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.