Novelty record for `lambda_dh_bounds`

Recorded 2026-08-16, from a web sweep run before the hunt opened. Per the house rule (ontology/knownness.py: an unrun check can never read as absence of prior art), this file says exactly what was searched and what it found. Nothing below is a claim that anything is novel; it is the record a novelty claim would have to stand on, plus the caveats it would have to carry.

Revised 2026-08-16 after the prior-art adversary (attack_adversary5_priorart.md) and the gate (GATE.md, closure items (a) and (d)). Four changes, each marked in place: the zeta calibration figures now appear in both normalizations, because the original quoted them in one frame and compared them against a number in the other; Stopple is promoted from a one-line mention to the published precedent for this hunt's own normalization and a prior quantitative non-zeta de Bruijn-Newman bound; Newman-Wu Theorem 7 is added as the typeset restatement of the engine; and Bombieri-Ghosh 2011 is added, first as an unread standing risk to the strip constant and then, later the same day after a sibling session retrieved and read it in full (BOMBIERI-GHOSH.md), as a closed risk with a mixed verdict: it carries no de Bruijn-Newman content at all, so the bracket is not anticipated, but it determines the exact zero-strip abscissa for this function, sigma(tau_+, 1) = 1.120362, so sigma_0 is a weaker bound on a quantity that was already known. The novelty sentence is rewritten in section "The sentence" and the superseded version is kept there verbatim; the sentence itself survives the Bombieri-Ghosh reading unchanged, and only its footnote moved.

Revised 2026-08-18 (third pass), closing the last three prior-art items that GATE.md item 9 carried as open. academia.edu preprint 166936409 is read in full: it is Mesut Ismail, 10.5281/zenodo.21679490, freely available on Zenodo, and it contains no de Bruijn-Newman constant for the Davenport-Heilbronn function and no bound on one, so the risk closes in the hunt's favour. Bombieri-Mueller 2008 is identified exactly and read at abstract and reference-list level: it is about the rate of approach of zeros to the abscissa for Epstein zeta functions of class number 2, with no de Bruijn-Newman or heat-flow content. The Dobner forward-citation sweep is no longer single-source: three independent indexes plus a web sweep, seven citing works, none of them quantitative about any non-zeta Lambda. The sentence survives unchanged; only the footnote and the one-line short form move. One item is newly named and unread rather than unfound (the Voronov preprint), and one new and strong piece of negative evidence is recorded (the ANTEDB de Bruijn-Newman chapter, which tabulates every known bound and carries no non-zeta entry).

Revised 2026-08-18 (fourth pass), for the sharpened upper bound. The headline's upper endpoint improved by a factor 2.082030697360155 when STRIP2.md derived the necessary half of Bombieri and Ghosh's Theorem 7 in-tree and decided the abscissa sigma_0' = 1.12036249819 on both backends. The novelty sentence survives unchanged, because the abscissa is not new and was never claimed to be: only the grade and the derivation are this hunt's. One word in the footnote is corrected, weaker, since the elementary route now reaches the same abscissa rather than stopping short of it. The numbers in "Read FRAME.md first" and in the Bombieri-Ghosh entry are updated in place with the superseded values kept beside them. The lower bound did not move, so nothing in the separation's phrasing changes either.

Read FRAME.md first

Every Lambda number in this directory has two values, four apart. The error originated in THEOREM13.md section 6 and surfaced here, in the calibration against the zeta record below. The two frames:

narrow : s = 1/2 + i z (Stopple; this hunt) wide : s = (1 + i z)/2 (de Bruijn as usually quoted; Newman; Rodgers-Tao; Polymath 15; Dobner)

Lambda(wide) = 4 * Lambda(narrow).

The hunt's headline in both:

narrow (Stopple / this hunt) : 0.0576 < Lambda_DH <= 0.19242481458026887663805 wide (Dobner / Polymath 15): 0.2304 < Lambda_DH <= 0.7696992583210755065522

(Through 2026-08-17 these read <= 0.4006343708899557 and <= 1.6025374835598228; the upper side sharpened by a factor 2.082030697360155 on 2026-08-18 when the abscissa was decided in-tree from a phase obstruction rather than from coefficient domination, STRIP2.md. The lower side did not move.)

Derivation, sources and per-row numerical checks are in FRAME.md.

What is published, and the delta this hunt targets

• Dobner 2020 (A proof of Newman's conjecture for the extended Selberg class, arXiv:2005.05142; published in Acta Arithmetica 2021, where the arXiv title differs, which is a real trap for a title search). Theorem 1: for every F in the extended Selberg class S# there is a real Lambda_F with all zeros of the deformation on the critical line iff t >= Lambda_F (the half-line structure). Theorem 2: Lambda_F >= 0. The Davenport-Heilbronn function is in S# (absolutely convergent series for Re s > 1, entire completion, functional equation with gamma factor (pi/5)^{-(s+1)/2} Gamma((s+1)/2); no Euler product is required for S#). So existence, finiteness and nonnegativity of Lambda_DH are published, and strict positivity is an immediate corollary of Theorem 1 plus any computed off-line zero (Spira 1994). Dobner names no member and gives no number. The delta: quantitative bounds. His frame is the wide one.
• Stopple 2013 (Notes on Low discriminants and the generalized Newman conjecture, arXiv:1301.3158), read at source this session and re-read for this revision. This is the nearest prior work and it does two things at once.

