A.19:5.2 - State Spaces & Comparability

Memory hook: Compare only values already in the same declared space or carried into one common space through an exact coordinate mapping. Reusing a predicate also requires the same semantic predicate. If the use also claims a relation between two exact F.17 local senses, cite an F.9 Bridge only after its predicate obtains and state the bounded-use claim and reliance separately. If the ReferencePlane changes, cite the applicable plane relation. Scope and window remain separate in either case.

This section supplies space projection, embedding, product, and two coordinate-comparability regimes. It does not perform a CPM comparison or a SelectorMechanism selection. A consumer that names a state or category cites the declared space and predicate, then keeps its own scope, evaluation, result, evidence, and work relations.

A CharacteristicSpace may be written abstractly as CS = ⟨I, basis⟩, where I indexes slots and basis is the ordered set of (Characteristic, Scale) bindings. A consumer-specific label for a space does not create another A.19 kind; the consumer instead states the exact use or relation position, entity, claim scope, context-slice membership, effective reference scheme and plane, and predicate relevant to that use.

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.

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.

A.19:5.2.3 - Predicate-use and state-assertion boundary

A.19 defines the space and CharacteristicSpacePredicate; it does not define a state assertion, applicability relation, dated evaluation work, gate, evidence relation, assurance result, or permission to act.

A consumer use recovers: the exact subject or input; any direct characteristic assignment or projection from that subject; the A.19 space and predicate; one set-valued U.ClaimScope; relevant A.2.6 U.ContextSlice membership; effective U.ReferenceScheme and reference plane; application or evaluation window; and, only when current, any obtaining F.9 Bridge with its separate bounded-use claim and reliance, plus any applicable plane relation. The consumer identifies the exact evaluation-operation application and its typed result under the applicable evaluation or assertion rule. A.10 provenance, G.11 currentness, measurement backing, assurance, and receiving-work disposition remain separate.

For a Ready claim requiring temperature below a cut and pressure above a cut, A.19 supplies the two declared coordinates, scales, normalization or coordinate-mapping basis, operators, cuts, polarity, and conjunction. The actual state assertion binds the pump, scope, slice, evaluation interval, inputs, result, and evidence use. Any semantic Bridge or plane relation needed by that use remains separate. Changing the evaluation interval does not change the predicate; changing either cut does.

Transporting a predicate into another space or transporting an assertion across spaces requires the exact Coordinate correspondence. Use an embedding only for point-injective structure-preserving transport; use a declared lossy mapping or projection when normalization discards distinctions. If the use relates two exact F.17 local senses and the F.9 predicate obtains, cite that Bridge and its separate bounded-use claim. If the ReferencePlane changes, cite the applicable plane relation. A scheme or plane difference alone establishes neither relation. If the required correspondence is absent, the current use is incomparable or unevaluable rather than approximately valid.

A.19:5.2.4 - Cross-reference-scheme and cross-plane comparability

A comparison across reference schemes or planes follows the relations the case actually needs. When it relates two exact F.17 local senses and the F.9 predicate obtains, cite that Bridge and a separate bounded-use claim; CL is optional evidence shorthand. A plane crossing cites its applicable plane relation. Keep the coordinate mapping and A.19.UNM instances explicit. A context, scheme, or plane difference alone establishes no Bridge or comparison admissibility, and a reverse comparison needs its own justified direction.

A comparison may reuse a predicate only when its complete by-value meaning is unchanged. When a coordinate mapping is needed, it must preserve every predicate component required by this use. If the reuse also relates two exact local senses through an obtaining Bridge, a separate bounded-use claim states that semantic use and any required reliance passes. CPM separately binds comparison scope, comparator, input values, effective reference plane, and evaluation window. The Bridge alone copies neither predicate content, scope, nor time, and a common label establishes none of them.

C.16 governs measurement uncertainty and calibration limits; A.3.3 governs prediction error and model applicability. Use B.3 or the direct assurance pattern when the use makes an assurance claim. Report the values as incomparable for the use when a critical coordinate lacks an admissible normalization or coordinate mapping; a separately needed semantic Bridge, bounded-use claim, or plane relation is absent; any required reliance does not pass; or the predicate, plane, scope, or window cannot be held fixed.

A.19:5.2.5 - Characteristic-Space Reference Chain

When evaluating a checklist, StateAssertion, gate, assurance argument, or decision through a declared CharacteristicSpace, keep the space-related references distinct:

declared Coordinates -> [normalization or quotient, when used] -> [indicator choice, when used] -> [order, topology, or distance overlay, when used] -> neighboring predicate evaluation, assertion, gate, assurance, or decision claim

Only the branches actually used are present. A.19 supplies the declared space and any named mapping, quotient, or overlay; the consumer supplies applicability, operation, result, and consequence. Co-implementation in software or records does not collapse these values.