RESULTS: two-sided bounds for the de Bruijn-Newman constant of the Davenport-Heilbronn function

Written 2026-08-16, after the gate (GATE.md) returned NOT YET with a five-item closure list, and after those repairs were made. This file is the hunt's written record. It is not a promotion: per hunts/README.md and MISSION.md, nothing in this directory is a result until the case log says how the hunt ended, and the gate's verdict on the artifacts as they stood was that the mathematics survived every attack while two defects stood in the sentences a reader would quote.

Vocabulary, per MISSION.md. Measured is one float route. Observed is a pattern in measured data. Decided is an enclosure whose exact endpoints settle a sign or an integer, stated with backend and precision. Cited is somebody else's theorem. A composite takes the weakest grade of its steps, and this file says so at every headline. The reserved enclosure vocabulary of zeta/rigor.py appears nowhere in this directory.

Machine-readable form of every number below: results.json.

Revised 2026-08-18, after the gate closed YES, by a hardening pass of three tasks run against the closed record. Every field they touch carries an update block with the superseded text kept beside it, per the repo's correction style. In one line each: the upper bound sharpened by the full factor 2.082, in-tree rather than by citation (section 3.6, STRIP2.md); M2 is no longer prose but a proved lemma with decided constants (section 5.1a, M2-LEMMA.md); and the three open prior-art items are closed or reduced, with the novelty sentence surviving unchanged (section 7.3 items 6, 7 and 9). Nothing broke. No decided lower-bound number moved, so the separation of SEPARATION.md is untouched.

0. The claim, in both frames

The de Bruijn-Newman constant is not a number attached to a function. It is a number attached to a function plus a choice of where the critical line is parameterised, and two conventions are in circulation, both refereed, neither announcing itself. Full derivation, sources and per-row numerical checks: FRAME.md.

narrow frame : s = 1/2 + i z (Stopple arXiv:1301.3158; this hunt; Newman-Wu's kernel) wide frame : s = (1 + i z)/2 (de Bruijn as usually quoted; Newman; Rodgers-Tao; Polymath 15; Dobner; zeta/heatflow.py)

Lambda(wide) = 4 * Lambda(narrow), Delta(wide) = 2 * Delta(narrow).

In the normalization of Stopple (arXiv:1301.3158), s = 1/2 + iz, in which Phi_DH(u) = 4 e^{3u/2} sum_{n>=1} n a_n exp(-pi n^2 e^{2u}/5) and H_t(z) = int_0^inf e^{t u^2} Phi_DH(u) cos(zu) du:

0.0576 < Lambda_DH <= 0.19242481458026887663805, 0.0576 = 36/625 exactly.

In the normalization of Newman, Rodgers-Tao, Polymath 15 and Dobner, s = (1 + iz)/2, where the same constant is four times larger:

0.2304 < Lambda_DH <= 0.7696992583210755065522, 0.2304 = 144/625 exactly.

The ratio upper/lower is 3.341 and is frame-free (FRAME.md section 6). It is the honest measure of how loose the bracket is.

Sharpened 2026-08-18. The superseded headline is kept here rather than deleted, and it was never wrong. Through 2026-08-17 this file read

narrow : 0.0576 < Lambda_DH <= 0.4006343708899557 wide : 0.2304 < Lambda_DH <= 1.6025374835598228 ratio : 6.955

from Delta = sigma_0 - 1/2 with sigma_0 = 1.3951361582351097210613..., the coefficient-domination abscissa of STRIP.md, still correct and still decided (sections 3.1 to 3.5 below, unaltered). The upper side is now reached instead by a phase obstruction in the two Euler products (STRIP2.md, strip2.py, section 3.6 below), which decides the abscissa sigma_0' = 1.12036249819 on both backends, a factor 2.082030697360155 better. The lower side did not move in any digit, so the separation corollary Lambda_DH > Lambda_zeta, which rests on the floor alone, is unaffected (SEPARATION.md). The bracket ratio falls from 6.955 to 3.341.

The two new decimals are exact, not outward roundings, because the abscissa is decided at an exact rational: Delta = 0.62036249819 exactly, so Delta^2/2 = 3848496291605377532761/20000000000000000000000 terminates and the wide value is exactly four times it. STRIP2.md section 5.2 displays the narrow value rounded outward to 22 decimals as 0.1924248145802688766381; that is the same bound one display ulp higher, and either may be quoted.

Grade of the headline: cited plus decided, and neither side is decided alone. The lower side is an enclosure-carrying integer count (decided) turned into an inequality by monotonicity from de Bruijn 1950 Theorem 13 at Delta = 0 (cited) and made strict by Dobner 2020 Theorem 1 (cited), and it additionally rests on one analytic step, M2, whose derivation is prose rather than an enclosure or a cited theorem (section 5). The upper side is a decided strip constant (both backends) fed to de Bruijn 1950 Theorem 13 (cited). Taking the weakest step, the bracket is a cited-theorem statement about decided constants, with one prose lemma inside the lower side.

Update 2026-08-18. M2 is no longer prose. M2-LEMMA.md states it as Lemma M2 and proves it, with every constant a reported Arb ball and every hypothesis a decided predicate, and m2_lemma.py exercises it by four routes and two falsification attacks. The composite grade of the bracket does not change, because the weakest step was and remains a citation: the proof's one non-elementary input is the evenness Phi_DH(-u) = Phi_DH(u), which is the cited functional equation already counted above. The last sentence should now read: the bracket is a cited-theorem statement about decided constants, with one proved lemma inside the lower side whose own weakest input is that same citation. The paragraph above is kept as written. Section 5 carries the same update in full.

Why publishing both columns is not pedantry. Read in the common wide frame, where the zeta literature lives (FRAME.md section 7):

Lambda_zeta <= 0.22 (Polymath 15, cited) Lambda_DH > 0.2304 (this hunt, decided modulo two cited theorems)

so the Davenport-Heilbronn constant sits above the best known upper bound for zeta. Stated in the narrow frame alone, 0.0576 sits below 0.22 and a reader draws the opposite conclusion. That single comparison is the reason the frame goes in the sentence every time.

That comparison is now carried as a named claim, the separation (SEPARATION.md; sources pinned at source in POLYMATH-PIN.md). The chain in full, every link graded:

0 <= Lambda_zeta cited (Rodgers-Tao Theorem 1) Lambda_zeta <= 0.22 cited (Polymath 15 Theorem 1.1, unconditional) 0.22 = 11/50 < 144/625 = 0.2304 exact rational arithmetic: 144 * 50 = 7200 > 6875 = 11 * 625 0.2304 = 4 * (36/625) < Lambda_DH decided (winding N = 1 at t = 36/625 narrow, python-flint 0.9.0 (Arb), 420 bits; second witness mpmath dps

1. modulo cited (Dobner Theorem 1) and the derived frame factor 4 Lambda_DH <= 0.7696992583210755065522 decided strip constant fed to cited de Bruijn 1950 Theorem 13 (not needed for the separation; this link read <= 1.6025374835598228 through 2026-08-17, and sharpening it moved nothing else in the chain)

hence, in the wide frame, Lambda_DH > Lambda_zeta unconditionally. Composite grade cited plus decided, weakest step cited; the decided link carries M2 (section 5; a proved lemma since 2026-08-18, M2-LEMMA.md, rather than the prose one this sentence originally named). Frame-invariant: in the narrow frame the chain reads 0 <= Lambda_zeta <= 0.055 < 0.0576 < Lambda_DH, the same inequality by the same cross-multiplication (36 * 200 = 7200 > 6875). The sanctioned novelty phrasing, its qualifier ("both constants nonnegative") and the function-field precedents that make the qualifier mandatory are in SEPARATION.md sections 5 and 6; the exact rational core is pinned by tests/test_lambda_dh_separation.py.

1. Instrument truth-telling

The instrument is instrument.py: exact rational inputs, an Arb ball for H_t(z) and H_t'(z) by rigorous adaptive integration of a truncated series with two hand-derived tail bounds. validate.py is its validation runner and validation.json its output.

1.1 What validation reports

From validation.json (backend python-flint 0.9.0 (Arb)), six checks after the gate's repairs, all_pass: true, total 160.0 s:

Checks 5 and 6 are the gate's closure item (e) landed. Check 5 was pointed at instrument.phi_ball, which no decision path calls; check 6 did not exist. The superseded text of each is kept at its check in validate.py rather than deleted.

