Foundation Dalembert Ultimate Is Symmetric Comparison

Symmetric comparison

In the Recognition Science framework, a cost function measures the price of a deviation. The declaration IsSymmetricComparison pins down one property of such a function: comparing x to 1 costs the same as comparing 1 to x. Formally, for any positive real number x, the cost F(x) equals F(1/x). This is the framework's definition of symmetric comparison, not an assumption about the world; it is what the word comparison means when the two sides of a ratio are interchangeable.

The declaration is one of three primitive requirements in the framework's foundational module. The other two are normalization, meaning no deviation costs zero, written F(1) = 0, and multiplicative consistency, meaning the cost of a product and a quotient combine through some fixed rule. Together these three form the minimal definition of a cost of comparison. The framework proves that any smooth, normalized, symmetric cost function with multiplicative consistency must equal J(x) = (x + 1/x)/2 - 1, and that its combination rule must be P(u,v) = 2uv + 2u + 2v. This is the theorem ultimate_inevitability in the machine-checked library of formal theorems.

The declaration itself does not establish uniqueness. It only states the symmetry condition. The proof that symmetry, normalization, and consistency force the unique cost function J comes from the larger theorem, not from IsSymmetricComparison alone. The framework also shows symmetry is essential: without it, comparison becomes directional, as the counterexample F(x) = x - 1 demonstrates. Dropping symmetry leaves room for cost functions that treat the two sides of a ratio differently, which the framework regards as not comparison at all.

What the declaration does not claim is broader than what it asserts. It does not define the full cost function, does not specify the combination rule, and does not by itself rule out alternative cost structures. Those results belong to the surrounding theorems in the same module. The declaration is a single, precise building block: the formal statement that a cost of comparison treats reciprocal ratios alike.

/-- Symmetry: F(x) = F(1/x). This is the DEFINITION of symmetric comparison. -/
def IsSymmetricComparison (F : ℝ → ℝ) : Prop :=
  ∀ x : ℝ, 0 < x → F x = F x⁻¹

/-- **THEOREM (Ultimate Inevitability)**

The three primitive requirements (symmetry, normalization, consistency)
plus regularity (smoothness, calibration) uniquely determine:
1. F = J
2. P = the RCL

There is no weaker foundation that still defines "cost of comparison."
-/
theorem ultimate_inevitability :
    -- The primitive requirements
    IsSymmetricComparison Cost.Jcost ∧
    IsNormalizedCost Cost.Jcost ∧
    HasMultiplicativeConsistency Cost.Jcost ∧
    -- The consequences (all proved)
    (∀ x : ℝ, 0 < x → Cost.Jcost x = (x + x⁻¹) / 2 - 1) ∧
    (∀ P : ℝ → ℝ → ℝ,
      (∀ x y, 0 < x → 0 < y → Cost.Jcost (x*y) + Cost.Jcost (x/y) = P (Cost.Jcost x) (Cost.Jcost y)) →
      ∀ u v, 0 ≤ u → 0 ≤ v → P u v = 2*u*v + 2*u + 2*v) := by
  refine ⟨?_, ?_, ?_, ?_, ?_⟩
  -- Symmetry
  · intro x hx
    show Cost.Jcost x = Cost.Jcost x⁻¹
    unfold Cost.Jcost
    rw [inv_inv]
    ring
  -- Normalization
  · show Cost.Jcost 1 = 0
    unfold Cost.Jcost
    norm_num
  -- Consistency (existence of P)
  · use fun u v => 2*u*v + 2*u + 2*v
    intro x y hx hy
    exact J_computes_P x y hx hy
  -- F = J (definitional)
  · intro x _
    simp only [Cost.Jcost]
  -- P uniqueness (from Unconditional)
  · exact rcl_unconditional

/-- Symmetry is NOT negotiable: without it, comparison is directional. -/
theorem symmetry_is_essential :
    ¬ IsSymmetricComparison (fun x => x - 1) := by
  intro h
  have := h 2 (by norm_num : (0 : ℝ) < 2)
  norm_num at this

What this page does not claim

IsSymmetricComparison alone does not define the cost function J. The declaration does not specify the combination rule P. The declaration does not rule out alternative cost structures by itself.

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

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