Foundation Ledger Composition To Jcost Satisfies Composition Law Of Composes Thr

The composition law

In the Recognition Science framework, a ledger is a discrete record of recognition events, and the cost of a recognition is a number assigned to each event. The framework asks how two costs combine when their events are composed, for instance by multiplying two quantities or dividing them. The answer is a specific equation: F(x·y) + F(x/y) = 2F(x)F(y) + 2F(x) + 2F(y), where F is the cost function and x and y are positive numbers.

This equation is not chosen freely. The framework's machine-checked library of formal theorems proves that any cost function satisfying five plain conditions must obey this composition law, and the law in turn forces the cost to be J(x) = (x + 1/x)/2 - 1. The declaration satisfiesCompositionLaw_of_composesThrough_rcl establishes one precise step: if a cost function composes through a specific two-point combiner, then it satisfies the composition law. The combiner, called rclCombiner, is defined as rclCombiner(u, v) = 2uv + 2u + 2v. The theorem states that composing costs through this combiner is exactly the same as satisfying the composition law.

The framework also proves that the cost J itself composes through this combiner, so the construction is not empty: J is a genuine fixed point of the whole setup. This means the composition law is consistent with the cost it forces. The declaration is a formal bridge: it connects the abstract idea of composability to the concrete equation, and it does so with no unproved assumptions in the framework's own logic.

What the declaration does not claim is broader. It does not claim that any cost function must compose through this combiner; that requires additional ledger-posting conditions. It does not claim that the composition law alone forces J; the full derivation needs the other four conditions (reciprocal symmetry, normalization, calibration, continuity). And it does not claim anything about the physical world, only about the mathematical structure of costs within the framework.

/-- If the cost composes through the RCL combiner, it satisfies the composition
law. -/
theorem satisfiesCompositionLaw_of_composesThrough_rcl (F : ℝ → ℝ)
    (h : CostComposesThrough F rclCombiner) :
    SatisfiesCompositionLaw F :=
  (satisfiesCompositionLaw_iff_rclCombiner F).mpr h

/-- **The composition law is the RCL combiner law on costs.**  `F` satisfies the
recognition composition law iff its symmetric two-point combination
`F (x·y) + F (x/y)` equals `rclCombiner (F x) (F y)`.  This is a pure
rearrangement: `rclCombiner u v = 2uv + 2u + 2v` is the composition-law RHS with
`u = F x`, `v = F y`. -/
theorem satisfiesCompositionLaw_iff_rclCombiner (F : ℝ → ℝ) :
    SatisfiesCompositionLaw F ↔
      ∀ x y : ℝ, 0 < x → 0 < y →
        F (x * y) + F (x / y) = rclCombiner (F x) (F y) := by
  unfold SatisfiesCompositionLaw rclCombiner
  constructor
  · intro h x y hx hy; rw [h x y hx hy]
  · intro h x y hx hy; rw [h x y hx hy]

/-- **`J` composes through the RCL combiner.**  The recognition cost
`J(x) = ½(x + x⁻¹) − 1` satisfies `J (x·y) + J (x/y) = rclCombiner (J x) (J y)`
for positive `x, y`.  This shows the ledger-composition setup is non-vacuous:
`J` is a fixed point of the composition law it forces. -/
theorem jcost_composesThrough_rclCombiner :
    CostComposesThrough Cost.Jcost rclCombiner := by
  intro x y hx hy
  have hx0 : x ≠ 0 := ne_of_gt hx
  have hy0 : y ≠ 0 := ne_of_gt hy
  unfold Cost.Jcost rclCombiner
  field_simp
  ring

What this page does not claim

It does not claim that any cost function must compose through the rclCombiner without additional ledger-posting conditions. It does not claim that the composition law alone forces the cost J, which requires all five conditions together. It does not claim any physical interpretation of the cost or the ledger.

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

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