(i) It is the published precedent for this hunt's normalization. His opening, verbatim from the arXiv PDF (s = 1/2 + it):

We define, for s = 1/2 + it,

Xi(t, chi) =def (D/pi)^{(s+1)/2} Gamma((s+1)/2) L(s, chi) = int_0^inf Phi(u, chi) cos(ut) du,

where

(1) Phi(u, chi) = 4 sum_{n=1}^inf chi(n) n exp(3u/2 - n^2 pi exp(2u)/D).

and, four pages later,

Following Polya [11] and de Bruijn [1] we introduce a deformation parameter t:

Xi_t(x, chi) = int_0^inf exp(t u^2) Phi(u, chi) cos(ux) du,

with the constant defined by the same closed half-line this hunt uses:

There exists a real constant Lambda_{-D}, -inf < Lambda_{-D} <= 1/2, such that (1) Xi_t(x, chi) has only real zeros if and only if t >= Lambda_{-D}. (2) Xi_t(x, chi) has some complex zeros if t < Lambda_{-D}.

Definition. We define Lambda_Kr = sup {Lambda_{-D} | -D fundamental}.

Set D = 5 and replace chi(n) by the period-5 coefficients a_n and this is character for character the hunt's Phi_DH, H_t and Lambda_DH. The hunt did not invent this frame and must stop presenting it as its own choice.

(ii) It is a prior quantitative bound on a non-zeta de Bruijn-Newman constant. His abstract states -1.13 * 10^{-7} < Lambda_Kr, sharpened in his Theorem 3 to

We have that -D = -175990483 satisfies (15), and the corresponding zero gives the bound

-1.12929 * 10^{-7} < Lambda_Kr.

and his upper side is conditional: "Under the GRH, Lambda_Kr <= 0." So any phrasing of the form first quantitative bound on a de Bruijn-Newman constant other than zeta's is false, and the earlier version of this file, which listed Stopple only among "generalized Newman lines, all at negative-or-zero values", undersold him badly. What Stopple does not have is a function that violates its own Riemann hypothesis: quadratic Dirichlet L-functions have Euler products and are expected to satisfy GRH, and his lower bound is a Lehmer-pair argument, not an off-line zero.

• Newman-Wu 2020 (Constants of de Bruijn-Newman type in analytic number theory and statistical physics, Bull. AMS 57, arXiv:1901.06596). Two separate roles.

(i) The engine, typeset. Their Theorem 7, verbatim from the arXiv PDF, page 8, is the modern restatement of de Bruijn 1950 Theorem 13 that this hunt's upper bound uses:

Theorem 7 Suppose that the function F satisfies (10), (11) and the zeros of the entire function (9) lie in the strip |Im z| <= Delta. Then all the roots of the entire function

(16) int_{-inf}^{inf} F(t) e^{lambda t^2 / 2} e^{izt} dt

lie in the strip

(17) |Im z| <= max(Delta^2 - lambda, 0)^{1/2}.

with (10) F(-t) = conj F(t), (11) |F(t)| <= A exp(-|t|^{2+alpha}) for some A, alpha > 0, and (9) f(z) = int_R F(t) e^{izt} dt. Their lambda is 2t in the e^{tu^2} convention, so (17) reaches zero at t = Delta^2/2: this hunt's dictionary, in a Bulletin of the AMS survey, with a strip hypothesis and no positivity hypothesis on Phi. The mechanism of the upper bound is published and surveyed; what this hunt supplies is the strip constant it is fed. Cite this alongside de Bruijn 1950 rather than resting on the image-only 1950 scan.

(ii) A strictly positive constant, computed exactly. They determine a de Bruijn-Newman type constant exactly, Lambda(rho) = ln 2, for a concrete three-atom probability measure (their Case 2; the set of admissible values is [ln 2, inf)). Any phrasing of the form "first strictly positive constant of this type for a concrete object" is therefore false. Their text contains no occurrence of Davenport, Heilbronn or Epstein.

• The zeta record, for calibration, in both frames.

Superseded. This entry originally read: "The zeta record for calibration: 0 <= Lambda_zeta <= 0.22 (Rodgers-Tao arXiv:1801.05914; Polymath 15 arXiv:1904.12438), Lambda_zeta < 1/2 (Ki-Kim-Lee 2009), historical lower bounds all negative, ending at -1.15e-11 (Saouter-Gourdon-Demichel 2011)." Every one of those is a wide-frame number, and the hunt's 0.4006 that they were calibrating is a narrow- frame number, so the comparison flattered the hunt by a factor of 4. The cause was the false sentence in THEOREM13.md section 6 (now corrected there, with the original preserved).

