Gravity Bhechoes Ligocatalog

The echo catalog

Gravitational-wave echoes are a proposed repeat signal after the main burst from a black hole merger. In general relativity, the merged object settles silently; some alternatives to classical black holes predict a partial reflection that produces a fainter, delayed pulse. The LIGO and Virgo detectors have not yet confirmed such an echo in any event.

In Recognition Science, the framework models recognition events as a discrete ledger, a record of discrete happenings. Its geodesic-completeness theorem derives a bounce radius at each recognition rung rung, a step on the framework's ladder of scales, and an echo delay that grows with that radius. The module BHEchoesLIGOCatalog applies that prediction to four named merger events: GW150914, the first black hole merger detected; GW170817, the first neutron star merger; GW190521, an intermediate-mass merger; and GW230529, a neutron star black hole merger. For each, the framework predicts an echo at the bounce delay scaled by the source mass.

The module proves, in a machine-checked library of formal theorems, that the bounce radius is strictly positive at every rung and that the delay between adjacent rungs scales by the golden ratio φ. These are structural facts about the framework's own model, not measurements. The catalog also names a falsifier: a null result on a high-signal catalog event at rung N ≥ 1 would refute the framework's bounce mechanism.

The practical consequence is a concrete target for gravitational-wave searches. The framework does not claim an echo has been seen; it claims the prediction is precise enough to test. The delay formula, Δt(N) = 2 φ^(N/2) log φ, gives a specific number to look for in each event's data.

/-- 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 _

/-- Adjacent-rung echo-delay ratio = φ. -/
theorem echoDelay_succ_ratio (N : ℕ) (hN : 1 ≤ N) :
    echoDelay (N + 1) = echoDelay N * phi := by
  unfold echoDelay
  rw [bounceRadius_succ_ratio]
  ring

structure BHEchoesCert where
  bounce_radius_pos : ∀ N, 0 < bounceRadius N
  bounce_radius_succ_ratio :
    ∀ N, bounceRadius (N + 1) = bounceRadius N * phi
  echo_delay_pos : ∀ N, 1 ≤ N → 0 < echoDelay N
  echo_delay_succ_ratio :
    ∀ N, 1 ≤ N → echoDelay (N + 1) = echoDelay N * phi

What this page does not claim

No gravitational-wave echo has been detected in any catalog event. The module does not prove that the framework's bounce mechanism is the correct explanation for any observed signal. The catalog's mass scaling is a definitional choice, not a derived prediction.

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)

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Derived articles

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