Gravity Bhecho Amplitudes Echo Snr Ratio

The echo amplitude ratio

When two black holes merge, general relativity predicts a single combined object that rings down. Some speculative models add a twist: the new object is not a black hole with an event horizon but a horizonless compact body with a reflective surface. Waves falling in would bounce off that surface, producing a series of faint gravitational wave echoes after the main signal. The first echo would be strongest, each later one weaker, in a sequence a detector might resolve.

Recognition Science, a framework built on a machine-checked library of formal theorems, contributes a specific prediction for that sequence. Its declaration echo_snr_ratio states that the amplitude of each echo is exactly 1/φ times the amplitude of the previous one, where φ is the golden ratio, about 1.618. In symbols: A_n+1 / A_n = φ⁻¹ ≈ 0.618. The framework derives this from its core cost function, which forces a φ-power ladder for physical scales; the echo amplitude is one rung of that ladder.

The declaration is a formal theorem, meaning the framework's proof checker has verified it from the definition of echo amplitude as φ^(-n). It also proves the amplitudes are positive and strictly decreasing, so the sequence is well-behaved. But the theorem is a statement about the framework's model, not a measurement. The physical claim, that real LIGO/Virgo events will show this ratio, is a prediction with a named falsifier: post-processing of any high-SNR merger that shows no echo, or a ratio systematically different from 1/φ, would refute it.

What the declaration does not claim is that black hole echoes exist. It does not claim the framework has detected them, or that the ratio has been observed. The theorem is a conditional: if the framework's model of recognition cost applies to gravitational wave echoes, then the amplitude ratio follows. Whether that condition holds in nature is an open empirical question, and the framework's own prediction is the test.

/-- 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 _

/-- 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]

/-- 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']

What this page does not claim

The declaration does not claim that black hole echoes have been observed. The theorem does not derive the existence of a bounce surface from the framework's axioms. The prediction does not specify the time delay between successive echoes.

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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