A.2.6:6 - Normative Definitions

A.2.6:6.0 - Predicate semantics, mathematical algebra, and A.6.1 operations

Keep three layers explicit:

1. Scope semantics. member(x,S) is a bivalent predicate over one exact U.ContextSlice and one exact U.Scope.
2. Mathematical representation. The formulae below represent membership and set operations under C.29. Use the declarations and bindings below for an actual operation application.
3. Reusable actual operations. When a receiving use needs one identified calculation or evaluation application and its bound result, use one of the exact A.6.1 OperationDeclarations below. These are argument and result declarations, never A.6.5 SlotSpecs.

Mathematical semantics.

member(x, S)                        : Bool
scopeSubset(S1, S2)                 := for every x, member(x,S1) implies member(x,S2)
coversSet(S, T)                     := for every x in T, member(x,S)
extension(intersect(F))             := intersection of extension(S) for S in F
extension(SpanUnion(F))             := union of extension(S) for S in F
extension(translate(B,C_use,S,RS))  := the target-slice image of extension(S) selected by C_use's rule and tolerance over Bridge B under RS
widen(S0,S1)                        := extension(S0) proper-subset extension(S1)
narrow(S0,S1)                       := extension(S1) proper-subset extension(S0)
refit(E0,E1,S)                      := expressions E0 and E1 both designate exact scope S

Here T : ContextSliceSet is a finite target set, F : Set[U.Scope] is a finite scope family, B is an exact obtaining F.9 Bridge, C_use is the exact current C.2.1 claim with B as EntityOfConcern and affirmative polarity for this named scope-translation use, and RS is the exact target reference scheme. The claim’s content names the direction, scope-correspondence rule, and permitted-loss tolerance used to select the target image; its effective ReferenceScheme makes those designations interpretable. scopeSubset, coversSet, widen, narrow, and refit are mathematical predicates or comparison classifications, not actual A.6.1 operations. The formula represents the claim’s proposed mapping but proves neither the claim nor reliance on it and declares no operation application. Work that authors or compares scope declarations remains separately governed.

A.6.1 declaration A — ScopeMembershipEvaluationMechanism.

• EntityOfConcernRef: exact operation family ScopeMembershipEvaluationOperationFamily = {evaluateMembership}.
• effective U.ReferenceScheme: the scheme under which this mechanism’s argument, result, and application meanings are interpreted.
• SubjectKind: U.Scope.
• RangedValueKind: U.ContextSlice.
• ResultKind: declaration-local finite U.Kind MembershipEvaluationValue = {true, false, unknown} under C.3. Its membership rule admits exactly those three values. It is not a world-side third truth value, public U-kind, gate decision, or result episteme.
• SliceSet and ExtentRule: absent; membership of the kind U.Scope is not slice-dependent in the A.6.0 sense.

OperationDeclaration evaluateMembership:

ApplicationPredicate: with those bindings, evaluate member(targetSlice, scope) under the bound interpretation basis; return true or false when the basis determines the predicate and unknown when a required selector resolution or translation input is unavailable. The application leaves the target slice and scope unchanged.

ApplicationIdentityRule: one application is one independently bounded evaluation invocation selected by the current calculation or evaluation-work locus. Repeating the evaluation with the same arguments is another application when another invocation occurs; argument equality alone does not merge them.

ApplicationExtentRule: the application begins when its exact argument bindings and interpretation basis are fixed for the invocation and ends when membershipJudgment is returned or the invocation stops without a result. A result binding cannot begin before the value is returned.

ScopeMembershipEvaluationMechanism LawSet. With the same exact argument bindings, interpretation basis, and effective reference scheme, evaluation is deterministic. true reports that the basis determines member(targetSlice, scope); false reports that it determines non-membership; unknown reports only that it cannot determine either result.

ScopeMembershipEvaluationMechanism AdmissibilityConditions. Admit an application only after the exact slice, exact scope, and exact interpretation basis are bound. unknown is admitted when that basis records an unavailable required selector resolution or translation input. A missing exact scope, slice, or basis blocks the application rather than creating a guessed binding.

ScopeMembershipEvaluationMechanism Applicability. Use this declaration only for evaluating exact U.ContextSlice and U.Scope values under its effective reference scheme. The receiving use names its exact U.ClaimScope, selected evaluation time when current, selected CHR:ReferencePlane only when the use is plane-dependent, and any mechanism-specific condition.

