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arXiv · 2607.23926

Balance Constants, Majority Cycles, and the Gold Partition Conjecture through Fourteen Elements

Anish Gupta
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Abstract

We determine the exact extremal balance data of all 1,338,193,159,771 unlabeled posets on fourteen elements. The least balance constant exceeding $1/3$ is $37/106$. The least over posets that are not nontrivial ordinal sums is $254/725$, attained by a ladder with broken rungs; this confirms a conjecture of Peczarski at order 14, while orders 12 and 13 reproduce De Loof, De Baets, and De Meyer. No balance constant lies in the gap above $1/3$ that Peczarski conjectures to be empty. Exactly 128 classes attain $1/3$, and every one is an ordinal sum of singletons and copies of the three-element poset with one relation, a family whose non-chain members are counted by $a(n)-1$, where $a(n)=a(n-1)+a(n-3)$. In the linear-extension-majority digraph the longest simple cycle has length 8, against 7 at order 13, and exactly 30 classes attain it; of the thirteen such classes whose witnesses the census retains, nine have a cycle spectrum containing no odd cycle at all. A second exhaustive pass over the same classes verifies Peczarski's Gold Partition Conjecture through fourteen elements, extending his order-11 frontier and implying in particular that the $1/3$-$2/3$ Conjecture holds through order 14. All arithmetic is exact and every extremal witness is recomputed by an independent program.

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BibTeXRIS

Anish Gupta. 2026-07-27. Balance Constants, Majority Cycles, and the Gold Partition Conjecture through Fourteen Elements. https://arxiv.org/abs/2607.23926

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