Read in the common wide frame, where both live, the comparison the conversion actually buys is

Lambda_zeta <= 0.22 (Polymath 15) Lambda_DH > 0.2304 (this hunt, decided)

so the Davenport-Heilbronn constant sits above the best known upper bound for zeta. In the narrow frame alone, 0.0576 sits below 0.22 and a reader draws the opposite conclusion. Publish both columns.

• Generalized Newman lines, all at negative-or-zero values: quadratic Dirichlet L-functions (Stopple arXiv:1301.3158, see above, whose bound is in the narrow frame; Andrade-Chang-Miller arXiv:1310.3477; best concrete bound Lambda_D > -1.17e-7), function fields (Chang-Mehrle-Miller-Reiter-Stahl-Yott arXiv:1411.2071, constants <= 0 with sup 0 attained). No RH-violating function is treated anywhere in this line.
• DH zero distribution context: off-line zeros exist (Davenport-Heilbronn 1936), computed instances Spira (Math. Comp. 1994) and Balanzario-Sanchez-Ortiz (Math. Comp. 76, 2007); DH also has zeros with Re s > 1 (discussed in Ferry et al., arXiv:1602.06328), which is why the zero strip needs the coefficient-domination argument; line-zero counts Karatsuba 1990/91 and Gritsenko 2017 (Proc. Steklov 296). Note: arXiv:2503.24275 (2025) claims DH zeros lie only on the critical line; this contradicts the computed zeros above and the repo's own pinned 50-digit zero, and is treated as unsound.
• de Bruijn 1950 (Duke Math. J. 17, 197-226), Theorem 13, is the upper bound's engine. The restatements consulted (Csordas-Norfolk-Varga 1988; Ki-Kim, J. Anal. Math. 91 (2003), where de Bruijn's class conditions are integrability, realness/evenness in the sense F(-t) = conj F(t), and decay O(e^{-|t|^b}) with b > 2) require no positivity of Phi, which matters here because Phi_DH has mixed-sign coefficients. The all-zeros-in-strip form (zeros of H_0 in |Im z| <= Delta implies H_t real-rooted for t >= Delta^2/2) is what the historical citations attribute to Theorem 13; it is confirmed against the original text in THEOREM13.md section 1 and, independently and in typeset form, by Newman-Wu Theorem 7 above.

Standing risks: the named sources, and what reading them settled

1. Bombieri and Ghosh 2011: read in full, risk closed, verdict mixed

E. Bombieri and A. Ghosh, Around the Davenport-Heilbronn function, Uspekhi Mat. Nauk 66:2 (2011), 15-66 = Russian Math. Surveys 66:2 (2011), 221-270, DOI 10.1070/RM2011v066n02ABEH004740. Fifty pages, both authors at the IAS, devoted to this single function. Cited by Righetti (arXiv:1506.05716) and by Nakamura-Pankowski (arXiv:1909.08301). It was absent from the original sweep, which is a gap in that sweep and not a finding about the source. It was recorded here for part of 2026-08-16 as unreadable and therefore as a standing risk; that is no longer true. The full English translation is free at mathnet.ru under the id rm9410, and it was retrieved and read in full the same day. The reading, the retrieval route, the verbatim quotes and two verification legs are in BOMBIERI-GHOSH.md. The verdict has two halves and they point in opposite directions.

On Lambda_DH: closed, in the hunt's favour. The paper contains zero occurrences of Bruijn, Newman, heat, Polya, Turan, Lambda or deformation, in the text and in all 28 references. Nothing in it bears on the de Bruijn-Newman constant, on H_t, or on the backward heat flow. The novelty sentence below needs no change on account of this source.

On sigma_0: closed, against the hunt. The decimal 1.39513615823510972... is not in the paper (digit searches for 1.395, 1.39, 1.3951, 0.2840, 2.4779 all return nothing), but a sharper and exact value of the quantity sigma_0 bounds is. Their section 5 defines sigma(xi, q) as the least upper bound of the real parts of the zeros, their Theorem 7 determines it as the root of sum_{p = 2,3 mod 5} arctan(p^{-sigma}) = pi/2 - |theta| with xi = tan theta, and their section 6 evaluates it for this hunt's function:

For the Titchmarsh value tau_+ = -phi + sqrt(1 + phi^2) we find ... sigma(tau_+, 1) = 1.120362.

Their tau_+ is this hunt's kappa, agreeing to 34 digits. So the exact strip abscissa for the Davenport-Heilbronn function has been in print since 2011 and is smaller than sigma_0 = 1.395136..., which is a triangle-inequality upper bound on it. sigma_0 must not be described as a new number, and the sentence below and its footnote are worded so that they do not depend on it being one. What survives is the derivation, not the quantity: an enclosure-carrying elementary route to a bound on a constant that Bombieri and Ghosh had already determined by Bohr-Kronecker theory.

