Step-Preserving Maps and the Impossibility of ContinuedFraction Representation: The CPD Paradox

This version (May 2026) includes additional theorems and revisions compared to the March 2026 preprint. In standard ZFC, simple continued fractions (SCF) provide unique representations forreal numbers, while generalized continued fractions (GCF) admit richer parametrization. This paper investigates the relationship between these two systems under computability and constructivity constraints. Classical real number theory presupposes representational closure through completeness axioms, yet constructive representationprocesses reveal strict incompatibilities. The paper concludes by exploring foundational implications for real number definability theory and proposes open problemsregarding representational hierarchies.