Literal words and twisted products
Abstract
Literal words and twisted products.
Theorem 1.1 (crossing substitution).
Lean statement: D5/S3/FiniteGroups/NikolovSegal/Equation47WordCoupling.crossing_substitution
Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Equation47WordCoupling.crossing_substitution (✓ std3). ∎
Source. Repository-derived.
Commentary.
Two distinct keys occurring with opposite signs in a literal crossing can be replaced by genuine scalar witnesses for a twisted product, while every other key is retained. All automorphisms are arbitrary and the word order is preserved.
Theorem 1.2 (solve extracted word).
Lean statement: D5/S3/FiniteGroups/NikolovSegal/Equation47WordCoupling.solve_extracted_word
Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Equation47WordCoupling.solve_extracted_word (✓ std3). ∎
Source. Repository-derived.
Commentary.
An actual D-step extraction and the literal twisted PRODUCT input solve the original word for every 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.