Gravity Bhecho Per Event Catalog

The echo prediction table

When two black holes merge, the violent collision rings like a struck bell. Some theories predict that the ring is not clean: a pulse of gravitational waves should bounce off the inner structure of the merged object and come back as a faint echo, delayed by a time that depends on the object's size. The recognition framework, which treats physical events as entries in a discrete cost ledger, turns this idea into a concrete prediction table for the four canonical LIGO/Virgo headline events: GW150914, GW170817, GW190521, and GW230529.

For each event, the catalog lists the source mass, a recognition rung (a level in the framework's discrete scaling ladder), and the predicted echo delay. The delays are not arbitrary: they follow a phi-power ladder, where φ is the golden ratio, approximately 1.618. The smallest event, GW170817 at about 2.7 solar masses, sits at rung 1 with a predicted delay of φ·log φ in the framework's native units. The largest, GW190521 at about 150 solar masses, sits at rung 10 with a delay of φ¹⁰·log φ. The catalog also records the echo frequency, simply the inverse of the delay, and proves that every delay and frequency is strictly positive.

The catalog is not a measurement. It is a falsifiable prediction table. The framework's library, a machine-checked collection of formal theorems, proves the internal consistency of the table: there are exactly four events, the delays are positive, and the rungs are ordered so that more massive events sit at higher rungs. The key consequence is that adjacent rungs have echo frequencies that differ by a factor of 1/φ. This means the catalog can be tested: a single high-signal merger whose echo does not appear at its predicted rung would falsify the ladder.

In plain terms, the catalog establishes a precise schedule for when echoes should arrive, if the framework's picture of gravity is right. It does not claim the echoes have been observed. It gives observers a specific set of times to look for, and a clear rule for what would count as a miss. That is the difference between a vague hope and a testable prediction.

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

/-- 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)

/-- Catalog count = 4. -/
theorem event_count : Fintype.card HeadlineEvent = 4 := by decide

/-- The two largest events (GW190521 and GW150914) sit at higher rungs
than the two smaller (GW170817, GW230529). -/
theorem rung_ordering :
    predictedRung .GW170817 < predictedRung .GW230529 ∧
    predictedRung .GW230529 < predictedRung .GW150914 ∧
    predictedRung .GW150914 < predictedRung .GW190521 := by
  refine ⟨?_, ?_, ?_⟩ <;> decide

What this page does not claim

No observed echo has been confirmed in LIGO/Virgo data by this catalog. The catalog does not derive the existence of black hole echoes from first principles. The specific rung assignments are chosen from a log-mass scaling rule, not derived within this module.

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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