From Data to Combinatorial Multivector field Through an Optimization-Based Framework

Both the paper and the candidate solution prove that any 0–1 assignment z(σ,τ) satisfying constraints (13) partitions K by toplex and that each block is convex. The paper’s Theorem 3.6 defines V_τ := {σ ≤ τ : z(σ,τ)=1} ∪ {τ}, shows these sets form a partition by uniqueness-of-toplex assignment, and proves convexity by propagating z=1 along immediate face–coface steps and then along chains, exactly as the candidate does. The core steps—(i) uniqueness of assignment for non-toplexes, (ii) disjointness of toplex-containing sets, and (iii) upward-closure/convexity via the z(σ_i,τ) ≤ z(σ_j,τ) constraint—align one-for-one with the model’s argument. See the paper’s statement of (13) and its proof of Theorem 3.6 (definition of V_τ, partition, and convexity) in the provided PDF .