Quasinormal Mode Frequency Ratio

τ-Formula

f₍₀,₁₎/f₍₁,₀₎ = ι_τ⁻¹ = (π+e)/2 ≈ 2.929

Derivation

Book V separates the ordinary Kerr primary ringdown scale from the proposed τ topological ratio. The Kerr primary mode depends on the remnant mass and spin. The τ claim is narrower: if the $T^2$ secondary ringdown readout is present, the leading-order frequency ratio cancels the common mass/spin scale and satisfies $f_{(0,1)}/f_{(1,0)} = ι_τ⁻¹$.

The ratio comes from the primitive outer and inner torus cycles with shape ratio $r/R = ι_τ$. Their eigenvalue ratio is $λ_{0,1}/λ_{1,0} = ι_τ⁻²$; taking the square root gives the frequency ratio $ι_τ⁻¹ ≈ 2.929$.

This is a released structural prediction, but it is not a claim that current ringdown data already detect a τ mode or validate the black-hole sector. It is a calibration and falsification surface for high-SNR ringdown spectroscopy.

Source

This prediction is derived in Book V, Chapters 41 and 50, with Registry anchor V.T168 and related ringdown refinement V.T223. See ERRATUM-005 for the 2026-05-15 correction of mass/spin wording and generated-page contamination.