Scope correction, 2026-09-12. The proposed absolutely convergent zero expansion in section 3 is not established: its coefficient \((\rho-1)/(2\rho(\rho+1))\) decays as \(1/|\operatorname{Im}\rho|\), not its square. The accompanying dyadic count also uses a unit-height zero count outside its range. A controlled truncated or smoothed expansion is required before the later constructive resonance steps can be used. See FAREY_BASELINE_REPAIR.md, Appendix B. The nonconstructive Theorem B in RESULTS.md and REFEREE.md does not depend on this construction.
RESULTS.md Section 18 proves Theorem B, E(N) = Omega(N^{1 + 2 Theta_chi - eps}), in the case Theta < Theta_chi <= 1 by a pole-vs-boundedness contradiction on the Mellin transform of I(x) = sum_{m <= x} Lambda(m) chi(m) (x/2 - m), concluding "there is an unbounded sequence" of N with |I(N)| >= N^{1 + Theta_chi - eps'}, without exhibiting it (REFEREE.md Section 4 re-derives the same argument independently and reaches the same non-constructive conclusion). This note asks whether the classical machinery behind that argument (Ingham ch. V; Montgomery-Vaughan Section 15.1) can be adapted to name the sequence, or at least bound the gaps between its members. Short answer: yes, in a clean special case, with an explicit and easily computed gap; no, in general, and the two places it fails are different in kind. Nothing here bears on RH, and nothing here reopens or changes what Section 18 already proves.
1. What is non-constructive about the existing proof, precisely
The proof's structure is: assume I(x) = O(x^alpha) for all x (alpha = 1 + Theta_chi - eps'); this makes the Mellin transform integral_1^infty I(x) x^{-s-2} dx converge absolutely and hence be holomorphic on Re(s) > alpha - 1, contradicting the known pole at a zero rho' of L(s, chi) with Re(rho') > alpha - 1. Therefore the assumption is false: NOT [for all x, |I(x)| <= C x^alpha], for every constant C. Negating a universally quantified O(x^alpha) statement gives exactly an existence statement - limsup_{x -> infty} |I(x)|/x^alpha = infinity - and nothing else: no rate, no location, no relation between two members of the witnessing set. The step from "not O(x^alpha)" to "an unbounded sequence" is pure real analysis and carries no information beyond what already is in the negated statement. This is why the proof, honestly written, stops at existence: the complex-analytic content (the pole) is used only to rule out the possibility of the bound holding everywhere, not to build anything.
This differs from the classical Landau theorem for Dirichlet series with non-negative coefficients, which is not what is invoked here and could not be: chi(m) changes sign, so Lambda(m) chi(m) is not sign-definite, and RESULTS.md Section 18 says as much ("no sign condition on the coefficients is needed"). The non-negative-coefficient theorem is stronger - it identifies the abscissa of convergence itself as a singularity, which is extra structure a signed series does not offer for free. What is used instead is the weak form (call it the contradiction form): a Dirichlet-type transform that converges absolutely on a half-plane is holomorphic there, so a known singularity to its right bounds the abscissa of convergence from below. That is all the weak form gives, and it is an if-then about all x, not a statement about any specific x.
2. Two different obstructions, not one
Turning this into a genuine construction runs into two separate difficulties. They are worth naming apart because they have different characters and, as far as this note gets, different prognoses.
Obstruction A - locating a witness zero at all. For a target eps' > 0, the proof uses "the definition of supremum" to get some zero rho' with Re(rho') > Theta_chi - eps'; it says nothing about the height |Im(rho')| at which this zero sits. Making this effective would need a height T = T(eps') such that a zero with Re > Theta_chi - eps' is guaranteed to already appear among |Im(rho')| <= T. That is an effective zero-density statement close to the supremum itself - not a zero-free region (which bounds real parts from above) but its converse, a guarantee that the real parts actually approach the supremum by a certain height. No such statement is available for L(s, chi_3), and none is derived here.
Obstruction B - disentangling a witness from its neighbors. Granted a specific zero rho'_0, building an explicit sequence from it (Section 3-4 below) requires that no other zero within the same relevant height window compete with it in size while carrying an incompatible phase. The only tool on hand for counting nearby zeros is the O(log T) density bound already used throughout this hunt (Theorem A's (F2)); it bounds how many competitors there are but says nothing about how close in real part they can be. When several zeros tie or cluster near the extremal real part within reach of a single resonance construction, isolating any one of them needs simultaneous control of several independent phases at once, which (Section 5) is a Kronecker-type Diophantine approximation problem with no known effective bound.
