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Unitary Diagonal Separation

Abstract

Actual unitary diagonal separation 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 (pairedEntries eq exponent).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalSeparation.pairedEntries_eq_exponent

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

Source. Repository-derived.

Commentary.

The parameterized pairedEntries eq exponent 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 (oddEntries eq exponent).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalSeparation.oddEntries_eq_exponent

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

Source. Repository-derived.

Commentary.

The parameterized oddEntries eq exponent 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 (pairedExponent bound).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalSeparation.pairedExponent_bound

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

Source. Repository-derived.

Commentary.

The parameterized pairedExponent bound 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 (odd positiveExponent bound).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalSeparation.odd_positiveExponent_bound

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

Source. Repository-derived.

Commentary.

The parameterized odd positiveExponent bound 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 (separated power weights).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalSeparation.separated_power_weights

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

Source. Repository-derived.

Commentary.

The parameterized separated power weights 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 (regular even diagonal).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalSeparation.regular_even_diagonal

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

Source. Repository-derived.

Commentary.

The parameterized regular even diagonal 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 (regular odd diagonal).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryDiagonalSeparation.regular_odd_diagonal

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

Source. Repository-derived.

Commentary.

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

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