Foundation Ledger Composition To Jcost Jcost Composes Through Rcl Combiner

The fixed point of the ledger

The recognition cost J is defined by the formula J(x) = ½(x + x⁻¹) − 1, a measure of how far a positive number x is from 1. The theorem jcost_composesThrough_rclCombiner states that this cost obeys a specific identity: for any positive x and y, J(x·y) + J(x/y) equals 2·J(x)·J(y) + 2·J(x) + 2·J(y). This is a pure algebraic fact, verified by expanding the definition and simplifying.

The identity is not arbitrary. It says the cost of a product and the cost of a ratio combine through a fixed binary operation, called the RCL combiner, applied to the individual costs. The theorem proves that J itself is a fixed point of this operation: when you feed two costs into the combiner, you get the cost of the combined inputs. This is the non-vacuity result. It shows the entire ledger-composition setup is consistent, because the cost that the framework derives is exactly the one that satisfies its own composition law.

In Recognition Science, the ledger is a discrete record of recognition events. The composition law is the statement that the cost of two events, when combined, is governed by a binary law of the two individual costs. The theorem establishes that the specific cost J satisfies this law. It does not, by itself, prove that J is the only cost with this property, nor does it derive the composition law from more basic principles. Those are separate results in the framework's library.

The practical consequence is that the cost function is not an isolated formula. It coheres with the ledger structure that the framework postulates. The theorem closes a loop: the cost that emerges from the framework's axioms is the same cost that satisfies the composition law the framework requires. This is a consistency check, not a derivation of new physics.

/-- **`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

This theorem does not prove that J is the only cost satisfying the composition law. This theorem does not derive the composition law from the ledger; it assumes it. This theorem does not establish any empirical prediction about the physical world.

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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