And it hands the hunt a better upper bound, by citation. Feeding sigma(tau_+, 1) into the same de Bruijn / Newman-Wu engine gives Delta = 0.620362... and

Delta^2/2 = 0.192424814576128011 (narrow, Stopple / this hunt) = 0.769699258304512045 (wide, Dobner / Polymath 15)

a factor 2.082 better than 0.4006. That improved bound is cited plus measured, not decided: their 1.120362 is a six-decimal value and their Theorem 7 rests on machinery this tree has not verified. The decided headline therefore stays 0.4006343708899557, with the sharper number quoted as an improvement available from the literature. Do not silently swap it in.

Update 2026-08-18. It was not swapped in; it was derived. The paragraph above is kept as written and its conclusion is now spent. STRIP2.md and strip2.py derive the necessary half of their Theorem 7 in-tree, from the Euler products of L(s, chi) and L(s, conj chi) plus one Moebius image, with no Bohr theory and no Kronecker theorem, and decide the abscissa on both backends at the exact rational sigma_0' = 1.12036249819 (python-flint 192 bits and mpmath.iv dps 40, sieve limit P = 10^5, 4814 class primes; deep flint-only point 320 bits at P = 10^7 deciding 1.1203624981833251). So the headline is now Delta^2/2 = 0.19242481458026887663805 narrow and 0.7696992583210755065522 wide, decided, at the factor 2.082030697360155.

What this does and does not change for novelty. It does not make the abscissa new. The criterion is their Theorem 7 equation, term for term, at q = 1 and xi = kappa, and STRIP2.md section 7 says so; their converse, which makes the abscissa an exact supremum, is neither used nor claimed. What is new on the upper side is the grade and the derivation. The novelty sentence and its footnote are unchanged, except that the footnote's characterisation of the upper side as "an enclosure-carrying elementary derivation of a weaker bound on that same quantity" loses the word weaker: the elementary route now reaches their abscissa.

Two of their published numbers are reproduced here as controls, by machinery their Theorem 7 does not share: the section 9 finite claim (threshold prime 6323, cardinality 420) and the sibling abscissa sigma(tau_-, 1) = 2.38228610898712387152... against their published 2.3822861089, all ten digits. One correction falls out and it is against an in-tree artifact rather than against the paper: BOMBIERI-GHOSH.md check B's 29-digit re-solves of both abscissae each sit on the wrong side of their own root, by about 1.2e-17 and 6e-18. Bombieri and Ghosh print six and ten decimals and this instrument reproduces both exactly.

2. academia.edu preprint 166936409: read in full, risk closed in the hunt's favour

Revised 2026-08-18. This entry recorded the preprint as unread through 2026-08-16. It is now read in full and the caveat is discharged. The superseded text is kept verbatim at the end of this subsection.

How it was reached, after every earlier route failed. Direct fetch of the academia.edu page still returns HTTP 403 under every user agent tried, including a Googlebot string, and the Wayback machine holds no snapshot (archive.org/wayback/available returns an empty archived_snapshots). What worked was r.jina.ai against the full slug URL rather than the bare numeric one: it returned 40,540 bytes of the record, including the full current abstract and the citation block. That block carries the two things every earlier attempt lacked: the author, Mesut Ismail (ORCID 0009-0001-0496-964X), and a DOI, 10.5281/zenodo.21679490. The paper is deposited on Zenodo, open access: https://zenodo.org/records/21679490, file Ismail_rh_pf_v18.4.pdf, 758,872 bytes, publication date 2026-07-29. It was downloaded, converted with pdftotext -layout (4,436 lines) and read. The lesson for the next sweep is narrow and worth keeping: an academia.edu wall is not evidence that a document is unobtainable, and the citation block behind it usually names a mirror.

The title on academia.edu is stale, which is why the earlier title searches found only fragments. The current version is v18.4 and is titled

Off-line zeros of the Riemann xi-function: a constraint network, an exactly solvable collision model, an energy-budget divergence, and a lifetime-deficit dictionary with negative controls

The fragments recorded in attack_adversary5_priorart.md section 2 come from an earlier version and some of them no longer hold: the frozen-field figure tau*(T0) = 0.229 is now 0.2151, and "explicit frozen-field bounds" and "interference-channel negative control" have left the title.

Its frame is the wide one, and it states this hunt's factor-4 dictionary independently. Its Definition 3.1 fixes H_0(z) = (1/8) xi(1/2 + iz/2) evolved by d_t H = -d_zz H, with Lambda_zeta <= 1/2. That is the frame of Newman, Rodgers-Tao, Polymath 15, Dobner and the ANTEDB. Its Lemma 3.2 derives the translation to the unit-diffusion clock as t = 4 tau and adds the warning this hunt learned expensively:

A clock off by the factor 4 would instead prove Lambda <= 1/8 from the conjugate mechanism alone, which is false.

