Repair of the sharp-cutoff character-weighted explicit formula

Status: useful intermediate lemma with exact hypotheses, plus a conditional strengthening of Theorem B. Grade: measured and derived, with hypotheses stated. The derivations below are ordinary analytic number theory; the numerical checks in section 9 are floating-point sanity checks and are not proof. Nothing here is kernel-checked and nothing here claims to be.

Review status. The first version of this file (commit 95ae36b) was attacked by an independent reviewer, whose findings are recorded verbatim in THEOREM_B_FORMULA_REVIEW.md. Five substantive objections (R1 to R5) were raised. This revision responds to all five; AUTHOR_RESPONSE.md lists each one, whether it was accepted or rejected, the mathematical reason, and the exact change made. The revision (commit 3006ef2) then received a targeted follow-up review, appended to the same file, which cleared the core formula (Proposition 3) and the conditional Theorem 4 and raised three residual findings F1 to F3, all in section 10. Those three exact edits were made afterwards and checked by the coordinator; no further reviewer run has occurred on them.

Date: 2026-09-20. Original commit: 95ae36b9c09c0cf28844e40e50f7fceac2f95cfb (base f402358c6c3f3c838605e71dd97cb6401a6963f0). Revised the same day after review.

0. Scope

This file repairs one object: the zero expansion of

I(x) = sum_{m <= x} Lambda(m) chi(m) (x/2 - m), chi = chi_3,

asserted at THEOREM_B_SEQUENCE.md section 3 as an absolutely convergent sum over the nontrivial zeros of L(s, chi_3), and identified as defective by FAREY_BASELINE_REPAIR.md Appendix B (2026-09-12).

It does not:

• prove RH, or bear on RH at all (docs/08);
• re-derive or modify Theorem B of RESULTS.md section 18, which is settled and stands unchanged;
• modify THEOREM_B_SEQUENCE.md, which is left as the record of the failed argument;
• reopen the hunt's completed numerical passes.

Landing note, 2026-10-21. The landing added two dated notes to THEOREM_B_SEQUENCE.md: a pointer to this file under its 2026-09-12 scope correction, and a correction in its section 6 about the inherited zero list (see the end of section 8). Its argument and the rest of its text are unchanged.

The new files the author wrote are this one, formula_repair_checks.py and its results JSON, plus AUTHOR_RESPONSE.md. The independent reviewer's record adds THEOREM_B_FORMULA_REVIEW.md, formula_review_checks.py and its results JSON, which the author did not edit. At landing one docstring line of formula_review_checks.py was changed to drop a machine-specific path; the script's header records that change, and nothing else in the reviewer's three files was changed.

What is claimed, in one sentence: the defective expansion is replaced by a truncated formula whose remainder is shape-explicit in x and T up to one absolute constant inherited from (F1), derived from a primary source checked for this revision; that formula supports the resonance argument under an explicitly stated isolation hypothesis, and the exact inequality still missing in the unconditional case is written down in section 10.

"Shape-explicit" is deliberate. C_1 below is the implied constant of Montgomery and Vaughan's Theorem 12.10 at q = 3; it is absolute and finite but no numerical value for it is computed or claimed here, so no bound in this file is numerically effective.

1. Setup and conventions

chi is the primitive real character mod 3, chi(1) = 1, chi(2) = -1, chi(0 mod 3) = 0. It is odd: chi(-1) = chi(2) = -1. Lambda is von Mangoldt.

psi(x, chi) = sum_{m <= x} Lambda(m) chi(m) psi_1(x, chi) = sum_{m <= x} Lambda(m) chi(m) (x - m) = integral_0^x psi(t, chi) dt I(x) = sum_{m <= x} Lambda(m) chi(m) (x/2 - m)

Endpoint convention. Every sum sum_{m <= x} is closed at x. Hence psi(., chi) and I are right-continuous, and I jumps by -Lambda(x) chi(x) x / 2 at each prime power x. Montgomery and Vaughan state the explicit formula for the midpoint normalization psi_0(x, chi) = (psi(x^+, chi) + psi(x^-, chi))/2 (verified against the primary source; see section 8). The two differ by Lambda(x) chi(x) / 2 at prime powers and nowhere else, and |Lambda(x) chi(x)/2| <= (log x)/2, which is inside the C_1 log x term of (F1). The right-continuous form used here is therefore legitimate as an upper bound of the stated shape, which is all Proposition 3 uses. No half-jump correction is applied anywhere in this file, and none is needed. (Step 1 of section 6 integrates psi, and the two normalizations differ on a finite set in any bounded range, so the integral does not see the difference at all; only Step 2 does.)

rho' = beta' + i gamma' runs over the nontrivial zeros of L(s, chi), with multiplicity m(rho'). Theta_chi = sup beta'. By the functional equation zeros pair as rho' <-> 1 - conj(rho'), so Theta_chi >= 1/2. Theta = sup Re rho over the zeros of zeta.

Lemma 1 (no real zeros). L(s, chi_3) > 0 for every real s > 0. In particular L(s, chi_3) has no zero in (0, 1), and every nontrivial zero has gamma' != 0.

Proof. For real s > 0 the partial sums of chi_3(n) are bounded by 1 and n^{-s} decreases to 0, so sum_{n >= 1} chi_3(n) n^{-s} converges by Dirichlet's test; it converges locally uniformly on Re s > 0 and so agrees there with the entire function L(s, chi_3) by analytic continuation from Re s > 1. Because the series converges, any subsequence of its partial sums has the same limit; taking S_{3k} and using chi_3(3k) = 0 gives the grouped form

L(s, chi_3) = sum_{k >= 0} [ (3k+1)^{-s} - (3k+2)^{-s} ].

Each bracket is positive for s > 0 since t -> t^{-s} is strictly decreasing, and the k = 0 bracket equals 1 - 2^{-s}. Hence L(s, chi_3) >= 1 - 2^{-s} > 0. QED

This is used in exactly one place: it lets the truncation |gamma'| <= T in Proposition 3 be written inclusively without ambiguity, because for chi_3 the index sets {|gamma'| <= T} and {0 < |gamma'| <= T} coincide. The reviewer was right that the first version of this file used the two notations interchangeably without saying why they agree. Check C7 in section 9 exhibits the grouped identity numerically. Lemma 1 is not needed for Proposition 3 itself: Proposition 3 is stated with the inclusive index set, which is what (F1) supplies whether or not a real zero exists.

