#A.19:5.2.1.3 Product – Combination CS₁ ⊗ CS₂ = CS⊗.

Preface node heading:a-19-5-2-1-3-product-combination-cs-cs-cs:20418

#Content

The product of two spaces CS₁ and CS₂ is a new space CS⊗ that effectively contains all slots of CS₁ and all slots of CS₂. If CS₁ has index set I₁ and basis slots {slot₁…} and CS₂ has I₂, then $CS⊗$ has index set $I_⊗ = I₁ ⊎ I₂$ (disjoint union) with each slot’s definition carried over from its original space. In practical terms, any state in the product space is a pair (x₁, x₂) where x₁ is a state of CS₁ and x₂ is a state of CS₂ (assuming the two spaces pertain to possibly different aspects or roles). Use cases: Product spaces allow modeling multi-role scenarios or bundling an entity’s state with some environmental or contextual state. For example, one might take a space of internal capability metrics and ⊗ with a space of external conditions to form a combined space for “readiness under conditions.” Note: When combining scores or coordinates from a product space, one must be mindful of scale incommensurability. Cross‑slot aggregation SHALL proceed only via a declared Γ‑fold (B.1) and, where needed, explicitly declared NormalizationMethods; naïve arithmetic is forbidden. The product operation itself doesn’t perform any aggregation; it only sets the stage.