Gravity Bhechoes Ligocatalog Bhechoes Cert

The echo certificate

Black-hole echoes are a proposed observational signature: after two black holes merge, the remnant might emit a series of weak, repeating gravitational-wave pulses instead of ringing down once. The Recognition Science framework derives a specific bounce radius r_min(N) = φ^(N/2) and echo delay Δt(N) = 2·r_min·log φ at each recognition rung N, where φ is the golden ratio. The declaration BHEchoesCert establishes that these quantities are strictly positive at every rung and that adjacent rungs scale by the golden ratio.

The certificate names four canonical merger events from the LIGO/Virgo catalog: GW150914 (first black-hole merger detection, about 65 solar masses), GW170817 (first neutron-star merger, about 2.7 solar masses), GW190521 (intermediate-mass black-hole merger, about 150 solar masses), and GW230529 (neutron-star black-hole merger, about 4.4 solar masses). For each, the framework predicts an echo at the bounce delay scaled by the source mass.

What the certificate does not claim is equally important. It does not assert that echoes have been observed; no LIGO/Virgo detection of echoes is claimed. It does not derive the echo amplitude or waveform, only the delay structure. The certificate is a structural permit, not a detection claim.

The falsification condition is explicit: a null result on a high-signal-to-noise catalog event with N ≥ 1 would falsify the RS bounce mechanism. This makes the prediction testable, not merely formal.

/-- 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

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

/-- 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 LIGO/Virgo echo detection is claimed. The certificate does not derive echo amplitude or waveform. The certificate does not establish that the bounce mechanism is the correct physical description of black-hole remnants.

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: