Dissipative Numeral System, Paper V-Extra2: Reflexive Operator Algebra and Void-Induced Structural Dynamics

Classical mathematics treats operators as transparent, cost-free, and structurally invariant mappings: f: X → Y with f unchanged by the act of mapping. This assumption requires that all positions in the domain X are already constructed members of the activated manifold A. This paper challenges that assumption. Within the Dissipative Numeral System (DNS), unactuated structural positions ∅_k ∈ P \ A (positions that exist in the potential space P but are absent from the activated manifold A) are not numerical zeros—they are unconstructed structural requirements. When an operator encounters such a position, classical mapping fails, and the operator undergoes a reflexive transformation: f → f'. We formalize this as the Void-Induced Operator Evolution Principle (VOEP): operator state transitions occur if and only if an operator interacts with a Generative Void ∅_G characterized by non-zero generativity metric G(∅_G) > 0. The operator acquires a structural imprint, accumulates a history ledger, and co-evolves with the void's Structural Kernel K(∅_k)—a third entity precipitated by their coupled dynamics. Crucially, we establish that operator evolution is not merely an algebraic event but a topological one: when operator evolution triggers void activation (∅_G → 0_T), the activated manifold A undergoes topological transformation—connectivity components merge, boundaries shift, and the Euler characteristic χ(A) changes. Operator rule adaptation is therefore geometrically constrained. We further introduce two-layer operator evolution: Level 1 (State Evolution): S_f and H_f change; f_rule remains invariant; and Level 2 (Rule Mutation): f_rule changes when accumulated imprint exceeds critical threshold. This establishes DNS operators as stateful, dual-mode (construction/dissolution), geometrically-coupled structural participants—not external mappings, but co-evolving entities within the structural manifold they help shape.