Gravity Bhecho Amplitudes

Echo amplitudes

When two black holes merge, some theoretical models predict that the new black hole's horizon is not a perfect trap. A small fraction of the gravitational wave energy can reflect off a structure near the horizon, producing a series of faint "echoes" after the main merger signal. The first echo is the strongest, the second is weaker, and each successive one should be quieter than the last. The open question is by how much.

Recognition Science gives a precise answer. In this framework, the golden ratio phi (about 1.618) appears as the unique self-similar scaling forced by the mathematics of recognition cost. The framework models each reflection as one step down a phi-ladder: the amplitude of echo number n is A_0 times phi to the power minus n. Written out, the first echo has amplitude A_0 times phi^(-1), about 0.618 times the primary signal; the second is phi^(-2), about 0.382; and so on. Each echo is attenuated by exactly the factor 1/phi relative to the one before it.

This is not a vague trend but a fixed ratio. The framework's machine-checked library of formal theorems proves that the amplitude of echo n+1 equals the amplitude of echo n multiplied by 1/phi, and that the signal-to-noise ratio between successive echoes is exactly 1/phi. The library also proves the amplitudes are positive, start at 1 for the primary signal, and strictly decrease. These results are assembled into a certificate, a bundle of formally verified properties that any observed echo train must satisfy if the framework's account holds.

The prediction is falsifiable. The framework states: post-processing of any high-SNR merger event that shows either no echo or an echo-amplitude ratio systematically different from 1/phi between successive echoes would refute the claim. This compounds with the existing catalog cert for per-event echo delays, so the amplitude prediction is not isolated but part of a broader set of testable consequences.

What this means for a reader: if LIGO or Virgo detects echoes in a merger event and the ratios between successive amplitudes cluster at 0.618, that is a striking confirmation of the framework's scaling law. If the ratios scatter or no echoes appear, the prediction fails on its own terms. Either way, the module turns a philosophical claim about recognition cost into a concrete number an experiment can check.

/-- 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]

/-- SNR ratio between successive echoes = 1/φ. -/
theorem echo_snr_ratio (n : ℕ) :
    echoAmplitude (n + 1) / echoAmplitude n = phi⁻¹ := by
  rw [echoAmplitude_succ_ratio]
  field_simp [(echoAmplitude_pos n).ne']

theorem echoAmplitude_pos (n : ℕ) : 0 < echoAmplitude n :=
  zpow_pos Constants.phi_pos _

What this page does not claim

No claim that gravitational-wave echoes have been observed in real LIGO or Virgo data. No claim that the phi-ladder amplitude law is derived from general relativity or any conventional gravitational theory. No claim that the framework predicts the absolute amplitude A_0 of the primary signal.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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