Indefinite Piecewise Continuation via Interface Expansions and Operator–Dual Summability

This note formalizes an “infinite continuation” mechanism for piecewise-defined functions via iterated interface expansions supported on nested barrier cores. A sharp operator-theoretic criterion is given: indefinite continuation remains compatible with a governing operator precisely while cumulative defect is summable in an appropriate operator–dual norm. This yields explicit collapse thresholds paralleling classical convergence phenomena (continued fractions, Egyptian/Farey expansions), expressed in operator-theoretic terms. This work was developed through sustained human–AI collaboration between the author and ChatGPT (OpenAI), with ChatGPT serving as an interactive reasoning partner in conceptual development, structural synthesis, and exposition. All mathematical claims, interpretations, and responsibility for correctness remain with the human author. Initial public release. Standalone theorem note establishing an operator–dual summability criterion governing indefinite piecewise continuation and collapse.