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Unitary Width Levi Geometry

Abstract

Actual unitary width levi 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 (leftEnd).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.leftEnd

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.leftEnd (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized leftEnd 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 (rightEnd).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.rightEnd

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.rightEnd (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized rightEnd 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 1.3 (mid).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.mid

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.mid (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized mid 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 (left ne right).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.left_ne_right

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

Source. Repository-derived.

Commentary.

The parameterized left ne right 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 (mid ne left).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.mid_ne_left

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

Source. Repository-derived.

Commentary.

The parameterized mid ne left 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 (mid ne right).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.mid_ne_right

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

Source. Repository-derived.

Commentary.

The parameterized mid ne right 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 (mid injective).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.mid_injective

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

Source. Repository-derived.

Commentary.

The parameterized mid 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 1.8 (interior eq mid).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.interior_eq_mid

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

Source. Repository-derived.

Commentary.

The parameterized interior eq mid 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 1.9 (Endpoints).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.Endpoints

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.Endpoints (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized Endpoints 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 (endpoints one).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.endpoints_one

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

Source. Repository-derived.

Commentary.

The parameterized endpoints one 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 1.11 (endpoints mul).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.endpoints_mul

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

Source. Repository-derived.

Commentary.

The parameterized endpoints 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 1.12 (endpoints inv).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.endpoints_inv

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

Source. Repository-derived.

Commentary.

The parameterized endpoints inv 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 (middle product).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.middle_product

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

Source. Repository-derived.

Commentary.

Middle blocks multiply on the actual two-endpoint stabilizer.

Definition 1.14 (centralEmbed).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.centralEmbed

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.centralEmbed (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized centralEmbed 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 1.15 (slCentral endpoints).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.slCentral_endpoints

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

Source. Repository-derived.

Commentary.

The parameterized slCentral endpoints 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 1.16 (centralEmbed endpoints).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.centralEmbed_endpoints

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

Source. Repository-derived.

Commentary.

The parameterized centralEmbed endpoints 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 1.17 (slCentral middle).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.slCentral_middle

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

Source. Repository-derived.

Commentary.

The parameterized slCentral middle 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 1.18 (centralEmbed middle).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.centralEmbed_middle

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

Source. Repository-derived.

Commentary.

The parameterized centralEmbed middle 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 1.19 (UpRadical).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.UpRadical

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.UpRadical (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized UpRadical 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 1.20 (DownRadical).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.DownRadical

Formalization. D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.DownRadical (✓ std3).

Source. Repository-derived.

Commentary.

The parameterized DownRadical 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 1.21 (upRadical positive).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.upRadical_positive

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

Source. Repository-derived.

Commentary.

The parameterized upRadical positive 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 1.22 (downRadical negative).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.downRadical_negative

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

Source. Repository-derived.

Commentary.

The parameterized downRadical negative 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 1.23 (upRadical endpoints).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.upRadical_endpoints

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

Source. Repository-derived.

Commentary.

The parameterized upRadical endpoints 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 1.24 (upRadical middle).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.upRadical_middle

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

Source. Repository-derived.

Commentary.

The parameterized upRadical middle 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 1.25 (slRadical of endpoints middle).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.slRadical_of_endpoints_middle

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

Source. Repository-derived.

Commentary.

Support recognition is proved from entries, not assumed as a group oracle.

Theorem 1.26 (upRadical of endpoints middle).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.upRadical_of_endpoints_middle

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

Source. Repository-derived.

Commentary.

The parameterized upRadical of endpoints middle 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 1.27 (central conjugate upRadical).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Unitary/UnitaryWidthLeviGeometry.central_conjugate_upRadical

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

Source. Repository-derived.

Commentary.

Arbitrary actual central SU conjugation preserves the genuine upper radical.

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