Write

c(rho') = (rho' - 1) / (2 rho' (rho' + 1)).

This is exactly the coefficient appearing in the defective formula (*), which is why the repair below is a repair of that formula rather than a replacement of it.

2. The defect, stated precisely

THEOREM_B_SEQUENCE.md section 3 asserts an absolutely convergent expansion

(*) I(x) = sum_{rho'} m(rho') c(rho') x^{rho' + 1} + (polar terms),

justifying absolute convergence by a dyadic count of zeros. Appendix B of FAREY_BASELINE_REPAIR.md records this as defective. Two distinct things are wrong, and it is worth separating them, because only one is fatal.

Defect 1 (fatal): the coefficient decays like 1/|gamma'|, not 1/|gamma'|^2.

For |gamma'| >= 1, using |rho' - 1| >= |gamma'|, |rho'| <= sqrt(2) |gamma'|, |rho' + 1| <= sqrt(5) |gamma'|,

|c(rho')| >= |gamma'| / (2 sqrt(10) |gamma'|^2) >= 1 / (7 |gamma'|),

and by the Riemann and von Mangoldt zero count for L(s, chi) in its two-sided form (fact (F2-low) below, Montgomery and Vaughan Corollary 14.6 together with the classical bound on S(T, chi)),

N(2T, chi) - N(T, chi) = (T / 2 pi) log(2 q T / pi e) + O(log qT) >> T log T

for T large. Each of those >> T log T zeros has |gamma'| <= 2T, so contributes at least 1/(14T) to sum |c(rho')|, and the dyadic block [T, 2T] contributes >> log T. Summing over log_2 T dyadic blocks,

sum_{0 < |gamma'| <= T} |c(rho')| >> log^2 T -> infinity.

Hence sum_{rho'} |c(rho')| = infinity and (*) is not absolutely convergent. This is the load-bearing error. The numerator rho' - 1 was dropped when the size of the coefficient was estimated; |c(rho')| ~ 1/(2|gamma'|), not ~ 1/|gamma'|^2.

Defect 2 (a real misstatement, but not the fatal one). The dyadic zero count is also misstated in that section. However, the conclusion it was invoked to support, namely sum_{rho'} 1 / |rho'(rho' + 1)| < infinity, is in fact true by the upper bound (F2-up) alone, so repairing the count does not rescue (*). Appendix B lumps these two; separating them is a refinement, and it identifies which one the repair must route around.

Consequence: the sum over zeros must be truncated, and truncation forces an error term that depends on both x and the height T. The whole content of the repair is that the error term can be made small relative to the resonance term, and under what conditions.

Defect 3 (a second, independent failure in the downstream argument). Even granted a truncated formula, THEOREM_B_SEQUENCE.md takes T = x^delta for a small fixed delta. The truncation error below is of size x^2 T^{-1} log^2(xT), which beats the witness term x^{1 + beta_0} only when T >> x^{1 - beta_0 + epsilon}. A fixed small delta fails whenever delta < 1 - beta_0, which is the generic case. The admissible range of T is stated explicitly in Proposition 3 and used in Theorem 4; check C6 in section 9 exhibits the crossover numerically.

3. FROZEN candidate proposition and hypotheses

Frozen before the derivations in sections 4 to 7. Nothing below this line was adjusted to make a later step work; where a derivation forced a change, that is recorded as a correction note rather than a silent edit. No such note was needed.

Hypotheses

(H-ISO). There is a nontrivial zero rho_0 = beta_0 + i gamma_0 of L(s, chi_3) with gamma_0 > 0 and multiplicity m_0 >= 1 such that

beta_c := sup { Re rho' : rho' nontrivial, rho' not in {rho_0, conj(rho_0)} } < beta_0.

Write g := beta_0 - beta_c > 0. Necessarily beta_0 = Theta_chi and the supremum is attained.

(H-ISO) implies GRH fails for L(s, chi_3). Under GRH every nontrivial zero has Re = 1/2, so beta_c = beta_0 and g = 0. This is not a defect of the hypothesis; it is the same regime in which Theorem B is informative at all (RESULTS.md section 18: Theorem B is unconditional but says nothing new unless RH fails for L(s, chi)).

(H-TRANSFER). Theta < Theta_chi. This is the regime in which Theorem A's transfer from I(N) to T(N) has room, and is the same regime RESULTS.md section 18 works in.

Proposition 3 (repaired truncated explicit formula) [candidate]

For all real x >= 4 and all real T >= 4,

I(x) = sum_{|gamma'| <= T} m(rho') c(rho') x^{rho' + 1} + E(x, T),

c(rho') = (rho' - 1) / (2 rho' (rho' + 1)),

with

|E(x, T)| <= C_1 x^2 T^{-1} log^2(xT) + 2 C_1 x log x + C_2,

where C_1 is the absolute implied constant of (F1) at modulus 3 and

C_2 = sum_{rho'} 4 m(rho') / |rho' (rho' + 1)| < infinity.

The index set is inclusive: every nontrivial zero with |gamma'| <= T is summed, counted with its multiplicity m(rho'), which is exactly what the residue computation behind (F1) produces. For chi_3 this set contains no real zero, by Lemma 1, so it coincides with {0 < |gamma'| <= T}; the proposition does not depend on that. Zeros occur in conjugate pairs and the truncation is symmetric, so the sum is real.

T is unrestricted above 4: no admissible-height condition is needed, because (F1) as verified in section 8 holds for every T >= 2 (Montgomery and Vaughan obtain this by choosing a nearby good height T_1 in [T, T+1] and bounding the intervening zeros; the statement the reader gets is uniform in T). All polar contributions (the mu_1 x term at s = 0, the log x + nu term at the s = -1 pole/trivial-zero collision, and the trivial-zero tail) are of size O(x) and are absorbed into the O(x log x) remainder rather than dropped; they are derived separately in section 5 and are available explicitly for any use that needs them.

Theorem 4 (conditional oscillation along a closed-form sequence) [candidate]

Assume (H-ISO) and (H-TRANSFER). Put c_0 = c(rho_0), phi_0 = arg c_0, and

x_n = exp( (2 pi n - phi_0) / gamma_0 ), N_n = floor(x_n).

Then, for all sufficiently large n,

I(x_n) = 2 m_0 |c_0| x_n^{1 + beta_0} ( 1 + O( x_n^{-g/2} log^2 x_n ) ),

and consequently

E(N_n) >> N_n^{1 + 2 Theta_chi}

along that sequence, with no epsilon loss, where x_{n+1} / x_n = exp(2 pi / gamma_0).

"Sufficiently large" is not decoration: the proof needs x_n >= 4 and needs the height T(x_n) = x_n^{1 - beta_0 + g/2} to satisfy T(x_n) >= max(4, |gamma_0|), so that Proposition 3 applies and the witness pair lies inside the truncation. Both hold for every n >= n_0(beta_0, gamma_0, g), and n_0 is computable from those three numbers.

This is stronger than Theorem B (Omega(N^{1 + 2 Theta_chi - eps}) along an unspecified sequence) in two ways: the epsilon is removed, and the sequence is written down. It is weaker in one, and the weakness is decisive: Theorem B is unconditional and this is not. The sequence is explicit only as a function of the hypothetical witness zero rho_0 supplied by (H-ISO); since (H-ISO) is an existence hypothesis and no zero is exhibited, no sequence of real numbers is exhibited either. Section 10 keeps that distinction.

4. Lemma 0: the decomposition

Lemma 0. For all x >= 0, I(x) = psi_1(x, chi) - (x/2) psi(x, chi).

Proof. Both sides are finite sums over m <= x. Writing S(x) = sum_{m <= x} Lambda chi(m) m,

I(x) = (x/2) psi(x, chi) - S(x), psi_1(x, chi) = x psi(x, chi) - S(x).

Subtract: psi_1 - (x/2) psi = x psi - S - (x/2) psi = (x/2) psi - S = I. QED

This identity is exact, carries no error, and respects the endpoint convention of section 1 since every term is the same closed sum. It is the whole reason the repair is cheap: I is a fixed linear combination of the first Riesz mean and the sharp sum, and (F1) covers both.

Check C1 in section 9 verifies it at every integer x <= 200 to 2.4e-58.

5. Polar structure, including the s = -1 collision

This section derives, rather than inherits, the polar terms. They do not enter the bound in Proposition 3 beyond O(x log x), but the brief requires them derived and they correct a specific error in the source document.

For Re s > 1, RESULTS.md section 18 and REFEREE.md (R7) give

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

The Mellin pair sends this to residues of G(s) x^{s+1}.

s = 1: no pole. L(1, chi) != 0 and L has no pole at 1 for non-principal chi, so L'/L is analytic there; the kernel factor (1 - s) vanishes. There is no x^2 main term, consistent with psi(x, chi) having no main term.

s = 0: simple pole. L(0, chi_3) = 1/3 != 0, so L'/L is analytic at 0 and the pole comes from the kernel's 1/s. Residue = mu_1 x with

mu_1 = -(1/2) (L'/L)(0, chi_3) = -0.474099413336...

(check C4, gap 2.6e-30 against the closed form).

s = -1: a DOUBLE pole, not a simple one. This is the correction. Because chi is odd, the completed function carries Gamma((s+1)/2), whose poles at s = -1, -3, -5, ... force trivial zeros of L(s, chi) at the odd negative integers, not the even ones. At s = -1 that simple trivial zero of L meets the kernel's own 1/(s+1), and L'/L contributes a further simple pole. The result is a double pole, so the residue is not a constant but

log x + nu.

Derivation. Put u = s + 1 and h(-1) = lim_{s -> -1} [ (L'/L)(s, chi) - 1/(s+1) ], so (L'/L)(s, chi) = 1/u + h(-1) + O(u). The kernel expands as

(1 - s) / (2 s (s + 1)) = (2 - u) / (2 u (u - 1)) = ( -1 - u/2 + O(u^2) ) / u,

and x^{s+1} = x^u = 1 + u log x + O(u^2). Multiplying,

G(s) x^{s+1} = 1/u^2 + ( log x + 1/2 + h(-1) ) / u + O(1),

so the double-pole coefficient is 1 and

nu = 1/2 + h(-1).

Closed form for nu. For chi_3 (real, primitive, odd, q = 3, root number 1) the completed function Lambda(s) = (q/pi)^{(s+1)/2} Gamma((s+1)/2) L(s, chi) satisfies Lambda(s) = Lambda(1-s). Logarithmic differentiation and the reflection give

(L'/L)(s, chi) = -log(q/pi) - (1/2) digamma((s+1)/2) - (1/2) digamma(1 - s/2) - (L'/L)(1-s, chi).

At s -> -1, digamma(u/2) = -2/u - euler + O(u) supplies the 1/u, and digamma(3/2) = 2 - euler - 2 log 2, so

h(-1) = euler + log(2 pi / 3) - 1 - (L'/L)(2, chi_3), nu = euler + log(2 pi / 3) - 1/2 - (L'/L)(2, chi_3) = 0.643831644563...

Check C3 confirms the double-pole coefficient is 1 to 2.7e-31, the residue linear term equals the closed-form nu to 6.4e-31, and the numeric h(-1) agrees with the functional-equation closed form to 3.0e-16.

The source document records this polar contribution as mu_1 x + mu_0, a constant. That is wrong: the s = -1 term is log x + nu, not a constant. It is harmless to the resonance argument (both are o(x)), but it is inherited elsewhere and should not be.

s = -(2k+1), k >= 1: simple poles at the remaining trivial zeros. Residue

-(k+1) x^{-2k} / (2 k (2k + 1)),

so the whole tail is O(x^{-2}) for x >= 2.

