Singular forcing and a counterexample to a coupled Caputo boundary-value existence theorem

This technical note examines the differential–integral equivalence and classical Caputo existence claim in Moumen et al., International Journal of Analysis and Applications 24 (2026), 286 (DOI: 10.28924/2291-8639-24-2026-286). An explicitly coupled constant-order example satisfies hypotheses (H1)–(H4), but its unique integral fixed point is not a solution of the classical Caputo differential system. Two analytic arguments establish nonexistence in the AC² domain, even for equations imposed almost everywhere. The note distinguishes this domain obstruction from the validity of the integral contraction estimate and discusses the limits of extending the inverse convention to singular forcings. Revised version 2 clarifies the domain qualification, displays an intermediate proof estimate, and expands the AI disclosure. The accompanying package includes the PDF, LaTeX source, a Python numerical consistency script, and its results. Author: Rafik Zeraoulia. Professional email: [email protected]. Both institutional affiliations appear in the note and creator metadata. AI assistance: OpenAI ChatGPT assisted with English improvements, some numerical checks, and mathematical drafting and algebraic reasoning. Responsibility for counterexamples, lemmas, theorems, proofs, interpretations, and all final mathematical claims rests solely with the author.