A.14:5 - Formal cores (normative semantics)

A.14:5.1 - PortionOf — metrical part of a measurable whole

Intent. Capture “some of the same stuff/extent”, governed by a measure that adds up.

Applicability. Any U.Holon that carries an extensive measure μ on the chosen scope (examples: mass, volume, length‑of‑text, byte size, wall‑time budget).

Primitive. PortionOf(x, y) means: x is a measured part of y of the same kind of stuff/content, allowing x = y; the strict case is ProperPortionOf under POR-2.

Axioms (A14‑POR‑*)

• POR‑1 (Partial order). PortionOf is reflexive, antisymmetric, transitive on its domain.
• POR‑2 (Metrical dominance). If x ProperPortionOf y then 0 < μ(x) < μ(y) for the agreed μ.
• POR‑3 (Additivity on disjoint portions). If PortionOf(x,y), PortionOf(z,y), and x ⟂ z (the two portions do not overlap), and their join is admitted under the same measure and boundary rule, then μ(x ⊔ z) = μ(x)+μ(z) and PortionOf(x ⊔ z,y). ProperPortionOf additionally requires the joined measure to remain strictly below μ(y); a join equal to the whole is PortionOf but not ProperPortionOf.
• POR‑4 (Kind integrity). x and y must share the same measure kind and unit (or a declared conversion).
• POR‑5 (Boundary compatibility). For physical wholes, the whole’s boundary encloses the union of its portions; cross‑boundary “leaks” are interactions, not portions.

Didactic tests.

• ✔ “5 kg from a 20 kg billet” — PortionOf.
• ✔ Two disjoint 5 kg cuts from the same 20 kg billet have a 10 kg join under the same mass unit and boundary rule; that join is still a ProperPortionOf the billet.
• ✔ “Pages 1–10 of the report” — PortionOf (μ = page or token count).
• ✘ “The pump module of the plant” as an integrated structural part — use ComponentOf for that claim, not PortionOf.
• ✘ “The Methods section of the paper” as a conceptual part of its argument — use ConstituentOf for that claim, not PortionOf.

Either object may also support a separate PortionOf claim when it satisfies the same-stuff/extent, measure and boundary conditions above.

A.14:5.2 - PhaseOf — temporal part of the same carrier

Intent. Capture “the same holon during a sub‑interval”, preserving identity through change.

Applicability. Any U.Holon that persists across time with a recognised carrier identity.

Primitive. PhaseOf(x, y) means: x is y restricted to a proper time interval.

Axioms (A14‑PHA‑*)

• PHA‑1 (Strict temporal parthood). PhaseOf is irreflexive, asymmetric, and transitive on proper temporal restrictions of one unchanged carrier. In particular, PhaseOf(y,y) is false: a whole-lifetime or self-reference is not a proper temporal part.
• PHA‑2 (Proper interval and same carrier). PhaseOf(x,y) requires the interval of x to be a proper sub-interval of y’s interval and the carrier-identity rule to hold throughout both. It does not require x to be a maximal cell of a partition.
• PHA‑3 (Nesting and overlap are allowed). Temporal restrictions of the same carrier may nest or overlap. A week may be part of a year-long phase, and a diagnostic window may overlap a calibration window. Those facts are not contradictions and do not by themselves select an aspect or partition.
• PHA‑4 (Selected partition is an additional claim). When a use needs exhaustive non-overlapping cells, declare one carrier, one interval to be covered, one analysis aspect or partition rule, and the selected family of PhaseOf values. Only cells of that same explicitly selected partition must be pairwise non-overlapping and jointly cover the declared interval. Another aspect or rule may select a different, overlapping family.
• PHA‑5 (Identity through change). Properties may vary between phases, but the carrier’s identity criteria hold continuously (e.g., same serial number, same legal identity, same theorem statement).
• PHA‑6 (Identity failure). If the carrier-identity criterion fails, do not assert PhaseOf across that boundary. Apply the subject’s identity and continuation rules; use B.2 only for a whole-reidentification claim and declare MHT only after its conditions hold.

Didactic tests.

• ✔ “PumpUnit#3 before calibration” — PhaseOf(Pump#3_pre, Pump#3).
• ✔ If PhaseOf(Pump#3@week-32, Pump#3@2026) and PhaseOf(Pump#3@2026, Pump#3), transitivity also gives PhaseOf(Pump#3@week-32, Pump#3). A high-vibration diagnostic window may overlap a calibration window for the same pump; neither is thereby a cell of one selected partition.
• ✔ “Specification episteme E during τ₂”, with the C.2.1 identity triple unchanged and a proper interval current — PhaseOf(E@τ₂, E). ✘ “Spec v2” — if a C.2.1 discriminator changed, identify another episteme and test EpistemeEditionRelation(E_v1,E_v2) separately; the label proves neither identity nor continuity.
• ✘ “Shift 1 of the same batch run” — use A.15.1 TemporalPartOf_work, EpisodeOf_work, OperationalPartOf_work, or another exact Work-part or occurrence relation whose predicate obtains.
• ✘ “Prototype vs. production unit” — identify each carrier and any separately established relation. The labels alone select neither PhaseOf, ComponentOf, ConstituentOf, nor MHT.

