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Unitary Involution Field

Abstract

Actual unitary involution field 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 (fixedField).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryInvolutionField.fixedField

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryInvolutionField.fixedField (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized fixedField 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 (mem fixedField).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryInvolutionField.mem_fixedField

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

Source. Repository-derived.

Commentary.

The parameterized mem fixedField 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 (finrank fixedField).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryInvolutionField.finrank_fixedField

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

Source. Repository-derived.

Commentary.

The parameterized finrank fixedField 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 (card field).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryInvolutionField.card_field

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

Source. Repository-derived.

Commentary.

The parameterized card field 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 (fixedAlgEquiv).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryInvolutionField.fixedAlgEquiv

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryInvolutionField.fixedAlgEquiv (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized fixedAlgEquiv 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 (involution eq pow).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryInvolutionField.involution_eq_pow

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

Source. Repository-derived.

Commentary.

The parameterized involution eq 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 (trace formula).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryInvolutionField.trace_formula

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

Source. Repository-derived.

Commentary.

The parameterized trace formula 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 (norm formula).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryInvolutionField.norm_formula

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

Source. Repository-derived.

Commentary.

The parameterized norm formula 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 (trace surjective).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryInvolutionField.trace_surjective

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

Source. Repository-derived.

Commentary.

The parameterized trace surjective 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 (norm units surjective).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryInvolutionField.norm_units_surjective

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

Source. Repository-derived.

Commentary.

The parameterized norm units surjective 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 (normOne).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryInvolutionField.normOne

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryInvolutionField.normOne (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized normOne 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 (mem normOne).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryInvolutionField.mem_normOne

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

Source. Repository-derived.

Commentary.

The parameterized mem normOne 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 (card normOne).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryInvolutionField.card_normOne

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

Source. Repository-derived.

Commentary.

The parameterized card normOne 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