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Unitary Exponent Avoidance

Abstract

Actual unitary exponent avoidance 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 (cyclic simultaneous avoidance).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryExponentAvoidance.cyclic_simultaneous_avoidance

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

Source. Repository-derived.

Commentary.

The parameterized cyclic simultaneous avoidance 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 (norm one simultaneous avoidance).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryExponentAvoidance.norm_one_simultaneous_avoidance

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

Source. Repository-derived.

Commentary.

The parameterized norm one simultaneous avoidance 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 (full unit simultaneous avoidance).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryExponentAvoidance.full_unit_simultaneous_avoidance

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

Source. Repository-derived.

Commentary.

The parameterized full unit simultaneous avoidance 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 1.4 (involutionUnit).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryExponentAvoidance.involutionUnit

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryExponentAvoidance.involutionUnit (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized involutionUnit 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 (involutionUnit val).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryExponentAvoidance.involutionUnit_val

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

Source. Repository-derived.

Commentary.

The parameterized involutionUnit val 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 (involutionUnit pow).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryExponentAvoidance.involutionUnit_pow

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

Source. Repository-derived.

Commentary.

The parameterized involutionUnit pow 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 (norm one correction).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryExponentAvoidance.norm_one_correction

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

Source. Repository-derived.

Commentary.

The parameterized norm one correction 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