composition_from_continuity
plain-language theorem explainer
Continuity of J on the positive reals ensures that J(xy) + J(x/y) evaluates to a finite real for any positive x and y. Ledger reconstruction work in the Closed Observable Framework cites this as the theorem form of R6. The proof is a one-line term construction that directly witnesses the existential quantifier with the sum itself.
Claim. Let $J : (0,∞) → ℝ$ be continuous. For all $x > 0$ and $y > 0$, there exists $v ∈ ℝ$ such that $J(xy) + J(x/y) = v$.
background
The Closed Observable Framework encodes positive-valued observables together with a ratio interface and conservation as structure fields. This module absorbs R1, R2, R5 and R6 as definitions rather than axioms, leaving only the Regularity Axiom to capture the finite-description content of C3. The local setting is Gap 1 of the axiom-closure plan, phases 1, 2 and 6.
proof idea
The proof is a term-mode one-liner that constructs the witness for the existential as the concrete sum J(x*y) + J(x/y) and closes the equality by reflexivity.
why it matters
The declaration turns R6 into a theorem inside the Closed Observable Framework, supplying the finiteness step required for ledger reconstruction. It aligns with the Recognition Composition Law by guaranteeing that the left-hand side is defined before any further identity is imposed. No downstream uses are recorded yet.
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