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

Quantitative Prediction Sheet: Tau Universe Framework & Tav Topology Theory — Exact Numerical Targets for Observational Tests Ernest C. Gatlin III | Independent Researcher | March 2026 This document provides the exact numerical values predicted by the Tau Universe Framework and Tav Topology Theory so that observers running survey pipelines can test the framework without reading the full theoretical derivation. Every entry is a precise number or inequality that can be checked against data. Companion to the Lagrangian paper: doi:10.5281/zenodo.19011211 The five core numbers: Compactification radius: τ = 7 h⁻¹ Mpc KK harmonic spacing in P(k): Δk = 1/7 h Mpc⁻¹ = 0.142857 h Mpc⁻¹ (exact, uniform) Lightest KK mode mass: m₁ = 6.4 × 10⁻³¹ eV Dark energy equation of state: w₀ = −1.000 (exact, all redshifts) Dark energy time variation: w_a = 0.000 (exact, no evolution) Contents: Fundamental scale parameters with unit conversions (natural units, physical Mpc, GeV⁻¹). KK harmonic wavenumber table: predicted P(k) feature locations for modes n = 1–10, in both h Mpc⁻¹ and Mpc⁻¹, with corresponding physical scales and regime labels (linear through non-linear). KK mass spectrum table: masses from m₁ through n = 10¹⁵, with Lyman-α threshold flagged at n ~ 10¹⁰. Dark energy predictions with explicit pass/fail criteria (fail if |w₀ + 1| > 0.05 at >3σ). Dark matter and cosmological parameter predictions (Ω_DM h² = 0.120, H₀ = 69.8 km/s/Mpc, ρ_DM/ρ_Λ ≈ 5). Null predictions: six things that must NOT be observed (no KK graviton at LHC, no sub-mm gravity deviation, no w ≠ −1, no DM annihilation signal, no large DM–nucleon cross-section, no DE time evolution). Tav junction signatures (longer-term, weaker): CMB alignment, LSS anisotropy, bispectrum, GW background anisotropy, lensing anomalies. Instrument-indexed lookup table: what Euclid, Rubin LSST, DESI, JADES, NANOGrav/IPTA, CMB-S4/LiteBIRD, XENON-nT/LZ, LHC, and SPHEREx should each check. Quick-reference card: the five numbers, the single most distinctive signature (uniformly spaced P(k) features with no ΛCDM analogue), and the single fastest falsifier (w₀ ≠ −1 at >3σ). Single most distinctive signature: Uniformly spaced features in the matter power spectrum P(k) at k_n = n × 0.14286 h Mpc⁻¹ with constant Δk. This pattern has no analogue in ΛCDM, warm dark matter, or single-field fuzzy dark matter. Single fastest falsifier: w₀ ≠ −1 at >3σ from Euclid or DESI kills the Casimir dark energy hypothesis. License: CC-BY-4.0 Contact: Ernest C. Gatlin III (@CLAYGATLIN on X) ORCID: 0009-0002-8197-4810