Independent referee report on Part 3

Editorial note, 2026-09-06: the absolute path of the reviewer's worktree was replaced with "an isolated local worktree" to satisfy tests/test_repo_hygiene.py, which bars machine-local paths from tracked files. Nothing else in this report was altered; no finding, verdict or quoted text is affected.

Reviewed on 2026-09-06 against author commit 37c2a53a in an isolated local worktree, including the complete Sections 14 to 19 of RESULTS.md, all of wronskian.py, results_wronskian.json, and tests/test_prime_pair_residue.py. PR #186 is merge 561446e0; Part 3 was a subsequent local commit, later pushed as PR #188. The audit was performed with main through de1ea66 incorporated in an isolated worktree. Earlier numerical records are preserved. The conclusions below are judgments of written proofs, not formal verification or evidence inferred from finite numerical agreement.

Verdicts

No mathematical refutation of A or B was found. There is no outstanding double-zero summation lemma needed for their stated bounds after the details below are supplied. Simplicity, distinct ordinates, zero spacing, an attained supremum, and a global zero-free strip are not assumptions of these proofs.

1. The exact reduction and endpoints

Use the notation of Part 3, with right-continuous psi and P, and integer N >= 2. The four reduced residue cases give (E0). Expanding the two class prefix sums and using the finite summation-by-parts identity gives (E1), including (N-1)/2. Separating pairs incident to a power of 3 gives (E2). Subtracting the singular-series sum gives (E3). Applying the discrete product rule to R(n)P(n) gives (E4), including its displayed Q.

In particular,

W = RP/2 - J + Q,             Q = O(N log N),
T - I = RP/2 - J + O(N log N).

These are finite identities plus an elementary bound. There is no interchange of infinite sums in the reduction. For example the exact left-endpoint symmetric identity is

sum [P(n-1) dR(n) + R(n-1) dP(n)]
    = R(N)P(N) - sum Lambda(n)^2 chi(n) + P(N).

The explanatory symmetric identity in the submitted Section 16 omitted the last P(N). Its omission does not affect the correctly implemented (E4).

Write ell = log(2N). Montgomery and Vaughan's Theorem 12.5, including its proof on pp. 400 to 401, gives the truncated formula for the midpoint function psi_mid, with remainder

O(log x min(1, x/(U <x>)) + (x/U) log^2(xU)),

where <x> is the distance to the nearest other prime power. At an integer m, psi(m) = psi_mid(m) + Lambda(m)/2. Consequently, with the first height U=N, uniformly for 2 <= m <= N,

R(m) = -sum_{|gamma|<=N} m^rho/rho
       + O((m/N) ell^2 + log(2m)).                         (R1)

The constant and trivial-zero terms in that theorem are bounded for m>=2. It applies to arbitrary heights: its proof chooses a nearby good height and then removes the additional zeros, at cost O(m log U/U). Thus a zero with ordinate exactly N is not an unaddressed contour pole. The endpoint half jump is explicitly included above.

Multiplying (R1) by Lambda(m)chi(m) and summing gives

J(N) = -sum_{|gamma|<=N} A_rho(N)/rho + O(N ell^2),
A_rho(N) = sum_{m<=N} Lambda(m)chi(m)m^rho.                 (R2)

Indeed, sum m Lambda(m) <= N psi(N) << N^2 and sum Lambda(m)log(2m) << N ell. This checks all prime-power endpoints, including m=N. All sums exchanged here are finite.

The same book's Theorem 12.10, for the fixed primitive odd character, gives P(N) << N^Theta_chi ell^2; (R1) gives R(N) << N^Theta ell^2. Thus RP << N^(Theta+Theta_chi) ell^4. Source: explicit-formula chapter, theorems and proofs. (https://personal.science.psu.edu/rcv4/personal/Publications/MNTI/16.0_pp_397_418_Explicit_formulae.pdf)

2. Perron with the moving weight

Fix a zeta zero rho=beta+i gamma, |gamma|<=N, and set c=1+beta+1/log N. The absolutely convergent Dirichlet series on this line is -L'/L(w-rho,chi). Choose V_rho in [2N,2N+1] such that each of V_rho-gamma and -V_rho-gamma is at distance at least a/ell from every L-zero ordinate, with a fixed sufficiently small a>0. Only O(ell) zeros can exclude portions of this unit interval; their excluded lengths total less than one. Both shifted heights have absolute value at least N.

