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Unitary Width Five Absorption

Abstract

Actual unitary width five absorption 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 (FiveFactor).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthFiveAbsorption.FiveFactor

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthFiveAbsorption.FiveFactor (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized FiveFactor 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 (five triangular core radicals).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthFiveAbsorption.five_triangular_core_radicals

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

Source. Repository-derived.

Commentary.

Move the five triangular Levi cores to the left. No arbitrary triangular factor is treated as a radical; each is decomposed by its genuine SU theorem.

Theorem 1.3 (higher rank five radicals).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthFiveAbsorption.higher_rank_five_radicals

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

Source. Repository-derived.

Commentary.

Genuine higher-rank reduction with exactly five ordered boundary radicals.

Theorem 1.4 (five factor absorb).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthFiveAbsorption.five_factor_absorb

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

Source. Repository-derived.

Commentary.

Absorption is constant length: arbitrary central SU normalizes BOTH actual radicals, so the five ordered slots retain their triangular orientations.

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