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Part IIUnitary General Even UProduct

Abstract

Actual part iiunitary general even uproduct 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 uniform even Levi prescribed product).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryGeneralEvenUProduct.actual_uniform_even_Levi_prescribed_product

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

Source. Repository-derived.

Commentary.

Actual type-A PRODUCT on the central even Levi. The entire prescribed D action is cancelled on ALL central elements; both scalar corrections and ONE ambient SU inner tuple precede all positive type-A targets.

Theorem 1.2 (actual uniform even P general prescribed product).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryGeneralEvenUProduct.actual_uniform_even_P_general_prescribed_product

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

Source. Repository-derived.

Commentary.

General diagonal-similitude/field even P PRODUCT. Real scalar/skew witnesses and exact central powers are transported; ONE ambient unitary inner tuple precedes every actual skew target.

Theorem 1.3 (actual uniform central even U prescribed product).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/PartIIUnitaryGeneralEvenUProduct.actual_uniform_central_even_U_prescribed_product

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

Source. Repository-derived.

Commentary.

The ENTIRE central even positive unitary U, with arbitrary genuine prescribed D/Phi actions. Derived even U=U2 P and both actual suppliers are consumed; ONE correction precedes ALL targets.

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