BU79 Residual Stratification and Task-Completion Boundary under B_U From Same-Layer Alignment and Higher-Order Retained Chains to Application-Level Standing

This paper develops BU79|Residual Stratification and Task-Completion Boundary under B_U as a general application-layer interface within the B_U sequence. Its central claim is that practical alignment cannot be treated as a universal demand for total residual elimination. In real tasks, engineering systems, research procedures, organizational cooperation, human interaction, and social operations, alignment first requires a clearly defined task-completion boundary. Only after the object, action, constraint, verification, cost, residual tolerance, time window, and human-confirmation gates are specified can a result obtain application-level standing. The paper begins from the residual stratification rule: r_n=e_n+u_n where e_n denotes same-layer technical error and u_n denotes cross-layer unclosed readout. The same-layer term can be compressed through estimation, modeling, measurement, computation, correction, testing, feedback, and rollback. The cross-layer term cannot be eliminated inside the current task layer. It must be stratified, gated, retained, escalated, reclassified, or settled only after a higher-order closure becomes available. This distinction prevents the backward-mirror mistake of treating all residuals as errors that should converge to zero. BU79 then extends this two-layer residual model into a three-way application judgment. A residual item may be a same-layer alignable item A_{\parallel}, a same-layer non-alignable item N_{\parallel}, or a higher-order retainable item R_H. Same-layer alignable items remain inside the current task boundary and enter correction or coordination. Same-layer non-alignable items require boundary reset, task split, rejection, or reclassification. Higher-order retainable items are not ordinary errors; they may become seeds of new projects, new capabilities, new research directions, new organizational structures, or new social and cultural forms. The paper introduces weak persistence morphism p^w as the application-layer connector that records how residuals continue, stop, or ascend. The same-layer continuation path p_A^w preserves task operation after correction. The boundary-stop path p_\bot^w records non-admissibility and prevents forced continuation. The higher-order retention path p_H^w carries unclosed but valuable readouts into a higher-order retained chain. In this sense, weak persistence morphisms do not assert strong identity or zero-cost equivalence; they preserve operational continuity under boundary, feedback, residual, and settlement conditions. BU79 applies this structure to both the material order and the consciousness order. In the material order, residuals may appear as measurement error, interface mismatch, production friction, material incompatibility, engineering failure, or emerging technical opportunity. In the consciousness order, residuals may appear as misunderstanding, organizational misfit, incompatible audiences, different business types, hidden semantic potential, promotion windows, career expansion, or seed-stage structures. The same classification holds: some residuals align at the same layer, some require boundary separation, and some must be retained for higher-order development. The final judgment is that application-level alignment is task completion inside a small-ontology boundary. Daily use, engineering deployment, scientific research, organizational cooperation, and Agentic AI all require different thresholds. Practical alignment therefore means deciding what is sufficient, under which boundary, at which resolution, with which cost, and with which residual-handling path.