The Self-Referential Remainder of the Syracuse Map

Clarification note for Serie I. Extracts from the quaternionic–affineframework (DOI 10. 5281/zenodo. 19702442) the exact residual obstructionto pointwise Collatz convergence: the self-referential remainderRT = −ΓT log 2 + AT, where ΓT is the co-length surplus and AT > 0is an additive correction generated by the orbit it measures. DecomposesRT into negative drift, endogenous arithmetic fluctuation, and nonlinearcorrection. Introduces the local gateway defect d₄ (q) and acontrol-theoretic reading via branch gains and state-zeros. No newconvergence theorem is proved; the document names and isolates the objectthat survives all known filters.