SEPARATION: Lambda_DH > Lambda_zeta, unconditionally

Written 2026-08-16, after the gate closed (GATE.md, verdict YES) and after a dedicated novelty adversary attacked the claim's phrasing. This page elevates the comparison that FRAME.md section 7 and NOVELTY.md carried as a calibration footnote to a named claim, at exactly the strength the adversary sanctioned and not more. Vocabulary per MISSION.md: measured is one float route, decided is an enclosure whose exact endpoints settle a sign or an integer, cited is somebody else's theorem; a composite takes its weakest grade.

Note added 2026-08-18, after a hardening pass sharpened the upper bound. The separation is untouched. It rests on the decided floor 144/625 = 0.2304 and on the cited Lambda_zeta <= 0.22, and neither moved in any digit. What moved is the last link of the chain below, which the separation does not use and which is printed only so the claim travels with the whole bracket: the wide-frame upper endpoint sharpened from 1.6025374835598228 to 0.7696992583210755065522 (narrow: 0.4006343708899557 to 0.19242481458026887663805), a factor 2.082030697360155, decided in-tree on both backends by the phase obstruction of STRIP2.md. Every occurrence of the old endpoint below is replaced and the superseded value is named where it stood. One other change touches this page only in the reader's favour: M2, named below as a prose lemma, is a proved lemma since 2026-08-18 (M2-LEMMA.md), so what the decided link inherits is a cited input rather than an unproven step. The claim, its phrasing, its qualifier and its grade are unchanged.

1. The claim

In the shared normalization of Newman, Rodgers-Tao, Polymath 15 and Dobner (s = (1+iz)/2, the wide frame of FRAME.md), the de Bruijn-Newman constant of the Davenport-Heilbronn function strictly exceeds the de Bruijn-Newman constant of the Riemann zeta function:

Lambda_DH > Lambda_zeta, unconditionally.

Composite grade: cited plus decided (weakest step cited), and the decided link carries the M2 lemma recorded in GATE.md (prose when this page was written; proved with decided constants since 2026-08-18, M2-LEMMA.md).

The chain, every link graded, with the sources pinned at source in POLYMATH-PIN.md:

0 <= Lambda_zeta cited: Rodgers-Tao arXiv:1801.05914, Theorem 1 ("One has Lambda >= 0"), pinned verbatim in POLYMATH-PIN.md section 2 Lambda_zeta <= 0.22 cited: Polymath 15 arXiv:1904.12438, Theorem 1.1 ("We have Lambda <= 0.22"), unconditional per their abstract, pinned verbatim in POLYMATH-PIN.md section 1 0.22 = 11/50 < 144/625 = 0.2304 exact rational arithmetic, cross-multiplied: 144 * 50 = 7200 > 6875 = 11 * 625, pinned by tests/test_lambda_dh_separation.py 0.2304 = 4 * (36/625) < Lambda_DH this hunt: winding count N = 1 for H_{36/625} (narrow frame) over a box whose interior has Im z >= 3/1024 in exact rationals, decided (python-flint 0.9.0 (Arb), 420 bits, winding_results.json; second witness mpmath dps 130, crosscheck_dhflow_results.json); strictness cited (Dobner arXiv:2005.05142 Theorem 1, closed half-line); frame factor 4 derived in FRAME.md section 3 Lambda_DH <= 0.7696992583210755065522 this hunt: 4 x the narrow Delta^2/2, exactly, from the phase-obstruction abscissa sigma_0' = 1.12036249819 decided on both backends at an exact rational (decided, strip2_results.json), fed to de Bruijn 1950 Theorem 13 (cited). This link read <= 1.6025374835598228 through 2026-08-17, from the coefficient- domination sigma_0 of strip_results.json, which is still correct and is retained

Assembled, in the wide frame where all four cited sources live:

0 <= Lambda_zeta <= 0.22 < 0.2304 < Lambda_DH <= 0.7696992583210755065522.

The last link is not needed for the separation; it is printed because the claim should travel with the whole bracket. It is also the only link the 2026-08-18 sharpening touched, which is why that sharpening changes nothing here: the separation is a statement about the floor.

2. The exact rational core, and frame invariance

The strict middle inequality is a comparison of two rationals and is checked by cross-multiplication, not by floats:

144/625 > 11/50 because 144 * 50 = 7200 > 6875 = 11 * 625.

