Foundation Multiplicative Recognizer L4 Multiplicative Recognizer Satisfies L4 P

The derived composition law

In the Recognition Science framework, a recognizer is a structure that assigns a cost to distinguishing one event from another; it is a discrete record of how hard a comparison is. A multiplicative recognizer is one whose event space is the positive real numbers, so events are compared by multiplication and division. The declaration multiplicativeRecognizer_satisfies_L4_polynomial proves that any such recognizer, when paired with a continuous comparison operator satisfying the Law of Logic, automatically satisfies the multiplicative form of the composition consistency condition (L4).

The condition in question is the d'Alembert form of route independence: for any positive x and y, the cost of comparing x times y with the cost of comparing x divided by y sums to a polynomial function of the two individual costs. The theorem establishes that a polynomial combiner of total degree at most two exists, is symmetric, and makes the equation hold for all positive inputs. This is not a hypothesis imposed on the recognizer; it is a derived consequence of the multiplicative structure and the comparator's laws.

The proof is short because the hard work lives in the comparator's route-independence field, which already supplies the polynomial combiner. The recognizer's geometric shape contributes nothing extra; the composition law is forced by the cost-functional structure on positive ratios. This closes a frontier identified in earlier work where (L4) was exposed as a substantive assumption. The theorem upgrades that assumption to a theorem, but only for the multiplicative event space.

The declaration does not claim that every recognizer satisfies (L4). That abstract claim is false: the equality-induced cost on the positive reals under multiplication refutes it. The theorem is conditional on the comparator satisfying the Law of Logic. It also does not derive the full cost function J(x) = (x + 1/x)/2 - 1; it only establishes the composition law in polynomial form. The exact form of the combiner is not pinned down beyond its degree and symmetry.

What the result changes is the status of (L4) in the multiplicative case. A condition that once had to be assumed can now be derived from the recognizer's structure and the comparator's laws. That makes the framework's logical foundation one step more automatic, and it gives the companion paper a single certificate to cite instead of separate pieces.

/-- **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

/-- A polynomial-degree-2 form of (L4): the combiner is a polynomial of
total degree at most two. This is the form the d'Alembert Inevitability
Theorem produces. -/
def MultiplicativeL4Polynomial (m : MultiplicativeRecognizer 𝒞) : Prop :=
  ∃ P : ℝ → ℝ → ℝ,
    (∃ a b c d e f : ℝ, ∀ u v, P u v = a + b*u + c*v + d*u*v + e*u^2 + f*v^2) ∧
    (∀ u v, P u v = P v u) ∧
    (∀ x y : ℝ, 0 < x → 0 < y →
      m.cost (x * y) + m.cost (x / y) = P (m.cost x) (m.cost y))

/-- **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

What this page does not claim

The abstract claim that every recognizer satisfies (L4) is false and is not claimed. The theorem does not derive the full cost function J(x) = (x + 1/x)/2 - 1. The polynomial combiner's exact coefficients are not determined by this theorem.

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.

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