Foundation Multiplicative Recognizer L4 L4 Derivable On Multiplicative Event Spa

The multiplicative composition law

A recognizer is a device that assigns a cost to each comparison it makes between events. A multiplicative recognizer is one whose events are positive real numbers, and whose cost is derived from a comparison operator that satisfies the Law of Logic, a set of conditions including symmetry and scale invariance. The declaration L4_derivable_on_multiplicative_event_space establishes that for any such recognizer, a composition law holds automatically: the cost of comparing a product plus the cost of comparing a quotient equals some polynomial function of the two individual costs. In symbols, for any positive x and y, cost(x·y) + cost(x/y) = P(cost(x), cost(y)), where P is a symmetric polynomial of degree at most two.

This is a conditional theorem, not a universal one. The abstract claim that every recognizer satisfies this composition law is false: the equality-induced cost, which assigns zero on the diagonal and a positive weight elsewhere, fails the law on the multiplicative event space. The theorem proves the law only when the recognizer is paired with a continuous comparison operator satisfying the Law of Logic. Under that pairing, the composition law is not an extra assumption but a derived consequence. The machine-checked library of formal theorems proves this for both the abstract existence form and the polynomial form.

The theorem also delivers companion results. The derived cost vanishes at the multiplicative identity, cost(1) = 0, and is symmetric under reciprocation, cost(x) = cost(1/x) for positive x. These follow from the comparator's Law of Logic certificate. The full certificate packages all these results together, showing that the geometric recognizer supplies the definitional conditions while the comparator supplies composition consistency.

What this does not claim is broader than what it does. It does not claim that composition consistency holds for recognizers on arbitrary event spaces, nor that the equality-induced cost satisfies the law, nor that the polynomial combiner P is unique. The theorem is scoped to the positive reals under multiplication with a continuous Law-of-Logic-satisfying comparator. The substantive content, the document notes, was always in the comparator's compositional structure, not in the recognizer's set-theoretic shape.

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

The polynomial form trivially gives the existence form. -/
theorem multiplicativeRecognizer_satisfies_L4
    (m : MultiplicativeRecognizer 𝒞) :
    MultiplicativeL4 m := by
  obtain ⟨P, _, _, hroute⟩ := multiplicativeRecognizer_satisfies_L4_polynomial m
  exact ⟨P, hroute⟩

/-- **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

/-- **(L1) Identity.** The derived cost vanishes at the multiplicative
identity. -/
theorem multiplicative_identity (m : MultiplicativeRecognizer 𝒞) :
    m.cost 1 = 0 := by
  show m.comparator 1 1 = 0
  exact m.laws.identity 1 (by norm_num)

/-- **(L2) Reciprocal symmetry.** The derived cost is symmetric under
reciprocation, a consequence of non-contradiction plus scale invariance. -/
theorem multiplicative_reciprocal_symmetry
    (m : MultiplicativeRecognizer 𝒞) :
    ∀ x : ℝ, 0 < x → m.cost x = m.cost (x⁻¹) := by
  intro x hx
  show m.comparator x 1 = m.comparator (x⁻¹) 1
  -- C(x, 1) = C(1, x) (non-contradiction) = C(x⁻¹, 1) (scale by x⁻¹)
  have hsymm : m.comparator x 1 = m.comparator 1 x :=
    m.laws.non_contradiction x 1 hx (by norm_num)
  have hxinv : (0 : ℝ) < x⁻¹ := inv_pos.mpr hx
  have hscale : m.comparator (x⁻¹ * x) (x⁻¹ * 1) = m.comparator x 1 :=
    m.laws.scale_invariant x 1 (x⁻¹) hx (by norm_num) hxinv
  -- (x⁻¹ * x) = 1 and (x⁻¹ * 1) = x⁻¹
  have hxx : x⁻¹ * x = 1 := inv_mul_cancel₀ (ne_of_gt hx)
  rw [hxx, mul_one] at hscale
  -- so C(1, x⁻¹) = C(x, 1)
  -- chain: C(x, 1) = C(1, x) (above), and C(1, x⁻¹) = C(x, 1) gives
  -- C(1, x) and C(1, x⁻¹) both equal C(x, 1)... use non-contradiction on x⁻¹
  have hsymm2 : m.comparator (x⁻¹) 1 = m.comparator 1 (x⁻¹) :=
    m.laws.non_contradiction (x⁻¹) 1 hxinv (by norm_num)
  rw [hsymm2, ← hscale]

What this page does not claim

The composition law holds for every recognizer on every event space. The equality-induced cost satisfies the composition law. The polynomial combiner P is unique. The theorem applies to recognizers whose event space is not the positive reals under multiplication.

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)

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