Foundation Dalembert Inevitability F Div Swap Of P Symmetric

The swap lemma

The d'Alembert equation is a classical functional equation that appears when a quantity's value at a product and at a ratio are tied to its values at the factors. In the Recognition Science framework, a cost is a function that measures the price of moving away from unity, and the framework's central question is which forms that cost can take. The declaration F_div_swap_of_P_symmetric is a proved lemma inside that investigation, and it establishes a precise implication: if the way two costs combine is symmetric, then the cost of a ratio equals the cost of its reciprocal.

More concretely, suppose a cost function F satisfies the consistency rule F(xy) + F(x/y) = P(F(x), F(y)) for some polynomial P. The lemma says that if P is symmetric, meaning P(u, v) = P(v, u) for all u and v, then F(x/y) = F(y/x) for all positive x and y. The proof is short and elegant: write the consistency rule for the pair (x, y) and for the pair (y, x), note that the first term on the left is the same in both because multiplication commutes, and use the symmetry of P to see that the right sides match. Subtracting the two equations leaves F(x/y) = F(y/x).

This lemma is the first link in a chain. A direct corollary, also proved, is that a cost function consistent with a symmetric combiner must itself be reciprocal-symmetric: F(x) = F(1/x) for all positive x. That symmetry is one of the five plain conditions from which the framework derives its unique cost function J(x) = (x + 1/x)/2 - 1. The lemma therefore does real work: it shows that a seemingly mild assumption about the combiner, symmetry in its two arguments, already forces a structural property of the cost itself, before any normalization or calibration is imposed.

The lemma is a theorem in the framework's machine-checked library of formal theorems, meaning its proof has been verified by a computer down to the axioms. It is not a claim about the physical world by itself. It does not say that any particular cost function exists, that the polynomial P must be of any specific form, or that the constant c in the eventual bilinear family equals 2. Those are separate results, established by other declarations in the same module. The lemma only says: given consistency and a symmetric P, the swap identity follows.

What this changes for a reader is the shape of the argument. The framework does not need to assume reciprocal symmetry of the cost as a separate axiom; it falls out of a more basic assumption about the combiner. That is the kind of reduction that makes the framework's uniqueness theorem persuasive, because it shows the cost's symmetry is not an extra choice but a consequence of a more elementary consistency condition.

/-- If the combiner `P` is symmetric and `F` is multiplicatively consistent with `P`,
then `F(x/y) = F(y/x)` for all `x,y>0`. -/
theorem F_div_swap_of_P_symmetric (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ)
    (hCons : HasMultiplicativeConsistency F P)
    (hSymP : ∀ u v, P u v = P v u) :
    ∀ x y : ℝ, 0 < x → 0 < y → F (x / y) = F (y / x) := by
  intro x y hx hy
  have hxy := hCons x y hx hy
  have hyx := hCons y x hy hx
  have hyx' : F (x * y) + F (y / x) = P (F x) (F y) := by
    -- rewrite y*x = x*y and use symmetry of P on the RHS
    simpa [mul_comm, hSymP (F y) (F x)] using hyx
  -- Compare the two consistency equations (same LHS first term and same RHS)
  linarith [hxy, hyx']

/-- If the combiner `P` is symmetric and `F` is multiplicatively consistent with `P`,
then `F` is reciprocal-symmetric: `F(x) = F(1/x)` for all `x>0`. -/
theorem F_symmetric_of_P_symmetric (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ)
    (hCons : HasMultiplicativeConsistency F P)
    (hSymP : ∀ u v, P u v = P v u) :
    IsSymmetric F := by
  intro x hx
  have h := F_div_swap_of_P_symmetric F P hCons hSymP x 1 hx one_pos
  simpa [div_one] using h

What this page does not claim

The lemma does not prove that any cost function exists. The lemma does not force the specific bilinear form with c = 2. The lemma does not by itself establish the uniqueness of the cost function J.

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

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