Adversary 1 (normalization and definition): findings

Scratch analysis, 2026-08-16. Written by an adversary tasked with breaking the claim 0.0576 < Lambda_DH <= 0.4006343708899557, not with defending it. All numbers below were produced by code written from the definitions in zeta/epstein.py, in /tmp/claude-0/-home-user-zeta-lab/ea2c50e5-c6c1-5be9-ac30-eda79b8fac85/scratchpad/ (a1_h0.py, mel2.py, mel3.py, a1_zeta_frame.py, dob2.py). Nothing in this directory was modified.

Summary

The interval survives as a statement about the object the claim itself defines. Every internal normalization check passed, several to 50 digits.

The interval does not survive as a statement about the object it is named after. In the normalization used by every source the hunt cites for it (de Bruijn 1950, Newman, Rodgers-Tao, Polymath 15, and Dobner 2020, whose Theorem 1 the strictness of the lower bound leans on), the de Bruijn-Newman constant of the Davenport-Heilbronn function is four times the number here:

hunt frame : 0.0576 < Lambda_DH <= 0.4006343708899557 Dobner frame : 0.2304 < Lambda_DH <= 1.6025374835598228

1. H_0 = Xi_DH, and (c, a) = (1, 1) - confirmed, not the defect

Independent quadrature of int_0^inf Phi_DH(u) cos(zu) du against zeta.epstein.completed_dh(1/2 + iz) at dps 40, nine points, real and complex:

Rival constants are excluded by orders of magnitude, at dps 30:

So the zeta-style H_0 = (1/8) Xi(z/2) reading is dead, as the hunt says.

2. The Mellin derivation - confirmed

With psi(x) = sum_{n>=1} n a_n exp(-pi n^2 x / 5):

• Theta transformation psi(1/x) = x^{3/2} psi(x), exact rational inputs, dps 50: relative defect 5.3e-57, 5.4e-57, 7.3e-57, 1.4e-56, 6.1e-57 at x = 3/10, 7/10, 1, 19/10, 4. This is exactly the evenness of Phi_DH, which is de Bruijn's hermitian hypothesis.
• No pole terms. F(s) = int_1^inf psi(x) [x^{(s+1)/2-1} + x^{(2-s)/2-1}] dx with no 1/s or 1/(s-1) residue terms reproduces completed_dh to 3.2e-32 (s=2), 2.5e-32 (s=1), 2.5e-32 (s=0), 1.6e-31 (s=1/2+5i), 4.9e-32 (s=4/5+3i), 8.6e-31 (s=-3). The zeta analogue of this split needs -1/s - 1/(1-s); DH needs nothing, because psi decays super-exponentially at both ends (measured: psi(0.02) = 8.03e-12, matching x^{-3/2} exp(-pi/(5x)) to six digits).
• The factor 4 and the e^{3u/2} follow from x = e^{2u}, dx/x = 2 du, x^{(s+1)/2} = e^{u(s+1)}, s = 1/2+iz, plus evenness converting the full-line integral to twice the half-line one. Verified numerically in section 1.

3. The defect: the frame is not the de Bruijn-Newman frame

Dobner, arXiv:2005.05142v2, equations (1), (2), (5), (6), read from the PDF:

(1) xi((1+iz)/2) = int_{-inf}^{inf} Phi(u) e^{izu} du (2) Phi(u) := 4 sum (2 pi^2 n^4 e^{9u} - 3 pi n^2 e^{5u}) e^{-pi n^2 e^{4u}} (5) Phi_F(u) := (1/2pi) int_{-inf}^{inf} xi^F((1+ix)/2) e^{-ixu} dx (6) xi_t^F((1+iz)/2) := int_{-inf}^{inf} e^{tu^2} Phi_F(u) e^{izu} du

and in his proof of Theorem 1, H(z) := xi^F((1+iz)/2).

The argument is (1+iz)/2 = 1/2 + i z/2. The hunt's is 1/2 + i z. So

z_Dobner = 2 z_hunt, and therefore t_Dobner = 4 t_hunt, Lambda_F (Dobner) = 4 * Lambda_DH (hunt).

The quadratic scaling is the hunt's own rule (THEOREM13.md section 6: "Strips scale by 1/a, times by a^2"), applied to a = 1/2.

Cross-check on zeta, both frames, verified numerically (dps 30, my own Phi in each frame against zeta.core.xi):

So the Delta^2/2 dictionary is correct and reproduces de Bruijn's 1/2 for zeta in zeta's classical frame. It is the hunt's own frame that is not that frame. In the hunt's frame the classical facts read

Lambda_zeta <= 1/8 (de Bruijn), 0 <= Lambda_zeta <= 0.055 (Newman, Polymath 15)

and NOVELTY.md's "zeta record for calibration: 0 <= Lambda_zeta <= 0.22 ... Lambda_zeta < 1/2" is quoted in the other frame without conversion.

