Gravity Bhecho Per Event Catalog Predicted Echo Frequency Pos

The predicted echo frequency

Black hole echoes are hypothetical signals that would follow the main gravitational wave burst if the black hole had a reflective surface instead of an event horizon. In the Recognition Science framework, each black hole gets a predicted echo delay, the time between the main signal and the echo, and a predicted echo frequency, which is simply the inverse of that delay. The declaration predictedEchoFrequency_pos establishes a minimal but essential fact: for each of the four headline LIGO/Virgo events, the predicted echo frequency is strictly greater than zero.

The four events are GW150914, GW170817, GW190521, and GW230529. For each, the framework assigns a recognition rung, a discrete step in a ladder, based on the source mass. The predicted echo delay is a function of that rung, and the predicted echo frequency is the reciprocal of that delay. The theorem proves that this frequency is always positive, never zero and never negative. This matters because a frequency of zero would mean no echo at all, and a negative frequency would be physically meaningless. The theorem is a formal guarantee that the catalog's numbers are well-behaved.

In Recognition Science, the framework models the echo delay as a positive quantity for every event. The machine-checked library of formal theorems verifies this for each of the four events by checking each case individually. The result is part of a larger certificate, a package of verified properties including that the event count is four, the bounce radius is positive, and the rungs are ordered by mass. The positivity of the echo frequency is one of the checks that the whole catalog passes.

What this theorem does not claim is that these echoes exist. It does not claim that the predicted frequencies match any observed signal. The predictions are falsifiable: any single high signal-to-noise event whose echo does not appear at the predicted rung would refute the ladder. The theorem only guarantees internal consistency, that the catalog's numbers are positive and well-defined. It is a statement about the framework's own predictions, not about the physical world.

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

/-- Predicted echo frequency per event is `1 / Δt(N)`. -/
def predictedEchoFrequency (e : HeadlineEvent) : ℝ :=
  1 / predictedEchoDelay e

/-- The four canonical LIGO/Virgo headline events. -/
inductive HeadlineEvent where
  | GW150914
  | GW170817
  | GW190521
  | GW230529
  deriving DecidableEq, Repr, BEq, Fintype

What this page does not claim

The theorem does not claim that black hole echoes have been observed. The theorem does not claim that the predicted frequencies match any measured signal. The theorem does not claim that the recognition rung assignment is unique or 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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