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Library / First Principles Framework (FPF) - Core Conceptual Specification
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A.6.2:4.2 - Direct signature components (A.6.0 alignment)

When repeated use needs a reusable formal declaration, an A.6.0 U.Signature(profile=FormalSubstrate) episteme may declare this local arrow family. Its direct declaration components are:

SubjectKind     = local formal type EpMorphism
RangedValueKind = admitted ordered-pair range over exact U.Episteme values satisfying the declared endpoint-kind constraints
ResultKind      = omitted; the arrow is the declared subject, not an operation result
Applicability   = selected formal substrate, admitted endpoint kinds, and arrow-family conditions

SubjectKind here is a type inside the selected formal substrate, not a durable FPF U-kind. Add SliceSet and ExtentRule only if one declared local type genuinely has slice-varying membership; do not use them to hide a use-specific suitability claim.

Vocabulary.

  • U.Episteme — the exact domain and codomain values.
  • EpMorphism — the local formal type of arrows in the selected substrate.
  • EntityOfConcernChangeMode = {preserve, retarget} — a local two-value classification of one arrow, derived from its resolved endpoint EntitiesOfConcern rather than a durable U-kind or a U.Characteristic.
  • Ep — the selected category whose objects are the admitted exact epistemes and whose arrows are the admitted EpMorphism values. Call it a category only when it contains the required identities and is closed under every declared composition.
  • EoCBase — the endpoint-only thin category used to compare EntityOfConcern identity. Its objects are the exact independently resolved EntitiesOfConcern represented in the substrate. Between every ordered pair of admitted objects A,B it has one formal endpoint arrow u_{A,B}; u_{A,A} is the identity, and composition follows endpoints. These arrows are not independently meaningful domain or world-side relations.
  • dom(f) and cod(f) — the exact endpoint epistemes; id_X and compose(g,f) — the declared identity and composition operations.
  • α : Ep -> EoCBase — the declared mapping on objects and arrows. α(X) is X’s exact EntityOfConcern after entityOfConcernRef(X) resolves it. For f : X -> Y, α(f) is the unique endpoint arrow u_{α(X),α(Y)}. It deliberately forgets f’s arrow rule; different Ep arrows with the same endpoint EntitiesOfConcern therefore have the same image.

For each arrow, recover the C.2.1 identity values of X and Y and state which identity-bearing values or ClaimGraph parts are preserved or differ. If the arrow rule uses a neighboring relation, name its exact predicate and participants on each side and state which endpoint facts it reads or compares. Equal or different endpoint profiles do not mean that the arrow changed a relation occurrence or made it obtain or cease; any actual relation change and producing application or Work remain under their direct patterns. SubjectRef remains only a legacy source projection; resolve it to the exact episteme and EntityOfConcern.

A claim that f is suitable for one exact use is a separate C.2.1 bounded-use assertion q; a current-case judgement separately tests exact facts against it. A.6.1 governs the application occurrence and its argument and result bindings when those facts are current; the applicable system and Work patterns govern the performing system and performed Work. The A.6.0 signature and the mathematical arrow f : X -> Y remain the reusable declaration and relation.

Laws and applicability. P0-P5 below govern the local arrow class. A.6.5 SlotSpecs enter only when an exact reusable direct-relation declaration is current; they are not fields of X, Y, or f.