Opened 2026-08-16. Nothing in this directory is a result until the case log in hunts/README.md says how it ended. Everything here obeys the probe discipline: the strongest words used are measured, observed and decided (an enclosure with an exact sign or an exact integer count); the reserved enclosure word belongs to zeta/rigor.py and appears nowhere in this directory.
Frame note added 2026-08-16 (
GATE.mdclosure item (a)). Nothing in the preregistration below is altered; this note only says which normalization its numbers are in, because the mission did not say and aLambdanumber means nothing without it. The deformation defined below sits ats = 1/2 + iz, which is Stopple's published frame (arXiv:1301.3158). de Bruijn as usually quoted, Newman, Rodgers-Tao, Polymath 15 and Dobner all sit ats = (1+iz)/2, where the same constant is four times larger. So the pre-registered prediction P5, "0.0575 < Lambda_DH <= 0.400634", reads "0.2300 < Lambda_DH <= 1.602537" there. Conversion table, derivation and per-row numerical checks:FRAME.md.
Post-registration note added 2026-08-16, after the gate closed (
GATE.md, verdict YES). Nothing in the preregistration below is altered. A corollary of the decided floor was recognized only after the gate: in the wide frame the floor0.2304 = 144/625exceeds Polymath 15's unconditionalLambda_zeta <= 0.22, soLambda_DH > Lambda_zetaunconditionally. This separation was not preregistered, was not a work package, and is recorded with its chain, its grade (cited plus decided) and its adversary-sanctioned phrasing inSEPARATION.md.
The question
hunts/flow_repair/ (closed, 2026-08-07) measured the backward-heat repair times of nine off-line quadruples of the Davenport-Heilbronn function and closed with the sentence "Nothing here bounds Lambda_DH from above". Its floor, Lambda_DH >= 0.0576518, is float grade: one route, no enclosures.
This hunt upgrades that loose end to theorem grade on both sides. In the normalisation flow_repair measured to be exact, with a_n the period-5 coefficients (1, kappa, -kappa, -1, 0),
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, H_0 = Xi_DH,
and Lambda_DH := inf{t : H_t has only real zeros}, which is a well-defined finite real number >= 0 by Dobner's theorem for the extended Selberg class (arXiv:2005.05142, Theorems 1 and 2; the half-line structure {t : all zeros real} = [Lambda_F, inf) is his Theorem 1).
What are the first quantitative two-sided bounds for Lambda_DH?
Positioning, so the claim is neither over- nor under-stated: existence, finiteness and Lambda_DH >= 0 are published (Dobner 2020, for all of S#). Strict positivity is an immediate corollary of Dobner's half-line structure plus any off-line zero (Davenport-Heilbronn 1936; computed by Spira 1994). What is absent from the literature, per the novelty sweep recorded in NOVELTY.md, is any number: no quantitative bound, upper or lower, for the de Bruijn-Newman constant of any RH-violating L-function. The deliverable is that number, from both sides.
Work packages
Independence note added 2026-08-16 (
GATE.mdclosure item (e)). The preregistration below is not altered; this note says what WP1's phrase "two independent winding routes" turned out to be worth, because the word independent was never measured when it was written. Measured withharness/independence.py, the two routes share the whole evaluator: independence radius 9 of 12 declared layers at the granularity ofindependence_decl.py, 8 of 11 at the gate's slightly coarser one, the same declaration either way. Their agreement is evidence about the two schemes that turn H-balls into an integer and about nothing else, and route 2 ran at one t and one box. The independence the claim rests on comes from two other legs, both landed here as runnable scripts:crosscheck_dhflow_winding.py(radius 0, N = 1 at both t) andcrosscheck_quadfree.py(radius 0, one reconvergent layer, a different kappa equation). Layer lists, the two repaired validation gaps and what is still not covered:INDEPENDENCE.md.
- WP1 (lower bound, decided). At the pre-registered t1 = 0.0575 (stretch 0.0576), an argument-principle count with every evaluation an Arb ball proves H_{t1} has a zero strictly off the real axis near the pair-5 site (centre Re z about 240.4165, the deepest measured quadruple). With Dobner's half-line structure this decides Lambda_DH > t1. Two independent winding routes: segment-argument tracking, and direct ball quadrature of H'/H.
