AI 輔助科學研究
Dual-Agent Research System Derives a New Lower Bound for the Grothendieck Constant From Repeated Failures, but the Key Pivot Still Came From Humans
Using reasoning models, coding agents, and file-based memory across roughly 240 research sessions, a team ultimately proposed—and the authors verified—`K_G ≥ 6π/11`. The system demonstrated strong local proof capabilities but did not independently recognize that its search had reached a bottleneck; research direction and decisions about which results to pursue still depended on roughly 40 human instructions.
A case study submitted on August 11 details how AI helped improve bounds on the Grothendieck constant, whose exact value has remained unknown since 1953. The constant describes the worst-case integrality gap between a bilinear combinatorial optimization problem and its semidefinite programming relaxation. The team ultimately narrowed the verified interval to `6π/11 ≤ K_G ≤ π/[2 log(1+√2)] − 3.47×10⁻⁴`, establishing for the first time that its first decimal digit is 7.
The research framework divided the work between two types of agents. GPT-5.5-Pro, and later GPT-5.6-Sol, selected research directions, derived arguments, and reviewed claims. Claude Code, paired first with Opus and later with Fable 5, maintained the repository, wrote experiments, retrieved literature, and orchestrated the reasoning models. No model state persisted between sessions; continuity instead relied on a problem statement, a single summary file, an append-only transcript, and experimental logs. At the end of each session, the system had to classify every claim as proved, numerically supported, conjectural, or heuristic.
Between June 16 and July 24, the system ran roughly 240 sessions, called reasoning models 2,091 times, consumed approximately 152 million tokens, and incurred an estimated API cost of $5,400. It initially searched repeatedly for a better upper bound, but merely recorded a recurring trade-off between nonlinear terms and the leading term without elevating those failures into a general obstruction. After session 18, human researchers instructed it to pivot to lower bounds and later directed it to synthesize the observed failure patterns. The agents then found an affine inequality for the relevant coefficients, reducing the high-dimensional problem to a one-dimensional Gaussian inequality and yielding `K_G ≥ 6π/11`.
The authors independently checked this lower bound. However, two stronger lower bounds and a subsequent upper bound also produced by the system have so far passed only internal machine verification and are not yet stated as theorems. An early numerical result was also withdrawn after an audit. The record is therefore significant not only for its mathematical result, but also for clearly delineating the current limits of long-horizon research agents: they are highly capable at local technical execution, while research judgment, synthesis of failures, and maintenance of global state remain fragile.