Human–AI mathematical research
This was not conventional authorship with occasional AI assistance. GPT‑5 performed much of the exploratory, generative, integrative, and adversarial work under sustained human research direction.
Human role
Leslie P. Polzer set the research goal, selected and redirected approaches, controlled the project’s standards, evaluated outputs, commissioned adversarial checks, integrated revisions, and takes responsibility for making the mathematical claim.
GPT‑5 role
The credibility problem
Large language models can invent citations, overlook quantifiers, smuggle in assumptions, mistake numerical evidence for proof, and produce circular arguments that read fluently. This project does not claim that scale, prompting, or confidence has eliminated those failure modes.
AI changed the scale and speed of the research process. It did not change the standard of mathematical evidence.
A statement is not accepted because GPT‑5 produced it, repeated it, or expressed confidence in it. Every claim must stand as ordinary mathematics.
The model’s central role is disclosed rather than reduced to generic ‘tool use.’ Human direction and responsibility are stated separately.
The same scale of assistance used to construct the argument is also used to attack it: searching for missing hypotheses, bad limits, false promotions, and circular dependencies.
Credibility ultimately requires experts who were not part of the development process to examine the written proof and reproduce its supporting work.
A useful framework
Terence Tao’s 2026 ICM lecture separates generating a proof from verifying it, explaining it, obtaining expert acceptance, and allowing the result to become part of the field. This project adopts that distinction rather than treating a generated manuscript as the finish line.
Tao also identifies a characteristic weakness of AI exposition: trivial passages can receive too much space while the genuinely novel steps are rushed or made to look frictionless. The proof guide therefore marks the load-bearing steps explicitly.
Tao, Mathematics in the age of AI ↗GPT‑5 accelerated exploration, drafting, integration, and adversarial search.
Experts, exact computations, source audits, and Lean test different failure modes.
Layered guides expose dependencies and mark the difficult steps instead of smoothing them away.
Independent specialists decide whether the written argument survives scrutiny.
Only the mathematical community can absorb, simplify, and eventually canonicalise a result.
Model disclosure
“AI-assisted” is too vague for meaningful provenance. The dominant model family was GPT‑5, an OpenAI reasoning model used for coding, reasoning, and agentic work across domains.
Naming the model does not transfer responsibility to OpenAI and does not imply endorsement by OpenAI. It makes the development history more exact.
Official GPT‑5 model documentation ↗