Retained literal variables and fibre counting
Abstract
Retained literal variables and fibre counting.
Theorem 1.1 (solve extracted word retaining).
Lean statement: D5/S3/FiniteGroups/NikolovSegal/ExtractionCounting.solve_extracted_word_retaining
Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/ExtractionCounting.solve_extracted_word_retaining (✓ std3). ∎
Source. Repository-derived.
Commentary.
The twisted PRODUCT input solves the original extracted word while preserving every unextracted variable and the final residual value.
Theorem 1.2 (solution fibre card lower bound).
Lean statement: D5/S3/FiniteGroups/NikolovSegal/ExtractionCounting.solution_fibre_card_lower_bound
Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/ExtractionCounting.solution_fibre_card_lower_bound (✓ std3). ∎
Source. Repository-derived.
Commentary.
For finite variable and scalar types, actual restriction to the unextracted variables is surjective. The real literal-word solution fibre has cardinality at least |S| raised to (|V| minus 2D).
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.