Foundation Multiplicative Recognizer L4 Multiplicative Recognizer Satisfies L4

The composition law

In mathematics, a functional equation is a rule that pins down a function by relating its values at different inputs. The d'Alembert equation, F(x+y) + F(x-y) = 2F(x)F(y), is a classical example; its continuous solutions are the familiar cosine and hyperbolic cosine functions. Recognition Science studies a multiplicative analogue, where the operation is multiplication instead of addition, and the equation takes the form F(xy) + F(x/y) = P(F(x), F(y)) for some two-variable function P. This is the composition consistency law, labeled (L4) in the framework's internal numbering.

The declaration multiplicativeRecognizer_satisfies_L4 establishes a conditional theorem: if a recognizer (a device that assigns a cost to comparing two events) uses the positive real numbers as its event space, and its comparison operator is continuous and satisfies the Law of Logic (a set of axioms including identity, symmetry, and scale invariance), then the derived cost function automatically satisfies the multiplicative form of (L4). The theorem provides a polynomial combiner P of degree at most two, meaning the composition law holds with P(u,v) = a + bu + cv + duv + eu² + fv² for some real coefficients a through f. This is a proved result in the framework's machine-checked library of formal theorems.

The importance is that (L4) was previously treated as a substantive hypothesis, an extra assumption a recognizer had to be given. This theorem shows that, under the multiplicative-structure condition, the hypothesis is redundant: it follows from the recognizer's basic setup. The framework's summary states that the geometric primitive (the recognizer) plus the cost-functional primitive (the Law of Logic on positive ratios) jointly force the composition law. The theorem also comes with companion results: the derived cost vanishes at the multiplicative identity (cost(1) = 0) and is symmetric under reciprocation (cost(x) = cost(1/x)).

What the declaration does not claim is broader. It does not prove that every recognizer satisfies (L4); in fact, the equality-induced cost on the positive reals under multiplication provably fails the condition. The theorem is scoped to recognizers paired with a continuous Law-of-Logic-satisfying comparator. It also does not derive the full Law of Logic from the recognizer alone; the comparator's axioms are assumed, not derived. The framework itself notes that the abstract claim "every recognizer satisfies (L4)" is false, and the honest conditional is what the theorem delivers.

In practical terms, the result means that when working with multiplicative event spaces, one less assumption needs to be checked. A researcher building a recognizer on positive ratios with a well-behaved comparator gets the composition law for free, and the polynomial form gives an explicit template for the combiner. This closes a frontier identified in earlier framework work, shifting (L4) from an assumed hypothesis to a derived consequence for this important case.

/-- **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

/-- **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⟩

/-- **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]

What this page does not claim

Every recognizer satisfies (L4); the equality-induced cost on positive reals is a counterexample. The Law of Logic axioms are derived from the recognizer; they are assumed as part of the structure. The full Law of Logic is derived from the recognizer alone; the comparator's axioms are assumed, not derived.

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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