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Part IIUnitary Radical Middle Field Product

Abstract

Actual part iiunitary radical middle field product 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 (Middle).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryRadicalMiddleFieldProduct.Middle

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryRadicalMiddleFieldProduct.Middle (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized Middle 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 1.2 (actual unitary radical middle field product).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryRadicalMiddleFieldProduct.actual_unitary_radical_middle_field_product

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

Source. Repository-derived.

Commentary.

Actual uniform original-s-power FIELD-action middle quotient PRODUCT. One determinant-one UNITARY h precedes ALL radical targets; witnesses are in V*, and the full corner is retained in P. The two boundary root pairs and central corner coverage remain later stages.

Theorem 1.3 (actual unitary middle residual).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryRadicalMiddleFieldProduct.actual_unitary_middle_residual

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

Source. Repository-derived.

Commentary.

The ACTUAL ordered residual after the middle V* PRODUCT: all middle row AND reflected column entries vanish. It stays unitary and in V; both exceptional root pairs and the corner remain unconstrained.

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