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Unitary Trace Root

Abstract

Actual unitary trace root 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 (antiDiagonal3).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRoot.antiDiagonal3

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRoot.antiDiagonal3 (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized antiDiagonal3 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 (traceRoot3).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRoot.traceRoot3

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRoot.traceRoot3 (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized traceRoot3 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 (traceRoot3 unitary).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRoot.traceRoot3_unitary

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRoot.traceRoot3_unitary (✓ std3). ∎

Source. Repository-derived.

Commentary.

The parameterized traceRoot3 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 1.4 (actual trace root exists).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryTraceRoot.actual_trace_root_exists

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

Source. Repository-derived.

Commentary.

The parameterized actual trace root exists 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