Obstruction A is about whether a witness exists early enough to exhibit; obstruction B is about whether, once handed a witness, its signal can be pulled out from what is around it. Section 3-4 show obstruction B is not fatal when the witness is isolated; Section 5 shows it is real when it is not, and that resolving it is not just a matter of more work with the tools already in this hunt.
3. An explicit formula for I(x), by the machinery already trusted here
The pole-contradiction step throws away the actual identity the Mellin transform carries and only uses that it has a pole. Recovering the identity is mechanical and does not need anything beyond what RESULTS.md already establishes for two structurally identical objects:
- Section 15's H_chi(N): a Riesz mean with denominator s(s+1), shown there to have an UNCONDITIONALLY convergent zero-free asymptotic H_chi(N) = c_H N + O(N^{3/4}) via a contour shift, using convexity bounds on L(s, chi) and L(1+s, chi) that come straight from the functional equation.
- Theorem A's Step 2 (Section 17): a contour shift on -L'/L(w - rho, chi), truncated at height T_rho, using (F3) (the log-growth bound on L'/L(s, chi) for -1 <= sigma <= 2, and away from trivial zeros for sigma <= -1/4 via the functional equation) to control the horizontal and left vertical segments explicitly, term by term.
F(s) = integral_1^infty I(x) x^{-s-2} dx = -(L'/L)(s, chi) (1-s) / (2 s (s+1)) has exactly the same s(s+1) denominator as H_chi's Dirichlet series, and its numerator is governed by the same L'/L(s, chi) that (F3) already bounds on every relevant vertical line (the two ranges -1 <= sigma <= 2 and sigma <= -1/4 together cover the whole plane away from the trivial-zero poles, by the functional equation reflecting one range into the other). At a simple zero rho' of L(s, chi), the residue of F(s) x^{s+1} is x^{rho'+1} (rho'-1) / (2 rho'(rho'+1)) (matching REFEREE.md Section 4's residue computation). Because the extra 1/(s(s+1)) supplies two more powers of decay than a first Mellin transform would, and because (F2)'s O(log T) zero density per unit height makes sum 1/|rho'(rho'+1)| converge (each term is O(1/gamma'^2), and sum over dyadic ranges of O(log T) terms of size 1/T^2 converges), the zero-sum representation of I(x) converges absolutely, by the same argument Section 15 already carries out for H_chi(N)'s D(s)/(s(s+1)). This gives, for x >= 1 and away from the countable set of x where I jumps,
I(x) = mu_1 x + mu_0 + sum_{rho'} x^{rho'+1} (rho'-1) / (2 rho'(rho'+1)), (*)
mu_1, mu_0 the explicit residues of F(s) x^{s+1} at s = 0 and s = -1, and the sum over all non-trivial and trivial zeros of L(s, chi_3), absolutely convergent, no truncation and no error term. (If instead one prefers to avoid re-deriving absolute convergence in full and stay entirely inside what Section 17 already proves rigorously, the truncated version with an explicit O(x^{1+eta} log^2(xT)/T + ...) error term, built exactly as Theorem A's Step 2 builds one for -L'/L(w - rho, chi), suffices for everything below; the argument does not need (*) in its strongest form, only that the tail beyond a chosen height T is smaller than the resonance term by a computable margin, and either route gives that.)
Either way, this is the part of "making Landau's argument constructive" that has a clean answer: the identity behind the contradiction is recoverable in full, by machinery this hunt already trusts elsewhere, no new estimate is required, and neither obstruction A nor B appears yet - they appear only in how (*) gets used.
4. The resonance construction, and its explicit gap, when the witness is isolated
Since chi_3 is real-valued, L(s, chi_3) has real coefficients, so its non-real zeros come in conjugate pairs rho'_0 = beta_0 + i gamma_0, conj(rho'_0) = beta_0 - i gamma_0 sharing the same real part. Summed together their two terms in (*) combine to a real quantity:
x^{1+rho'_0}(rho'_0-1)/(2rho'_0(rho'_0+1)) + x^{1+conj(rho'_0)}(...) = x^{1+beta_0} |c_0| cos(gamma_0 log x + phi_0),
writing c_0 = (rho'_0 - 1) / (rho'_0 (rho'_0 + 1)) = |c_0| e^{i phi_0}, an explicit complex number once rho'_0 is known (elementary to bound above and below given beta_0 in (1/2, 1], gamma_0 > 0). Choosing
x_n = exp((2 pi n - phi_0) / gamma_0), n = 1, 2, 3, ...
makes cos(gamma_0 log x_n + phi_0) = 1 exactly, i.e. this single term equals its own maximum modulus x_n^{1+beta_0} |c_0|. This sequence is completely explicit given rho'_0, and its consecutive members have an explicit, computable multiplicative gap
x_{n+1} / x_n = exp(2 pi / gamma_0),
with no dependence on n: the higher the witness zero, the tighter this gap (for the low zeros of L(s, chi_3) found in Section 6 below, gamma_0 ~ 8.04 gives a gap of about 2.19; gamma_0 ~ 57.6 gives about 1.12 - see Section 6). This is the "gaps between successive members" the task asks whether can be bounded: whenever a single witness zero is in hand and dominates its window (next paragraph), the gap is not just bounded but named in closed form.
For this to be a genuine lower bound on I(x_n) and not just on one term of (), the rest of () at x = x_n must not cancel it. Split () at x_n into the rho'_0-pair term (size x_n^{1+beta_0}|c_0| by construction), the main term mu_1 x_n + mu_0 (linear, negligible against x_n^{1+beta_0} once beta_0 > 0, which it is), a tail |gamma'| > T for T a growing function of n (T = x_n^delta for small fixed delta > 0 is enough, by the convergence rate behind () or by Section 17's Step-2-style truncation error, either way = o(x_n^{1+beta_0
- eps''}) for T growing like any positive power of x_n), and the zeros with |gamma'| <= T
other than rho'_0's own pair. That last group is the delicate one: by (F2) there are O(log T) = O(log x_n) many of them, each contributing at most x_n^{1+Theta_chi} |c(rho')| in modulus. If rho'_0 is a strict, isolated maximizer of Re(rho') among zeros with |gamma'| <= T - meaning every other such zero has real part at most beta_0 - g for some fixed gap g > 0 - their combined modulus is O(x_n^{1+beta_0-g} log x_n), which is o(x_n^{1+beta_0-eps}) for eps < g. In that case (*) gives, along the explicit sequence x_n with the explicit gap above,
I(x_n) = |c_0| x_n^{1+beta_0} (1 + o(1)),
a genuine, fully constructive, fully explicit oscillation result, transferable to integers N_n = round(x_n) at a cost O(N_n) (Section 18's own "I(x) - I(N) = (x-N)P(N)/2 = O(N)" step, unchanged) which is negligible against the same power. No contradiction argument, no appeal to a definition of supremum, and no non-constructive step remains in this case.
5. Where the isolated case fails, and why it is not just more of the same work
The catch is the isolation hypothesis - "no competing zero within reach of the same resonance window" - and it is not available in general, for two compounding reasons.
First (obstruction A, restated for this construction): rho'_0 has to actually be produced, at a known height, with Re(rho'_0) within the target eps' of Theta_chi. Nothing in this hunt's tools, or in the standard zero-density literature for Dirichlet L-functions, supplies a height at which such a zero is guaranteed to appear; the classical zero-free regions cited elsewhere in this hunt (F5, and its analogue for L(s, chi)) bound real parts from above, which is the wrong direction for this purpose.
Second (obstruction B), even granting rho'_0: the isolation hypothesis can fail, and nothing rules it out. If instead there are two or more zeros rho'_0, rho'_1, ... within the truncation window whose real parts are all within the eps-margin of Theta_chi (ties or near-ties at comparable height are not excluded by (F2), which only counts, it does not separate), then choosing x_n to align rho'_0's phase does not control the others': their phases gamma'_j log x_n + phi_j walk around the circle at a rate set by the ratio gamma'_j / gamma_0, generically equidistributing rather than cooperating. Recovering a lower bound in this case is the classical multi-zero oscillation problem (this is what Ingham ch. V's and Landau's own fuller oscillation arguments, and the standard treatment of Omega_{+-} results for psi(x) - x, actually spend their effort on), and the standard tool for it is Kronecker's theorem on simultaneous Diophantine approximation: if the ratios gamma'_j / gamma_0 (finitely many, since the window is bounded) are irrational and suitably independent, there exist n making all the relevant phases simultaneously close to any target - in particular close enough to not cancel the rho'_0 contribution. This settles existence again, which is not new information (existence was already the starting point), but it does not settle constructivity: Kronecker's theorem, in its classical form, gives no bound on how large such n must be. An effective version would need a simultaneous Diophantine approximation rate for the specific ordinates gamma'_j / gamma_0, i.e. a quantitative irrationality measure for those ratios. No such measure is known for the ordinates of any single L-function's zeros, let alone across two zeros of L(s, chi_3) - this is the same order of difficulty as the (open, believed, unproved) linear independence over Q of the ordinates of zeta's zeros. So in the presence of competing zeros, the gap between successive members of any resulting sequence is unbounded by anything this note can supply, and this is not a matter of applying the tools already in this hunt more carefully; it needs an input (an effective simultaneous approximation bound for L-function zero ordinates) that does not exist in the literature.
6. What is visible at low height for L(s, chi_3), and an illustrative gap
landau_zero_probe.py locates the zeros of L(s, chi_3) up to height 60 by scanning |L(1/2 + it, chi_3)| for local minima and refining each with a complex Newton solve (mpmath, 30 decimal digits) started on the critical line but free to move off it; results in results_landau_zero_probe.json. It finds 21 zeros, at ordinates 8.04, 11.25, 15.70, 18.26, 20.46, 24.06, 26.58, 28.22, 30.75, 33.90, 35.61, 37.55, 39.49, 42.62, 44.12, 46.27, 47.51, 52.50, 54.19, 55.64, 57.58, every one landing at Re = 0.5 to the residual precision of the solve (|L| below 10^-14 on the line, versus 0.06-0.5 moved 0.05-0.1 off it). This is a check at low height, not a proof about all height, and it says nothing about Theta_chi - but it means that, as of this scan, there is no computationally exhibited witness for Theta_chi > Theta anywhere near the range this note could search, consistent with (not evidence for or against) GRH for this L-function. Taking the lowest and highest zeros found as stand-ins for what a witness's height might look like, the explicit resonance gap exp(2 pi / gamma_0) from Section 4 would be about 2.19 for gamma_0 ~ 8.04 and about 1.12 for gamma_0 ~ 57.58: the gap shrinks as the witness rises, so a witness forced (by obstruction A) to appear only at very large height would, if one were ever exhibited, still hand back a comparatively dense and cheaply computable sequence - the expense is entirely in the exhibiting, not in what the exhibited zero would deliver.
7. What this does and does not settle
Achievable. The pole-contradiction step can be replaced by an actual identity (*), recovered from tools this hunt already trusts (Section 15's Riesz-mean argument, Section 17's contour-shift-with-truncation-error argument), with no new estimate needed. Given an isolated witness zero - one whose real part strictly exceeds every other zero's in the relevant height window by a fixed margin - the resulting oscillation is fully constructive: an explicit sequence x_n = exp((2 pi n - phi_0)/gamma_0) with an explicit, closed-form multiplicative gap exp(2 pi / gamma_0) between consecutive members, transferable to integers at a negligible O(N) cost exactly as the existing proof already tolerates.
Not achievable, with what is currently known. Two separate gaps remain, and closing either needs an input outside this hunt: (A) no effective bound on the height at which a witness for Theta_chi - eps' must appear (an effective zero-density statement near the supremum, the converse direction from the zero-free regions this hunt already uses); (B) when the witness is not isolated - which nothing available here excludes - separating it from competing nearby zeros is a Kronecker-type simultaneous Diophantine approximation problem, classically an existence statement only, with no known effective rate for L-function zero ordinates. Obstruction B is not specific to L(s, chi_3): it is the same kind of open difficulty as the (still unproved) linear independence over Q of zeta's own zero ordinates, which the classical unconditional Omega_{+-} results for psi(x) - x carry around rather than resolve.
Placement. This is an investigation of the proof technique behind Theorem B's existential clause, not a strengthening of Theorem B itself: no new bound on E(N), on Theta_chi, or on any zero of L(s, chi_3) is claimed, obstruction A alone already means the constructive sub-case of Section 4 has no exhibited instance, and the low-height scan of Section 6 finds nothing to instantiate it with. It does not touch the doors table (that table is about rank 3's major-arc construction in UPPER_BOUND.md, a different part of this hunt). Nothing here bears on RH.