Cost Jcost Logic Satisfies Composition Law L

The composition law

A recognition event is a discrete record of one thing being matched against another. The framework's central object is the cost of that event, a positive number measuring how expensive the match is. The declaration SatisfiesCompositionLawL states, in the framework's logic, that any legitimate cost function F must obey the equation F(x·y) + F(x/y) = 2F(x)F(y) + 2F(x) + 2F(y) for all positive x and y. This is the forced composition law: the cost of recognizing a product and a quotient together is completely determined by the costs of recognizing each factor separately.

The equation is not chosen freely. It is one of five plain conditions that any cost function must meet, and together those conditions force the unique form J(x) = (x + 1/x)/2 - 1. The declaration here is a transport mirror: it states the law on the framework's recovered reals, and the library proves that this statement reduces to the already-verified real-number version under the standard map. That reduction is a theorem in the machine-checked library of formal theorems, so the composition law holds in the recovered setting exactly because it holds in the real setting.

What the declaration does not claim is just as important. It does not assert that every function satisfying the law is a valid cost; the law is a necessary condition, not a sufficient one. It does not claim that the cost function is unique from this law alone; uniqueness requires all five conditions together. And it does not claim anything about the physical world: the composition law is a formal statement about the framework's cost function, and any connection to measured physics is a separate empirical check, not part of this declaration.

/-- Recognition Composition Law on recovered reals for a cost function. -/
def SatisfiesCompositionLawL (F : LogicReal → LogicReal) : Prop :=
  ∀ x y : LogicReal, (0 : LogicReal) < x → (0 : LogicReal) < y →
    F (x * y) + F (x / y)
      = fromReal 2 * F x * F y + fromReal 2 * F x + fromReal 2 * F y

/-- Transported RCL: a recovered-real composition law becomes the existing
real composition law under `toReal`. -/
theorem compositionLawL_to_real {F : LogicReal → LogicReal}
    (hF : SatisfiesCompositionLawL F) :
    Cost.FunctionalEquation.SatisfiesCompositionLaw (transportCost F) := by
  intro x y hx hy
  unfold transportCost
  have hxL : (0 : LogicReal) < fromReal x := by
    rw [lt_iff_toReal_lt, toReal_zero, toReal_fromReal]; exact hx
  have hyL : (0 : LogicReal) < fromReal y := by
    rw [lt_iff_toReal_lt, toReal_zero, toReal_fromReal]; exact hy
  have hxy : fromReal x * fromReal y = fromReal (x * y) := by
    rw [eq_iff_toReal_eq]
    simp [toReal_fromReal]
  have hdiv : fromReal x / fromReal y = fromReal (x / y) := by
    rw [eq_iff_toReal_eq]
    simp [toReal_fromReal]
  have hL := hF (fromReal x) (fromReal y) hxL hyL
  rw [hxy, hdiv] at hL
  have h := congrArg toReal hL
  simpa [toReal_add, toReal_mul, toReal_div, toReal_fromReal] using h

What this page does not claim

The composition law alone does not force the cost function to be J(x) = (x + 1/x)/2 - 1. The declaration does not assert that every function satisfying the law is a valid cost function. The declaration makes no claim about measured physical constants or experimental data.

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/Cost/JcostLogic.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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