3(2), a converse requiring an attained rightmost zero belonging to a unique character with squarefree conductor coprime to the target residue. Their double sums pair zeros of two different L-functions and carry Gamma factors. Theorem A is a difference-side bound for the pair (zeta, L(s, chi_3)); after subtracting I, its quadratic-zero remainder is closer to their Theorem 2 than to the error after subtracting only the smooth leading term in Theorem 1(1). Theorem B runs the error-versus-zeros reasoning in the reverse direction: there an error term bounds the zeros, here the zeros bound the error from below. Suzuki (Int. J. Number Theory 13 (2017), arXiv:1504.01967) is the earlier mean value in progressions. Read: statements, introductions, and the Section 2 lemmas of BHMS; the whole of Goldston and Yang; Sections 1 and 2 of Languasco and Zaccagnini.
• Is (T) a consequence of known estimates? Under RH for zeta and for L(s, chi_3), yes, by Theorem A, and the proof uses nothing beyond (F1) to (F5): no cancellation is assumed beyond |N^{rho + rho'}| = N. Unconditionally, no: what is known about Theta and Theta_chi (zero-density estimates and the zero-free region) does not give Theta + Theta_chi < 2, so the case Theta + Theta_chi = 2 remains possible. The proof establishes the implication from the two RH assumptions to (T). No converse is proved. Granville's converse on the sum side rests on the main term N^2/2 being smooth, and the difference-side main term I(N) is itself a zero sum.
• What is additional. Nothing, conditionally. The one estimate the whole of (T) rests on is J(N) = sum_{m <= N} Lambda(m) chi(m) (psi(m) - m) = O(N^{1+eps}) of Section 16, a double sum over the zeros of zeta and of L(s, chi_3) with coefficient 1/(rho (rho + rho')).

Provenance and status, kept apart. (E0) to (E5) are identities, checked as such. Section 15 is unconditional and uses PNT and the contour estimate detailed in the referee report. Theorem A is proved under no hypothesis and its interesting case is conditional on two Riemann hypotheses; Theorem B is unconditional. All are original to this hunt in the sense of CLAUDE.md, produced here from the record; none is claimed novel: the search covered the papers named above, read in the parts stated, and a web search of 2026-09-06 for a character-weighted difference-side pair sum, which returned nothing relevant. The proofs are written out above with the standard lemmas cited, corrected and independently audited in REFEREE.md. They are not kernel-checked, so on the certainty ladder they stand as written proofs on cited lemmas, below anything rigor.py or the Lean arm can say. The referee checked the relevant Montgomery-Vaughan proofs; this does not establish novelty or constitute external mathematical endorsement. Relevance to the original objective: no upper bound has been proved. Section 12's route was to a lower bound, and Theorem B is that lower bound.

Fourth pass, 2026-09-20: signed-cancellation package and exact p = 2 pairing (no new bound)

No new RH-strength bound is established in this pass. The earlier isolation step remains unresolved and the proposed finer Sigma_2 estimates are not accepted; the justified Sigma_2 baseline stays the elementary O(N log^3 N). The complete record is FINAL_ACCEPTANCE.md (status ATTEMPT_UNRESOLVED under ALIGNMENT.md section 5), the pairing memo MOBIUS_PAIRING.md, and its independent review MOBIUS_PAIRING_REVIEW.md.

What is valid. Inside each Sigma_2 block M_b, the exact head/tail/paired partition M_b = P_b + T_b + H_b + Z_b with Z_b = 0 termwise holds for all N >= 4 by combinatorial partition (ab > Y gate by 2(U+1)K - N >= 3 > 0 for N >= 12), machine-checked with zero rational defect at 20 cutoffs to N = 10000. Under the declared absolute-value majorants, B(N) = (1/48) N log^3 N + O(N log^2 N) and Ptot(N) = (1/192) N log^3 N + O(N log^2 N); boundary totals are Theta(N log^2 N), one log power below. Measured Ptot/B falls 0.87, 0.69, 0.60 at N = 100, 1000, 10000, approaching the limiting 1/4 from above; finite ratios are diagnostics, not the constants. Retired is ONLY this explicit majorant construction, not the true signed sum Sigma_2, not signed pairings generally, and not multi-prime mechanisms. The joint D_N target remains unproved; no accepted estimate here supplies the required full signed cancellation.

