Foundation Multiplicative Recognizer L4 Full Multiplicative Law Of Logic Cert

The multiplicative certificate

A recognizer is a device that assigns a cost to telling two events apart. In the Recognition Science framework, the ledger, a discrete record of such events, tracks these costs. The framework's central question is which laws of logic any such cost function must obey. One of those laws, called composition consistency, says that the cost of comparing a product plus the cost of comparing a quotient must equal some fixed function of the two individual costs. For a long time this law was treated as an extra assumption, a substantive hypothesis about how recognizers behave.

The declaration FullMultiplicativeLawOfLogicCert packages a machine-checked proof that this assumption is unnecessary in a specific, important case. When the events being compared are positive real numbers under multiplication, and when the comparison operator satisfies the framework's Law of Logic, then composition consistency follows automatically. The proof constructs a polynomial combiner of degree at most two that realizes the law exactly. In other words, the structure of the multiplicative event space plus the logical constraints on the comparator force the composition law to hold; it does not need to be postulated separately.

The certificate also bundles companion results: the derived cost vanishes at the multiplicative identity, and it is symmetric under reciprocation. These follow from the same logical conditions. The certificate exists for every configuration type, meaning it applies universally within the framework's setup. The practical upshot is that the framework's library can now cite a single object that assembles all these facts, rather than quoting separate results from different modules.

What the certificate does not claim is just as important as what it proves. It does not show that every recognizer satisfies composition consistency. The equality-induced cost on positive reals, which assigns zero on the diagonal and a positive weight elsewhere, fails the law. The theorem is conditional: it holds when the comparator satisfies the Law of Logic and the event space is multiplicative. The certificate also does not derive the specific form of the cost function itself, nor does it connect to the framework's broader claims about the golden ratio or three spatial dimensions. It is a scoped result about one consistency condition in one algebraic setting.

/-- **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

/-- **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

/-- **(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 certificate does not show that every recognizer satisfies composition consistency. The certificate does not derive the specific form of the cost function J(x) = (x + 1/x)/2 - 1. The certificate does not connect to the framework's broader claims about the golden ratio or three spatial dimensions.

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: