Constants Derivation C Derived Eq Codata

The speed identity

The speed of light in a vacuum, c = 299792458 meters per second, is exact by the modern SI definition of the meter. The declaration c_derived_eq_codata proves that in any Recognition Science unit system, the derived speed equals this exact value. A unit system here means a choice of a time unit, a length unit, and the golden ratio, with one consistency condition: the product of the speed and the time unit must equal the length unit. The theorem states that for any such system, the ratio of length to time is always c_codata.

The proof is a direct algebraic consequence of that consistency condition. If c times tau equals ell, and tau is nonzero, then dividing both sides by tau gives c = ell / tau. The machine-checked library of formal theorems records this as c_derived_eq_codata, with a companion lemma c_derived_pos showing the derived speed is positive. This is not a numerical approximation or a fitted value; it is an identity that follows from the definition of the unit system itself.

What the declaration does not do is derive the numerical value 299792458 from first principles. That number enters as c_codata, a definition that mirrors the SI standard. The framework proves that whatever unit system you choose, as long as it satisfies the consistency condition, the derived speed will match this defined constant. The physical content is the relationship, not the origin of the meter's length.

In Recognition Science, this identity anchors the derivation of other constants. The library also proves that the derived gravitational constant G_derived equals the CODATA value, and that tau0, the base time unit, relates to the Planck time by a factor of the square root of pi. These results depend on the same style of consistency argument. The speed identity is the load-bearing first step: it fixes the ratio of length to time, so that subsequent derivations of hbar and G have a stable foundation.

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 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

def c_codata : ℝ := 299792458

What this page does not claim

The numerical value 299792458 is not derived from the framework's axioms; it is a defined constant matching the SI standard. The theorem does not show that the meter or the second are unique choices; it only constrains their ratio once a consistent system is chosen. No claim is made that the speed of light varies across unit systems; the theorem states it is invariant under the consistency condition.

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.

Derived articles

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