Fourteen-Column Words with a Long Gap
Abstract
Rotating a long gap to the root reduces its score bound to positive compositions of fourteen.
Definition 1.1 (Cyclic rotation of a gap word).
Formalization. D5/S3/Combinatorics/GeneralizedPetersen/ZeroForcingThreeGapLong.rotateWord (✓ std3).
Source. Repository-derived.
Acknowledgement. Arnav Krishnan (2026). A correction to the Zero Forcing Number of the Generalized Petersen Graphs P(n,3). DOI: 10.48550/arXiv.2607.19412. URL: https://arxiv.org/abs/2607.19412v1.
Commentary.
The rotated word starts at r and retains the cyclic order of the gaps.
Definition 1.2 (Rotation as an index equivalence).
Formalization. D5/S3/Combinatorics/GeneralizedPetersen/ZeroForcingThreeGapLong.rotation (✓ std3).
Source. Repository-derived.
Acknowledgement. Arnav Krishnan (2026). A correction to the Zero Forcing Number of the Generalized Petersen Graphs P(n,3). DOI: 10.48550/arXiv.2607.19412. URL: https://arxiv.org/abs/2607.19412v1.
Commentary.
Cyclic addition by r is an equivalence of Fin c; its inverse subtracts r.
Theorem 1.3 (A long gap bounds the ten-slot score).
Proof. Machine-checked in Lean as D5/S3/Combinatorics/GeneralizedPetersen/ZeroForcingThreeGapLong.long_gap_score (✓ std3). ∎
Source. Repository-derived.
Acknowledgement. Arnav Krishnan (2026). A correction to the Zero Forcing Number of the Generalized Petersen Graphs P(n,3). DOI: 10.48550/arXiv.2607.19412. URL: https://arxiv.org/abs/2607.19412v1.
Commentary.
A positive cyclic word with five through eight entries, total fourteen, and an entry at least eight has T at most four times its length plus five. The rooted positive-composition family contains twenty-two words.
References
- Truth anchor:
D5/S3/Combinatorics/GeneralizedPetersen/ZeroForcingThreeGapLong.long_gap_score - Truth anchor:
D5/S3/Combinatorics/GeneralizedPetersen/ZeroForcingThreeGapLong.rotateWord - Truth anchor:
D5/S3/Combinatorics/GeneralizedPetersen/ZeroForcingThreeGapLong.rotation - Dependency: D5/S3/Combinatorics/GeneralizedPetersen/ZeroForcingThreeGapMaskBridge