Upper bounds for the de Bruijn-Newman constant
the de Bruijn-Newman constant
- steps
- 3
- by AI
- 0
- since
- 1950
The line is the frontier over time, falling as the bound comes down: lower is better here. Filled dots are steps that moved it; muted dots are results that did not. Orange dots are catalog entries, results with AI in the loop. Hollow dots are candidates under review and never move the line.Grey dots along the top edge are results from before the quantity had a number, placed at the worst end because they have no value on this axis. Dots that would overlap are nudged sideways a few pixels. Hover a dot for its value and attribution.
About this frontier
For each real the entire function is the heat-flow deformation of the Riemann function, and de Bruijn and Newman showed there is a finite constant such that the zeros of are all real exactly when . The Riemann hypothesis is equivalent to , and Rodgers and Tao proved Newman's complementary conjecture in 2018, so the constant is pinned below at zero and the open question is how far the upper bound can be pushed toward it. This frontier tracks that upper bound. Reaching would prove the Riemann hypothesis; nothing here claims to be near it.
Every step, newest first
| Date | Value | Who | Model | Status | Source |
|---|---|---|---|---|---|
| 20 Aug 2026 | Polymath15's Theorem 1.2 at exact parameters X = 6000000185827, t0 = 129/800, y0^2 = 87677/2500000, with fail-closed Arb interval certificates and a sealed manifest. Unconditional in form: no RH beyond the finite Platt-Trudgian height. Not peer reviewed; its referee report is an adversarial AI panel that calls itself no substitute for human review and leaves three items needing sign-off. | Jude Gomila, with Claude and ChatGPT/Codex | Claude; ChatGPT and Codex (versions undisclosed) | candidate unreviewed | entry |
| 3 Jul 2026 | A certified unconditional bound on a versioned Zenodo record, CC BY 4.0. Zenodo refused automated access from here, so this row is transcribed from the artifact lock in the 0.1787854 package, which pins the DOI, the date and SHA-256 hashes of both files. Unreviewed. | Mosaic Intelligence | – | candidate | source ↗ |
| May 2020 | best Not a separate paper: the same Polymath15 criterion, fed by Platt and Trudgian's verification of RH to height 3e12. The linked source is that verification; the 0.2 figure is stated on the Polymath project wiki and in Tao's commentary, which could not be opened from here. This is the standing reviewed record. | D.H.J. Polymath's criterion at Platt and Trudgian's verified height | – | historical | source ↗ |
| May 2019 | The Polymath15 project, which built an effective theory of the heat flow and instantiated it. This is the paper every later step uses: its Theorem 1.2 is the barrier criterion the 2026 claims plug exact parameters into. | D.H.J. Polymath (Polymath15) | – | historical | source ↗ |
| 2009 | Strict inequality, sharpening de Bruijn without moving the number. The flat stretch on this chart from 1950 to 2009 is real: fifty-nine years with no numerical improvement. | Haseo Ki, Young-One Kim and Jungseob Lee | – | historical | source ↗ |
| 1950 | De Bruijn's original bound, from the paper that introduced the deformation. Classical; cited here rather than opened. | Nicolaas de Bruijn | – | historical | source ↗ |
Steps below 0.22 are the live race; the two 2026 rows are unreviewed claims and are drawn hollow, so the standing reviewed record on this staircase is 0.2. Two rows could not be opened from here and are marked in their notes: the 0.2 figure, whose primary statement is on the Polymath project wiki, and the 0.1875 claim, whose Zenodo record refused automated access and which is transcribed from the artifact lock in the 0.1787854 package. The 1950 and 2009 rows are classical and were not independently opened. Everything else was read at source.