Lexicographic ordering and full indexed formulation of optimal bases

Determine whether the conjectured lexicographic ordering of the dual-feasible optimal bases and the full indexed formulation associated with Conjecture 5.10 of Berend, Ernst, Kontorovich, and Kumar hold for the extremal problem of maximizing the probability that all n k-wise independent Bernoulli variables equal one.

Background

The paper proves a reflection principle for interior feasibility sets and for transitions between optimal supports when k is even. Under the node involution s mapsto n-1-s, with the distinguished node n fixed, a transition at p is mapped to a reversed transition at 1-p, preserving its multiplicity. This establishes the reflection assertion associated with Conjecture 5.11 of Berend, Ernst, Kontorovich, and Kumar.

The authors explicitly limit the scope of their result: they do not establish the conjectured lexicographic ordering of the bases or the complete indexed formulation associated with Conjecture 5.10. Thus, determining whether those additional structural claims are valid remains outside the results proved in the paper.

References

We make no assertion about the conjectured lexicographic ordering of the bases or the full indexed formulation ofConjecture 5.10; our reflection is the explicit set map~eq:star.

eq:star:

J∗={n−1−s:s∈J},I∗=J∗∪{n}.J^*=\{n-1-s:s\in J\},\qquad I^*=J^*\cup\{n\}.

— Reflection of optimal supports for k-wise independent bits  (2609.25595 - Hermann, 22 Sep 2026) in Section 3, immediately after Corollary 3.1, in the paragraph concluding the discussion of exceptional points and local transitions