Author response to THEOREM_B_FORMULA_REVIEW.md

• Artifact: THEOREM_B_FORMULA_REPAIR.md, with formula_repair_checks.py and results_formula_repair_checks.json.
• Reviewed form: 95ae36b9c09c0cf28844e40e50f7fceac2f95cfb. This response describes the revision committed after it.
• The review file and the reviewer's own checks (formula_review_checks.py, results_formula_review_checks.json) were not edited, not re-run and not committed by the author.
• Review record: original review at 95ae36b; targeted follow-up review at 3006ef2 (appended to the review file), which cleared the core formula (Proposition 3) and the conditional Theorem 4 and raised residual findings F1 to F3 in section 10. The three exact edits answering F1 to F3 (closeout at the end of this file) were checked by the coordinator afterwards; no further reviewer run has occurred on them.

Summary

Load-bearing formula: unchanged. The coefficient c(rho') = (rho'-1)/(2 rho'(rho'+1)), the remainder shape C_1 x^2 T^{-1} log^2(xT) + 2 C_1 x log x + C_2, and every numerical check value are as in 95ae36b. What changed is the index set, multiplicity notation, thresholds, provenance and the scope of several claims.

Source verification for (F1)

The reviewer could not reach Montgomery and Vaughan and marked (F1) as resting on the RESULTS.md section 17 paraphrase. For this revision the primary statement was read: MNT I Theorem 12.10 (chapter 12, "Explicit formulae", pp. 397 to 418, author-hosted copy at Penn State, read 2026-09-20). Checked against what Proposition 3 needs:

1. Hypotheses. c > 1 constant, x >= c, T >= 2, chi primitive mod q > 1. No restriction T <= x and no admissible-height condition. The implied constant depends on c only, so the bound is uniform in x and T.
2. Normalization. The theorem is for the midpoint form psi_0(x, chi); the half-jump at prime powers is absorbed as the file already said.
3. Index set and multiplicity. The sum runs over nontrivial zeros with |gamma| <= T, inclusive of any real zero, each counted with multiplicity.
4. Trivial zeros. For odd chi (kappa = 1) the trivial-zero terms are as used in section 5.
5. Error term. The printed error carries min(1, x/(T<x>)) and log^2(3xT). Using min(1, .) <= 1, log^2(3xT) <= 4 log^2(xT) for xT >= 3, and the odd-character collapse of the log term gives exactly the shape Proposition 3 uses.

Consequence: Proposition 3 does not need to be made conditional on (F1), and neither its coefficient nor its remainder shape changes. No numerical value of C_1 is claimed, so the remainder is shape-explicit and not effective.

(F2-low) was checked in the same way against chapter 14 (Theorem 14.5, Corollary 14.6). One gap remains and is stated in the artifact: the bound S(T, chi) << log(q(T+2)) was not located in the extracted chapter text and is cited from the standard literature.

R1. Proposition 3 drops a possible real zero

Accepted. The reviewer is right that the proof carried |gamma'| <= T and the statement narrowed to 0 < |gamma'| <= T without justification; a real zero would contribute a term of order up to x^2 that E does not absorb.

Change: yes. Both remedies were applied, with the inclusive statement as the load-bearing one:

• Proposition 3, section 6 Steps 1 to 3 and C_2 now use sum_{|gamma'| <= T} m(rho') ..., inclusive, multiplicity explicit, matching (F1) as verified above.
• New Lemma 1 (section 1): L(s, chi_3) > 0 for real s > 0, proved by grouping the series as sum_k [(3k+1)^{-s} - (3k+2)^{-s}], each bracket positive, giving L(s) >= 1 - 2^{-s} > 0, with the grouping justified by Dirichlet's test and analytic continuation. For chi_3 the inclusive and exclusive index sets therefore coincide, but Proposition 3 does not depend on Lemma 1.
• rho' = 0 and rho' = -1 are excluded from C_2 as the reviewer noted (L(0) = 1/3, -1 is trivial).
• Check C7 illustrates Lemma 1 numerically against a tail bound proved on the spot. It is labelled NOT PROOF.

