Effective 4D and 5D Lagrangians for the Tau Universe Framework and Tav Topology Theory: Unifying Dark Energy and Dark Matter from a Single Geometric Scale

We tested the core quantitative predictions of the Tau Universe Framework — a compact fifth dimension of radius τ = 7 h−1 Mpc — against public data from JWST JADES (DR3 and DR4, 7,775 galaxies) and DESI DR1 (BGS, ~4 million galaxies). Three independent tests were performed: (1) Berard residual clustering at the 1.054 scale in τ-modulated comoving distances, (2) Rayleigh phase uniformity of distance residuals, and (3) a search for a uniform Kaluza-Klein comb at Δk = 0.142857 h Mpc−1 in the galaxy power spectrum monopole. All tests returned null results. No significant excess, phase clustering, or periodic feature was detected in any dataset. The DR4 Berard and Rayleigh tests were pre-registered. These results constrain the amplitude of any topological imprintfrom a compact S1 in the probed redshift and wavenumber regimes. License: CC-BY-4.0 Contact: Ernest C. Gatlin III (@CLAYGATLIN on X) ORCID: 0009-0002-8197-4810