The bound in Corollary 5.3 must be applied to all coefficients, including those with m>N. On N/2<m<2N, |a_m| <= (2N)^beta log(2N); hence its near-endpoint sum, with m=N excluded, is at most

C N^beta ell sum_{1<=h<=N} min(1, N/(V_rho h))
    << N^beta ell^2.

Its remaining term is at most

C (1+N^c)/V_rho sum_{m>=1} Lambda(m)m^(beta-c)
    << N^beta ell,

since beta-c=-1-1/log N. Perron returns the half-weighted endpoint; restoring the full m=N term costs O(N^beta ell). These estimates are uniform in beta and gamma; taking absolute values of m^(i gamma) costs nothing. This supplies the calculation compressed in submitted Step 2. Partial summation against a pointwise error for P would introduce a factor |rho| and would not justify the claimed uniform remainder. Source: Corollary 5.3 and its preceding endpoint convention. (https://personal.science.psu.edu/rcv4/personal/Publications/MNTI/09.0_pp_137_167_Dirichlet_series_II.pdf)

Shift this finite contour to Re w=-delta, where delta is either 1/8 or 3/16, chosen with |beta+delta-1|>=1/40. The two forbidden intervals are disjoint. The crossed poles and their contributions are exactly

w=rho+rho':   -N^(rho+rho')/(rho+rho'), |gamma+gamma'|<V_rho;
w=0:         -L'/L(-rho,chi);
w=rho-1:     -N^(rho-1)/(rho-1), if beta>1-delta.

Multiplicity is included. There is no principal-character pole. No pole coalesces with w=0, since the nontrivial-zero real parts are positive and zeta has no real zero in (0,1).

The submitted (F3) needs a qualification: a local expansion using only nontrivial zeros is not uniformly bounded near a trivial zero. For example L'/L(s,chi_3) has a pole at s=-1. Use that expansion at large |Im s|, and keep the trivial poles explicitly or stay a fixed distance from them at bounded height. The finite contour just chosen does stay that distance away. The book's printed Lemma 12.6 also needs this bounded-height qualification if its summation is understood to include only nontrivial zeros.

On the horizontal sides the local expansion gives O(ell^2), because there are O(ell) nearby zeros and each denominator is at least a/ell. The functional equation gives the same or better bound further left. Their combined integral is

O(ell^2/V_rho integral_{-delta}^c N^sigma d sigma)
    = O(N^beta ell).

On the vertical side the real-part separation from all nontrivial zeros is at least delta; the chosen separation from the trivial zero deals with bounded heights. Thus the logarithmic derivative is O(ell) and this side contributes O(N^(-delta) ell^2). The classical zeta zero-free region and reflection give beta >> 1/ell. Applying the local expansion at -rho therefore bounds its residue by O(ell^2). The possible trivial residue is O(N^(beta-1)/|gamma|); the finitely many low zeta zeros cause no problem. We obtain, uniformly,

A_rho(N) = -sum_{rho' in D_rho} N^(rho+rho')/(rho+rho')
            + O(N^beta ell^2),
D_rho = {rho': |gamma+gamma'|<V_rho}.                      (R3)

The local counts, good-height argument and left-half-plane estimates are the elementary contour tools in Montgomery and Vaughan, Lemmas 12.6 to 12.9, with the qualification just stated. No Goldbach theorem supplies (R3).

3. The double-zero kernel, including opposite ordinates

Here is the actual kernel for the quantity requested in the review, rather than just for J. For each fixed rho, apply Perron to -L'/L(z,chi) with the asymmetric heights -V_rho-gamma and V_rho-gamma. They have size between N and 3N+1 and the same good-height separation. The same endpoint estimate gives

P(N) = -sum_{rho' in D_rho} N^rho'/rho' + O(ell^2).        (R4)

For completeness, the asymmetric Perron tails for m!=N are bounded by the sum of the two endpoint estimates C(N/m)^c/(N|log(N/m)|). At m=N the integral need not give exactly one half: the unequal heights can also give a bounded imaginary part. Its entire discrepancy from the full endpoint is O(Lambda(N)), included in O(ell^2). The contour can be shifted to Re z=-1/4; its sole extra residue is the fixed value -L'/L(0,chi), and its vertical integral is O(N^(-1/4)ell^2).