1.2 What the gate found that validation does not have purchase on

This subsection is as load-bearing as the previous one and is stated at the same prominence, per the gate's own instruction.

1. The two hand-derived tail bounds are outside validation's reach. Adversary 4 (attack_adversary4_instrument.md section 2e) measured them at the parameters validate.py uses: the series tail is about 9.47e-210 and the u tail about 1.68e-160 at the decision point, against a delivered ball radius of 1.49e-124 and a float-reference allowance of 1e-135. Both sit 36 to 41 orders of magnitude below the ball radius. validation.json's containment rows would read contains_float: true even if _u_tail were understated by a factor of 1e25 and _series_tail_finite by 1e75. The eleven containment rows establish that the integrator and the series are right. They establish nothing about the tails.

The tails themselves were audited, independently and for the first time, by that adversary: 36 cases for the omega tail, 12 for the series-truncation tail, 11 for the integral tail beyond U, all dominating, with the discriminating case at the actual decision parameters clearing the true tail by a factor of only 2.28. That is real evidence and it is measured, not decided.

1. The only mpmath.iv cross-leg pointed at code no decision path calls. phi_ball appears in instrument.py and validate.py and nowhere else; _H_core carries its own inline copy of the theta-recurrence series, and the two copies were never checked against each other. Adversary 4 section 3 demonstrated this with a planted fault in a copy of the module: dropping the factor n from the n = 4 (mod 5) coefficient inside _H_core alone leaves phi_ball bit-identical while moving H at the box corner from 6.2078e-82 to 1.6608e-07, a shift of 75 orders of magnitude that both winding routes would inherit identically.

1. Four of validation's twenty-two component checks are decided by the widening allowance rather than by the data. On the four real-z rows the imaginary component's eps/radius is about 1.14e+25 (7.1e+28 at prec 600), and the unwidened comparison fails, because the dps-150 float reference returns a spurious imaginary part around 2.7e-154 while the instrument's imaginary ball has radius 8.78e-161 and correctly contains 0. This is the reference's round-off and not an instrument defect, and the docstring anticipates it, but the reported boolean does not show it.

1.2b What the repairs bought, and what they left blind

All three gaps in section 1.2 are now repaired in validate.py and the artifact in section 1.1 is the repaired run. INDEPENDENCE.md section 5 records both repairs in full, and two of its numbers belong here because they bound what the repairs are worth:

• The iv cross-leg now has measured purchase, with one named blind spot. instrument._truncated_integrand was hoisted out of _H_core as a module-level factory, unchanged in operations, order and precision, so the callable the integrator receives is reachable; the refactor moved no number (H_ball at z = 240.4165 + 0.02i, t = 23/400, prec 420 returns midpoint 4.3658433958660405e-83 and radius 1.4894837452822593e-124, bit-identical to the values already in validation.json). Planting a fault in the recurrence, seeding w = q^2 instead of q^3 so every term carries the wrong power of q, is caught at 6 of the 7 points. The point that misses is u = 5/2, where exp(-pi e^{2u}/5) is about e^{-93} and the truncated sum is its own first term to well below the interval width. A check at large u sees only n = 1, and that is the blind spot to know about.
• The tail check has a stated blindness radius: 8.02. That is the smallest domination ratio over all eighteen rows, so a bound deflated by more than about a factor of 8 is caught at one of these points and a bound deflated by less is not. The stress regime the gate asked for is included, |Im z| up to 50, which is where the |cos(zu)| <= cosh(yu) <= e^{yU} step would fail first if it were wrong, and it does not. One row is a deliberate refusal: _u_tail returns nothing when its own hypothesis cannot be decided, and _H_core then raises rather than integrating.

Both checks are measured and both are necessary rather than sufficient, exactly like winding.measured_h2_guard. Passing does not make a bound right; failing proves one wrong. Reporting the blindness radius rather than only the lesion threshold is docs/25's standing consequence, and it is the same lesson control 5 teaches in the other direction.

1.3 What has a second implementation, and what does not

kappa is well covered: the Arb self-duality linear solve (kappa_ball, 500 bits), the zeta.epstein mpmath solve behind KAPPA_REF, adversary 4's Gauss sum route from tau(chi) for the odd character mod 5 (a different equation, overlapping at 200, 400 and 600 bits), adversary 3's independent linear solve at s = 2.3 + 1.7i (dps 50), and FRAME.md's own route, agreeing to 32 digits. Bombieri and Ghosh's tau_+ = -phi + sqrt(1 + phi^2) is the same constant and agrees to 34 digits (BOMBIERI-GHOSH.md section 2).

Not covered by any second implementation before the gate: _H_core's inline series (checked only against a float route), the three tail bounds at the rho = 1 call site, default_U, and acb.integral itself.

The one genuinely independent evaluator leg is adversary 4's quadrature-free route, since landed in this directory as crosscheck_quadfree.py with output crosscheck_quadfree_results.json: kappa from the Gauss sum of the odd character mod 5 (a different equation, not merely a different implementation), H_0 by Hurwitz zeta with no quadrature at all, H_t by Taylor in t from dH/dt = -d^2H/dz^2 with an explicit truncation remainder. At K = 100, prec 900, remainder ball 3.0012585847746115e-158 against the instrument's radius 1.1628170864163578e-124, it agrees with instrument.H_ball at 8 of 8 points on the boundary of the very box that decides N = 1 at t = 23/400, zero mismatches, with a min_measured_agreement_digits of 40.32. Both sides are enclosures and the independent one is about 34 orders of magnitude tighter, so the overlap is a real agreement rather than a wide ball swallowing a narrow one.

That artifact also corrects a number in the gate's own summary, which is worth recording because it runs against the hunt's interest: the gate wrote that this route reproduces the instrument "to about 22 significant digits", and that was the width of the printed comparison rather than a measurement. The measured relative agreement, an upper bound for |taylor - instrument| over a lower bound for |instrument|, is 40.32 digits at worst over the eight points.

2. The lower bound

2.1 What was decided

winding.py (route 1: segment-argument winding), backend python-flint 0.9.0 (Arb), prec 420 bits, exact rational contour, from winding_results.json:

Both boxes lie strictly inside the open upper half-plane in exact rational arithmetic: the t2 box has Im z >= 3/1024 > 0 throughout. min_ball_margin_digits is 40.32 and 39.87; min_chord_margin_digits is 0.02 for both, and section 5 explains why that second number is not a health metric.

decided_floor_t = 36/625. The conjugate rectangle in the lower half-plane carries the same decided count by H_t(conj z) = conj(H_t(z)), which the kappa ball plus real coefficients establish (winding_results.json, symmetry_note).

2.2 By which routes

Four counts reached the same integer, and they are not four independent ones. The measured radius, per gate closure item (e), replaces the phrase "two independent winding routes" that MISSION.md WP1 and prediction P2 used:

• Route 1, winding.py, decided N = 1 at t = 23/400 and at t = 36/625.
• Route 2, winding_quad.py (ball quadrature of H'/H, degree-48 panel-local Taylor models, prec 420 with Taylor coefficients at prec 700), decided N = 1 at t = 23/400 on an overlapping but different box (Re in [2400953/10000, 2405953/10000], Im in [1/250, 3/50]). Its winding ball is 1 +/- 1.28e-12. Route 2 was run only at t = 23/400, not at the headline t = 36/625.
• Routes 1 and 2 share 9 of 12 declared layers, independence radius 9, per independence_results.json: the whole evaluator is one implementation run twice, and their agreement is evidence about the bookkeeping that turns H-balls into an integer and about nothing upstream of it. The gate's own coarser declaration reported 8 of 11, and independence_decl.py re-runs that granularity to show the two differ by one layer split in two and by nothing else (matches_gate_8_of_11: true). The load-bearing invariant does not depend on the granularity: the routes duplicate none of the evaluator. The standing caveat is harness/independence.py's own and is not a formality: it measures a declaration, not the code, and an undeclared shared layer is precisely the fault the structure cannot see.
• The genuinely independent legs are the gate's own, now landed here rather than living in a scratch directory. crosscheck_dhflow_winding.py (route 3) is an argument-principle count sharing no code with the instrument: hunts/flow_repair's DHFlow composite Gauss-Legendre evaluator, mpmath floats at dps 130, uniform dense boundary sampling with continuous-argument tracking refined until every step is under 1 radian. From crosscheck_dhflow_results.json: 5814 DHFlow nodes, 256 boundary evaluations per box, maximum consecutive argument step 0.2854607296912167 and 0.39463362905733557, total argument 6.283185307179587 and N = 1.0000000000000002 on both published boxes, t = 23/400 and t = 36/625, using route 1's own boxes as recorded in winding_results.json; agrees_at_both_t: true, 348.2 s. The minimum |H_t| on the boundary was 2.4488139927472482e-84 at t = 23/400 and 8.644114156488054e-85 at t = 36/625, which is the scale that makes the count hard and is why the instrument needs balls rather than floats. Measured radius against route 1: 0 shared layers. That is the first and only second witness the headline value 0.0576 has, and it is measured, float grade, by construction: it cannot decide an integer and does not claim to, its job is to catch a wrong one. One layer it does not duplicate: its kappa comes from zeta.epstein.kappa, the mpmath form of the same self-duality linear solve, so its agreement is not evidence about the equation that defines kappa. Route 4, crosscheck_quadfree.py (section 1.3), supplies that leg by deriving kappa from the Gauss sum instead, and shares with route 1 only the Arb backend, reconvergent, radius 0.

