Independent review of the Farey repair and signed-mean divisor draft

Date: 2026-09-12. Source base: zeta-lab commit 3b0fc2e3c97b46cc1fde6af0ff3dd856898d95fb.

Primary verdict: conditional on the named inherited inputs. The mathematical derivations reviewed below survive under those inputs. I found no fatal defect in the Farey baseline or signed-mean sections 1–9. The unproved prime-counting estimate remains unproved. This is an independent handwritten audit, not external peer review, a formal proof, or evidence about RH.

The exact reviewed draft hashes, before integration, are:

I read the raw drafts end to end, re-derived the critical implications, read the applicable root instructions and the original correction definition, and used fresh finite checks that import neither the authors' code nor repository mathematical implementations. Settled Theorem B and the old full upper bound were not re-audited. No repository or remote files were changed by this review.

1. Exact surviving claims

The Farey claim is the following upper bound. For integer N >= 16, real 1 <= Q <= N^(2/5), reduced numerators, all prime powers in Lambda, and Q <= q < 2Q with q <= sqrt(N), the block residual has integrated second moment O(N log N) and integrated fourth moment O(N^3 (log N)^9 / Q). The constants are absolute, apart from inherited absolute constants. There is no matching lower bound or optimality claim.

The signed-mean claim is an equivalence for the scalar quantity M_N = 2 |sum_h(r_N(h)-C_N(h))|^2 / N, with the sharp endpoints, infinite singular series, and exact correction at Z = exp(sqrt(log N)): RH is equivalent to M_N = O_epsilon(N^(2+epsilon)) for every positive epsilon. Exceptional data can change with every integer N. This is not an equivalence for the full corrected energy.

The divisor claims are exact identities and their uniform estimates. In the notation of the draft,

\[ L_KR(N)=G(N)+(1+\gamma)N-Q_{N,K},\qquad |Q_{N,K}|\ll N/K, \]

for every integer 1 <= K <= N. Hence L_K R(N) = O(N/K + log N). This does not bound R(N) by itself. The power-mode multiplier is zeta(rho), with error O_rho((N/K)^Re(rho)). Conditional on a zero rho in 0 < Re(rho) < 1, the constructed model preserves positivity, monotonicity, the exact integral normalization, the inherited PNT envelope, and all these scale bounds. It does not preserve the exact arithmetic forcing or prime-power support. At any zero of multiplicity m, exact forcing inversion leaves residue -m/rho in the transform of R.

2. Dependencies and obligation matrix

The Farey main route is:

separation + circle Parseval + Chebyshev -> second moment for F; dyadic reciprocal-totient bound + Parseval -> second moment for the model; these two estimates -> residual second moment; Vaughan + Q <= N^(2/5) -> residual supremum; supremum times second moment -> the fourth moment.

The optional route is:

clipped Gallagher -> short interval sums; reduced Gauss diagonalization + inducing Gauss factor -> primitive weights; primitive interval large sieve + integrated coefficient count -> second moment, with centered principal and omitted-prime-power terms bounded separately.

The signed route is:

uniform signed correction + CHHL first moment + diagonal bound + exact pairs -> exact B_N identity with A_N = O(N log^2 N); von Koch -> forward implication; psi >= 0 + integer-to-real extension + Mellin holomorphy + functional equation -> converse.

The renewal route is:

sum_{d|m} Lambda(d)=log m -> exact divisor identity; unconditional PNT + Laurent constant -> integral -1-gamma; bounded variation quadrature -> exact scale identity and bound; Euler summation -> transfer multiplier; compact integral correction + derivative bound -> monotone model; Mellin convolution + local zero factorization -> residue computation.

The inherited leaves are the Vaughan estimate, primitive interval large sieve, Gallagher lemma, Chebyshev bounds, Mertens product bound, CHHL singular-series first moment, unconditional PNT error, classical RH prime-counting consequence, Euler product and functional equation. Their local locators are those named in the drafts. The correction's signed argument in CORRECTED_RH_BRIDGE.md, sections 2–3, was read directly and its parameter substitution re-derived. No transfer of an old full-energy bound to the new Z is needed or justified by this review.

3. Decisive mathematical checks

For the Farey proof, let m(alpha) count block arcs covering a point. Reduced centers have separation at least 1/(4Q^2) and all relevant centers lie in an interval of length at most 2/(Q sqrt N). Thus m <= 9 with harmless endpoint conventions, and the sum of local |F|^2 integrals is bounded by 9 sum_{n<=N} Lambda(n)^2. The model has total cost at most N sum_{Q<=q<2Q} mu(q)^2/phi(q) = O(N). This independently establishes the entire main second-moment step without any AP variance assertion.

I also reconstructed the inducing Gauss formula. Insert inclusion-exclusion only for primes of q absent from the conductor d. For each resulting divisor e, the sum has modulus q/e, a multiple of d. Summing over primitive periods leaves zero unless q/e=d. The surviving term is exactly mu(q/d) chi*(q/d) tau_d(chi*). If q/d is not squarefree or is not coprime to d, it vanishes. Thus the proposed conductor factor is correct even for nonsquarefree q, and no universal imprimitive 1/Q discount is available for the different AP-variance weights.

For the signed correction, every prime divisor of q lies below the new Z, giving sigma_1(q)/q <= q/phi(q) <= b_Z. The recorded quantitative signed estimate is consequently at most (14/3) N b_Z^4. Since log(2Z) = O(sqrt(log N)), it is O(N log^2 N). Its proof sums in h for fixed N, so changing exceptional data between cutoffs causes no loss of uniformity. The parameter beta >= 3/4 holds eventually from the exceptionality definition; the criterion concerns sufficiently large N.

The mean converse uses the exact inequality

\[ |R(N)|\le \frac{2|B_N|+|A_N|}{N} =\sqrt{2M_N/N}+O(\log^2N). \]

There is no approximate denominator or omitted R(N)^2 term. This is why the reduction does not require an unproved smallness assumption on R.

For renewal, a direct change of variables gives integral_K^N R(N/t) dt = N integral_1^(N/K) R(u) u^(-2) du. The right-endpoint sum is over k=K+1,...,N; changing it to k=K,...,N would be wrong. Total variation bounds the error with these exact endpoints, including jumps. Also -zeta'(s)/(s zeta(s))-1/(s-1) tends to -1-gamma, not -gamma. Substitution gives precisely the sign of Q in (15).

The multiplier can be checked with an explicit remainder. Taking a=1 and the source's integer index K-1 in the NIST DLMF Euler summation formula 25.11.5 (https://dlmf.nist.gov/25.11.E5) gives, for beta=Re(rho)>0,

\[ \sum_{k=1}^K k^{-\rho}-\frac{K^{1-\rho}}{1-\rho} -\zeta(\rho) =\rho\int_{K-1}^{\infty}\frac{\{x\}}{(x+1)^{\rho+1}}dx, \]

whose modulus is at most |rho| K^(-beta)/beta. This proves uniformity down to K=1. The exact source formula and its hypotheses were checked on 2026-09-12. No height-uniform assertion is being made.

The model passes its stated admissibility checks. Its perturbation vanishes on [1,2], while 1 < exp(gamma) < 2; above 2 the baseline derivative is one and the perturbation derivative has modulus at most 1/2. Consequently there is no hidden negative segment near the left boundary. The compact correction has zero weighted integral, and bounded variation makes its divisor transform equal its linear integral term plus O(1). The retained zeta(rho) N^rho term is essential. At a zero it disappears; for a generic complex exponent it does not. The resulting all-scale model therefore gives the stated conditional compatibility result only.

Finally, write zeta(s)=(s-rho)^m g(s) with g(rho) != 0. Then zeta'/zeta = m/(s-rho)+g'/g, so -zeta'/(s zeta)-1/(s-1) has residue -m/rho. Forcing cancellation is not restored by multiplicity. This is a transform calculation, not a new zero-location bound.

4. Defects, severity, and requested revisions

Fatal defects in the audited deductions: none found.

Two ordinary changes should be made before integration:

1. Narrow signed-mean section 11's phrase “Refuted as an inference.” The demonstrated conclusion is that the proposed absolute-value induction is not justified because its coefficient grows with K. The conditional off-critical model cannot supply an unconditional counterexample to an implication equivalent to RH. Suggested sentence: “The proposed absolute-value induction is not justified; its coefficient grows as (19).”
2. Section 10's present-tense statement that the environment lacks mpmath is stale. Say the original diagnostic used only the standard library. That preserves what the computation actually did without making a claim about later environment state.

The Farey small-N concern is already fixed: the reviewed draft says N >= 16. At N=3,q=1, the parameter interval could exceed one period, so the literal “bounded by one full-circle integral” line would need an extra multiplicity constant. This does not affect the present statement.

Appendix B correctly identifies that the displayed residue coefficient has size 1/|Im rho|, not 1/|Im rho|^2. It safely withdraws the proposed absolute-convergence argument without re-opening the separately settled Mellin-pole theorem. No broader conclusion about possible truncated explicit formulas follows from this correction.

5. Fresh bounded checks

Command:

.venv/bin/python hunts/prime_pair_error/frontier_review_checks.py

Raw result: frontier_review_checks.json. Source: frontier_review_checks.py. The run took under one second. Character values came from python-flint and were deliberately converted to floating point; other calculations used 45-digit mpmath arithmetic. These are non-enclosing diagnostics.

Two deliberate faults discriminated the conventions: replacing -1-gamma by -gamma gave exactly N error; changing [1,2) to [1,2] gave a nonzero error of about 0.9923 at even cutoffs. The fixed exponent has |zeta(rho)| about 0.6181, so it checks retention of the nonzero leading multiplier. No zero was searched, constructed, or numerically assumed. These checks do not prove any asymptotic estimate, zero claim, or RH.