Surviving claim. Proposition 3 as stated, with an inclusive, multiplicity-weighted zero sum, derived from the verified (F1); and, for chi_3, no real zero in (0, 1).

R2. Divergence cited from an upper-bound-only fact

Accepted. An upper bound on zero density is consistent with convergence and cannot give sum |c(rho')| = infinity.

Change: yes. Defect 1 (section 2) now uses (F2-low), the two-sided count from MNT I Theorem 14.5 and Corollary 14.6, to get N(2T, chi) - N(T, chi) >> T log T; with the per-zero bound |c(rho')| >= 1/(7 |gamma'|) for |gamma'| >= 1 (re-derived by the reviewer and unchanged) a dyadic sum gives sum_{|gamma'| <= T} |c| >> log^2 T, hence divergence. Wherever only an upper bound is used (convergence of C_2, the competitor bound in section 7(b)) the text now cites (F2-up) and says an upper bound suffices. "Asymptotically" survives only where both bounds are cited (section 10, point 1: bounded above and below by constant multiples of log^2 T).

Surviving claim. sum |c(rho')| diverges, so the expansion (*) of THEOREM_B_SEQUENCE.md is refuted as stated. The refutation rests on Corollary 14.6 (read in the primary source) plus the classical S(T, chi) bound (cited, page not inspected).

R3. (MISSING) not sufficient as stated

Accepted, and the reviewer's remedy was checked rather than adopted wholesale.

The margin actually needed in step (e) is not 1 - Theta > 0. Step (e) must absorb the Theorem A error O(N^{Theta + Theta_chi} log^4 N) against the signal N^{1 + beta_0}, so the exact condition is

1 + beta_0 > Theta + Theta_chi.

Under (H-ISO), beta_0 = Theta_chi and this reduces to Theta < 1, which is where the original 1 - Theta > 0 came from. Under (MISSING) the witness is only nonextremal, beta_0 >= Theta_chi - eta, and then:

• (H-TRANSFER) (Theta < Theta_chi) gives only Theta < 1 in effect and does not imply the margin for arbitrary eta: with beta_0 = Theta_chi - eta the margin is 1 - Theta - eta, which is negative once eta > 1 - Theta. Adding (H-TRANSFER) to (MISSING), as the review suggested first, is therefore not sufficient on its own.
• A sufficient restriction is eta <= 1 - Theta, since then 1 + beta_0 >= 1 + Theta_chi - eta >= Theta + Theta_chi. The reviewer's Theta <= 1 - 2 eta also suffices but is stronger than needed.
• Alternatively the exact margin can be carried as an explicit hypothesis.

Separately, epsilon_1 = g/2 in step (a) is defined from (H-ISO)'s gap and is meaningless under (MISSING); the epsilon_1 supplied by (MISSING) is used instead, and the truncation error is rebalanced with it.

Change: yes. Section 10 replaces "go through verbatim" with Adaptation 1 (epsilon_1 from (MISSING); surviving main term (3 delta / 4) . 2 m_0 |c_0| x_n^{1 + beta_0} (1 + o(1))) and Adaptation 2 (exact margin, sufficient restriction eta <= 1 - Theta, and an explicit sentence that (H-TRANSFER) does not supply the margin for arbitrary eta).

Surviving claim. Under (MISSING) plus either the stated margin or eta <= 1 - Theta, E(N_n) >> N_n^{1 + 2 Theta_chi - 2 eta} along an infinite sequence. (MISSING) itself is unproved, so this is a statement of what would suffice, not a result.

R4. "Explicit sequence" overclaims

Accepted.

Change: yes. Theorem 4 is retitled from "conditional constructive oscillation" to "conditional oscillation along a closed-form sequence", and now says the sequence is explicit only as a function of the hypothetical witness zero rho_0 supplied by (H-ISO), and that since no zero is exhibited no sequence of real numbers is exhibited. Section 10 now states that (MISSING) is an existence statement, that everything derived from it is existential, and that no constructive or effective claim is made. The two remaining sentences in section 10 that spoke of a "constructive statement" or "recovering constructiveness" were reworded to match.

