Plaquette-Deviation Functionals, Non-Abelian Excess Channels, and Finite-Size Scaling Diagnostics in SU(2) Lattice Yang–Mills Theory

We introduce a gauge-invariant plaquette-deviation functional V for SU(2) lattice gauge fields onfinite four-dimensional hypercubic lattices and prove the exact algebraic identity SW = β4 V with theWilson action. This identity allows one to formulate action-based and plaquette-based diagnosticsin a common language without invoking any continuum approximation. We then study a controlledtopologically trivial two-lump family consisting of instanton-like and anti-instanton-like lattice fieldswith variable relative color orientation. To separate genuinely non-Abelian effects from those alreadypresent in a commuting sector, we introduce an Abelianized baseline obtained by projection ontoa fixed Cartan direction and define three excess observables: the non-Abelian excess ∆VNA =VnonAb − VAb, the interaction excess ∆Vint = Vtot − V1 − V2, and the differential interaction excess∆∆Vint = ∆V nonAbint − ∆V Abint . Across the tested family, ∆VNA remains positive and the interactionchannel depends strongly on relative color orientation, showing that non-Abelian structure modifiescancellation patterns in a way absent from the Abelianized comparison system. We further examinefinite-size scaling along fixed-lattice-unit and fixed-fraction trajectories. Raw excess observables andplaquette-density normalizations do not exhibit an observable-independent extrapolation pattern,whereas selected dimensionless ratios display substantially more stable behavior. These results donot prove the Yang–Mills mass gap and do not establish the non-existence of a continuum theory.They do, however, isolate a controlled finite-lattice non-Abelian mechanism and show that supportfor naive continuum extrapolation is highly observable-dependent rather than self-evident.