In IndisputableMonolith.Foundation.ConstantDerivations, the Planck length in RS-native units is defined as planck_length_rs := sqrt (ℏ_rs * G_rs / c_rs^3). The theorem planck_length_eq_one proves this quantity equals 1.
The proof proceeds by direct substitution using supporting results in the same module:
- By c_rs_eq_one,
c_rs = 1(from the definitionsℓ₀ = 1andτ₀ = 1). - By G_ℏ_product,
G_rs * ℏ_rs = 1(which follows fromG_rs = φ^5andℏ_rs = φ^{-5}).
Substituting these into the definition yields sqrt(1 * 1 / 1^3) = sqrt(1) = 1, which is discharged by sqrt_one. This establishes that the RS-native Planck length coincides with the fundamental length scale ℓ₀ = 1.