Foundation Ledger Comparison To Composition Has Multiplicative Consistency Iff E

The factorization condition

In the Recognition Science framework, a ledger is a discrete record of events, and a cost is a number assigned to a comparison between two observable states. The declaration hasMultiplicativeConsistency_iff_exists_composesThrough states a plain equivalence: a cost function F has the property called multiplicative consistency if and only if there exists some binary combining function P such that F(x·y) + F(x/y) = P(F x, F y) for all positive x and y. In other words, the symmetric combination of two costs can be expressed as a function of those two costs alone.

The declaration does not prove that such a combining function P exists for any particular cost. It only establishes that the two statements are logically interchangeable: asking for multiplicative consistency is the same as asking for the existence of a combiner. The proof is immediate from the definition, since multiplicative consistency is defined to be exactly this existential statement. The declaration is a reformulation, not a discovery of a new property.

What the declaration does not claim is that the combining function is unique, that it has any particular form, or that it exists at all for a given cost. It also does not claim that any cost satisfying the condition must be the specific function J(x) = (x + 1/x)/2 - 1. That forcing result is a separate theorem that requires additional hypotheses, such as the cost being reciprocal, normalized, calibrated, and continuous. The declaration merely unpacks the definition of multiplicative consistency into an equivalent existential form.

In Recognition Science, this equivalence matters because it shows that a seemingly analytic assumption, the existence of a factorization combiner, is nothing more than a well-definedness condition on the ledger. The framework's library proves this equivalence as a formal theorem, with zero axioms beyond the standard three. The practical consequence is that when a cost's symmetric combination depends only on the single-point costs, the factorization input is automatically satisfied, and no separate analytic hypothesis is needed.

/-- `HasMultiplicativeConsistency F` is definitionally `∃ P, CostComposesThrough F P`:
the factorization input is exactly "the cost composes through some combiner". -/
theorem hasMultiplicativeConsistency_iff_exists_composesThrough (F : ℝ → ℝ) :
    DAlembert.Ultimate.HasMultiplicativeConsistency F ↔
      ∃ P : ℝ → ℝ → ℝ, CostComposesThrough F P :=
  Iff.rfl

What this page does not claim

The declaration does not prove that any particular cost function has multiplicative consistency. The declaration does not claim the combining function P is unique or has a specific form. The declaration does not imply that a cost with multiplicative consistency must equal the J function.

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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