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.