Dissipative Numeral System, Paper V: Structural Algebra — The Multidimensional Structural State Space

Previous version (v1.1) of Paper V defined DNS numbers as vectors in a unified structural state space with a single shared origin for all structural directions. This revision (v2.3) refines that foundation. We show that DNS numbers are not elements of a single vector space, but are distributed across replica algebraic systems — independent structural algebras indexed by suit cc and level kk. Each replica has its own origin 0c,k0c,k, basis, and internal rules. Cross-replica operations are not directly defined; they require normalized projection to a reference comparison space LrefLref, rather than direct linear combination. Key innovations of this revision: Replica Algebraic Systems: Independent structural algebras with distinct origins Reference Comparison Space LrefLref: A task-dependent observation frame, not a new universal space Normalized Projection Πc,k∗=Nc,k∘Πc,kΠc,k∗=Nc,k∘Πc,k: Cross-replica comparison without forcing homogeneity Aggregation Operator ALrefALref: Generalizes cross-replica aggregation without assuming linearity Replica Type Classification: Vector-like, semiring-like, monoid-like, and ordered Structural Logarithm: Local vs global distinction; local logs are not additive Two Statistical Modes: Structureless vs structure-preserving aggregation Classical Arithmetic as Special Case: Single replica, zero cost, identity projection This revision resolves the question of how structurally different systems (different suits, different levels) can be compared and combined without forcing them into a single homogeneous space. Classical arithmetic is recovered as the special case of a single replica with zero cost and identity projection. Version 1.x of this paper remains valid as the single-replica limit of the revised framework.