Theorem 13 (de Bruijn 1950) and Dobner (2020), pinned

Source pinning for WP2 (upper bound) and for the Dobner half-line structure the lower bound leans on. Everything quoted below was read from the sources named; the de Bruijn quotes are transcribed from the image-only publisher scan (https://pure.tue.nl/ws/files/1769368/597490.pdf, DOI 10.1215/S0012-7094-50-01720-0) by visual page reads, as close as a visual read allows, with math ASCII-normalized. Page numbers are the printed journal pages (Duke Math. J. 17 (1950), 197-226). All numbers in section 6 are measured, with backend and precision stated; nothing in this file is an enclosure.

Corrected 2026-08-16 (GATE.md closure item (a)). Section 6 asserted that this hunt's deformation and Dobner's "share Lambda exactly". They differ by a factor of exactly 4. The false sentence is preserved inside the correction box in section 6, together with the derivation that replaces it and a record of what it was routed into. Every Lambda number in this directory now carries its frame: this hunt sits at s = 1/2 + iz (Stopple, arXiv:1301.3158), the Newman line sits at s = (1+iz)/2, and Lambda(that frame) = 4 Lambda(this one). Conversion table: FRAME.md.

1. Verbatim statements, with page numbers

de Bruijn's standing hypotheses: condition (1.4), p. 197

Suppose that the function F(t) of the real variable t satisfies

(1.4) F(t) integrable over -inf < t < inf; F(t) = (F(-t))*, -inf < t < inf; F(t) = O(e^{-|t|^b}) for t -> +-inf, b > 2.

(The * indicates the conjugate imaginary.)

Properties (alpha) and (beta) of a strong universal factor, p. 199

A function S(t) of the real variable t, satisfying S(t) = (S(-t))* will be called a strong universal factor if it joins properties (alpha) and (beta) below, for any function F(t) satisfying (1.4).

(alpha) If the roots of (1.5) lie in a strip |Im z| <= Delta (Delta > 0), then those of int_{-inf}^{inf} F(t)S(t)e^{izt} dt lie in a strip |Im z| <= Delta_1, where Delta_1 < Delta, Delta_1 independent of F(t).

(beta) If F(t) is such that, for any eps > 0, all but a finite number of roots of (1.5) lie in the strip |Im z| <= eps, then the function int_{-inf}^{inf} F(t)S(t)e^{izt} dt has only a finite number of non-real zeros.

(Here (1.5) is int_{-inf}^{inf} F(t)e^{izt} dt, p. 197.) Also on p. 199:

The functions e^{gamma t^2}, gamma > 0, also turn out to have property (alpha), but it is doubtful whether they have property (beta).

Theorem 10, p. 204 (the class conditions Theorem 13 references)

THEOREM 10. Let b be a number > 2, and let the real or complex function F(t) be integrable over -inf < t < inf and satisfy

(3.4) F(t) = (F(-t))* for all real values of t,

(3.5) F(t) = O(e^{-|t|^b}) (t -> +-inf).

Then the trigonometric integral

(3.6) f(z) = int_{-inf}^{inf} F(t)e^{izt} dt

represents a real integral function of order < 2.

Theorem 11, p. 204 (exponential-sum factors; both zero-count forms)

THEOREM 11. Let F(t) satisfy the conditions of the preceding theorem and suppose that the roots of the function S(t) = sum_{-M}^{M} a_k e^{k lambda t}, a_k* = a_{-k}, a_M != 0, lambda > 0, lie on the imaginary axis. Then we have: If the roots (all but a finite number of the roots) of (3.6) lie in the strip |Im z| <= Delta then the roots (all but a finite number of the roots) of the real integral function

(3.7) int_{-inf}^{inf} F(t)S(t)e^{izt} dt

lie in the strip |Im z| <= {Delta^2 - (1/2)M lambda^2}^{1/2} if Delta > lambda((1/2)M)^{1/2}, and are real if Delta <= lambda((1/2)M)^{1/2}.

Theorem 12, p. 205

THEOREM 12. If S(t) = prod_1^N (xi_k e^{lambda_k t} + xi_k* e^{-lambda_k t}), where |xi_k| = 1, lambda_k > 0, k = 1, 2, ..., N, and the roots (all but a finite number of the roots) of (3.6) lie in the strip |Im z| <= Delta, then the roots (all but a finite number of the roots) of (3.7) lie in the strip |Im z| <= {Max (Delta^2 - sum_1^N lambda_k^2, 0)}^{1/2}.

