R-layer Mode Theory (RMT) Appendix U: Effective Layer Definition and Tension-Field Geometry in Layered Expansion Cosmology

This appendix provides a mathematically precise and observationally applicable definition of “layers” in Layered Expansion Cosmology (RLMT). As stated in the manuscript, “we model them as effective layers defined by smooth window functions of a tension field T on a spatial hypersurface.” We introduce level-set based layer definitions, smooth window functions, and observational kernels that describe realistic probes such as TRGB/SN Ia, PTA, CMB, and strong lensing. These tools allow cosmological observables to be expressed as layer-resolved effective averages, making explicit how different probes weight different regions of the tension field. The formalism is designed to be fully compatible with the observational traceability developed in RLMT Volumes XLI and XLI‑A‑R, and with the falsifiability thresholds used in the LAYER‑X SMEX Phase A concept study. It provides a robust mathematical basis for interpreting RLMT predictions in terms of real data, clarifying how ambiguities such as “center vs boundary” measurements are controlled by a small number of parameters. This framework establishes the foundation for future developments, including general geometries, SU(2) topological structures, and dynamical tension-field evolution to be explored in subsequent RLMT volumes such as Volume XCIII.