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Canonical Prefix Count

Abstract

The connected source matchings reduce to canonical prefix pairs, whose finite fibers give the polynomial count.

Theorem 1.1 (Count canonical pairs).

Lean statement: D5/S3/Combinatorics/SemiMeanderSecondDiagonal/Stage5.canonical_card

Proof. Machine-checked in Lean as D5/S3/Combinatorics/SemiMeanderSecondDiagonal/Stage5.canonical_card (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Hunter Hogan (2026). OEIS A400429, semi-meanders by crossings and winding number. URL: https://oeis.org/A400429.

Commentary.

The canonical prefix pairs split into finite families whose cardinalities can be counted directly.

Theorem 1.2 (Polynomial arithmetic).

Lean statement: D5/S3/Combinatorics/SemiMeanderSecondDiagonal/Stage5.canonical_count_arithmetic

Proof. Machine-checked in Lean as D5/S3/Combinatorics/SemiMeanderSecondDiagonal/Stage5.canonical_count_arithmetic (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Hunter Hogan (2026). OEIS A400429, semi-meanders by crossings and winding number. URL: https://oeis.org/A400429.

Commentary.

The canonical family count simplifies to the second-diagonal polynomial after substituting n minus four.

References