Ultra log-concavity for increasing families generated by singletons and pairs

Unrefereed research candidate. In one sentence: this paper proves Xie and Xu's Conjecture 4.1 that the numbers of dependent k-vertex sets of a graph are ultra log-concave, in a stronger weighted form. For a graph on n vertices, call a set of vertices dependent if it contains at least one edge, and let dk count the dependent k-sets. Horrocks (J. Combin. Theory Ser. B, 2002) proved that dk is log-concave. Xie and Xu (Electron. J. Combin. 33(4), P4.9, 2026) conjectured the stronger ultra log-concave form: dk/C(n,k) is log-concave, so the size of a random dependent set is at least as concentrated as a binomial distribution. The main theorem proves this for every increasing family generated by singletons and pairs (graphs with loops) and arbitrary nonnegative element weights: k(n−k)Ak2 ≥ (k+1)(n−k+1)Ak−1Ak+1. The route is a two-colouring theorem (the binomially normalised number of k-sets S with S and its complement both dependent is symmetric and unimodal, strengthening Horrocks 2000), a marked-downset lemma from Katona's intersection-shadow theorem, a critical-element double count, closure under conditioning, and the Kahn–Neiman antipodal-pairs criterion. The paper also proves a weighted strengthening of Xie–Xu Conjecture 4.4 for additive 2-matroids of rank at most four or on at most seven elements (the general conjecture remains open, with an explicit eight-element obstruction to the method), shows that witnesses of size three break the theorem (a five-element family with counts (0,1,4,10,5,1) gives 96 < 120), and derives ULC of noncover-complex face numbers and a variance bound for Bernoulli variables conditioned to hit an edge. The archive contains the manuscript (PDF and LaTeX source), a standard-library exact-integer check script (41,657 families, 124,971 weighted checks, 258,132 marked comparisons; valid under python -O) with its recorded output, an exhaustive six-element 2-matroid enumeration in C++, supplementary notes, the claim and prior-art records, a rating-free internal review summary, a claim-by-claim AI index, a bibliography from Crossref and arXiv records, CI and SHA-256 sums. Assurance: written proofs; the computations are finite checks, not proofs; internal AI-assisted review only (not peer review, independent reproduction or formal verification); bounded prior-art search; Horrocks (2002) read in full and compared. Creator: Anonymous. Text and data CC0-1.0; code MIT. Version 0.2.0-candidate corrects the prior-art record. Version 0.1.0-candidate (10.5281/zenodo.23935650) credited Horrocks (2002) from its abstract only. Horrocks proves pk2 ≥ pk−1pk+1 + pk, unweighted and unnormalised, which does not imply ultra log-concavity. His proof sorts pairs of sets by intersection and union and reduces to doubly dependent counts. That method is now credited as a precursor of this paper's conditioning and antipodal-pairs route. The binomially normalised two-colouring inequality, the marked-downset lemma and the critical-element count are not in his paper. No theorem, proof or computation changed.