v4.0 Mathematical Substrate — Core Equations and Derivations of the Tau Universe Framework & Tav Topology Theory

This document extracts and archives the core mathematical equations and derivations of the Tau Universe Framework and Tav Topology Theory as of April 2026. It consolidates results from three primary sources: (i) the Eective 4D and 5D Lagrangians paper (Zenodo DOI: 10.5281/zenodo.19011211), (ii) the Revised Lagrangian Summary (March 2026), and (iii) Volume 4 of the Tau Universe CodexMathematical Extensions: From Logistic Resonance to Tav Topology (Amazon KDP, April 3, 2026). All results are governed by a single geometric parameter: the compact fth-dimensional radius τ ≈ 7 h−1 Mpc. Falsiable predictions are mapped onto Euclid, Rubin LSST, DESI, CMB-S4, and PTA observational windows for the 20262030 era. This substrate chains to the Quantitative Prediction Sheet (DOI: 10.5281/zenodo.19323070), the Scientic Pre-Registration (DOI: 10.5281/zenodo.18155027), and the Null Results paper (DOI: 10.5281/zenodo.19322831). This v4.0 substrate consolidates the mathematical heart of Volume 4 (Amazon KDP, April 3, 2026) with all prior Lagrangian results. It is the executable resonance graph for all 2026–2030 observational tests and remains fully falsifiable.