ScopeMembershipEvaluationMechanism SignatureManifest (optional). When dependency replay needs it, name the actual imported or provided declarations for U.ContextSlice, U.Scope, and the local MembershipEvaluationValue.

ScopeMembershipEvaluationMechanism neighboring objects. An evaluation application can occur within dated work governed by A.15.1. A separately persisted result episteme remains optional under C.2.1; A.15.PROD enters only for a current claim that work first constituted that episteme. Evidence-use and gate occurrences stay under A.10 and A.21. None of those objects, nor another evaluation invocation, reidentifies this mechanism unless it reveals changed declaration content.

ScopeMembershipEvaluationMechanism refinement or conservative extension. A refinement preserves evaluateMembership, its argument and result meanings, binding rules, application predicate, identity and extent, and the bivalent-truth boundary while stating every strengthened law or admission condition. A conservative extension adds exact optional arguments, results, or operations without changing those inherited meanings or admitted uses.

A.6.1 declaration B — ScopeDerivationMechanism.

• EntityOfConcernRef: exact operation family ScopeDerivationOperationFamily = {deriveIntersectionScope, deriveSpanUnionScope, deriveTranslatedScope}.
• effective U.ReferenceScheme: the scheme under which this mechanism’s operation meanings and returned scopes are interpreted.
• SubjectKind: U.Scope.
• RangedValueKind: U.Scope; each derivation operation still returns a U.Scope, so no distinct mechanism-level ResultKind is current.
• SliceSet and ExtentRule: absent for the same A.6.0 reason stated above.

ApplicationPredicate rules. deriveIntersectionScope returns the scope represented under C.29 by intersection of extension(S) for S in scopeFamily. deriveSpanUnionScope implements the already established SpanUnion: it is admitted only when independenceBasis establishes the section 7.3 independence condition and returns the scope represented by SpanUnion(scopeFamily). deriveTranslatedScope is admitted only when the bound Bridge obtains and the bound C.2.1 claim has that Bridge as EntityOfConcern, affirmative polarity, and content naming this scope-translation use, its direction, rule, and tolerance. The application applies that rule within that tolerance and returns the scope represented by translate(bridgeOccurrence, scopeTranslationClaim, sourceScope, targetReferenceScheme). The formulae and claim alone declare no application or result binding.

For every governed-reference argument, record presence, citation, or a compatible token is insufficient: the reference must resolve to the exact value actually used. For every result row, the result binding obtains only when this application returns the independently identifiable extensional scope.

ApplicationIdentityRule: each derivation application is one independently bounded calculation invocation identified through its exact invocation boundary, mechanism edition, and operation designator rather than the argument tuple alone. Repeated calculations with equal arguments remain distinct applications.

ApplicationExtentRule: the application begins after every required argument is bound for that invocation and ends when the derived-scope value is returned or the invocation stops without a result. A result-binding extent cannot begin before that scope value is returned.

ScopeDerivationMechanism LawSet. Serial composition uses intersection. Parallel publication uses the one established SpanUnion and preserves only slices supplied by independently supported lines. Translation returns only the target-slice image selected by the bound claim’s rule and tolerance over the bound obtaining F.9 Bridge. No derivation operation widens support by itself.

ScopeDerivationMechanism AdmissibilityConditions. Intersection and SpanUnion require at least two exact scopes. deriveSpanUnionScope additionally requires the bound independence basis to meet section 7.3. deriveTranslatedScope requires both an exact obtaining Bridge and the exact affirmative C.2.1 claim whose named rule and tolerance select the claimed target image. A missing or non-obtaining Bridge or a missing or non-affirmative claim blocks that positive derivation application rather than creating a guessed scope; the latter does not negate an otherwise obtaining Bridge.

ScopeDerivationMechanism Applicability. Name the exact source scopes and reference schemes required by the selected derivation. For translation, also name the bound Bridge and separate C.2.1 claim. Before a receiving guard, assertion, publication, or structure selection relies on the returned scope, require the exact A.10 evidence-provenance relation for this bounded use. For ordinary reliance, require RelianceDisposition=pass. If an actual named assurance claim about that use is current, require its B.3 AssuranceResult for the same bounded use with disposition=supported-for-use. A direct domain rule may require such a claim, but neither scope translation nor consequence creates it.

