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Component colour and exact support counts

Abstract

Component colour and exact support counts.

Theorem 1.1 (normalized connected residual colour).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Equation47ValueSequence.normalized_connected_residual_colour

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Equation47ValueSequence.normalized_connected_residual_colour (✓ std3). ∎

Source. Repository-derived.

Commentary.

On a connected powered component, the residual compressed colour list embeds in the bound for the actual number of vertices.

Theorem 1.2 (normalized component residual colour).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Equation47ValueSequence.normalized_component_residual_colour

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Equation47ValueSequence.normalized_component_residual_colour (✓ std3). ∎

Source. Repository-derived.

Commentary.

For an arbitrary already contracted true component, the root residual uses that component own cardinality in its colour bound.

Theorem 1.3 (normalized component residual pair count).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/Equation47ValueSequence.normalized_component_residual_pair_count

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/NikolovSegal/Equation47ValueSequence.normalized_component_residual_pair_count (✓ std3). ∎

Source. Repository-derived.

Commentary.

Each component loses exactly its own n minus one forest labels from its genuine nonbase value support.

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.

References