3a. The false sentence

THEOREM13.md section 6:

Dobner's deformation xi_t^F((1+iz)/2) = int e^{tu^2} Phi_F(u) e^{izu} du is the hunt's H_t up to the constant factor 2 (Phi even turns the full-line integral into 2 int_0^inf ... cos(zu) du), so the two share zeros and share Lambda exactly.

The factor 2 is right and the conclusion is wrong: the substitution also carries z -> z/2. Working it out, Dobner's Phi_F(u) = Phi_DH(2u) and

xi_t^F((1+iz)/2) = H_{t/4}(z/2).

They share zeros only after the z-rescaling, and they do not share Lambda; they differ by exactly 4.

Phi_F(u) = Phi_DH(2u) is measured, not merely derived. Evaluating Dobner's (5) directly for DH, Phi_F(u) = (1/pi) int_0^R Xi_DH(x/2) cos(xu) dx with R = 55 at dps 15 (truncation floor about 1e-9):

The u = 0.15 row separates the hypotheses: Dobner's Phi_F agrees with Phi_DH(2u) to the truncation floor and misses Phi_DH(u) by 0.206, i.e. by 2e8 times the numerical noise.

This sentence contradicts the paragraph immediately above it in the same section, which correctly notes that zeta's classical convention has a = 1/2 while DH's has a = 1, and that a rescaling moves t quadratically.

3b. The numbers

sigma_0 (flint, outward) = 1.3951361582351097210613588712 ... 9375 Delta = sigma_0 - 1/2 = 0.8951361582351097210613588712 (hunt frame, a = 1) Delta^2/2 = 0.40063437088995569444695475 (matches the claim) Delta_Dobner = 2 Delta = 1.7902723164702194421227177424 Delta_Dobner^2/2 = 1.6025374835598227777878190108 = 4 x the claim lower endpoint 4 x 0.0576 = 0.2304

Cross-check that the whole hunt lineage sits in the a = 1 frame: pair 1 of hunts/flow_repair/NOTES.md has y0 = 0.30851718245663738555, and y0^2/2 = 0.04759142593549104 reproduces that table's "naive y0^2/2" column 0.0475914 exactly. In Dobner's frame the same quadruple sits at y0_D = 0.61703436491327477 and its naive landing is 0.19036570374196417.

4. Attacks that did NOT break anything

• Pole terms. None needed; verified at six s including s = 1 and s = -3.
• The factor 4 / the e^{3u/2}. Correct.
• Evenness / hermitian symmetry. Correct to 5e-57.
• Decay. psi super-exponential at both ends; b = 3 > 2 stands.
• Delta^2/2 vs Delta^2/8 vs 2 Delta^2. The rule is right; it reproduces de Bruijn's classical 1/2 for zeta in zeta's classical frame.
• S# membership of DH, checked one by one against Dobner's verbatim (i)-(iii) (extracted from the PDF, section 2):
• not identically zero: f(2) != 0 (|f(2)| >= 1 - 0.2666).
• (i) sum |a_n| n^{-sigma} <= zeta(sigma) < inf for sigma > 1 since |a_n| <= max(1, kappa) = 1.
• (ii) m = 0; f entire (coefficient sum over a period is 0), finite order (a finite linear combination of order-1 Dirichlet L-functions).
• (iii) gamma(s) = alpha s^m (s-1)^m Q^s Gamma(omega s + mu) with m = 0, alpha = (5/pi)^{1/2} in C\{0}, Q = (5/pi)^{1/2} > 0, omega = 1/2 > 0, mu = 1/2 with Re mu = 1/2 >= 0; and xi^F(s) = xi^F(1-s), measured defect ~1e-50.
• Dobner's derived consequence "a_n nonzero for more than one n": holds. So Theorems 1 and 2 do apply, the real-rooted set is a closed half-line, and the strict lower bound is licensed. No defect here. The half-line structure is invariant under t -> 4t, so it transfers to the hunt's frame; only the value of Lambda does not.

5. What I would change

1. State the frame in the headline, every time: "in the normalization H_0 = Xi_DH, i.e. s = 1/2 + iz", and publish both pairs of numbers.
2. Delete or repair the "share Lambda exactly" sentence in THEOREM13.md section 6.
3. Convert the zeta calibration figures in NOVELTY.md, or move the whole hunt into Dobner's frame.

Note what the conversion buys rather than costs: in the common frame the lower bound reads Lambda_DH > 0.2304, which is above the current best upper bound for zeta (0.22, Polymath 15). Stated in the hunt's frame, 0.0576 sits below 0.22 and the comparison reads backwards.