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Unitary Width Rank Three

Abstract

Actual unitary width rank three 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 (rankThree prepare first).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRankThree.rankThree_prepare_first

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

Source. Repository-derived.

Commentary.

One actual lower root creates a nonzero first coordinate when necessary. Isotropy excludes a row supported on the middle coordinate.

Theorem 1.2 (rankThree clear last).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRankThree.rankThree_clear_last

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

Source. Repository-derived.

Commentary.

A unit last pivot is cleared by one genuine lower short root, retaining its central quadratic term in every characteristic.

Theorem 1.3 (rankThree four factor).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRankThree.rankThree_four_factor

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

Source. Repository-derived.

Commentary.

Actual SU3 has four alternating unitriangular factors over EVERY finite quadratic field, including F4/F9. Relative norm and trace are used in the constructed roots; no low-field exception or decomposition input occurs.

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