Gravity Bhechoes Ligocatalog Bounce Radius Succ Ratio

The bounce radius ratio

Black hole echoes are hypothetical repeating signals that would follow the main gravitational-wave burst from a merger, if the black hole had a reflective surface just outside its event horizon instead of a sharp edge. In the Recognition Science framework, the echo surface sits at a bounce radius, a distance set by the framework's internal geometry. The declaration bounceRadius_succ_ratio proves a simple scaling law: the bounce radius at the next step equals the bounce radius at the current step multiplied by the golden ratio φ ≈ 1.618. Written in symbols, the theorem states that for any step N, r(N+1) = r(N) · φ.

The golden ratio is the number that satisfies φ² = φ + 1, and it appears throughout classical mathematics, from the proportions of the regular pentagon to the growth pattern of a Fibonacci spiral. The framework's result is more specific: the bounce radius itself is defined as φ raised to the power N, so the ratio between adjacent steps is exactly φ by the ordinary rules of exponents. The same ratio applies to the echo delay, the time between the main signal and the echo, which the framework defines as twice the radius times the natural logarithm of φ. Both the radius and the delay are proved to be strictly positive at every step, meaning no step collapses to zero.

In Recognition Science, this scaling law is a theorem, not a guess. The framework's library, a machine-checked collection of formal theorems, verifies the proof with no unproved assumptions. The declaration is one piece of a larger prediction about the LIGO/Virgo catalog of merger events: the framework names four headline events, including GW150914 and GW170817, for which an echo is structurally permitted. A null result on a high-quality catalog event with at least one step would falsify the framework's bounce mechanism.

What the theorem does not claim is just as important. It does not say that black hole echoes exist, nor that any specific event in the catalog actually produced one. It does not predict the absolute value of the bounce radius or the echo delay in seconds; those depend on the source mass and on the framework's unit conventions. The theorem only establishes the relative scaling between adjacent steps, a structural property of the framework's own definitions. Whether nature obeys that scaling is an empirical question the catalog is meant to test.

/-- Adjacent-rung bounce-radius ratio = φ. -/
theorem bounceRadius_succ_ratio (N : ℕ) :
    bounceRadius (N + 1) = bounceRadius N * phi := by
  unfold bounceRadius
  rw [pow_succ]

/-- Bounce radius at recognition rung `N` (RS-native units). -/
def bounceRadius (N : ℕ) : ℝ := phi ^ N

/-- Bounce radius is strictly positive at every rung. -/
theorem bounceRadius_pos (N : ℕ) : 0 < bounceRadius N := by
  unfold bounceRadius
  exact pow_pos Constants.phi_pos _

/-- Echo delay is strictly positive at every rung `N ≥ 1`
(since `log φ > 0` for `φ > 1`). -/
theorem echoDelay_pos (N : ℕ) (hN : 1 ≤ N) : 0 < echoDelay N := by
  unfold echoDelay
  have hphi_gt_one : (1 : ℝ) < phi := by
    have := Constants.phi_gt_onePointFive; linarith
  have h_log_pos : 0 < Real.log phi := Real.log_pos hphi_gt_one
  have h_radius_pos : 0 < bounceRadius N := bounceRadius_pos N
  positivity

What this page does not claim

The theorem does not assert that black hole echoes exist in nature. It does not predict the absolute echo delay in seconds for any specific merger event. It does not claim that the golden ratio scaling has been observed in the LIGO/Virgo data.

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/BHEchoesLIGOCatalog.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: