Gravity Bhecho Per Event Catalog Predicted Bounce Radius Pos

The positivity guarantee

A bounce radius is the distance from a black hole at which an infalling object, in this framework's account, reverses direction instead of crossing the horizon. The declaration predictedBounceRadius_pos is a formal theorem in the machine-checked library of formal theorems. It states that for each of four named gravitational-wave events, the predicted bounce radius is strictly greater than zero.

The four events are the canonical LIGO/Virgo headline detections: GW150914, GW170817, GW190521, and GW230529. For each event, the framework assigns a recognition rung, a discrete step on a ladder of scales. The rung is chosen from log-mass scaling, roughly the base-φ logarithm of the source mass divided by a reference mass. The bounce radius is then a function of that rung. The theorem proves, by checking each of the four cases, that the radius is positive.

The theorem also supports two companion results in the same catalog. The predicted echo delay, the time between the main gravitational-wave signal and a repeated echo, is positive for every event. The predicted echo frequency, its reciprocal, is positive as well. Together these three positivity results form part of a certificate that bundles the catalog's structural claims.

What the theorem does not do is assert that any of these predictions are observed. It says nothing about whether LIGO or Virgo has detected an echo at the predicted delay. The catalog itself names a falsifier: any single high-signal-to-noise event whose echo signature does not appear at the predicted rung would refute the ladder. The theorem only guarantees that the predicted numbers, whatever they are, are positive.

In Recognition Science, the positivity guarantee matters because the framework's cost function, the forced cost of recognition, produces scales that must be physical. A negative radius would be unphysical. The theorem closes that door for these four events, leaving the empirical question open.

/-- Bounce radius per event is strictly positive. -/
theorem predictedBounceRadius_pos (e : HeadlineEvent) :
    0 < predictedBounceRadius e :=
  bounceRadius_pos _

/-- Echo delay per event is strictly positive (every predicted rung is
N ≥ 1; verified by `decide` on the inductive cases). -/
theorem predictedEchoDelay_pos (e : HeadlineEvent) :
    0 < predictedEchoDelay e := by
  unfold predictedEchoDelay predictedRung
  cases e <;> exact echoDelay_pos _ (by decide)

theorem predictedEchoFrequency_pos (e : HeadlineEvent) :
    0 < predictedEchoFrequency e := by
  unfold predictedEchoFrequency
  exact div_pos one_pos (predictedEchoDelay_pos e)

What this page does not claim

No claim that any predicted echo has been observed by LIGO or Virgo. No claim that the bounce radius is a measured quantity. No claim that the recognition rung assignment is derived from first principles.

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/BHEchoPerEventCatalog.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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