Algebra Cost Algebra Canonical Recognition Cost System Cost Inv

The cost invariant

The declaration canonicalRecognitionCostSystem_cost_inv establishes a symmetry property of the framework's central cost function. In plain terms, it proves that the cost of recognizing one positive number relative to another is the same whether you compare them in one order or the reversed order. If the cost of moving from a to b is some amount, then the cost of moving from b to a is exactly that same amount. This is a reciprocity or mirror symmetry: the recognition process does not care about direction.

The cost function itself, written J(x), is defined as J(x) = ½(x + 1/x) − 1 for any positive number x. The declaration's content follows from the more basic theorem J_reciprocal, which states that J(x) = J(1/x) for all x > 0. Since dividing by x is the same as multiplying by its reciprocal, the cost of a ratio and its inverse are equal. The framework's library, a machine-checked collection of formal theorems, records this as a proved result, not a conjecture.

What the declaration does not claim is broader. It does not assert that all costs are symmetric under every transformation, only under the reciprocal map. It does not say that the cost function is the only one with this property, nor that the reciprocal symmetry alone determines the form of J. Those are separate results. The declaration is a narrow algebraic fact within a larger theory, and its scope is exactly the equality of cost for a positive number and its reciprocal.

/-- **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

/-- The J-cost function: the unique cost satisfying the Recognition Composition Law.
    J(x) = ½(x + x⁻¹) − 1 -/
noncomputable def J (x : ℝ) : ℝ := Jcost x

What this page does not claim

The declaration does not claim that all costs are symmetric under every transformation. The declaration does not claim that the reciprocal symmetry alone determines the form of J. The declaration does not claim that J is the only cost function with this property.

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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Derived articles

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