Special affine graph transforms and commutator defects of position-decorated graph laplacians

Abstract

A finite-dimensional theory of special-affine transforms is developed for position-decorated spectral graphs. A Hermitian graph Laplacian generates a normalized frequency observable, while a normalized diagonal decoration supplies the position observable. Their ordered exponentials define graph Weyl operators, which are coupled to Iwasawa and three-shear graph linear-canonical factors. The resulting transforms are unitary and covariant under decorated graph isomorphisms, including monomial gauge-permutation equivalence for Hermitian directed Laplacians. The commutator generator is proved to be Hilbert–Schmidt orthogonal to the Laplacian commutant, yielding an exact defect formula and a weighted-Laplacian eigenproblem for selecting the position decoration. Quantitative Baker–Campbell–Hausdorff estimates control local Weyl, Egorov, homogeneous, and full special-affine composition errors. For odd-prime cycles, quantized exponentials of centered position and momentum matrices recover the clock-shift system and its exact finite Weil covariance. Frame criteria, canonical Parseval preconditioning, and a reproducible sensor-graph experiment quantify reconstruction accuracy, noise amplification, conditioning trade-offs, and factorization dependence. The theory therefore connects finite operator obstructions with computable graph-transform design.