Gravity Echo Reflection Coefficient Echo Phase Separation Succ

Phase separation in echoes

In the study of gravitational wave echoes, the phase separation between successive echoes is the extra phase a wave accumulates with each round trip inside the barrier. The Recognition Science framework models this barrier as a ledger, a discrete record of events, where each rung, a single crossing layer, reflects the same fraction of the wave's energy. The framework's machine-checked library of formal theorems proves a simple recurrence: the phase separation at echo n+1 equals the phase separation at echo n plus a constant phase per rung, log φ, where φ is the golden ratio.

The golden ratio, approximately 1.618, is the unique positive solution to φ² = φ + 1. This single equation forces the energy partition at each rung: 1 = φ⁻¹ + φ⁻², meaning about 61.8% of the energy is reflected and 38.2% is transmitted. The reflection amplitude, the square root of the reflected fraction, is φ⁻¹. The theorem echoPhaseSeparation_succ states this recurrence, and its proof is a direct algebraic consequence of the definition of phase separation as n times log φ. It is a structural result: no fitting parameter or dimensional analysis enters; the golden ratio's defining equation determines the barrier's scattering behavior.

This recurrence means the phase separation grows linearly with echo number, a geometric progression in the phase domain. The framework's library also proves that the echo amplitude decays geometrically, with each echo smaller than the last by a factor of φ⁻¹. Together, these two facts characterize the echo train: amplitudes fall off by a fixed ratio while phases advance by a fixed step. The phase separation theorem is one half of this complete description, and it holds for any barrier satisfying the self-similarity condition, where each rung reflects the same amplitude.

What the theorem does not claim is that this model describes actual astrophysical black holes. The framework's barrier is an idealized structure, and the physical bridge from recognition events to real gravitational wave echoes remains an open target. The theorem establishes a mathematical consequence of a specific model, not an empirical prediction verified against observations. The echo delay, Δt_echo = (log φ) / (2πf_ringdown), is a derived quantity within the model, not a measured value.

theorem echoPhaseSeparation_succ (n : ℕ) :
    echoPhaseSeparation (n + 1) = echoPhaseSeparation n + phasePerRung := by
  unfold echoPhaseSeparation
  push_cast
  ring

/-- 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 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 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)))]

What this page does not claim

The theorem does not establish that real black holes produce echoes with golden-ratio phase separation. The model does not fit any observed gravitational wave data; it is a mathematical consequence of a definitional choice. The physical bridge from recognition events to gravitational wave echoes is not proved and remains open.

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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