FRAME: which normalization a Lambda number is stated in

Written 2026-08-16, as closure item (a) of GATE.md.

This page exists because the hunt got the frame wrong once, in the one sentence a reader would quote. THEOREM13.md section 6 said the hunt's deformation and Dobner's "share Lambda exactly". They do not. They differ by a factor of exactly 4, and every headline number in this directory therefore has two values, not one.

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. Two conventions are in circulation, both standard, both used by refereed papers on exactly this constant, and neither of them announces itself:

narrow frame : s = 1/2 + i z (Stopple; 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).

Everything below is derived rather than recalled, and every row was checked by code written for this page (frame_check.py, frame_deform2.py, frame_zeta.py in this session's scratchpad, sharing no code with instrument.py). Grades follow MISSION.md: measured is one float route, decided is an enclosure with an exact endpoint sign or an exact integer, cited is somebody else's theorem.

1. The four conventions, as printed

1a. Narrow: Stopple, arXiv:1301.3158, read at source

Stopple's opening, verbatim from the arXiv PDF (his page 1, with s = 1/2 + it):

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

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

where

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

and his page 4, verbatim:

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

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

so that for t = 0, Xi_0(x, chi) is just Xi(x, chi).

and his Lambda, verbatim:

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

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

Set D = 5 and replace chi(n) by the period-5 Davenport-Heilbronn coefficients a_n = (1, kappa, -kappa, -1, 0), and this is character for character the hunt's own pair

Phi_DH(u) = 4 e^{3u/2} sum_{n>=1} n a_n exp(-pi n^2 e^{2u}/5), H_t(z) = int_0^inf e^{t u^2} Phi_DH(u) cos(zu) du,

including the closed half-line that defines Lambda. Note (D/pi)^{(s+1)/2} = (pi/5)^{-(s+1)/2} at D = 5: the same Gamma factor the hunt uses. The hunt's frame is published, refereed, and is Stopple's. It was never the hunt's own invention, and the hunt should stop presenting it as a private choice.

1b. Wide: Dobner, arXiv:2005.05142, and the Newman line

Dobner's equations (1), (2), (5), (6), as transcribed in THEOREM13.md section 1 and re-read by adversary 1:

(1) xi((1+iz)/2) = int_{-inf}^{inf} Phi(u) e^{izu} du (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^{t u^2} Phi_F(u) e^{izu} du

The argument is (1+iz)/2 = 1/2 + i z/2, so this z is twice the narrow frame's. Rodgers-Tao (arXiv:1801.05914) and Polymath 15 (arXiv:1904.12438) sit here too, with H_0(z) = (1/8) Xi(z/2) and Xi(T) = xi(1/2 + iT); so does this repository's own zeta/heatflow.py, whose module docstring states H_0(z) = (1/8) Xi(z/2) in its equation (3).

1c. de Bruijn's own multiplier

de Bruijn 1950 Theorem 13 (transcribed in THEOREM13.md section 1) uses

g(z) = int_{-inf}^{inf} F(t) e^{(1/2) lambda^2 t^2} e^{izt} dt, all roots real as soon as lambda >= Delta.

This is a statement about the multiplier, not about the critical line, so it is orthogonal to the narrow/wide split and applies inside either. Matching e^{(1/2) lambda^2 u^2} to e^{t u^2} gives t = lambda^2 / 2, hence the all-real threshold t >= Delta^2 / 2 used by both Stopple's frame and Dobner's, with each frame's own Delta.

Newman and Wu (Bull. AMS 57, 2020, arXiv:1901.06596) restate the same theorem as their Theorem 7, verbatim from the arXiv PDF, page 8:

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

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

lie in the strip

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

with their (10) F(-t) = conj F(t) and their (11) |F(t)| <= A exp(-|t|^{2+alpha}). Their lambda is 2t in the e^{tu^2} convention, so (17) reaches zero at t = Delta^2/2 again.

