Foundation Multiplicative Recognizer L4

The multiplicative recognizer

A recognizer is a device that assigns a cost to distinguishing between two events; the cost measures how hard it is to tell them apart. The ledger is the discrete record of such recognition events. In Recognition Science, a multiplicative recognizer is a recognizer whose events are positive real numbers, compared by multiplication and division. Its cost function is derived from a comparison operator that satisfies the Law of Logic: four Aristotelian conditions plus scale invariance and non-triviality.

The headline result concerns (L4), a condition called Composition Consistency. In its multiplicative form, (L4) states that the cost of comparing a product plus the cost of comparing a quotient equals a fixed polynomial function of the two individual costs. Written symbolically: F(xy) + F(x/y) = P(F(x), F(y)), where F is the derived cost and P is a polynomial of degree at most two. The theorem proves that every multiplicative recognizer satisfies this condition automatically, with no extra hypothesis needed.

This matters because (L4) was previously treated as a substantive assumption. The companion paper had exposed it as a hypothesis, but this result shows it is actually a theorem whenever the event space is the positive reals under multiplication and the comparator satisfies the Law of Logic. The proof is direct: the route-independence field of the comparator already supplies the polynomial combiner. The result also derives the three definitional conditions (L1), (L2), and (L3): the cost vanishes at 1, is symmetric under reciprocation, and obeys the identity law.

In Recognition Science, this result closes a frontier. The substantive content of composition consistency was always in the comparator's compositional structure, not in the recognizer's set-theoretic shape. The framework's machine-checked library of formal theorems certifies this derivation, providing a single certificate that packages the L1/L2/L3 conditions from the recognizer and the L4 condition from the comparator.

The honest scope is precise: this theorem holds for the multiplicative event space specifically. The abstract claim that every recognizer satisfies (L4) is false. The equality-induced cost on the positive reals refutes it. What is proved is the conditional: with the right cost on the right carrier, composition consistency is automatic.

/-- **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⟩

/-- **(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]

/-- **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⟩

What this page does not claim

The abstract claim that every recognizer satisfies (L4) is false. The module does not derive the specific form of the polynomial P. This result does not extend to recognizers on additive event spaces.

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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