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Unitary Width Transpose

Abstract

Actual unitary width transpose 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 (transpose).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthTranspose.transpose

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthTranspose.transpose (✓ std3).

Source. Repository-derived.

Commentary.

Transposition really stays inside the same actual SU carrier.

Theorem 1.2 (transpose entry).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthTranspose.transpose_entry

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

Source. Repository-derived.

Commentary.

The parameterized transpose 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 (transpose transpose).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthTranspose.transpose_transpose

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

Source. Repository-derived.

Commentary.

The parameterized transpose 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 1.4 (transpose mul).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthTranspose.transpose_mul

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

Source. Repository-derived.

Commentary.

The parameterized transpose mul 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 (transpose inv).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthTranspose.transpose_inv

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

Source. Repository-derived.

Commentary.

The parameterized transpose inv 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 1.6 (positive transpose iff).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthTranspose.positive_transpose_iff

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

Source. Repository-derived.

Commentary.

The parameterized positive transpose iff 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 (positive inv).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthTranspose.positive_inv

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

Source. Repository-derived.

Commentary.

The parameterized positive inv 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 (negative inv).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthTranspose.negative_inv

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

Source. Repository-derived.

Commentary.

The parameterized negative inv 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 1.9 (positive mul).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthTranspose.positive_mul

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

Source. Repository-derived.

Commentary.

The parameterized positive mul 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 1.10 (negative mul).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthTranspose.negative_mul

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

Source. Repository-derived.

Commentary.

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

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