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Unitary Sylow Flag

Abstract

Actual unitary sylow flag 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 (prefix action).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowFlag.prefix_action

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

Source. Repository-derived.

Commentary.

Forward intrinsic flag relation uses only literal unitriangular entries.

Theorem 1.2 (paired detect).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowFlag.paired_detect

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

Source. Repository-derived.

Commentary.

The actual paired unitary root detects its leading adjacent coordinate.

Theorem 1.3 (short detect a).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowFlag.short_detect_a

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

Source. Repository-derived.

Commentary.

The parameterized short detect a 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 (short detect b).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowFlag.short_detect_b

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

Source. Repository-derived.

Commentary.

The parameterized short detect b 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 (shortRoot mem Uplus).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowFlag.shortRoot_mem_Uplus

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

Source. Repository-derived.

Commentary.

Actual short roots are upper unitriangular when the three positions are ordered.

Theorem 1.6 (prefix succ iff).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitarySylowFlag.prefix_succ_iff

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

Source. Repository-derived.

Commentary.

The whole standard coordinate flag is intrinsic to actual unitary U. Central long roots use a nonzero anti-fixed scalar; central short roots use actual trace surjectivity. This includes characteristic two and every rank.

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