The bridge sentence before Theorem 13, p. 205

Theorem 11 proves the statements (alpha) and (beta) made in the introduction concerning strong universal factors. Although it will not be used in this paper, we shall prove here that also the functions e^{(1/2)lambda^2 t^2}, lambda^2 > 0, have property (alpha). We do not yet know whether they have or have not property (beta).

Theorem 13, p. 205 (the statement WP2 leans on)

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}.

Proof. By Theorem 12, the roots of g_N(z) = int_{-inf}^{inf} F(t) (cosh lambda t/N)^{N^2} e^{izt} dt lie in the strip (3.8). Owing to Theorem 7 it is now sufficient to prove that g_N(z) -> g(z) uniformly in any finite region. Now this follows from (3.5) and from the fact that for mu^2 > lambda^2 we have

(3.9) e^{-(1/2)mu^2 t^2} (cosh lambda t/N)^{N^2} -> e^{-(1/2)mu^2 t^2 + (1/2)lambda^2 t^2},

uniformly in -inf < t < inf. (3.9) results from the inequality cosh y <= e^{(1/2)y^2}, -inf < y < inf, whence (cosh lambda t/N)^{N^2} <= e^{(1/2)lambda^2 t^2}.

Internal consistency check on the transcription: each factor xi e^{lambda t} + xi* e^{-lambda t} in Theorem 12 contracts the squared strip by lambda^2, and (cosh lambda t/N)^{N^2} is N^2 such factors with parameter lambda/N, total contraction N^2 (lambda/N)^2 = lambda^2, matching (3.8); and (cosh(lambda t/N))^{N^2} -> e^{(1/2)lambda^2 t^2} matches the multiplier in the statement. The transcribed exponent (1/2)lambda^2 t^2 is therefore forced by the surrounding mathematics, not just by the visual read.

Theorem 14, p. 205 (context: the Delta = 0 Polya case)

THEOREM 14. Let F(t) satisfy the conditions of Theorem 10 and suppose that the roots of (3.6) lie in the strip |Im z| <= Delta. Let phi(z) be a real integral function of genus 0 or 1 (that is, a function of the type (3.1)), with real roots only. Then the roots of

(3.10) int_{-inf}^{inf} F(t)phi(it)e^{izt} dt

lie in the strip |Im z| <= Delta also.

Dobner, arXiv:2005.05142v2, "A proof of Newman's conjecture for the extended Selberg class"

Theorems 1 and 2 (from the paper's Section 1; text extracted from the arXiv PDF):

Theorem 1. For every F in S#, there is a real number Lambda_F such that all the zeros of xi_t^F lie on the critical line if and only if t >= Lambda_F.

Theorem 2. Lambda_F >= 0 for every F in S#.

His deformation (his equations (5)-(6)): Phi_F(u) := (1/2pi) int_{-inf}^{inf} xi^F((1+ix)/2) e^{-ixu} dx and xi_t^F((1+iz)/2) := int_{-inf}^{inf} e^{t u^2} Phi_F(u) e^{izu} du.

Dobner's restatement of de Bruijn's Theorem 13 (his Theorem 3, Section 2):

Theorem 3 (De Bruijn [9, Thm. 13], cf. Polya [16, Thm. 1]). Let phi: R -> C be an integrable function satisfying phi(u) = conj(phi(-u)) and phi(u) = O(e^{-|u|^b}) for some b > 2, and let G(z) := int_{-inf}^{inf} phi(u)e^{izu} du. For any t > 0, let G_t(z) := int_{-inf}^{inf} e^{tu^2} phi(u)e^{izu} du. If all the roots of G lie in the strip |Im z| <= Delta, then all the roots of G_t lie in the strip

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

S# membership conditions as Dobner states them (his Section 2): F not identically zero and

(i) (Dirichlet series) F has a Dirichlet series representation F(s) = sum_{n=1}^{inf} a_n / n^s which converges absolutely for all s with Re s > 1.

(ii) (Meromorphic continuation) (s-1)^m F(s) is an entire function of finite order for some nonnegative integer m.