Surviving claim. Theorem 4 removes the epsilon of Theorem B and gives a closed-form sequence in terms of gamma_0, phi_0, conditionally on (H-ISO) and (H-TRANSFER). Nothing constructive or effective is claimed, conditionally or otherwise.

R5. Clarity bundle

Per instruction only items bearing on truth or hypotheses were acted on.

• (a) Accepted, changed. "Explicit two-parameter remainder" is now "shape-explicit in x and T up to one absolute constant", with a sentence that no value of C_1 is claimed and nothing is effective.
• (b) Accepted, changed. Multiplicity m(rho') is now explicit in (F1), (F2) and the proof of Proposition 3, and (F1)'s multiplicity convention was confirmed in the primary source.
• (c) Accepted, changed. Theorem 4 now reads "for all sufficiently large n", with the threshold stated: x_n >= 4 and T(x_n) >= max(4, |gamma_0|), which hold for n >= n_0(beta_0, gamma_0, g). A matching threshold paragraph was added to section 7.
• (d) Accepted, changed. The + N log N in the step (e) transfer bound is now attributed to (E3) of RESULTS.md section 15, not to Theorem A.
• (e) Accepted, changed. Section 9 no longer calls C5 "the direct numerical statement of Defect 1"; it now says C5 illustrates the size estimate at 21 critical-line ordinates and that divergence is the analytic argument resting on (F2-low). The same paragraph now says C6 compares an error bound with the signal, so it shows the estimate cannot establish resonance and says nothing about the actual error.
• (f) Declined, no change. The source's c_0 = (rho'-1)/(rho'(rho'+1)) and the repair's c = (rho'-1)/(2 rho'(rho'+1)) differ by the factor 2 that a conjugate pair supplies: 2 |c| = |c_0|, so both documents give the same pair sum |c_0| x^{1+beta_0} cos(...). The reviewer confirms the repair is self-consistent. This is a normalization difference with no truth or hypothesis at stake, so under the brief it is left alone.

Additional change not raised by the review

Check C8 was added to answer the review's remaining uncertainty about the 21 inherited ordinates: it confirms |L(1/2 + i gamma)| <= 1.57e-14 at every listed ordinate and compares the count (21) with the Riemann and von Mangoldt prediction to height 60 (22.61). It records that completeness is not proved. Nothing load-bearing depends on the list.

Revised disposition

• Lemma 0, Lemma 1, the section 5 polar structure and Proposition 3: derived, with exact hypotheses (primitive odd chi_3 mod 3; x >= 4, T >= 4; one absolute constant C_1 from (F1)). Proposition 3 is a useful intermediate lemma with exact hypotheses.
• Theorem 4: derived, conditional on (H-ISO) and (H-TRANSFER).
• The unconditional variant in section 10: an unresolved attempt with a missing analytic input, (MISSING), plus the stated margin or eta <= 1 - Theta.
• Section 9 checks: measured, NOT PROOF.
• Nothing here is enclosure-carrying or kernel-checked, and nothing is numerically effective.

Checks run for this revision

formula_repair_checks.py (main venv, about 3 s): C1 to C8 all pass; results in results_formula_repair_checks.json. The reviewer's scripts were not run.

Closeout of follow-up findings F1 to F3 (review of 3006ef2)

All three are in section 10; none touches Proposition 3 or Theorem 4. Each was accepted as the reviewer stated it.

Status after closeout: Proposition 3 is a useful intermediate lemma with exact hypotheses; Theorem 4 stays conditional on (H-ISO) and (H-TRANSFER); the unconditional extension in section 10 remains unresolved, pending (MISSING) plus the margin. The cited-not-inspected S(T, chi) bound caveat stands unchanged and is not needed for the core formula. These edits were checked by the coordinator; no further reviewer run has occurred.