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Literal crossings before reduction

Abstract

Literal crossings before reduction.

Theorem 1.1 (balanced nontrivial crossing).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/BalancedCrossingLift.balanced_nontrivial_crossing

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

Source. Repository-derived.

Commentary.

A balanced word with nontrivial free-group reduction has an actual literal crossing; the reduction crossing lifts to a sublist of the original word.

Theorem 1.2 (crossing balanced).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/BalancedCrossingLift.crossing_balanced

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

Source. Repository-derived.

Commentary.

ADCBE crossing surgery preserves balance of every remaining key.

Theorem 1.3 (crossing support card).

Lean statement: D5/S3/FiniteGroups/NikolovSegal/BalancedCrossingLift.crossing_support_card

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

Source. Repository-derived.

Commentary.

Deleting the crossing keys lowers support by exactly two.

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