2.3 The exact logic from a nonreal zero to Lambda_DH > t

Let t2 = 36/625. The decided count says H_{t2} has exactly one zero in a rectangle whose interior lies in Im z >= 3/1024 > 0, so H_{t2} has a zero that is not real, so H_{t2} does not have only real zeros.

Weak inequality, with no appeal to Dobner. THEOREM13.md section 4c: if all zeros of H_{t0} are real, then Phi_{t0}(u) = e^{t0 u^2} Phi_DH(u) still satisfies Theorem 10's conditions (integrable, hermitian, O(e^{-|u|^{b'}}) for any 2 < b' < 3), so Theorem 13 applies to it with Delta = 0 and gives all zeros of H_t real for every t > t0, since max(0 - lambda^2, 0)^{1/2} = 0. Rodgers-Tao attribute the same monotonicity to Polya. Contrapositive: since H_{t2} is not all-real, no t <= t2 is all-real. Hence

Lambda_DH >= t2 = 36/625 = 0.0576.

Strictness, and only here, from Dobner. Dobner 2020 Theorem 1 states that {t : xi_t^F has only real zeros} = [Lambda_F, inf), closed at the left end. Without closedness the all-real set could be (t2, inf) and Lambda_DH would equal t2. Dobner's theorem is a statement in the wide frame. The conversion, derived in THEOREM13.md section 6 and FRAME.md section 3 from his own equations (5) and (6), is

Phi_F(u) = Phi_DH(2u), xi_t^F((1+iz)/2) = H_{t/4}(z/2),

and t -> t/4 is an increasing bijection of the real line, so it carries closed half-lines to closed half-lines. Hence {t : H_t all real} = [Lambda_F/4, inf) is closed at the left in this frame too, and Lambda_DH = t2 would force H_{t2} all-real, contradiction. Therefore

Lambda_DH > 36/625 = 0.0576 (narrow), Lambda_F > 144/625 = 0.2304 (wide, since Lambda_F = 4 Lambda_DH).

S# membership of the Davenport-Heilbronn function, which is what licenses Dobner's theorem, is verified condition by condition in THEOREM13.md section 5 and independently against Dobner's verbatim (i)-(iii) by adversary 1 section 4. No Euler product is required by S#, and the function genuinely has none (a_6 = a_1 = 1 while a_2 a_3 = -kappa^2), which is exactly why it can violate its own Riemann hypothesis.

What the frame error did and did not damage. The earlier text asserted that this hunt's deformation and Dobner's "share Lambda exactly". That sentence was false by a factor of 4 and is preserved, inside a correction box, in THEOREM13.md section 6. It did not damage the strictness argument, because only the increasing-bijection property of t -> t/4 is used there. What it would have licensed is quoting Dobner's Lambda_F for this function as this hunt's number, four times too large, and what it did cause is the zeta calibration in NOVELTY.md, now repaired.

2.4 The margin, honestly

The binding constraint on route 1 is not ball precision. min_ball_margin_digits is 40.32, while min_chord_margin_digits is 0.02, so the chord-clearance versus Taylor-tube inequality clears by a factor of about 1.05. That inequality depends on M2, and M2 is where section 5 recorded a standing blind spot. That blind spot is closed at source as of 2026-08-18 (M2-LEMMA.md); the binding constraint itself is unchanged, and so is every number in this paragraph.

3. The upper bound

Read 3.6 first if you want the bound of record. Sections 3.1 to 3.5 derive and decide the coefficient-domination abscissa sigma_0 = 1.3951361582351097210613... and are kept exactly as written on 2026-08-16. Nothing in them is wrong and nothing in them is altered. They are no longer the headline: section 3.6 records a second, independent and sharper route to a zero-free half-plane, decided on both backends, which supersedes the constant of 3.1 to 3.5 and reuses their sections 3.1(d) and 3.1(e) unchanged. STRIP.md and STRIP2.md stand in the same relation, and both are kept for the same reason: they are two valid derivations and a reader should be able to see both.

3.1 The strip argument

STRIP.md and strip.py. With g(sigma) = sum_{n>=2} |a_n| n^{-sigma} - 1 for sigma > 1:

(a) Uniqueness. Every term with a_n != 0 is strictly decreasing in sigma, and 0 < kappa < 1 is decided by the Arb ball, so g is strictly decreasing and continuous on (1, inf), tends to +inf as sigma -> 1+ and to -1 as sigma -> inf. Exactly one root sigma_0.

(b) No zeros of f for Re s > sigma_0. Strict monotonicity gives sum_{n>=2} |a_n| n^{-sigma} < 1, so |f(s)| >= 1 - sum_{n>=2} |a_n| n^{-sigma} > 0. The bound depends on Re s alone, so there is no large-height escape route (adversary 3 section 2.5).

(c) The boundary-equality case, Re s = sigma_0 exactly. This is the case a triangle-inequality argument usually leaves open, and it is closed exactly. Suppose f(sigma_0 + it) = 0. Then W = sum_{n>=2} a_n n^{-s} = -1 with |W| = 1 = sum_{n>=2} |a_n| n^{-sigma_0}, so the triangle inequality is an equality and every nonzero term is a strictly negative real multiple of a common unit vector: n^{-it} = -1 when a_n > 0 and n^{-it} = +1 when a_n < 0. Take n = 3 (a_3 = -kappa < 0): 3^{-it} = 1. Take n = 4 (a_4 = -1 < 0): 4^{-it} = 1. Take n = 12 (12 = 2 mod 5, a_12 = kappa > 0): 12^{-it} = -1. But 12^{-it} = 3^{-it} 4^{-it} = 1. Contradiction, and t = 0 fails the same constraints. Adversary 3 section 2.2 confirms this as airtight and adds that it is not load-bearing: step (b) at the decided rational sigma* = 1.3951361582351097210613589375 already delivers the closed strip Theorem 13 wants.

(d) The gamma factor and the reflection. (pi/5)^{-(s+1)/2} is entire and nonvanishing; Gamma((s+1)/2) is nonvanishing with simple poles only at s = -1, -3, -5, ..., all at Re s <= -1. So on Re s >= sigma_0 > 1 the gamma factor is analytic and nonvanishing and F = gamma f has no zero there; F entire with F(s) = F(1-s) reflects this to Re s <= 1 - sigma_0. Every zero of F lies in the open strip 1 - sigma_0 < Re s < sigma_0.

(e) The trivial zeros, and why the claim is about F and not f. 1/gamma has a simple zero at each s_m = -(2m+1), so f vanishes there: f has trivial zeros at s = -1, -3, -5, ..., outside the strip, forced by the functional equation exactly as zeta's are and shifted to the odd negative integers because the character is odd. "All zeros of f lie in the strip" would therefore be false. The decided statement is about F, whose zeros are exactly the nontrivial zeros of f. Adversary 3 section 2.4 checked this has teeth rather than being a formality: measured at dps 50, f(-1) = 1.02e-51, f(-3) = 3.92e-51, f(-5) = 4.89e-50, f(-7) = 1.22e-48 while F(-1) = F(2) = 1.77952795928, F(-3) = 4.29503671195, F(-5) = 16.9670072265 are finite and nonzero, so the zeros of f at those points are simple and cancel the gamma poles exactly. A double zero at any s_m would put a zero of F outside the strip and collapse the upper bound; it cannot happen, because F(s_m) = F(1 - s_m) and 1 - s_m = 2m + 2 sits in Re s >= 2 where g(2) < 0 already keeps f away from zero.

