Forts for the innerNext Anchor

Abstract

An ordered family assigns a disjoint fort to every seven-vertex set with the specified first force.

Theorem 1.1 (A complete ordered fort family).

$$\exists rows\in \mathrm{List}\left(\mathrm{Nat}\times \mathrm{Nat}\right),\mathrm{familyOrderOK}\left(rows\right)=\mathrm{true}\wedge \left(\forall row\in \mathrm{Nat}\times \mathrm{Nat},row\in rows\Rightarrow \left(\mathrm{candidateShapeOK}\left(\mathrm{innerNext},\mathrm{fst}\left(row\right)\right)=\mathrm{true}\wedge \mathrm{rowOK}\left(row\right)=\mathrm{true}\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/Combinatorics/GeneralizedPetersen/ZeroForcingThreeFortData5.certifiedFamily (✓ 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 innerNext family has 7315 strictly increasing candidate masks. Each row satisfies candidateShapeOK for this anchor and rowOK, whose fortOK and disjointOK tests certify a nonempty fort disjoint from the candidate.

References

• Truth anchor: D5/S3/Combinatorics/GeneralizedPetersen/ZeroForcingThreeFortData5.certifiedFamily
• Dependency: D5/S3/Combinatorics/GeneralizedPetersen/ZeroForcingThreeFiniteCore