Frame invariance: divide every Lambda by 4 to reach the narrow frame (Stopple arXiv:1301.3158, s = 1/2 + iz, this hunt's own):

0 <= Lambda_zeta <= 0.055 < 0.0576 = 36/625 < Lambda_DH <= 0.19242481458026887663805,

(the last endpoint read 0.4006343708899557 through 2026-08-17) and the middle comparison is the same cross-multiplication, because the factor 4 cancels: 0.055 = 11/200 and 36 * 200 = 7200 > 6875 = 11 * 625. The inequality Lambda_DH > Lambda_zeta reads identically in both frames, as an inequality between constants of one frame must. Both computations are pinned by tests/test_lambda_dh_separation.py.

3. Why the quantitative floor is load-bearing

Strict positivity alone does not separate. Lambda_DH > 0 is an immediate corollary of Dobner's Theorem 1 plus any computed off-line zero (Spira 1994), and it has been available since 2020 without anyone displaying it; but Lambda_DH > 0 is compatible with Lambda_DH <= 0.22 and so decides nothing against zeta's window. The separation exists only because the decided floor is quantitative and happens to clear Polymath 15's bound: 0.2304 > 0.22, with 144/625 - 11/50 = 13/1250 = 0.0104 to spare in the wide frame. Had Polymath 15's bound come out above 0.2304, or the decided floor below 0.22, this page would not exist. The direction of future movement is asymmetric: any further sharpening of the zeta upper bound only widens the separation, so what the claim is hostage to is not progress but the correctness of the cited 0.22 itself, and of the decided floor.

Note added 2026-08-18, and it is headroom rather than a change. The 2026-08-18 prior-art sweep surfaced Tao, Trudgian and Yang's ANTEDB (teorth.github.io/expdb), whose chapter 18 tabulates the de Bruijn-Newman bound history and records Platt-Trudgian 2021, Lambda_zeta <= 0.2, sharper than the <= 0.22 this chain cites. The chain is deliberately left on Polymath 15's 0.22, because that is the bound pinned verbatim at source in POLYMATH-PIN.md and a claim should stand on the source it actually checked. The direction is the favourable one: 0.2304 > 0.2 as well, so the separation survives the sharper bound with more room, and the asymmetry argued above is confirmed rather than tested. zeta.heatflow.lambda_facts() carries 0.2 as the current record and 0.22 as superseded, which is the in-tree ground truth for this.

4. What it means, and what it does not

What it means. Lambda_zeta is confined to [0, 0.22] (Rodgers-Tao; Polymath 15). The Davenport-Heilbronn function's constant sits strictly above that entire window: the backward-heat flow time this counterexample needs before all its zeros become real exceeds the whole remaining uncertainty window for zeta's own constant. The flow-time failure margin of the counterexample is not merely positive; it is larger than everything zeta still has left to resolve.

What it does not mean. Nothing here is evidence about the Riemann hypothesis (docs/08; house rule). The separation is a decided property distinguishing DH from zeta through DH's already-known off-line zeros, which is gate-3 framing (MISSION.md WP4): the same pipeline pointed at zeta produces no positive floor, because no off-line zero of zeta is known, so the number separates the two functions only through what was already known to separate them. It quantifies the difference; it does not discover it. Nor is the inequality surprising: Lambda_zeta <= 0 is equivalent to RH while Lambda_DH > 0 follows from DH's off-line zeros, so under RH the separation is trivially expected. The content is that it now holds unconditionally, with a decided quantitative gap, whether or not RH is true.

5. The adversary's verdict and caveats

A dedicated novelty adversary attacked the phrasing "first proven strict inequality between the de Bruijn-Newman constants of two Dirichlet series" on 2026-08-16. Its verdict, verbatim:

VERDICT: the claim DIES as phrased; the result survives under a repaired phrasing.

Its kills and clearances, quoted verbatim from its report (the report's final sentence arrived truncated mid-word at "The ln 2 e"; the truncation is recorded rather than reconstructed, and the ln 2 case it was evidently reaching is already carried in NOVELTY.md: Newman-Wu determine a constant of this type exactly, ln 2, for a three-atom measure, not a Dirichlet series):

Kill 1: Andrade-Chang-Miller, arXiv:1310.3477 (2013), read in full this session (PDF fetched and read page by page). Their L-functions L(s, chi_D) = sum_{f monic} chi_D(f) |f|^{-s} (their eq. 3.1-3.2) are literally Dirichlet series, and the paper itself calls Lambda_D a "De Bruijn-Newman constant" (their Section 3.2). They prove:

• Lemma 3.18 / Theorem 1.9: an exact closed form,

Lambda_{D_p} = log(|a_p(D)|/(2 sqrt p)) for deg D = 3 (a_p the Frobenius trace of the elliptic curve y^2 = D(T)).

• Remark 3.10: D = T^3 + T over F_3 has Lambda_D = -infinity,

exactly.

• Appendix A: D = T^5+T^4+T^3+2T+2 over F_5 has

Lambda_D ~= -0.189, exact as the log of a root of an explicit quartic.

