Recognition Event Cost is Nonnegative

1. Setting

Recognition Event Cost is Nonnegative is anchored in Foundation.RecognitionForcing. 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)

$$ J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y) $$

Recognition Composition Law.

3. Prediction or structural target

• Structural target: Foundation.RecognitionForcing 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 Foundation.RecognitionForcing..self_recognition_zero_cost.

/-- Recognition events with ratio = 1 are cost-free. -/
theorem self_recognition_zero_cost (e : LedgerForcing.RecognitionEvent) :
    e.ratio = 1 → recognition_cost e = 0 := by
  intro h
  simp only [recognition_cost, h, LedgerForcing.J]
  norm_num

/-- Non-trivial recognition has positive cost.
    Uses the fact that J(x) = (x + 1/x)/2 - 1 ≥ 0, with = 0 iff x = 1. -/
theorem nontrivial_recognition_positive_cost (e : LedgerForcing.RecognitionEvent)

5. What is inside the Lean module

Key theorems:

• self_recognition_zero_cost
• nontrivial_recognition_positive_cost
• recognition_is_cost_structure
• recognition_unique
• cost_minima_are_recognition
• global_minimum_is_self_recognition
• stability_forces_recognition
• recognition_necessary
• recognition_forcing_complete
• ledger_is_minimal_recognition_tracker
• cost_to_recognition_bridge

Key definitions:

• recognition_cost
• Observable
• ObservableExtractionMechanism
• RecognitionStructure
• recognition_from_extraction
• Configuration
• config_to_recognition
• JStableStructure

6. Derivation chain

• self_recognition_zero_cost - Self-recognition zero cost
• nontrivial_recognition_positive_cost - Nontrivial recognition positive cost
• cost_nonneg - Cost nonnegative

7. Falsifier

A negative-cost recognition event would refute cost_nonneg; a zero-cost nontrivial event would refute nontrivial_recognition_positive_cost.

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.

11. Why this belongs in the derivations corpus

The corpus is organized around load-bearing consequences, not around file names. This entry is included because Foundation.RecognitionForcing contributes a reusable theorem or definitional bridge that other pages can cite. Keeping the page public gives readers a stable URL, a JSON record, and a direct path into the Lean theorem page. If the entry becomes redundant with a stronger derivation later, the current slug should be retired rather than silently rewritten; the replacement should absorb its anchors and preserve the audit history.