Pith. sign in

Explain the theorem planck_length_eq_one from IndisputableMonolith.Foundation.ConstantDerivations.

Big AI job. Grok 4.3 reads the canon and writes a Lean-grounded derivation; usually 20 seconds to 2 minutes. Your answer will appear below.
confidence: high in recognition cached

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 definitions ℓ₀ = 1 and τ₀ = 1).
  • By G_ℏ_product, G_rs * ℏ_rs = 1 (which follows from G_rs = φ^5 and ℏ_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.

cited recognition theorems

recognition modules consulted

The Recognition library is at github.com/jonwashburn/shape-of-logic. The model is restricted to the supplied Lean source and instructed not to invent theorem names. Treat output as a starting point, not a verified proof.