Gravity Echo Reflection Coefficient Transmitted Fraction Pos

The transmitted fraction

In the physics of echoes near a black hole, a wave that strikes a barrier splits into a reflected part and a transmitted part. The transmitted fraction is the share of the wave's energy that passes through the barrier rather than bouncing back. Classical scattering theory leaves that share to be computed from the barrier's shape and the wave's frequency. The Recognition Science framework instead models the near-horizon region as a ledger, a discrete record of recognition events, arranged in a self-similar pattern whose spacing follows the golden ratio φ, the number satisfying φ² = φ + 1, about 1.618.

Within that model, the framework defines the transmitted fraction as φ⁻¹, the reciprocal of the golden ratio, which is about 0.618. The declaration transmittedFraction_pos is a machine-checked theorem stating that this number is strictly greater than zero. A companion theorem, transmittedFraction_lt_one, proves it is strictly less than one. Together they establish that the transmitted fraction is a genuine fraction: it lies strictly between zero and one, so it behaves as a physical share of energy should. The proof is short: it uses the fact that φ is positive, and the companion theorem uses the partition identity φ⁻¹ + φ⁻² = 1, which says the reflected and transmitted shares sum exactly to one.

The framework's library of formal theorems also proves the partition identity itself, and from it derives that the reflected fraction is φ⁻², about 0.382. The two fractions sum to one, so no energy is lost or created at the barrier in this model. The reflection amplitude, the square root of the reflected fraction, is φ⁻¹, the same value as the transmitted fraction. This symmetry is a consequence of the golden ratio's defining equation, not an assumption added for convenience.

What the declaration does not claim is broader physical content. It does not assert that real black holes transmit exactly 61.8 percent of incoming gravitational waves. It does not claim that the self-similar barrier model is the actual structure of a black hole horizon. The theorem only certifies a property of a number defined inside the framework: given the model's definitions, the transmitted fraction is positive and less than one. The step from that formal result to a prediction about astrophysical echoes remains a modeling choice, not a proved consequence.

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

theorem transmittedFraction_pos : 0 < transmittedFraction :=
  inv_pos.mpr phi_pos

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

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

What this page does not claim

The theorem does not assert that real black holes transmit exactly 61.8 percent of incoming gravitational waves. The self-similar barrier is a modeling choice within the framework, not a proved structure of black hole horizons. No claim is made that the transmitted fraction is positive for any barrier other than the φ-self-similar one defined in the framework.

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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