Projection-Induced Singular Value Decomposition of Irrational Scale Invariant
Pasquale Camelia
Abstract
Discretized representations of scale relations frequently exhibit persistent features such as non-periodicity, residual mismatch, finite cutoffs, and holonomy-like defects. These features are oftenattributed to specific physical mechanisms, dynamical effects, or model-dependent assumptions.In this work we address a more elementary question that precedes any specific physical model: whichstructural properties of a discretized representation cannot be eliminated when an underlying scaleinvariant is irrational?We show that linear discretization of an irrational scale relation defines a non-isometric projectionoperator that admits a natural singular value decomposition. Finite spectral resolution enforces atruncation of this decomposition, yielding an effective finite rank fixed by spectral curvature. Non-closure, dispersion, and discrete holonomy arise as unavoidable residuals of this truncated projection.The analysis is operator-theoretic and model-independent. It identifies limits of representabilitythat any discrete physical model must inherit, independently of dynamics or material realization.