Check C2 confirms the parity: |L(-1)| = 1.5e-62, |L(-3)| = 3.3e-62, while |L(-2)| = 0.2222, and L(0) = 1/3 exactly. Trivial zeros at the odd negatives only, as an odd character requires.

Nontrivial zeros. -L'/L has residue -m(rho') at rho', so the residue of G(s) x^{s+1} there is

m(rho') (1 - rho') / (2 rho' (rho' + 1)) . (-1) . x^{rho'+1} = m(rho') c(rho') x^{rho'+1},

which is the coefficient named in section 1. Multiplicity is carried through unchanged.

6. Proof of Proposition 3

Fact (F1), at modulus 3, right-continuous form, every T >= 2 and every x >= 4, absolute constant C_1, with the index set inclusive and zeros counted with multiplicity:

psi(x, chi) = - sum_{|gamma'| <= T} m(rho') x^{rho'} / rho' + E_0(x, T), |E_0(x, T)| <= C_1 ( x T^{-1} log^2(xT) + log x ).

Section 8 derives this from the primary source (Montgomery and Vaughan Theorem 12.10), which was inspected for this revision; the derivation is three crude majorizations and is written out there rather than assumed. Multiplicity is written explicitly here because the first version of this file carried it in Proposition 3 but suppressed it in this step.

Step 1: integrate. psi(t, chi) = 0 for t < 2, so psi_1(x, chi) = integral_2^x psi(t, chi) dt. Integrating the truncated sum term by term (a finite sum),

integral_2^x t^{rho'} dt / rho' = ( x^{rho'+1} - 2^{rho'+1} ) / ( rho' (rho' + 1) ),

hence

psi_1(x, chi) = - sum_{|gamma'| <= T} m(rho') ( x^{rho'+1} - 2^{rho'+1} ) / ( rho'(rho'+1) ) + E_1, |E_1| <= C_1 ( x^2 log^2(xT) / (2T) + x log x ).

Step 2: the sharp term. Multiply (F1) by -x/2:

-(x/2) psi(x, chi) = (x/2) sum_{|gamma'| <= T} m(rho') x^{rho'} / rho' + E_2, E_2 = -(x/2) E_0(x, T),

so that |E_2| <= (x/2) C_1 ( x T^{-1} log^2(xT) + log x ) = C_1 ( x^2 log^2(xT) / (2T) + (x log x)/2 ).

Step 3: add, using Lemma 0. The coefficient of x^{rho'+1} is

1/(2 rho') - 1/( rho'(rho'+1) ) = ( (rho'+1) - 2 ) / ( 2 rho'(rho'+1) ) = (rho' - 1) / ( 2 rho'(rho'+1) ) = c(rho').

This is the identity that makes the repair a repair: the two routes recombine into exactly the coefficient the defective formula named. (Check C5 verifies the identity at 21 located zeros to 1.2e-63.)

The leftover constant from Step 1 is

B(T) = sum_{|gamma'| <= T} m(rho') 2^{rho'+1} / ( rho'(rho'+1) ),

and since 0 < beta' < 1 gives |2^{rho'+1}| = 2^{beta'+1} <= 4,

|B(T)| <= sum_{rho'} 4 m(rho') / |rho'(rho'+1)| = C_2 < infinity,

uniformly in T. Finiteness: for |gamma'| >= 1, |rho'| >= |gamma'| and |rho' + 1| >= |gamma'|, so the terms are <= 4 m(rho')/gamma'^2, and (F2-up) gives convergence by dyadic summation. For |gamma'| < 1 there are finitely many zeros (F2-up), each with |rho'| >= delta_0 > 0 because L(0, chi_3) = 1/3 != 0 and L is analytic near 0.

Collecting, with E = E_1 + E_2 + B(T),

|E(x, T)| <= C_1 x^2 T^{-1} log^2(xT) + 2 C_1 x log x + C_2.

QED (Proposition 3).

Reality. c(conj(rho')) = conj(c(rho')) and x^{conj(rho')+1} = conj(x^{rho'+1}), and the truncation |gamma'| <= T is symmetric, so the sum is real, as I(x) must be.

Where the polar terms went. They are all O(x) or smaller (section 5) and are absorbed in the 2 C_1 x log x slack. Proposition 3 could instead be stated with mu_1 x, log x + nu and the trivial-zero tail written out and a correspondingly smaller remainder; nothing below needs that, and stating it this way keeps the constant honest.

7. Proof of Theorem 4

Assume (H-ISO). Set epsilon_1 = g/2 and, for each x, the height

T = T(x) = x^{1 - beta_0 + epsilon_1}.

Note beta_0 >= 1/2 (functional equation), so T <= x and log(xT) <= 2 log x for x >= 4.

Threshold. Proposition 3 needs x >= 4 and T(x) >= 4, and step (c) needs the witness pair inside the truncation, that is T(x) >= |gamma_0|. Since 1 - beta_0 + g/2 > 0, both hold for all x >= x_0 with x_0 = x_0(beta_0, gamma_0, g) explicit. Everything below is asserted for x >= x_0 only, and "for all sufficiently large n" in Theorem 4 means x_n >= x_0.

(a) Truncation error. From Proposition 3,

C_1 x^2 T^{-1} log^2(xT) <= 4 C_1 x^{1 + beta_0 - epsilon_1} log^2 x,

and 2C_1 x log x + C_2 = o(x^{1+beta_0}) since beta_0 > 0.

(b) Competitor sum. For |gamma'| >= 1,

|c(rho')| <= ( |1 - beta'| + |gamma'| ) / ( 2 |gamma'|^2 ) <= 1 / |gamma'|,

and the finitely many zeros with |gamma'| < 1 contribute a bounded amount. So by (F2-up), counting with multiplicity,

sum_{|gamma'| <= T} m(rho') |c(rho')| << log^2 T << log^2 x.

Therefore

| sum_{rho' not in {rho_0, conj(rho_0)}, |gamma'| <= T} m(rho') c(rho') x^{rho'+1} | <= x^{1 + beta_c} sum_{|gamma'| <= T} m(rho') |c(rho')| << x^{1 + beta_0 - g} log^2 x.

