Rigorous Rearrangement Tensor Renormalization in Four Dimensions

We present an exact, truncation-free real-space tensor renormalization construction in four dimensions for normalized eight-leg Hilbert–Schmidt tensors in a neighborhood of the trivial fixed-point tensor. The construction combines a Type-0 gauge step, a 2^4 preliminary blocking transformation, source and source-free primitive decompositions, occurrence-specific tagged channels, and an exact connected-cluster cancellation mechanism adapted to four-dimensional incidence geometry. The resulting scale-four renormalization map satisfies the superlinear estimate ‖a′‖₂ ≤ t ‖a‖₂^(32/31) for sufficiently small Hilbert–Schmidt perturbations. The argument includes explicit control of source-free realizations, support-completed tagged cloning, cluster power counting, open-network Hilbert–Schmidt contraction bounds, normalization, and iteration. Version 4.0 strengthens the functional-analytic and algebraic details of the construction. In particular, it makes the oriented Type-0 bond action explicit, gives an explicit source-free realization, formulates tagged cloning through occurrence-specific Hilbert-space isometries, sharpens the support-completion hypotheses, and uses a read-2/Finner argument for the open grey-block Hilbert–Schmidt estimate. The accompanying supplementary material provides additional derivations and structural details. A reproducibility package contains finite combinatorial, geometry, power-counting, and regression checks used to audit the construction. The results concern the stated high-temperature tensor-renormalization problem. They do not constitute a proof of the four-dimensional continuum Yang–Mills existence and mass-gap problem.