Gravity Bhecho Amplitudes Echo Amplitude Strictly Decreasing

Echo amplitudes and their decay

In the study of gravitational waves, a black hole echo is a hypothesized secondary signal that follows the main merger event, thought to arise from a reflection off a structure near the event horizon. The Recognition Science framework formalizes a specific amplitude model for these echoes. The declaration echoAmplitude_strictly_decreasing establishes that, within this model, the amplitude of each successive echo is strictly smaller than the one before it. This is a theorem in the framework's machine-checked library of formal theorems, carrying no unproven assumptions.

The model defines the amplitude of the n-th echo as A_n = A_0 · φ^(-n), where φ is the golden ratio, approximately 1.618. This means the first echo has amplitude A_0, the second A_0/φ, the third A_0/φ², and so on. The theorem proves that A_(n+1) is always less than A_n, confirming that the echoes form a strictly decreasing sequence. The ratio between successive amplitudes is exactly 1/φ, about 0.618. This ratio is the framework's structural prediction for the signal-to-noise ratio between consecutive echoes in any LIGO/Virgo event.

The framework's model also proves that the amplitude of the primary signal, echo number zero, is exactly 1, and that all amplitudes remain positive. These properties are bundled into a certificate structure that the framework presents as the complete amplitude prediction for black hole echoes. The framework treats this as a prediction to be tested against observations, not as a derived law of physics.

In Recognition Science, this amplitude model is one piece of a larger account. The framework derives the golden ratio from its foundational cost function, and here applies it to echo amplitudes. The strictly decreasing property is a direct consequence of the chosen amplitude formula, not an independent physical discovery. The framework's library proves the mathematics of this decay, but whether real black holes produce echoes at all, and whether their amplitudes follow this ratio, remains an empirical question.

/-- 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]

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

/-- 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 existence of black hole echoes in real gravitational wave data is not established by this theorem. The framework does not claim that the golden-ratio amplitude decay is the only possible model for echo attenuation. This theorem does not prove that any observed LIGO/Virgo event contains 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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