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Unitary Width Rank Two

Abstract

Actual unitary width rank two 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 (rankTwoCarrierEquiv).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRankTwo.rankTwoCarrierEquiv

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRankTwo.rankTwoCarrierEquiv (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized rankTwoCarrierEquiv 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 (rankTwo four factor).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRankTwo.rankTwo_four_factor

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

Source. Repository-derived.

Commentary.

Absolute four-factor width on actual SU2 over every finite quadratic field. There is no field-size or characteristic cutoff.

Theorem 1.3 (trivial four factor).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRankTwo.trivial_four_factor

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

Source. Repository-derived.

Commentary.

Degenerate ranks remain literal actual SU groups.

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