Permutation Symmetry Selection Rules and Ultraviolet Transfer Suppression in Graph Laplacian Propagators

Abstract

Permutation Symmetry Selection Rules and Ultraviolet Transfer Suppression in Graph Laplacian Propagators We study transfer between distinct sources in positive, second-order scalar theories whose mass mixing is a graph Laplacian. A ten-vertex shared-anchor graph carries an S₃ × S₄ permutation symmetry. Its two- and three-dimensional protected spaces are inequivalent irreducible representations, so Schur’s lemma forbids anchor-to-protected and inter-protected two-point transfer. These zeros persist for general symmetry-invariant kinetic and mass matrices and, in an invariant vacuum with a symmetry-preserving regulator, to all perturbative orders. Specified reintegration-edge mass perturbations open the channels at p⁻⁶ and p⁻¹⁰ for canonical sitewise kinetics. The powers follow from standard mass-insertion moments; unlike the exact symmetric zeros, they are not protected by permutation symmetry alone. An explicit invariant kinetic-mixing counterexample separates the two statements. We introduce scalar probe interactions on the benchmark graph and calculate a finite one-loop mixed quartic threshold whose internal lines must traverse the protected channel. Spectral positivity and a two-field Schur complement distinguish signed cross-source residues from a higher-derivative single-field propagator. The results provide a worked symmetry-controlled transfer mechanism and its limits, without altering diagonal ultraviolet behavior or completing gravity. Related work: The manuscript cites the ECOF foundation paper as the origin of the research program (doi:10.5281/zenodo.16944712). The graph construction and its transfer results are developed in this paper. Reproduction code is supplied in the linked software release. ECOF foundation and persistent links Research-program foundation: Kate Lynn Blatt (2025), The Energy Consciousness Observer Framework: Foundations of Observer-Coupled Physics — Toward a Unified Field for General Relativity’s Mass–Energy Corrections.ECOF foundation, all versions/latest: 10.5281/zenodo.16944711.Foundation version cited in the manuscript: 10.5281/zenodo.16944712.This paper, all versions/latest: 10.5281/zenodo.23046113. Research context: The study is motivated in part by the model-building questions discussed by F. Quevedo at ICHEP 2026, cited in the manuscript. It connects graph theory and quantum field theory through a worked scalar-transfer model and compares that construction with dimensional deconstruction, clockwork and string-theory-related gauge quivers. The result concerns specified transfer channels; quantum-gravity completion remains outside its demonstrated scope. Reproduction software. The matching Python package is available as a separate Software record: Graph verification software — 10.5281/zenodo.23111499. The package reproduces the calculations within this paper’s stated scope. Related studies in the ECOF research program. CP-even information flow · CP-odd information flow · Graph transfer and ultraviolet suppression · Higgs weight concentration (Higgs code). Version 0.11. Supersedes v0.10. Adds nine explanatory subsections covering resolvent remainders, spectral residues, symmetry blocks, perturbations, kinetic normalization and matching. Clarifies the protected subspace versus the full degenerate eigenspace; results are unchanged. Equations and references are consistently numbered. Method attribution. The expanded derivations use established matrix-function, Krylov and functional matching methods: Higham (2006), Antoulas (2005), and Henning, Lu and Murayama (2018).