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Unitary Width Radical Normalization

Abstract

Actual unitary width radical normalization 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 (slCentral transpose).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRadicalNormalization.slCentral_transpose

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

Source. Repository-derived.

Commentary.

The parameterized slCentral transpose 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 (centralEmbed transpose).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRadicalNormalization.centralEmbed_transpose

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

Source. Repository-derived.

Commentary.

The parameterized centralEmbed transpose 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 (centralEmbed positive).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRadicalNormalization.centralEmbed_positive

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

Source. Repository-derived.

Commentary.

The parameterized centralEmbed positive 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 (centralEmbed negative).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRadicalNormalization.centralEmbed_negative

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

Source. Repository-derived.

Commentary.

The parameterized centralEmbed negative 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 (central conjugate downRadical).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRadicalNormalization.central_conjugate_downRadical

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

Source. Repository-derived.

Commentary.

The actual opposite radical is normalized by every actual central SU.

Theorem 1.6 (upRadical one).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRadicalNormalization.upRadical_one

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

Source. Repository-derived.

Commentary.

The parameterized upRadical one 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 (downRadical one).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRadicalNormalization.downRadical_one

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

Source. Repository-derived.

Commentary.

The parameterized downRadical one 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 1.8 (positive levi radical).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRadicalNormalization.positive_levi_radical

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

Source. Repository-derived.

Commentary.

The accepted actual U=Levi*radical decomposition, in the literal SU carrier.

Theorem 1.9 (negative levi radical).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRadicalNormalization.negative_levi_radical

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

Source. Repository-derived.

Commentary.

Genuine opposite decomposition, with the Levi moved to the left by proved normalization.

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