Order Preservation Under Real Extension: Admissibility Conditions, Counterexamples, and the Distinction Between Iteration and Input Limits

This independent mathematical note examines the distinction between the existence of a strictly increasing function, the existence of a real extension, and the preservation of strict increase under extension. Through elementary constructions and counterexamples, the note establishes that every strictly increasing real-valued function on the positive integers admits a continuous, strictly increasing extension to the real line; that agreement on an originating domain does not determine monotonicity across all extensions; and that certain infinite ordered subsets obstruct strictly increasing real extensions. The analysis also distinguishes limits under repeated function iteration from limits obtained by varying the input. The mathematical results are independently established through their stated definitions, constructions, and proofs. The publication does not claim a new general extension theorem. The accompanying System Reference identifies the originating scientific records for the Structural Identification Method (SIM), Structural Stability Science (SSS), and Dimensional Human Field Theory (DHFT), including admissibility conditions, causal structure, and minimal stability geometry. The mathematical propositions do not modify or extend the originating scientific architecture. Foundational Source Records Tonucci, C. (2026). Dimensional Human Field Theory (DHFT): Architectural Identification of a Closed Explanatory System. Zenodo.https://doi.org/10.5281/zenodo.19075948 Tonucci, C. (2026). Structural Identification Method: Deriving Invariant Mechanics from Somatic Detection. Zenodo.DOI: 10.5281/zenodo.19337749 Tonucci, C. (2026). Admissibility Conditions: Valid System Description in Dimensional Human Field Theory (DHFT). Zenodo.DOI: 10.5281/zenodo.19572188 Technical Orientation & Source Architecture Structural Stability Science (SSS) — GitHub Wikihttps://github.com/Cara-Tonucci/Structural-Stability-Science/wiki