And

No primitive can be derived from the other four. Each of the five primitives (Point, Line, Angle, Curve, Enclosure) is independent and irreducible.

The four sorts (G, T, R, C) classify all well-formed expressions without overlap or gap. Every well-formed expression belongs to exactly one sort.

The 11 Σ_UL operations are closed over the set of well-formed expressions. Applying any operation to well-formed inputs always yields a well-formed output.

Every well-formed expression uses finitely many applications of primitives and operations. No infinite construction is required to express any concept in UL.

Each operation signature specifies the sorts of its inputs and output. The sort of the result is fully determined by the operation and the sorts of its arguments.

The sort system and operation system are mutually consistent: no operation can produce a sort violation, and no sort restriction blocks a structurally valid operation.

The abstract and embed operations form a dual pair: abstracting an embedded expression recovers the original, and embedding an abstracted expression recovers the original, up to canonical equivalence.

Any concept expressible in any natural or formal language can be represented as a well-formed UL expression using the five primitives and eleven operations.

Universal Language can describe its own syntax, semantics, and proof system entirely within its own formalism. UL is self-describing.

The dimensional assignments to primitives (0D Point, 1D Line, 2D Angle, 3D Curve, 4D Enclosure) are consistent with all operations and sort assignments.

The quantify operation (the sole T2 operation) is well-defined over all T1 expressions and always produces a well-formed T2 expression.

There is a one-to-one correspondence between geometric constructions and semantic meanings in UL. Form and meaning are identical — a construction's meaning IS its geometry.