Clay Millennium Prize Problems (2000)

Clay Mathematics Institute, Paris, 24 May 2000. Seven problems, one million dollars each. Official list: The Millennium Prize Problems.

Status cutoff 2026-08-30. "Resolved" means CMI has accepted a solution. Six remain open. Poincaré is the only prize that has been awarded (announced 2010; Perelman declined).

This notebook already has Hilbert, Landau, Smale, and Simon lists. Riemann is Hilbert 8(a) and Smale 1. Navier–Stokes is Smale 15. P vs NP is Smale 3. Landau 3 (Legendre) in problems/landau-legendre/ is an RH-conditional explicit gap bound, not the Clay problem.

#	Problem	Status	Note
1	Birch and Swinnerton-Dyer	Open	Rank equals analytic rank, with the usual leading-term formula. Many individual curves are checked; the conjecture is not.
2	Hodge	Open	Hodge classes on a non-singular complex projective variety are rational linear combinations of algebraic cycle classes. No finite-handle record in the style of this notebook.
3	Navier–Stokes existence and smoothness	Open	Also Smale 15. 3D incompressible; Fefferman's Clay write-up. 2D is known. Tao's averaged model blows up; that does not settle the true equations.
4	P vs NP	Open	Also Smale 3. Circuit lower bounds and the τ-conjecture (Smale 4) are neighbouring finite questions; they are not a resolution.
5	Poincaré	Resolved (yes)	Perelman 2002–03, Ricci flow with surgery. CMI prize 2010.
6	Riemann hypothesis	Open	Also Hilbert 8(a) and Smale 1. First campaign for this list: see below.
7	Yang–Mills existence and mass gap	Open	Construct quantum Yang–Mills on R^4 with a mass gap. No finite-handle record here.

Order follows CMI's page, not Hilbert numbering.

First campaign: Riemann

Among the six open Clay problems, this is the one with a living finite handle. A dent here is a verified finite improvement of a published explicit record, not a proof of RH.

Published records to replay before claiming anything (arXiv is the record; blogs and GitHub audits are leads):

Rodgers–Tao: the de Bruijn–Newman constant satisfies Λ ≥ 0 (arXiv:1801.05914, Forum Math. Pi 2020). RH is equivalent to Λ ≤ 0.
Polymath 15 instantiated an effective heat-flow criterion for an upper bound. With the Platt–Trudgian verified-height input this has been used to get Λ ≤ 0.2 (Polymath 15 itself printed 0.22; later instantiations used the larger height). A later unofficial claim of 0.1787854 is a lead, not a citation, until it is on arXiv and independently replayed.
Platt–Trudgian: RH verified for all zeros with imaginary part at most 3,000,175,332,800 (arXiv:2004.09765, Bull. LMS 2021).
Landau 3 in this notebook is RH-conditional (δ ≥ 0.22525) and does not move the Clay problem.

Current folder: problems/riemann-hypothesis/. The first computation replayed the published $0.2$ arithmetic and fresh parts of the off-arXiv $0.1787854$ lead. The full finite producer and analytic review remain, so the published explicit window is still $0\leq\Lambda\leq0.2$ .

A failed search is residue, not a lower bound.