Closure as Projection under Bounded Non-Closure: Reconstruction of Wasan as an Operational System

This paper formalizes closure as a projectional operation under bounded non-closure, where residual difference persists and becomes observable through projection. Mathematical figures and relations are treated as projection-dependent appearances rather than intrinsic structures. Wasan is reconstructed as an operational realization of this framework, demonstrating a trace-dependent, non-commutative update system in which closure is achieved locally without global termination. Numerical approximations and geometric constructions are shown to produce locally stabilized non-closure. An appendix presents a trefoil-like projection of an unknot, illustrating that apparent topological complexity may arise from indexing and projection rather than entanglement.