Algebra Cost Algebra Defect Dist No Global Quasi Triangle

The local bound and its limit

The cost function J(x) = ½(x + x⁻¹) − 1 measures how far a positive number x sits from 1, the identity of multiplication. It is zero at x = 1, grows as x moves away in either direction, and treats x and 1/x alike: J(x) = J(1/x). This function is the primitive object of Recognition Science, where it is called the cost, a discrete record of how much recognition a comparison demands. The framework proves J is the unique function satisfying five plain conditions, a theorem checked in its machine-checked library of formal theorems.

For any two positive numbers x and y, the framework defines the defect distance as J(x/y), the cost of their ratio. This quantity behaves like a distance in one important way: it is zero exactly when x equals y, and it is symmetric because J(x/y) = J(y/x). But it is not a true metric, because it can violate the triangle inequality. The declaration defectDist_no_global_quasi_triangle establishes precisely this: there is no constant C such that J(x/z) ≤ C·(J(x/y) + J(y/z)) holds for all positive x, y, z. In plain language, the cost of comparing two far-apart values cannot always be bounded by the sum of costs along an intermediate stop.

The framework does prove a local version, defectDist_quasi_triangle_local: when the ratios x/y and y/z stay within a bounded range, a quasi-triangle bound does hold with a specific constant. This is the honest shape of the result: the defect distance is locally well-behaved, but its global failure is not an accident of the proof. The declaration shows the failure is forced by the algebraic structure itself, not by a poor choice of constant.

What the declaration does not claim is that the defect distance is useless as a global measure. It remains a valid pseudometric on the positive reals, and its local bound is enough for many comparisons. The global failure means only that the cost algebra does not embed into a metric space with the usual triangle inequality. This is a structural fact about the framework's primitive, not a defect in it.

/-- 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

/-- **Defect distance**: d(x,y) = J(x/y) measures the "cost of deviation"
    between two positive reals.

    Properties:
    - d(x,x) = 0 (identity)
    - d(x,y) = d(y,x) (symmetry, from J reciprocity)
    - d(x,y) ≥ 0 (non-negativity) -/
noncomputable def defectDist (x y : ℝ) : ℝ := J (x / y)

/-- **Defect distance**: d(x,y) = J(x/y) measures the "cost of deviation"
    between two positive reals.

    Properties:
    - d(x,x) = 0 (identity)
    - d(x,y) = d(y,x) (symmetry, from J reciprocity)
    - d(x,y) ≥ 0 (non-negativity) -/
noncomputable def defectDist (x y : ℝ) : ℝ := J (x / y)

/-- **Defect distance**: d(x,y) = J(x/y) measures the "cost of deviation"
    between two positive reals.

    Properties:
    - d(x,x) = 0 (identity)
    - d(x,y) = d(y,x) (symmetry, from J reciprocity)
    - d(x,y) ≥ 0 (non-negativity) -/
noncomputable def defectDist (x y : ℝ) : ℝ := J (x / y)

What this page does not claim

The defect distance is a metric on the positive reals; it is only a pseudometric, and the global triangle inequality fails. The local quasi-triangle bound holds for all positive x, y, z; it requires the ratios to stay within a bounded range. The global failure of the quasi-triangle inequality is a defect in the framework; it is a proved structural property of the cost 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/Algebra/CostAlgebra.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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