rungPhaseDelay
plain-language theorem explainer
rungPhaseDelay supplies the per-rung phase delay on the recognition lattice as the natural logarithm of phi. Gravitational-wave modelers working on black-hole ringdown echoes in Recognition Science cite it when computing the two-way travel time across the bounce region. The definition is introduced as a direct one-line assignment of Real.log phi.
Claim. The per-rung phase delay on the recognition lattice is defined as $log phi$, where $phi$ is the golden ratio.
background
In the Black-Hole Echoes from RS Bounce module the classical Schwarzschild singularity is replaced by a bounce at finite radius $r_min = ell_P cdot phi^{N/2}$ in RS-native units. The echo delay is then $Delta t = 2 r_min log phi$ with $c=1$. rungPhaseDelay supplies the factor $log phi$ that appears in this delay formula and in the cumulative echo amplitude series with ratio $1/phi$ (BlackHoleEchoesCert).
proof idea
The definition is a one-line assignment that sets rungPhaseDelay to the real logarithm of phi.
why it matters
This definition supplies the phase factor that enters the echo delay formula $Delta t = 2 r_min cdot rungPhaseDelay$, which is used in echoDelay and BlackHoleEchoesCert to certify bounce properties and geometric damping. It fills the per-rung phase delay role in the RS bounce model, enabling the prediction of phi-delayed echo trains in LIGO/Virgo ringdowns. The empirical falsifier is the presence or absence of the echo signature in actual gravitational-wave data.
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