Proposition 8.4 and normalized-value inputs
Abstract
Proposition 8.4 and normalized-value inputs.
Theorem 1.1 (proposition8 4).
Lean statement: D5/S3/FiniteGroups/NikolovSegal/CrossingKernelAPI.proposition8_4
Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/CrossingKernelAPI.proposition8_4 (✓ std3). ∎
Source. Repository-derived.
Commentary.
Balance, compressed colour bound n and support at least n+2D give a genuine D-step extraction, exactly 2D distinct selected keys, exact support loss, and a final residual avoiding those keys.
Theorem 1.2 (extraction from owner counts).
Lean statement: D5/S3/FiniteGroups/NikolovSegal/CrossingKernelAPI.extraction_from_owner_counts
Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/CrossingKernelAPI.extraction_from_owner_counts (✓ std3). ∎
Source. Repository-derived.
Commentary.
The proved normalized signed count formula and actual colour bound instantiate the literal extraction theorem on the actual root residual.
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.