Hunt #61 (hunts/lambda_dh_bounds/) produced, so far as the literature search recorded in hunts/lambda_dh_bounds/NOVELTY.md reaches, the first quantitative bounds, from either side, on the de Bruijn-Newman constant of a Dirichlet series with a Riemann-type functional equation whose Riemann hypothesis is false, and one corollary that is sharper than the bracket itself: the constant strictly exceeds zeta's, unconditionally. The hedge and the scope are both load-bearing and are NOVELTY.md's own adopted wording: Stopple (arXiv:1301.3158) already has an unconditional quantitative bound on a non-zeta constant of this type, so "first quantitative bounds on the Davenport-Heilbronn function" would be false as stated. This page is the reading course for that record. The evidence lives in the hunt directory; every number here is pinned by tests/test_lambda_dh_separation.py or by the hunt's own result files.
1. The object, and the frame trap
For zeta, the de Bruijn-Newman constant Lambda is defined through the backward heat deformation of the Riemann Xi function, and Rodgers-Tao proved Lambda >= 0 ("if RH is true, it is only barely so"), while Polymath 15 proved Lambda <= 0.22, since sharpened to Lambda <= 0.2 by Platt and Trudgian (2021), which is the current record zeta.heatflow.lambda_facts() carries. The hunt's chain below is stated against Polymath 15's 0.22 because that is the bound it pinned at source (hunts/lambda_dh_bounds/POLYMATH-PIN.md); the sharper 0.2 only widens the gap the chain needs. The Davenport-Heilbronn function F is the canonical counterexample: a Dirichlet series with zeta's functional equation shape and real coefficients whose own Riemann hypothesis is false (Davenport-Heilbronn 1936; computed off-line zeros: Spira 1994, Balanzario and Sanchez-Ortiz 2007). Its deformation
Phi_DH(u) = 4 e^{3u/2} sum_{n>=1} n a_n exp(-pi n^2 e^{2u}/5), H_t(z) = int_0^inf e^{t u^2} Phi_DH(u) cos(zu) du,
with a_n the period-5 pattern (1, kappa, -kappa, -1, 0), defines Lambda_DH := inf{t : H_t has only real zeros}, well defined and finite by Dobner's theorem for the extended Selberg class.
The trap, which cost this hunt a correction cycle and is now its most transferable lesson: the literature uses two normalizations, and the same constant is four times larger in one than in the other. The hunt's frame (s = 1/2 + iz, published precedent Stopple arXiv:1301.3158) is the narrow one; de Bruijn as usually quoted, Newman, Rodgers-Tao, Polymath 15 and Dobner all sit at s = (1+iz)/2, the wide one. Two refereed papers quote de Bruijn's 1/2 across that frame change unconverted (neither statement is false, since the narrow implies the wide, but the reader is not told). The conversion is derived and numerically checked row by row in hunts/lambda_dh_bounds/FRAME.md.
2. The bracket
In the wide frame of Rodgers-Tao and Polymath 15:
0.2304 < Lambda_DH <= 0.7696992583210755065522
and in the hunt's narrow frame the same statement reads 0.0576 < Lambda_DH <= 0.19242481458026887663805, with 0.0576 = 36/625 exactly. The bracket ratio, which is frame-free and is the honest measure of how loose the two sides are, is 3.341.
The lower side is an argument-principle count: at t = 36/625 the deformation still has a zero strictly off the real axis near the deepest measured quadruple (height about 240.4), decided by ball-arithmetic winding counts (python-flint Arb, 420 bits) with a second witness sharing no code layer (mpmath at dps 130), made strict by Dobner's closed half-line.
