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Unitary Integer Weights

Abstract

Actual unitary integer weights 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 (positiveWeight).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.positiveWeight

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.positiveWeight (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized positiveWeight 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 (weightSum).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.weightSum

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.weightSum (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized weightSum 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 (positiveWeight bounds).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.positiveWeight_bounds

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

Source. Repository-derived.

Commentary.

The parameterized positiveWeight bounds 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 (weightSum bounds).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.weightSum_bounds

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

Source. Repository-derived.

Commentary.

The parameterized weightSum bounds 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 (evenWeight).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.evenWeight

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.evenWeight (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized evenWeight 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 (evenWeight sum).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.evenWeight_sum

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

Source. Repository-derived.

Commentary.

The parameterized evenWeight sum 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 (evenWeight nonzero).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.evenWeight_nonzero

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

Source. Repository-derived.

Commentary.

The parameterized evenWeight nonzero 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 (evenWeight injective).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.evenWeight_injective

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

Source. Repository-derived.

Commentary.

The parameterized evenWeight injective 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 (evenWeight bound).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.evenWeight_bound

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

Source. Repository-derived.

Commentary.

The parameterized evenWeight 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.

Definition 1.10 (pairedExponent).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.pairedExponent

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.pairedExponent (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized pairedExponent 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 1.11 (oddExponent).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.oddExponent

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.oddExponent (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized oddExponent 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 1.12 (mixed weight nonzero).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.mixed_weight_nonzero

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

Source. Repository-derived.

Commentary.

The parameterized mixed weight nonzero 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 1.13 (pairedExponent injective).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.pairedExponent_injective

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

Source. Repository-derived.

Commentary.

The parameterized pairedExponent injective 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 1.14 (odd positiveExponent injective).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryIntegerWeights.odd_positiveExponent_injective

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

Source. Repository-derived.

Commentary.

The parameterized odd positiveExponent injective 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