Return Theory (RT) - Geometric Consistency of Scale-Locking: Background Curvature Normalization and the Inevitability of the Matter-Era Envelope

This document establishes that the redshift envelope q(z) ∝ (1 + z)^{-3/2} within Return Theory (RT) is not phenomenological, not data-fitted, and not an arbitrary modeling choice. The exponent emerges as a structural consequence of geometric consistency once non-orthogonal shell-time foliations are embedded into a homogeneous FLRW background possessing a unique invariant curvature scale. Return Theory operates at the level of observational reconstruction. It introduces no new dynamical fields and leaves Einstein’s field equations, curvature tensors, and null geodesics unchanged. Observable deviations arise solely through the projection structure relating intrinsic shell-time T to reconstructed observer-time t. The document formalizes the geometric consistency condition requiring invariance of foliation tilt normalization under homogeneous background rescaling. It proves that the tilt scalar q must be normalized by the unique invariant curvature scale H^{-1}. Any alternative normalization leads to epoch-dependent reinterpretation of geometric tilt and induces implicit additional structure incompatible with FLRW covariance. Under matter domination, where H(z) ∝ (1 + z)^{3/2}, scale-locking inevitably yields: q(z) ∝ (1 + z)^{-3/2} The matter-era exponent therefore arises from geometric inevitability rather than phenomenological fitting. This work integrates consistently with: Return Theory (RT): A Geometric, Projection-Based Cosmological Framework – Theory, Numerical & Global Fits doi:10.5281/zenodo.17857410 A Minimal Time-Reconstruction Correction and Its Impact on Cosmological Data Consistency doi:10.5281/zenodo.18182643 Return Theory remains a geometric theory of observational reconstruction operating prior to dynamical evolution and fully compatible with General Relativity.