3.2 The decided sigma_0, on both backends

From strip_results.json. Intervals are outward-rounded decimal strings containing the exact rational bisection endpoints.

kappa is decided at 500 bits in [0.284079043840412296028291832393126169091088088, 0.284079043840412296028291832393126169091088089], inside KAPPA_REF +/- 1e-39 by exact rational comparison. Delta^2/2 < 0.4007 is decided on both backends against 4007/10000; g(2) < 0 is decided on both. 79 flint sign decisions and 46 iv decisions, every bisection endpoint an exact rational, every sign settled by an enclosure excluding zero, an undecided sign a loud abort. The backend intervals overlap and both sit inside the mission's scouted 1.39513615823511 +/- 5e-15. Whole run 1.03 s.

In the wide frame Delta doubles to 1.7902723164702194421227177424 and Delta^2/2 quadruples to the decided interval [1.6025374835598227777878190108, 1.6025374835598227777878192480] (FRAME.md section 6).

An independent recomputation from Hurwitz zeta at dps 50, sharing no code with strip.py, returned sigma_0 = 1.3951361582351097210613588973265388 (adversary 3 section 2.7), inside the flint bracket; adversary 5 section 7 returned 1.39513615823510972106135889733 by a third route; the gate's own fourth route returned 1.3951361582351097210613588973 with g decided negative at the bracket's upper endpoint (-1.28e-25) and positive at its lower (+8.3e-26).

3.3 de Bruijn Theorem 13, as pinned

THEOREM13.md section 1 transcribes Theorem 13 (Duke Math. J. 17 (1950), p. 205) from the image-only publisher scan by visual page reads:

THEOREM 13. If F(t) satisfies the conditions of Theorem 10, and if all the roots of (3.6) lie in the strip |Im z| <= Delta, then all the roots of g(z) = int_{-inf}^{inf} F(t) e^{(1/2)lambda^2 t^2} e^{izt} dt lie in the strip (3.8) |Im z| <= {Max (Delta^2 - lambda^2, 0)}^{1/2}.

All zeros, in both hypothesis and conclusion. The "(all but a finite number of the roots)" parenthetical appears in Theorems 11 and 12 on the same and facing pages and is absent from Theorem 13. Adversary 3 section 2.1 re-read the same scan independently and reports Theorems 10 to 14 matching word for word. Two typeset restatements corroborate: Dobner's Theorem 3 (|Im z| <= max(Delta^2 - 2t, 0)^{1/2}) and Newman-Wu 2020 Theorem 7 (|Im z| <= max(Delta^2 - lambda, 0)^{1/2} for the multiplier e^{lambda t^2/2}). Kill condition 1 of MISSION.md is not triggered.

Theorem 10's class conditions are the complete hypothesis set: integrability, hermitian symmetry F(t) = (F(-t))*, and decay O(e^{-|t|^b}) with b > 2. There is no positivity condition, which matters because Phi_DH has mixed-sign coefficients. Phi_DH satisfies all three: it is real, so hermitian symmetry reduces to evenness, which is the functional equation F(s) = F(1-s) in disguise (THEOREM13.md section 5 item 2; measured by the gate to relative 1.0e-34, by FRAME.md to 2.0e-41 at u = 0.1, and by adversary 1 to 5.3e-57 in the theta form); and |Phi_DH(u)| <= 4 e^{3u/2} e^{-(pi/5) e^{2u}} S(0) with S(0) = 1.32368007594847 gives b = 3 > 2 (calibration.json, phi_dh_decay_domination).

Writing the multiplier as e^{t u^2} sets t = lambda^2/2, so the all-real threshold is t >= Delta^2/2. This is a statement about the multiplier and is frame-free, because Lambda/Delta^2 is invariant under the z-rescaling that separates the two frames (FRAME.md section 2). A bound of the form Lambda <= Delta^2/2 is a statement about a ratio; a bound of the form Lambda <= 0.4006 is not.

3.4 The Delta^2/2 calibration, derived rather than recalled

calibrate_theorem13.py, output calibration.json. All measured, one route each. Route 0 first checks that the e^{tu^2} multiplier under the integral and the finite polynomial series both satisfy the backward heat equation dG/dt = -d2G/dz2 (worst relative residual 0.0 for the quadrature route at dps 30, 4.195e-06 for float64 central differences at h = 1e-5, consistent with the h^2 truncation), which is what licenses calibrating the integral multiplier with polynomials.

• Route A, bare conjugate pair p(z) = z^2 + Delta^2: 2t*/Delta^2 is 1.0 at Delta = 0.6, 1.0000000000000 at Delta = 0.895136 (the DH strip's own Delta, landing at t* = 0.40063422924800), 1.0 at Delta = 1.2. A claimed factor Delta^2/8 is violated by a factor of 4 and 2 Delta^2 is slack by 4.
• Route B, cosine polynomial cos z + c: ratios 0.8366, 0.9839, 0.99967 at c = 2.0, 1.05, 1.001, climbing to 1 as c -> 1+. The constant 1/2 is sharp in this family.
• Route C, pair plus distant real zeros: ratios 0.9329, 0.99506, 0.99980 at A = 5, 20, 100. Spectator zeros only accelerate the landing.

Adversary 3 section 2.8 reproduced the same dictionary independently at D = 0.3, 0.6, 0.895136, 1.0, 2.0, all ratios 1.0000000000, and refuted D^2/8 and 2D^2 each by a factor of 4.

3.5 The upper bound itself

All zeros of H_0 = Xi_DH satisfy |Im z| < Delta (strictly, section 3.1), hence a fortiori |Im z| <= Delta, so Theorem 13 gives all zeros of H_t real for t >= Delta^2/2, hence Lambda_DH <= Delta^2/2. The headline decimal 0.4006343708899557 sits above the decided flint upper endpoint 0.4006343708899556944469548120, by 5.553e-18 (adversary 3 section 2.7), so the inequality is safe and the rounding is outward. It is a rounding, not an exact value, and should be read as one.

3.6 The sharpened bound of record: a phase obstruction

STRIP2.md and strip2.py, added 2026-08-18; decided values in strip2_results.json. This section supersedes the constant of 3.1 to 3.5, not their argument.

Why coefficient domination is weak, quantified. Section 3.1(b) replaces each n^{-it} by an independent worst case, so it is the free-phase relaxation and needs the L1 coefficient mass below 1. The phases are not free: n^{-it} is determined by its values at the primes, multiplicatively. Section 3.1(c) already uses that once qualitatively, in the 12 = 3 * 4 boundary argument. Both sides of the slack, decided at 192 bits:

At the true abscissa the relaxation still asks for a coefficient mass of 3.94 to be below 1; at sigma_0, where it finally concludes, the primes can supply only 1.526 radians against the 2.588 a zero needs. One fact from either side, and the factor 2.082 in Delta^2/2 is what it costs.

The obstruction. With chi the odd primitive character mod 5 and A = (1 - i kappa)/2, a_n = A chi(n) + conj(A) conj(chi)(n) for every n (decided: five acb residual balls containing 0 with radius below 1e-40, chi from flint's own Dirichlet character table), so for Re s > 1

f(s) = A L(s, chi) + conj(A) L(s, conj chi), f(s) = 0 <=> R(s) := L(s, chi)/L(s, conj chi) = -conj(A)/A = exp(i (pi + 2 arctan kappa)).

Both Euler products converge absolutely and neither vanishes there, so this is an equivalence and not merely an implication. In the product R(s) = prod_p (1 - conj(chi)(p) p^{-s})/(1 - chi(p) p^{-s}) only the primes with chi(p) = +-i, that is p = 2, 3 mod 5, contribute anything: for those, with u = chi(p) p^{-s} and |u| = p^{-sigma}, the factor is (1+u)/(1-u), whose argument is bounded by 2 arctan(p^{-sigma}). That bound is a Moebius image of a disc: the image has centre C = (1+r^2)/(1-r^2) and radius rho = 2r/(1-r^2), and arcsin(rho/C) = 2 arctan r. Since C^2 - rho^2 = 1, the argument-maximising point has modulus exactly 1, so the |R| = 1 constraint that (3.2) also demands is free there and no modulus-phase trade sharpens this further. Hence

if Theta(sigma) := sum_{p = 2,3 mod 5} 2 arctan(p^{-sigma}) < tau, then f has no zero on the line Re s = sigma,

and Theta is a sum of strictly decreasing positive terms, so one decided sigma closes the whole half-plane. Sections 3.1(d) and 3.1(e), the gamma factor and the trivial zeros, then carry it to F unchanged.

