Gravitational-Wave Cycle-Delay Time

τ-Formula

t_outer = 4GM·ι_τ⁻¹/c³; t_inner = 4GM·ι_τ/c³

Derivation

For a $T^2$-topology black hole of mass $M$, Book V defines two cycle-delay readouts: $t_{\mathrm{outer}} = 4GMι_τ^{-1}/c^3$ and $t_{\mathrm{inner}} = 4GMι_τ/c^3$. The separation is $Δt = t_{\mathrm{outer}} - t_{\mathrm{inner}} = 4GM(ι_τ^{-1} - ι_τ)/c^3$.

These are topology-readout candidates, not exotic-compact-object reflective-surface echoes. They remain released predictions because their absolute scale and ratio can be searched for in high-SNR ringdown data, but no current public result validates the τ black-hole sector.

Source

This prediction is derived in Book V, Chapters 41, 50, and 51, with Registry anchor V.T169 and ratio refinement V.D243. See ERRATUM-005 for the 2026-05-15 correction of echo terminology and generated-page contamination.