The Remaining Inequality for the Yang–Mills Mass Gap: Equivalent Forms, Proved Cases, a Sharp Multiscale Criterion, and SU(3) Lattice Evidence

A Multiscale Criterion for the Yang–Mills Mass Gap This study formulates the Yang–Mills mass-gap problem as a multiscale Poincaré inequality. On a finite lattice, a ground-state transformation identifies the Hamiltonian spectral gap with the optimal Poincaré constant of the Scretching–Planck ground-state measure (P_0=|Ψ_0|^2). Connected Euclidean-time correlations of gauge-invariant observables decay at the spectral-gap rate when the observable overlaps the lowest excitation. The continuum objective is a positive lower bound in physical units, uniform as the lattice spacing tends to zero and the volume tends to infinity. The study also examines the quantum Hamiltonian (H_g=-12Δ+12g^2x^2y^2). Despite its noncompact classical flat valleys, the model has a positive spectral gap for every (g>0). An exact dilation gives the scaling (Δgap(g)=g2/3Δgap(1)). The fractional power illustrates how transverse quantum confinement can produce a gap in a model with classically flat directions. The proposed renormalization-group criterion depends on two estimates. Lemma M requires that each factor-of-two blocking step incur at most an (O(g_k^4)) relative loss in the rescaled Poincaré constant. Lemma IR requires that the effective theory reach a strong-coupling region where a positive gap can be established. Along a weak-coupling trajectory with (g_k^2 k⁻¹), quartic losses are summable, allowing a positive infrared bound to survive the iteration. Losses proportional to (g_k^2) would instead make the resulting product bound vanish. Complementary SU(3) Cabibbo–Marinari heat-bath simulations at (β=5.7,\ 5.8,) and (6.0) estimate the string tension and obtain variational upper bounds in the smeared scalar glueball channel. Results are expressed in Sommer units and compared with the reference values (r_0√σ≈1.16) and (r_0m_{0⁺⁺}≈4.21). The finite-lattice identity, the flat-valley model, and the simulations isolate distinct parts of the argument. Lemma M, Lemma IR, and the Osterwalder–Schrader construction of the continuum theory remain open for four-dimensional Yang–Mills.