No prime counting anywhere. The head is summed exactly from a sieve; the tail is closed by the Euler products of zeta and of L(., chi5) themselves, through T1 - Tchi = 2 Q + (E_chi - E_1) with the even-k terms cancelling exactly because chi5(p)^2 = 1. Only odd k >= 3 survives, giving eps3 = (2/3) P^{1-3 sigma}/((3 sigma - 1)(1 - P^{-2 sigma})). Bounding the two E separately would leave an O(P^{1-2 sigma}) error instead: at P = 10^5 and sigma = 1.12036249819 that is 2.52e-07 against 4.42e-13, a factor 5.7e+05.

The decided numbers.

with 4814 class primes on both legs, overlapping intervals, and margin tau - Theta_up(sigma_0') = 5.12e-11. Two-sided enclosure of the root itself (flint, from bisecting Theta_lo as well): [1.1203624981832156488068, 1.1203624981833869487332], width 1.71e-13, which is the head/tail systematic and not the ball precision. Hence

sigma_0' = 1.12036249819 = 112036249819/100000000000 (exact rational) Delta = 0.62036249819 (exact) Delta^2/2 = 0.19242481458026887663805 (narrow, exact) = 0.7696992583210755065522 (wide, exact)

A flint-only deep point (320 bits, P = 10^7, 332442 class primes, one evaluation, 5.9 s) decides sigma = 1.1203624981833251, giving 0.1924248145761280189989039 narrow and 0.7696992583045120759956154 wide.

The obvious sharpening, tried and decided useless. Regrouping the series into period-5 blocks and applying the mean value theorem gives |B_k| <= |s|[3(5k+1)^{-sigma-1} + kappa(5k+2)^{-sigma-1}], correct and O(n^{-sigma-1}) per block. The |s| = sqrt(sigma^2 + t^2) is not an artifact: both pairs in a period share the midpoint 5k + 5/2 and their first-order terms add, B_k = s(3 + kappa)(5k + 5/2)^{-s-1}(1 + O(|s|/n)), so a trapezoid correction only refines the constant and the next order carries |s|^2. Measured (mpmath dps 40), the bound tracks |B_k| to 1.0062 when |s| << n and overshoots by 1104 when |s| >> n. The hybrid split gives a decided but height-restricted strip that climbs back to sigma_0: sigma_block(T) = 1.19585459 (T = 10), 1.33701478 (100), 1.37495619 (1000), 1.38762602 (10^4), 1.39224106 (10^5), limit 1.3951361582. de Bruijn's theorem consumes a half-plane statement, so this cannot feed it, and beyond T = 10 it is already worse than the phase bound.

Eight controls, each of which aborts the run rather than downgrading a claim (STRIP2.md section 6): the character decomposition and the phase target as acb residual balls (decided); zeta.epstein.dh_f against A L + conj(A) Lbar at three complex points with large imaginary part, to 1.2e-41 (measured); the phase lemma against 4001-point circle samples at four radii (measured); Bombieri and Ghosh's section 9 finite claim recomputed from the exact sieve, threshold prime 6323 and cardinality 420, both matching (decided, and it shares no machinery with their Theorem 7); the Euler-product tail identity against an explicit partial tail (measured); L(., chi5) by the Hurwitz combination against flint's acb.dirichlet_l (decided); the iv enclosures of zeta, L and arctan containing the corresponding Arb balls at nine points (decided); and Theta enclosures at P = 10^3, 10^4, 10^5 intersecting, common width 1.34e-12 (decided). Plus a control at the sibling Titchmarsh root: the identical head, tail bound and bisection with only the phase target changed decide 1/kappa = 3.52014702134020199243... against their published tau_- = -3.520147021340, and sigma(tau_-, 1) = 2.38228610898712387152... against their published 2.3822861089, hitting all ten digits. Nothing in the instrument was built around that constant.

Relation to Bombieri and Ghosh, stated exactly. Their Theorem 7 at q = 1 and xi = kappa is Theta(sigma) = tau, term for term. Their necessary half is what is derived above, from the Euler product and one Moebius image, with no Bohr theory and no Kronecker theorem; their converse, which makes the abscissa an exact supremum rather than an upper bound, is not used and not claimed. What is new here is the grade and not the number. BOMBIERI-GHOSH.md set the two conditions for adopting the sharper constant at the decided rung, that Theorem 7's hypotheses be checked in-tree and the equation re-solved with outward rounding; both are now met, and the constant no longer needs adopting from the literature at all because the inequality it supplies is derived here. One correction falls out, and it is against an in-tree artifact rather than against the paper: at P = 10^7 and 320 bits, 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 for tau_+ and 6e-18 for tau_-. Bombieri and Ghosh print six and ten decimals and this instrument reproduces both exactly.

Honest ceiling. The headline sits 6.8e-12 above the decided lower end of the root enclosure at P = 10^5 and the deep point about 4e-17 above it; both error sources fall like P^{1-3 sigma}, so accuracy here is essentially free and the strip constant is no longer where the looseness is. What is left on the upper side is (i) the converse, which this argument does not establish, so if Bombieri and Ghosh's converse holds Delta cannot be improved at all; and (ii) the de Bruijn engine plus the sparsity of the extreme zeros, which is the whole of the remaining bracket factor 3.34: with Delta = 0.62036249819 the engine returns 0.19242481458 narrow while the deepest measured DH zeros reach |Im z| = 0.347, which would give 0.0602 against a decided floor of 0.0576.

Grades. kappa, tau, the Theta enclosures, sigma_0' and Delta^2/2 are decided (python-flint 192 bits and mpmath.iv dps 40 at P = 10^5; deep point flint 320 bits at P = 10^7). The phase obstruction is exact elementary analysis on top of them, using the Euler products of two Dirichlet L-functions on Re s > 1, F entire with F(s) = F(1-s), and the classical nonvanishing of Gamma. Lambda_DH <= Delta^2/2 is cited plus decided, weakest step cited (de Bruijn 1950 Theorem 13), unchanged from 3.5.

4. The census and prediction P4

census.py, output census_results.json. Everything in the census is measured, not decided: there are no enclosures anywhere in it (its own summary.grades.counts).

Coverage: 25 windows spanning heights 412.05 to 600.0, all_windows_closed: true. Accounting n_strip = n_line + 2 * n_quadruples: 179 = 167 + 2*6, true. Six off-line quadruples found, at gamma = 440.4845, 520.9438, 531.2797, 548.9068, 566.5097, 595.0234, with depths y0 = 0.2089, 0.0159, 0.3470, 0.2295, 0.2866, 0.0829.

P4 verdict: held, observed. No pair in (412.05, 600.0) beats the flow_repair landing floor 0.05765184034869543 (its pair 5 at gamma 240.4047, measured). The one candidate whose naive y0^2/2 exceeds the floor is gamma = 531.27972689652 with y0 = 0.34695380309204904 and naive 0.060188470740018166. It lands at 0.05033975468118168 by the N-body ODE null control (float grade; the instrument matched the Xi-plane contour measurement within 0.04 percent on flow_repair pairs 1 to 5) and at 0.048403 by the fitted shave model, 12 to 16 percent below the floor.

A budget honesty note carried from the artifact. The Xi-plane contour-moment measurement at gamma 531 was skipped: it needs dps about 206 and about 27000 quadrature nodes per flow evaluation, projected at 35 to 60 minutes, more than the whole census budget. The screening question was decided by the ODE route at the precision that route was validated to. The candidate needs a shave below 4.4 percent to beat the floor and every surveyed pair of comparable depth shaves 12.5 to 15.6 percent.

The census does not feed the headline. The decided floor 36/625 = 0.0576 comes from the winding count at the pair-5 site, and it sits slightly below flow_repair's measured float floor 0.0576518, which is the expected direction: a decided value is the largest rational the enclosures will carry, not the measured landing time.

5. The controls, including the M2 blind spot

controls.py, output controls_results.json, backend python-flint 0.9.0 (Arb), geometry from winding_results.json t1_run.

controls_1_to_4_pass: true. all_pass: false. Reporting only controls_1_to_4_pass would be the flattering read and is not the headline.

