Native Optimal-Completion Fibers, Real-Time Pushdown Recognition, and Protected Paired-Session Comparisons

The paper develops the connection between optimal-completion geometry and physical pushdown realization, building on our earlier work on shortest-completion fibers, simplicial realization, and protected-obligation structures. It asks whether prescribed topological complexity can be realized within the actual shortest-completion solutions of a counting system, without replacing the original source problem by an artificial enlarged construction. The paper establishes native realization results for finite simplicial complexes, shows that scalar folding preserves the exact optimal fiber, and connects these geometric constructions to deterministic one-stack comparison under explicitly controlled session boundaries. Its main novelty is the simultaneous preservation of source fidelity, optimal-solution geometry, and operational realizability, demonstrating that rich topological structures can arise directly from genuine minimum-completion problems while carefully distinguishing reset-controlled comparison services from unrestricted global pushdown recognition.