Gravity Bhecho Per Event Catalog Predicted Echo Delay Pos

Predicted echo delays

Gravitational-wave echoes are hypothetical repeating signals that some theories expect after the main chirp of a black hole merger. The LIGO/Virgo detectors have not confirmed any such echo, but the Recognition Science framework's machine-checked library of formal theorems records what its model would predict for four headline events: GW150914, GW170817, GW190521, and GW230529. For each event the library stores a source mass, a recognition rung (a discrete index from the framework's scaling law), and a predicted echo delay Δt(N) in dimensionless units.

The specific declaration predictedEchoDelay_pos proves a narrow but essential fact: for every one of the four events, the predicted echo delay is strictly greater than zero. The proof runs by checking each of the four cases and citing a general theorem that echo delays are positive whenever the rung is at least 1, which all four rungs are. The companion declaration predictedEchoFrequency_pos proves the corresponding frequency, defined as 1/Δt, is also positive. These are not measurements and not detections; they are internal consistency checks on the framework's own prediction table.

The predicted delays themselves follow a striking ladder. For GW170817 (mass ~2.7 M☉) the delay is φ·log φ, where φ is the golden ratio. For GW230529 (~4.4 M☉) it is φ²·log φ; for GW150914 (~65 M☉) it is 47·log φ; for GW190521 (~150 M☉) it is φ¹⁰·log φ. Adjacent rungs differ in echo frequency by the fixed ratio 1/φ, a consequence of the framework's golden-ratio scaling. The library also proves the rungs are ordered by mass: the two smaller events sit at lower rungs than the two larger ones.

In Recognition Science, this catalog is a falsifiable prediction. The framework states that any single high-signal-to-noise event whose echo signature does not appear at its predicted rung would refute the ladder. The library does not claim that any echo has been observed, that the delays match any actual LIGO/Virgo measurement, or that the framework's model of black hole interiors is physically correct. It claims only that the prediction table is internally consistent: the delays are positive, the frequencies are positive, and the rungs are ordered by mass.

/-- 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)

/-- 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 gravitational-wave echo has been observed at any of these delays. The predicted delays are not compared against any measured LIGO/Virgo echo detection. The framework's black hole interior model is not asserted to be physically correct.

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)

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.

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