Gravity Echo Reflection Coefficient Echo Damping Factor Eq Reflection Amplitude

The echo damping identity

In the study of gravitational echoes, a key question is how quickly successive signals fade. The Recognition Science framework answers this with a single number: the echo damping factor, which is the ratio of one echo's amplitude to the next. The framework's machine-checked library of formal theorems proves that this damping factor equals the reflection amplitude at a single barrier, and that both equal the reciprocal of the golden ratio, φ⁻¹, where φ satisfies φ² = φ + 1.

The golden ratio, approximately 1.618, is the unique positive number whose square is one more than itself. Its reciprocal, about 0.618, appears here because the framework models the near-horizon region as a self-similar barrier with rungs. At each rung, energy splits between reflected and transmitted parts according to the golden-ratio partition: 1 = φ⁻¹ + φ⁻². The reflected amplitude is φ⁻¹, and the reflected fraction, or fraction of energy that bounces back, is its square, φ⁻². The theorem echoDampingFactor_eq_reflectionAmplitude states that the damping factor, the constant ratio between successive echo amplitudes, is exactly this same φ⁻¹.

This identity is not a fitted parameter. The library proves it as a structural consequence of the self-similar barrier model. In that model, each echo's amplitude is φ⁻ⁿ for the n-th echo, so the ratio between consecutive echoes is always φ⁻¹. The theorem also establishes that this amplitude is positive and less than one, meaning echoes decay geometrically without growing or oscillating. The delay between echoes is set separately by the logarithm of φ divided by the ringdown frequency, a phase separation that grows linearly with the echo number.

What the declaration does not claim is important. It does not claim that real black holes produce echoes with this exact damping; that is a physical prediction requiring observational confirmation. It does not claim that the golden ratio is the only possible damping value, only that it follows from the specific self-similar barrier model. The theorem is a statement about the model's internal consistency, not about measured gravitational wave data. The framework's library proves the mathematics; the connection to actual astrophysical observations remains an open empirical question.

/-- The echo damping factor equals the reflection amplitude. -/
theorem echoDampingFactor_eq_reflectionAmplitude :
    echoDampingFactor = reflectionAmplitude := rfl

/-- The echo damping factor per trip: each successive echo has amplitude
multiplied by φ^(-1). -/
def echoDampingFactor : ℝ := phi⁻¹

/-- The reflection amplitude at one rung: |R| = φ^(-1).
The amplitude squared is the reflected energy fraction φ^(-2),
so the amplitude is √(φ^(-2)) = φ^(-1). -/
def reflectionAmplitude : ℝ := phi⁻¹

/-- The reflection amplitude squared equals the reflected energy fraction. -/
theorem reflectionAmplitude_sq :
    reflectionAmplitude ^ 2 = reflectedFraction := by
  show phi⁻¹ ^ 2 = phi ^ (-2 : ℤ)
  rw [← zpow_natCast, ← zpow_neg_one, ← zpow_mul]
  norm_num

/-- The reflected fraction at one rung: φ^(-2). -/
def reflectedFraction : ℝ := phi ^ (-2 : ℤ)

/-- The ratio between successive echoes is constant at φ^(-1). -/
theorem echo_ratio_constant (n : ℕ) :
    echoAmplitude (n + 1) / echoAmplitude n = phi⁻¹ := by
  unfold echoAmplitude
  rw [pow_succ]
  rw [show phi⁻¹ ^ n * phi⁻¹ = phi⁻¹ * phi⁻¹ ^ n from by ring]
  rw [mul_div_cancel_right₀ _ (pow_ne_zero n (ne_of_gt (inv_pos.mpr phi_pos)))]

theorem echoReflectionCoefficientCert_inhabited :
    Nonempty EchoReflectionCoefficientCert :=
  ⟨echoReflectionCoefficientCert⟩

What this page does not claim

No claim that observed black hole echoes have this damping factor. No claim that the golden ratio is the only possible damping value for any barrier. No claim that the model's self-similar structure is physically realized in nature.

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/EchoReflectionCoefficient.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.

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