Recognition Length lambda_rec

1. Setting

The recognition length is the positive root of the balance equation tying curvature cost to ledger closure. The point of the Lean module is not just a number; it proves uniqueness of the positive balanced root.

2. Equations

(E1)

$$ \lambda_{\mathrm{rec}}>0,\qquad \mathrm{balance}(\lambda_{\mathrm{rec}})=0 $$

Recognition length as unique balance root.

3. Prediction or structural target

• lambda_rec: predicted unique positive root (length); empirical Planck-scale bridge. Source: RS Planck-scale matching

This entry is one of the marquee derivations. The numerical or formal target is explicit, and the falsifier identifies the failure mode.

4. Formal anchor

The primary anchor is Constants.LambdaRecDerivation..lambda_rec_is_root.

theorem lambda_rec_is_root : K lambda_rec = 0 := by
  unfold K lambda_rec ell0
  simp only [one_pow, div_one]
  ring

theorem lambda_rec_unique_root (lambda : ℝ) (hlambda : lambda > 0) :
    K lambda = 0 ↔ lambda = lambda_rec := by
  unfold K lambda_rec ell0
  simp only [one_pow, div_one]

5. What is inside the Lean module

Key theorems:

• lambda_0_pos
• lambda_0_sq
• balance_at_lambda_0
• balance_unique_positive_root
• lambda_rec_is_root
• lambda_rec_unique_root
• lambda_rec_is_forced
• angular_deficit_value
• total_curvature_gauss_bonnet
• kappa_normalized_eq_one
• J_curv_derivation
• balance_determines_lambda

Key definitions:

• J_bit_normalized
• J_curv
• totalCost
• balanceResidual
• lambda_0
• K
• Q3_vertices
• Q3_faces

6. Derivation chain

• lambda_rec_is_root - lambda_rec is a balanced root
• lambda_rec_unique_root - Unique positive root
• lambda_rec_is_forced - Forced by balance
• total_curvature_gauss_bonnet - Total curvature (Gauss-Bonnet)
• G_derivation_chain_complete - G derivation chain complete

7. Falsifier

A second positive recognition-length root, or a failure of the positive root to satisfy the curvature balance equation, refutes the 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 lambda-rec-derivation, start with the primary Lean anchor Constants.LambdaRecDerivation.lambda_rec_is_root. 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.