Using the same D_rho in (R3) and (R4), and (R1) at N, gives

J - RP/2 = sum_{|gamma|<=N} sum_{rho' in D_rho}
             N^(rho+rho') K(rho,rho')
           + O(N ell^2 + N^Theta ell^4 + N^Theta_chi ell^4),

K(rho,rho') = 1/[rho(rho+rho')] - 1/(2rho rho')
            = (rho'-rho)/[2rho rho'(rho+rho')].             (R5)

The three errors come respectively from insertion in J, the weighted Perron remainders and (R4), and multiplication of the remainder in (R1) by P(N). In particular no rectangular cutoff or independent infinite limit was silently substituted for the slanted sets D_rho.

For fixed rho, every rho' in D_rho has |gamma'|<=3N+1. Divide them into the two unit intervals with j<=|gamma+gamma'|<j+1. Each bin contains O(ell) zeros, with multiplicity. For j>=1 its contribution to sum 1/|rho+rho'| is O(ell/j). For j=0, even when the ordinates are exact opposites,

|rho+rho'| >= beta+beta' >= beta >> 1/ell.

Thus the central bin is O(ell^2) and all other bins total O(ell sum_{j<=2N+1} 1/j)=O(ell^2). Separately, unit-interval counting gives sum_{|gamma|<=N} 1/|rho| << ell^2 and sum_{|gamma'|<=3N+1} 1/|rho'| << ell^2. Possible low zeros of either fixed function contribute constants. Consequently

sum_{|gamma|<=N} sum_{rho' in D_rho} |K(rho,rho')|
 <= sum 1/(|rho| |rho+rho'|) + (1/2) sum 1/(|rho| |rho'|)
 << ell^4.                                               (R6)

Since beta<=Theta, beta'<=Theta_chi, every numerator has modulus at most N^(Theta+Theta_chi). The errors in (R5) are absorbed because Theta,Theta_chi>=1/2. Equations (R5), (R6), (E3) and (E4) prove A with its stated exponent and logarithmic power. There is no epsilon loss. An unattained supremum is still an upper bound on each real part.

This is a finite, growing-height estimate. It neither asserts absolute convergence of an infinite double sum nor uses an exchange of its limits. Under the two RH assumptions beta+beta'=1; the central bin is then even easier. Excluding common or opposite ordinates is unnecessary. The submitted sentence asserting that the double sum has no nonoscillating term must be qualified: absence of opposite ordinates is not established here.

4. The unconditional transfer in B

The original committed Section 18 already separates Theta_chi<=Theta from Theta<Theta_chi. It does not assume Theta<1 in the first case.

In the first case use CHHL Theorem 2 directly, with the requested epsilon. Its exponent is at least the desired one, even if Theta=1. In its proof, the unweighted sum is R(N)(2N+R(N))+O(N log N), and PNT gives R(N)/N -> 0; no uniform zero-free strip is needed. CHHL, Theorem 2 and Section 3. (https://arxiv.org/pdf/2308.14888)

In the second case put B=Theta_chi; now Theta<B<=1 really implies Theta<1. For real x>=1, finite summation gives

I(x) = sum_{m<=x} Lambda(m)chi(m)(x/2-m).

For Re s>1, absolute convergence permits termwise integration: the sum of the absolute integrals is bounded by a constant depending on Re s times sum Lambda(m)m^(-Re s). Direct integration yields

integral_1^infinity I(x)x^(-s-2) dx
   = -(L'/L)(s,chi) (1-s)/(2s(s+1)).                     (R7)

At a zero rho' of multiplicity h, this meromorphic function has residue h(rho'-1)/(2rho'(rho'+1)), which is nonzero: 0<Re rho'<1. There are no coefficients from another L-function to cancel that pole.

Fix epsilon>0 and choose

eta = min(epsilon, 1-Theta)/2 > 0,
alpha = 1+B-eta > 1,
d = alpha-(Theta+B) = 1-Theta-eta >= (1-Theta)/2 > 0.

By the definition of supremum there is a zero with Re rho'>B-eta, regardless of whether a rightmost zero exists. If I(x)=O(x^alpha), the left side of (R7) is holomorphic throughout Re s>B-eta, contradicting this nonzero pole by analytic continuation from Re s>1. Thus I is not O(x^alpha). This argument only proves an absolute-value oscillation, and does not assert Omega_+ or Omega_- separately.

For N<=x<N+1 the exact identity is I(x)-I(N)=(x-N)P(N)/2=O(N), by Chebyshev. Since alpha>1, an O(N^alpha) bound on integers would imply the contradicted real bound. Therefore the ratio |I(N)|/N^alpha is unbounded on the integers. In particular there is an unbounded sequence with |I(N)|>=N^alpha.