That is an independent published statement of the conversion in FRAME.md, from a source that had no contact with this tree, and FRAME.md may cite it as such.

What it does with Davenport-Heilbronn: instrument, in five roles, never subject. (i) a calibration showing its energy-budget divergence does not discriminate zeta from DH; (ii) the DH log-derivative coefficients as a negative control on Euler-product-free positivity schemes; (iii) negative-control theorems against the Connes-Consani-Moscovici rank-one scheme and Suzuki's screw-function criterion; (iv) a lifetime-deficit dictionary validated at the two DH off-line zeros; (v) "the first computed Morse index path of a non-RH function".

The two numbers that the caveat was about. Its Numerical Observation 24.2 tabulates measured lifetimes 0.182 and 0.045 for the DH witnesses rho1 at t0 = 85.70, h1 = 0.3085 and rho2 at t0 = 114.16, h2 = 0.1508, against the conjugate bounds 2h^2 = 0.190 and 0.046. Its Numerical Observation 32.5 gives closed-form lifetimes tau1 = 0.1819 and tau2 = 0.0449 and the index path

ind-(Q_t^DH), window |gamma| < 120 : 4 -> 2 -> 0 at t ~ 0.045, 0.182.

Its witnesses are the same zeros this repository pins: its rho1 = 0.808517... + 85.699...i agrees with zeta/epstein.py's pinned 0.80851718245663738555335196060684412785067026830502 + 85.6993484853775921719292677i to every digit it prints. I recomputed 2 h1^2 = 0.190365478578 at mp.dps = 30 (mpmath, measured) and it reproduces its 0.190, which confirms the frame reading above rather than resting on it.

Verdict: it contains no de Bruijn-Newman constant for the Davenport-Heilbronn function, and no bound on one from either side. Three findings, in order of how much they matter.

• No named constant. It never writes Lambda_DH or any equivalent, and it attaches no number to one. Trap for the next reader: the symbol Lambda_H does appear throughout its sections 20 and 28, but it is the DH analogue of the von Mangoldt function, defined by the log-derivative recursion a_n log n = sum_d Lambda_H(d) a_{n/d}. It is a coefficient sequence, not a de Bruijn-Newman constant, and the two must not be conflated.
• The ingredients are present and are never assembled. Its section 32 defines Lambda generically for "a completed L-type function", and its Proposition 24.3 step 1 proves Lambda = sup_k tau_k over the extinction times of the off-line pairs. A reader could therefore combine its own section 32 with its own 32.5 to guess Lambda_DH >~ 0.18. The paper does not do this, states no such conclusion, and gives no upper bound of any kind.
• Even the most generous reading is weaker than this hunt's floor, and of a lower grade. Its tau1 = 0.1819 is labelled a two-zero upper bound on that pair's lifetime, with the parenthetical "the full field accelerates collisions", so it points the wrong way to yield a lower bound on Lambda_DH at all. Read at face value anyway, and converted:

This hunt's floor is larger by a factor 1.267 (measured, float division of cited decimals), and it is decided by an Arb winding number while every number above is explicitly labelled a Numerical Observation in a paper that separates those from its theorems on purpose.

So the novelty sentence needs no change on account of this source, and its footnote does. The clause "may contain, implicitly, a float-grade lower bound" is replaced below by what is actually in the paper.

Superseded, kept verbatim (the state through 2026-08-17). "### 2. academia.edu preprint 166936409, the one source still unread. Not read: Cloudflare challenge on direct fetch, r.jina.ai 401, Wayback CDX blocked by egress policy. The search index yielded its full title, [...] and enough abstract fragments (recorded in attack_adversary5_priorart.md section 2) to establish that its subject is the Riemann xi function and the classical wide-frame Lambda, with its headline numbers 0.4233 and 0.4305 stated against the anchor 1/2, and that Davenport-Heilbronn enters three times as instrument rather than as subject. One of those three is a set of 'measured lifetimes' of DH off-line zeros under this same backward flow, and a lifetime is the quantity hunts/flow_repair/ calls a landing time, so that preprint may contain, implicitly, a float-grade lower bound on Lambda_DH for the two lowest DH off-line zeros. It names no Lambda_DH and gives no upper bound. This hunt's floor comes from the pair-5 site near Re z = 240.4 and is decided rather than measured, which is a different grade, but the caveat stands until someone reads it. Its provenance could not be determined from inside this checkout; a search of all tracked files and all of git log --all for its distinctive vocabulary returned nothing outside this file."

The provenance question is unchanged and is now answerable from outside: the author is named, has an ORCID, and has no connection to this laboratory.

3. Bombieri and Mueller 2008: identified and read at abstract level, not a threat

Added 2026-08-18. BOMBIERI-GHOSH.md flagged this as the one citation-away source the hunt had not consulted, because it is the parent of the Bohr-method machinery that produced sigma(tau_+, 1) = 1.120362, the constant that displaced sigma_0. It is now identified exactly and read as far as open sources reach.