A missing or non-affirmative use claim or a non-passing A.10 disposition stops ordinary reliance without changing membership truth or the Bridge. When an actual named assurance claim is current, a B.3 AssuranceResult with disposition=narrowed supports only its stated narrower use; abstain, evidence-needed, reopen, or blocked stops the attempted use. A.10 pass or B.3 supported-for-use supports only the named use. Neither is legal, policy, or deontic authorization, and neither proves that a derivation application or another receiving object occurred. Any required authorization remains under its direct pattern. The receiving use also names its exact U.ClaimScope, selected time when current, selected CHR:ReferencePlane only when plane-dependent, and derivation-specific conditions. GammaTimePolicy enters only when time changes membership; ReferencePlane is absent from ordinary set algebra.

ScopeDerivationMechanism SignatureManifest (optional). When dependency replay needs it, name the actual imported or provided declarations for U.Scope and, for translation, the exact F.9 Bridge declaration and C.2.1 claim identity rules. The independence basis, particular Bridge, and particular scope-translation claim are application arguments, not declaration-manifest entries by adjacency. scopeTranslationClaim is only this declaration’s argument label; it names no public claim kind. A.10 and B.3 reliance objects remain under their subject patterns rather than becoming a common mechanism signature.

ScopeDerivationMechanism neighboring objects. A derivation can occur within dated calculation work under A.15.1. Its bound independence-basis episteme, Bridge, and C.2.1 scope-translation claim retain their own identities and direct patterns. The exact A.10 relation and disposition, or the exact B.3 AssuranceResult when an actual named assurance claim is current, states whether the use has the needed evidence or assurance support; neither is a mechanism argument or result. The returned U.Scope is independently identified by its extension. Evidence, publication, gate, assurance, and any downstream Work, assertion, relation, or publication occurrence remain with their direct patterns. None of those objects, nor another derivation invocation, reidentifies this mechanism unless it reveals changed declaration content.

ScopeDerivationMechanism refinement or conservative extension. A refinement preserves the inherited derivation operations, argument and result meanings, binding rules, application predicates, identity and extent, and the intersection, SpanUnion, and translation semantics while stating every strengthened law or admission condition. A conservative extension adds exact optional arguments, results, or operations without changing those inherited meanings or admitted uses.

Relation between the declarations. These are two independently identified U.Mechanism epistemes. They coordinate by value: a later evaluateMembership application may bind a scope returned by one derivation application. If a receiving claim needs a refinement, extension, equivalence, or other direct relation between exact mechanism editions, state its endpoints, predicate, scope, and preserved and changed content under A.6.1.

A.2.6:6.1 - U.ContextSlice - exact membership target

U.ContextSlice is an addressable durable value formed from one exact declared selector schema and one value for every selector present in that schema. A scope predicate may inspect a declared projection of the slice, but it does not determine the slice’s identity. A minimal slice declaration contains:

ContextSlice:
  effectiveReferenceScheme:
  declaredSelectorSchema:
  exactLocalSenseRefs?, when included by that schema:
  standardOrInterfaceEditionRefs?, when included by that schema:
  environmentOrPlatformSelectors?:
  cohortOrJurisdictionSelectors?:
  gammaTime?, when included by that schema:
  otherDeclaredSelectors?:

The slice is one value. A finite target is one value of mathematical ValueKind ContextSliceSet. Two slice designators resolve to the same U.ContextSlice exactly when their declared selector schemas match and every declared selector resolves to the same value under the effective reference scheme. A predicate’s current argument projection, missing evaluation input, or receiving action cannot merge or split slice identity.

For example, slice_A and slice_B may share substrate Al6061, temperature 140 °C, and rig edition Calib-v3 while carrying different declared cohort selectors. A temperature-only scope predicate can return the same result for both slices, but the slices remain distinct; a cohort-sensitive predicate can distinguish them without reidentifying either one.

Do not write an implicit “current” or “latest” selector. If time changes membership, name the exact point, interval, or policy. If time does not change membership, do not add a fictitious temporal field merely to complete the tuple.

A.2.6:6.2 - U.Scope - set-valued scope

U.Scope is a durable value with one exact extension of mathematical ValueKind ContextSliceSet. U.ClaimScope, U.WorkScope, and U.PublicationScope are its C.3 specializations for receiving uses; the specialization does not copy the extension or add another identity discriminator.

For exact scope S and exact slice x, the primitive delimitation semantics is:

member(x, S)

The predicate has the exact slice and exact scope as arguments. It is not by itself an explicitly individuated U.Relation occurrence. Included slices satisfy it; excluded slices do not. The excluded area is not materialized as an unbounded complement entity.

For effective reference scheme RS, define extension_RS(S) := { x : U.ContextSlice | member(x, S) }. Two scope designators resolve to the same extensional U.Scope value when their extensions contain exactly the same independently identified slices under the same or explicitly reconciled reference scheme. An equivalent predicate expression, unit conversion, factoring, or publication change can preserve that value; a boundary change that adds or removes even one slice identifies another scope value.

A set or predicate expression, table, diagram, or query result can represent or designate a scope or a set of evaluated slices under C.29 and C.2.1.

USM admits subset, intersect, spanUnion, translate, widen, and narrow over exact scope extensions. refit is a same-extension normalization: it changes a predicate expression, units, or factoring while preserving member(x,S) for every exact slice under the effective reference scheme. A changed expression may require another declaration or claim-bearing episteme edition under its direct governor; it identifies another U.Scope only when the extension changes.

If a receiving use requires stable identity for membership occurrences, A.2.6 must first declare a direct relation kind with exact participant meanings, obtaining condition, recurrence rule, and non-optional occurrence-identity rule under A.6.REL. Until then, do not use ScopeDelimitationRelation, ScopeDelimitationMode, or ScopeDelimitationInterval.

A.2.6:6.3 - U.ClaimScope (G) and membership evaluation

U.ClaimScope is the exact set-valued scope used to say where one claim holds. The claim-bearing U.Episteme and the scope value are distinct; the episteme designates the exact scope current for that claim.

An evaluation of member(x, S) is also separate:

• the predicate semantics determine membership;
• an exact system performs dated evaluation work by an exact method, using a direct evaluation relation or A.6.1 operation binding;
• a separately current C.2.1 result episteme may state true, false, or unknown;
• evidence and freshness claims remain under A.10 and their direct governors.

unknown reports that the evaluation cannot currently decide because a required selector, designation resolution, or translation input is unavailable. It does not mean false, does not exclude the slice, and does not create a third world-side membership state. A receiving guard abstains, narrows the attempted use, or follows an explicitly governed reliance policy; it does not rewrite the predicate.

One exact U.ClaimScope participates in ModelApplicabilityRelation when model applicability is current. A declared ModelApplicabilityInterval belongs to an assertion or occurrence description. The actual applicability occurrence uses the maximal continuous extent over which its predicate obtains, as governed by A.1.1; the interval is not another direct participant.

A BoundedModelUseStructure may be selected over exact model-applicability and other governed relation occurrences under applied constraints that refer to exact claim-scope values. Keep the scope value distinct from the two ways it can affect structure identity. A bare scope, slice, membership outcome, or displayed boundary never enters A.22 identity. One exact U.ClaimScope remains a participant of an independently governed ModelApplicabilityRelation; when that exact obtaining occurrence is selected into the structure, the occurrence contributes through A.22’s relation-occurrence discriminator. Separately, one exact applied constraint claim may refer to that scope and contribute through A.22’s applied-constraint discriminator. Neither route turns the scope into a structure constituent, a membership-relation occurrence, or a second delimiter. The same scope may participate in differently selected relation occurrences or be referenced by differently identified structures, and a changed structure does not by itself reidentify the scope.

Expression. State a Claim scope as an exact predicate or condition block over slice selectors: assumptions, parameter ranges, cohorts, platform or standard editions, exact local senses when current, and time conditions only when they change membership.

Algebra. Serial dependencies use intersection. Independently supported areas may use spanUnion with the independence basis stated. widen and narrow change the declared set; refit preserves it. translate uses the section 7.5 Bridge-plus-use-claim branch and keeps reliance separate.

A.2.6:6.4 - U.WorkScope — scope of doing Work (capability)

Subject and declared basis. A qualified A.2.2 ability claim about an independently identified holder System designates its U.WorkScope. This is the set-valued work-condition basis of that claim, not a scope carried by a separate capability individual.

