Gravity Echo Reflection Coefficient Echo Reflection Coefficient Cert Inhabited

The certified echo coefficient

When a gravitational wave strikes a potential barrier near a black hole's horizon, part of it reflects back. The question is how much. In the Recognition Science framework, the answer is not a free parameter: the reflection coefficient is forced to be the reciprocal of the golden ratio, about 0.618. The declaration echoReflectionCoefficientCert_inhabited is a machine-checked proof that this coefficient is mathematically consistent: it shows that a certificate object exists, one that bundles together all the required properties of the reflection process.

The certificate is a structure containing seven facts. It states that the golden ratio's defining equation φ² = φ + 1 implies the energy partition 1 = φ⁻¹ + φ⁻², meaning the reflected and transmitted fractions sum to one. It states that the reflection amplitude equals φ⁻¹, that its square equals the reflected fraction, and that successive echo amplitudes decay by the constant ratio φ⁻¹. It also states that this amplitude is positive and less than one, and that the phase accumulated per rung is positive. The theorem echoReflectionCoefficientCert_inhabited proves that a certificate with all these properties exists, built from the underlying lemmas in the framework's machine-checked library of formal theorems.

The classical mathematics here is the golden ratio itself, known since antiquity as the extreme and mean ratio. Its defining equation φ² = φ + 1 has the positive solution φ = (1 + √5)/2 ≈ 1.618. The reciprocal φ⁻¹ ≈ 0.618 satisfies φ⁻¹ + φ⁻² = 1, a partition that the framework interprets as the energy split at each rung of a self-similar barrier. The reflection coefficient |R|² = φ⁻² ≈ 0.382 is the fraction of energy reflected at a single rung, and the amplitude |R| = φ⁻¹ ≈ 0.618 is its square root.

What the declaration does not claim is that this coefficient has been measured in any gravitational-wave experiment. It is a structural theorem about a mathematical model, not an empirical prediction. The framework models the near-horizon structure as a φ-self-similar barrier, and the proof shows that within that model the reflection coefficient is forced. Whether real black holes exhibit such echoes, and at what amplitude, remains an open observational question. The theorem also does not claim that the golden ratio itself is derived from physics; it takes φ as given and shows what follows from its defining equation.

In Recognition Science, this result is one link in a chain: the same golden ratio that forces the cost function J(x) = (x + 1/x)/2 − 1 also forces the echo structure at a self-similar barrier. The payoff is that the echo prediction is not a dimensional-analysis estimate but a consequence of a single structural assumption. The reader can now see that the reflection coefficient is not adjustable within the framework; it is pinned down by the mathematics of φ.

theorem echoReflectionCoefficientCert_inhabited :
    Nonempty EchoReflectionCoefficientCert :=
  ⟨echoReflectionCoefficientCert⟩

/-- 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)))]

/-- 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

/-- Phase per rung is positive (since φ > 1). -/
theorem phasePerRung_pos : 0 < phasePerRung := by
  unfold phasePerRung
  exact Real.log_pos phi_gt_one

What this page does not claim

No gravitational-wave experiment has measured this reflection coefficient. The golden ratio itself is not derived from physics in this declaration; it is taken as given. The theorem does not claim that real black holes exhibit echoes, only that the model forces the coefficient if the barrier is φ-self-similar.

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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