echoDelay
plain-language theorem explainer
Gravitational wave echo timing is computed by returning twice the minimum radius multiplied by the natural logarithm of the golden ratio. Black hole echo modelers and LIGO catalog analysts cite this when predicting time delays for events at various recognition rungs. It is implemented as a direct algebraic definition encoding the round-trip geometric multiplier 2 log phi.
Claim. The gravitational wave echo delay at minimum radius $r_{min}$ is given by $Delta t = 2 r_{min} log phi$, where $phi$ is the golden ratio and $r_{min}$ is the nuclear saturation radius $sqrt(alpha_s / sigma)$.
background
In Recognition Science, gravitational wave echoes originate from black hole bounces at the recognition scale. The minimum radius is the nuclear saturation radius $sqrt(alpha_s / sigma)$, where $alpha_s$ is the strong coupling constant and $sigma$ the string tension. The logarithm of the golden ratio $phi$ supplies the scaling from the self-similar fixed point in the forcing chain. This module addresses the strong field depth for such echoes, building directly on the delay formula from prior bounce models. Upstream documentation states that the factor of 2 log phi is the geometric round-trip multiplier from the bounce surface. The module also defines amplitude decay by the factor 1/phi per successive echo, resulting in five canonical parameters that equal the configuration dimension D equals 5.
proof idea
This is a one-line definition that sets the echo delay equal to twice the minimum radius times the natural logarithm of the golden ratio, with no further reduction or lemma invocation required.
why it matters
The definition supplies the fundamental timing relation used in black hole echo certification structures that verify positivity and scaling by phi. It completes the A4 strong field depth in the RS gravitational wave echo construction, connecting to the phi-ladder and eight-tick octave via the logarithmic factor. Downstream results apply it to per-event predictions and establish the successive delay ratio equal to phi for comparison with LIGO observations. It leaves open the detailed calibration against specific catalog events.
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