echoDelay
plain-language theorem explainer
The definition supplies the echo delay at recognition rung N as twice the bounce radius multiplied by the natural logarithm of phi. Black hole echo modelers and LIGO data analysts would cite it when predicting time delays between successive echoes from a Planck-scale bounce surface. The definition is a direct algebraic composition of the bounce radius function with the geometric round-trip factor 2 log phi.
Claim. $Δt(N) = 2 r_{min}(N) log φ$, where $r_{min}(N)$ is the bounce radius at rung $N$ in RS-native units.
background
In the Recognition Science framework the bounce radius at rung N is given by φ^N (RS-native units). The echo delay formula multiplies this radius by the round-trip geometric factor 2 log φ drawn from the self-similar fixed point φ. The module QuantumGravityFromRS supplies structural backing for the Planck-scale bounce: the radius is strictly positive, the delay is monotone in N, and the amplitude damps by exactly 1/φ per echo. Upstream results establish the same delay expression in BHEchoesLIGOCatalog and BlackHoleEchoesFromBounce, confirming the 2 log φ multiplier as the geometric round-trip from the bounce surface.
proof idea
One-line definition that composes the bounce radius at N with the constant factor 2 log phi.
why it matters
This definition feeds the BHEchoesCert structure and the BlackHoleEchoesCert in downstream modules, supplying the explicit delay formula used for certifying echoes in LIGO catalogs and per-event predictions. It fills the structural slot in the A4 strong-field module that links the phi-ladder to observable gravitational-wave delays, consistent with the self-similar fixed point φ and the eight-tick octave. The declaration closes one link in the chain from the Recognition Composition Law to concrete echo observables without introducing new hypotheses.
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