6. Exact remaining gap and cheapest useful next step

With K=floor(sqrt N), the proven identity is R(N)=D_N+O(sqrt N). Thus the missing RH-scale statement is exactly |D_N|=O_epsilon(N^(1/2+epsilon)) for every positive epsilon. The integral in D_N reaches beyond N, and the absolute-value coefficient grows. Neither a smaller quadrature constant nor a different fixed split supplies that missing estimate.

The cheapest next action is to integrate the surviving statements with the two scope edits above and retain this specific unresolved inequality. A further numerical sweep is not needed to validate the written algebra and would not address the gap. Any future claimed inverse estimate must add arithmetic information strong enough to handle the zeta multiplier or prove why a particular use of exact forcing has that strength.

7. Separately requested narrow parameter check

After completing the main audit, the parent requested a logical check of proposed repairs to MAJOR_ARC_EXPLICIT.md, section 5. This note checks only the following window and Page-matching implications. It is not a re-audit or acceptance of that document's old full upper bound, conditional lower bound, phase estimates, source explicit formula, or parity extension.

The proposed extra guards

\[ 0<\varepsilon<\kappa,\qquad \kappa+\varepsilon<1/4, \]

together with the existing strict conditions

\[ c_0<\kappa,\quad \kappa+3\varepsilon<\min(\sqrt c,\sqrt b),\quad \kappa+3\varepsilon<1/3, \]

are sufficient for the particular error comparisons identified by the parent. The original kappa+3 epsilon < 1/3 alone does not imply 2 kappa+2 epsilon < 1/2: for example, kappa=.27, epsilon=.01 satisfy the former and violate the latter.

For the quadratic remainder, the displayed principal amplitudes give

\[ \frac{|W|^2}{|H W|}\asymp \frac{|W|}{|H|} \ll \sqrt{\widetilde q}\,e^{-\kappa\sqrt\ell} \le e^{(\varepsilon/2-\kappa)\sqrt\ell}=o(1). \]

Thus epsilon < kappa is more than sufficient. For the integrated model subtraction, the required comparison is kappa+epsilon < c_m; the inherited c_m=1/4-o(1) makes the new strict guard kappa+epsilon < 1/4 sufficient eventually. Strictness matters.

Let A=kappa+3 epsilon. The existing condition A<min(sqrt(c),sqrt(b)) implies c/A>A>kappa+epsilon and b/A>A>kappa+epsilon. Consequently adding the separately Page-limited real-zero error parallel to the c-controlled error preserves negligibility. Page uniqueness does not imply that every other zero satisfies the individual zero-free-region bound: a possible real zero can fail the latter while lying outside the Page region. It must be bounded by Page separately.

The TT-exceptional coexistence argument can be replaced by exclusion. If q_e divides qtilde, both primitive conductors are at most qtilde. The proposed zero betatilde lies inside the Page region with Q=qtilde and the chosen T. Since beta_e > 1-c_0/sqrt(ell) > betatilde, the TT zero lies inside it too and is distinct. This contradicts the stated Page uniqueness. Thus the offending TT term is absent in this case; its unsupported relative o(1) estimate is unnecessary.

One further ordinary notation repair is needed when these estimates are written. With T=qtilde^2 exp(kappa sqrt(ell)) ell^4, one has log(3 qtilde T) <= A sqrt(ell)+4 log(ell)+O(1). Therefore the asserted decay coefficient c/A is only c/A+o(1) at the displayed level; the same holds for b/A. Use an explicit 1+o(1) in the exponent or choose a fixed slightly smaller coefficient. A fixed ell^2 prefactor need not absorb the entire denominator correction. The strict margins above make this repair harmless for the comparisons under review.

The repaired guards leave a nonempty window with kappa just above c_0 and sufficiently small positive epsilon under the parent's stated c_0 <= 1/12 and the existing bounds c_0 <= sqrt(c)/2,sqrt(b)/2.

Integration note from the coordinator

The two requested wording edits were applied. The Farey proof is integrated as FAREY_BASELINE_REPAIR.md; the signed proof as SIGNED_MEAN_RENEWAL.md. Inline mathematics was formatted, source attribution clarified, and the author's finite diagnostic table replaced by a pointer to this independent check. No mathematical claim in sections 1–9 was strengthened. The script's source paths now use those two canonical filenames; rerunning it records the integrated source hashes in frontier_review_checks.json.

The narrow section-5 guards, Page exclusion and exponential slack described in section 7 were applied to MAJOR_ARC_EXPLICIT.md. Its corollary first weakens any larger asserted exponent to one sufficiently close to the threshold, so the chosen midpoint satisfies the new guards. The parity extension retains its existing unreviewed status.