(iii) (Functional Equation) Let m be the order of the pole of F at s = 1 (or let m = 0 in the case where F has no pole). There exists a function gamma(s) of the form gamma(s) := alpha s^m (s-1)^m Q^s prod_{i=1}^{k} Gamma(omega_i s + mu_i) where alpha in C \ {0}, Q > 0, omega_i > 0, and mu_i in C with Re mu_i >= 0 such that xi^F(s) := gamma(s) F(s) satisfies xi^F(s) = xi^F(1-s).

No Euler product appears anywhere in (i)-(iii).

2. Class conditions, summarized

Theorem 13's full hypothesis set is exactly Theorem 10's conditions plus the strip condition:

1. F integrable over the real line.
2. Hermitian symmetry F(t) = (F(-t))* (equivalently: the transform f is real on the real axis). For a real F this is evenness.
3. Decay F(t) = O(e^{-|t|^b}) for some b > 2.
4. All roots of f(z) = int F(t)e^{izt} dt lie in |Im z| <= Delta.

There is no positivity condition, no monotonicity condition, no condition on the sign or shape of F beyond (1)-(3), and F is allowed to be complex-valued. Order and genus are conclusions, not hypotheses: Theorem 10 concludes order < 2, which is what feeds Theorems 6-9 inside the proofs.

3. Restatements consulted

• Dobner 2020 (arXiv:2005.05142v2, his Theorem 3, quoted above): the all-zeros form, for the class "integrable, hermitian, O(e^{-|u|^b}) with b > 2", multiplier e^{tu^2}, contraction Delta^2 -> Delta^2 - 2t. This is the restatement in the hunt's own normalization.
• Ki-Kim 2003 (J. Anal. Math. 91, 369-387; from the MathSciNet review MR2037415, scratchpad kikim.txt): their Theorem 2.2 is the all-but-finitely-many weakening for genus 1* functions: > Theorem 2.2. Let f(z) be a real entire function of genus 1* having order > not greater than 2, and, if the order is 2, let f have type alpha for > some alpha < inf. Let Delta >= 0 and lambda > 0 be constants such that > lambda alpha < 1/4. If, for each eps > 0 all but a finite number of the > zeros of f lie in the strip |Im z| <= Delta + eps, then e^{-lambda D^2} > f(z) is also of genus 1* and all but a finite number of its zeros lie in > the strip |Im z| < sqrt(max{Delta^2 - 2 lambda, 0}) + eps. Further, if > Delta^2 < 2 lambda, then all but a finite number of the zeros of > e^{-lambda D^2} f(z) are real and simple. Same contraction constant (their lambda is the hunt's t), weaker hypothesis, weaker conclusion. This is a separate, later theorem; it is not what de Bruijn's Theorem 13 says.
• Rodgers-Tao (arXiv:1801.05914, published intro text): with their H_t(z) = int_0^inf e^{tu^2} Phi(u) cos(zu) du and H_0(z) = (1/8)Xi(z/2): > From results of Polya [22] it is known that Ht has purely real zeroes for > some t then Ht' has purely real zeroes for all t' > t. De Bruijn showed > that the zeroes of Ht are purely real for t >= 1/2. (Quoted as printed, including the elided "if".) The 1/2 here is exactly Delta^2/2 for Delta = 1, the strip of H_0(z) = (1/8)Xi(z/2) after the z/2 rescaling doubles Xi's strip of 1/2; see section 5.
• Csordas-Norfolk-Varga 1988 (Numer. Math. 52, 483-497): not reachable through this session's proxy (Springer paywall; GDZ and EuDML lookups came back empty), and Ki-Kim-Lee 2009 (Adv. Math. 222) is not on arXiv. Not quoted; the three restatements above triangulate the same statement, one of them (Dobner) being the exact theorem the hunt cites for the half-line structure anyway.

4. The three questions, answered

(a) All zeros, or all but finitely many?

All zeros, in both hypothesis and conclusion. Theorem 13 reads "if all the roots of (3.6) lie in the strip |Im z| <= Delta, then all the roots of g(z) ... lie in the strip (3.8)". The parenthetical "(all but a finite number of the roots)" variants appear in Theorems 11 and 12 (exponential-sum factors) but are absent from Theorem 13 as printed; the all-but-finitely-many Gaussian-multiplier statement is Ki-Kim 2003 Theorem 2.2 (genus 1*), a separate and later result. Dobner's Theorem 3 restates de Bruijn's Theorem 13 in the all-zeros form.

