#A.19:5.2.1.2 Embedding – Injection ι : CS₁ ↪ CS₂.

Preface node heading:a-19-5-2-1-2-embedding-injection-cs-cs:20407

#Content

An embedding is a structure-preserving injection from one space CS₁ into another space CS₂. It consists of two parts: (a) an injective slot correspondence from CS₁ to CS₂, and (b) (only where needed) cited normalization instances that make the correspondence semantically safe. Formally, let CS₁ have basis I₁ and CS₂ have I₂. An embedding declares an injective function m: I₁ → I₂ that identifies each slot of CS₁ with a corresponding slot in CS₂.

For each slot i ∈ I₁ where the scale/unit differs from the target slot m(i) in CS₂, the embedding MUST cite a NormalizationMethodInstanceId (per A.19.UNM) that re‑expresses values from ValueSet(slot_i) into ValueSet(slot_{m(i)}) within the declared invariants and validity window. The embedding does not define normalization semantics; it only references the required instances.

Intuitively, an embedding says: “Any coordinate tuple from CS₁ can be interpreted as a coordinate tuple in CS₂, possibly after converting units or re‑scaling, and without losing any information except what the declared NormalizationMethods intentionally coarse‑grain.” If there is no loss at all (NormalizationMethods are identity or strict conversions), the embedding is essentially an inclusion of one space into a larger one; if there is some information loss (e.g., converting a fine‑grained scale to a coarse one), that loss is explicit in the NormalizationMethodDescription. Locality:

Embeddings are defined within a single U.BoundedContext (i.e., both CS₁ and CS₂ are in the same context). Using an embedding across contexts requires an Alignment Bridge (see F.9) and MUST be declared via the relevant mechanism’s A.6.1 Transport clause (BridgeId + channel + ReferencePlane(src,tgt) + any CL^plane; no implicit crossings).

Normalization declaration duties (MUST): Each cited NormalizationMethodInstanceId MUST satisfy the declaration/admissibility obligations owned by A.19.UNM (incl. method‑class token and validity window). If such normalization artifacts are used for gating or assurance, their evidence/calibration backing and waiver rules are governed by C.16 (MM‑CHR). In other words, you cannot assume one context’s space fits into another’s without an explicit Bridge; any attempt to do so must treat it as a cross‑context alignment with potential loss.