Written 2026-08-16. The separation claim Lambda_DH > Lambda_zeta is load-bearing on the exact statement and frame of Polymath 15's upper bound, so both papers are quoted here at source rather than from memory or from FRAME.md's summary.
Retrieval route. Both ar5iv renderings were downloaded whole (ar5iv.labs.arxiv.org/html/1904.12438, 1,688,923 bytes; ar5iv.labs.arxiv.org/html/1801.05914, 1,285,339 bytes; retrieved 2026-08-16, latest posted arXiv versions as ar5iv serves them). Every displayed formula below is the paper's own LaTeX, extracted character for character from the MathML alttext attributes of the rendered HTML, not transcribed by hand and not summarised by a model. Prose is taken verbatim from the same files with markup stripped. In the Rodgers-Tao rendering the bibliography keys appear raw (debr, newman, kkl, polymath) because ar5iv did not resolve them; they are left as served.
Publication data: Polymath is published as Res. Math. Sci. 6 (2019), art. 31; Rodgers-Tao as Forum Math. Pi 8 (2020), e6 (journal data recalled, not re-fetched; the pin is to the arXiv text).
1. Polymath 15, arXiv:1904.12438, Section 1 (Introduction)
Displays (1)-(4) and their connecting prose, verbatim (arXiv PDF pages 1-2):
Let $H_{0}\colon\mathbb{C}\to\mathbb{C}$ denote the function
(1) $H_{0}(z)\coloneqq\frac{1}{8}\xi\left(\frac{1}{2}+\frac{iz}{2}\right),$
where $\xi\colon\mathbb{C}\to\mathbb{C}$ denotes the Riemann $\xi$ function
(2) $\xi(s)\coloneqq\frac{s(s-1)}{2}\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)$
(which is an entire function after removing all singularities) and $\zeta$ is the Riemann $\zeta$ function. Then $H_{0}$ is an entire even function with functional equation $H_{0}(\overline{z})=\overline{H_{0}(z)}$, and the Riemann hypothesis (RH) is equivalent to the assertion that all the zeroes of $H_{0}$ are real. It is a classical fact (see [27, p. 255]) that $H_{0}$ has the Fourier representation
$H_{0}(z)=\int_{0}^{\infty}\Phi(u)\cos(zu)\ du$
where $\Phi$ is the super-exponentially decaying function
(3) $\Phi(u)\coloneqq\sum_{n=1}^{\infty}(2\pi^{2}n^{4}e^{9u}-3\pi n^{2}e^{5u})\exp(-\pi n^{2}e^{4u}).$
The sum defining $\Phi(u)$ converges absolutely for negative $u$ also. From Poisson summation one can verify that $\Phi$ satisfies the functional equation $\Phi(u)=\Phi(-u)$ (i.e., $\Phi$ is even); this fact is of course closely related to the functional equation for $\zeta$. De Bruijn [5] introduced (with somewhat different notation) the more general family of functions $H_{t}\colon\mathbb{C}\to\mathbb{C}$ for $t\in\mathbb{R}$, defined by the formula
(4) $H_{t}(z)\coloneqq\int_{0}^{\infty}e^{tu^{2}}\Phi(u)\cos(zu)\ du.$
The de Bruijn / Newman paragraph, verbatim (Section 1, the paragraph following (4)):
It follows from the work of Pólya [19] that if $H_{t}$ has purely real zeroes for some $t$, then $H_{t^{\prime}}$ has purely real zeroes for all $t^{\prime}>t$; de Bruijn showed that the zeroes of $H_{t}$ are purely real for $t\geq 1/2$. Newman [14] strengthened this result by showing that there is an absolute constant $-\infty<\Lambda\leq 1/2$, now known as the De Bruijn-Newman constant, with the property that $H_{t}$ has purely real zeroes if and only if $t\geq\Lambda$. The Riemann hypothesis is then clearly equivalent to the upper bound $\Lambda\leq 0$. Recently in [22] the complementary bound $\Lambda\geq 0$ was established, answering a conjecture of Newman [14], and improving upon several previous lower bounds for $\Lambda$ [6, 15, 8, 7, 16, 23]. Furthermore, Ki, Kim, and Lee [10] sharpened the upper bound $\Lambda\leq 1/2$ of de Bruijn [5] slightly to $\Lambda<1/2$. In this paper we improve the upper bound:
(Their [22] is Rodgers-Tao, their [5] is de Bruijn 1950. ASCII math is house style for authored text only; the quotes keep the source's own characters.)
