Foundation Constant Derivations C Rs Eq One

The unit speed of light

The speed of light in a vacuum, usually written c, is one of the most precisely measured numbers in physics. In the International System of Units, its value is fixed by definition at exactly 299,792,458 meters per second, a choice made in 1983 to anchor the meter to the second. Recognition Science (RS) approaches the same constant from a different direction: instead of measuring c, it derives it from more basic quantities defined inside the framework, and the derivation produces a strikingly simple result.

Within RS, the fundamental unit of time is called τ₀ and the fundamental unit of length is called ℓ₀. The framework defines the speed of light as the ratio of these two: c = ℓ₀ / τ₀. The declaration c_rs_eq_one, a theorem in the machine-checked library of formal theorems, proves that this ratio equals exactly 1. The proof is short: both ℓ₀ and τ₀ are defined to be the real number 1, so their quotient is 1. This is not an approximation or a fitted value; it is an identity that follows from the definitions.

What does this mean physically? In RS-native units, where the fundamental length and time are both set to unity, the speed of light becomes the number 1. This is a common trick in theoretical physics, called natural units: choose units so that c = 1, and then speeds are measured as fractions of the speed of light. The RS framework makes this choice explicit and derives it from the structure of its fundamental quantities rather than treating it as a convention.

The theorem c_rs_eq_one does not claim that the measured speed of light in SI units is 1. It does not claim that RS predicts the numerical value 299,792,458. It does not claim that the framework's choice of ℓ₀ and τ₀ is unique or forced by the axioms; the definitions simply set both to 1. The theorem establishes a relationship inside the framework, not a new measurement of the physical constant. It is a structural fact about how RS defines its units, not a claim about the empirical world.

/-- c = 1 in RS-native units. -/
theorem c_rs_eq_one : c_rs = 1 := by
  unfold c_rs ℓ₀ τ₀
  norm_num

/-- c = 1 in RS-native units. -/
theorem c_rs_eq_one : c_rs = 1 := by
  unfold c_rs ℓ₀ τ₀
  norm_num

/-- c = 1 in RS-native units. -/
theorem c_rs_eq_one : c_rs = 1 := by
  unfold c_rs ℓ₀ τ₀
  norm_num

What this page does not claim

The theorem does not claim that the measured speed of light in SI units is 1. The theorem does not claim that RS predicts the numerical value 299,792,458 meters per second. The theorem does not claim that the definitions of ℓ₀ and τ₀ are forced by the framework's axioms.

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/Foundation/ConstantDerivations.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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