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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.

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 constructive level of a compound expression is always greater than or equal to the levels of its constituent parts. Composition never decreases level.

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.

Applying the negate operation twice to any well-formed expression returns the original expression. Negation is its own inverse.

The Erlangen Program hierarchy applied to UL captures all geometrically meaningful distinctions between expressions. No finer classification is needed.

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.

Every invertible well-formed expression has a unique inverse under the invert operation. The set of invertible expressions forms a group under composition.