Update 2026-08-18. Control 5's verdict field read BLIND SPOT through 2026-08-17 and now reads SENSITIVITY MEASURED (not a pass). Not one number in the table below moved. What changed is what the table is evidence about: M2 is now a proved lemma (M2-LEMMA.md), so deflating it no longer stands in for "the derivation might be wrong" but for "the implementation might be wrong", which is narrower and still live. It is still not a pass: the expectation as written is still unmet, because the detector returns integers under the lesion instead of refusing, and all_pass stays false. Section 5.1 below is the 2026-08-17 text, kept as written, with the changes gathered in 5.1a after it.

5.1 Control 5, in full

M2 is the uniform bound for |H_t''| on the box. It is the single load-bearing analytic step in the lower-bound route whose derivation is prose rather than an enclosure or a cited theorem: its numerical ingredients are ball-computed (M2_upper_float = 1.1886642645115153e-78 at the t1 box, 1.137082903400534e-78 at t2, prec 300, 800 panels), but the shifted-contour argument that assembles them is written out in winding.py and nowhere checked. Controls 1 to 4 all hold M2 fixed: two move geometry, two move precision.

The lesion holds the t1 box, t = 23/400, prec 420, instrument and subdivision rule fixed and changes only the number M2:

wrong_answer_onset_factor: 75, n_wrong_and_silent: 3.

Why it matters. M2 too large costs only compute. M2 too small silently licenses a segment whose true argument variation exceeds pi; the branch resolution Delta = Arg q is then wrong by 2 pi, which lands in the sum as a whole unit of winding. The failure mode is a wrong integer with status decided, which is the one output this routine promises never to produce.

The perverse metric. min_chord_margin_digits is log10(dist(0, chord) / tube radius) minimised over accepted segments. Deflating M2 shrinks the tube, so the ratio grows: the correct run reports 0.02, the wrong N = 0 run at factor 100 reports 0.11, and at factor 1000 reports 1.11. The detector's own health metric reads about five times healthier exactly as the answer becomes wrong, and it may not be used as a guard on M2. min_ball_margin_digits is no better here: it moves from 40.32 to 42.61 across the same lesion, because the wrong runs accept fewer and larger segments and never sample the tight ones.

The countermeasure now in place, winding.measured_h2_guard: M2 must dominate a directly measured sup |H_t''| sampled on the box by quadrature of the defining integral, on flow_repair's DHFlow rule, sharing no code with the shifted-contour derivation. On the honest M2 it passes with measured_sup_absH2 = 2.1357367685579024e-80 at the t1 box against M2 = 1.1886642645115153e-78, a ratio of 55.66 (53.88 at t2). winding.py's main() refuses the floor when the guard fails.

What is still blind, and it is three things. (1) The guard is a finite grid of a smooth function; a peak between nodes is invisible to it. (2) The guard is float grade, so it cannot upgrade M2 from prose to decided; only an in-tree or Lean derivation of the shifted-contour step would do that. (3) The ordering that makes the guard useful on this box is luck, not structure: the guard trips once M2 falls below the measured sup, at deflation past 55.7, while the first wrong integer observed appears at deflation 75. Nothing guarantees that ordering on another box, and a derivation wrong by a factor under 55 would pass the guard and could still be wrong.

This is recorded, not repaired. It is a standing blind spot in the lower-bound route, and any statement of the bound carries it. Note also, in fairness to the artifact's own honesty, that the docstring previously described the M2 cushion as "three digits of slack" against the measured sup, when the measured factor is 55.7; that overstatement was named by the gate and corrected in winding.py.

5.1a What changed on 2026-08-18, and what did not

Section 5.1 above is kept exactly as written on 2026-08-17. Four of its claims are now false and are corrected here rather than edited there.

"whose derivation is prose rather than an enclosure or a cited theorem" and "nowhere checked". Both false since M2-LEMMA.md. The bound is Lemma M2 there, proved: differentiation under the integral sign with an explicit dominating function, Cauchy's theorem on the shifted contour with the far side bounded rather than asserted to vanish, the vertical-leg cancellation, separate proofs of both majorants for the theta-like sum, and the panel-plus-tail split. Every constant of the proof is a reported Arb ball and every hypothesis is a decided predicate, so no step of the proof rests on an unverified numerical claim. Four routes exercise it (m2_lemma.py): an independent re-implementation, an unshifted majorant that needs neither Cauchy's theorem nor the evenness, a pointwise Arb enclosure of H_t'', and this section's float guard. Two attacks stand beside them: the H_t'' identity against second central differences of H_ball (worst relative gap 7.66e-07 against h^2 = 9.54e-07), and both majorants against a sharp truncated enclosure at 24 probe points, with no refutation.

"it cannot upgrade M2 from prose to decided". The guard still cannot, and that sentence was about the guard. But the quantity the guard measures is now decided by a different route: an Arb enclosure of H_t'' at the guard's own sup point reproduces 2.1357367685579024e-80 to all 17 digits, and a 433-point grid gives a decided sup |H_t''| >= 2.1358117413634282e-80 at t1 and >= 2.1139544551457620e-80 at t2. The cushion is therefore decided as well: at most 55.65 at t1 and 53.79 at t2, against the measured 55.66 and 53.88. The published numbers stand and move only in the third digit.

"the ordering that makes the guard useful on this box is luck, not structure". Partly answered. The trip point is the cushion, and the cushion is a structural constant of the bound rather than a property of these rectangles: M2 depends on x_lo only through e^{-x_lo v} with v = pi/4 - 1/256 against the strip half-width pi/4, which is the fastest rate any shift of this contour can carry, and across a 40-unit span of Re z over which |H_t''| falls by 14 orders of magnitude the decided cushion stays between 33 and 204. What is still not proved is the ordering itself on an arbitrary box, and the wrong-answer onset depends on the subdivision rule as well as on M2.

What is left, stated at its true size. The proof needs one cited classical input, the evenness Phi_DH(-u) = Phi_DH(u), which is Hecke's theta transformation plus F(s) = F(1-s) transported and is section 6's assumption 5 already. It is load-bearing for M2 and for nothing else in the route, and there is no numerical substitute: the quantity that must vanish is identically zero, so enclosing it to 1e-78 would take of order 1e78 subdivisions. The proof is written prose plus decided arithmetic, at the hardened rung, not kernel-checked, and it has been read by no human. And the detector still cannot see a corrupted M2 by itself, which is why the guard and its refusal path stay in place.

5.2 Rival framing, carried from WP4

The same pipeline pointed at zeta yields no positive floor: the lower bound needs a box around a zero strictly off the real axis of the corresponding H_t, and no off-line zero of zeta is known, so every box this detector could honestly place for zeta would decide N = 0, as the displaced-box lesion does for an empty window. The number this hunt produces separates DH from zeta only through the already-known off-line zeros (Davenport-Heilbronn 1936; computed by Spira 1994). Nothing here is evidence about the Riemann hypothesis, which is flow_repair's P5 moral restated. zeta.epstein.battery referees structural claims that purport to explain RH by distinguishing zeta from an RH-violating rival; this hunt makes no such claim, and its object of study is that rival.

6. The pre-registered predictions, settled

7. The gate

GATE.md, adjudicated 2026-08-16 after four adversary reports and one prior-art report, verdict NOT YET. Every defect it accepted was reproduced in that session and every claim credited to it recomputed there.

7.1 What the adversaries found

• Adversary 1 (normalization). The interval survives as a statement about the object the claim defines, and does not survive as a statement about the object it is named after: Lambda(Dobner) = 4 Lambda(hunt). H_0 = Xi_DH with (c, a) = (1, 1) confirmed at nine points, rivals excluded by orders of magnitude; the Mellin derivation, the factor 4, the e^{3u/2}, evenness to 5.3e-57, and S# membership all confirmed. The false sentence in THEOREM13.md section 6 named and its consequences traced.
• Adversary 3 (upper bound). The inequality survives. Theorem 13 re-read from the original scan and matching word for word; the triangle-equality case airtight; the trivial-zero trap worked out and holding; sigma_0 recomputed independently to 30 digits; the Delta^2/2 dictionary recalibrated and Delta^2/8 and 2 Delta^2 each refuted by a factor of 4. One major defect reported: the same normalization error.
• Adversary 4 (instrument and independence). The claim survives this lane: neither tail bound could be broken in 59 tests, and a route to H_t sharing no layer with the instrument below Arb itself agrees at 8 of 8 box-boundary points with a ball 34 orders tighter. The independence story does not survive: 8 of 11 shared layers, three of five validation checks with no purchase on what is neither a library call nor cross-checked, and a planted fault demonstrating the iv leg's blindness.
• Adversary 5 (prior art). No source states a numerical bound, from either side, on a de Bruijn-Newman constant of this function or of any RH-violating function. Four corrections required, including Stopple as the published precedent for this hunt's own frame and as a prior quantitative non-zeta bound, Newman-Wu Theorem 7 as the typeset engine, and Bombieri-Ghosh 2011 as an unread high-probability risk to the strip constant.

