Foundation Ledger Comparison To Composition Jcost Combination Cost Determined

The cost-determined condition

In Recognition Science, a ledger (a discrete record of events) assigns a strictly positive observable value to each state. The comparison between two states is the ratio of their observables, a positive number. A cost function J assigns a nonnegative number to each such ratio, measuring the recognition effort. The declaration jcost_combinationCostDetermined establishes that for the specific cost function J(x) = (x + 1/x)/2 - 1, the symmetric combination J(x·y) + J(x/y) depends only on the pair of single-point costs (J(x), J(y)), never on the particular ratio values x and y themselves. In plainer terms: if two different ratio pairs produce the same individual costs, their symmetric combinations are equal.

This property, called cost-determined, is a well-definedness condition. It means the combination is a function of the costs alone, not of the underlying ratios. The theorem proves this is equivalent to the existence of a binary combiner P such that J(x·y) + J(x/y) = P(J(x), J(y)). That equivalence is the d'Alembert factorization input, a standard functional equation form. The declaration demonstrates non-vacuity: the condition is not empty, because J itself satisfies it. The proof constructs the combiner explicitly as rclCombiner, showing J composes through it.

The significance is that the factorization combiner is not an arbitrary analytic assumption. It exists exactly when the symmetric combination is cost-determined, a condition native to the ledger. Combined with other results, a comparison cost that is reciprocal, normalized, calibrated, and continuous, whose combination is cost-determined through a ledger-posting combiner, is forced to equal J. The declaration does not claim that J is the only cost function satisfying the condition, nor does it claim the condition holds for arbitrary cost functions. It also does not claim that the combiner P is unique; the theorem asserts existence, not uniqueness.

/-- **`J`'s combination is cost-determined.**  Since `J` composes through
`rclCombiner`, its symmetric combination is a function of the single-point costs,
so the factorization-existence condition is non-vacuous. -/
theorem jcost_combinationCostDetermined :
    CombinationCostDetermined Cost.Jcost :=
  (hasMultiplicativeConsistency_iff_costDetermined Cost.Jcost).mp
    ⟨rclCombiner, jcost_composesThrough_rclCombiner⟩

/-- **Factorization existence ⇔ cost-determined combination.**  A comparison cost
admits a binary combiner `P` with `F(x·y) + F(x/y) = P(F x, F y)` (the d'Alembert
`HasMultiplicativeConsistency` factorization input) **iff** its symmetric
combination is cost-determined.  So the combiner is not an arbitrary analytic
assumption: it exists exactly when the combination is a function of the costs
alone.  The reverse direction builds `P` by choosing, for each cost pair in the
image, a witnessing ratio pair; cost-determinedness makes the choice irrelevant. -/
theorem hasMultiplicativeConsistency_iff_costDetermined (F : ℝ → ℝ) :
    DAlembert.Ultimate.HasMultiplicativeConsistency F ↔ CombinationCostDetermined F := by
  constructor
  · rintro ⟨P, hP⟩ x₁ y₁ x₂ y₂ hx₁ hy₁ hx₂ hy₂ hFx hFy
    rw [hP x₁ y₁ hx₁ hy₁, hP x₂ y₂ hx₂ hy₂, hFx, hFy]
  · intro hdet
    classical
    refine ⟨fun u v =>
      if h : ∃ p : ℝ × ℝ, 0 < p.1 ∧ 0 < p.2 ∧ F p.1 = u ∧ F p.2 = v
      then F (h.choose.1 * h.choose.2) + F (h.choose.1 / h.choose.2)
      else 0, ?_⟩
    intro x y hx hy
    have hex : ∃ p : ℝ × ℝ, 0 < p.1 ∧ 0 < p.2 ∧ F p.1 = F x ∧ F p.2 = F y :=
      ⟨(x, y), hx, hy, rfl, rfl⟩
    show F (x * y) + F (x / y) =
      (if h : ∃ p : ℝ × ℝ, 0 < p.1 ∧ 0 < p.2 ∧ F p.1 = F x ∧ F p.2 = F y
        then F (h.choose.1 * h.choose.2) + F (h.choose.1 / h.choose.2)
        else 0)
    rw [dif_pos hex]
    obtain ⟨hp1, hp2, hpx, hpy⟩ := hex.choose_spec
    exact (hdet hex.choose.1 hex.choose.2 x y hp1 hp2 hx hy hpx hpy).symm

What this page does not claim

The theorem does not claim that J is the unique cost function satisfying the cost-determined condition. The theorem does not claim that the combiner P is unique for a given cost function. The theorem does not claim that the cost-determined condition holds for arbitrary cost functions, only for 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/LedgerComparisonToComposition.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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