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Unitary Width Row Clearing

Abstract

Actual unitary width row clearing 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/UnitaryWidthRowClearing.middle

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRowClearing.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 (clear last row).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRowClearing.clear_last_row

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

Source. Repository-derived.

Commentary.

A unit last pivot permits one genuine lower-SU operation to clear the entire last row. The central quadratic corner is retained in every characteristic.

Theorem 1.3 (last row reduction).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRowClearing.last_row_reduction

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

Source. Repository-derived.

Commentary.

Four alternating actual SU operations put any target into the genuine last-row stabilizer, uniformly in rank and field.

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