Foundation Logic As Functional Equation Non Contradiction And Scale Imply Recipr

Two laws that force symmetry

A comparison operator is a rule that takes two positive quantities and returns a real number measuring the cost of comparing them. The framework Recognition Science studies such operators as the primitive structure from which physical laws are derived. Its machine-checked library of formal theorems proves that if a comparison operator satisfies two plain conditions, then it must treat the two inputs symmetrically: the cost of comparing x with y equals the cost of comparing y with x.

The first condition, non-contradiction, states that the comparison cost is symmetric in its arguments: C x y = C y x. The second, scale invariance, states that multiplying both inputs by the same positive factor leaves the cost unchanged: C (lam * x) (lam * y) = C x y. The theorem non_contradiction_and_scale_imply_reciprocal derives symmetry of the derived cost function, which is the cost of comparing any positive quantity with the fixed reference 1.

This is a small but load-bearing step in the framework's chain. The symmetry of the derived cost is one of the hypotheses needed for the later theorem that forces the unique cost function J(x) = (x + 1/x)/2 - 1. That uniqueness result, in turn, leads to the golden ratio and the framework's derivation of three spatial dimensions. The symmetry theorem is a lemma that makes the larger derivation possible.

What the declaration does not claim is just as important. It does not claim that non-contradiction and scale invariance alone determine the full cost function. Other conditions, such as route independence and calibration, are needed for that. It also does not claim that any actual physical comparison operator satisfies these laws; it states a conditional implication: if an operator satisfies them, then it is symmetric.

/-- A comparison operator on positive reals takes two positive quantities and
returns a real-valued cost of comparing them. The four Aristotelian
constraints below are the structural content of comparison being a
well-posed operation. -/
abbrev ComparisonOperator := ℝ → ℝ → ℝ

/-- **Translation lemma 2 (Non-contradiction + Scale invariance ⇒ Reciprocity)**:
If a comparison operator is single-valued under argument reordering and
depends only on ratios, then the derived cost function is invariant under
inversion of its argument: F(x) = F(1/x).

The chain of equalities:
  F(x) = C(x, 1)                       definition of derivedCost
       = C(1, x)                       non-contradiction
       = C(x⁻¹·1, x⁻¹·x)               scale invariance (multiply both args by x⁻¹)
       = C(x⁻¹, 1)                     simplify (x⁻¹·1 = x⁻¹, x⁻¹·x = 1)
       = F(x⁻¹)                        definition of derivedCost
-/
theorem non_contradiction_and_scale_imply_reciprocal
    (C : ComparisonOperator)
    (hNC : NonContradiction C)
    (hSI : ScaleInvariant C) :
    IsSymmetric (derivedCost C) := by
  intro x hx
  have hxinv : (0 : ℝ) < x⁻¹ := inv_pos.mpr hx
  have hx_ne : (x : ℝ) ≠ 0 := ne_of_gt hx
  -- Step 1: C(x, 1) = C(1, x) by non-contradiction.
  have h1 : C x 1 = C 1 x := hNC x 1 hx one_pos
  -- Step 2: scale invariance with x' = 1, y' = x, λ = x⁻¹ gives
  --   C(x⁻¹·1, x⁻¹·x) = C(1, x), so C(1, x) = C(x⁻¹·1, x⁻¹·x).
  have h2 : C 1 x = C (x⁻¹ * 1) (x⁻¹ * x) :=
    (hSI 1 x x⁻¹ one_pos hx hxinv).symm
  -- Step 3: simplify x⁻¹·1 = x⁻¹ and x⁻¹·x = 1.
  have h3 : C (x⁻¹ * 1) (x⁻¹ * x) = C x⁻¹ 1 := by
    rw [mul_one, inv_mul_cancel₀ hx_ne]
  show derivedCost C x = derivedCost C x⁻¹
  unfold derivedCost
  exact h1.trans (h2.trans h3)

/-- **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

Non-contradiction and scale invariance alone do not determine the full cost function; other conditions are required. The theorem does not assert that any actual physical comparison operator satisfies these laws, only that if one does, it is symmetric.

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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