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

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.

arXiv comment16 pages, 3 figures. Version 2 adds an exhaustive order-14 balance and linear-extension-majority census, including extremal balance constants, the equality locus, and cycle spectra; the Gold Partition result is unchanged. Code, data, and archived computational artifacts: https://doi.org/10.5281/zenodo.21696940