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Unitary Width Root Operations

Abstract

Actual unitary width root operations 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 (Positive).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRootOperations.Positive

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRootOperations.Positive (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized Positive 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 (Negative).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRootOperations.Negative

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRootOperations.Negative (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized Negative 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 1.3 (positive one).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRootOperations.positive_one

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

Source. Repository-derived.

Commentary.

The parameterized positive one 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 (negative one).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRootOperations.negative_one

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

Source. Repository-derived.

Commentary.

The parameterized negative one 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 1.5 (pairRoot).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRootOperations.pairRoot

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRootOperations.pairRoot (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized pairRoot 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 (pairRoot matrix).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRootOperations.pairRoot_matrix

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

Source. Repository-derived.

Commentary.

The parameterized pairRoot matrix 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 (pairRoot positive).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRootOperations.pairRoot_positive

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

Source. Repository-derived.

Commentary.

The parameterized pairRoot positive 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 (pairRoot negative).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRootOperations.pairRoot_negative

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

Source. Repository-derived.

Commentary.

The parameterized pairRoot negative 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 (mul pairRoot entry).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRootOperations.mul_pairRoot_entry

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

Source. Repository-derived.

Commentary.

Exact row operation for the two simultaneous actual unitary transvections.

Theorem 1.10 (row isotropic).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthRootOperations.row_isotropic

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

Source. Repository-derived.

Commentary.

Literal last-row isotropy, derived from actual fixedness and the group inverse.

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