1d. Newman-Wu's own kernel is narrow, which is a trap

Newman-Wu's equation (7) is

Phi(u) = sum_{n>=1} (4 pi^2 n^4 e^{9u/2} - 6 pi n^2 e^{5u/2}) e^{-pi n^2 exp(2u)}

and their (19) is H_lambda(z) = int_R e^{lambda t^2} Phi(t) e^{izt} dt. Measured here (frame_zeta.py, mpmath dps 30, my own quadrature against zeta.core.xi):

So Newman-Wu's kernel sits in the narrow frame, alongside Stopple's, while Rodgers-Tao's and Dobner's sit in the wide one. Applying (*) of section 2 with Phi_NW(u) = 2 Phi_RT(u/2), their (19) satisfies H_lambda^{NW}(z) = 8 H_{4 lambda}^{RT}(2z), so the constant their (19) defines is a quarter of the classical de Bruijn-Newman constant that Polymath 15 bounds by 0.22. This is worth stating because of what it implies for the sentence on their page 9:

Moreover, since the roots of xi lie in the strip |Im z| <= 1/2, de Bruijn's quantitative bound (17) gives an upper bound Lambda_DN <= 1/2.

In the narrow frame Delta = 1/2 is right, and (17) applied to it gives all roots real at lambda = Delta^2 = 1/4, which in their own e^{lambda t^2} kernel of (19) is Lambda_DN <= 1/8. The quoted 1/2 is the wide-frame number. It is not false, because 1/8 implies 1/2; it is four times weaker than what their own Theorem 7 delivers in their own normalization.

Stopple carries the same classical 1/2 in the same narrow frame. His page 4, verbatim:

Since Phi(u, chi) has doubly exponential decay, [1, Theorem 13] applies to Xi_t(x, chi) and we have an analog of the theorem of de Bruijn for the Riemann zeta function: (1) For t >= 1/2, Xi_t(x, chi) has only real zeros.

For a quadratic Dirichlet L-function the Euler product forces all nontrivial zeros into 0 <= Re s <= 1, and Gamma((s+1)/2) cancels the trivial zeros rather than adding any, so in his own frame Delta = 1/2 and Theorem 13 gives t >= 1/8. Again 1/2 is implied and so not false, and again it is the wide-frame number.

Two refereed sources, both about this exact constant, both carrying the classical number across a frame change without converting it. That is the whole reason this page exists, and it is why every number in this directory now travels with its frame attached.

2. The scaling law, derived

Let Phi be any admissible kernel, a > 0, c != 0, and set

Phitilde(u) := (c/a) Phi(u/a), Htilde_t(z) := int_0^inf e^{t u^2} Phitilde(u) cos(zu) du.

Substitute u = a v, du = a dv:

Htilde_t(z) = int_0^inf e^{t a^2 v^2} (c/a) Phi(v) cos(z a v) * a dv = c int_0^inf e^{(a^2 t) v^2} Phi(v) cos((a z) v) dv = c H_{a^2 t}(a z). (*)

Three consequences, all immediate from (*) and none of them recalled:

1. Zeros. Htilde_t(z) = 0 iff H_{a^2 t}(a z) = 0, so the zero set of Htilde_t is 1/a times the zero set of H_{a^2 t}. Since a is real and positive, real zeros map to real zeros and Im scales by 1/a.
2. Strip. Delta_tilde = Delta / a.
3. Time. {t : Htilde_t all real} = {t : H_{a^2 t} all real} = (1/a^2) {s : H_s all real}, so Lambda_tilde = Lambda / a^2.

Hence

z -> a z gives Lambda -> Lambda / a^2, Delta -> Delta / a,

and therefore Lambda / Delta^2 is invariant, which is exactly why the de Bruijn threshold Delta^2/2 can be quoted frame-free while Lambda and Delta separately cannot. A bound of the form Lambda <= Delta^2/2 is a statement about a ratio; a bound of the form Lambda <= 0.1924 is not.

Verified at a = 1/2, 2, 13/10 with c = 3/7: relative defects 2.2e-29, 0.0, 0.0. Numbers and method in section 5.

3. Narrow to wide, the explicit conversion

Apply (*) with a = 1/2 and c = 1/2, i.e. Phi_F(u) = Phi_DH(2u):

int_0^inf e^{t u^2} Phi_F(u) cos(zu) du = (1/2) H_{t/4}(z/2),

and Dobner's (6) is the full-line integral, which is twice the half-line one because Phi_F is even. Therefore

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

and for the constants,

xi_t^F all real <=> H_{t/4} all real <=> t/4 >= Lambda(narrow) <=> t >= 4 Lambda(narrow),

Lambda(Dobner) = 4 Lambda(hunt).

