Hubble Constant h

τ-Formula

h = 2/3 + ι_τ²/17 = 0.6735

The denominator $17 = W_3(3) = a_3 + a_4 + a_5 = 13 + 3 + 1$ is the sum of three consecutive partial quotients of the continued-fraction expansion $CF(ι_τ) = [0; 2, 1, 13, 3, 1, 1, 1, 42, …]$ starting at index 3 (Lean-certified at Tau.CF.w3_at_3); the base value $2/3$ is the matter-dominated Einstein-de Sitter late-time limit. Both numbers are structural — fixed once $ι_τ = 2/(π+e)$ is posited — not fitted. See the result page for the full derivation.

Derivation

tter 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.