Gravity Black Hole Echoes Si Bounce Radius Si Strict Mono

The bounce radius ladder

A black hole is a region of space where gravity is so strong that nothing, not even light, can escape. Some theoretical models suggest that instead of a silent event horizon, a black hole might have a surface that briefly reflects infalling matter or energy, producing faint, repeating signals called echoes. The declaration bounceRadius_SI_strict_mono is a formal, machine-checked statement about one such model: it proves that the proposed radius of this reflective surface, called the bounce radius, increases by a fixed ratio at each step of a ladder.

Specifically, the theorem proves that for any non-negative integer N, the bounce radius at step N is strictly less than the radius at step N+1. The radius is defined as planckLength_SI multiplied by φ^N, where φ is the golden ratio, approximately 1.618. The Planck length is a tiny unit of distance built from the speed of light, the gravitational constant, and Planck's constant. The theorem shows this sequence is strictly increasing: each rung of the ladder is larger than the one before it. It also proves a two-step identity: the radius at step N+2 equals the radius at step N multiplied by φ², meaning the ladder grows by a factor of φ² every two steps.

This is a structural result about the model's internal algebra, not a physical prediction. The declaration is part of a larger library that converts a native unit system into SI units. The radius formula is a definition, and the monotonicity is a proved theorem about that definition. The library explicitly states that it does not close the physical black-hole echo mechanism. The old event-horizon escape mechanism is recorded as rejected, and the horizon-consistent exterior mechanism is marked as open. In plain terms: the theorem establishes that if you accept the model's definition of the bounce radius, then the ladder of radii is strictly ordered. It does not establish that such a surface exists, that echoes are real, or that the radius formula is correct.

The one physical anchor is the measured value of the gravitational constant G, which enters through the Planck length. The speed of light and Planck's constant are exact in the SI-2019 system. The theorem itself is axiom-clean, with zero unresolved proofs and no new axioms beyond the standard ones. The practical consequence is that the model's internal structure is self-consistent: the radii do not cross, overlap, or reverse order. This is a necessary condition for any physical model, but it is far from sufficient. A model can be internally consistent and still be wrong about nature.

theorem bounceRadius_SI_strict_mono (N : ℕ) :
    bounceRadius_SI N < bounceRadius_SI (N + 1) := by
  unfold bounceRadius_SI
  rw [pow_succ]
  have hN : 0 < phi ^ N := pow_pos phi_pos N
  have hℓ : 0 < planckLength_SI := planckLength_SI_pos
  have hphi : 1 < phi := one_lt_phi
  nlinarith [mul_pos hℓ hN]

/-- 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

theorem bounceRadius_SI_two_step (N : ℕ) :
    bounceRadius_SI (N + 2) = bounceRadius_SI N * phi ^ 2 := by
  unfold bounceRadius_SI
  rw [pow_add]
  ring

structure BlackHoleEchoesSICert where
  planckTime_SI_pos : 0 < planckTime_SI
  planckLength_SI_pos : 0 < planckLength_SI
  planckTime_SI_sq :
    planckTime_SI ^ 2 = hbar_SI * G_SI / c_SI ^ 5
  planckLength_SI_sq :
    planckLength_SI ^ 2 = hbar_SI * G_SI / c_SI ^ 3
  planckLength_SI_eq_planckTime_mul_c :
    planckLength_SI = planckTime_SI * c_SI
  bounceRadius_SI_pos : ∀ N : ℕ, 0 < bounceRadius_SI N
  bounceRadius_SI_two_step :
    ∀ N : ℕ, bounceRadius_SI (N + 2) = bounceRadius_SI N * phi ^ 2
  bounceRadius_SI_strict_mono :
    ∀ N : ℕ, bounceRadius_SI N < bounceRadius_SI (N + 1)
  echoDelay_SI_def :
    ∀ N : ℕ, echoDelay_SI N = (2 * bounceRadius_SI N / c_SI) * Real.log phi
  echoDelay_SI_eq_planckTime_form :
    ∀ N : ℕ, echoDelay_SI N = 2 * planckTime_SI * phi ^ N * Real.log phi
  echoDelay_SI_pos : ∀ N : ℕ, 0 < echoDelay_SI N
  echoDelay_SI_two_step :
    ∀ N : ℕ, echoDelay_SI (N + 2) = echoDelay_SI N * phi ^ 2
  echoDelay_SI_strict_mono :
    ∀ N : ℕ, echoDelay_SI N < echoDelay_SI (N + 1)
  echoDelay_SI_sq :
    ∀ N : ℕ, (echoDelay_SI N) ^ 2 =
      4 * (hbar_SI * G_SI / c_SI ^ 5) * phi ^ (2 * N) * (Real.log phi) ^ 2
  echoDampingRatio_SI_band :
    (0.617 : ℝ) < echoDampingRatio_SI ∧ echoDampingRatio_SI < 0.622

What this page does not claim

This does not claim that black hole echoes exist in nature. This does not claim that the bounce radius formula is physically correct. This does not claim that the theorem provides a falsifiable prediction for LIGO or Virgo.

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)

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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