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Part IIUnitary Radical Torus

Abstract

Actual part iiunitary radical torus 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 (unitaryRadical).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryRadicalTorus.unitaryRadical

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryRadicalTorus.unitaryRadical (✓ std3).

Source. Repository-derived.

Commentary.

Definition6.6(ii), with the actual V support and true Steinberg law.

Theorem 1.2 (actual unitary radical regular torus values).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryRadicalTorus.actual_unitary_radical_regular_torus_values

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

Source. Repository-derived.

Commentary.

Genuine rank-independent-support torus kernel for V*. Separation is required ONLY on first-row/last-column roots, not every matrix root. ONE actual determinant-one unitary h precedes ALL targets and all witnesses remain in V*, including the noncommutative corner coordinate.

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