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Part IIUnitary Odd Levi Decomposition

Abstract

Actual part iiunitary odd 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.

Definition 1.1 (embed).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddLeviDecomposition.embed

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddLeviDecomposition.embed (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized embed 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 odd levi entry).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddLeviDecomposition.actual_odd_levi_entry

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

Source. Repository-derived.

Commentary.

The parameterized actual odd levi entry 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 odd levi unitary).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddLeviDecomposition.actual_odd_levi_unitary

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

Source. Repository-derived.

Commentary.

The parameterized actual odd levi unitary 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 1.4 (actual odd levi upper).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddLeviDecomposition.actual_odd_levi_upper

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

Source. Repository-derived.

Commentary.

The parameterized actual odd levi upper 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 1.5 (actual odd unitary U decomposition).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddLeviDecomposition.actual_odd_unitary_U_decomposition

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

Source. Repository-derived.

Commentary.

Actual odd U=U2 P, with genuine vector/central constraints constructed from every actual target; no decomposition or coordinate-law input.

Theorem 1.6 (actual odd levi field action).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddLeviDecomposition.actual_odd_levi_field_action

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

Source. Repository-derived.

Commentary.

The parameterized actual odd levi field action 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 1.7 (actual odd levi ambient diagonal inner).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryOddLeviDecomposition.actual_odd_levi_ambient_diagonal_inner

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

Source. Repository-derived.

Commentary.

Actual ambient determinant-one unitary H realizes every diagonal action on the odd type-A Levi, before any group target.

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