Gravity Black Hole Echoes Si Black Hole Echoes Sicert Inhabited

The SI echo certificate

Black hole echoes are a hypothetical signal: after a black hole merger, the new hole might ring not just once but repeatedly, like a bell that hums several times after a strike. The Recognition Science framework has a mathematical model of such echoes built on a ladder of radii spaced by the golden ratio φ ≈ 1.618. The declaration blackHoleEchoesSICert_inhabited is a theorem in the framework's machine-checked library of formal theorems. It proves that this ladder model can be expressed in SI units, the international system of meters, seconds, and kilograms, without any free parameters.

The core formulas are simple. The bounce radius at rung N is r_min(N) = φ^N in Planck units, where a Planck length is about 1.6 × 10⁻³⁵ meters. The echo delay, the time between successive echoes, is Δt = 2 r_min log φ. The factor 2 is a geometric two-way traversal factor for a bounce, not something the framework forces. The damping ratio, how quickly each echo fades, is 1/φ ≈ 0.618. The certificate bundles these into a single structure and proves basic properties: the radii and delays are positive, they grow strictly with N, and each delay is φ² times the delay two rungs earlier.

The SI conversion uses two exact constants from the 2019 SI redefinition, the speed of light c and the reduced Planck constant ħ, plus one measured constant, Newton's gravitational constant G. The Planck time and length are defined as √(ħG/c⁵) and √(ħG/c³). The certificate proves the squared forms of these quantities, avoiding square roots in the algebra, and shows that the delay squared equals 4(ħG/c⁵)φ^(2N)(log φ)². The damping ratio is dimensionless, so it is already SI-invariant.

What the certificate does not do is close the physical mechanism. The framework's own documentation states that the old event-horizon escape mechanism is rejected and the horizon-consistent exterior mechanism remains open. These SI formulas are not a LIGO/Virgo falsifier until such a mechanism exists. The certificate is a structural theorem about unit conversion, not an observation of echoes from real black holes.

theorem blackHoleEchoesSICert_inhabited : Nonempty BlackHoleEchoesSICert :=
  ⟨blackHoleEchoesSICert⟩

/-- Bounce radius in SI at rung gap `N`: `r_min(N) = ℓ_Planck_SI · φ^N`.
This is the meter-scale lift of `BlackHoleEchoesFromBounce.bounceRadius N
= φ^N` (which is dimensionless in Planck units). -/
def bounceRadius_SI (N : ℕ) : ℝ := planckLength_SI * phi ^ N

/-- Echo delay in SI: `Δt = (2 · r_min / c) · log φ`. -/
def echoDelay_SI (N : ℕ) : ℝ :=
  (2 * bounceRadius_SI N / c_SI) * Real.log phi

theorem echoDampingRatio_SI_eq : echoDampingRatio_SI = 1 / phi := rfl

theorem echoDelay_SI_sq (N : ℕ) :
    (echoDelay_SI N) ^ 2 =
      4 * (hbar_SI * G_SI / c_SI ^ 5) * phi ^ (2 * N) * (Real.log phi) ^ 2 := by
  rw [echoDelay_SI_eq_planckTime_form]
  have hphi_pow : phi ^ N * phi ^ N = phi ^ (2 * N) := by
    rw [show (2 * N : ℕ) = N + N from by omega, pow_add]
  have h_expand :
      (2 * planckTime_SI * phi ^ N * Real.log phi) ^ 2
        = 4 * planckTime_SI ^ 2 * (phi ^ N * phi ^ N) * (Real.log phi) ^ 2 := by
    ring
  rw [h_expand, hphi_pow, planckTime_SI_sq]

What this page does not claim

The certificate does not prove that real black holes produce echoes. The factor 2 in the delay formula is not forced by the Recognition Science framework; it is a geometric choice. The formulas are not a falsifier for LIGO or Virgo until a horizon-consistent exterior echo mechanism exists.

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

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