This paper details the mathematical and methodological foundations of the Tav (Tau Advanced Version) framework, an extension of the Tau Universe model. We focus on the 5D topological structure R3,1 × S1, multi-scale fractional compactification, Casimir-derived dark energy, partial computability of Kaluza-Klein (KK) towers, and statistical validation using recent datasets like JWST JADES DR3 (data released May 2024) and DESI 2024. The model resolves cosmological tensions (e.g., H0 and S8) with zero free parameters by treating the universe as a computable prefix of an undecidable 5D manifold. We introduce novel Tau Methods, including Yuan 2n3m filtering for harmonic isolation and Prefix Windowing for computability cutoffs. Predictions for Euclid DR1 (expected October 2026) are provided, emphasizing detectable harmonic spikes at kn = n/τ ≈ n × 0.143 h Mpc−1.

Supplementary Material: Robustness testing of Yuan 2ⁿ3ᵐ filter (Significance Matrix + synthetic power spectrum plots at 8% noise). Demonstrates 3.5–3.8× clarity gain and >5σ recovery in JADES-like noise.