Component movement and residual support budgets
Abstract
Component movement and residual support budgets.
Theorem 1.1 (component movement value bound).
Lean statement: D5/S3/FiniteGroups/NikolovSegal/Equation47ComponentBudget.component_movement_value_bound
Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Equation47ComponentBudget.component_movement_value_bound (✓ std3). ∎
Source. Repository-derived.
Commentary.
The movement sum on one actual powered component is at most twice its number of genuine nonbase value coordinates.
Theorem 1.2 (normalized component typeII residual budget).
Lean statement: D5/S3/FiniteGroups/NikolovSegal/Equation47ComponentBudget.normalized_component_typeII_residual_budget
Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Equation47ComponentBudget.normalized_component_typeII_residual_budget (✓ std3). ∎
Source. Repository-derived.
Commentary.
For a component of n at least two vertices, movement at least (4+2D)n leaves residual support of at least n+2D+1 after its n minus one forest links.
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.