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Unitary Width Pivot

Abstract

Actual unitary width pivot 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 (first).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthPivot.first

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthPivot.first (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized first statement describes the actual unitary geometry or prescribed ordered-product construction. Its Lean telescope retains every field, rank, involution, input and equality hypothesis shown.

Definition 1.2 (last).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthPivot.last

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthPivot.last (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized last statement describes the actual unitary geometry or prescribed ordered-product construction. Its Lean telescope retains every field, rank, involution, input and equality hypothesis shown.

Definition 1.3 (interior).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthPivot.interior

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthPivot.interior (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized interior 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 1.4 (rev first).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthPivot.rev_first

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

Source. Repository-derived.

Commentary.

The parameterized rev first 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 1.5 (rev last).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthPivot.rev_last

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

Source. Repository-derived.

Commentary.

The parameterized rev last 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 1.6 (first ne last).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthPivot.first_ne_last

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

Source. Repository-derived.

Commentary.

The parameterized first ne last 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 1.7 (interior ne first).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthPivot.interior_ne_first

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

Source. Repository-derived.

Commentary.

The parameterized interior ne first 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 1.8 (interior ne last).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthPivot.interior_ne_last

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

Source. Repository-derived.

Commentary.

The parameterized interior ne last 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 1.9 (interior ne rev).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthPivot.interior_ne_rev

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

Source. Repository-derived.

Commentary.

The parameterized interior ne rev 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 1.10 (prepare interior pivot).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthPivot.prepare_interior_pivot

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

Source. Repository-derived.

Commentary.

Actual arbitrary SU, not an isotropic-vector or generation premise. One upper and one lower operation create a noncentral interior pivot. A short root handles the odd central-coordinate case using true trace surjectivity.

Theorem 1.11 (unit last pivot).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthPivot.unit_last_pivot

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

Source. Repository-derived.

Commentary.

A literal unit last pivot is obtained in three alternating actual SU slots. This count is absolute. No fixed flag, orbit transitivity, or decomposition is assumed.

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