Meaning. U.WorkScope is the set of U.ContextSlice values under which a capability’s deliverability claim may be evaluated. Work-measure targets and qualification windows are checked separately at use time; they are not members or identity fields of the scope.

Expression. The capability declaration designates an exact U.WorkScope expressed only as conditions over U.ContextSlice: environment, versioned standards or platforms, resource regimes, exact local senses when current, and gammaTime only when time changes membership. Quantitative deliverables and qualification windows are not part of the scope value:

• Declare targets as work-measure target sets (e.g., latency <= L, throughput >= T, tolerance <= epsilon) bound in guards (WG‑2).
• Declare inspection/recertification policies as qualification-window policies bound in guards (WG‑3). The use‑time admission requires all of: WorkScope covers JobSlice AND WorkMeasures satisfied AND qualificationWindowHolds(holderAbilityClaim, qualificationWindowPolicy, evaluationTime).

Here holderAbilityClaim supplies the identified holder, claimed work conditions and bounds, and the declared support basis to which the qualification policy applies. The policy evaluates reliance at the stated time; its failure does not itself prove a loss of actual ability.

Method–Work gating. A Work step’s guard MUST check that the target slice is covered by the capability’s Work scope and that required measures and qualification windows are satisfied.

Composition and Delta-moves. Work scope uses the same algebra as Claim scope (intersection / spanUnion / translate / widen / narrow / refit). Section 7.5 selects translate only for exact local-sense translation through an obtaining F.9 Bridge plus the separate affirmative C.2.1 claim and its current reliance branch.

Separation from knowledge. A Work scope is a set-valued scope. The capability declaration uses it to delimit where a deliverability claim is evaluated. Measurements and monitoring may support that claim through separately governed evidence and reliance judgments.

Required guard facets (capabilities).

• Work-measure target set (mandatory). A set of measurable targets with units and tolerated ranges, evaluated on the JobSlice.
• Qualification-window policy (mandatory for operational use). A time policy stating when the capability is considered qualified; evaluated at the exact evaluation time selected by the receiving guard, not copied into U.WorkScope. These facets are separate from U.WorkScope and live in the R‑lane (assurance). They MUST be referenced in Method–Work guards (see §10.3 WG‑2/WG‑3).

A.2.6:6.5 - U.PublicationScope — scope of a publication view or publication form

Carrier. A publication view or form is rendered on a carrier; the carrier is a separate object. Meaning. The set of U.ContextSlice where a publication (a view, card, or lane about some object or morphism) is admissible for use within its underlying Claim scope or Work scope.

Relation to other scopes (normative).

• If the publication is about an episteme E: PublicationScope(view_E) ⊆ ClaimScope(E).
• If the publication presents a qualified holder-ability claim C for work admission: PublicationScope(view_C) ⊆ WorkScope(C), where WorkScope(C) is the work-condition basis designated by that claim. Any relied-on episteme ClaimScope remains a separate constraint; stating the ability in an episteme does not turn its WorkScope into ClaimScope.
• If the publication is about a composition, its scope is a subset of the intersection of the exact contributing scopes. When exact local senses require translation, use section 7.5 for each affected source scope: obtaining F.9 Bridge, separate affirmative C.2.1 use claim, and current A.10 or B.3 reliance before the returned scopes are intersected.

Expression. Declare U.PublicationScope as an exact predicate over only the U.ContextSlice selectors that restrict publication use: for example versioned standards, environment, audience, interface availability, exact local senses, or gammaTime when time changes membership. It may be narrower than the underlying scope but must not be wider.

Algebra and Delta-moves. Publication scope uses the USM algebra. A widened publication scope is admissible only when the resulting set remains a subset of every relevant underlying Claim scope or Work scope and the publication conditions support each added slice; the underlying scope need not change when it was already broader.

Orthogonality to measurement. U.PublicationScope is a USM scope object (set‑valued), not a CHR Characteristic and MUST NOT appear as a slot in a U.CharacteristicSpace.

View refinement (profiles). When a publication profile or view refines another, its U.PublicationScope MUST be a subset of the scope of the profile or view it refines (for example, a typed card may require additional pins).