• Appendix B: a table of seven named D over F_3 with seven distinct

numerical lower bounds.

• Lemma 3.11: Lambda_D < 0 strictly whenever the zeros are simple.

Kill 2: Chang-Mehrle-Miller-Reiter-Stahl-Yott, arXiv:1411.2071 (2014) (abstract fetched verbatim): "In contrast with previous work, we are able to exhibit specific L-functions for which Lambda_D = 0, and thereby prove a stronger statement: max_{L in F} Lambda_L = 0." So exactly determined constants, including exact zeros, for named Dirichlet series are published.

Consequence: strict orderings between exactly determined de Bruijn-Newman constants of two named Dirichlet series (same field, same normalization) have been immediate one-line corollaries of published statements since 2013-2014, e.g. Lambda_{T^3+T} = -infinity < -1.44e-1 <= Lambda_{T^3+2T+1} inside ACM's own tables, or -infinity < 0 = Lambda_D across ACM/CMMRSY. Even an ordering against zeta is composable from print since 2013: Lambda_{T^3+T} = -infinity < Lambda_zeta (finite by Newman 1976). Nobody displays these inequalities because in function fields they are trivial (RH is a theorem there), but a referee will produce them in one paragraph. The unqualified phrase is indefensible.

What did NOT kill, verified this session:

• Stopple 1301.3158 (ar5iv full text queried): no comparison between

Lambda_Kr and zeta's constant anywhere; Lambda_Kr is a sup over discriminants not including zeta; his bounds are negative/ GRH-conditional. No strict ordering between two named constants.

• ACM/CMMRSY number-field side: conjectures only; no positive

constant, nothing beating 0.22.

• **Newman-Wu 1901.06596, full text extracted with pdftotext and

grepped: the survey **never compares constants across objects; 0.22 appears only as zeta's own bound (lines 21, 78, 1005 of extracted text); no occurrence of Davenport, Heilbronn, or any cross-object ordering.

6. The sanctioned phrasing

What the kills remove is the unqualified word "first" over all Dirichlet series: in function fields, where the Riemann hypothesis is a theorem, every constant in print is exactly determined and non-positive (with -infinity and exact 0 attained), so strict orderings there, including orderings against zeta through the -infinity examples, are one-line corollaries of statements published since 2013-2014, displayed by nobody because they are trivial. What survives every named kill is the qualifier that both constants are nonnegative, which no function-field pair and no -infinity composite can satisfy. The claim, at the strength the adversary's evidence sanctions:

The strict inequality Lambda_DH > Lambda_zeta holds unconditionally, by Lambda_zeta <= 0.22 (Polymath 15, cited) against Lambda_DH > 0.2304 = 144/625 (this hunt, decided modulo two cited theorems), in the shared normalization of Newman, Rodgers-Tao, Polymath 15 and Dobner. So far as the searches recorded in NOVELTY.md and in this page reach, it is the first strict inequality between de Bruijn-Newman constants of two Dirichlet series in which both constants are nonnegative. It is not the first strict inequality between such constants outright: in function fields such orderings are immediate corollaries of exact determinations published since 2013-2014 (Andrade-Chang-Miller arXiv:1310.3477; Chang-Mehrle-Miller-Reiter-Stahl-Yott arXiv:1411.2071).

Do not drop "so far as the searches reach". Do not drop "in which both constants are nonnegative". Do not display the chain without the frame.

7. Grade, in full

Weakest-step accounting, per MISSION.md's vocabulary contract:

Composite: cited plus decided. The separation inherits, through its decided link, the M2 blind spot recorded in GATE.md (the one prose analytic step no cross-route exercises, whose lesion table shows wrong silent integers from deflation 75); and, through its cited links, the correctness of Polymath 15's Theorem 1.1, Rodgers-Tao's Theorem 1 and Dobner's Theorem 1 as published, none of which is verified in-tree. It is not a claim this directory proves on its own, and per MISSION.md it is not promoted anywhere outside this hunt.

Update 2026-08-18. The inherited M2 exposure is narrower than the paragraph above describes, and the paragraph is kept as written. M2 is proved in M2-LEMMA.md, with every constant a reported Arb ball and every hypothesis a decided predicate, and four routes plus two falsification attacks stand behind it. What the decided link now inherits is (i) the one cited classical input the proof needs, the evenness Phi_DH(-u) = Phi_DH(u), and (ii) the fact that the detector still cannot see a corrupted M2 by itself, so the lesion table and its wrong silent integers survive as a measure of sensitivity to an implementation fault. The cited links are unchanged, and they are now the weakest steps without qualification. Composite grade unchanged: cited plus decided.