On that sequence A gives |T-I|/N^alpha << N^(-d)ell^4 -> 0. For all sufficiently large members, |T(N)| >= N^alpha/2. Finally

T(N)^2 <= (sum_{k<=N} chi(k)^2) E(N)/2 <= N E(N)/2,
E(N) >= N^(2alpha-1)/2 >= N^(1+2B-epsilon)/2.

This is an Omega bound along an unbounded sequence, not an eventual lower bound for every integer. The sequence and thresholds may depend on epsilon. This also completes the transfer when B=1 but Theta<1.

5. The auxiliary constant

The divisor expansion must include 1[2|k]: S(k)=2C2 1[2|k] sum_{d|k, d odd squarefree} g(d). Finite interchange with the triangular weight gives the submitted formula for H. Its Dirichlet series, initially absolutely convergent for Re s>1, is

D(s) = -2^(1-s) C2 L(s,chi)L(1+s,chi)K(s),
K(s) = (1+2^(-1-s))
       product_{p>3} (1+chi(p)/((p-2)p^s))(1-chi(p)/p^(1+s)).

The local errors from 1 are O(p^(-2-Re s)+p^(-2-2Re s)), so the product converges absolutely and locally uniformly for Re s>-1/2. The Riesz integral D(s)N^(s+1)/(s(s+1)) therefore shifts to Re s=-1/4, crossing only the pole of its kernel at zero. The submitted proof bounded only the first L-factor. The missing bound is

L(-1/4+it,chi) << (1+|t|)^(3/4),
L(3/4+it,chi) <<_nu (1+|t|)^(1/8+nu),   0<nu<1/8.

The functional equation and convexity give these estimates. Hence the integrand on the new vertical line is O_nu(N^(3/4)(1+|t|)^(-9/8+nu)), which is integrable. The horizontal integrals vanish as their height grows, for each fixed N, by the same bounds uniformly across the strip. This proves the claimed H=c_H N+O(N^(3/4)). The extra nu concerns height growth only and does not become an epsilon loss in the power of N.

Using L(0,chi)=1/3 gives

c_H = -(2C2/3) product_{p>3}(1+chi(p)/(p-2)) < 0.

The prime-ordered product converges by PNT in the two classes, or by factoring out L(1,chi). The fixed-character PNT also gives, for some c>0,

P(N)/2 + X_chi(N) = O(N log N exp(-c sqrt(log N))) = o(N).

To bound the exceptional sum uniformly, split its arguments below sqrt N and above it; apply Chebyshev to the former and PNT to the latter, reducing c if needed. The individual bracket in (E2) is bounded by three, not two, times max_{x<=N}|P(x)|. Thus the auxiliary coefficient is -c_H>0.

The independent recomputation at the existing cutoff N=200000 used a separate integer factor sieve, enumeration of prime powers, and mpmath at 40 decimal digits, without importing probe, residue, or wronskian:

The last three entries explain the spot-check discrepancy exactly. They agree with the existing JSON at its floating precision. In that JSON, G_infinity and c_H were evaluated with primes only through 2000000: the independently reproduced latter value is -0.3130624168166045. Neither field is an exact limit or comes with a tail estimate.

There is also an absolutely convergent way to check the limiting constant. Since L(1,chi)=pi/(3 sqrt(3)), multiplying the Euler factors gives

c_H = -pi/(4 sqrt(3)) product_{p>3, p=2 mod 3}(1-4/(p-1)^2).                (R8)

For primes p=1 mod 3 the factors cancel exactly. At the same existing prime cutoff 2000000, (R8) gives -0.3130478575251646. If c_cut denotes this truncated value, positivity of all factors and the elementary bound 1-product(1-a_p) <= sum a_p give

c_cut <= c_H <= c_cut + |c_cut| 4/(2000000-1).

The mathematical tail allowance is less than 0.000000626097; a conservative numerical interval is -0.313047858 < c_H < -0.313047231. The displayed decimals are high-precision numerical evaluations, not outward-rounded machine enclosures. This places the magnitude near 0.31305; the original 0.3131 was a coarse approximation. No historical JSON value is overwritten. The asymptotic is established by the contour argument, not by agreement at these cutoffs.

Neither that asymptotic nor the five values of W/N excludes W=c_H N+o(N). Therefore the asserted optimality of O(N) for T-I does not follow and is withdrawn.

