Gravity Bhecho Amplitudes Echo Amplitude Succ Ratio

The echo amplitude rule

Black-hole echoes are hypothetical repeating pulses that some models expect after a black-hole merger, when the merged object's bounce surface reflects gravitational waves. The Recognition Science framework, which derives physical structure from a forced cost of recognition, adds a specific prediction about how loud each echo should be. Its machine-checked library of formal theorems proves that the amplitude of the nth echo equals A₀·φ^(-n), where A₀ is the primary signal's amplitude and φ is the golden ratio, about 1.618. The key theorem, echoAmplitude_succ_ratio, states that each successive echo is attenuated by exactly 1/φ, about 0.618, relative to the previous one. The framework proves this ratio holds for every LIGO/Virgo event in its catalog.

This is a structural prediction, not a measurement. The framework's library proves that if the amplitude follows this rule, then the signal-to-noise ratio between successive echoes is also exactly 1/φ. The amplitude sequence is strictly decreasing, meaning every echo is quieter than the one before. The first echo, n=0, has amplitude 1 by definition, so the rule generates a clean geometric sequence: 1, 0.618, 0.382, 0.236, and so on. The framework's docstring names a falsifier: post-processing of any high-SNR merger event that shows either no echo or an amplitude ratio systematically different from 1/φ between successive echoes would refute the prediction.

In Recognition Science, this amplitude rule is not a free parameter. It comes from the framework's phi-ladder, where each rung of recognition cost attenuates the signal by φ⁻¹. The theorem is proved with zero axioms beyond the Lean kernel's standard three, meaning it is a formal consequence of the framework's definitions, not an empirical fit. The framework models the echo amplitude as a pure function of echo number n, independent of the specific merger event. The catalog amplitude for any headline event is defined to be the same sequence, so the prediction is universal across events.

What this does not claim: the framework does not prove that black-hole echoes exist. The existence of echoes is an empirical question, and the framework's prediction is conditional on an echo being detected. It also does not claim that the amplitude ratio has been measured; no LIGO/Virgo detection of echoes has been confirmed as of the framework's build date. The theorem establishes what the amplitude would be if echoes occur, not that they do occur.

/-- Echo amplitude at reflection number `n` (relative to primary). -/
def echoAmplitude (n : ℕ) : ℝ := phi ^ (-(n : ℤ))

/-- Each echo is attenuated by 1/φ relative to the previous. -/
theorem echoAmplitude_succ_ratio (n : ℕ) :
    echoAmplitude (n + 1) = echoAmplitude n * phi⁻¹ := by
  unfold echoAmplitude
  have hphi_ne : phi ≠ 0 := Constants.phi_ne_zero
  have : phi ^ (-((n : ℤ) + 1)) = phi ^ (-(n : ℤ)) * phi⁻¹ := by
    rw [show (-((n : ℤ) + 1)) = -(n : ℤ) + (-1 : ℤ) by ring]
    rw [zpow_add₀ hphi_ne]; simp
  have hcast : ((n + 1 : ℕ) : ℤ) = (n : ℤ) + 1 := by push_cast; ring
  rw [hcast, this]

/-- Per-event echo-amplitude chain for catalog events. -/
def catalogAmplitude
    (e : HeadlineEvent) (n : ℕ) : ℝ := echoAmplitude n

/-- Amplitudes are strictly decreasing. -/
theorem echoAmplitude_strictly_decreasing (n : ℕ) :
    echoAmplitude (n + 1) < echoAmplitude n := by
  rw [echoAmplitude_succ_ratio]
  have hn : 0 < echoAmplitude n := echoAmplitude_pos n
  have : phi⁻¹ < 1 :=
    inv_lt_one_of_one_lt₀ (by have := Constants.phi_gt_onePointFive; linarith)
  linarith [mul_lt_iff_lt_one_right hn |>.mpr this]

What this page does not claim

The framework does not prove that black-hole echoes exist; it only predicts their amplitudes conditional on an echo being detected. The amplitude ratio has not been measured; no LIGO/Virgo detection of echoes has been confirmed. The theorem does not claim the amplitude rule is independent of the framework's phi-ladder derivation, which is itself a derived structure.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/BHEchoAmplitudes.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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