Normalization and conversion: de Bruijn's multiplier is e^{(1/2)lambda^2 t^2} and his conclusion strip is {Max(Delta^2 - lambda^2, 0)}^{1/2}, so all zeros are real as soon as lambda >= Delta. Writing the multiplier as e^{t u^2} (the hunt's and Dobner's normalization) sets t = lambda^2/2, the contraction becomes sqrt(max(Delta^2 - 2t, 0)), and the all-real threshold is

t >= Delta^2 / 2.

Consequence for the hunt: kill condition 1 of MISSION.md is not triggered. The upper bound route Lambda_DH <= Delta^2/2 stands, conditional on WP2 deciding that all zeros of H_0 = Xi_DH lie in |Im z| <= Delta (that is WP2's strip computation, not this file's claim).

(b) Positivity of Phi?

No positivity is required anywhere. The complete hypothesis list is quoted in section 1: integrability, hermitian symmetry (3.4), decay (3.5) with b > 2, and the strip. F may even be complex-valued. This matches the expectation recorded in the mission: realness/evenness plus O(e^{-|t|^b}) decay with b > 2 plus integrability suffice.

(c) Monotonicity: once real, always real?

de Bruijn states no numbered monotonicity theorem, but the Delta = 0 case of Theorem 13 delivers it immediately, and the hypotheses survive the deformation. Concretely: suppose all zeros of H_{t0}(z) = int e^{t0 u^2} Phi(u) e^{izu} du are real, with Phi satisfying Theorem 10's conditions for some b > 2. The deformed integrand Phi_{t0}(u) = e^{t0 u^2} Phi(u) is still integrable, still hermitian, and still O(e^{-|u|^{b'}}) for any 2 < b' < b (for |u| large, t0 u^2 - |u|^b <= -|u|^{b'}), so Theorem 13 applies to Phi_{t0} 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 (their [22]; quote in section 3); either attribution suffices and both predate Dobner.

What this buys the lower-bound logic: an off-line zero of H_{t1} plus Theorem-13 monotonicity alone already gives the weak bound Lambda_DH >= t1 (no t <= t1 can be real-rooted, else t1 would be), with no appeal to Dobner. The strict inequality Lambda_DH > t1 still uses Dobner's Theorem 1, because his half-line {t : all zeros real} = [Lambda_F, inf) is closed at the left end: without closedness the real-rooted set could be (t1, inf) and Lambda_DH would equal t1. So: weak lower bound independent of Dobner, strictness via Dobner, exactly as the vocabulary contract's "decided modulo Dobner's Theorem 1" phrasing anticipated.

How Dobner's theorem reaches this frame (added 2026-08-16; the earlier route ran through section 6's false sentence and is corrected there). His Theorem 1 is a statement in his frame: {t : xi_t^F all real} = [Lambda_F, inf), closed at the left. Section 6 derives 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), also closed at the left, which is everything the strictness argument needs. What does not transfer is the value: Lambda_F = 4 Lambda_DH. The strictness of the lower bound was never at risk from the error; only the number's label was.

5. Hypothesis check for Phi_DH

Phi_DH(u) = 4 e^{3u/2} sum_{n>=1} n a_n exp(-pi n^2 e^{2u}/5), with a_n the period-5 coefficients (1, kappa, -kappa, -1, 0) and kappa real, derived in zeta/epstein.py by a linear solve on every call (kappa = 0.2840790438..., pinned by tests/test_epstein.py::test_kappa_matches_pinned_reference and re-derived independently at four base points).