- WP2 (upper bound, cited theorem + decided strip). de Bruijn 1950 (Duke Math. J. 17, Theorem 13): if all zeros of H_0 lie in |Im z| <= Delta then H_t has only real zeros for t >= Delta^2 / 2. The exact statement and hypotheses are to be pinned from the original text and its restatements (Csordas-Norfolk-Varga 1988; Ki-Kim 2003) before use, and the strip-to-time factor Delta^2/2 is to be re-derived numerically in-tree (polynomial flow calibration), never recalled. The strip itself, |Im z| <= Delta = sigma_0 - 1/2 with sum_{n>=2} |a_n| n^{-sigma_0} = 1, is decided by directed-rounding interval arithmetic on both backends; scouted value sigma_0 = 1.39513615823511, giving Delta^2/2 = 0.400634.
- WP3 (census, optional). Screen heights above flow_repair's list for deeper quadruples; any pair landing later than 0.0576518 raises the floor.
- WP4 (controls). Lesion: a deliberately mis-centred or on-axis box must fail loudly or return undecided, never a wrong integer. Precision response: every enclosure must shrink when precision rises. Rival framing: the same pipeline pointed at zeta produces no positive floor (no off-line zero is known for zeta), so the number separates DH from zeta only through the already-known off-line zeros; this echoes flow_repair's P5 moral and claims nothing about RH.
Pre-registered predictions
- P1: the winding count at t1 = 0.0575 over the pair-5 box decides N = 1 in the open upper half-plane box, on the first budget, with >= 30 digits of sign margin on every boundary segment.
- P2: the two winding routes agree exactly (both decide N = 1).
- P3: sigma_0 lands in [1.3949, 1.3954] on both backends and the two-backend intervals overlap.
- P4: no surveyed pair beats 0.0576518 below height 600 (flow_repair's data says depth, not height, drives landing times, and the strip caps depth).
- P5: the headline lands as 0.0575 < Lambda_DH <= 0.400634, a ratio of about 7 between the two sides.
Scope
This hunt may write: hunts/lambda_dh_bounds/, figures/, one new docs/NN-*.md if the run earns it, a case-log entry in hunts/README.md, and a pinning test under tests/ (precedent: tests/test_frontier_math_clean_kill.py).
This hunt may not write: zeta/, ontology/, harness/, lean/, and may not promote its own claim into README.md, ROADMAP.md or HANDOFF.md as an established finding.
id: lambda_dh_bounds
question: What are the first quantitative two-sided bounds for the de Bruijn-Newman constant Lambda_DH of the Davenport-Heilbronn function?
frontier: lower 0.0576518 measured float-grade by flow_repair (nine quadruples, no enclosures), upper absent everywhere; Dobner 2020 gives existence, finiteness and Lambda_DH >= 0 with no numbers
proposed_attack: Arb ball argument-principle winding count at t1 = 0.0575 for the lower side; de Bruijn 1950 Theorem 13 with a directed-rounding zero-strip constant sigma_0 for the upper side
dead_routes:
- reading an O(1) residue as an off-line zero without a completeness check (Hunt 2, withdrawn)
- float-grade landing times as bounds (flow_repair already measured them; they decide nothing)
- mp.diff on internally rounded functions (returns exactly 0, pinned by regression tests)
required_oracles:
- Arb ball arithmetic winding counts that must enclose an exact integer
- directed-rounding interval arithmetic on both in-tree backends for the strip constant
- cross-route agreement between H_0 quadrature and zeta.epstein.completed_dh
- published theorems cited with hypotheses checked (de Bruijn 1950, Dobner 2020)
kill_conditions:
- the all-zeros-in-strip form of de Bruijn Theorem 13 cannot be confirmed from the original text or a reliable restatement, in which case the upper bound is withdrawn and the hunt reports the lower side alone
- a winding enclosure fails to bracket an integer after the subdivision budget, in which case no lower-bound claim is made at that t
- any enclosure that fails to shrink when working precision rises, which marks the instrument as the artifact
- any source found to already bound Lambda_DH quantitatively, in which case the novelty claim is withdrawn and the delta restated against that source
agents_may:
- build instruments inside this directory
- run ball-arithmetic computations and record enclosures with their precisions
- fetch and read the cited papers
- write measured and decided values into RESULTS.md and results.json
agents_may_not:
- modify zeta or ontology or harness or lean
- use the reserved enclosure vocabulary of zeta/rigor.py anywhere in this directory
- promote this hunt's claim into repo-level status files
- treat a float measurement as a decided factVocabulary contract
Measured: one float route. Observed: a pattern in measured data. Decided: an Arb or interval enclosure whose exact endpoints settle a sign or an integer count, stated with backend and precision. The composite headline takes the weakest grade of its steps and says so: the lower bound is decided modulo Dobner's Theorem 1; the upper bound is a cited theorem applied to a decided strip constant.