Gravity Bhecho Amplitudes Bhecho Amplitude Cert

Echo amplitude certificate

Black-hole echoes are hypothetical repeating signals that would follow the main gravitational-wave burst from a merger, if the remnant has a reflective surface instead of a sharp event horizon. The Recognition Science framework's BHEchoAmplitudeCert is a formal structure that bundles six proven properties of these echo amplitudes into a single certificate. The central prediction: each successive echo is attenuated by a factor of 1/φ, where φ is the golden ratio, giving amplitudes A_n = A_0 · φ^(-n) for echo number n.

The certificate's six fields are all machine-checked theorems. They state that every amplitude is positive, the primary echo has amplitude 1, each step multiplies by φ^(-1), the ratio of successive amplitudes is exactly 1/φ ≈ 0.618, the sequence strictly decreases, and the same properties hold for every catalog event. The structure packages these as a single object that any downstream analysis can consume without rechecking each property.

In Recognition Science, this is the amplitude companion to the existing echo-delay certificate. The delay certificate predicts when echoes arrive; this one predicts how strong they are. The framework models the attenuation as one rung of recognition cost, where the golden ratio appears as the unique self-similar scaling forced by the framework's cost function. The structural prediction is that the signal-to-noise ratio of the nth echo relative to the (n-1)th echo is exactly 1/φ for every LIGO/Virgo event.

The certificate does not claim that black-hole echoes exist. It is a conditional prediction: if echoes are observed, their amplitudes should follow this decay. The stated falsifier is any high-SNR merger event showing either no echo or an amplitude ratio systematically different from 1/φ between successive echoes. The certificate is a formal packaging of a prediction, not evidence that the prediction is true.

/-- 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']

/-- 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 certificate does not prove that black-hole echoes exist. The certificate does not establish that any observed LIGO/Virgo event actually shows echoes. The certificate does not derive the golden ratio from black-hole physics; it imports it from the framework's cost function.

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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