Gravity Bhechoes Ligocatalog Bounce Radius Pos

The bounce radius

The declaration bounceRadius_pos is a formal theorem in the machine-checked library of the Recognition Science framework. It states that for any natural number N, the value of the bounce radius at that rung is strictly greater than zero. In the framework's model, a rung is a step in a discrete recognition sequence, and the bounce radius is a length scale associated with that step. The theorem is a simple consequence of the definition of the radius as phi raised to the power N, where phi is the golden ratio, which is itself greater than zero.

This positivity result is a foundational property of the framework's geometric model. It ensures that the bounce radius, and consequently the echo delay derived from it, is never zero or negative. The echo delay is defined as twice the bounce radius times the natural logarithm of phi, and the library also proves this delay is strictly positive for any rung N greater than or equal to one. These are internal consistency results within the framework's own definitions.

In Recognition Science, this theorem underpins a specific prediction about gravitational wave events. The framework's geodesic-completeness theorem gives the bounce radius and echo delay at each rung, and the library names four canonical LIGO/Virgo merger events (GW150914, GW170817, GW190521, GW230529) for which this bounce-echo prediction is structurally permitted. The prediction is that each event carries an echo at the bounce delay scaled by the source mass. The theorem bounceRadius_pos itself, however, only establishes the positivity of the radius; it does not assert that such echoes exist or have been observed.

The honest scope of the theorem is narrow. It proves a mathematical property of a defined quantity within the framework, not a physical fact about the universe. The framework's own documentation states that a null result on a high signal-to-noise catalog event would falsify the bounce mechanism. This theorem is a necessary but not sufficient condition for the prediction to hold; it guarantees the model is well-formed, but it cannot guarantee the model matches reality.

/-- 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 _

/-- 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 _

/-- 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 _

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

This theorem does not claim that any gravitational wave echo has been detected. This theorem does not claim that the bounce radius model is physically correct. This theorem does not claim that the golden ratio is the only possible scaling for the bounce radius.

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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