Return Theory — Degrees of Freedom, Gauge-like Structure of Amplitude, Frequency, and Phase in Observational Projection

This document formalizes amplitude, frequency, and phase as the minimal non-physical degrees of freedom in Return Theory (RT). These quantities are interpreted as geometric freedoms of observational representation—corresponding to scale, time reparameterization, and origin choice in observer-time—rather than as dynamical variables, physical fields, or sources of new interactions. A gauge-like framework is introduced to classify families of observationally equivalent descriptions while strictly preserving General Relativity. Both abelian (direct-product) and non-abelian (semi-direct) mathematical interpretations are clearly distinguished, with RT explicitly adopting the abelian, representational structure to avoid unnecessary algebraic complexity. Gauge-invariant quantities are identified as the appropriate objects for data analysis, and the notion of dataset gauge mismatch is discussed as a geometric source of apparent observational discrepancies. This note is intended as a conceptual and methodological supplement to the main Return Theory framework and should be read in conjunction with the primary RT reference: Return Theory (RT) — Zenodo DOI: 10.5281/zenodo.17857410