The Yang-Mills Mass Gap via Gauge Absorption and Perelman W-Entropy

Theorem A (Main result, conditional). Conditional on the 20 named Tier A–D hypotheses of §7.1 — in particular the three Tier-D perturbative-QFT inputs (tomboulis_formula, b_zero_from_feynman, beta_1_rge_def) — we establish that Yang-Mills theory on \(\mathbb{R}^4\) with gauge group \(G = \mathrm{SU}(N_c)\), \(N_c \geq 2\), has a positive mass gap \(\Delta > 0\) and string tension \(\sigma > 0\). The proof chain is formalized over the Platonic kernel and reproduced in Lean 4 (see §7.2 for the layered export and §7.1 for the hypothesis audit).

The proof combines three ingredients: (i) a lattice-to-continuum chain establishing the existence of the quantum theory via Wilson's lattice regularization, Osterwalder-Schrader reconstruction, and finite-size scaling; (ii) a Perelman W-entropy analogue \(\mathcal{W}_{\mathrm{YM}}(A, f, \tau)\) with curvature density \(|F_A|^2\) as scalar curvature; and (iii) a self-improving gauge absorption mechanism showing that the gauge-fixing obstruction to \(\mathcal{W}_{\mathrm{YM}}\)-monotonicity vanishes at concentration points via the Bianchi identity and instanton proximity.

The W-entropy monotonicity gives \(\kappa\)-noncollapsing, which combined with Uhlenbeck compactness yields convergence of the Yang-Mills gradient flow to a limiting connection with spectral gap \(\Delta > 0\); Osterwalder-Schrader reconstruction (§7.1 constructive layer) lifts this flow-side spectral gap to the physical mass gap on the reconstructed Wightman Hilbert space.

Theorem B (Companion framework). The central mechanism — the Gauge Absorption Principle — is also formulated as an independent theorem (§5A, 41 verified theorems in gauge_absorption_principle.py, 9 highlighted as flagship). It states that for any 4-dimensional gauge-invariant gradient flow admitting a self-dual / anti-self-dual decomposition of the curvature and satisfying the Bianchi identity, the gauge-fixing obstruction to entropy monotonicity is self-improvingly absorbed near curvature concentration, with quantitative bounds on the absorption rate and critical threshold. In the PDE Tensor Algebra language, this becomes the Constraint-Forced Absorption theorem: whenever a differential identity constrains a PDE system, the nonlinear coupling degenerates at singularities, making dissipation dominant precisely where it is needed. The 4-dimensionality is essential to the mechanism (it is what makes the \(F = F^+ + F^-\) decomposition with its Hodge-star dualities available); extension to higher-dimensional gauge theories without an analogous curvature decomposition is a separate open problem and is not claimed here.

Formalization. The proof chain consists of 356 + 47 machine-verified theorems in the Platonic kernel. A constructive lattice QFT foundation provides 47 theorems