The upper side feeds a decided zero-strip constant to de Bruijn 1950 Theorem 13, whose all-zeros form was transcribed from the original text and corroborated against two typeset restatements. That constant is where the hunt improved on itself after the gate closed, and the two derivations are worth seeing together because the second explains why the first was loose. The first bounded the zeros by coefficient domination, sum |a_n| n^{-sigma} < 1, giving sigma_0 = 1.39513615823511 on both backends. That step replaces every phase n^{-it} by an independent worst case, and the phases are not independent: they are determined multiplicatively by their values at the primes. Used quantitatively, that gives a strictly better criterion. Writing f as a combination of the two Dirichlet L-functions of the odd character mod 5, a zero with Re s > 1 forces a specific total argument on their ratio, while each prime p = 2, 3 mod 5 can supply at most 2 arctan(p^{-sigma}) radians of it (a Moebius image of a disc, whose argument-maximising point turns out to have modulus exactly 1, so nothing further is available from trading modulus against phase). The abscissa where the primes run out is decided on both backends at the exact rational sigma_0' = 1.12036249819, a factor 2.082030697360155 better in Delta^2/2, with the sum's tail closed by the Euler products themselves so that no prime-counting estimate enters anywhere. Both derivations are kept: STRIP.md for the first and STRIP2.md for the second.
Two honest notes on that improvement. The equation the second route solves is Bombieri and Ghosh's Theorem 7 at their parameter values, term for term; only its necessary half is used, and that half is derived in-tree, so the number is theirs and the grade is the hunt's. And the obvious alternative sharpening was tried and recorded as failing: regrouping the series into period-5 blocks and applying the mean value theorem is correct but carries a factor |s|, so it gives a height-restricted strip that climbs back to sigma_0 as the height grows, and de Bruijn's theorem consumes a half-plane statement.
Grade, per the certainty ladder: the composite is a decided computation glued to cited theorems, and it takes the weakest step's grade. The analytic lemma inside the lower bound (the derivative bound called M2) was prose with a measured guard and a recorded blind spot through 2026-08-17; it is now proved in M2-LEMMA.md, with every constant a reported ball and every hypothesis a decided predicate, exercised by four routes and two falsification attacks. Its single non-elementary input is the evenness of Phi_DH, which is the functional equation transported and was already an acknowledged citation. What survives of the blind spot is narrower and is still disclosed: the detector cannot see a corrupted M2 by itself, its own health metric moves the wrong way across the lesion, and the lesion table is published for that reason.
3. The separation, which is the quotable part
Polymath 15's bound and this hunt's floor sit in the same frame, and they cross:
0 <= Lambda_zeta <= 0.22 < 0.2304 = 144/625 < Lambda_DH.
(Polymath 15's 0.22 is the pinned link; Platt-Trudgian's sharper 0.2 makes the same chain hold with more room, 0.2304 > 0.2, and is not needed for it.)
So Lambda_DH > Lambda_zeta, unconditionally: the counterexample's failure margin, measured in flow time, exceeds zeta's entire remaining uncertainty window. The rational core is exact (144/625 against 11/50 cross-multiplies to 7200 > 6875) and the inequality is frame-invariant. Positivity alone could not have given this: Lambda_DH > 0 is a two-line corollary of Dobner plus Spira and is compatible with the whole of zeta's window, 0.22 or 0.2 alike. The quantitative floor is what crosses the published bound. The separation rests on the floor and on the cited zeta bound alone, so the 2026-08-18 sharpening of the upper side left it untouched in every digit.
Scope of the novelty claim, as adversarially narrowed in hunts/lambda_dh_bounds/SEPARATION.md: in the function-field analogue of the flow, exactly determined and trivially ordered Newman constants have been in print since 2013-2014 (Andrade-Chang-Miller; Chang-Mehrle-Miller-Reiter- Stahl-Yott), all negative or zero because RH is a theorem there. So far as the recorded searches reach, this is the first strict inequality between such constants in which both are nonnegative, and the first proof that any of them strictly exceeds zeta's.
4. What prior art owns
- Stopple (arXiv:1301.3158): the hunt's normalization, verbatim, and a prior quantitative non-zeta bound (negative side, GRH-conditional upper).
- Dobner (arXiv:2005.05142): existence, finiteness, nonnegativity, and the closed half-line, for the whole extended Selberg class, with no numbers.
