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Part IIUnitary Levi Decomposition

Abstract

Actual part iiunitary levi decomposition supplies the parameterized matrix, finite-field or ordered-product identities consumed by the whole-SU K5 width and intrinsic prescribed-product proof paths. All original hypotheses and actual carriers are retained; this source provides no oracle for bare automorphism recognition.

Theorem 1.1 (actual steinberg central levi).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryLeviDecomposition.actual_steinberg_central_levi

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryLeviDecomposition.actual_steinberg_central_levi (✓ std3). ∎

Source. Repository-derived.

Commentary.

The parameterized actual steinberg central levi statement describes the actual unitary geometry or prescribed ordered-product construction. Its Lean telescope retains every field, rank, involution, input and equality hypothesis shown.

Theorem 1.2 (actual steinberg radical mem).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryLeviDecomposition.actual_steinberg_radical_mem

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryLeviDecomposition.actual_steinberg_radical_mem (✓ std3). ∎

Source. Repository-derived.

Commentary.

The parameterized actual steinberg radical mem statement describes the actual unitary geometry or prescribed ordered-product construction. Its Lean telescope retains every field, rank, involution, input and equality hypothesis shown.

Theorem 1.3 (actual levi radical middle).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryLeviDecomposition.actual_levi_radical_middle

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryLeviDecomposition.actual_levi_radical_middle (✓ std3). ∎

Source. Repository-derived.

Commentary.

The literal middle block after multiplying by an arbitrary radical factor. No commutation of the two factors is used.

Theorem 1.4 (actual unitary ambient U decomposition).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryLeviDecomposition.actual_unitary_ambient_U_decomposition

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryLeviDecomposition.actual_unitary_ambient_U_decomposition (✓ std3). ∎

Source. Repository-derived.

Commentary.

Exact p255 unitary decomposition on the actual central SU(n) and actual V*=V intersect Fix(Steinberg). The fixed-point facts are DERIVED from the original target; no off-block/unitary coverage is assumed.

The results concern special-unitary matrix groups over finite fields. Their geometric setting is Nikolov and Segal, On finitely generated profinite groups, II: Products in quasisimple groups, Annals of Mathematics 165 (2007), 239-273, DOI 10.4007/annals.2007.165.239. Prescribed-product statements retain their field-size, rank and automorphism hypotheses.

References