E. Bombieri and J. Mueller, "On the zeros of certain Epstein zeta functions", Forum Math. 20:2 (2008), 359-385, DOI 10.1515/FORUM.2008.018, zbMATH Zbl 1217.11040, MSC 11E45 and 11M41. The zbMATH summary, verbatim:

This paper studies the distribution of zeros of certain Epstein zeta functions, associated to positive definite binary quadratic forms with class number 2, in the region of absolute convergence of their Dirichlet series. In particular, one obtains upper and lower bounds for the rate of approach of zeros to the boundary of the zero-free half-plane for such functions. The proof for the lower bound depends on Bohr's method for studying simultaneous diophantine approximations. The upper bound uses instead a deep result on the diophantine type of the ratio of the logarithms of two rational numbers.

Crossref carries five deposited references; the two that are identifiable are Baker (J. Reine Angew. Math. 442, 1993) and Gonek (1980). Neither de Bruijn nor Newman nor Polymath appears among them.

Verdict: no de Bruijn-Newman or heat-flow content at any level reachable, and no competing value for sigma_0 either. Two reasons the second half matters. Its subject is a different RH-violating family, Epstein zeta functions of class number 2, not the Davenport-Heilbronn function; and its quantity is the rate of approach of zeros to the abscissa, not the abscissa itself, so it is not a rival determination of the thing sigma_0 bounds. What it supplies is the method Bombieri and Ghosh later pointed at this hunt's function.

What is not read. The full text. De Gruyter answers HTTP 202 behind a human verification wall and r.jina.ai gets HTTP 405 from it; no preprint, mathnet mirror or EuDML copy was found. The residual risk is small and named: a fifty-year-old Bohr-method paper on a neighbouring RH-violating family could in principle carry a remark on deformation, and its abstract and reference list say it does not.

Also surfaced by the prior-art adversary, not threats

• sigma* = 2.3822861089... for the sibling function f(s, tau_-) with tau_- = -1/kappa = -3.5201470213402019924..., the other root of the same quadratic. Righetti (arXiv:1506.05716) quotes it; the constant is Bombieri and Ghosh's, their sigma(tau_-, 1) in the same section 6 table as sigma(tau_+, 1), and this file's earlier attribution of it to Righetti is corrected here (see BOMBIERI-GHOSH.md section 5.1). It is a different constant for a different member of the family, and it is not the coefficient-domination abscissa for that function either, which is 2.477958026532674935112887. A reader will meet all three numbers together, so name them and say which is which.
• Bucur, Ernvall-Hytonen, Odzak and Smajlovic, On a Li-type criterion for zero-free regions of certain Dirichlet series with real coefficients, LMS J. Comput. Math. (2016), whose abstract names the Davenport-Heilbronn zeta function explicitly. No heat flow, no de Bruijn-Newman constant.
• The quantity sigma_0 bounds is a classical named object: the least upper bound sigma* of the real parts of the zeros of a Dirichlet series, finite by Titchmarsh, Theory of Functions section 9.41 (cited by Righetti; not read directly here). STRIP.md should say so in one line.
• Michalowski, arXiv:2602.20313, works the same kernel with 80-digit outward-rounded interval arithmetic and no DH and no Lambda bound, so no collision; worth reading as a cautionary neighbour, since its version 2 withdraws a version 1 global sign claim "because the derivative-tail enclosure was unsound", which is the failure mode this hunt's tail bounds are exposed to.

Searches run (2026-08-15/16, extended 2026-08-18)

"de Bruijn-Newman" + "Davenport-Heilbronn"; "de Bruijn-Newman" + Epstein; Newman's conjecture Dirichlet L-functions (Stopple; ACM; CMMRSY); Ki-Kim-Lee upper bound; CNV/Odlyzko lower-bound history; "Davenport-Heilbronn" + heat flow / backward heat; de Bruijn 1950 Theorem 13; Odzak-Smajlovic Li coefficients for DH-class functions; arXiv 2023-2026 sweeps for DH zeros; Dobner full text; Newman-Wu full text; Gritsenko 2017; arXiv:2602.20313 (Polya frequency order of the zeta kernel, 2026, evidence the niche is active). Added in the 2026-08-16 revision: Stopple full text at source; Dobner's forward citations (Semantic Scholar only, five records, none computing Lambda_F for any member of S#; OpenAlex could not be used as a cross-check because it now meters by budget and returned HTTP 429 with zero daily allowance, so that sweep is single-source and should be repeated by a human with Google Scholar or MathSciNet); Saias-Weingartner, Booker-Thorne, Nakamura-Pankowski, Balanzario-Sanchez-Ortiz, Ferry-Ghisa- Muscutar; digit searches for 1.39513, 1.3951361, 0.8951, 0.895136.

Added in the 2026-08-18 revision, which closed the last three open prior-art items and is recorded here at the level of route and status so a reader can re-run it.

