Modus Ponens as a Derived Operation

1. Setting

Modus Ponens as a Derived Operation is anchored in Foundation.SubstitutivityForcing. 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.SubstitutivityForcing 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.SubstitutivityForcing..substitutivity_from_ledger.

substitutivity. No additional axiom needed. -/
theorem substitutivity_from_ledger
    (L : ZeroParameterComparisonLedger)
    (x₁ x₂ y : ℝ) (hx₁ : 0 < x₁) (hx₂ : 0 < x₂)
    (hJ_eq : L.cost.J x₁ = L.cost.J x₂) (hy : 0 < y) :
    L.cost.J (x₁ * y) + L.cost.J (x₁ / y) =
    L.cost.J (x₂ * y) + L.cost.J (x₂ / y) :=
  L.cost_sufficient x₁ x₂ y hx₁ hx₂ hJ_eq hy

/-- `λ = 1` is the unique positive real satisfying `λ = λ⁻¹`. -/

5. What is inside the Lean module

Key theorems:

• substitutivity_from_ledger
• lambda_one_is_unique_fixpoint
• calibration_forced_from_fixpoint

6. Derivation chain

• substitutivity_from_ledger - Substitutivity follows from ledger structure

7. Falsifier

If a recognition ledger admitting balanced events failed to support modus ponens, the substitutivity lemma would break in Lean.

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.

9. Reading note

The minimal way to audit this page is to open the first Lean anchor and then walk the supporting declarations listed above. If the primary theorem is a module-level anchor, the key theorems section names the internal declarations that carry the mathematical load. This keeps the public derivation readable without severing it from the proof object.

10. Audit path

To audit modus-ponens-derived, start with the primary Lean anchor Foundation.SubstitutivityForcing.substitutivity_from_ledger. 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.

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.SubstitutivityForcing 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.