Gravity Echo Reflection Coefficient Transmitted Fraction Lt One

The transmitted fraction

In the study of gravitational wave echoes, a key quantity is the transmitted fraction, the portion of an incoming wave's energy that passes through a barrier rather than reflecting back. The Recognition Science framework models the near-horizon region of a black hole as a self-similar barrier, a structure that looks the same at every scale, with each layer, or rung, splitting energy in the same proportion. The declaration transmittedFraction_lt_one establishes, as a proved theorem, that this transmitted fraction is always less than one. In plain terms, no barrier built from these self-similar rungs lets the entire wave through; some energy always reflects back.

The theorem is a formal consequence of the golden ratio's defining equation, φ² = φ + 1. The framework defines the transmitted fraction as φ⁻¹, approximately 0.618, and the reflected fraction as φ⁻², approximately 0.382. These two numbers sum to exactly one, a partition of energy that the framework derives from the golden ratio's self-similar structure. The theorem transmittedFraction_lt_one then follows directly: since the reflected fraction is positive, the transmitted fraction must be less than one. This is not a numerical fit or a dimensional-analysis estimate; it is a forced consequence of the self-similarity assumption, verified in a machine-checked library of formal theorems.

The result belongs to a larger derivation of echo properties. Each rung reflection reduces the echo amplitude by a factor of φ⁻¹, so the nth echo has amplitude φ⁻ⁿ, a geometric decay. The framework also derives the round-trip phase accumulation per rung as log φ. These pieces together predict a specific pattern of echoes with decreasing amplitude and increasing delay, all tied to the golden ratio. The theorem about the transmitted fraction is the foundational piece that guarantees the echo amplitude never grows and that some signal is always reflected.

What the theorem does not claim is that this specific barrier model is the correct description of real black holes, nor that gravitational wave echoes have been observed. The framework's derivation is a structural result: given a self-similar barrier with golden-ratio spacing, these are the forced consequences. Whether real astrophysical systems exhibit such structure is an empirical question, not a theorem. The framework presents this as a prediction, with the observation of echoes as the named falsifier. The theorem itself is a statement about a mathematical model, not about the physical universe.

theorem transmittedFraction_lt_one : transmittedFraction < 1 := by
  have : 0 < reflectedFraction := reflectedFraction_pos
  linarith [partition_complete]

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

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

/-- The partition is complete: reflected + transmitted = 1. -/
theorem partition_complete :
    reflectedFraction + transmittedFraction = 1 := by
  unfold reflectedFraction transmittedFraction
  rw [add_comm]
  exact phi_energy_partition

/-- 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)))]

/-- Echo n has amplitude proportional to φ^(-n). -/
def echoAmplitude (n : ℕ) : ℝ := phi⁻¹ ^ n

What this page does not claim

The theorem does not claim that real black holes have self-similar barriers. The theorem does not claim that gravitational wave echoes have been observed. The theorem does not claim that the transmitted fraction applies to any barrier other than the φ-self-similar one.

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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