Geometric Bounds and Conditional Tensor Renormalization in Four Dimensions

Version 6.0 was published with the intention of making the results of my research, including its limitations, unsuccessful approaches, and open questions, publicly available to anyone who might find them useful for further scientific investigation.

This work investigates mathematically controlled tensor-renormalization constructions on four-dimensional lattices with countably infinite-dimensional Hilbert-space tensor legs.

The reorganized manuscript places its principal mathematical results first: an explicit four-dimensional alternating ABAB geometry with an all-scale distance-gain estimate; conditional contraction bounds for multisite interactions; anchored joint-source transport; and a weighted Hilbert–Schmidt-to-Walsh embedding.

The accompanying appendices preserve the original v5.3.1 rearrangement construction, including its proposed superlinear estimate with exponent 32/31 and a technical prefactor correction. They also document lattice-field applications involving Wilson SU(2), scalar φ⁴ theory, restricted Regge geometry, numerical benchmarks, and reproducible verification procedures.

The work distinguishes exact mathematical identities, conditional results, finite computational checks, and unresolved proof obligations. A general autonomous four-dimensional tensor RG and continuum field-theoretic constructions remain open.

Version 6.0 — Editorial reorganization: The original mathematical constructions and results are retained and presented in a theorem-first structure. The main text emphasizes the mathematical contributions, while archival derivations, physical examples, and verification materials are organized into appendices. This revision introduces no additional mathematical claims beyond those established in the preceding version.