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Conditional all-simple small-large extension

Abstract

Conditional all-simple small-large extension.

Theorem 1.1 (uniform twisted at max of large scalar one).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/UniformTwistedSmallLarge.uniform_twisted_at_max_of_large_scalar_one

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/UniformTwistedSmallLarge.uniform_twisted_at_max_of_large_scalar_one (✓ std3). ∎

Source. Repository-derived.

Commentary.

At fixed M and C, assume only the literal large-simple q=1 scalar PRODUCT input with genuine pre-target corrections. Small simple groups use proved bounded cardinality coverage; larger simple groups use Lemma 4.1. Identity padding yields a common length max(M,C+1).

Theorem 1.2 (uniform all simple twisted of large scalar one).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/UniformTwistedSmallLarge.uniform_all_simple_twisted_of_large_scalar_one

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/UniformTwistedSmallLarge.uniform_all_simple_twisted_of_large_scalar_one (✓ std3). ∎

Source. Repository-derived.

Commentary.

If some M and C satisfy the still-open uniform large-simple scalar-one PRODUCT premise, a positive common twisted length exists for every finite noncommutative simple group. The uniform length precedes every group, arbitrary automorphism tuple and target.

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 8, Lemma 8.3 and Proposition 8.4, Section 9, Proposition 9.1 (pages 223-226), and 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 and Lemma 4.1 (pages 247-248). These are formal adaptations and conditional consequences of published mathematics, with elementary finite-set growth supporting bounded-small coverage; no originality claim or redistribution of the papers is made. Large-simple scalar PRODUCT existence, quasisimple central covers, all-length Proposition 10.2, uniform width and restricted Burnside bounds, and full strong completeness remain unproved.

References