- de Bruijn 1950 Theorem 13 (restated by Newman-Wu 2020 Theorem 7): the strip-to-time engine of the upper bound.
- Bombieri and Ghosh (Russian Math. Surveys 66:2, 2011), read in full during the gate: no de Bruijn-Newman or heat-flow content, so the bracket is not anticipated; but their Theorem 7 determines the exact least upper bound of the real parts of the zeros of this function (1.120362), so any originality claim for the hunt's strip abscissa is withdrawn. The hunt's second route now reaches that same abscissa and decides it, which changes the grade and not the ownership: their equation is reproduced term for term, only its necessary half is used, and their converse (which makes the abscissa an exact supremum rather than an upper bound) is neither used nor claimed. Two of their published numbers serve as controls on the new instrument: the finite claim that the smallest set of primes p = 2, 3 mod 5 with sum arctan(1/p) > pi/2 ends at 6323 with 420 elements, and the sibling abscissa 2.3822861089 to all ten digits they print, neither of which the instrument was built around. Their tau_+ equals the lab's kappa, and the gate's check of their golden closed form (kappa = sqrt(1 + phi^2) - phi, phi the golden ratio, from the quadratic kappa^2 + 2 phi kappa - 1 = 0) is decided on both backends in
hunts/lambda_dh_bounds/KAPPA-CLOSED-FORM.md, tying the counterexample's mixing constant to the conductor-5 golden-field story ofhunts/golden_control/. - Newman-Wu (Bull. AMS 2020): a strictly positive constant of this type, computed exactly, for a three-atom measure; orderings against a measure's constant are not frame-invariant and are not claimed.
5. Honest scope
Nothing here is evidence about RH. The separation distinguishes the counterexample from zeta only through its already-known off-line zeros, which is gate-3 framing, not a mechanism. The zeta side of the chain rests on Polymath 15's refereed computer-assisted bound, not re-verified in this tree. The preprint that once stood as the unread risk (academia.edu 166936409) has since been traced to an open-access deposit and read in full: it contains no de Bruijn-Newman constant for this function and no bound on one. What is left unread is smaller and named in the gate: the full text of Bombieri and Mueller 2008, behind a publisher wall, and one Dobner-citing preprint behind Cloudflare. The novelty language is conditioned on the recorded searches by standing instruction.
What a skeptical referee should attack first has changed as the record hardened, and the gate re-ranks it honestly rather than declaring the work finished. It is no longer M2 and no longer the looseness of the upper side, because both were spent. It is the citations, which are now unambiguously the weakest steps: de Bruijn's Theorem 13 in its all-zeros form, transcribed from an image-only scan; Dobner's Theorem 1; the evenness of Phi_DH; and, for the separation, Polymath 15's Theorem 1.1. After them comes the plainest caveat this page can offer: the hunt's two pieces of new mathematics, the phase obstruction and Lemma M2, are prose plus decided arithmetic at the hardened rung, written by agent sessions, attacked by scripts from the same sessions, and read by no human. The full record is hunts/lambda_dh_bounds/GATE.md; the verdict there is a publication candidate pending external verification, which is not ours to award.
6. Where to read
hunts/lambda_dh_bounds/: MISSION.md (preregistration), GATE.md (the adjudication), SEPARATION.md (the corollary with per-link grades), FRAME.md (the two normalizations), RESULTS.md and results.json (all claims, keyed and graded), STRIP.md and STRIP2.md (the two strip derivations, weaker and sharper), M2-LEMMA.md (the derivative bound, proved), THEOREM13.md (the engine, transcribed), NOVELTY.md (the search record and the sanctioned sentence), BOMBIERI-GHOSH.md, KAPPA-CLOSED-FORM.md, POLYMATH-PIN.md (verbatim sources), INDEPENDENCE.md (measured route independence), and the four adversary write-ups. Reproduction is one command per instrument, listed in GATE.md.