Rigorous Rearrangement Tensor Renormalization in Four Dimensions

We construct a truncation-free real-space tensor renormalization-group transformation in four dimensions for countable-leg tensors of finite Hilbert-Schmidt norm in a neighborhood of the trivial high-temperature tensor. The construction follows the rigorous two-dimensional program of Kennedy and Rychkov and the three-dimensional rearrangement construction of Ebel. A direct transfer of the three-dimensional proof is obstructed in four dimensions because the cubic exclusion step uses the fact that a cube vertex has only three internal neighbours; in a four-dimensional hypercube the critical local inequality 2 + 2 > 3 becomes 2 + 2 = 4. We replace this geometry-specific exclusion mechanism by a finite tagged cluster-cancellation construction. For each connected false-compatible primitive family, the complete finite tensor-valued realization sum is cloned componentwise into fresh orthogonal index channels and cancelled exactly. The construction avoids an implicit copy operator, preserves Hilbert-Schmidt control, and gives a cluster exponent of at least 17/16. Generalizing the source/sink estimates from the three-dimensional cubic block to the sixteen-vertex four-dimensional hypercube yields, with c = 1/31: a = 16/31b = 32/31 The resulting normalized renormalized perturbation satisfies ||a’||_2 <= t epsilon^(32/31) for sufficiently small epsilon. Iterated normalized maps therefore converge super-exponentially to the reference high-temperature tensor. The accompanying Supplementary Material provides additional proof details and adversarial consistency checks. The reproducibility package contains the finite cluster-bookkeeping, power-counting, black/grey tiling, and abstract tagged-algebra audit code and results. This is an infrared/high-temperature tensor-renormalization result. It does not constitute a continuum Yang-Mills construction or a proof of the Yang-Mills mass gap. Preprint version 1.0.