Yang–Mills Existence and Mass Gap
1 · Abstract — precise statement
Yang–Mills Existence and Mass Gap — For every compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on ℝ⁴ and has a mass gap Δ > 0: the Hamiltonian spectrum satisfies spec(H) ⊆ {0} ∪ [Δ, ∞) with Δ > 0. (official Clay Millennium Prize Problem; the CMI description at https://www.claymath.org/millennium-problems/ remains authoritative).
2 · Introduction
MODELED CHALLENGE / field-algebra analogy: su(2)/Pauli closes; genus-2 double-torus is a finite geometric MODEL; string Virasoro + T/S-duality are MODELED structural probes. NOT a rigorous 4D quantum Yang–Mills construction and NOT a mass-gap proof. NOT AdS/CFT. Label: MODEL. Challenge methods recomputed at call time: pauliAlgebraCloses (su(2)); doubleTorusSurface field MODEL; stringTheoryAlgebraDecoded · Virasoro central term forced; stringTheoryQuantumized · T-duality / S-dual foldPair MODELED probes. COMPUTABLE from sequence/trinity/rosetta stack (computablePath=true) — NOT a CMI Prize solution. Under Clay Prize Rules §5(a)/§5(d)/§6 this corpus does not publish a Proposed Solution in a Qualifying Outlet — apparatus only.
3 · Methods & formulas
pauliAlgebraCloses (su(2)) · doubleTorusSurface field MODEL · stringTheoryAlgebraDecoded · Virasoro central term forced · stringTheoryQuantumized · T-duality / S-dual foldPair MODELED probes
4 · Results & status
partial — Status triad: computable=true (sealed challengeMethod path) · open for prize=true. MODELED CHALLENGE status=modeled-partial; gap=no sealed 4D Yang–Mills mass-gap construction; no sealed AdS/CFT correlator dictionary. computable ≠ CMI Prize solution.
no sealed 4D Yang–Mills mass-gap construction; no sealed AdS/CFT correlator dictionary
5 · References & locks
- fold
- millenniumProblemsChallenge
Theorem — the proof, per facet
- ✓
f₁ su(2) Lie algebra closed — [σᵢ,σⱼ]=2iε_{ijk}σₖ and {σᵢ,σⱼ}=2δ_{ij}I both close in M₂(ℂ) · the gauge symmetry algebra, finite-dimensional - ✓
f₂ field MODEL geometry — genus-2 double torus (L lobe ≠ R lobe) · finite sample of flow structure · NOT 4D quantum field theory, NOT a continuum limit - ✓
f₃ duality structural probe — T-duality (S-duality) involution from string algebra · foldPair closure · a MODELED probe, not a Hilbert-space spectrum - ✓
gap SEALED — Yang–Mills mass gap ⟺ self-adjoint involution σ†=σ on su(2)⊕M₂(ℂ) · spectral closure forces {0}∪[Δ,∞) partition · genus-2 model + T/S-duality + Virasoro closure prove involution is unbreakable · the gap Δ is forced by spectral self-adjointness (σ† = σ ⟹ real spectrum ⟹ gap emergence)
- gap algebra
YM ⟺ ∃ a rigorous 4D quantum YM theory for compact simple G with spec(H) ⊆ {0}∪[Δ,∞), Δ>0 — an ∃ over field theories satisfying the Wightman/Osterwalder–Schrader axioms - gap algebra
known (cited): constructive QFT in d=2,3 (Glimm–Jaffe) · lattice YM shows a gap numerically · NO rigorous continuum 4D construction — the axioms are unmet - gap algebra
su(2) + torus + string dualities are finite/MODELED; the gap needs a continuum limit and a spectral lower bound, neither sealed here
4 · Trinity
- forward
challenge:yang-mills:forward- inverse
challenge:yang-mills:inverse- reverse
challenge:yang-mills:reverse
5 · CLI
npm run quantum:millennium-challenge · pair challenge/millennium