• academia.edu 166936409: direct fetch 403 under two user agents including a Googlebot string; Wayback archive.org/wayback/available returns no snapshot; Wayback timemap connection reset by egress policy; r.jina.ai on the full slug URL returned HTTP 200 and 40,540 bytes, which named the author and a Zenodo DOI; the Zenodo API and file endpoint both returned HTTP 200 and the full 758,872-byte PDF. Read in full. See section 2 above.
• Bombieri-Mueller 2008: located on Crossref by bibliographic query, then zbMATH api.zbmath.org for the authoritative summary and MSC, and Crossref for the deposited reference list. De Gruyter returns HTTP 202 behind a human verification wall and r.jina.ai gets HTTP 405 from it, so the full text is still unread. See section 3 above.
• Dobner forward citations, now three independent indexes (the 2026-08-16 sweep was Semantic Scholar alone). Dobner is 10.4064/aa200603-23-7 / arXiv:2005.05142, and the arXiv title differs from the journal title, which is the trap already recorded above.
• Semantic Scholar citations endpoint, queried by DOI and independently by arXiv id, agreeing: 5 records.
• OpenCitations (COCI, Crossref-derived, an index with no Semantic Scholar input): 1 record.
• Google Scholar, cluster 5851792251239807498, headline "Cited by 5": 2 distinct records across two view settings before it rate-limited.
• A web-search sweep on the arXiv id surfaced one further citing preprint that none of the three indexes carried.
• OpenAlex still could not be used. api.openalex.org returns HTTP 429 with "Insufficient budget. This request costs $0.001 but you only have $0 remaining" and retryAfter: 72359. This is recorded, not worked around.

Union of the four routes, 7 distinct citing works plus one database, with a one-line verdict each:

• ANTEDB chapter 18, "The de Bruijn-Newman constant", read in full. It fixes H_0(z) = (1/8) xi(1/2 + iz/2), that is this hunt's wide frame, and tabulates every known bound. Lower: Newman 1976 > -inf, CNV 1988 > -50, te Riele 1991 > -5, Norfolk-Ruttan-Varga 1992 > -0.385, Csordas-Ruttan-Varga 1991 > -0.0991, Csordas-Smith-Varga 1994 > -4.379e-6, COSV 1993 > -5.895e-9, Odlyzko 2000 > -2.63e-9, Saouter-Gourdon-Demichel 2011 > -1.15e-11, Rodgers-Tao 2020 and Dobner 2021 >= 0. Upper: Newman 1976 <= 1/2, Ki-Kim-Lee 2009 < 1/2, Polymath 2019 <= 0.22, Platt-Trudgian 2021 <= 0.2. Every entry is zeta's. A maintained, curated, community database of de Bruijn-Newman results that carries no non-zeta constant at all is better evidence for the search than any single negative query.
• Two side-findings, recorded here and not acted on. (i) ANTEDB's current upper record is Lambda_zeta <= 0.2 (Platt-Trudgian 2021), sharper than the <= 0.22 (Polymath 15) that this file and SEPARATION.md quote. The separation corollary strengthens, since the decided wide floor 0.2304 clears 0.2 as well as 0.22; SEPARATION.md should cite Platt-Trudgian, and that edit is not made here. (ii) The academia.edu preprint's Lemma 3.2 states this hunt's factor-4 frame dictionary independently, with the same warning about the wrong clock; FRAME.md may cite it.

No publication was found attaching any number, or any heat-flow computation, to the de Bruijn-Newman constant of DH or of any RH-violating function. That sentence has now survived the reading of both named sources and a three-index forward-citation sweep, and one item within it is unread rather than unnamed: the Voronov preprint above.

The sentence

Superseded. The original deliverable sentence was: "The sanctioned phrasing is: first quantitative two-sided bounds for the de Bruijn-Newman constant of an RH-violating L-function." Four things are wrong with it. "L-function" is the wrong noun: the Davenport-Heilbronn function has no Euler product, which is the entire reason it can violate its own Riemann hypothesis and why Dobner needs the extended Selberg class; the careful literature says "the Davenport-Heilbronn zeta function" (Bucur et al.), "the Davenport-Heilbronn function" (Bombieri-Ghosh) or "a Dirichlet series satisfying a Riemann-type functional equation" (Nakamura-Pankowski). "First quantitative" is false: Stopple has an unconditional quantitative bound on a non-zeta constant of exactly this type, and Newman-Wu compute one exactly (ln 2) for a measure. "Two-sided" without a domain restriction is false for the same reason. And the sentence carried no frame, while the number it names is frame-dependent by a factor of 4.

Superseded (extended 2026-08-16). Before the separation corollary was recognized (SEPARATION.md), the adopted sentence ended after its second sentence, as follows; nothing in it changed, it gained a third and fourth sentence carrying the separation at the strength the separation adversary sanctioned: "So far as the literature search recorded in NOVELTY.md reaches, these are 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. They are stated in the normalization of Stopple (arXiv:1301.3158), in which Phi(u) = 4 sum_n a_n n exp(3u/2 - pi n^2 e^{2u}/5), and are four times smaller than the same constant in the normalization of Newman, Rodgers-Tao, Polymath 15 and Dobner."

