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Unitary Diagonal Geometry

Abstract

Actual unitary diagonal geometry 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 (hermitianForm).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.hermitianForm

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.hermitianForm (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized hermitianForm 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 1.2 (adjoint).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.adjoint

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.adjoint (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized adjoint 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 (diagonalSL).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.diagonalSL

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.diagonalSL (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized diagonalSL 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 (diagonal hermitian).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.diagonal_hermitian

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

Source. Repository-derived.

Commentary.

The parameterized diagonal hermitian 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 (diagonal conjugation entries).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.diagonal_conjugation_entries

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

Source. Repository-derived.

Commentary.

The parameterized diagonal conjugation entries 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 1.6 (pairSwap).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.pairSwap

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.pairSwap (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized pairSwap 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 (pairedEntries).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.pairedEntries

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.pairedEntries (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized pairedEntries 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 (pairedEntries hermitian).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.pairedEntries_hermitian

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

Source. Repository-derived.

Commentary.

The parameterized pairedEntries hermitian 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 (prod zpow base).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.prod_zpow_base

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

Source. Repository-derived.

Commentary.

The parameterized prod zpow base 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 (prod pairedEntries).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.prod_pairedEntries

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

Source. Repository-derived.

Commentary.

The parameterized prod pairedEntries 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 1.11 (oddSwap).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.oddSwap

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.oddSwap (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized oddSwap 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 1.12 (oddEntries).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.oddEntries

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.oddEntries (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized oddEntries 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 1.13 (prod oddEntries).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.prod_oddEntries

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

Source. Repository-derived.

Commentary.

The parameterized prod oddEntries 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 1.14 (oddEntries hermitian).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.oddEntries_hermitian

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

Source. Repository-derived.

Commentary.

The parameterized oddEntries hermitian 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 1.15 (evenDiagonal).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.evenDiagonal

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.evenDiagonal (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized evenDiagonal 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 1.16 (oddDiagonal).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.oddDiagonal

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.oddDiagonal (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized oddDiagonal 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 1.17 (evenDiagonal unitary).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.evenDiagonal_unitary

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

Source. Repository-derived.

Commentary.

The parameterized evenDiagonal 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 1.18 (oddDiagonal unitary).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalGeometry.oddDiagonal_unitary

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

Source. Repository-derived.

Commentary.

The parameterized oddDiagonal 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.

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