A.19:5.2.2 - Comparability of States (two admissible regimes)

A label such as Ready, Authorized, or Degraded is a consumer-side category, not a space or comparison result. Its subject pattern states the predicate and evaluation use. Comparing two coordinate states depends on the declared spaces, mappings, scales, and comparison scope; A.19 permits only the following two coordinate regimes.

A.19:5.2.2.1 Coordinatewise comparability (≼_coord)

Two states can be compared coordinatewise only under strict conditions. Essentially, we require the states to be expressed in the same measurement space, with the same units and scales, and using the same state definitions. Formally, coordinatewise comparison is allowed only if all of the following hold:

• Same space. Both coordinate values lie in the same CharacteristicSpace by value. Similar names, shared storage, or a common model-use label are insufficient.

• Scale congruence. For each slot being compared, the scale type, unit, and polarity orientation are identical. For example, if comparing temperature values, both must be on the same scale (say, °C on an interval scale with “higher = hotter” orientation). No unit mismatches or differing interpretations can be present.

• Predicate and use congruence. When comparison depends on a category predicate, both values use the same CharacteristicSpacePredicate by value. CPM still states the exact comparison scope, comparator, reference plane, and evaluation window; A.19 does not infer them from matching labels.

When these conditions are met, one can define a coordinatewise preorder over states. Common patterns include:

• Dominance: For a given set of “higher is better” slots, we say state x ≼_coord state y if and only if for every relevant slot a, the coordinate a(x) \le a(y) (after orienting all slots to the declared polarity for that slot). In other words, y is as good or better on all enforced criteria. This defines a Pareto-like ordering (often partial, not total).

• Predicate-region inclusion: Predicate-defined categories denote regions of the declared space. Compare their regions by inclusion: Region(P) subseteq Region(Q) means every point satisfying P also satisfies Q. For example, speed > 120 and accuracy > 95% implies speed > 100 and accuracy > 90%. Two points satisfying the same category predicate need not be ordered; ordering those points requires the separately declared coordinate comparator.

By default, no comparability is assumed unless proven. If any of the above congruence conditions fails, one must not fall back to ad-hoc comparisons (like matching by name or normalizing without declaration). Either switch to a normalization-based regime or declare the states incomparable.

A.19:5.2.2.2 Normalization‑based comparability (≼_normalization)

When two state vectors do not meet the strict conditions for coordinatewise comparison (e.g. they come from different spaces, or the “same” Characteristics are measured on different scales or units), the only sanctioned way to compare them is: normalize, then compare.

Concretely: if we have state x in CS₁ and state y in CS₂, a normalization‑based comparison is permitted only if the model can cite a set of NormalizationMethodInstanceId(s) under a chosen UNM (per A.19.UNM) that lands the relevant coordinates of x into CS₂ (or lands both into a declared common target space). The base results are NCVs with their preserved/lost distinctions. Use an ≡_UNM class only when the receiving query is recoverable from that class; an inherited operation needs its separate compatibility and availability argument under A.19.UNM.

Comparability rule (normalize-then-compare). We say x ≼_{normalization} y only if, after applying the cited normalization instances to produce a representation of x in CS₂ (or a common target), the mapped state can be compared coordinatewise under ≼_coord. In other words, we never compare raw x and y; we compare after mapping into a common, well-typed space.

If a normalization use also spans different reference schemes or planes, keep the decisions separate. The A.19.UNM instance supplies the coordinate mapping. Cite an F.9 Bridge only when the use relates two exact F.17 local senses and its direct predicate obtains; state the bounded-use claim and reliance separately, with CL only as optional evidence shorthand. A ReferencePlane crossing cites its applicable plane relation. CPM supplies the comparison scope and evaluation window, and B.3 enters only for an actual assurance use. None of these relations or consequences follows from the scheme or plane difference alone.

Inspectability. Each normalization instance used for comparison is recoverable through its A.19.UNM declaration. C.16 governs measurement and calibration backing. When values differ in scale, reference scheme, or plane, keep the normalization, any independently obtaining semantic Bridge with its separate use claim, any applicable plane relation, and their limitations explicit.

Mnemonic: Never compare before both values are carried into the same well-typed space; never claim the same predicate, scope, plane, or window merely from matching labels.