Code, data and certificates for: The crossing numbers of W4 + S_n and W4 + W_n.

Code, data and exact certificates for the computer-assisted proofs that cr(W4 + S_n) = 6Z(n) + 3n + 2⌊n/2⌋ + 1 (n ≥ 1) and cr(W4 + W_n) = 6Z(n) + 3n + 2⌊n/2⌋ + 8 (n ≥ 3), settling the case m = 4 of the conjectures of Staš, Fortes and Švecová (Axioms 13 (2024), 427). It contains the enumeration of all 2140 good drawings of K_{1,1,2,2}, the star-cost and partner tables, 1298 rational certificates, and a standalone verifier that needs only the Python standard library (./verify.sh). It also contains the logs of the independent re-computations.Version 1.1 fixes a gap in the case n = 4 of Theorem 2, which an external reader found: partners missing from the cost-8 tables are now included, and two independent searches confirm the bound. It also adds an exact audit of Theorem 2 and a check script that covers the whole proof.