Encyclopedia Constants Constants Derivation Tau0 Sq Eq

ARTICLE 4 claims 4 theorems

Constants Derivation Tau0 Sq Eq

A single equation ties a fundamental time unit to Planck time and π, but it does not derive that unit from scratch.

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The meaning of tau0_sq_eq

The declaration tau0_sq_eq is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It establishes that the square of a fundamental time unit, called tau0 (the basic recognition tick, a discrete record of events), equals a specific expression built from the measured values of the reduced Planck constant, the gravitational constant, the speed of light, and π. In plain terms, it says that if you take the measured values of these physical constants, the framework's basic time unit is not arbitrary: its square is fixed by them. The theorem is a formal statement in the framework's code, and it is marked as proven with no proof holes.

The equation itself is tau0² = (hbar_codata * G_codata) / (π * c_codata³). This is not a new physical law. It is a definitional identity within the framework: the framework defines tau0 in terms of these measured constants. The theorem's role is to confirm that this definition is consistent and that it matches the framework's own foundation. The library also proves that this tau0 is related to the Planck time by tau0 = planck_time / sqrt(π), and that the framework's derived values for G and hbar, when computed from tau0, agree exactly with the measured CODATA values. These are internal consistency checks, not independent derivations of the constants themselves.

What tau0_sq_eq does not claim is that the framework has explained why the fundamental constants have the values they do. The constants hbar, G, and c are taken as inputs, measured by experiment, and the framework builds its time unit from them. The theorem does not predict the values of these constants; it uses them. It also does not claim that tau0 is the smallest possible unit of time in the physical universe, nor that time is discrete in a way that contradicts established physics. The theorem is a statement about the framework's internal definitions and their consistency with measured data, not a claim about the ultimate nature of time.

The practical consequence is that the framework can express its other derived constants, such as G and hbar, in terms of its own time unit and the speed of light. This gives the framework a self-consistent set of units, but it does not reduce the number of free parameters in physics. The measured values of the constants still come from experiment. The theorem is a piece of internal bookkeeping that makes the framework's unit system coherent, not a new discovery about the universe.

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
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
THEOREM G_relation_satisfied · IndisputableMonolith/Constants/Derivation.lean
G_relation_satisfied · IndisputableMonolith/Constants/Derivation.lean:137 
/-- **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
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 predict the values of hbar, G, or c; it uses their measured values as inputs. The theorem does not claim that tau0 is the smallest possible unit of time in the physical universe. The theorem does not derive the fine-structure constant or any other dimensionless physical constant.

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)

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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