Foundation Closed Observable Framework Composition From Continuity

What continuity alone buys

In mathematics, a function is continuous when small changes to its input produce small changes to its output. The declaration composition_from_continuity starts from that plain idea and draws a precise conclusion. For any function J defined on the positive real numbers, if J is continuous on that domain, then for any two positive numbers x and y, the sum J(xy) + J(x/y) is always a finite real number. The proof is immediate: the sum is a real number by construction, and continuity is not even needed to see it. The theorem's real content is that the expression is well-defined, not that it has any special value.

The statement belongs to a larger project within Recognition Science, a framework that models physical structure as a discrete record of events, called a ledger. The framework asks what a comparison function must look like if it is to measure mismatch between two observations. Earlier results in the same file force the function to be symmetric under swapping the two observations, so that J(x) equals J(1/x), and to assign zero cost when the two observations are identical, so that J(1) equals 0. The present theorem is the next step in that chain: it shows that a continuous comparison function can always be combined with itself, by adding the cost of a product and the cost of a quotient, without ever leaving the real numbers.

The theorem does not claim that the sum J(xy) + J(x/y) has any particular value, nor that it satisfies a functional equation. It does not assert that J is convex, differentiable, or calibrated. Those properties are tracked separately in the framework as explicit obligations, not derived from continuity. The declaration also does not prove that a continuous J exists that meets the symmetry and normalization conditions; it only states a conditional fact about any J that is continuous. In that sense the result is a small but necessary link: it guarantees that the composition operation is available, leaving the harder question of what the composition must equal to other theorems.

/-- **R6 as theorem**: Compositional closure follows from continuity.
If J is continuous on R_{>0}, then J(xy) + J(x/y) is finite. -/
theorem composition_from_continuity
    (J : ℝ → ℝ)
    (hJ_cont : ContinuousOn J (Set.Ioi 0))
    (x y : ℝ) (hx : 0 < x) (hy : 0 < y) :
    ∃ v : ℝ, J (x * y) + J (x / y) = v :=
  ⟨J (x * y) + J (x / y), rfl⟩

What this page does not claim

The theorem does not assign a value to the sum J(xy) + J(x/y). The theorem does not prove that a continuous J exists satisfying the symmetry and normalization conditions. The theorem does not derive convexity, calibration, or any functional equation from continuity alone.

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/ClosedObservableFramework.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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