Gravity Echo Reflection Coefficient Echo Reflection Coefficient Forced

The forced echo coefficient

When a wave meets a barrier, part of it bounces back and part passes through. The reflection coefficient is the fraction of energy that returns. In general relativity, the region near a black hole's event horizon can act as such a barrier, and some quantum gravity models predict that it produces a series of echoes in the gravitational-wave signal. The question is what fraction of the wave's energy each bounce sends back.

The classical answer depends on the detailed shape of the barrier. But there is a special case where the barrier is self-similar: each layer, or rung, looks like the previous one scaled down by a constant factor. If that factor is the golden ratio φ, which satisfies φ² = φ + 1 and equals (1 + √5)/2 ≈ 1.618, then the mathematics forces the answer. The energy at each rung splits according to 1 = φ⁻¹ + φ⁻², meaning the reflected fraction is φ⁻² ≈ 0.382 and the transmitted fraction is φ⁻¹ ≈ 0.618. The reflected amplitude, the square root of the reflected fraction, is φ⁻¹ ≈ 0.618.

This is not a dimensional-analysis estimate. It is a theorem in a machine-checked library of formal proofs, with no fitting parameters and no additional assumptions beyond the self-similar structure. The theorem echo_reflection_coefficient_forced states that the ratio of successive echo amplitudes is constant at φ⁻¹, that the energy partition sums to one, and that the reflection amplitude lies strictly between zero and one. Each successive echo is therefore quieter by the same factor, and the delay between echoes grows by a fixed phase per rung, equal to the natural logarithm of φ.

In Recognition Science, this result is part of a larger chain where the golden ratio emerges from the cost of recognition events. But the echo theorem itself does not depend on that framework. It is a standalone statement about a self-similar barrier: if the barrier has that scaling, the reflection coefficient is forced. The framework's contribution is to argue that such barriers arise naturally from its principles, not to change the mathematics of the reflection.

What the theorem does not do is predict that real black holes produce echoes. That would require the physical bridge from the mathematical model to actual astrophysical objects, which remains an open question. The theorem also does not determine the absolute delay between echoes; that depends on the ringdown frequency, which is not part of the proof. It establishes the relative structure: the geometric decay and the constant phase step, given the self-similar barrier.

/-- The golden-ratio energy partition: at a self-similar barrier with
scale ratio φ, energy splits into reflected fraction φ^(-2) and
transmitted fraction φ^(-1).

The proof uses only φ² = φ + 1 (the defining equation of the golden ratio).
Dividing through: 1 = φ^(-1) + φ^(-2). -/
theorem phi_energy_partition :
    phi⁻¹ + phi ^ (-2 : ℤ) = 1 := by
  have hne : phi ≠ 0 := phi_ne_zero
  have hsq : phi ^ 2 = phi + 1 := phi_sq_eq
  have hphi_pos := phi_pos
  have h1 : phi * phi⁻¹ = 1 := mul_inv_cancel₀ hne
  have h2 : phi ^ 2 * phi ^ (-2 : ℤ) = 1 := by
    rw [← zpow_natCast, ← zpow_add₀ hne]
    norm_num
  nlinarith [sq_nonneg (phi * (phi⁻¹ + phi ^ (-2 : ℤ)) - phi)]

/-- 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)))]

/-- 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

/-- Echo amplitudes form a geometric series with ratio φ^(-1). -/
theorem echo_geometric (n m : ℕ) (hnm : n ≤ m) :
    echoAmplitude m = phi⁻¹ ^ (m - n) * echoAmplitude n := by
  unfold echoAmplitude
  rw [← pow_add]
  congr 1
  omega

What this page does not claim

The theorem does not predict that real black holes produce gravitational-wave echoes. The theorem does not determine the absolute time delay between echoes, which depends on the ringdown frequency. The theorem does not prove that the golden ratio arises from physical principles; that is a separate framework claim.

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)

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