Interval Spectra of Small Finite Posets: Four Invariants Through Seven Points and an Eight-Point Obstruction

Abstract

Scientific summary For a finite partially ordered set (poset), the interval-size distribution counts comparable pairs according to the number of elements strictly between them. We prove that, for posets with at most seven elements and a fixed cardinality, this complete distribution is determined by four statistics: the number of comparable pairs, the number of cover relations, and the first two raw moments of interval size. The bound is sharp. We construct two eight-element posets with the same four statistics but different interval-size distributions. Adding isolated elements extends this non-uniqueness to every larger finite cardinality. We also derive an exact identity for the change in the interval polynomial under the insertion of a single element into an arbitrary finite poset, together with update formulas for the four statistics. The reconstruction theorem is proved by finite combinatorial arguments. Reproducible computations independently check the small-order classifications and the insertion identity. Plain-language summary A partially ordered set describes which elements precede others, while allowing some pairs to remain incomparable. For two elements in a given order, we can count how many elements lie between them. These counts reveal features of the order that are not captured by the total number of relations alone. This paper asks whether the entire distribution of these counts can be recovered from just four summary measurements. The answer is yes for structures of up to seven elements, when the total number of elements is known. At eight elements, however, two structures can agree on all four measurements and still have different interval-size distributions. Why it matters Mathematical and physical models often compress complicated structures into a small number of observables. Our result identifies an exact limit to one such compression: four simple order statistics retain the full interval-size distribution through seven elements, but cease to do so in general at eight. The one-element insertion identity also gives a direct way to calculate how these observables change when an order is modified. Interval counts are used in causal set theory, but the results here are combinatorial. They do not imply that the poset itself is uniquely determined, nor do they establish Lorentzian realizability or continuum-limit behavior. Connection to related preprints This work complements two earlier preprints by the author: Renewal structure and critical dimension of causal set d'Alembertians and Incomplete phase cancellation in the causal set path sum over layered orders. Those studies use interval-layer information in discrete operators and causal-set path sums. The present paper examines a distinct mathematical question: how much of that interval information is retained by four summary statistics. Its results do not establish consequences for the dynamics studied in the earlier works. Reproducibility The accompanying manuscript source, finite-poset classification data, and verification programs allow readers to reproduce the reported small-order calculations and test the one-element insertion identity. The mathematical proof does not depend on the exhaustive checks.