Gravity Black Hole Echoes Si Planck Length Si Eq Planck Time Mul C

The Planck bridge

The Planck length and the Planck time are the scales at which quantum effects and gravity meet. The Planck length, about 1.6 x 10^-35 meters, is the distance light travels in one Planck time, about 5.4 x 10^-44 seconds. The declaration planckLength_SI_eq_planckTime_mul_c is a theorem in the framework's machine-checked library of formal theorems. It proves that the Planck length equals the Planck time multiplied by the speed of light, a relation that holds by definition in standard physics.

The proof works from the standard definitions. The Planck time is the square root of hbar times G divided by c to the fifth power. The Planck length is the square root of hbar times G divided by c to the third power. The theorem shows these two expressions are consistent: the length is the time times c. The proof is a formal derivation, not a new physical claim. It converts the framework's native unit system, where the speed of light is 1, into SI units with meters and seconds.

In Recognition Science, this identity is the dimensional bridge for a model of black hole echoes. The model describes a bounce radius at rung N as the Planck length times phi to the N, where phi is the golden ratio. The echo delay at rung N is twice that radius divided by c, times the natural log of phi. Using the bridge, the delay becomes two times the Planck time times phi to the N times the log of phi. The framework proves this compact form as echoDelay_SI_eq_planckTime_form.

The declaration does not claim that black hole echoes exist. The physical mechanism for such echoes is open or rejected in the framework's own records. The SI formulas are not a test for LIGO or Virgo until a horizon-consistent exterior echo mechanism exists. The theorem only establishes the dimensional conversion of a formal model. The damping ratio, about 0.619, and the per-rung phase delay are dimensionless and unaffected by the SI lift.

What the declaration changes is practical. It lets the framework's formal results be stated in ordinary units, so a reader can compare them with measured physics. The identity itself is a consequence of the standard definitions of the Planck units, not a new law. The framework's contribution is the formal proof and the clean algebraic form, not a new relation between space and time.

/-- Geometric relation: `planckLength_SI = planckTime_SI · c_SI`. -/
theorem planckLength_SI_eq_planckTime_mul_c :
    planckLength_SI = planckTime_SI * c_SI := by
  unfold planckLength_SI planckTime_SI
  -- √(ℏG/c³) = √((ℏG/c⁵)·c²) = √(ℏG/c⁵) · √(c²) = √(ℏG/c⁵) · c (since c > 0)
  rw [show hbar_SI * G_SI / c_SI ^ 3 =
        (hbar_SI * G_SI / c_SI ^ 5) * c_SI ^ 2 by
        have hc : c_SI ≠ 0 := ne_of_gt c_SI_pos
        field_simp]
  rw [Real.sqrt_mul
        (le_of_lt
          (div_pos (mul_pos hbar_SI_pos G_SI_pos) (pow_pos c_SI_pos 5)))]
  rw [Real.sqrt_sq (le_of_lt c_SI_pos)]

/-- 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

What this page does not claim

The theorem does not prove that black hole echoes occur in nature. The theorem does not derive the speed of light or the Planck units from first principles. The theorem does not provide a falsifiable prediction for gravitational wave observatories.

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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Derived articles

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