Three counterexamples from fall 2026, each found by working with large language models: what happened, and what worked.
The question. At Route 81 (Syracuse University, 26 September 2026), Jen Biermann spoke on graphs and their cover ideals. In their paper, Biermann, Chau, Kara, O'Keefe, Skelton, Sosa Castillo and Vien conjecture that every connected graph G on n vertices satisfies
|reg(R/J(G)) − deg h(R/J(G))| ≤ ⌈n/2⌉ − 2,
where J(G) is the cover ideal (arXiv:2602.10376, Conjecture 12.4). The conjecture comes down to a bound on how many times −1 can be a root of the graph's independence polynomial.
What happened. I really liked the talk, and the question was nice and explicit: one clean inequality about graphs. I just really wanted to know if it was true. The talk was in the afternoon, and that night I was still thinking about it, so we set up a big search for a counterexample, starting from graph databases. By 2 a.m., in the hotel, we had an explicit one: a connected graph on 99 vertices and 1,056 edges with independence polynomial
(1 + x)⁶⁶ + 33x(1 + x)⁵⁰.
So −1 is a root 50 times, more than half the number of vertices, and the edge ring has depth 2. That breaks the conjectured bound by one. The construction also scales up to an infinite family where the excess grows without bound, so no added constant can rescue the inequality.
→ Ninety-nine vertices and a fifty-fold root at −1
The question. With the conjecture false, how large can the multiplicity of −1 actually get? Bhardwaj, Chau, Ikram, Lather, Nanjangud and Venugopal study this number directly. They pin down most of what can happen and reduce the rest to their Question 5.8 (arXiv:2607.08480).
What happened. Next we tried to prove a refined bound. It kept failing. Each time we modified the conjecture, a new computation broke it, until the counterexample appeared: a family of connected graphs of depth 2 in which the multiplicity of −1 is a fraction 1 − 2⁻ᵇ of the number of vertices, for any b. So the multiplicity can be pushed as close to the size of the graph as you like. These graphs also give new violations of Conjecture 12.4, and they realize pairs in the gap that Question 5.8 asks about.
The construction uses one large independent set and two cliques of "centres", with each centre joined to an interval of a cyclically ordered ground set. It is almost certainly well known somewhere in the literature. The LLMs helped find the example and connect the dots.
What worked: massive targeted computations, slowly refining the bound, and disproving each conjecture we made until the construction leapt out.
→ Graphs whose multiplicity at −1 approaches their size
The question. For an irreducible representation ρ of SL₂(ℤ) of dimension d, the vector-valued modular forms form a free module of rank d over ℂ[E₄, E₆]. Equivalently, the corresponding vector bundle on the moduli stack of elliptic curves splits as a sum of line bundles, and the weights of a free basis give its splitting type. The three-term inequality, conjectured by Franc–Mason, Franc, and Franc–Rayan, says that no weight has more generators than its two neighbours together:
mₖ ≤ mₖ₋₂ + mₖ₊₂.
What happened. Cameron Franc asked me to work on this in the first year of my postdoc. We made some minor progress, but nothing substantial. I came back to it with modern tools, this time with an eye for a counterexample. The goal was to build one. The LLM helped with the strategy and did the linear algebra to build the transition functions: running the usual correspondence backwards, starting from a matrix of modular forms for the modular derivative and reading the representation off its solutions.
Then the counterexample: an irreducible 13-dimensional ρ with profile (1,1,1,1,1,3,1,1,1,1,1). That is three generators in weight 10 against one each in weights 8 and 12. It is given by an exact rational matrix, and its irreducibility is certified exactly. Along the way we also found a short, computer-free proof that the inequality holds in every dimension up to 12, so 13 is the first dimension where it fails.