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Unitary Rank Two Geometry

Abstract

Actual unitary rank two geometry 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 (J).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryRankTwoGeometry.J

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryRankTwoGeometry.J (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized J 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 (J eq).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryRankTwoGeometry.J_eq

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

Source. Repository-derived.

Commentary.

The parameterized J eq 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 (exists antifixed).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryRankTwoGeometry.exists_antifixed

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

Source. Repository-derived.

Commentary.

The parameterized exists antifixed 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 (adjoint mul).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryRankTwoGeometry.adjoint_mul

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

Source. Repository-derived.

Commentary.

The parameterized adjoint mul 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 (hermitian iff inverse).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryRankTwoGeometry.hermitian_iff_inverse

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

Source. Repository-derived.

Commentary.

The parameterized hermitian iff inverse 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 (inverse matrix).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryRankTwoGeometry.inverse_matrix

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

Source. Repository-derived.

Commentary.

The parameterized inverse 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 (hermitian iff entries).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryRankTwoGeometry.hermitian_iff_entries

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

Source. Repository-derived.

Commentary.

The parameterized hermitian iff entries 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 1.8 (specialUnitary).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryRankTwoGeometry.specialUnitary

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryRankTwoGeometry.specialUnitary (✓ std3).

Source. Repository-derived.

Commentary.

Actual SU2 for the Hermitian anti-diagonal, as a subgroup of native SL2.

Theorem 1.9 (mem specialUnitary).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryRankTwoGeometry.mem_specialUnitary

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

Source. Repository-derived.

Commentary.

The parameterized mem specialUnitary 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