Balance and support after leaf elimination
Abstract
Balance and support after leaf elimination.
Theorem 1.1 (normalized value residual sign count).
Lean statement: D5/S3/FiniteGroups/NikolovSegal/Equation47ValueSequenceBalance.normalized_value_residual_sign_count
Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Equation47ValueSequenceBalance.normalized_value_residual_sign_count (✓ std3). ∎
Source. Repository-derived.
Commentary.
After eliminating nonroots, each retained nonbase variable occurs once with each sign in its own component root residual and nowhere else.
Theorem 1.2 (normalized value residual support).
Lean statement: D5/S3/FiniteGroups/NikolovSegal/Equation47ValueSequenceBalance.normalized_value_residual_support
Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Equation47ValueSequenceBalance.normalized_value_residual_support (✓ std3). ∎
Source. Repository-derived.
Commentary.
The exact support is the set of genuine nonbase value coordinates outside the eliminated forest labels in the specified component.
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.