Requirement by requirement
The Jaffe–Witten description was the guideline for the proof’s claim structure. This guide separates the exact formal statement from its broader technical expectations and shows where each enters the construction.
Unconditional proof claimed in Volumes I–III. Independent expert review ongoing.
Read the official problem description ↗Formal prize statement
For any compact simple gauge group G: construct a nontrivial quantum Yang–Mills theory on ℝ⁴, satisfying an accepted axiomatic framework, with a strictly positive mass gap.
Faithful paraphrase. Follow the link above for the official wording.
The quantifier is universal, not SU(2) or SU(3) only.
The theorem is stated for every compact connected non-abelian group with simple Lie algebra, with a dedicated transfer layer for finite central quotients and global forms.
Audit that no group-specific estimate is silently promoted to all groups.
A finite lattice or four-torus is not the endpoint.
Finite Wilson measures pass through thermodynamic and continuum limits; marked gauge-invariant observables are carried through the limit to prevent collapse to a trivial theory.
Uniform estimates, cofinal extraction, Euclidean restoration, and nontriviality.
Existence must be at least as strong as the cited Wightman or Osterwalder–Schrader frameworks.
The Euclidean construction targets temperedness, covariance, symmetry, reflection positivity, and clustering, followed by OS reconstruction of the Wightman theory.
Every reconstruction hypothesis must belong to the same limiting family.
Local fields should correspond to gauge-invariant polynomials in curvature and its covariant derivatives.
A Wilson-to-curvature operator dictionary identifies the reconstructed local fields with Yang–Mills observables rather than an unspecified gapped QFT.
Operator renormalisation and preservation of the dictionary through the limit.
Correlations should match asymptotic freedom and perturbative renormalisation at short distance.
The multiscale trajectory is tied to the asymptotically free Wilson theory and its continuum normalisation, rather than introduced only in the infrared.
β-function normalisation, local singularities, OPE, and stress-tensor treatment.
The reconstructed Hamiltonian must have no spectrum in (0, Δ) for some 0 < Δ < ∞.
Uniform centered transfer contraction is promoted by spectral calculus to a lower bound on the Hamiltonian over the orthogonal complement of the vacuum.
The estimate must control the whole non-vacuum sector, not one correlator.
The gap sits above the vacuum and implies exponential decay for centered local correlations.
Ergodicity and reconstruction identify the invariant vacuum; the same spectral estimate supplies exponential clustering below the gap scale.
Vacuum uniqueness, centering, and the exact decay-to-spectrum implication.
This is a map of the manuscript’s claim and dependency structure—not an assertion that Clay, a journal, or the mathematical community has validated any row.
Direct questions
The project should answer it with inspectable work, not status performance.
That cannot be settled by branding, confidence, or a FAQ. The useful test is whether the manuscript addresses the exact problem, exposes its dependencies, identifies its hardest steps, survives specialist attempts to break it, and can be independently checked. The project is organised to make those tests possible; it does not ask readers to accept a result because its author says so.
The research director does not claim an academic pedigree that he does not have. That reasonably raises the prior level of scrutiny. It is not a mathematical refutation. The response is a lower reliance on authority and a higher reliance on explicit statements, source traces, reproducible computations, formal checks, and expert review. Acceptance must come from people qualified to assess the mathematics.
No. The site presents a claimed proof under independent review. Clay recognition, journal publication, community digestion, and canonical acceptance are external later stages and cannot be declared by the project.
No. Volumes I–III claim the unconditional proof. Lean is an additional confidence-building project: it checks selected load-bearing implications and prevents silent changes of scope, but it does not replace the manuscript’s analytic arguments or determine independent acceptance.
No. It would be strong independent evidence that the constructed theory has the right spectral content. It cannot repair a logical gap. A mismatch, however, could falsify the physical identification or expose a normalization or limit error.