Foundation Multiplicative Recognizer L4 Full Multiplicative Law Of Logic Cert In

The multiplicative certificate

The declaration fullMultiplicativeLawOfLogicCert_inhabited is a machine-checked theorem inside the Recognition Science framework's library of formal theorems. It states that a certain certificate object exists for every configuration type. The certificate bundles four separate results about a recognizer, a device that assigns a recognition cost to pairs of events: the cost vanishes at the identity, it is symmetric under reciprocation, and it satisfies a composition law in two forms, one abstract and one using a polynomial combiner of degree at most two.

The classical mathematics here is the d'Alembert functional equation. For positive real numbers x and y, the composition law reads F(xy) + F(x/y) = P(F(x), F(y)), where F is the derived cost and P is a symmetric polynomial of total degree at most two. This is the multiplicative form of route independence: the cost of comparing a product and a quotient equals a fixed polynomial function of the component costs. The framework proves that whenever a recognizer lands in the positive reals and its comparison operator satisfies the Law of Logic, a continuous operator obeying four Aristotelian conditions plus scale invariance, this equation holds automatically. The polynomial P is not chosen freely; it comes from the operator's route-independence field.

In Recognition Science, this result upgrades a previously open hypothesis. An earlier module had exposed composition consistency as a substantive assumption on recognizers. This theorem shows that assumption is unnecessary when the event space is the positive reals under multiplication and the comparator satisfies the Law of Logic. The certificate exists without any extra hypothesis beyond the multiplicative recognizer structure itself. The framework's summary states the substantive content was always in the comparator's compositional structure, not in the recognizer's set-theoretic shape.

The declaration does not claim that every recognizer satisfies composition consistency. That abstract claim is false: the equality-induced cost on positive reals refutes it. The theorem is conditional on the multiplicative event space and the Law-of-Logic comparator. It also does not derive the specific form of the cost function J(x) = (x + 1/x)/2 - 1; that uniqueness result lives elsewhere in the library. This certificate only packages the composition law for the multiplicative case, leaving the general recognizer question open.

theorem l4DerivableCert_inhabited (𝒞 : Type*) :
    Nonempty (L4DerivableCert 𝒞) :=
  ⟨l4DerivableCert 𝒞⟩

/-- **L4 is automatic in the polynomial form for any multiplicative recognizer.**

The route-independence field of `SatisfiesLawsOfLogic` already provides the
polynomial-degree-2 combiner satisfying the multiplicative L4. -/
theorem multiplicativeRecognizer_satisfies_L4_polynomial
    (m : MultiplicativeRecognizer 𝒞) :
    MultiplicativeL4Polynomial m := by
  obtain ⟨P, hpoly, hsymm, hroute⟩ := m.laws.route_independence
  refine ⟨P, hpoly, hsymm, ?_⟩
  intro x y hx hy
  exact hroute x y hx hy

/-- **The (L4) substantive hypothesis is derivable in the multiplicative case.**

This is the headline theorem: the route-independence condition that the
companion paper exposed as a hypothesis (`RecognizerComposition`) is in
fact a theorem under the multiplicative-event-space structure. -/
theorem L4_derivable_on_multiplicative_event_space
    (m : MultiplicativeRecognizer 𝒞) :
    ∃ P : ℝ → ℝ → ℝ,
      ∀ x y : ℝ, 0 < x → 0 < y →
        m.cost (x * y) + m.cost (x / y) = P (m.cost x) (m.cost y) :=
  multiplicativeRecognizer_satisfies_L4 m

What this page does not claim

The equality-induced cost on positive reals satisfies composition consistency. The specific cost function J(x) = (x + 1/x)/2 - 1 is derived in this module. Every recognizer, regardless of event space, satisfies composition consistency.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/MultiplicativeRecognizerL4.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: