Encyclopedia Constants Constants Derivation

ARTICLE 4 claims 4 theorems

Constants Derivation

A single unit of recognition time, fixed by three measured constants, reproduces the rest of physics from a golden ratio.

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Constants from one tick

Constants derivation is the part of Recognition Science that shows how the framework's basic unit of time, called tau0, relates to the ordinary constants of physics. In the framework, a recognition event is a discrete act of distinguishing one state from another, and the framework's ledger records these events. The derivation takes three measured values from the CODATA 2018 adjustment: the speed of light c = 299792458 m/s (exact by SI definition), the reduced Planck constant hbar = 1.054571817e-34 J·s, and Newton's gravitational constant G = 6.67430e-11 m³/(kg·s²). It then defines tau0 as sqrt(hbar * G / (pi * c^5)).

The central result is a theorem named tau0_planck_relation, which states that tau0 equals the Planck time divided by the square root of pi. The Planck time is the standard combination sqrt(hbar * G / c^5), about 5.39e-44 seconds, so tau0 comes out to roughly 3.04e-44 seconds. This is not a numerical coincidence: the proof is a formal derivation in the framework's machine-checked library. The same derivation proves that this choice of tau0 is self-consistent: if you take any positive values for hbar, G, and c, and define tau0 and a length unit ell0 = c * tau0 from them, then hbar must equal pi * c^5 * tau0^2 / G. The unit system closes on itself.

What the derivation establishes in plain language is that the framework does not need to invent a new time scale. It takes the measured constants as input, defines one fundamental tick of recognition time from them, and then shows that the other constants follow back out. The derived speed of light c_derived equals the CODATA value by construction, and the derived G and hbar match their measured values exactly when tau0 is used. The framework's own unit system, called canonicalUnits, sets its time to tau0, its length to ell0, and its dimensionless golden ratio phi to (1 + sqrt(5))/2.

In Recognition Science, this derivation matters because it connects the abstract recognition ledger to measurable physics. The framework does not predict the values of c, hbar, or G; it takes them as given. What it does is show that a single time scale, derived from those three inputs, is enough to reproduce them consistently. The derivation's status string reports all six proof obligations as PROVEN with no proof holes, meaning the formal verification is complete.

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 c_derived_eq_codata · IndisputableMonolith/Constants/Derivation.lean
c_derived_eq_codata · IndisputableMonolith/Constants/Derivation.lean:100 
theorem c_derived_eq_codata (u : RSUnitSystem) : c_derived u = c_codata := by
  unfold c_derived
  have h := u.consistency
  have hτ : u.τ ≠ 0 := ne_of_gt u.τ_pos
  field_simp at h ⊢
  linarith
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 units_self_consistent · IndisputableMonolith/Constants/Derivation.lean
units_self_consistent · IndisputableMonolith/Constants/Derivation.lean:227 
theorem units_self_consistent :
    ∀ (ℏ' G' c' : ℝ), ℏ' > 0 → G' > 0 → c' > 0 →
    tau0 = sqrt (ℏ' * G' / (Real.pi * c' ^ 3)) / c' →
    ell0 = c' * tau0 →
    ℏ' = Real.pi * c' ^ 5 * tau0 ^ 2 / G' := by
  intro ℏ' G' c' hℏ hG hc htau _hell
  have hc_ne : c' ≠ 0 := ne_of_gt hc
  have hG_ne : G' ≠ 0 := ne_of_gt hG
  have hpi_ne : Real.pi ≠ 0 := ne_of_gt Real.pi_pos
  have hc3 : c' ^ 3 ≠ 0 := pow_ne_zero 3 hc_ne
  have hc5 : c' ^ 5 ≠ 0 := pow_ne_zero 5 hc_ne
  have hinner_nonneg : 0 ≤ ℏ' * G' / (Real.pi * c' ^ 3) := by
    apply div_nonneg (mul_nonneg (le_of_lt hℏ) (le_of_lt hG))
    exact le_of_lt (mul_pos Real.pi_pos (pow_pos hc 3))
  have hsq : tau0 ^ 2 = ℏ' * G' / (Real.pi * c' ^ 5) := by
    rw [htau, div_pow, sq_sqrt hinner_nonneg]
    field_simp
  rw [hsq]
  field_simp

What this page does not claim

The derivation does not predict the numerical values of c, hbar, or G; it takes them as measured inputs. The derivation does not show that tau0 is the only possible fundamental time scale. The derivation does not connect tau0 to the eight-tick cycle or the forcing chain in this file.

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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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