A.19:5.2.1 - CS Operators (notation-neutral, reference-scheme-local)

To enable model composition, define operations on CharacteristicSpaces independently of notation. Every operation states its effective U.ReferenceScheme and reference plane. Those values locate the operation but create no correspondence. When a use relates two exact F.17 local senses, test the direct F.9 predicate and cite the Bridge only when it obtains; state the bounded-use claim and any reliance separately. A ReferencePlane crossing cites its applicable plane relation. A scheme or plane difference alone establishes neither relation.

A.19:5.2.1.1 - Subspace — projection

For a space CS_I with basis I and a subset S, the projection pi_S^I : CS_I -> CS_S keeps the Coordinates in S and discards the others. The type-correct laws are pi_I^I = identity_CS_I and, for T subseteq S subseteq I, pi_T^S after pi_S^I = pi_T^I. A projection preserves an order, topology, or other structure only when that fact follows from the named overlays; projection alone makes no such promise.

A.19:5.2.1.2 - Embedding and lossy mapping

An embedding iota : CS_1 -> CS_2 is point-injective and preserves every structure named by its declaration. It gives an injective slot correspondence and an injective value map for each corresponding slot. Identity maps and exact, reversible unit conversions can support an embedding when they preserve the declared Scale meaning. The declaration states its domain, image, preserved structures, and any A.19.UNM instances used.

A coarse-graining, binning, many-to-one normalization, or dropped-coordinate operation is not an embedding. Declare it as a lossy mapping or projection, state the preserved and lost distinctions, and let each consumer decide whether that loss is admissible for its comparison, prediction, gate, or assurance use. When the use relates two exact F.17 local senses and the F.9 predicate obtains, cite that Bridge and a separate bounded-use claim. A ReferencePlane change instead cites its applicable plane relation. The coordinate mapping, semantic relation, plane relation, and C.16 calibration or measurement backing remain separate.

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

The product of two spaces CS₁ and CS₂ is a new space CS⊗ whose basis is the disjoint union of both bases, so even same-named slots retain their source identity. Its state is a pair (x₁, x₂). For example, a product can combine internal capability Coordinates with external-condition Coordinates for a readiness use. The product does not aggregate them: any cross-slot scale aggregation uses a declared Gamma fold under A.19.ULSAM and any needed A.19.UNM normalization. Use B.1 when a separate holonic-composition claim is made.