Planck Length in RS Units

1. Setting

Planck Length in RS Units is anchored in Constants.Derivation. The page is not a loose explainer: it is a public map from the Recognition Science forcing chain into one Lean-checked declaration bundle. The primary anchor determines what is proved, and the surrounding declarations show how the result is used.

2. Equations

(E1)

$$ J(x)=\frac12(x+x^{-1})-1,\qquad \varphi^2=\varphi+1 $$

Shared constant-forcing backbone.

3. Prediction or structural target

• Planck length: predicted phi-derived (voxel); empirical 1.616255e-35 m. Source: Computed from RS-native G, hbar, c

This page is currently a structural derivation. Where the claim has direct empirical content, the prediction table gives the measurable target; otherwise the claim is a formal bridge inside the Lean canon.

4. Formal anchor

The primary anchor is Constants.Derivation..planck_length.

def planck_length : ℝ := sqrt (hbar_codata * G_codata / c_codata ^ 3)
def planck_time : ℝ := sqrt (hbar_codata * G_codata / c_codata ^ 5)
def planck_mass : ℝ := sqrt (hbar_codata * c_codata / G_codata)

lemma planck_length_pos : 0 < planck_length := by
  unfold planck_length
  exact sqrt_pos.mpr (div_pos (mul_pos hbar_codata_pos G_codata_pos) (pow_pos c_codata_pos 3))

lemma planck_time_pos : 0 < planck_time := by

5. What is inside the Lean module

Key theorems:

• c_codata_pos
• hbar_codata_pos
• G_codata_pos
• c_codata_ne_zero
• hbar_codata_ne_zero
• G_codata_ne_zero
• tau0_pos
• tau0_ne_zero
• inner_pos
• inner_nonneg
• tau0_sq_eq
• ell0_pos

Key definitions:

• c_codata
• hbar_codata
• G_codata
• tau0
• ell0
• RSUnitSystem
• canonicalUnits
• c_derived

6. Derivation chain

• planck_length - Planck length definition
• planck_length_pos - Planck length positive
• planck_gate_identity - Planck gate identity
• planck_gate_normalized - Planck gate normalized

7. Falsifier

A precision measurement outside the stated RS interval, after checking SI calibration and systematic error, refutes this constant-level derivation.

8. Where this derivation stops

Below this page the chain reduces to the RS forcing sequence: J-cost uniqueness, phi forcing, the eight-tick cycle, and the D=3 recognition substrate. If any upstream theorem changes, this page must be versioned rather than patched silently. The published URL is stable, but the version field is the contract.

10. Audit path

To audit planck-length-from-rs, start with the primary Lean anchor Constants.Derivation.planck_length. Then inspect the theorem names listed in the module-content section. The page is intentionally built so the public explanation is not a substitute for the proof object; it is a map into it. The mathematical dependency is the same in every case: reciprocal cost fixes J, J fixes the phi-ladder, the eight-tick cycle fixes the recognition clock, and the domain theorem listed above supplies the last step. If that last step is empirical, the falsifier section names what observation would break it. If that last step is formal, a Lean-checkable counterexample is the relevant failure mode.