7.2 What was repaired

One repair went beyond the list. BOMBIERI-GHOSH.md: the source the gate called "the unresolved risk" and "unread and paywalled" was retrieved and read in full the same day (mathnet.ru id rm9410, not umn9410, and www.mathnet.ru rather than mi.mathnet.ru, which 503s). Its verdict is mixed and is stated in section 8.

7.3 What remains open

Stated at least as prominently as the positive results, per the house rule.

1. M2 was a standing blind spot (section 5.1). Closed 2026-08-18; see section 5.1a. It is proved in M2-LEMMA.md with decided constants and exercised by four routes, and the cushion is decided at 55.65 and 53.79. What remains open is narrower and is stated here rather than dropped: the proof needs one cited classical input, the evenness Phi_DH(-u) = Phi_DH(u) (section 6 assumption 5), which is load-bearing for M2 and has no numerical substitute; the proof is written prose plus decided arithmetic, not kernel-checked, and has been read by no human; and the detector still cannot see a corrupted M2 by itself, so the lesion of section 5.1 still measures a live sensitivity to an implementation fault. Superseded text, kept verbatim: "It is prose, it is load-bearing on the lower side, its lesion produces a wrong integer with status decided, and the guard that now covers it is measured, finite-grid, and useful on this box by luck rather than by structure. The lower bound carries this."
2. The two validation repairs have stated blindness radii, and they are not large. The iv cross-leg misses a planted recurrence fault at u = 5/2, because a check at large u sees only the n = 1 term (1 of 7 points). The tail-domination check has a blindness factor of 8.02: a bound deflated by less than about a factor of 8 passes it. Both are measured and neither can upgrade the thing it checks.
3. Route 2 never ran at the headline t = 36/625, only at t = 23/400, and on its own recipe box rather than route 1's, because route 1's output file did not exist when it ran. So even the bookkeeping agreement of P2 is one t and one box wide, and the stretch value that carries the published floor has no route-2 witness at all. The second witness at the headline t is the DHFlow count of crosscheck_dhflow_winding.py, which is float grade at mpmath dps 130, not an enclosure, and it says so: it cannot decide an integer and does not claim to, its job is to catch a wrong one.
4. The upper bound is visibly loose, by three independent measures. Largely closed 2026-08-18 (section 3.6): the strip constant is no longer where the looseness is, and the bracket ratio falls from 6.955 to 3.341. What remains open is narrower and is stated here rather than dropped. (i) The converse: the phase argument bounds the supremum of the real parts of the zeros from above and does not show it is attained, so if Bombieri and Ghosh's converse holds Delta cannot be improved at all, and if it does not, this argument would not see the improvement. (ii) The engine: the deepest measured DH zeros reach |Im z| = 0.347 against Delta = 0.620362..., which would give 0.0602 against the decided floor 0.0576, so essentially the whole remaining bracket factor is de Bruijn's theorem plus the sparsity of the extreme zeros. Superseded text, kept verbatim: "The phase-minimum refinement inside the hunt's own materials already gives 0.38710055 at M = 12 (adversary 3, section 3, float grade); the deepest measured DH zeros reach |Im z| = 0.347 against Delta = 0.895; and the same coefficient domination applied to zeta returns 3.0191480758 in the wide frame where the truth is 1/2, a factor of 6.04 that the Euler product buys zeta and this function cannot."
5. A sharper strip constant is available by citation and has not been adopted. CLOSED 2026-08-18, and closed the way this item asked for. It required their hypotheses checked in-tree and the arctan equation re-solved with outward rounding before the headline could move. Both were done, by an in-tree derivation of the necessary half of their Theorem 7 from the Euler products (section 3.6), so the constant is now decided on both backends rather than cited plus measured, and nothing was silently swapped: the superseded headline is printed beside the new one in section 0 and the old derivation is kept intact in sections 3.1 to 3.5. Superseded text, kept verbatim: "Bombieri and Ghosh's sigma(tau_+, 1) = 1.120362 gives Delta = 0.620362... and Delta^2/2 = 0.192424814576128011 (narrow), 0.769699258304512045 (wide), a factor 2.082 better. It is cited plus measured, not decided: their value is a six-decimal Mathematica number and their Theorem 7 rests on Bohr-Kronecker machinery this tree has not verified. Adopting it needs their hypotheses checked in-tree and the arctan equation re-solved with outward rounding. Do not silently swap the headline." One consequence runs the other way and is recorded rather than buried: the 18-digit figure in that superseded text is derived from BOMBIERI-GHOSH.md's own 29-digit re-solve, which the new instrument decides is high by about 1.2e-17; the decided replacement from the deep point is 0.1924248145761280190 narrow, agreeing to 17 digits.
6. One source bearing on the claim is still unread: academia.edu preprint
7. CLOSED 2026-08-18, in the hunt's favour. It is Mesut Ismail, DOI 10.5281/zenodo.21679490, open access on Zenodo (Ismail_rh_pf_v18.4.pdf, 758,872 bytes, 2026-07-29), reached through r.jina.ai against the full slug URL after direct fetch returned 403 again and the Wayback machine held no snapshot. It was downloaded, converted and read in full: its subject is the classical wide-frame Lambda_zeta, and Davenport-Heilbronn appears only as instrument and negative control. It contains no Lambda_DH, no bound on one from either side, and no claim about one. Two traps for the next reader are recorded in NOVELTY.md section 2: the symbol Lambda_H does appear in it, as the DH analogue of the von Mangoldt function and not as a de Bruijn-Newman constant; and its two DH off-line lifetimes, tau1 = 0.1819 and tau2 = 0.0449 in the wide frame, are labelled upper bounds inside a Numerical Observation, which is the wrong direction to bound Lambda_DH from below. Read at face value anyway, 0.1819 wide is 0.045475 narrow, below this hunt's decided floor by a factor 1.267.
8. One sweep is single-source: the forward-citation search on Dobner ran on Semantic Scholar alone, because OpenAlex returned HTTP 429 with zero daily allowance. CLOSED 2026-08-18: it was re-run across Semantic Scholar (5 records, queried by DOI and arXiv id separately and agreeing), OpenCitations/COCI (1), Google Scholar (2 before rate limiting) and a web sweep that surfaced one preprint none of the three indexes carried. The union is 7 distinct citing works and none attaches a quantitative Lambda to any non-zeta object. OpenAlex still returned HTTP 429 with a $0 daily budget and was not worked around. One citing item is now 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. Stronger than any single query, the sweep also turned up Tao, Trudgian and Yang's ANTEDB (teorth.github.io/expdb), whose chapter 18 is The de Bruijn-Newman constant and tabulates the complete known bound history from Newman 1976 onward: every entry is zeta's.
9. One hypothesis discharge is a bounded-range check standing in for an unbounded-range claim. THEOREM13.md section 5.3 asserts the decay margin g(u) = (pi/5) e^{2u} - (3/2)u - u^3 is increasing on [2, 30] from a 201-point grid. The fact itself is immediate from the artifact's own domination, and one line settles it (g'(u) = (2 pi/5) e^{2u} - 3/2 - 3u^2, which is 68.6 against 13.5 at u = 2), so this is a discharge gap and not a doubt.
10. Bombieri-Mueller 2008, On the zeros of certain Epstein zeta functions, the parent of the constant that displaced sigma_0, has not been consulted. Reduced 2026-08-18, not closed. It is identified exactly: E. Bombieri and J. Mueller, Forum Math. 20:2 (2008), 359-385, DOI 10.1515/FORUM.2008.018, Zbl 1217.11040, MSC 11E45 and 11M41, and it was read at abstract and reference-list level. Per the zbMATH summary it bounds the rate of approach of zeros to the boundary of the zero-free half-plane for Epstein zeta functions of class number 2, by Bohr's method for the lower side and a diophantine-type result for the upper; its five deposited references name neither de Bruijn nor Newman. So: no de Bruijn-Newman or heat-flow content, and no rival value for the quantity the strip constant bounds either, since it is a different family and a different quantity. Its full text is still unread (De Gruyter answers HTTP 202 behind a human-verification wall, r.jina.ai gets HTTP 405, no mirror found), so this is a small named residual rather than a closed item.
11. The two winding routes do not use the same box. They overlap and both decide N = 1, which is arguably better than identical boxes, but a reader should not have to diff two JSON files to learn it.
12. Classical facts used without in-tree proof: Gamma has no zeros and only the simple poles at s = -1, -3, ...; F is entire with F(s) = F(1-s) (structural, measured in-tree to defect about 1e-50 and test-pinned).
13. Residual float-grade steps, none load-bearing: the dps-112 locating pass that places the boxes, the whole WP3 census, and calibration.json's numerical re-derivation of the Delta^2/2 dictionary. All three are labelled measured in the artifacts.
14. There is no second rigorous integrator. acb.integral has no counterpart in mpmath's iv context, and the in-tree precedent zeta.rigor.enclose_weil_functional is flint-only and says so. Route 4 is the answer to this and it is a route rather than a backend: it reaches H_t with no quadrature at all and agrees to the instrument's full resolution. What no leg supplies is a second integrator.
15. M2 is exercised by no cross-route. Routes 3 and 4 evaluate H; neither computes a uniform second-derivative bound. This is item 1 seen from the independence side (INDEPENDENCE.md section 6). Closed 2026-08-18: m2_lemma.py adds an independent re-implementation of the bound, an unshifted majorant that needs neither Cauchy's theorem nor the evenness, and a pointwise Arb enclosure of H_t''. Note that the independence radii published in INDEPENDENCE.md are unchanged, because the new routes were not declared to harness/independence.py: they add evidence about M2 without moving any published radius, and this file does not restate one as though they had.
16. A declaration is not an attestation. Nothing in harness/independence.py verifies that the layer lists are complete, and an undeclared shared layer is exactly the fault the structure cannot see. independence_decl.check_anchors pins ten attribute names against drift, which catches a rename that would make a layer name a fiction; ten anchors are not a completeness proof.

