Foundation Logic As Functional Equation Logic Non Contradiction L To Real

Transporting symmetry

A comparison operator takes two positive numbers and returns a third. In the Recognition Science framework, such an operator is a candidate for the ledger, a discrete record of recognition events, and one of the laws it must satisfy is non-contradiction: the result of comparing x with y equals the result of comparing y with x. The declaration nonContradictionL_to_real is a transport theorem. It states that if a comparison operator over the framework's recovered reals satisfies this symmetry condition, then its transported counterpart over ordinary real numbers satisfies the corresponding symmetry condition there.

The transport works through a bridge function that converts recovered reals to ordinary reals. The theorem's proof is a direct application of this bridge: it takes the symmetry hypothesis, applies the conversion to both sides of the equation, and concludes the real-valued symmetry. This is a structural preservation result, not a new discovery about the real numbers themselves. It says that a property stated in one formal setting remains true when moved to another, provided the bridge respects the relevant operations.

The significance is that it feeds into a larger uniqueness result. When all four laws (identity, non-contradiction, scale invariance, and non-triviality) hold for a recovered-real operator, the transport theorems together show the real-valued counterpart satisfies the same laws. The framework's library then applies its existing theorem for real numbers, which forces the derived cost function to have a specific polynomial form. The symmetry transport is one of the four pillars that make this forcing argument go through in the recovered-real setting.

What the declaration does not claim is equally important. It does not assert that non-contradiction alone forces the cost function; the uniqueness requires all four laws together. It does not say anything about the physical content of the recovered reals or about what the cost function means empirically. It is purely a formal bridge: a statement about how a symmetry condition behaves under a translation between two mathematical structures. The theorem's value is architectural, ensuring that a property verified in one setting is not lost when moving to another.

theorem nonContradictionL_to_real (C : ComparisonOperatorL) (h : NonContradictionL C) :
    LogicAsFunctionalEquation.NonContradiction (transportComparison C) := by
  intro x y hx hy
  unfold transportComparison
  have hxL : (0 : LogicReal) < fromReal x := by
    rw [lt_iff_toReal_lt, toReal_zero, toReal_fromReal]; exact hx
  have hyL : (0 : LogicReal) < fromReal y := by
    rw [lt_iff_toReal_lt, toReal_zero, toReal_fromReal]; exact hy
  exact congrArg toReal (h (fromReal x) (fromReal y) hxL hyL)

/-- Transport a recovered-real comparison operator to the existing real
comparison-operator surface. -/
def transportComparison (C : ComparisonOperatorL) :
    LogicAsFunctionalEquation.ComparisonOperator :=
  fun x y => toReal (C (fromReal x) (fromReal y))

/-- The recovered-real Law of Logic transports to the existing real theorem
surface. -/
theorem lawsL_to_real {C : ComparisonOperatorL} (h : SatisfiesLawsOfLogicL C) :
    LogicAsFunctionalEquation.SatisfiesLawsOfLogic (transportComparison C) :=
  h.transported_real_laws

/-- RCL is forced for recovered-real logic, by transport through the existing
real theorem. -/
theorem RCL_is_unique_functional_form_of_logicL
    (C : ComparisonOperatorL) (h : SatisfiesLawsOfLogicL C) :
    ∃ (P : ℝ → ℝ → ℝ) (c : ℝ),
      DAlembert.Inevitability.HasMultiplicativeConsistency
        (LogicAsFunctionalEquation.derivedCost (transportComparison C)) P ∧
      (∀ u v, P u v = 2*u + 2*v + c*u*v) :=
  LogicAsFunctionalEquation.RCL_is_unique_functional_form_of_logic
    (transportComparison C) (lawsL_to_real h)

What this page does not claim

Non-contradiction alone forces the cost function; all four laws are required. The theorem assigns any physical or empirical meaning to the recovered reals or the cost function. The transport theorem proves the uniqueness result for recovered reals independently; it relies on the existing real-number 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/LogicAsFunctionalEquationLogic.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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