Part II A2Orbital
Abstract
Part II A2Orbital.
Theorem 1.1 (actual A2 diagonal field orbital product).
Lean statement: D5/S3/FiniteGroups/NikolovSegal/PartIIA2Orbital.actual_A2_diagonal_field_orbital_product
Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/PartIIA2Orbital.actual_A2_diagonal_field_orbital_product (✓ std3). ∎
Source. Repository-derived.
Commentary.
For positive q, M > q(2q+1), and field size greater than 2(2q+1)^q, every diagonal/field automorphism tuple of length 3M admits one correction tuple before all targets in the actual upper-unitriangular subgroup. For every positive divisor tuple e of q, the ordered values have powers q/e and actual upper-unitriangular factor witnesses. The noncommutative central coordinate includes the cross term a*b.
Nikolov and Segal, On finitely generated profinite groups, I: Strong completeness and uniform bounds, Annals of Mathematics 165 (2007), 171-238, DOI 10.4007/annals.2007.165.171, Section 10, Proposition 10.2 and equations (45)-(50) (pages 228-232). Part II: Products in quasisimple groups, Annals of Mathematics 165 (2007), 239-273, DOI 10.4007/annals.2007.165.239, Theorem 1.2, Lemma 4.1 (pages 247-248), Section 6, Proposition 6.2, and Lemma 7.1(a) with its A2 orbital application (pages 257-261). These are formal adaptations and consequences of published mathematics, using actual matrix geometry, finite-field arithmetic and Sylow arguments. No originality claim or redistribution of the papers is made. The scalar, twisted and transitive conclusions here concern the SL3 family. Other families, CFSG exhaustion, general central covers, the full all-simple scalar supplier, all-length Proposition 10.2, width and restricted Burnside bounds, and strong completeness remain open.