Gravity Echo Reflection Coefficient

The golden-ratio barrier

The gravity echo reflection coefficient is a number that describes how much of an incoming wave bounces back at a boundary. In the Recognition Science framework, the near-horizon region of a black hole is modeled as a stack of identical barriers, each one self-similar to the whole. At every boundary, the energy of the wave splits between a reflected part and a transmitted part. The reflection coefficient is the fraction of energy that comes back.

The framework's key result is that this coefficient is not a free parameter. It is forced to be the inverse of the golden ratio φ, which satisfies φ² = φ + 1. The reflection amplitude, the square root of the reflected energy fraction, is φ⁻¹ ≈ 0.618. The transmitted fraction is φ⁻¹ as well, and the two add to exactly 1: φ⁻¹ + φ⁻² = 1. This is the golden-ratio energy partition, and it follows directly from the defining equation of φ, with no fitting and no dimensional analysis.

In the framework's account, a wave crossing the barrier accumulates a phase of log φ per rung. Each successive echo arrives with an amplitude multiplied by φ⁻¹, so the nth echo has amplitude φ⁻ⁿ. The delay between echoes is (log φ) / (2π f_ringdown), where f_ringdown is the ringdown frequency. The echoes form a geometric series, with each one quieter than the last by the same factor.

The machine-checked library of formal theorems proves that this structure is consistent: the ratio between successive echo amplitudes is constant, the reflection amplitude squared equals the reflected fraction, and the reflection amplitude lies strictly between 0 and 1. The theorem echo_reflection_coefficient_forced packages all of this into a single certificate, showing that the golden ratio's defining equation acts as the barrier's scattering matrix.

In Recognition Science, this means the echo prediction from quantum-gravity papers is not a rough estimate. It is a forced consequence of the substrate's self-similar structure at golden-ratio spacing. The framework models the near-horizon recognition structure as a φ-self-similar potential barrier, and the reflection coefficient falls out of the mathematics. This is a structural theorem: the framework proves the coefficient must be φ⁻¹, given its model of the 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

What this page does not claim

This module does not prove that real black holes produce echoes; it proves a mathematical consequence of a model. The reflection coefficient being φ⁻¹ is derived from the model's self-similarity assumption, not from a measurement. The framework does not yet provide a physical derivation of the barrier's existence.

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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