Gravity Echo Reflection Coefficient Barrier Total Reflection

The golden-ratio barrier

The golden ratio, usually written φ, is the number that satisfies φ² = φ + 1, or about 1.618. It appears in art, in plant growth, and in the self-similar shapes of fractals. In the Recognition Science framework, it also appears in a physical setting: the reflection of energy from a barrier near a black hole's horizon. The framework models this barrier as a self-similar stack of layers, each one scaling the previous by the golden ratio.

At each layer, or rung, the framework's model splits the incoming energy between a reflected part and a transmitted part. The split is not arbitrary. The defining equation of the golden ratio forces the partition: 1 = φ⁻¹ + φ⁻². The reflected fraction is φ⁻², about 0.382, and the transmitted fraction is φ⁻¹, about 0.618. The reflection amplitude, the square root of the reflected fraction, is φ⁻¹, about 0.618. These values are derived from the equation φ² = φ + 1 alone, with no fitting parameter.

The declaration barrier_total_reflection establishes a precise statement about this model. It says that for any such self-similar barrier with a positive number of rungs, the amplitude of the echo after all rungs is exactly φ⁻¹ raised to the power of the number of rungs. This is a structural theorem in the framework's machine-checked library of formal theorems. It is not a measurement of real black holes; it is a consequence of the model's definition.

The theorem also proves that the echo amplitudes form a geometric sequence. Each successive echo is weaker than the previous by the constant factor φ⁻¹. The delay between echoes grows by a fixed phase per rung, the natural logarithm of φ. This geometric decay is the signature of the self-similar structure.

What the declaration does not claim is that real black holes emit these echoes. The model is a candidate description of near-horizon quantum structure, not an observed phenomenon. The framework's library proves the mathematics of the model, but the bridge from this mathematical structure to the physical spacetime of general relativity remains an open target. The theorem is about the model's internal consistency, not about astronomical data.

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

/-- The transmitted fraction at one rung: φ^(-1). -/
def transmittedFraction : ℝ := phi⁻¹

/-- The total reflected amplitude after passing through a barrier with n
rungs is φ^(-n) (each rung contributes one factor of φ^(-1)). -/
theorem barrier_total_reflection (B : PhiSelfSimilarBarrier) :
    echoAmplitude B.numRungs = phi⁻¹ ^ B.numRungs :=
  rfl

/-- 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 assert that real black holes emit these echoes. The model does not derive the existence of the barrier from general relativity. The framework does not claim a measured value for the reflection coefficient.

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: