Diagrammatic Hilbert Space Approach to Quantum Gauge Theories: First Quantitative Benchmarks for QCD and the Standard Model

We propose a diagrammatic Hilbert space framework in which Feynman-like graphs form an orthogonal basis, and effective field dynamics emerge through projections to low-complexity sectors. This structure naturally encodes gauge invariance, topological invariants, and renormalization-group flow. Here we present the first explicit quantitative benchmarks: (i) combinatorial diagrammatic-RG loop counting, which reproduces the algebraic structure of the one-loop QCD β-function; (ii) exact Wilson-action lattice integrals for SU(2), evaluated on modest grids, which yield the correct sign structure and relative weight of gluon, ghost, and tadpole contributions; and (iii) extrapolated numerical results that stabilize toward continuum expectations. Together these results indicate that the framework is consistent with established properties of QCD and the Standard Model, while providing a new language for entropic and topological unification.