23 Proven Theorems

Every well-formed UL expression has a unique decomposition into the five primitives. There is no ambiguity in decomposition — meanings are always traceable to concrete geometric operations.

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

The constructive levels form a well-ordered set. Every non-empty collection of expressions has a member of minimal 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 compose operation is associative: composing (f ∘ g) ∘ h yields the same result as f ∘ (g ∘ h) for all composable well-formed expressions.

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.

Every well-formed expression belongs to exactly one symmetry class determined by its transformation group under the Erlangen Program hierarchy.

For each symmetry class, there exists a set of geometric invariants that are preserved by all transformations in the corresponding group.

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

The axioms governing each primitive are independent: no axiom can be derived from the remaining axioms.

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 well-formed expression has a unique normal form obtained by exhaustive application of reduction rules. Equivalent expressions reduce to the same normal form.

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.

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

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.