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Part IIUnitary Radical Trace Coordinates

Abstract

Actual part iiunitary radical trace 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.

Theorem 1.1 (actual radical unitary of trace).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryRadicalTraceCoordinates.actual_radical_unitary_of_trace

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

Source. Repository-derived.

Commentary.

Literal coordinate unitary criterion for the constructed radical. The column relation and corner trace are exactly those of the TRUE Steinberg inverse action; no unitary-recognition premise is used.

Theorem 1.2 (actual unitary radical row lift).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryRadicalTraceCoordinates.actual_unitary_radical_row_lift

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

Source. Repository-derived.

Commentary.

Every genuine first-row quotient parameter has an ACTUAL unitary radical lift. The corner is constructed by surjective relative trace, including characteristic two and arbitrary ranks.

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