Foundation Dalembert Ultimate Symmetry Is Essential

The indispensable symmetry

Symmetry is the property that comparing A to B is the same as comparing B to A. In the Recognition Science framework, a cost function (a rule that assigns a nonnegative number to the deviation between two states) must be symmetric: F(x) = F(1/x). The declaration symmetry_is_essential proves a minimal counterexample: the function F(x) = x - 1, which is not symmetric, fails the requirement. The proof is one line: for x = 2, F(2) = 1 but F(1/2) = -1/2, so the two sides differ. This is not a deep theorem; it is a sharp reminder that symmetry is not negotiable.

The declaration is part of a larger result called ultimate_inevitability. That theorem shows that any smooth, symmetric, normalized cost function with multiplicative consistency is uniquely the J function, J(x) = (x + 1/x)/2 - 1. The symmetry requirement is one of three primitive conditions, alongside normalization (F(1) = 0) and consistency (F(xy) + F(x/y) = P(F(x), F(y)) for some combiner P). The declaration symmetry_is_essential isolates the symmetry condition and shows that dropping it allows a function like x - 1, which is not a cost of comparison at all. The theorem is proved in a machine-checked library of formal theorems, so the reasoning is verified by a computer kernel.

What the declaration does not claim is broader. It does not claim that symmetry alone forces the J function; that requires the other conditions. It does not claim that the counterexample x - 1 is a meaningful cost function; it is only a demonstration that without symmetry, the defining property fails. The declaration does not assert that all comparison must be symmetric in a philosophical sense; it only states that within the framework's definition of cost, symmetry is a necessary component. The proof is a negative result: it shows what happens when symmetry is absent, not what happens when it is present.

The practical consequence is that any attempt to build a theory of comparison without symmetry will fail the framework's standard. This is a guardrail, not a discovery. It tells a reader that the framework's foundation is not a loose collection of assumptions but a tight set of requirements, each of which is essential. The declaration is a small but load-bearing piece of that foundation.

/-- 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

/-- **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

What this page does not claim

The declaration does not claim that symmetry alone forces the J function. The declaration does not claim that the counterexample x - 1 is a meaningful cost function. The declaration does not assert that all comparison must be symmetric in a philosophical sense.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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