A.14:5.3 - AspectOf — bearer-dependent structural part under one named facet rule

Intent. State that one identified bearer-dependent part is an aspect of its bearer without turning a Characteristic, viewpoint, representation, concern, partition, or time window into a part.

Participants and qualifier. x and y occupy the U.Holon parthood domain: x is the aspect and y its bearer. Each must already satisfy its applicable holon-kind and identity rule; AspectOf does not grant systemness, agency, or independent-whole status. The qualifier f names the facet rule used in this occurrence; it does not introduce a universal U.Facet kind.

Primitive. AspectOf(x, y; f) means: x is the bearer-dependent structural part of y distinguished under facet rule f. The notation shows the required qualifier; the public sentence may remain “x is an aspect of y under the f rule.”

Obtaining conditions and properties (A14-ASP-*).

• ASP-1 (Identified occurrence). Name x, y, f, the relation occurrence, and the aspect-identity rule. The facet rule states what distinguishes x from the rest of y and what change preserves or ends this aspect.
• ASP-2 (Structural dependence). AspectOf(x,y;f) implies ut:StructPartOf(x,y), x != y, and asymmetry for that occurrence. It implies none of ComponentOf, ConstituentOf, PortionOf, PhaseOf, collection belonging, or independent systemhood.
• ASP-3 (Facet-local and non-transitive). An occurrence under f gives no occurrence under another facet. AspectOf is not assumed transitive through another bearer or facet; state every relied-on relation directly.
• ASP-4 (Bearer and aspect identity). If the bearer is reidentified, or f and the aspect-identity rule no longer identify x, the old occurrence ends. A changed view, diagram, name, or measurement does not by itself change the world-side occurrence.
• ASP-5 (Neighbor boundary). A measured quality routes to C.16/A.19; a viewpoint or view to E.17.0/E.17.1; a representation or projection to C.29 or its direct projection pattern; a selected temporal window to PhaseOf or C.27.TA; a selected partition to the pattern governing that structure. None creates AspectOf by selection alone.

Didactic tests.

• ✓ In Reactor-7, the thermal-boundary rule distinguishes ThermalEnvelope-7 as the connected enclosure of insulation panels, seals, and boundary interfaces that constrains heat transfer across the reactor boundary. ThermalEnvelope-7 is identified by the continuing enclosure under that rule, not by a fixed panel list. AspectOf(ThermalEnvelope-7, Reactor-7; thermal-boundary) obtains while that enclosure and Reactor-7 continue. Replacing one panel under the same rule preserves the aspect; dismantling the enclosure, replacing the facet rule with a different boundary, or reidentifying Reactor-7 ends the occurrence. A changed temperature reading or dashboard view does not.
• ✗ “Safety is an aspect of the design” when safety is only a Characteristic, concern, viewpoint, or heading. Recover that actual claim first.
• ✗ “Pump-7 during warm-up is its thermal aspect.” Use PhaseOf for the proper temporal restriction and A.19 for a measured thermal Characteristic when those claims obtain.

A.14:5.4 - CT2R-LOG and Compose-CAL handshake

• A direct structural parthood claim is usable without this assurance handshake. If the publication elects B.3.5 or a named current requirement demands it, link the claim through tv:groundedBy to its applicable current C.2.1 Γ_m.sum or Γ_m.slice construction-trace episteme and declare validationMode=axiomatic. The direct relation pattern decides whether the occurrence obtains and how it is identified; the relevant entity pattern decides identity through change. The trace only reports that basis.
• AspectOf uses one current C.13 slice trace when that assurance branch is elected. The trace names the aspect, bearer, facet rule, relation occurrence, and identity conditions; it creates none of them.
• PhaseOf is temporal parthood and shall not be grounded through Γ_m. Its assurance follows the same-carrier and proper-interval criteria, the separately declared selected-partition rule when one is claimed, and Γ_time ordering (B.1.4).
• A collection’s own belongs-to relation remains distinct from constructive parthood (CC-MEM-2). State its participants, what makes it obtain, and whether later belonging is the same occurrence or a new one under the collection’s pattern. A direct claim needs no B.3.5 fields. If B.3.5 assurance is elected, link validationMode=axiomatic to one current C.13 set trace that reports the relation that already obtains. The trace supports neither a ComponentOf inference nor a universal prohibition on separately grounded parthood.

Two quick identity tests apply before relying on a trace. The same listed constituents can form a different whole when their direct assembly relations or rule differ. Conversely, a permitted constituent replacement can preserve the same whole. An equal input list, a repeated trace, or validationMode=axiomatic decides neither case.