8. Honest scope and novelty

8.1 The sentence

Adopted verbatim from NOVELTY.md, which adopted it from the gate:

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 word "L-function" is not used: 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 phrase "first quantitative" is not used: Stopple has an unconditional quantitative bound on a non-zeta constant of exactly this type (-1.12929e-7 < Lambda_Kr), and Newman and Wu determine one exactly (ln 2) for a three-atom measure. "So far as the search reaches" is not droppable, and neither is the frame.

The mandatory footnote is in NOVELTY.md and is not to be compressed. Its substance: existence, finiteness and nonnegativity are Dobner's, for all of S#, with no member named and no number given; the upper bound's mechanism is de Bruijn's 1950 theorem, surveyed as Newman-Wu Theorem 7, applied to a strip computed here; and the strip constant is not new either.

8.2 What Bombieri and Ghosh cost the claim, and what they hand it

BOMBIERI-GHOSH.md, read in full 2026-08-16, 50 pages, English translation, free at mathnet.ru. The verdict has two halves pointing in opposite directions.

On Lambda_DH: closed, in the hunt's favour. 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 needs no change on account of it.

On sigma_0: closed, against the hunt. 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; their section 6 evaluates it for this very function at sigma(tau_+, 1) = 1.120362, and their tau_+ is this hunt's kappa, agreeing to 34 digits. sigma_0 = 1.395136... must not be described as a new number. It is a triangle-inequality upper bound on a quantity these authors determined exactly, and theirs is sharper. What survives is the derivation and not the quantity: an enclosure-carrying elementary route, from a two-line argument that invokes no Bohr-Kronecker theory, to a weaker bound on a constant already known.

Update 2026-08-18. The last four words are no longer right and the paragraph is kept as written. The elementary route no longer stops short of their abscissa: section 3.6 reaches it, decided on both backends, from the Euler products and one Moebius image. Their Theorem 7 has moved from being the source of a number this hunt could not use to being the standard the hunt's decided number is measured against, and two of their published constants are now reproduced as controls by machinery their Theorem 7 does not share. The withdrawal of the originality claim is unaffected in either direction: the quantity was determined in 2011, and reaching the same quantity by a different argument does not make it new.

Two independent verification legs were run against their paper. Their section 9 claim that the smallest prime set with sum arctan(p^{-1}) > pi/2 for p = 2, 3 mod 5 ends at 6323 with 420 primes reproduces exactly, by direct prime summation sharing no code (decided, a finite exact prime set and integer answers). Their Theorem 7 roots reproduce to every published digit at sigma(0,1) = 1.06702646637238936888624138422, sigma(tau_+,1) = 1.12036249818332508773010350311, sigma(tau_-,1) = 2.38228610898712386578711039387 and four Table 1 entries (measured, and honestly a reproduction of their method rather than an independent route).

They also supply context worth carrying: for small xi the zeros with Re s > 1 are extraordinarily rare, and for xi = 0 a search of 1/2 <= sigma < 1.2, 0 <= t < 10000 found 5,358 zeros and no zero with real part above 1. The Davenport-Heilbronn function proper sits at the hard end of that spectrum, which is consistent with the looseness recorded in section 7.3 item 4. And an attribution is corrected: sigma* = 2.3822861089... for the sibling root tau_- = -1/kappa is Bombieri and Ghosh's constant, not Righetti's; he quotes it.

8.3 Scope

• Nothing here is evidence about the Riemann hypothesis. The lower bound exists only because this function has known off-line zeros, which is what makes it a counterexample function in the first place (docs/08, docs/09).
• The bracket is a statement about Lambda_DH in a named frame, and about nothing else. The strip abscissa and the bracket ratio are the frame-free quantities: sigma_0' = 1.12036249819 and 3.341 since 2026-08-18, sigma_0 = 1.395136... and 6.955 before it.
• The upper side is a cited theorem applied to a decided constant, not a new mechanism, and the constant it is fed is not new either. The lower side is a decided integer count turned into an inequality by two cited theorems, with one lemma (M2) inside it, proved in M2-LEMMA.md since 2026-08-18 and prose before that; the lemma's own weakest input is a citation, so the composite grade is unchanged.
• Added 2026-08-18. The sharpened upper side is still a cited theorem applied to a decided constant, and the constant is still not new: the criterion it solves is Bombieri and Ghosh's Theorem 7 equation, term for term. What is new on that side is the grade and the derivation, an elementary in-tree route to their necessary half from the Euler products, with their converse neither used nor claimed. The abscissa sigma_0' must not be described as a new number, for exactly the reason sigma_0 must not.
• Grades, per the vocabulary contract: sigma_0, sigma_0', Delta, Delta^2/2 in both frames, g(2) < 0, kappa, tau, the Theta enclosures, the 6323 / 420 finite claim, sigma(tau_-, 1), the decided sup |H_t''| lower bounds and cushions, and the winding integers N = 1 at both t are decided, with backends and precisions stated. The census, the calibration, the locating pass, the M2 float guard, three of STRIP2.md's eight controls and every cross-check ratio in section 1 are measured. de Bruijn Theorem 13, Dobner Theorems 1 and 2, Newman-Wu Theorem 7, Bombieri-Ghosh Theorem 7, Polymath 15 Theorem 1.1, Rodgers-Tao Theorem 1 and the evenness Phi_DH(-u) = Phi_DH(u) are cited.
• Per MISSION.md, this hunt may not promote its claim into README.md, ROADMAP.md or HANDOFF.md as an established finding, and this file does not.

Artifacts: MISSION.md, FRAME.md, STRIP.md, STRIP2.md, M2-LEMMA.md, THEOREM13.md, NOVELTY.md, BOMBIERI-GHOSH.md, KAPPA-CLOSED-FORM.md, POLYMATH-PIN.md, SEPARATION.md, INDEPENDENCE.md, GATE.md, the four attack_adversary*.md reports, and the JSON outputs validation.json, calibration.json, strip_results.json, strip2_results.json, m2_lemma_results.json, census_results.json, winding_results.json, winding_quad_results.json, controls_results.json, independence_results.json, crosscheck_quadfree_results.json and crosscheck_dhflow_results.json. Machine-readable claims: results.json.