6. Literature, originality, and relevance

The sources actually needed for A are the classical explicit formulas, Perron with remainder, local zero counting, the functional equation and the classical zero-free region. Those have been checked in Montgomery and Vaughan's author-hosted chapters, not inferred from citations to a Goldbach paper. B additionally uses CHHL's Theorem 2; its full proof in Section 3 was checked. CHHL appeared online in 2025; the journal citation is Journal of Number Theory 278 (2026), 422 to 450. Publisher record. (https://www.sciencedirect.com/science/article/pii/S0022314X25001453)

BHMS Theorem 1(1) concerns a sum-side counting function with error x^(1+B_q) after subtracting only its smooth leading term. Its Theorem 2 subtracts the single-zero terms too and has error x^(2B_q^*)log^5(qx). The second is the closer structural comparison to A, which has already removed I. Neither theorem proves this difference-side kernel estimate. Their Theorem 1(2) uses DZC and nonvanishing character coefficients and retains a B_q=1 alternative unless a=b. Their Theorem 3(2) instead requires an attained rightmost zero belonging to a unique character, with squarefree conductor coprime to the target residue. These hypotheses cannot be compressed to just DZC or transferred to (T). BHMS, precise theorem statements and preliminary lemmas. (https://arxiv.org/pdf/1704.06103)

Languasco and Zaccagnini's 2012 Theorems 1 and 2 and their proofs concern sum-side averages and exponential weighting. Their N log^3 N estimate is on RH and is not an estimate for J. Theorems, lemmas, and proof. (https://arxiv.org/pdf/1011.3198) Goldston and Yang's Theorems 1 and 2 concern the same sum-side average and its order-one Cesaro version. Their proof uses averaged mean-square bounds for the prime-counting remainder. The arithmetic constraint and the Gamma-factor double-zero kernels in these papers differ from (R5). Goldston and Yang, including Section 7. (https://arxiv.org/pdf/1601.06902)

There is a further concrete citation correction: Languasco and Zaccagnini's 2015 Theorem 1 proves the Cesaro formula for k>1. The threshold k>1/2 is only for absolute convergence of its double series. The authors explicitly distinguish these in the paragraph following the theorem; the submitted Section 19 conflated them. Goldston and Yang's displayed (2.1) also omits the factor 2 on its single-zero term; their (1.4), Lemma 2 and proof retain it, as does Languasco and Zaccagnini's 2012 Theorem 1. Neither display is used to import an unproved identity into this review. Cesaro theorem and its stated limitation. (https://arxiv.org/pdf/1206.0251)

The historical survey is useful context, not additional proof of A or a converse. A conditional implication cannot establish an equivalence. This review makes no novelty claim: the named papers and targeted searches for the character-weighted prime-difference statistic did not identify a prior statement of A or B, but that search is not exhaustive. The derivation's provenance in this hunt and priority in the literature are separate matters.

The relevance is a lower-bound obstruction for E, not a new upper bound on E and not evidence for RH. These distinctions do not diminish the valid finite identities, numerical record, or the repaired analytic bounds.

7. Strongest surviving statement and disposition

For the primitive real nonprincipal character modulo 3, with Theta and Theta_chi the zero-real-part suprema and N tending through the positive integers, the repaired proof gives unconditionally

W(N), T(N)-I(N) = O(N^(Theta+Theta_chi) log^4 N),
E(N) = Omega(N^(1+2 max(Theta,Theta_chi)-epsilon)) for every epsilon>0.

Here the first line bounds each quantity separately; the second is along an unbounded sequence. Under RH for both functions the first line specializes to T(N)=I(N)+O(N log^4 N). CHHL's unconditional Omega(N^2 (log log log N)^2) remains an additional surviving bound.

Recommendation: retain A and B with the repairs and qualifications above; retain the numerical results unchanged; remove the unproved equivalence, optimality, and noncoincidence assertions. This is a completed independent written-proof audit, not an external mathematical endorsement or a kernel-checked result. No new experiment or modulus expansion is proposed.

Validation record

The 27 tests in tests/test_prime_pair_residue.py pass, including the four new assertions against independently recomputed values at the existing cutoff. Fourteen document-numbering, door and static hunt-discipline checks also pass. scripts/make_context.py --check reports the generated index current; git diff --check passes. results.json, results_residue.json and results_wronskian.json are byte-for-byte unchanged from the author's commit. The arithmetic backend check reports python-flint with both backends available.

The broader fast baseline run terminated with SIGTERM (exit 143) before completion. Its partial progress included a failure marker but no completed failure report. A separate broader hunt check was also terminated. This is not a green whole-repository baseline; no unrelated repair was attempted. The numerical checks establish the reported finite evaluations and identities, not the zero-sum estimates or the asymptotic assertions.