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Unitary Trace Root Conjugation

Abstract

Actual unitary trace root conjugation 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 (traceRoot3 conjugation).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRootConjugation.traceRoot3_conjugation

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRootConjugation.traceRoot3_conjugation (✓ std3). ∎

Source. Repository-derived.

Commentary.

The parameterized traceRoot3 conjugation 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 (threeEntries).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRootConjugation.threeEntries

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRootConjugation.threeEntries (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized threeEntries 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 (prod threeEntries).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRootConjugation.prod_threeEntries

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

Source. Repository-derived.

Commentary.

The parameterized prod threeEntries 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 1.4 (threeDiagonal).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRootConjugation.threeDiagonal

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRootConjugation.threeDiagonal (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized threeDiagonal 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 (threeEntries unitary).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRootConjugation.threeEntries_unitary

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

Source. Repository-derived.

Commentary.

The parameterized threeEntries 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 1.6 (threeDiagonal unitary).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRootConjugation.threeDiagonal_unitary

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

Source. Repository-derived.

Commentary.

The parameterized threeDiagonal 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 1.7 (regular unitary trace root3).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRootConjugation.regular_unitary_trace_root3

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRootConjugation.regular_unitary_trace_root3 (✓ std3). ∎

Source. Repository-derived.

Commentary.

The parameterized regular unitary trace root3 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