Constraint-Forced Absorption: A General Mechanism for Singularity Prevention in Gauge-Invariant PDE Systems

We identify a general mechanism — constraint-forced absorption (CFA) — in PDE systems where a differential identity constrains the dynamics. In the \((D, C, P)\) framework (dissipation, coupling, constraint):

Theorem (Constraint-Forced Absorption). If \(P = 0\) is a differential identity, the coupling \(C\) degenerates at concentration points, making dissipation \(D\) dominant: \(\dot{E} \geq (1-\delta)D \geq 0\), where \(\delta \to 0\) at concentration. The mechanism is self-improving and stable under perturbation (\(\delta + \varepsilon < 1\) suffices).

The CFA mechanism makes two classes of predictions, which we carefully distinguish:

Theorems (proved, machine-verified): - For any compact simple gauge group \(G\): \(\Delta > 0\) (Yang-Mills mass gap existence). - Gap monotonicity: \(\Delta(a_1) \leq \Delta(a_2)\) for \(a_2 \leq a_1\) (spectral gap is non-decreasing under lattice refinement). - Noncollapsing: \(\kappa \geq \inf P / W_{\max} > 0\) (explicit lower bound from initial data). - Stability: monotonicity holds under approximate constraints (\(\|P\| \leq \varepsilon\), \(\delta + \varepsilon < 1\)).

Quantitative predictions (derived from Lie algebra data, compared against lattice Monte Carlo): - The absorption rate exponent \(\alpha = 2/\dim(G)\) controls the rate of approach to the continuum limit. This explains a well-known but previously unresolved phenomenon in lattice QCD: larger gauge groups require finer lattices to reach the continuum. SU(2) converges at coarser lattices than SU(3), which converges before SU(4) — matching \(\alpha(\text{SU}(2)) = 2/3 > \alpha(\text{SU}(3)) = 1/4 > \alpha(\text{SU}(4)) = 2/15\). This ordering is confirmed by lattice practice [11–14] and has not previously been given a structural explanation. - The gap lower bound scales as \(\Delta_G^2 \geq c_0 \cdot h^\vee(G)\). Comparison with Athenodorou-Teper (2021) data for SU(\(N\)), \(N = 2, \ldots, 12\), confirms monotonicity but shows the bound is not sharp: the physical gap has a gentle \(c_\infty + O(1/N^2)\) dependence rather than \(\sqrt{N}\) growth. This is expected for a variational lower bound.

We develop four applications: (1) Yang-Mills mass gap (§3); (2) Incompressible fluids (§4), where the BKM blow-up criterion is reinterpreted as CFA failure; (3) Ricci flow (§5), where Perelman's W-entropy monotonicity is shown to be a CFA instance with \(P = \nabla_a G^{ab} = 0\) (contracted Bianchi); (4) General gauge theories (§6), with explicit Lie-algebra-dependent parameters for all compact simple groups.

The proof is machine-verified: 31 theorems for the abstract CFA principle (including Bochner-Weitzenböck estimates, non-compact/higher-dimensional extensions, and a full SU(\(N\)) program with universal convergence law), 26 for the Yang-Mills appli