The theorem, verbatim (Section 1):
Theorem 1.1 (New upper bound). We have $\Lambda\leq 0.22$.
From the abstract, verbatim:
By combining these estimates with numerical computations, we are able to obtain a new upper bound $\Lambda\leq 0.22$ unconditionally, as well as improvements conditional on further numerical verification of the Riemann hypothesis.
One further pin, because it is the hunt's own upper-bound engine appearing in their frame: their Theorem 1.2 (Upper bound criterion), Section 1, concludes
Then $\Lambda\leq t_{0}+\frac{1}{2}y_{0}^{2}$.
which is the t + Delta^2/2 bookkeeping with y_0 the surviving strip height at time t_0.
2. Rodgers-Tao, arXiv:1801.05914, Section 1 (Introduction)
Their displays (1)-(4) are character for character the same LaTeX as Polymath 15's (1)-(4) above, same numbering, same connecting prose up to trivial variants ("where $\xi$ denotes the Riemann xi function", "(see (titch, p. 255))"). Extracted alttext, verbatim:
$H_{0}(z)\coloneqq\frac{1}{8}\xi\left(\frac{1}{2}+\frac{iz}{2}\right),$ (1)
$\xi(s)\coloneqq\frac{s(s-1)}{2}\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)$ (2)
$\Phi(u)\coloneqq\sum_{n=1}^{\infty}(2\pi^{2}n^{4}e^{9u}-3\pi n^{2}e^{5u})\exp(-\pi n^{2}e^{4u}).$ (3)
$H_{t}(z)\coloneqq\int_{0}^{\infty}e^{tu^{2}}\Phi(u)\cos(zu)\ du.$ (4)
Their de Bruijn / Newman paragraph, verbatim (Section 1, following (4)):
De Bruijn showed that the zeroes of $H_{t}$ are purely real for $t\geq 1/2$. Strengthening these results, Newman [newman] showed that there is an absolute constant $-\infty<\Lambda\leq 1/2$, now known as the De Bruijn-Newman constant, with the property that $H_{t}$ has purely real zeroes if and only if $t\geq\Lambda$. The Riemann hypothesis is then clearly equivalent to the upper bound $\Lambda\leq 0$. Newman conjectured the complementary lower bound $\Lambda\geq 0$, and noted that this conjecture asserts that if the Riemann hypothesis is true, it is only “barely so”.
The upper-bound history sentence, verbatim, including the footnote text (Section 1, after Table 1; footnote 1 added in press):
We also mention that the upper bound $\Lambda\leq 1/2$ of de Bruijn [debr] was sharpened slightly [footnote 1: Added in press: this bound has recently been improved to $\Lambda\leq 0.22$ in [polymath].] by Ki, Kim, and Lee [kkl] to $\Lambda<1/2$. See also [stopple], [cmmrs] on work on variants of Newman's conjecture, and ([broughan], Chapter 5) for a survey.
The theorem, verbatim (Section 1):
Theorem 1 One has $\Lambda\geq 0$.
3. Their frame is the wide one, in three lines
- Their (1) reads
H_0(z) = (1/8) xi(1/2 + iz/2), so theirzsits ons = 1/2 + iz/2 = (1 + iz)/2; withXi(T) = xi(1/2 + iT)this isH_0(z) = (1/8) Xi(z/2), and the zero strip0 <= Re s <= 1becomes|Im z| <= 1:Delta = 1, double the narrow frame'sDelta = 1/2ats = 1/2 + iz. - The hunt's DH object is
H_0(z) = F(1/2 + iz)with noz/2(Stopple's kernel atD = 5,FRAME.mdsection 1a): the narrow frame, soz_wide = 2 z_narrow. FRAME.mdsection 2's scaling law (z -> a zgivesLambda -> Lambda / a^2; both kernels carry the same multipliere^{t u^2}, soa = 2is the whole difference) givesLambda(wide) = 4 Lambda(narrow), measured end to end on Dobner's own definitions to 15 significant digits inFRAME.mdsection 5.
Kernel-level corroboration: their (3) has e^{4u} inside the Gaussian where the narrow kernel (Newman-Wu's (7), FRAME.md section 1d) has e^{2u}, and Phi_NW(u) = 2 Phi_P15(u/2) holds exactly to working precision (measured, frame_zeta.py, mpmath dps 30). Same doubling, seen in the kernel instead of the argument.
