Encyclopedia Constants Constants Derivation Tau0 Matches Foundation

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Constants Derivation Tau0 Matches Foundation

A single number, tau0, is defined as the base unit of time in Recognition Science, and a machine-checked proof confirms it is consistent with the framework's own definitions.

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The base time unit

In physics, a unit of time is a standard interval used to measure duration. The SI second is defined by the frequency of a cesium atom's radiation. Recognition Science, a framework that derives physical structure from a ledger of recognition events, introduces its own base unit of time, called tau0. This value is not chosen arbitrarily; it is defined by a specific formula that combines the measured values of the reduced Planck constant, the gravitational constant, and the speed of light.

The declaration tau0_matches_foundation is a theorem in the framework's machine-checked library of formal theorems. It proves that the definition of tau0, as written in the code, is exactly equal to the expression formed from those three physical constants. The proof is a direct computation: it unfolds the definitions and confirms the equality by reflexivity. This means the statement is true by definition, not by a chain of derived arguments. It establishes that the framework's internal definition of its base time unit is consistent with its own stated formula.

This consistency is a foundational check. It verifies that the code's definition of tau0 matches the intended mathematical expression. The theorem does not, however, claim that tau0 is the Planck time. A separate theorem, tau0_planck_relation, proves a different relationship: tau0 equals the Planck time divided by the square root of pi. The tau0_matches_foundation theorem is a narrower statement, confirming only the internal self-consistency of the definition.

What this means for the reader is that the framework's base unit of time is a well-defined quantity, anchored to the accepted measured values of fundamental constants. The theorem acts as a guard against errors in the definition itself. It does not prove that this unit has any physical significance beyond the framework, nor does it derive the value of tau0 from first principles. It simply certifies that the definition is internally coherent and matches the formula it claims to represent.

THEOREM tau0_matches_foundation · IndisputableMonolith/Constants/Derivation.lean
tau0_matches_foundation · IndisputableMonolith/Constants/Derivation.lean:249 
theorem tau0_matches_foundation :
    tau0 = sqrt ((1.054571817e-34 : ℝ) * (6.67430e-11 : ℝ) /
           (Real.pi * (299792458 : ℝ) ^ 3)) / (299792458 : ℝ) := by
  unfold tau0 hbar_codata G_codata c_codata
  rfl
THEOREM tau0_matches_foundation · IndisputableMonolith/Constants/Derivation.lean
tau0_matches_foundation · IndisputableMonolith/Constants/Derivation.lean:249 
theorem tau0_matches_foundation :
    tau0 = sqrt ((1.054571817e-34 : ℝ) * (6.67430e-11 : ℝ) /
           (Real.pi * (299792458 : ℝ) ^ 3)) / (299792458 : ℝ) := by
  unfold tau0 hbar_codata G_codata c_codata
  rfl
THEOREM tau0_planck_relation · IndisputableMonolith/Constants/Derivation.lean
tau0_planck_relation · IndisputableMonolith/Constants/Derivation.lean:169 
/-- **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

What this page does not claim

The theorem does not prove that tau0 is the Planck time. The theorem does not derive the value of tau0 from the framework's axioms. The theorem does not assign any physical meaning to tau0 outside the framework.

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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