Dissipative Numeral System, Paper V-Extra1: Structural Algebra — DNS Paper V-Extra1

This supplement establishes the foundational distinction that DNS Paper V left implicit: the difference between an actuated structure holding zero payload (0_T) and an unactuated structural position that has never been constructed (∅). We formalize this distinction through a Tri-State Partition of all structural positions, grounded in a two-layer spatial model that distinguishes the potential generatable space from the activated index manifold. The paper's central contribution is the Operator-Imprinting Thesis: in DNS, operators are not transparent, cost-free mappings. They are structural participants whose own states are modified when interacting with incomplete structures. This imprinting is not a degradation but a feedback mechanism—the imprinted operator carries back a Structural Kernel, a blueprint that accumulates across repeated interactions and eventually precipitates into actual structure. We demonstrate the framework through a complete worked example (converting 15000 through a heterogeneous dissipative base), then extend the formalism to DNS Structural Tensors, where ∅ marks positions that are not merely zero-valued but absent from the activated index manifold. Finally, we position this work within the broader Reset Dynamics framework, showing how microscopic structural potentials provide the algebraic foundation for macroscopic residual fields, Residuum algebra, and cohomological reconfiguration.