Algebra Cost Algebra Canonical Recognition Cost System Domain

The domain of costs

The cost function J(x) = ½(x + x⁻¹) − 1 measures the cost of a recognition event, the discrete record of a comparison in the framework's ledger. The declaration canonicalRecognitionCostSystem_domain states that the positive domain, the set of inputs on which this cost is defined, is exactly the set of positive real numbers, written as Set.Ioi 0. It is a definitional identity: the theorem's proof is the reflexivity of equality, meaning the statement is true by construction of the domain itself.

This theorem does not claim that every positive real number is a physically realizable cost, nor does it say anything about the values the cost function takes. It only fixes the domain of discourse for the cost algebra: the multiplicative monoid of positive reals (ℝ₊, ·, 1) on which J acts as a pseudometric. The cost function itself is nonnegative, zero at the identity, and symmetric under reciprocals, as separate theorems in the same module establish. The domain theorem is the stage, not the play.

In the broader architecture of the framework, this domain statement is a small but load-bearing piece. It ensures that the cost algebra is built on the correct carrier set before the Recognition Composition Law, the equation J(xy) + J(x/y) = 2·J(x)·J(y) + 2·J(x) + 2·J(y), is imposed. The composition law, together with calibration and continuity, forces the unique form of J in a separate theorem. The domain theorem does no forcing itself; it merely declares where the forcing happens.

The theorem is a definitional reflexivity, so it carries no empirical content. It does not predict any measured value, does not derive the fine-structure constant, and does not assert that the cost function is defined on any larger set such as all real numbers or complex numbers. Its scope is narrow: the positive reals are the domain, and that is all.

/-- The canonical recognition cost system uses the positive reals as state space. -/
theorem canonicalRecognitionCostSystem_domain :
    PositiveDomain = Set.Ioi 0 := rfl

/-- **Non-negativity**: All costs are non-negative on ℝ₊. -/
theorem J_nonneg (x : ℝ) (hx : 0 < x) : 0 ≤ J x :=
  Jcost_nonneg hx

/-- **Normalization**: The multiplicative identity has zero cost. -/
theorem J_at_one : J 1 = 0 := Jcost_unit0

/-- **Reciprocal symmetry**: Cost is invariant under inversion.
    This is the algebraic encoding of "double-entry": every ratio x
    and its reciprocal 1/x carry the same cost. -/
theorem J_reciprocal (x : ℝ) (hx : 0 < x) : J x = J x⁻¹ :=
  Jcost_symm hx

/-- **Cost-composition**: The binary operation on costs induced by the RCL.
    Given two "cost levels" a = J(x) and b = J(y), the composed cost is:
    a ★ b = 2ab + 2a + 2b = 2(a+1)(b+1) − 2

    This captures how costs combine under multiplication of ratios. -/
noncomputable def costCompose (a b : ℝ) : ℝ := 2 * a * b + 2 * a + 2 * b

What this page does not claim

This theorem does not claim that every positive real number is a physically realizable cost. This theorem does not derive the fine-structure constant or any measured physical constant. This theorem does not assert the cost function is defined on all real numbers or complex numbers.

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

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