Co-PEER: Auditing Mathematics with Proofs of Error

A precise counterexample can reveal a recurring failure pattern, motivate a correction, and lead to a theorem about an entire family. We introduce Co-PEER, a workflow for developing and checking such counterexamples, alongside BugBank, a collection of 147 dossiers from 137 source units. Each dossier connects a source claim and its assumptions to a counterexample, a Lean-checked proof, and a separate assessment of whether the objection applies to the source. We also introduce a novel benchmark based off BugBank. On 100 texts known to contain an error, the best-performing model selects a line marked incorrect as its first choice in 73.0% of cases, compared with 18.0% for the strongest rule-based baseline. In author-led mathematical follow-ups with AI-assisted proofs, we extend individual witnesses into complete characterizations and sharp bounds. We characterize and count frozen six-colorings of doubled odd cycles, where no vertex can legally change color: for odd cycle length $n\ge 5$, they exist exactly when three divides $n$, yielding $90\cdot 2^n$ colorings for labeled vertices and colors. We also prove that $9/22$ is the optimal limiting fraction of ordered triples $(x,y,x+y)$ with three different colors within a two-threshold integer-coloring family, exceeding the conjectured bound of $2/5$.