Gravity Black Hole Echoes Si Echo Delay Si Eq Planck Time Form

The SI echo delay formula

In general relativity, a black hole's event horizon is a one-way boundary: nothing that crosses it can come back out. Some speculative models replace the horizon with a surface that reflects, so a pulse of gravitational waves could bounce off it and return, producing a faint "echo" after the main merger signal. The Recognition Science framework has a formal, machine-checked library of theorems that explores one such rung-based model. Its declaration echoDelay_SI_eq_planckTime_form (the name is a technical label, not a physical law) proves a compact algebraic identity: the echo delay in seconds equals 2 times the Planck time times φ^N times the natural logarithm of φ.

The Planck time is the tiny interval built from the speed of light, the gravitational constant, and the reduced Planck constant; it is about 5.39 × 10⁻⁴⁴ seconds. The factor φ is the golden ratio, about 1.618, and N is a nonnegative integer labeling the rung of the model. The theorem states that the delay for rung N is exactly that product. It also proves two immediate consequences: the delay is always positive, and it grows by a factor of φ² when N increases by two. These are pure algebraic facts about the defined quantity, not measurements of real black holes.

The substance of the declaration is a unit conversion. The model originally worked in Planck units, where the delay came out as a pure number. This theorem lifts that number into seconds using the SI-2019 exact values for the speed of light and the reduced Planck constant, plus a single measured value for the gravitational constant. The result is a clean, sqrt-free formula in SI units. The framework's library proves the identity with zero unproved assumptions and no new axioms, meaning the algebra is airtight.

What the declaration does not claim is just as important. It does not claim that black hole echoes exist, that this model describes the real universe, or that any observatory can detect the predicted signal. The physical mechanism that would produce an echo outside a horizon is explicitly marked as open or rejected in the framework's own files. The theorem is a statement about a formal model, not a prediction about nature. It is a structural result for SI conversion only, as its own documentation states.

In plain terms, the theorem says: if you accept this rung-based model and its definitions, then the echo delay has this exact SI form. It is a piece of internal consistency, not a ticket to a LIGO detection. The library's own status line calls it a structural theorem for SI conversion only. That is the honest scope of the declaration.

/-- Compact form: `echoDelay_SI(N) = 2 · planckTime_SI · φ^N · log φ`.
Uses `planckLength_SI = planckTime_SI · c_SI`. -/
theorem echoDelay_SI_eq_planckTime_form (N : ℕ) :
    echoDelay_SI N = 2 * planckTime_SI * phi ^ N * Real.log phi := by
  unfold echoDelay_SI bounceRadius_SI
  rw [planckLength_SI_eq_planckTime_mul_c]
  have hc : c_SI ≠ 0 := ne_of_gt c_SI_pos
  field_simp

theorem echoDelay_SI_pos (N : ℕ) : 0 < echoDelay_SI N := by
  rw [echoDelay_SI_eq_planckTime_form]
  have h_log : 0 < Real.log phi := Real.log_pos one_lt_phi
  have h_phi_pow : 0 < phi ^ N := pow_pos phi_pos N
  have h_pt : 0 < planckTime_SI := planckTime_SI_pos
  have h2 : (0 : ℝ) < 2 := by norm_num
  positivity

theorem echoDelay_SI_two_step (N : ℕ) :
    echoDelay_SI (N + 2) = echoDelay_SI N * phi ^ 2 := by
  rw [echoDelay_SI_eq_planckTime_form, echoDelay_SI_eq_planckTime_form,
      pow_add]
  ring

What this page does not claim

The declaration does not claim that black hole echoes exist in nature. It does not claim that the rung model is a correct description of real black holes. It does not claim any observable prediction that a current or near-future detector could test.

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: