The exhaustive-pairs question for caterpillar principal specializations

Bingham, Celano, Commins, Orellana, Pan and Simone, Principal specializations of the t-chromatic symmetric function, arXiv:2610.31785v1, Remark 3.11, second question:

And if so, do they continue to be the only pairs of compositions which do?

The assertion is that for every n at least one and every two source-admissible positive compositions of n, unequal ordinary second and identical ordinary third principal specializations imply that their unordered graph-isomorphism pair is the displayed base pair {(5,4,3),(4,5,3)} or {(5,j,j,4,3),(4,j,j,5,3)} for some positive integer j. All graph relabelings, reversals and pair exchanges are included. There is no size cutoff. The first all-j question is a separate result.

Definition 2.1 admits (1), and every nonempty positive composition whose two endpoint parts are at least two, including length-one stars. The actual graph has one spine vertex per part and a_i−1 pendant leaves at the ith spine vertex. Definitions 2.4, 2.7 and 2.9 and equation (2.1) give the ordinary specialization as the sum over all proper colorings into {0,...,k−1} of X raised to the sum of vertex colors. Surjectivity, weights, the t refinement and numerical evaluation of X are absent.

The source found only its listed pairs through total size sixteen. A pair outside the full listed graph-isomorphism closure refutes the unrestricted assertion, without requiring a classification of all other pairs.

The listed pairs do not exhaust the phenomenon. The counterexample is

$$A=(2,2,2,2,2,1,4,3),\qquad B=(2,2,3,2,3,4,2).$$

Both compositions have total size eighteen and admissible endpoints. This counterexample refutes the assertion over its full universal domain.

Use the original Vertex, edgeRel, caterpillar, ps, alpha and beta definitions from BinghamCaterpillarPs3Family. Its coloring factorization accounts for every actual proper coloring exactly once. For spine color c, a part m contributes

$$X^c\left(\sum_{0\le d<k,\ d\ne c}X^d\right)^{m-1}.$$

Sum products of these weights over all consecutive unequal spine colors. Finite polynomial normalization gives

$$P_2(A)=X^8+X^{10},\qquad P_2(B)=2X^9,\qquad P_3(A)=P_3(B).$$

The coefficient of degree eight separates the second polynomials. The source's complex coefficient polynomials are defined as the same actual coloring sums over C[X]. A map-of-sums identity and injectivity of the coefficient map Z[X] to C[X] transfer both comparisons.

A vertex with two distinct neighbors must be on the spine, since a pendant vertex has only its incident spine vertex as a neighbor. All eight spine vertices of A have two distinct neighbors, including the internal vertex whose part is one. An isomorphism from A to a listed member would induce an injection from eight indices into its spine indices. The base members have three indices; every positive-j member has five, including j=1. Cardinality therefore excludes all such graph isomorphisms. Since either order of a listed pair would require A to be isomorphic to a listed member, the full unordered pair closure is excluded. Lemma 2.2 and its length hypotheses are not used.

A discrepancy in any coefficient of the actual third coloring polynomials, equality of the second polynomials, or an actual graph isomorphism placing the witness pair in any listed class would invalidate this counterexample. Agreement only at finitely many values of X would not establish the needed identity.

D5/S3/Combinatorics/Graph/PrincipalSpecialization/BinghamCaterpillarUniquenessRefutation.result negates the full exhaustive-pairs assertion. The coloring-sum identities give unequal second and equal third ordinary specializations over the source's complex coefficient ring. The nonleaf invariant excludes A from every base and positive-j family member, and therefore excludes the witness from the full unordered graph-isomorphism closure.

theorem; the exhaustive-pairs assertion is false.

The earlier first-question theorem, that every positive-j listed pair exhibits the phenomenon, is compatible with this refutation. A classification of all pairs, the smallest counterexample size, other infinite families, four-color questions and t-refined analogues remain unresolved here. None follows from this finite counterexample alone.

The bounded source and literature screen for the external question did not locate an earlier settlement. Citation and third-party code searches were incomplete. This does not establish exhaustive absence, originality, priority or minimality.