Algebra Cost Algebra Defect Dist Quasi Triangle Local

A local bound on cost distance

The declaration defectDist_quasi_triangle_local is a proved theorem inside the Recognition Science framework's machine-checked library of formal theorems. It concerns the cost function, which assigns a nonnegative real number to any positive real input, with the convention that the cost of 1 is 0. The theorem establishes a local quasi-triangle bound: when two positive numbers x and y are both confined to a bounded interval away from zero, the cost of their ratio J(x/y) is bounded above by a constant multiple of the sum of the individual costs J(x) and J(y), plus a constant term. In plain terms, it says that the cost distance between two points cannot explode when the points themselves stay within a fixed, bounded region.

The bound is called "local" because it holds only under the hypothesis that the inputs lie in a bounded ratio range, not globally for all positive reals. The theorem is a formal consequence of the Recognition Composition Law, the single algebraic primitive of the framework, which states that J(xy) + J(x/y) = 2·J(x)·J(y) + 2·J(x) + 2·J(y). The declaration is part of the cost algebra module, which develops the algebraic structure of the cost function, including its reciprocal symmetry J(x) = J(1/x), its zero at unity, and the shifted operation A • B = 2AB that forms a commutative monoid on the interval [1/2, ∞).

The theorem does not claim that the cost function satisfies a global triangle inequality, nor does it assert that the bound is sharp or that it holds without the boundedness condition. It also does not establish that the cost function is a metric on the entire positive real line. The declaration is a technical lemma within a larger formal development, not a standalone physical law. Its role is to support later results in the cost algebra hierarchy, which in turn feed into the framework's derivation of the golden ratio, the eight-tick cycle, and three spatial dimensions.

For a reader outside the framework, the practical upshot is that the cost distance between two positive numbers, defined as J(x/y), behaves in a controlled way over bounded regions. This is a regularity property that makes the cost function amenable to further analysis, even though it is not globally Lipschitz or globally bounded. The theorem is one of several proved statements in the cost algebra module that together characterize the algebraic and analytic behavior of the cost function.

/-- **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 theorem does not assert a global triangle inequality for the cost function on all positive reals. The theorem does not claim that the bound is sharp or that it holds without the boundedness condition. The theorem does not establish that the cost function is a metric on the entire positive real line.

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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