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 remainderR_T = −Γ_T log 2 + A_T, where Γ_T is the co-length surplus and A_T > 0is an additive correction generated by the orbit it measures. DecomposesR_T into negative drift, endogenous arithmetic fluctuation, and nonlinearcorrection. Introduces the local gateway defect d_4(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.