Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

Part IIUnitary Type ALevi

Abstract

Actual part iiunitary type alevi 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/PartIIUnitaryTypeALevi.embed

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

Source. Repository-derived.

Commentary.

Actual type-A Levi embedding, native Fin(2d) anti-diagonal coordinates.

Theorem 1.2 (actual typeA levi entry).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryTypeALevi.actual_typeA_levi_entry

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

Source. Repository-derived.

Commentary.

The parameterized actual typeA 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 typeA levi unitary).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryTypeALevi.actual_typeA_levi_unitary

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

Source. Repository-derived.

Commentary.

The parameterized actual typeA 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 typeA levi upper).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryTypeALevi.actual_typeA_levi_upper

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

Source. Repository-derived.

Commentary.

Actual positive type-A U maps into the positive UNITARY U. Reversal of the second half is essential and is proved in the index law.

Theorem 1.5 (actual typeA levi field action).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryTypeALevi.actual_typeA_levi_field_action

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

Source. Repository-derived.

Commentary.

Every finite-field automorphism restricts to the literal field action on the type-A Levi, on ALL group elements. Commutation with the quadratic involution is DERIVED from its actual Frobenius power, not assumed.

Theorem 1.6 (actual typeA levi ambient diagonal inner).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryTypeALevi.actual_typeA_levi_ambient_diagonal_inner

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

Source. Repository-derived.

Commentary.

The actual p255 ambient H induces ALL diagonal actions on the type-A Levi inside central SU(2d). ONE genuine determinant-one ambient unitary correction is constructed before ALL full Levi group targets.

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