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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.