1. Realness. Every term is real (kappa real, exponentials real), so Phi_DH: R -> R. Hermitian symmetry (3.4) therefore reduces to evenness.
2. Evenness = the functional equation in disguise. Three lines: the completed function F(s) = (pi/5)^{-(s+1)/2} Gamma((s+1)/2) f(s) satisfies F(s) = F(1-s) (that identity is what defines kappa in zeta/epstein.py; measured defect < 1e-25 in tests/test_epstein.py::test_functional_equation_defect_below_1e25). Setting Xi_DH(z) = F(1/2 + iz) gives Xi_DH(-z) = F(1/2 - iz) = F(1 - (1/2 + iz)) = F(1/2 + iz) = Xi_DH(z), so Xi_DH is even. Phi_DH is (1/(2pi) times) the Fourier transform of Xi_DH restricted to the real line, and the Fourier transform of an even function is even; conversely Phi even forces Xi even, which at s = 1/2 + iz is exactly F(s) = F(1-s). Measured in-tree: hunts/flow_repair/NOTES.md records the raw-series evenness defect, relative <= 4.2e-51 for |u| <= 0.5 at dps 50.
3. Decay, dominated explicitly. For u >= 0, |Phi_DH(u)| <= 4 e^{3u/2} e^{-(pi/5)e^{2u}} S(u) with S(u) = sum_n n exp(-pi(n^2-1)e^{2u}/5) (using |a_n| <= 1), and S(u) <= S(0) = 1.3237 (measured, mpmath dps 30). The margin g(u) = (pi/5)e^{2u} - (3/2)u - u^3 is >= 23.30 at u = 2 and increasing on [2, 30] (measured on a 201-point grid, dps 30; calibration.json, phi_dh_decay_domination), so |Phi_DH(u)| = O(e^{-|u|^3}): condition (3.5) holds with b = 3 > 2, and by evenness on both tails.
4. Integrability. Continuous on R plus the b = 3 tail bound: integrable.
5. Order/genus. Not hypotheses; Theorem 10 concludes order < 2 for H_0 = Xi_DH. No positivity claim is made or needed (see 4b).

S# membership of DH (for Dobner's Theorems 1-2)

• (i) Absolute convergence for Re s > 1: |a_n| is periodic (1, kappa, kappa, 1, 0) and bounded by 1, so sum |a_n| n^{-sigma} <= zeta(sigma) < inf for sigma > 1; measured partial sums (mpmath nsum, dps 30): 1.7809 at sigma = 1.05, 1.4131 at 1.5, 1.2154 at 2.0 (calibration.json, sharp_class_dirichlet_sum).
• (ii) Entire of finite order, m = 0: the period-5 coefficient sum is 1 + kappa - kappa - 1 + 0 = 0, so f has no pole at s = 1 (tests/test_epstein.py::test_dh_f_is_entire_no_pole_at_1).
• (iii) Functional equation of the S# shape: gamma(s) = (pi/5)^{-(s+1)/2} Gamma((s+1)/2) = alpha Q^s Gamma(omega s + mu) with alpha = (pi/5)^{-1/2}, Q = (5/pi)^{1/2} > 0, omega = 1/2, mu = 1/2 (Re mu >= 0), m = 0, and F(s) = gamma(s) f(s) = F(1-s). The gamma factor is derived and pinned in zeta/epstein.py (the docstring records that the completing factor is Gamma((s+1)/2), not Gamma(s/2)) and exercised by tests/test_epstein.py (functional-equation and completed-function tests).
• No Euler product is required by S#, and DH genuinely has none: a_6 = a_1 = 1 while a_2 a_3 = -kappa^2 (tests/test_epstein.py::test_no_euler_product_coefficients_not_multiplicative).

So Dobner's Theorems 1-2 apply to DH: Lambda_DH exists, the real-rooted set is the closed half-line [Lambda_DH, inf), and Lambda_DH >= 0.

6. Normalization dictionary and numeric calibration

Dictionary

This table is about the multiplier, and only about the multiplier. All four rows say the same thing once the kernels are matched, and all four are frame-free: the threshold t >= Delta^2/2 is a statement about the ratio t/Delta^2, which the z-rescaling below leaves invariant. The other axis of normalization, where the critical line is parameterised, is a separate question and is the subject of the next two subsections and of FRAME.md. Confusing the two axes is what produced the correction below.

Dobner's frame is not this frame: the factor is 4

Correction, 2026-08-16 (GATE.md closure item (a)). This section previously ended its dictionary with the sentence

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 false. The substitution also carries z -> z/2, which the z-rescaling paragraph of this same section (kept unchanged, at the end) already said moves t quadratically: the false sentence contradicted the paragraph it sat next to. The correct statement is Phi_F(u) = Phi_DH(2u) and xi_t^F((1+iz)/2) = H_{t/4}(z/2), so Lambda(Dobner) = 4 Lambda(hunt). The sentence is kept here rather than deleted because two claims in this directory were routed through it and a reader is entitled to see what they were routed through.

