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.