echoAmplitude
plain-language theorem explainer
Recognition Science models black hole echo amplitudes as a geometric sequence damped by the golden ratio. The definition sets the amplitude of the nth echo to phi raised to minus n. Gravitational wave researchers would use this to predict signal-to-noise ratios between successive echoes in LIGO data. The implementation is a direct real-number exponentiation.
Claim. The amplitude of the nth echo is given by $A_n = phi^{-n}$, where $phi$ denotes the golden ratio.
background
The phi-ladder encodes recognition costs through successive powers of the golden ratio phi, the unique positive solution to x = 1 + 1/x. In this module, echo amplitudes are attenuated by one rung per reflection, yielding the factor phi^{-1} per step. This builds directly on the echo amplitude definition from GravitationalWaveEchoFromRS, which expresses the same quantity as the reciprocal of phi to the power k. The module context is black-hole echo amplitudes with phi-ladder damping for gravitational wave events, compounding with the per-event catalog.
proof idea
This is a direct definition that assigns to each natural number n the real value obtained by raising the golden ratio constant to the power of negative n.
why it matters
The definition supplies the core formula for the BHEchoAmplitudeCert structure, which asserts positivity, primary unity, and the exact ratio phi^{-1} for successive amplitudes. It completes the amplitude prediction in the gravity module, aligning with the phi fixed point and the structural SNR ratio of 1/phi. This supports the falsifier test for high-SNR merger events showing echo ratios deviating from 1/phi.
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