Note, and it is the reason the unconditional case is hard. That competitor bound used beta_c < beta_0 and nothing else. The weighted mass sum_{|gamma'| <= T} m(rho') |c(rho')| is of order log^2 T: at most, by (F2-up), and at least, by (F2-low) and the computation in Defect 1. So it grows with x and the growth is not an artifact of a lossy upper bound. The set of zeros that can compete with the witness is not finite and does not stay finite as x -> infinity. Section 10 returns to this.

(c) Witness. The conjugate pair rho_0, conj(rho_0) contributes

2 Re [ m_0 c_0 x^{1 + rho_0} ] = 2 m_0 |c_0| x^{1 + beta_0} cos( gamma_0 log x + phi_0 ).

At x = x_n = exp( (2 pi n - phi_0)/gamma_0 ) the cosine is exactly 1. Hence

I(x_n) = 2 m_0 |c_0| x_n^{1+beta_0} + O( x_n^{1+beta_0-g} log^2 x_n ) + O( x_n^{1+beta_0-g/2} log^2 x_n ) = 2 m_0 |c_0| x_n^{1+beta_0} ( 1 + O( x_n^{-g/2} log^2 x_n ) ),

using epsilon_1 = g/2 <= g. The successive ratio is x_{n+1}/x_n = exp(2 pi / gamma_0). Also c_0 != 0, since c(rho') = 0 only at rho' = 1, which is not a zero.

(d) Integer transfer. For N <= x < N+1 no new prime power enters, so I(x) - I(N) = (x - N) psi(N, chi)/2 = O(N), negligible against N^{1+beta_0} since beta_0 > 0. This is the same step RESULTS.md section 18 uses.

(e) To E(N). Theorem A (RESULTS.md section 17, unconditional) reads

T(N) = I(N) + O( N^{Theta + Theta_chi} log^4 N ).

The + N log N that the first version of this file attached to Theorem A is not part of Theorem A; it appears in RESULTS.md section 18, where Theorem A is combined with the section 15 estimate (E3). Adding it only weakens the statement, so nothing downstream breaks, but the attribution was wrong and is corrected here. Step (e) needs only Theorem A's own error term.

Under (H-TRANSFER), Theta < Theta_chi = beta_0 <= 1, so Theta < 1 and Theta + Theta_chi < 1 + beta_0 with margin 1 - Theta > 0. Hence |T(N_n)| >= (1 + o(1)) 2 m_0 |c_0| N_n^{1+beta_0}, and Cauchy-Schwarz as in section 18 (E(N) >= 2 T(N)^2 / N, using sum_{k <= N} chi(k)^2 <= N) gives

E(N_n) >= (1 + o(1)) 8 m_0^2 |c_0|^2 N_n^{1 + 2 beta_0} >> N_n^{1 + 2 Theta_chi}.

QED (Theorem 4).

What Theorem 4 does not say. It is conditional on (H-ISO), which implies GRH fails for chi_3, and on (H-TRANSFER). It is not evidence for or against either. It does not improve the unconditional Theorem B. It says nothing about the upper bound E(N) = O(N^2 log^2 N), which is the direction the hunt's original objective needed.

8. External inputs

Used as published. (F1) and (F2-low) were checked against the primary source for this revision, because the reviewer could not reach it and the first version of this file relied on an inherited paraphrase.

(F1) Montgomery and Vaughan, Multiplicative Number Theory I, Theorem 12.10. Verified 2026-09-20 against the author-hosted chapter (R. C. Vaughan, Penn State, chapter 12, "Explicit formulae", pp. 397 to 418). The theorem as printed: let c > 1 be a constant; suppose x >= c, T >= 2, and chi is a primitive character mod q > 1. Then

psi_0(x, chi) = - sum_{rho, |gamma| <= T} x^rho / rho

• (1/2) log(x - 1) - (chi(-1)/2) log(x + 1) + C(chi) + R(x, T; chi), C(chi) = (L'/L)(1, chi) + log(q / 2 pi) - C_0, R(x, T; chi) << (log x) min(1, x / (T <x>)) + (x/T) log^2(q x T),

where <x> is the distance from x to the nearest prime power other than x itself, and psi_0 is the midpoint normalization. Five things needed checking and all five check out for chi_3:

1. Hypotheses. x >= c > 1 and T >= 2, nothing more. Take c = 2. There is no restriction T <= x and no admissible-height condition; Lemma 12.7 (a nearby T_1 in [T, T+1] with L'/L(sigma +- i T_1, chi) << log^2 qT uniformly for -1 <= sigma <= 2) is what buys that, and it is internal to the proof.
2. Normalization. psi_0 is the midpoint form, as section 1 says; the half-jump is <= (log x)/2.
3. Index set. |gamma| <= T, inclusive, over all nontrivial zeros; multiplicity enters intrinsically because the sum arises as residues of -(L'/L)(s, chi) x^s / s. Montgomery and Vaughan's proof of Theorem 14.5 explicitly guards against a real zero in (0,1), so a real zero is a live possibility in general and the inclusive index set is the honest one. Lemma 1 above settles the question for chi_3 specifically.
4. Trivial zeros. The same proof carries kappa = 0 or 1 as chi(-1) = 1 or -1, and the trivial-zero tail sum_{1 <= k < (K+kappa)/2} x^{kappa-2k}/(2k-kappa). For odd chi this puts the trivial zeros at the odd negative integers, independently confirming section 5.
5. Error shape. min(1, x/(T<x>)) <= 1 always, so the first term is <= log x uniformly in x, including at prime powers where <x> could otherwise be small. For q = 3 and x, T >= 2, log(3xT) <= 2 log(xT), so log^2(qxT) <= 4 log^2(xT). For odd chi the two logarithms collapse: -(1/2)log(x-1) + (1/2)log(x+1) = (1/2) log((x+1)/(x-1)), which is <= (1/2) log 3 for x >= 2, and C(chi_3) is an absolute constant. Both are <= const . log x for x >= 2.

