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Unitary Antidiagonal Coordinates

Abstract

Actual unitary antidiagonal coordinates 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 (evenLabel).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryAntidiagonalCoordinates.evenLabel

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryAntidiagonalCoordinates.evenLabel (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized evenLabel 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 (evenLabel reflection).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryAntidiagonalCoordinates.evenLabel_reflection

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

Source. Repository-derived.

Commentary.

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

Definition 1.3 (oddLabelFn).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryAntidiagonalCoordinates.oddLabelFn

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryAntidiagonalCoordinates.oddLabelFn (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized oddLabelFn 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 (oddLabelFn injective).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryAntidiagonalCoordinates.oddLabelFn_injective

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

Source. Repository-derived.

Commentary.

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

Definition 1.5 (oddLabel).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryAntidiagonalCoordinates.oddLabel

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryAntidiagonalCoordinates.oddLabel (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized oddLabel 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 (oddLabel reflection).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryAntidiagonalCoordinates.oddLabel_reflection

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

Source. Repository-derived.

Commentary.

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

Definition 1.7 (antiDiagonal).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryAntidiagonalCoordinates.antiDiagonal

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryAntidiagonalCoordinates.antiDiagonal (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized antiDiagonal 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 (transport antidiagonal torus).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryAntidiagonalCoordinates.transport_antidiagonal_torus

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

Source. Repository-derived.

Commentary.

The parameterized transport antidiagonal torus 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 (regular even antidiagonal torus).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryAntidiagonalCoordinates.regular_even_antidiagonal_torus

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

Source. Repository-derived.

Commentary.

The parameterized regular even antidiagonal torus 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 (regular odd antidiagonal torus).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryAntidiagonalCoordinates.regular_odd_antidiagonal_torus

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

Source. Repository-derived.

Commentary.

The parameterized regular odd antidiagonal torus 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