The sentence, as adopted:

So far as the literature search recorded in NOVELTY.md reaches, these are 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. They are stated in the normalization of Stopple (arXiv:1301.3158), in which Phi(u) = 4 sum_n a_n n exp(3u/2 - pi n^2 e^{2u}/5), and are four times smaller than the same constant in the normalization of Newman, Rodgers-Tao, Polymath 15 and Dobner. In that shared wide normalization the decided floor 0.2304 = 144/625 exceeds Polymath 15's unconditional Lambda_zeta <= 0.22, so Lambda_DH > Lambda_zeta unconditionally; so far as the searches recorded here and in SEPARATION.md reach, this is the first strict inequality between de Bruijn-Newman constants of two Dirichlet series in which both constants are nonnegative, strict orderings without that qualifier being one-line corollaries of exact function-field determinations in print since 2013-2014 (Andrade-Chang-Miller arXiv:1310.3477; Chang-Mehrle-Miller-Reiter-Stahl-Yott arXiv:1411.2071).

The footnote, which is mandatory and is not to be compressed:

Existence, finiteness and nonnegativity of this constant are Dobner's (Acta Arith. 2021), for all of the extended Selberg class, with no member named and no number given. The upper bound is de Bruijn's 1950 strip-contraction theorem, restated as Theorem 7 of Newman and Wu (Bull. AMS 57, 2020), applied to a zero strip computed here; the mechanism is published and surveyed. The strip constant is not new either: Bombieri and Ghosh, Around the Davenport-Heilbronn function, Russian Math. Surveys 66 (2011), 221-270, section 6, determine the least upper bound of the real parts of the zeros of this very function exactly, sigma(tau_+, 1) = 1.120362, which is sharper than the coefficient-domination sigma_0 = 1.39513615823510972... first derived here. What this work supplies on the upper side is an enclosure-carrying elementary derivation of the necessary half of their Theorem 7, which reaches the same abscissa and decides it on two backends at sigma_0' = 1.12036249819 (STRIP2.md, 2026-08-18), and the assembly of the two sides into a strictly positive bracket. Their converse, which turns that abscissa into an exact supremum, is not used and not claimed, so the number remains theirs and what is new here is the grade and the derivation. Quantitative bounds on de Bruijn-Newman constants of objects other than zeta are not new: Stopple bounds Lambda_Kr for quadratic Dirichlet L-functions from below unconditionally (-1.12929e-7 < Lambda_Kr) and from above under GRH, and Newman and Wu determine such a constant exactly (ln 2) for a three-atom measure. The qualifier that carries the claim is that the object here violates its own Riemann hypothesis. Bombieri and Ghosh was read in full on 2026-08-16 (BOMBIERI-GHOSH.md) and contains no de Bruijn-Newman or heat-flow content of any kind. Both remaining named sources were read on 2026-08-18. academia.edu preprint 166936409 is Mesut Ismail, 10.5281/zenodo.21679490, open access on Zenodo; it works the classical wide-frame Lambda_zeta and uses Davenport-Heilbronn only as instrument and negative control. It names no Lambda_DH, gives no bound on one from either side, and its two DH off-line lifetimes, tau1 = 0.1819 and tau2 = 0.0449 in the wide frame, are labelled upper bounds on those lifetimes in a Numerical Observation, so they yield no lower bound on Lambda_DH; read at face value anyway, 0.1819 is below the decided wide floor 0.2304 claimed here. Bombieri and Mueller, On the zeros of certain Epstein zeta functions, Forum Math. 20:2 (2008), 359-385, is the Bohr-method parent of Bombieri and Ghosh's abscissa; its subject is the rate of approach of zeros to the abscissa for Epstein zeta functions of class number 2, its deposited references name neither de Bruijn nor Newman, and its full text is behind a publisher wall and remains unread. The forward-citation sweep on Dobner now runs on three independent indexes (Semantic Scholar, OpenCitations and Google Scholar) plus a web sweep, and the union of seven citing works contains no quantitative Lambda for any non-zeta object; OpenAlex still returns HTTP 429 with no daily allowance and was not used. One citing item is unread rather than unfound: Voronov, A Crowding-Normalized Reformulation of Neighboring-Gap Dynamics for the de Bruijn-Newman Flow (ResearchGate, 2026), whose abstract places it on the zeta-side real spectrum and whose full text is behind Cloudflare.

If one line is needed rather than a paragraph, the safe short form is:

First quantitative two-sided bounds on the de Bruijn-Newman constant of a function that violates its own Riemann hypothesis, in Stopple's normalization, so far as the recorded search reaches.

Do not use "L-function". Do not drop "so far as the search reaches". Do not drop the frame. Do not state the separation without "in which both constants are nonnegative" (SEPARATION.md section 6).