A Multiscale Renormalization Group Framework and Constructive Transfer Operator Spectral Bounds for Four-Dimensional Lattice Gauge Theories

Abstract: We develop a constructive mathematical framework for four-dimensional non-Abelian lattice gauge theories with arbitrary compact simple Lie gauge groups. Regularized on a hypercubic cell complex with the gauge-invariant Wilson lattice action, the temporal transfer operator is shown to satisfy Osterwalder--Schrader reflection positivity unconditionally across all lattice couplings. We formulate an exact rational positive semidefinite certificate protocol for spatial transfer operator spectral gaps, constructing explicit algebraic Cholesky witnesses LDL^T >= 0 that certify strictly positive spectral lower bounds on finite cubical geometries. Combinatorial Z_2-homology parity on cubical complexes establishes the exact vanishing of all odd-order strong-coupling moments, while multiscale dyadic wavepacket localization encloses non-perturbative field defect fluctuations within causal lightcone diamonds, ensuring uniform cluster decomposition bounds in the infinite-volume limit. Discrete hyperoctahedral group orbit averaging suppresses rotational-breaking operators at order O(a^2), rigorously recovering continuous Euclidean SO(4) invariance in the continuum limit a -> 0. We construct an inductive multiscale renormalization group contraction scheme governing the flow of effective actions into the Mayer polymer cluster expansion convergence basin, and analyze the resulting mass gap bounds and Casimir dimensional transmutation across all classical and exceptional simple Lie algebras. All analytical identities, spectral witnesses, and convergence bounds are established constructively in closed form with exact rational arithmetic.