Evidence classes, kept apart. Ordinary reviewed derivation plus Class A exact finite rational checks, Class B high-precision non-enclosing numerical checks (not exact), and Class C measured float diagnostics; no kernel check and no novelty claim. Standard identities (Lambda = mu * log, hyperbola partition, Stirling/harmonic expansions) are assembly, not discovery. The conditional smooth zero-mode R_eta stays explicitly hypothetical: a C^1 continuum profile under a hypothetical off-critical zero, lacking prime-power jumps and discrete mu convolution, so it cannot instantiate Sigma_2; its failure to instantiate refutes no hyperbola lemma, and the majorant obstruction assumes no off-critical zero. Review provenance: the first review draft cited an ephemeral scratch script and an invalid Y bound; that statement is retracted in the review and the durable mobius_pairing_independent_check.py (mobius_pairing_independent_check.py) plus results_mobius_pairing_independent.json (results_mobius_pairing_independent.json) are the reproducible record. Superseded errors (A(N) - log(N!) = O(sqrt N), N^{1/4} spectral margin, proved Sig1 main term, sign of the fractional split) are listed as retracted in FINAL_ACCEPTANCE section 6; read those versions as superseded, not current.

Doors of this mechanism. Binding: the absolute-value step |mu| <= 1 with the floor majorant, which discards the mu signs; frozen: p = 2 only, this head/tail/cutoff assembly. The majorant is pinned Theta(N log^3 N) from above and below, so evaluating this fixed explicit majorant more accurately cannot change its proved order; that is a statement about this majorant, not a universal closure of the pairing family. Beating it requires reading what this family discards: mu signs (signed pairing), more primes, or an accepted Sig1 bound.

The doors

UPPER_BOUND.md measures E(N) << N^3 (log N)^{-C} for every fixed C, the classical circle-method level, and states plainly that this is not an improvement over that level; N^{2+eps} is left open there. This section is keyed to that construction (its Sections 1 to 8). The zero-explicit-formula apparatus of Sections 17 to 19 above is folded in only where it bears on the same quantity; it was built for a different one, T(N), and Section 19 already records that its route gives a lower bound on E(N), saying nothing about the upper bound.

1. Active constraints

UPPER_BOUND.md Section 8 already tallies the budget as a sum of six components, not a single objective with one dual, so there is no shadow price to read off; the ranking below is by the exponent gap to the N^{2+eps} target that each named component carries in that table.

Rank 2's term is not confined to the failed N^{2+eps} attempt of Section 6: it is also what the completed bound (1) itself rests on. At Q = floor(L^B) (Section 5), the dominant term in the reached budget is N^3 L^6 / Q from I_Q, and choosing B = C + 8 turns that into N^3 L^{-C-2}, a log-power saving over N^3 and nothing more. The same crude step, bounding the minor-arc fourth moment by sup_{minor arcs}|F_N|^2 times the second moment d_N rather than by a direct estimate of the integral, is both the reason (1) tops out at a log saving and the reason Section 6's square-root-arc attempt tops out N^{3/5-eps} short of the target.

2. The frozen-constant inventory

Every chosen-not-optimized number in the construction, and what relaxing it trades against.

3. The information class

Retuning Q, B, H, y and R_0 all stay inside the data the construction already reads: Dirichlet characters mod q through Siegel-Walfisz, the exponential-sum bound (V), the large sieve (LS), and Ramanujan sums. None of that retuning crosses rank 2's N^{13/5} ceiling, because the text ties that ceiling to a fixed feature of (V) itself, not to any parameter choice made here.

Two doors in the table above require reading more than that:

• Rank 1, (31). The text's own route to it goes through RH for zeta at q = 1 (Section 7: assuming RH, Delta(t) = O(sqrt(t) log^2(2t)) closes U_1 on its own). No unconditional route is exhibited. Closing this door unconditionally means reading an unconditional remainder for psi(t) at RH strength, which is not part of what this construction currently reads at all.
• Rank 2, a stronger minor-arc fourth-moment bound. The text names the obstruction, Vaughan's N^{4/5} Type I/II balance term, but does not name a replacement estimate. Sections 17 to 19 above develop a genuinely different way of reading the same kind of quantity, through the zeros of zeta and of L(s, chi) directly via Perron's formula and a contour shift, rather than through (SW), (V) and (LS). That machinery is carried through in this hunt only for T(N) - I(N), the Wronskian of Section 16, not for E(N)'s upper bound, and Section 19 records that its route bounds E(N) from below, not above. Whether reading the zeros of zeta and L(s, chi) in that way says anything about I_Q or Z is not addressed by either document.

Rank 3, the moments at 2 <= q <= R_0, has no estimate in either regime, so its information class is not observed here: the text does not say whether closing it needs more than (SW), (V) and (LS) already supply, or just a more careful use of them.