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.