Collecting 2 and 5 gives (F1) as used in section 6, with an absolute C_1, for x >= 2 and T >= 2. The load-bearing conclusion: Proposition 3 does not need to be made conditional, and neither its coefficient nor the shape of its remainder changes. What changed is the index set, the multiplicity notation, and this provenance.

• (F2-up) Zero counting, upper bound: the number of zeros of L(s, chi) with ordinate in [t, t+1] is O(log(|t|+2)); hence sum_{0 < |gamma'| <= T} m(rho')/|gamma'| << log^2 T. RESULTS.md section 17. Used for the convergence of C_2 and for the competitor bound in section 7(b). An upper bound suffices for both.
• (F2-low) Zero counting, two-sided: Montgomery and Vaughan, MNT I, Theorem 14.5 and Corollary 14.6 (verified 2026-09-20 against chapter 14, "Zeros", pp. 452 to 462): N(T, chi) = (T/2 pi) log(qT / 2 pi) - T/2 pi + S(T, chi) - S(0, chi) - chi(-1)/8 + O(1/(T+1)) for primitive chi mod q > 1, where S(T, chi) = (1/pi) arg L(1/2 + iT, chi), and zeros with gamma = 0 or gamma = T are counted with weight 1/2. Combined with the classical S(T, chi) << log(q(T+2)) this gives N(2T, chi) - N(T, chi) >> T log T. Caveat, stated because it is the one gap in this verification: Corollary 14.6 was read in the primary source, but the bound on S(T, chi) was not located in the extracted chapter text and is cited from the standard literature rather than from an inspected page. The divergence in Defect 1 rests on that citation. Nothing else in this file uses (F2-low).
• Theorem A, RESULTS.md section 17: T(N) = I(N) + O(N^{Theta+Theta_chi} log^4 N). Unconditional. Used only in step (e). The additional N log N term seen in RESULTS.md section 18 comes from (E3) of section 15, not from Theorem A.
• Theorem B, RESULTS.md section 18. Quoted for comparison. Unchanged.
• (R7), REFEREE.md section 4: independent re-derivation of the Mellin transform G(s). The transform is used as the starting point of section 5 only.
• Landau's oscillation theorem, in the direction not requiring a sign condition (Ingham ch. V; Montgomery and Vaughan section 15.1). Cited in section 10 as the classical substitute for the missing inequality; not used in any proof above.
• Functional equation for a primitive odd real character (Gamma((s+1)/2), root number 1 for chi_3), used in section 5 to derive h(-1) in closed form.

chi_3 zero ordinates used in the numerical checks come from results_landau_zero_probe.json (21 ordinates in (0, 60], from a sign-change scan of step 0.25), produced by an earlier pass of this hunt. They are used only to test identities, never as evidence for a claim. The reviewer noted that their provenance had not been audited. Check C8 audits it two ways: it re-evaluates |L(1/2 + i gamma)| at each listed ordinate, and it compares the count to what (F2-low) predicts. Neither is a completeness proof. A scan of step 0.25 can miss a close pair, and C8 cannot detect that; it establishes only that the listed ordinates are near genuine zeros and that the list is not obviously short. No claim in this file depends on the list being complete.

Correction, 2026-10-21 (at landing). The paragraph above understates the gap and misdescribes the scan. The inherited list misses a zero of L(s, chi_3) at gamma = 50.3751386506, inside its gap from 47.514 to 52.497. That zero is isolated, not one of a close pair. The probe is not a sign-change scan: landau_zero_probe.py keeps the local minima of |L(1/2 + it)| on a step-0.25 grid that fall below 0.5 and refines each by a Newton solve, as THEOREM_B_SEQUENCE.md section 6 says. Near this zero the grid minimum is 0.538 at t = 50.5, above that cutoff, so it was never refined. Counting it gives 22 zeros to height 60 against the predicted 22.61, so C8's deficit of -1.61 becomes -0.61. Three routes agree on the zero. The independent verification of this landing found it with a step-0.01 sign-change scan of the Hardy-type function in mpmath (22 zeros on (0, 60]). PARI/GP 2.17.2, which shares no code with mpmath, returns 22 zeros up to height 60 from lfunzeros(lfuncreate(-3), 60): the 21 listed ordinates and this one. Ball arithmetic in python-flint 0.9.0 encloses the completed function Lambda(s) = (3/pi)^{(s+1)/2} Gamma((s+1)/2) L(s, chi_3), which is real on the critical line because the root number is 1 (section 5), as [-8.1504288e-19 +/- 1.7e-27] at t = 50.37 and [7.6373067e-19 +/- 1.4e-27] at t = 50.38, so a zero with 50.37 < t < 50.38 exists. These were run by hand at landing (RUNS.md) and no test pins them. The existence of this unlisted zero is hardened; that there are exactly 22 zeros to height 60 is measured by two routes and not proved. The paragraph above is left as written. Nothing in this file uses the list's completeness: C5, C8 and the reviewer's coefficient identity read only the 21 listed ordinates, and their recorded results stand.

9. Reproducible checks. NOT PROOF.

hunts/prime_pair_error/formula_repair_checks.py, results in hunts/prime_pair_error/results_formula_repair_checks.json. Eight checks, mp.dps = 60, about three seconds total. Every check is a finite floating-point evaluation of an identity at finitely many points; none of them proves anything, and the script says so in its header.

C5 illustrates the size estimate behind Defect 1 at 21 ordinates (taken on the critical line) and nothing more; the divergence itself is the analytic argument of section 2, which rests on the two-sided count (F2-low). C6 illustrates Defect 3: with the source document's height the error bound exceeds the signal, so that estimate cannot establish resonance, and the admissible range removes the obstruction. It says nothing about the size of the actual error. C7 illustrates Lemma 1 and does not prove it; note in particular that the grouped series converges slowly for small s, so C7 asks only that the truncation gap stay inside a tail bound proved on the spot, not that it be tiny. C8 answers the reviewer's provenance question in the two directions a cheap check can answer it, and records in the results JSON that it does not establish completeness.