What it would have caused, and what it did cause. It did not damage the strictness of the lower bound: t -> t/4 is an increasing bijection of the time axis, so Dobner's closed half-line {t : all zeros real} = [Lambda_F, inf) transfers to this frame as a closed half-line, and section 4c's use of it stands unaltered. What it would have licensed is quoting Dobner's Lambda_F for DH as this hunt's number, which is four times too large. What it did cause is one file down: NOVELTY.md calibrated the upper bound 0.4006 against the zeta record 0.22 and 1/2 with no conversion, which are 0.055 and 1/8 here, so the comparison flattered the hunt by exactly that factor and, worse, concealed that in the common frame the DH lower bound 0.2304 sits above the best published upper bound for zeta. Both are repaired; the full conversion table is FRAME.md.

The derivation. Dobner's (5) and (6), quoted in section 1, are

Phi_F(u) := (1/2pi) int_{-inf}^{inf} xi^F((1+ix)/2) e^{-ixu} dx, xi_t^F((1+iz)/2) := int_{-inf}^{inf} e^{t u^2} Phi_F(u) e^{izu} du.

His argument is (1+ix)/2 = 1/2 + i x/2, while this hunt's is 1/2 + i z. So xi^F((1+ix)/2) = Xi_DH(x/2), and by the inversion that defines Phi_DH from Xi_DH,

Phi_F(u) = (1/2pi) int_{-inf}^{inf} Xi_DH(x/2) e^{-ixu} dx = (1/pi) int_{-inf}^{inf} Xi_DH(y) e^{-2iyu} dy (y = x/2) = Phi_DH(2u).

Now substitute u = v/2 in his (6), using that Phi_F is even:

xi_t^F((1+iz)/2) = 2 int_0^inf e^{t u^2} Phi_DH(2u) cos(zu) du = int_0^inf e^{(t/4) v^2} Phi_DH(v) cos((z/2) v) dv = H_{t/4}(z/2).

At matched arguments the two deformations are the same function; what differs is the label on the time axis. Since t -> t/4 is an increasing bijection,

xi_t^F has only real zeros <=> H_{t/4} has only real zeros <=> t >= 4 Lambda_DH(this frame),

so

Lambda(Dobner frame) = 4 * Lambda(this frame), Delta(Dobner frame) = 2 * Delta(this frame).

Measured, this session, by code sharing nothing with instrument.py (scratchpad frame_check.py, frame_deform2.py; kappa from a linear solve on F(s) = F(1-s), f from Hurwitz zeta at r/5, and Dobner's Phi_F computed only from his own (5) by direct Fourier inversion, never by substituting Phi_DH(2u)):

and the deformation identity itself, checked end to end at four (z, t) with Phi_F again taken only from (5):

(mpmath dps 28; Phi_F on 50 composite Gauss-Legendre nodes, each value a separate Fourier inversion of Xi_DH(x/2) out to x = 80; the node set's own quality against the analytically known 2 int_0^inf Phi_F cos(3u) du = Xi_DH(3/2) is 7.6e-16, which is the floor for this table. The t = 0 endpoint xi_0^F((1+iz)/2) = Xi_DH(z/2) reproduces to 7.5e-15, 7.6e-16 and 1.6e-12 at z = 1, 3, 9.) The correct conversion holds to 15 digits; the reading the false sentence licensed is wrong in the second significant digit.

Neither frame is more correct. This one is Stopple's, published and refereed (arXiv:1301.3158, his equation (1) and his Xi_t(x, chi) at D = 5); that one is Dobner's, Newman's, Rodgers-Tao's and Polymath 15's. What is not allowed is a number without its frame. See FRAME.md.

z-rescaling (unchanged)

z-rescaling scales t quadratically: if Htilde_0(z) = c H_0(a z), i.e. Phitilde(u) = (c/a) Phi(u/a), then Htilde_t(z) = c H_{a^2 t}(a z). Strips scale by 1/a, times by a^2... so Lambda/Delta^2 is invariant. Zeta's convention H_0(z) = (1/8) Xi(z/2) has a = 1/2: Xi's strip 1/2 becomes Delta = 1 and the bound Delta^2/2 = 1/2 is exactly de Bruijn's classical "real for t >= 1/2" (Rodgers-Tao quote, section 3). DH needs no conversion: flow_repair measured H_0 = c Xi_DH(a z) with |c - 1| = 4.2e-42, |a - 1| = 2.0e-18 (hunts/flow_repair/NOTES.md), so (c, a) = (1, 1), Delta is taken directly as sigma_0 - 1/2 (scouted float value sigma_0 = 1.39513615823511, giving Delta^2/2 = 0.400634; deciding that strip is WP2's job, not this file's).

