Foundation Logic As Functional Equation Law Of Logic Forces Recognition Composit

The forced composition law

A comparison operator, a function that assigns a real-valued cost to comparing any two positive quantities, can be thought of as a bookkeeping rule for a ledger, a discrete record of recognition events. The Recognition Science framework asks what such an operator must look like if it is to behave like logic itself. The answer, proved in its machine-checked library of formal theorems, is that six plain conditions force the comparison cost to combine according to one specific formula.

The six conditions are named after classical logical principles. Identity says comparing a quantity with itself costs nothing. Non-contradiction says comparing x with y costs the same as comparing y with x. Excluded middle, in this setting, is a continuity requirement: the cost varies smoothly as the quantities vary. Scale invariance says multiplying both quantities by the same factor leaves the cost unchanged. Route independence says the cost of a compound comparison does not depend on the order in which you combine its parts, and it is captured by a composition law written in symbols as derivedCost(x·y) + derivedCost(x/y) = P(derivedCost x, derivedCost y), where P is a symmetric polynomial of degree at most two. Non-triviality says the cost is not identically zero.

The theorem law_of_logic_forces_recognition_composition_law proves that any operator satisfying these six laws has a derived cost whose composition law is forced to be P(u, v) = 2u + 2v + c·u·v for some real constant c. This is the unique functional form of logic under the stated conditions. The result is a theorem in the framework's library, derived from the six laws without additional assumptions. A second theorem, law_of_logic_forces_canonical_cost, goes further: if the derived cost also satisfies a composition law and a calibration condition, then it must equal the canonical cost J(x) = (x + 1/x)/2 - 1.

What the declaration does not claim is broader necessity. It does not claim that every conceivable comparison operator must satisfy these six laws; they are hypotheses, not tautologies. It does not claim that the constant c is pinned down by logic alone; c remains free at this stage. It does not claim that the canonical cost J is the only possible cost function, only that it is the unique one under the added smoothness and calibration assumptions. The theorem establishes a conditional uniqueness: if you accept the six laws, the composition law is forced.

The payoff is that a large piece of structure follows from a small set of logical-looking axioms. The framework uses this forced composition law as a stepping stone: from it, the golden ratio emerges as a self-similar scaling, and from that an eight-tick recognition cycle and three spatial dimensions are derived in later theorems. For a reader, the immediate consequence is that the cost of recognition is not a free choice once the laws of logic are accepted; it is a derived object with a single undetermined parameter.

/-- **Law of Logic forces the Recognition Composition Law**:
the continuous positive-ratio Law of Logic forces the route-independence
combiner to have the RCL form `P(u,v) = 2u + 2v + c*u*v`. -/
theorem law_of_logic_forces_recognition_composition_law
    (C : ComparisonOperator) (hLogic : LawOfLogic C) :
    ∃ (P : ℝ → ℝ → ℝ) (c : ℝ),
      HasMultiplicativeConsistency (derivedCost C) P ∧
      (∀ u v, P u v = 2*u + 2*v + c*u*v) := by
  exact RCL_is_unique_functional_form_of_logic C hLogic

/-- Expanded formula for the continuous positive-ratio Law of Logic. -/
theorem lawOfLogic_iff (C : ComparisonOperator) :
    LawOfLogic C ↔
      Identity C ∧
      NonContradiction C ∧
      ExcludedMiddle C ∧
      ScaleInvariant C ∧
      RouteIndependence C ∧
      NonTrivial C := by
  constructor
  · intro h
    exact ⟨h.identity, h.non_contradiction, h.excluded_middle,
      h.scale_invariant, h.route_independence, h.non_trivial⟩
  · rintro ⟨hId, hNC, hEM, hSI, hRI, hNT⟩
    exact
      { identity := hId
        non_contradiction := hNC
        excluded_middle := hEM
        scale_invariant := hSI
        route_independence := hRI
        non_trivial := hNT }

/-- **Law of Logic forces the canonical cost**:
under the canonical RCL normalization and unit log-curvature calibration, the
unique continuous positive-ratio cost satisfying the Law of Logic is
`J(x) = ½(x + x⁻¹) - 1`. -/
theorem law_of_logic_forces_canonical_cost
    (C : ComparisonOperator) (hLogic : LawOfLogic C)
    [Cost.FunctionalEquation.AczelSmoothnessPackage]
    (hRCL : Cost.FunctionalEquation.SatisfiesCompositionLaw (derivedCost C))
    (hCalib : Cost.FunctionalEquation.IsCalibrated (derivedCost C)) :
    ∀ x : ℝ, 0 < x → derivedCost C x = Cost.Jcost x := by
  exact J_is_unique_cost_under_logic C hLogic hRCL hCalib

What this page does not claim

The six laws of logic are not claimed to be tautologies; they are hypotheses. The constant c is not claimed to be determined by logic alone. The canonical cost J is not claimed to be the only cost function without the added smoothness and calibration assumptions.

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/LogicAsFunctionalEquation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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