pith. machine review for the scientific record. sign in
Applied HYPOTHESIS Gravity v5

Black-Hole Echoes from RS Bounce

Predicted echo amplitudes in post-merger gravitational wave data

Predicted echo amplitudes in post-merger gravitational wave data.

Equations

[ c_{\mathrm{grav}}=c_{\mathrm{EM}},\qquad G_{\mathrm{RS}}=\varphi^5/\pi ]

Shared RS gravity bridge.

Derivation chain (Lean anchors)

Each row links to the corresponding Lean 4 declaration in the Recognition Science canon. A resolved anchor has a green check; an unresolved anchor flags a registry/canon mismatch.

  1. 1 BH echoes from bounce module checked
    IndisputableMonolith.Gravity.BlackHoleEchoesFromBounce Open theorem →
  2. 2 Echo amplitudes module checked
    IndisputableMonolith.Gravity.BHEchoAmplitudes Open theorem →
  3. 3 LIGO catalog module checked
    IndisputableMonolith.Gravity.BHEchoesLIGOCatalog Open theorem →
  4. 4 Per-event catalog module checked
    IndisputableMonolith.Gravity.BHEchoPerEventCatalog Open theorem →

Narrative

1. Setting

Black-Hole Echoes from RS Bounce is anchored in Gravity.BlackHoleEchoesFromBounce. The page is not a loose explainer: it is a public map from the Recognition Science forcing chain into one Lean-checked declaration bundle. The primary anchor determines what is proved, and the surrounding declarations show how the result is used.

2. Equations

(E1)

$$ c_{\mathrm{grav}}=c_{\mathrm{EM}},\qquad G_{\mathrm{RS}}=\varphi^5/\pi $$

Shared RS gravity bridge.

3. Prediction or structural target

  • Structural target: Gravity.BlackHoleEchoesFromBounce must keep resolving in the Lean canon, and all downstream pages that cite this anchor must continue to type-check.

This page is currently a structural derivation. Where the claim has direct empirical content, the prediction table gives the measurable target; otherwise the claim is a formal bridge inside the Lean canon.

4. Formal anchor

The primary anchor is Gravity.BlackHoleEchoesFromBounce..

5. What is inside the Lean module

Key theorems:

  • bounceRadius_pos
  • bounceRadius_zero
  • bounceRadius_two_step
  • bounceRadius_strict_mono
  • rungPhaseDelay_pos
  • rungPhaseDelay_band
  • echoDelay_pos
  • echoDelay_scaling
  • echoDelay_two_step
  • echoDampingRatio_pos
  • echoDampingRatio_lt_one
  • echoDampingRatio_band

Key definitions:

  • bounceRadius
  • rungPhaseDelay
  • echoDelay
  • echoDampingRatio
  • cumulativeEchoAmplitude
  • BlackHoleEchoesCert
  • blackHoleEchoesCert

6. Derivation chain

7. Falsifier

Null results for echoes at predicted amplitudes across LIGO/Virgo merger events refute the echo hypothesis.

8. Where this derivation stops

Below this page the chain reduces to the RS forcing sequence: J-cost uniqueness, phi forcing, the eight-tick cycle, and the D=3 recognition substrate. If any upstream theorem changes, this page must be versioned rather than patched silently. The published URL is stable, but the version field is the contract.

9. Reading note

The minimal way to audit this page is to open the first Lean anchor and then walk the supporting declarations listed above. If the primary theorem is a module-level anchor, the key theorems section names the internal declarations that carry the mathematical load. This keeps the public derivation readable without severing it from the proof object.

10. Audit path

To audit black-hole-echoes, start with the primary Lean anchor Gravity.BlackHoleEchoesFromBounce. Then inspect the theorem names listed in the module-content section. The page is intentionally built so the public explanation is not a substitute for the proof object; it is a map into it. The mathematical dependency is the same in every case: reciprocal cost fixes J, J fixes the phi-ladder, the eight-tick cycle fixes the recognition clock, and the domain theorem listed above supplies the last step. If that last step is empirical, the falsifier section names what observation would break it. If that last step is formal, a Lean-checkable counterexample is the relevant failure mode.

Falsifier

Null results for echoes at predicted amplitudes across LIGO/Virgo merger events refute the echo hypothesis.

References

  1. lean Recognition Science Lean library (IndisputableMonolith)
    https://github.com/jonwashburn/shape-of-logic
    Public Lean 4 canon used by Pith theorem pages.
  2. paper Uniqueness of the Canonical Reciprocal Cost
    Washburn, J.; Zlatanovic, B.
    Axioms (MDPI) (2026)
    Peer-reviewed paper anchoring the J-cost uniqueness theorem.
  3. spec Recognition Science Full Theory Specification
    https://recognitionphysics.org
    High-level theory specification and public program context for Recognition Science derivations.

How to cite this derivation

  • Stable URL: https://pith.science/derivations/black-hole-echoes
  • Version: 5
  • Published: 2026-05-14
  • Updated: 2026-05-15
  • JSON: https://pith.science/derivations/black-hole-echoes.json
  • YAML source: pith/derivations/registry/bulk/black-hole-echoes.yaml

@misc{pith-black-hole-echoes, title = "Black-Hole Echoes from RS Bounce", author = "Recognition Physics Institute", year = "2026", url = "https://pith.science/derivations/black-hole-echoes", note = "Pith Derivations, version 5" }