Landing notes, 2026-10-21. The C6 row's 112 -> 0.073 gives the first and last tabulated values. At T = x^{0.45} the ratio is 112.5, 284.0, 452.3, 286.7, 28.8 and 0.0727 at x = 10^4, 10^8, 10^16, 10^32, 10^64, 10^128 (results_formula_repair_checks.json), so it rises to about 452 before it falls, and "decays" describes the last rows, not every row. The C8 row's count deficit of -1.61 includes one missing zero, not a close pair, and counting it leaves -0.61: see the correction at the end of section 8. Both rows are left as written.

10. The exact remaining gap

Theorem 4 is conditional. What is missing for an unconditional statement of the same shape is one inequality, and it can be written down exactly.

(MISSING). There exist epsilon_1 > 0, delta in (0, 1), a nontrivial zero rho_0 = beta_0 + i gamma_0 with beta_0 > Theta_chi - eta for a given small eta > 0, and a sequence x_{n_k} -> infinity (as k -> infinity) along which, for every k, BOTH the phase condition cos(gamma_0 log x_{n_k} + phi_0) >= 1 - delta/4 AND, with T_{n_k} = x_{n_k}^{1 - beta_0 + epsilon_1}, the competitor bound

| sum_{0 < |gamma'| <= T_{n_k}, rho' not in {rho_0, conj(rho_0)}} m(rho') c(rho') x_{n_k}^{1 + rho'} | <= (1 - delta) . 2 m_0 |c_0| x_{n_k}^{1 + beta_0}

hold.

The earlier wording (an unbounded sequence x_n with the competitor bound for infinitely many n) was insufficient: the unboundedness could come entirely from indices where the bound fails, so the good indices need not escape to infinity. The form above requires the good subsequence itself to tend to infinity, which is what the o(1) terms, the growth of T_{n_k} and the integer transfer below all use.

What (MISSING) buys, and at what cost. The first version of this file said sections 7(a), 7(d) and 7(e) then "go through verbatim". They do not, and the reviewer was right to stop there. Two adaptations are needed, and neither is free.

Adaptation 1, the truncation error (7a). Section 7 fixes epsilon_1 = g/2, a quantity defined by (H-ISO), which is absent here. Use instead the epsilon_1 > 0 that (MISSING) supplies. With T_{n_k} = x_{n_k}^{1 - beta_0 + epsilon_1} one has exactly log(x_{n_k} T_{n_k}) = (2 - beta_0 + epsilon_1) log x_{n_k}, so for every k with x_{n_k} >= 4 and T_{n_k} >= 4 (all large k)

C_1 x_{n_k}^2 T_{n_k}^{-1} log^2(x_{n_k} T_{n_k}) = C_1 (2 - beta_0 + epsilon_1)^2 x_{n_k}^{1 + beta_0 - epsilon_1} log^2 x_{n_k} = o( x_{n_k}^{1 + beta_0} )

for each fixed epsilon_1 > 0. The factor 4 of section 7 is not inherited: there it came from T <= x, which the (H-ISO) gap supplies, and (2 - beta_0 + epsilon_1)^2 exceeds 4 whenever epsilon_1 > beta_0. Combining with the (1 - delta) competitor bound and the (1 - delta/4) phase bound leaves a surviving main term

|I(x_{n_k})| >= (3 delta / 4) . 2 m_0 |c_0| x_{n_k}^{1 + beta_0} (1 + o(1)).

So 7(a) adapts in two lines, but it does so with (MISSING)'s epsilon_1, not 7's.

Adaptation 2, the transfer (7e), and this one is a genuine restriction. Step (e) needs

1 + beta_0 > Theta + Theta_chi

and nothing weaker: that is the exact margin at which Theorem A's error N^{Theta + Theta_chi} log^4 N stays below the signal N^{1 + beta_0}. Section 7 gets it from beta_0 = Theta_chi exactly, which is (H-ISO)'s doing. Here beta_0 is only known to exceed Theta_chi - eta, so the margin is implied by

1 + Theta_chi - eta >= Theta + Theta_chi, that is, eta <= 1 - Theta.

(H-TRANSFER) does not supply this for arbitrary eta. Theta < Theta_chi gives Theta < 1, hence 1 - Theta > 0 and the permissible range of eta is non-empty; but eta is quantified before Theta in the statement of (MISSING), and for eta > 1 - Theta the transfer step fails even though (H-TRANSFER) holds. So either

• state the margin 1 + beta_0 > Theta + Theta_chi as an additional hypothesis of the unconditional statement, or
• restrict to eta <= 1 - Theta (equivalently, Theta <= 1 - eta), which is sufficient and which Theta < 1 makes non-vacuous for every Theta, without making it checkable for a given eta.

With either, and keeping the strict margin 1 + beta_0 > Theta + Theta_chi, (MISSING) yields E(N_{n_k}) >> N_{n_k}^{1 + 2 Theta_chi - 2 eta} along the corresponding floor integers N_{n_k} = floor(x_{n_k}), unconditionally on (H-ISO). This is a conditional implication from the unproved (MISSING); no stronger unconditional result is inferred. (H-ISO) implies (MISSING) with delta close to 1 for all large n, which is precisely how Theorem 4 is proved, and in that case beta_0 equals Theta_chi, so eta may be taken to be 0 and Adaptation 2 is vacuous.

(MISSING) is an existence statement. It asserts that some zero, some epsilon_1, some delta and some unbounded sequence exist. It exhibits none of them. Any statement derived from it is therefore existential too. Only the phase condition is explicit in gamma_0 and phi_0 (which are not known numbers); the sequence x_{n_k} satisfying it is purely existential, and no closed form is supplied under (MISSING). The closed form x_n = exp((2 pi n - phi_0)/gamma_0) belongs to Theorem 4's conditional construction under (H-ISO), not to (MISSING). No constructive or effective claim is made here.

Why (MISSING) is hard, stated honestly.

1. The competing set grows. Section 7(b) and Defect 1 together establish that sum_{|gamma'| <= T} m(rho') |c(rho')| is bounded above and below by constant multiples of log^2 T, so the number of zeros that can interfere with the witness at height T_n is unbounded as n -> infinity. This is a direct consequence of Defect 1: because |c| ~ 1/(2|gamma'|) rather than 1/|gamma'|^2, the weighted mass diverges. THEOREM_B_SEQUENCE.md treats the competitors as a fixed finite obstruction; they are not.
2. Kronecker or Weyl simultaneous approximation controls the phases of any fixed finite set of ordinates {gamma'_1, ..., gamma'_k} along a sequence of x. It does not control a set whose size grows with x, and no uniformity in k is available from that direction without further input.
3. Without (H-ISO) there may be zeros with beta' arbitrarily close to beta_0 at arbitrarily large height, so no fixed power saving x^{-g} separates witness from competitors.
4. The classical way past exactly this obstruction is Landau's oscillation theorem, or an Ingham-type positive-kernel argument: instead of controlling the competitors, one contradicts analyticity of the Mellin transform in a half-plane. That is what RESULTS.md section 18 already does, and it is why Theorem B carries an epsilon and an unspecified sequence. The non-constructivity is not an accident of the write-up; it is what that method yields. An actually exhibited sequence would need an exhibited witness zero (which (H-ISO) only posits) or a genuinely new mechanism, and I do not have one.

Claims from the source document that remain unproved and are not asserted here. The "any delta > 0" aside on the truncation height (refuted as stated; see Defect 3 and C6); the existence of an isolated witness zero (assumed as (H-ISO), not proved); effective phase alignment against the full competitor set (this is exactly (MISSING)).

11. Disposition

Proposition 3 is a useful intermediate lemma with exact hypotheses, and I believe it is correct as stated. Its hypotheses are: x >= 4, T >= 4, chi = chi_3, and (F1) in the form verified against Montgomery and Vaughan Theorem 12.10 in section 8. Its conclusion is the displayed identity with the remainder bound, C_1 absolute but not numerically evaluated here and C_2 finite by (F2-up). It replaces a divergent formula with a convergent one carrying a remainder of explicit shape in x and T, it recovers the same zero coefficients c(rho'), and it fixes the admissible height range that the source argument got wrong.

The one place a reviewer should look hardest is section 8's reduction of Montgomery and Vaughan's error term to C_1(x T^{-1} log^2(xT) + log x). That reduction is crude by design (it throws away min(1, x/(T<x>)) entirely), and if it is wrong, Proposition 3's remainder is wrong with it. Everything else in the proof is elementary.

It is not a complete repair of the sequence argument. The payoff standard in the brief asks for a usable error regime with all competitors handled. Proposition 3 supplies the error regime; the competitors are handled only under (H-ISO). By the brief's own standard this is therefore "a formula with valid error but insufficient for resonance", which is a useful intermediate lemma, not a complete sequence repair. Theorem 4 is the best justified intermediate result I can support, and it is explicitly conditional.

The rest of the sequence argument is an unresolved attempt, and the missing input is analytic, not bookkeeping. Section 10 names it: an inequality bounding the full competitor sum at height T_n against the witness term, for infinitely many n, without an isolation hypothesis. The obstruction is that the competing set grows like log^2 T in weighted mass (both bounds now established), so no fixed finite phase-approximation argument reaches it. I do not have that input and do not claim a route to it.

Grades, per the repository's certainty ladder:

• Lemma 0, Lemma 1, section 5 polar structure, Proposition 3: derived, hypotheses stated, reviewed at 95ae36b and again, targeted, at 3006ef2; the follow-up cleared them.
• Theorem 4: derived and conditional. Its composite grade is that of its weakest step, which is its hypothesis (H-ISO); it is a conditional theorem and is called one.
• Section 10's unconditional variant: unresolved attempt, with the missing analytic input written out and the margin 1 + beta_0 > Theta + Theta_chi stated exactly.
• Section 9: measured, float grade, one route, small finite ranges.
• Nothing here is enclosure-carrying or kernel-checked, and no claim in this file is numerically effective.

No novelty is claimed. No literature search for prior art on this specific repair was run; the ingredients are standard and the write-up makes no originality claim beyond "this hunt produced this". The only literature contact made for this revision was to the primary source for (F1) and (F2-low), recorded in section 8.

Review record. Reviewed once, in the 95ae36b form, by an independent reviewer (THEOREM_B_FORMULA_REVIEW.md). That review cleared Lemma 0, the section 5 residue computations (re-derived by an independent contour method), the structure of Theorem 4's steps (a) to (e), the Defect 3 framing, and the attribution spot-checks it ran. It raised five substantive objections. All five are answered in AUTHOR_RESPONSE.md: four were accepted and the artifact changed; the fifth was accepted in part, with the part rejected argued there rather than quietly dropped. The review's own open item, that the primary source for (F1) had not been reached, is closed in section 8. The revision at 3006ef2 received a targeted follow-up review (appended to the same file), which confirmed the primary (F1) reduction and cleared Proposition 3 and the conditional Theorem 4, and raised three residual findings F1 to F3 in section 10. The three exact edits answering them (exact truncation constant, subsequence quantifiers in (MISSING), explicitness wording) were made afterwards and checked by the coordinator; no further reviewer run has occurred. The classical bound S(T, chi) << log(q(T+2)) (used only for (F2-low), the divergence claim) remains cited, not inspected, as stated in section 8; it is not needed for the core formula.

12. Reproducibility note

From the repository root, with the venv created per CLAUDE.md Setup:

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

The script touches no network, writes only its own results JSON, and needs no ball-arithmetic backend. It was first run on 2026-09-20 from a worktree that had no venv of its own, with the main checkout's interpreter (Python 3.13.14, mpmath 1.3.0); that worktree's system python3 (3.10, mpmath 1.4.1) was not used. At landing, 2026-10-21, it was re-run with Python 3.12.3 and mpmath 1.3.0, took about 6 s of wall time, and reproduced results_formula_repair_checks.json byte for byte. This note and the file list in section 0 were edited at landing, to drop a machine-specific path and to name the reviewer's files. Dated landing notes were also added to sections 0, 8 and 9; the one in section 8 corrects what C8 says about the inherited zero list. No derivation, check or recorded result in this file changed.