(Read the last four words of that paragraph narrowly. "DH needs no conversion" says only that H_0 equals Xi_DH with no constant and no z-rescaling, so Delta may be read straight off the s-plane strip. It does not say that this frame agrees with Dobner's; a = 1/2 is exactly the rescaling that separates them, and it is the same a that turns zeta's Delta = 1/2 into Delta = 1 two sentences earlier.)

Calibration of the factor Delta^2/2 (derived, never recalled)

Script: calibrate_theorem13.py in this directory; raw output: calibration.json. All numbers measured (mpmath/numpy floats, one route each; precision per block). Route 0 first checks the frame: the e^{tu^2} multiplier under the integral and the finite polynomial series sum_k (-t)^k/k! p^{(2k)} both satisfy the backward heat equation dG/dt = -d2G/dz2 (measured residual 0 at dps 30 for the quadrature route; 4.2e-6 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, evolved by the exact finite series, landing time by 60-iteration bisection (float64, imag tolerance 1e-10):

The middle row is the DH strip's own Delta: the landing lands on the mission's scouted 0.400634 to all printed digits. The factor is 1/2 exactly for the bare pair: a claimed factor Delta^2/8 would be violated (t* is four times later) and 2 Delta^2 would be slack by a factor 4.

• Route B, cosine polynomial H_0(z) = cos z + c, flowed to e^t cos z + c (dps 30; Delta measured as |Im| of the zero found by findroot near pi + i, not recalled; the complex pair tracked by Newton continuation, landing by 60-iteration bisection):

The ratio is always <= 1 (Theorem 13 is an upper bound on the landing time) and climbs to 1 as c -> 1+: the constant 1/2 is sharp in this family, so no smaller factor (in particular Delta^2/8) can be a theorem, and any larger one (2 Delta^2) is not sharp.

• Route C, pair plus distant real zeros (z^2 + 1)(z^2 - A^2) (float64, bisection as in A): 2 t*/Delta^2 = 0.9329, 0.99506, 0.99980 for A = 5, 20, 100. Repulsion from the extra real zeros only accelerates the landing, and the bound is approached as the spectators recede: consistent with Delta^2/2 as a sharp universal threshold.

7. Verdict

The upper-bound route stands. de Bruijn (1950) Theorem 13 (p. 205) is the all-zeros form in both hypothesis and conclusion; its only class conditions are Theorem 10's (p. 204): integrability, hermitian symmetry, O(e^{-|t|^b}) decay with b > 2, no positivity; Phi_DH satisfies every one of them (section 5); the strip-to-time factor converts to t >= Delta^2/2 in the hunt's e^{tu^2} normalization and was re-derived numerically on three families (section 6), landing at Delta^2/2 exactly on the bare pair and sharply in the c -> 1 cosine limit. Kill condition 1 of MISSION.md is not triggered. The remaining load on WP2 is the strip itself: decide that all zeros of Xi_DH lie in |Im z| <= sigma_0 - 1/2, with interval arithmetic on both backends. The lower-bound logic gains a Dobner-independent weak form (section 4c); strictness of Lambda_DH > t1 still cites Dobner's Theorem 1, whose S# hypotheses DH meets (section 5), applied through the conversion t -> t/4 derived in section 6 rather than through the frame identification that section used to assert.

And the verdict carries a frame. Delta^2/2 = 0.4006343708899557 is the threshold in the normalization s = 1/2 + iz (Stopple's, arXiv:1301.3158). In the normalization s = (1+iz)/2 (de Bruijn as usually quoted, Newman, Rodgers-Tao, Polymath 15, Dobner) the same threshold is 1.6025374835598228, because Delta doubles there. Neither is more correct; a number without its frame is what is wrong. See FRAME.md.

Update 2026-08-18. The two numbers above are the coefficient-domination ones and are superseded as the headline, though not as arithmetic: they are still what Delta = 0.895136... gives in the two frames. The strip constant was sharpened in-tree on 2026-08-18 to Delta = 0.62036249819, giving Delta^2/2 = 0.19242481458026887663805 narrow and 0.7696992583210755065522 wide (STRIP2.md). Nothing in this file changes on account of it: Theorem 13 is the engine either way, its transcription and hypothesis checks are unaffected, and the frame point this paragraph makes is unaffected. The conversion t -> t/4 of section 6 is what carries the new constant between the frames, exactly as it carried the old one.