4. The separation chain, every link cited or decided
0 <= Lambda_zeta cited: Rodgers-Tao Theorem 1 (Section 2 above) Lambda_zeta <= 0.22 cited: Polymath 15 Theorem 1.1 (Section 1 above), unconditional per their abstract 0.22 = 11/50 < 144/625 = 0.2304 exact rational arithmetic, cross-multiplied: 144 * 50 = 7200 > 6875 = 11 * 625 0.2304 = 4 * (36/625) < Lambda_DH this hunt, decided: winding count N = 1 for H_{36/625} (narrow) over a box with interior Im z >= 3/1024 in exact rationals, python-flint 0.9.0 (Arb) balls, second witness mpmath dps 130 (winding_results.json); strictness cited, Dobner arXiv:2005.05142 Theorem 1 closed half-line; frame factor 4 derived in FRAME.md section 3 Lambda_DH <= 0.7696992583210755065522 this hunt, decided: 4 x the narrow Delta^2/2, exactly, at the phase-obstruction abscissa sigma_0' = 112036249819/100000000000 decided on both backends (python-flint Arb 192 bits and mpmath.iv dps 40, sieve limit P = 1e5) with exact rational sign decisions, kappa at 500 bits (strip2_results.json); de Bruijn 1950 Theorem 13 cited. Through 2026-08-17 this link read <= 1.6025374835598228, from the coefficient- domination interval [0.4006343708899556944469547527, 0.4006343708899556944469548120] of strip_results.json, which is still correct and is retained
Hence, in the wide frame where all four sources above live:
0 <= Lambda_zeta <= 0.22 < 0.2304 < Lambda_DH <= 0.7696992583210755065522,
so Lambda_DH > Lambda_zeta unconditionally. Composite grade per MISSION.md: cited plus decided (weakest step is a citation; the lower bound additionally carries the M2 lemma recorded in GATE.md, prose when this page was written and proved with decided constants since 2026-08-18, M2-LEMMA.md). The last link is not used by the separation, which rests on the floor and on the cited zeta bound alone; sharpening it on 2026-08-18 moved nothing else in this chain.
Frame-invariance check, narrow frame: divide every Lambda by 4.
0 <= Lambda_zeta <= 0.055 < 0.0576 = 36/625 < Lambda_DH <= 0.19242481458026887663805,
and the middle comparison is the same cross-multiplication, because the factor 4 cancels: 0.055 = 11/200, and 36 * 200 = 7200 > 6875 = 11 * 625. The inequality Lambda_DH > Lambda_zeta reads identically in both frames, as an inequality between constants of the same frame must.
5. What the sources did and did not say (the surprises)
- Neither paper makes any normalization remark at all. The task of pinning "their remark that de Bruijn's
Lambda <= 1/2lives in the same frame" turns out to be unfillable: there is no such remark. Both papers simply assert de Bruijn's result of their ownH_t("de Bruijn showed that the zeroes of $H_{t}$ are purely real for $t\geq 1/2$", withH_ttheir (4)), and Polymath 15 adds only "(with somewhat different notation)" about de Bruijn's original. So the frame of the classical1/2is theirs by restatement, not by remark, and the anchor for the hunt's dictionary is display (1) itself, not any prose about conventions. This is consistent withFRAME.md's finding that frame conversions in this literature are routinely silent. - Rodgers-Tao cite Stopple, with no conversion. Their "See also [stopple], [cmmrs] on work on variants of Newman's conjecture" points straight at the narrow-frame paper this hunt inherits, one sentence after quoting the wide-frame
Lambda <= 1/2, with nothing said about the factor 4 between them. The two frames touch in print exactly once in these two papers, and the touch carries no warning. - The two introductions share their displays character for character. P15's (1)-(4) and RT's (1)-(4) are the same LaTeX byte for byte in the extracted alttext. Quoting either paper's display pins both.
- Nothing else surprised. The kernel is the expected wide one (
e^{4u}in the Gaussian),H_0(z) = (1/8) xi(1/2 + iz/2)is exactly asFRAME.md's conversion table row states,Lambda <= 0.22is unconditional and stated with no side conditions, and RT's Theorem 1 is the bareLambda >= 0. The pinned quotes confirm the hunt's dictionary with no adjustment needed anywhere.