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

Preface node heading:a-19-5-2-1-2-embedding-injection-cs-cs:28582

#What this page is

This is generated FPF reference text from the specification preface or supporting sections. It helps interpret FPF; it is not FPF Reference product documentation.

#Methodology

Use it to understand how the specification wants to be read, then return to a route, pattern, or work packet for active work. Cite generated IDs only when the wording changes the task decision.

#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 or 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:

An embedding whose endpoints share semantic interpretation remains local to their declared reference scheme and plane. A cross-scheme or cross-plane embedding requires an F.9 Bridge with exact endpoints, preserved and lost meaning, applicable use, CL value, and any receiving assurance consequence; the relevant A.6.1 operation application cites that Bridge explicitly.

Normalization declaration duties (MUST): Each cited NormalizationMethodInstanceId satisfies A.19.UNM declaration and admissibility obligations, including method-class token and validity window. C.16 governs calibration and measurement backing when relied on. Normalization alone does not license a change of reference scheme, plane, predicate, scope, or evaluation window; each changed boundary needs its direct declaration or Bridge.

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Last Updated: 2026-07-30 — upstream FPF commit 308edacf (github.com/ailev/FPF)