# A recent experience with ChatGPT 5.5 Pro (Gowers)

Tim Gowers — Fields medalist, combinatorics — recounts feeding ChatGPT 5.5 Pro a problem from Mel Nathanson's 2026 *Diversity, Equity and Inclusion for Problems in Additive Number Theory*. Over a little under two hours of model time across several prompts, the LLM produced a polynomial upper bound on $N(h,k)$ — the minimal $N$ such that all sumset sizes in $\mathcal{R}(h,k)$ are achievable using $k$-element subsets of $\{0, \dots, N\}$ — improving Isaac Rajagopal's earlier exponential bound to $O(k^{10h^3})$.

Rajagopal, a current MIT undergrad and the author of the original framework, contributed the guest section evaluating the result. His verdict: *"As far as I can tell, this idea is completely original."*

This is significant inside [[anti-llm-discourse]] for an inverse reason — it sits outside the cluster as a positive case. Gowers explicitly notes Rajagopal said the proof "looked correct, not just at line-by-line level but at the level of ideas."

## What the model actually did

Three nested rounds:

1. **$h=2$ case.** ChatGPT thought 17 minutes, returned a quadratic upper bound (best possible), wrote it as a LaTeX preprint in 2 minutes. The trick: replace Nathanson's powers-of-2 Sidon set with a known efficient one of quadratic diameter.
2. **Restricted sumset.** Same trick, no trouble.
3. **General $h$.** Tighten Rajagopal's argument. Improved from exponential-in-$k$ to exponential-in-$k^{1/2+\varepsilon}$ (16:41), then to polynomial-in-$k$ (13:33 + 9:12 verification + 31:40 writeup). Rajagopal forwarded the preprint and said the polynomial bound was "almost certainly correct" at the level of ideas.

## The original idea

Rajagopal's prior framework built sets $A$ as combinations of two geometric series $S = \{0, 1, m, \dots, m^{\ell-2}\}$ and $T = \{1, m, \dots, m^{\ell-1}\}$. Each is a $B_h$ set (one whose $h$-fold sumset hits the maximum size $\binom{h+\ell-1}{h}$) or fails to be by a structurally controlled deficit. But $S$ and $T$ have *exponentially large* elements — the bound on $N(h,k)$ from the construction was exponential.

Rajagopal asked, via Gowers, whether sets $G, H$ of polynomial-sized elements could reproduce the $B_h$ structure of $S, T$. He didn't know. ChatGPT returned a construction:

$$G = \{0, u_1, \dots, u_r, m u_1, \dots, m u_r\}, \quad H = \{u_1, \dots, u_r, m u_1, \dots, m u_r\}$$

where $\{u_1, \dots, u_r\}$ is an $h^2$-dissociated set (no nontrivial additive relation among $\leq h^2$ elements) built by finite-field methods (Bose–Chowla 1963 style). Element diameter: polynomial in $r$.

Rajagopal's after-the-fact intuition: $S, T$ contain ~$\ell$ relations of the form $mx = y$; $G, H$ contain ~$\ell/2$. The $h^2$-dissociation kills all other low-order relations. So $G, H$ retain exactly the relevant additive structure of $S, T$ while collapsing diameter from exponential to polynomial. *The idea to use $h^2$-dissociated sets to control relations of order $\leq h$ is what Rajagopal calls original.*

Substituting $G, H$ for $S, T$ in the rest of Rajagopal's construction yields $N(h,k) \leq O(k^{10h^3})$.

## Gowers' read

Equivalent of "a perfectly reasonable chapter in a combinatorics PhD." Not amazing, leans on Rajagopal, but the extension is a non-trivial contribution that would take a beginning PhD student weeks of work — reading Rajagopal's paper, looking for non-optimal places, learning the algebraic techniques.

The implication that drives the rest of the post: **the bar for a problem you can give a beginning PhD student has just risen.** "Gentle problems" — the historical tool for getting a new mathematician started — are now LLM-solvable. The lower bound for a meaningful contribution is now: prove something *in collaboration with LLMs* that LLMs can't do alone.

Two qualifications Gowers offers:

1. PhD students can also use LLMs. The bar isn't "what LLMs can't do," it's "what LLMs+humans can do that LLMs alone can't."
2. He doesn't know how this generalizes outside combinatorics. Combinatorics is problem-driven (reason backward from a question). Forward-reasoning fields — which observation is interesting? — may not be vulnerable in the same way yet.

## On where the result should live

Gowers refuses both "AI slop" and "publish in a journal." His suggestion: a moderated AI-result repository. Moderation by human-mathematician certification, ideally proof-assistant-verified, ideally answering a question already in the literature. Workload for moderators is the open problem; he warns against AI-assisted moderation for obvious recursive reasons.

> "Had the result been produced by a human mathematician, it would definitely have been publishable, so I think it would be wrong to describe it as AI slop."

## What this means for mathematics as a vocation

The thought experiment Gowers proposes: a mathematician solves a major problem via long LLM exchange where the mathematician guides, the LLM has the main ideas. Would we count that as a major achievement *of the mathematician*? "I don't think we would."

So why still do the work? His answer: solving hard problems gives you insight into the problem-solving process, which is what makes you good at AI-collaborative problem solving. People who have themselves solved problems are likely better at vibe-collaborating with LLMs, the way good arithmeticians notice when calculator output looks off. The reward is no longer immortality-by-theorem; it's the durable skill of recognizing when the LLM-assisted answer feels wrong.

Same shape as [[average-is-all-you-need]] (when output becomes cheap, attention moves to the upstream judgment that's still scarce). Same direction as [[building-syntaqlite-ai]] (vibe-coding fails for those who don't already know what good looks like). The implication: high-skill humans get more leverage from LLMs; the people for whom LLMs were going to "democratize" research get less than the headline suggests.

## Where this sits next to the anti-LLM cluster

This piece is the thing [[anti-llm-discourse]] notes is missing — *"a serious technical analysis of what would have to be true for LLM-assisted dev to work well."* Gowers' analogue isn't dev; it's research mathematics. He provides the missing positive case, with a Rajagopal-certified original idea attached.

It doesn't refute the engineering / craft objections in [[no-silver-bullet-llms]] or [[peril-of-laziness-lost]] — those are about LLMs producing bloat in long-running codebases, not about LLMs proposing one idea in two hours. Different domain, different failure mode.

It does complicate the pessimism in [[building-syntaqlite-ai]] / [[i-will-never-use-ai]]. If the LLM can produce an "original and clever" mathematical idea — verified by the original author of the framework being extended — the strong version of "LLMs can't reason, only recombine" is harder to defend in this case. Gowers himself notes that "quite a lot of perfectly good human mathematics consists in putting together existing knowledge and proof techniques" — the LLM-vs-human boundary is fuzzier than the public discourse usually allows.

## Cross-references

- Author entity: [[gowers-blog]]
- Concept: [[llm-mathematical-research]] (case studies + structural implications for research-bar)
- Anti-LLM cluster context: [[anti-llm-discourse]], [[i-am-an-ai-hater]], [[i-will-never-use-ai]], [[no-silver-bullet-llms]]
- Skill-leverage framing: [[average-is-all-you-need]], [[building-syntaqlite-ai]], [[skill-atrophy-supervision-paradox]]
- Adjacent: [[claude-emotion-concepts]] (model internals), [[neurosymbolic-ai]] (AI + formal verification, relevant to Gowers' moderation suggestion)