Constants Derivation Tau0 Planck Relation

The Planck time bridge

The Planck time is a unit of time built from the three most fundamental constants in physics: the speed of light c, the gravitational constant G, and the reduced Planck constant hbar. Its standard definition is the square root of hbar times G divided by c to the fifth power. It is roughly 5.39 times 10 to the minus 44 seconds, an interval so short that it is often described as the smallest time that has physical meaning in quantum gravity. The Recognition Science framework, a system that derives physical structure from a forced cost of recognition, defines its own base unit of time, called tau0, and then proves a precise relationship between the two.

The relationship is a result in the framework's machine-checked library of formal theorems. The result, named tau0_planck_relation, states that tau0 equals the Planck time divided by the square root of pi. In symbols: tau0 = planck_time / sqrt(pi). This is not an approximation and not a fitted value. It is an exact equality that follows from the definitions of tau0 and the Planck time, both of which are built from the CODATA reference values for c, hbar, and G. The framework's own time unit is therefore not an arbitrary scale; it is anchored to the standard Planck time by a simple, fixed factor.

What this result does not claim is just as important as what it proves. It does not claim that the framework derives the value of the Planck time from first principles. The Planck time is defined using the measured CODATA values, and the result only establishes the ratio between the framework's tau0 and that defined quantity. It also does not claim that the framework derives the fine-structure constant or any other dimensionless coupling constant. The result is a bridge between two unit systems, not a derivation of a fundamental constant from nothing.

The practical consequence is that the framework's unit system is not disconnected from conventional physics. When the framework produces a value in its own units, that value can be translated into seconds or meters through this exact relation. The bridge works in both directions: the framework's time unit is a fixed multiple of the Planck time, and the framework's length unit is fixed by the speed of light times that time. This makes the framework's internal constants commensurable with the standard system of units, which is a necessary step for any empirical comparison.

The result also feeds into a larger consistency check. The framework proves that its derived value for G, computed from its own tau0 and the CODATA hbar and c, exactly equals the CODATA G. This is a separate result, G_relation_satisfied, and it confirms that the unit system closes on itself. The tau0_planck_relation is the hinge that makes that closure possible.

/-- **Theorem**: τ₀ = t_P / √π

This relation shows τ₀ is the Planck time divided by √π. -/
theorem tau0_planck_relation : tau0 = planck_time / sqrt Real.pi := by
  unfold tau0 planck_time
  have hc : c_codata ≠ 0 := c_codata_ne_zero
  have hpi : Real.pi ≠ 0 := ne_of_gt Real.pi_pos
  have hpi_pos : 0 < Real.pi := Real.pi_pos
  have hc_pos : 0 < c_codata := c_codata_pos
  have hinner_pos : 0 < hbar_codata * G_codata := mul_pos hbar_codata_pos G_codata_pos
  have hsqrt_pi_pos : 0 < sqrt Real.pi := sqrt_pos.mpr hpi_pos
  have hsqrt_pi_ne : sqrt Real.pi ≠ 0 := ne_of_gt hsqrt_pi_pos
  have hc3_pos : 0 < c_codata ^ 3 := pow_pos hc_pos 3
  have hc5_pos : 0 < c_codata ^ 5 := pow_pos hc_pos 5
  have hinner5_nonneg : 0 ≤ hbar_codata * G_codata / c_codata ^ 5 :=
    le_of_lt (div_pos hinner_pos hc5_pos)
  have hc3 : c_codata ^ 3 ≠ 0 := pow_ne_zero 3 hc
  have hc5 : c_codata ^ 5 ≠ 0 := pow_ne_zero 5 hc
  have hinner3_div_pos : 0 < hbar_codata * G_codata / (Real.pi * c_codata ^ 3) :=
    div_pos hinner_pos (mul_pos hpi_pos hc3_pos)
  have hinner3_div_nonneg : 0 ≤ hbar_codata * G_codata / (Real.pi * c_codata ^ 3) :=
    le_of_lt hinner3_div_pos
  -- Strategy: show both sides equal by direct calculation
  -- LHS = sqrt(ℏG/(πc³))/c
  -- RHS = sqrt(ℏG/c⁵)/sqrt(π)
  -- Show: LHS² = RHS² and both are positive
  have hlhs_pos : 0 < sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata :=
    div_pos (sqrt_pos.mpr hinner3_div_pos) hc_pos
  have hrhs_pos : 0 < sqrt (hbar_codata * G_codata / c_codata ^ 5) / sqrt Real.pi :=
    div_pos (sqrt_pos.mpr (div_pos hinner_pos hc5_pos)) hsqrt_pi_pos
  have hlhs_sq : (sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata) ^ 2 =
                 hbar_codata * G_codata / (Real.pi * c_codata ^ 5) := by
    rw [div_pow, sq_sqrt hinner3_div_nonneg]
    field_simp
  have hrhs_sq : (sqrt (hbar_codata * G_codata / c_codata ^ 5) / sqrt Real.pi) ^ 2 =
                 hbar_codata * G_codata / (Real.pi * c_codata ^ 5) := by
    rw [div_pow, sq_sqrt hinner5_nonneg, sq_sqrt (le_of_lt hpi_pos)]
    field_simp
  have hsq_eq : (sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata) ^ 2 =
                (sqrt (hbar_codata * G_codata / c_codata ^ 5) / sqrt Real.pi) ^ 2 := by
    rw [hlhs_sq, hrhs_sq]
  have hlhs_nonneg : 0 ≤ sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata :=
    le_of_lt hlhs_pos
  have hrhs_nonneg : 0 ≤ sqrt (hbar_codata * G_codata / c_codata ^ 5) / sqrt Real.pi :=
    le_of_lt hrhs_pos
  have hsqrt_lhs : sqrt ((sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata) ^ 2) =
                   sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata :=
    sqrt_sq hlhs_nonneg
  have hsqrt_rhs : sqrt ((sqrt (hbar_codata * G_codata / c_codata ^ 5) / sqrt Real.pi) ^ 2) =
                   sqrt (hbar_codata * G_codata / c_codata ^ 5) / sqrt Real.pi :=
    sqrt_sq hrhs_nonneg
  calc sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata
      = sqrt ((sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata) ^ 2) := hsqrt_lhs.symm
    _ = sqrt ((sqrt (hbar_codata * G_codata / c_codata ^ 5) / sqrt Real.pi) ^ 2) := by rw [hsq_eq]
    _ = sqrt (hbar_codata * G_codata / c_codata ^ 5) / sqrt Real.pi := hsqrt_rhs

def planck_time : ℝ := sqrt (hbar_codata * G_codata / c_codata ^ 5)

/-- **Theorem**: G_derived tau0 hbar_codata c_codata = G_codata -/
theorem G_relation_satisfied :
    G_derived tau0 hbar_codata c_codata = G_codata := by
  unfold G_derived
  rw [tau0_sq_eq]
  have hℏ : hbar_codata ≠ 0 := hbar_codata_ne_zero
  have hc : c_codata ≠ 0 := c_codata_ne_zero
  have hpi : Real.pi ≠ 0 := ne_of_gt Real.pi_pos
  have hc5 : c_codata ^ 5 ≠ 0 := pow_ne_zero 5 hc
  field_simp

What this page does not claim

The framework does not derive the value of the Planck time from first principles. The result does not derive the fine-structure constant or any other dimensionless coupling constant. The result does not claim that tau0 is the Planck time itself; it is the Planck time divided by the square root of pi.

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

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