Note what the conversion is not: it is not a claim that the two deformations have different zeros. At matched arguments they are the same function. What differs is the label on the time axis, and a constant defined as an infimum over that axis inherits the label.

4. The conversion table

z_n is the narrow-frame variable (s = 1/2 + i z_n), z_w the wide one (s = (1 + i z_w)/2); z_w = 2 z_n.

The last row is the one that stays put under a change of frame, because it is the ratio statement of section 2.

5. Numerical verification of every row

All of this was run for this page, at dps 25 to 40, in code that shares nothing with instrument.py: kappa from a linear solve on F(s) = F(1-s), f from Hurwitz zeta at r/5, Phi_DH from the raw series, and Dobner's Phi_F from his own definition (5) by direct Fourier inversion, never by substituting Phi_DH(2u).

kappa, own route: 0.28407904384041229602829183239312617 against the repo's pinned KAPPA_REF ...39312651, agreeing to 32 digits at dps 60 internal. Functional-equation defect F(s) - F(1-s) at s = 5/2, -3/2, 1/2 + 4i, 4/5 + 3i: 0.0, 0.0, 9.4e-42, 2.6e-42 relative. Grade: measured.

Row: H_0 = Xi_DH in the narrow frame (dps 40, own quadrature vs own F(1/2 + iz)):

Row: evenness of Phi_DH (the hermitian hypothesis of Theorem 13, and the step winding.py needs), dps 40, relative |Phi(u) - Phi(-u)|: 2.0e-41 at u = 0.1, 4.5e-41 at 0.5, 3.3e-41 at 1.0. Grade: measured; the reason is the functional equation F(s) = F(1-s), written out in THEOREM13.md section 5, item 2.

Row: Phi_F(u) = Phi_DH(2u), Dobner's Phi_F computed only from his (5), by inversion of xi^F((1+ix)/2) = Xi_DH(x/2) truncated at R = 140, dps 40. The truncation floor is set by the Gamma factor: |Xi_DH(y)| decays like exp(-pi y/4) (measured: 1.86e-6, 4.81e-14, 8.87e-19, 4.17e-23 at y = 20, 40, 55, 70), and the inversion runs in x = 2y, so the neglected tail is about exp(-pi R/8). Measured directly, by varying R at fixed u = 0.3 and dps 30:

which is the exp(-pi R/8) law, so R = 140 supports about 24 digits and R = 80 about 13. At R = 140, dps 40:

The u = 0 row cannot separate the hypotheses and is included only as a sanity check. The other three separate them by twenty orders of magnitude. Grade: measured.

Row: the deformation identity xi_t^F((1+iz)/2) = H_{t/4}(z/2), checked end to end. Dobner's (6) is evaluated from his (5) alone, on a composite Gauss-Legendre node set (5 panels, 10 nodes each, on [0, 1.6]) whose 50 Phi_F values are each a separate Fourier inversion at R = 80, dps 28. The node set's own quality, measured against the analytically known 2 int_0^inf Phi_F(u) cos(3u) du = Xi_DH(3/2), is 7.6e-16, so that is the floor for this row.

and the t = 0 endpoint xi_0^F((1+iz)/2) = Xi_DH(z/2): relative 7.5e-15 at z = 1, 7.6e-16 at z = 3, 1.6e-12 at z = 9. So the correct conversion is confirmed to 15 significant digits while the reading the false sentence licensed is wrong in the second significant digit. Grade: measured.

Row: the scaling law (*) of section 2, both sides by independent quadrature, c = 3/7, z = 2, t = 3/10, dps 28:

Grade: measured.

Row: the zeta frames (frame_zeta.py, dps 30, against zeta.core.xi): int_0^inf Phi_RT cos(zu) du = (1/8) Xi(z/2) to 1.8e-42 - 5.2e-42; int_R Phi_D e^{izu} du = Xi(z/2) to the same; 2 int_0^inf Phi_NW cos(zu) du = Xi(z) to 2.9e-42 - 7.1e-42 while missing (1/8)Xi(z/2) by 0.09 to 0.87. Grade: measured.

6. The headline numbers, in both frames

The bounds themselves are unchanged; only their labels are. The lower side rests on a decided argument-principle count (winding_results.json, decided_floor_t = 36/625) plus Dobner's closed half-line; the upper side is Delta^2/2 with Delta = sigma_0 - 1/2 and sigma_0 decided on both backends (strip_results.json).

Note that the abscissa and the ratio are the frame-free quantities, and the ratio is the honest measure of how loose the bracket is. No outward rounding is involved in this row: the abscissa is decided at an exact rational, so Delta^2/2 terminates. STRIP2.md section 5.2 displays the narrow value rounded outward to 22 decimals as 0.1924248145802688766381, which is the same bound one display ulp higher.

Superseded 2026-08-18, kept because it is still correct. Through 2026-08-17 the headline came from the coefficient-domination abscissa of STRIP.md instead, and that row read:

The sharpening is a factor 2.082030697360155 and it is in-tree: the abscissa is derived from a phase obstruction in the Euler products of the two Dirichlet L-functions whose combination f is, and decided on both backends (python-flint 192 bits and mpmath.iv dps 40, sieve limit P = 10^5, 4814 class primes), with a flint-only deep point at 320 bits and P = 10^7 deciding 1.1203624981833251. The lower bound did not move, so the separation of SEPARATION.md, which rests on the floor alone, is unaffected.

The relation to the literature, restated exactly. Bombieri and Ghosh (Russian Math. Surveys 66 (2011), section 6, read in full 2026-08-16, see BOMBIERI-GHOSH.md) determine the exact least upper bound of the real parts of the zeros of this function, sigma(tau_+, 1) = 1.120362. The criterion STRIP2.md decides is their Theorem 7 at q = 1 and xi = kappa, term for term; only its necessary half is used, and that half is derived here 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. So the number is theirs and the grade is this hunt's, and neither sigma_0 nor sigma_0' may be described as a new number.

A row this file used to carry, and the correction it needed. Through 2026-08-17 the table above had a "B-G sigma(tau_+, 1) = 1.120362 (cited, six decimals)" row printing 0.192424814576128011... narrow and 0.769699258304512045... wide. GATE.md's closure log flagged that those 18 digits came from BOMBIERI-GHOSH.md's own 29-digit re-solve rather than from the six cited decimals (which alone give 0.192424505522). strip2.py now decides that the re-solve itself sits about 1.2e-17 above its own root, so the correct value from the deep point is 0.1924248145761280190 narrow and 0.7696992583045120759956154 wide, agreeing with the old row to 17 digits. The headline above is the slightly more conservative two-backend P = 10^5 value, and it is exact.

7. The zeta record, in both frames

NOVELTY.md originally quoted these in the wide frame while calibrating a narrow-frame number against them, which flattered the comparison by exactly the factor 4. Both columns, so that cannot recur:

And the comparison the conversion actually buys, stated in the common wide frame where both live:

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

so the Davenport-Heilbronn constant sits above the best known upper bound for zeta, hence Lambda_DH > Lambda_zeta. Stated in the narrow frame alone, 0.0576 sits below 0.22 and a reader draws the opposite conclusion. That is the single most important reason to publish both columns rather than pick one.

Grade of that comparison, per MISSION.md: the DH side is decided modulo Dobner's Theorem 1 (an enclosure-carrying integer count plus a cited closed half-line); the zeta side is cited (Polymath 15). A composite takes its weakest step, so Lambda_DH > Lambda_zeta is a cited-plus-decided statement and not a claim this directory proves on its own.

8. What to write, every time

• Give the frame with the number: "in the normalization of Stopple (arXiv:1301.3158), s = 1/2 + iz".
• Print both values, or print one and give the factor 4 and its direction in the same sentence.
• Never quote a zeta record without converting it into the frame of the number it is being compared with.
• Lambda / Delta^2 and sigma_0 are the frame-free quantities. Prefer them when the point is a comparison rather than a value.