Foundation Logic As Functional Equation Logic Transport Comparison

The transport bridge

In mathematics, a transport result shows that a property proved in one setting also holds when the objects are moved into another setting through a structure-preserving map. The declaration transportComparison is exactly such a bridge. It takes a comparison operator defined on recovered reals, a number system built from the framework's recognition ledger, and moves it into the ordinary real numbers. The map works by converting each ordinary real into its recovered-real counterpart, applying the comparison there, and converting the result back. This is a definition, a choice of how to connect the two worlds, not a theorem about the physical world.

The point of building the bridge is to show that the laws of logic, once proved in one setting, carry over intact. The framework's library proves that if a comparison operator on recovered reals satisfies four structural conditions, identity, symmetry, scale invariance, and non-triviality, then its transported version satisfies the corresponding laws on ordinary reals. The proof is a chain of small lemmas, each verified by a machine-checked library of formal theorems. The final theorem states that any such operator must take a specific functional form, a forced shape that the framework derives from its axioms. This is a statement about the internal consistency of the framework's mathematics, not a claim about how human logic works in practice.

The declaration does not claim that recovered reals are the same as ordinary reals, nor that the transport map is an isomorphism. It does not claim that the structural conditions are the only possible laws of logic, only that within this framework they are sufficient. It also does not claim that the forced functional form applies to any comparison operator outside the framework's setting. The transport is a precise, limited tool: it extends a proof from one domain to another, and it leaves the question of what logic fundamentally is completely open.

What the bridge changes is practical. It lets the framework reuse a large body of verified results about ordinary reals without re-proving them from scratch. Instead of duplicating every theorem, the library can transport them. This is a significant economy of effort, and it is the kind of result that makes a formal system tractable. The reader can now see how a carefully built mathematical structure can connect to a familiar one, and what the limits of that connection are.

In Recognition Science, this transport is one step in a larger program. The framework models reality as a ledger of recognition events, and it derives physical constants from the cost of recognition. The transport result is a technical tool within that program, not a headline claim. It is the kind of careful, limited result that gives the larger program its credibility, because it shows that the framework's mathematics is precise enough to be connected to standard analysis.

/-- 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

The transport map is not an isomorphism between recovered reals and ordinary reals. The structural conditions are not claimed to be the only possible laws of logic. The forced functional form does not apply to comparison operators outside the framework's setting.

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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