An Efficient, Lean 4 Verified Necessary and Sufficient Condition for Membership in Pedigree Polytopes

The set $P_n$ consists of the characteristic vectors of pedigrees, which are combinatorial encodings in one-to-one correspondence with Hamiltonian cycles. For $n\ge 4$ and ${\tau }_n=\left(\genfrac{}{}{0px}{}{n}{3}\right)-1$, the Membership Problem for the Pedigree Polytope (M3P) asks whether a given vector $X\in {\mathbb{Q}}^{{\tau }_n}$ belongs to $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}(P_n)$. This paper gives a complete account, with the full proof statements and machine-verified Lean~4 certificates, of the necessary and sufficient (\NS) condition established in Chapter~5 of \emph{Pedigree Polytopes: New Insights on Computational Complexity of Combinatorial Optimisation Problems} (Arthanari, 2023): membership of $X$ in $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}(P_n)$ is characterised by the feasibility, at every stage $k=4,\dots ,n-1$, of a recursively constructed layered network, together with, when needed, a multicommodity flow problem $\mathrm{M}\mathrm{C}\mathrm{F}(k-1)$ achieving its maximum possible flow $z_{\max }$. We present the construction, the \NS{} condition itself, and the proofs of this characterisation, together with the resulting polynomial-time algorithm for M3P. Both directions of the \NS{} condition — sufficiency and necessity (the multicommodity-flow feasibility argument of the book’s Chapter~5) — are machine-verified in Lean~4/Mathlib4 with zero unresolved $\mathtt{\text{sorry}}$s in either dependency chain. The \NS{} characterisation therefore follows directly from the two machine-checked implications. The complete Lean~4 project is publicly available. A companion introductory article (arXiv:2507.09069) presents the same material for a broader readership, without the Lean~4 verification; the present paper adds full statements, proof sketches, and the complete machine-checked certificates.