RS Lambda_rec Matches the Planck Length

1. Setting

RS Lambda_rec Matches the Planck Length is anchored in Constants.PlanckScaleMatching. 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

• Structural target: Constants.PlanckScaleMatching must keep resolving in the Lean canon, and all downstream pages that cite this anchor must continue to type-check.

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.PlanckScaleMatching..planck_gate_identity.

    The Planck gate identity: π · ℏ · G = c³ · λ_rec². -/
theorem planck_gate_identity :
    Real.pi * hbar * G = c^3 * lambda_rec^2 := by
  unfold G lambda_rec hbar c ell0 cLagLock tau0 tick
  simp only [one_pow, mul_one]
  have hpi : Real.pi ≠ 0 := Real.pi_pos.ne'
  have hphi5 : phi ^ (-(5 : ℝ)) ≠ 0 := (Real.rpow_pos_of_pos phi_pos _).ne'
  field_simp [hpi, hphi5]

/-- Equivalent form: c³λ²/(πℏG) = 1. -/

5. What is inside the Lean module

Key theorems:

• J_eq_Jcost
• J_exp_eq_cosh
• J_bit_eq_cosh
• J_bit_pos
• J_bit_explicit
• J_bit_eq_phi_minus
• J_bit_bounds
• Q3_faces
• Q3_vertices
• J_curv_zero
• J_curv_nonneg
• lambda_rec_from_Jbit_pos

Key definitions:

• J
• J_bit_val
• cube_faces
• cube_vertices
• J_curv
• lambda_rec_from_Jbit
• solid_angle_per_octant
• num_octants

6. Derivation chain

• planck_gate_identity - Planck gate identity
• lambda_rec_over_ell_P - lambda_rec / ell_P
• one_over_sqrt_pi_approx - 1/sqrt(pi) approximation

7. Falsifier

A determination that lambda_rec / ell_P differs from the canonical RS bridge beyond the certified bound refutes planck_gate_identity.

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-scale-matching, start with the primary Lean anchor Constants.PlanckScaleMatching.planck_gate_identity. 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.