Negative elementary coefficients and fixed-template obstructions for three-legged spiders

We give an explicit negative elementary coefficient for the chromatic symmetric function of S(3A+1,3k,2) whenever k ≥ 4 and k ≤ A ≤ 2k. The coefficient indexed by 4⁴ 3^(A+k−4) is a negative multiple of a quartic, whose positivity on the full closed parameter region is certified by a polynomial with 25 positive coefficients. We also identify two limitations of coefficient templates. For S(17,12,2), every partition with parts only 3 and 4 has a positive coefficient, although the coefficient indexed by 5⁴ 4³ is −31920. More generally, for fixed k and any fixed finite multiset of parts at least 4, adjoining sufficiently many parts 3 gives a positive coefficient. An explicit infinite family satisfies the earlier necessary criteria considered here and is excluded by our interval theorem.