Baryonic Tully-Fisher Relation (20 Galaxies)

τ-Formula

τ-BTFR slope = 3.991 (zero free parameters)

Derivation

Ninety-five percent of the universe is missing. That is the orthodox verdict: $27%$ is dark matter, $68%$ is dark energy, and the $5%$ that we observe is all that ordinary physics explains. After decades of direct detection experiments (XENON1T, LZ, PandaX), no dark matter particle has been found. After two decades of theoretical effort, the cosmological constant problem—a mismatch of $120$ orders of magnitude between the quantum vacuum prediction and the observed value—remains the worst quantitative failure in the history of science.

This chapter demonstrates that the dark sector dissolves within Category $τ$. The dissolution is not speculative: it rests on five quantitative results, each derived from the master constant $ι_τ = 2/(π + e)$ with zero free parameters.

• Flat rotation curves: the master formula $v^4 = G M_b c^2/(2_τ)$ reproduces NGC 3198 at $0.6%$ and passes a 20-galaxy survey at $0.067$ dex RMS (V.T85, V.D258).

• Dark energy density: $Ω_Λ = κ_D(1 + ι_τ^3) = 0.6849$, matching Planck at $+269$ ppm (V.T234).

Source

This prediction is derived in the Numerical Physics Ledger (Chapter 63 — dark-sector), Books IV–V of Panta Rhei.