The two Millennium problems
Checked against the official Clay documents on 2026-09-05. These are compact mathematical restatements; the linked originals govern details.
Navier–Stokes: existence and smoothness
Source: Charles L. Fefferman, pp. 1–2, equations (1)–(11), alternatives A–D.
For viscosity \(\nu>0\), velocity \(u\), pressure \(p\), and prescribed force \(f\):
\[ \partial_tu+(u\cdot\nabla)u=\nu\Delta u-\nabla p+f, \qquad \nabla\cdot u=0, \qquad u(x,0)=u_0(x). \]
There are three spatial dimensions and \(t\geq0\). Initial velocity is smooth and divergence-free.
Whole space: \(x\in\mathbb R^3\). For every spatial multi-index \(\alpha\) and nonnegative integers \(m,K\), require finite constants with
\[ |\partial_x^\alpha u_0(x)|\leq C_{\alpha K}(1+|x|)^{-K}, \qquad |\partial_x^\alpha\partial_t^m f(x,t)| \leq C_{\alpha mK}(1+|x|+t)^{-K}. \]
An admissible global solution has smooth \(u,p\) and uniformly bounded energy:
\[ \sup_{t\geq0}\int_{\mathbb R^3}|u(x,t)|^2\,dx<\infty. \]
Periodic case: \(u_0,f,u\) are unit-periodic spatially; \(u,p\) are globally smooth. Replace the decay assumptions by smooth periodic \(u_0\) and
\[ |\partial_x^\alpha\partial_t^m f(x,t)|\leq C_{\alpha mK}(1+t)^{-K}. \]
Proving any one alternative suffices:
| Alternative | Domain | Required conclusion |
|---|---|---|
| A | Whole space | Every admissible \(u_0\), with \(f=0\), admits a global solution above. |
| B | Periodic | Every admissible \(u_0\), with \(f=0\), admits a global solution above. |
| C | Whole space | Some admissible \(u_0,f\) admit no global solution above. |
| D | Periodic | Some admissible \(u_0,f\) admit no global solution above. |
Thus breakdown alternatives permit forcing; existence alternatives specify zero forcing.
The forced side is the current comparison track. OpenAI’s public announcement of 8 September 2026 reports a proof of the forced breakdown alternatives C/D and links a paper and formalization. See the announcement and our source companion. The repository records an attributed first reading of that artifact; independent proof verification and prize adjudication are separate statuses. The unforced existence alternatives A/B retain their distinct hypotheses. B19 records the related multiscale programme for other fluid equations.
Yang–Mills: quantum existence and mass gap
Source: Arthur Jaffe and Edward Witten, §§3–5, pp. 5–7.
For every compact simple gauge group \(G\), construct a nontrivial quantum Yang–Mills theory on \(\mathbb R^4\) with a positive mass gap. Here four dimensions describe spacetime.
Existence requires axiomatic strength at least that of the cited Wightman or Osterwalder–Schrader formulations. The physical description includes a Hilbert space \(\mathcal H\), Poincaré covariance, positive energy, a vacuum \(\Omega\) unique up to phase, and locality, subject to the full axiomatic requirements in the source.
Local quantum observables must correspond, with renormalisation subtleties, to gauge-invariant polynomials in curvature and covariant derivatives. Short-distance correlations must match asymptotic freedom and perturbative renormalisation, including the prescribed stress tensor and operator product behaviour.
Mass gap: for the physical Hamiltonian \(H\),
\[ H\Omega=0,\qquad H\geq0,\qquad \operatorname{spec}(H)\cap(0,\Delta)=\varnothing \quad\text{for some }\Delta>0. \]
The supremum of such \(\Delta\) must be finite: \(0<m<\infty\). Thus the lower edge of nonvacuum energies is strictly positive and finite.
The construction concerns the continuum quantum theory in infinite volume. Confinement and an isolated one-particle state are listed as further questions beyond the stated existence and mass-gap target.