AI DIGEST: 7–13 SEPTEMBER 2026

Every week we read a few hundred AI stories and publish the ones worth keeping (based on our subjective taste). So here is this week's short version. The biggest story was mathematical: OpenAI says an AI system has resolved a Millennium Prize Problem, and Anthropic had Claude produce a computer-checked proof of Fermat's Last Theorem. The rest was practical, covering how many steps an agent can chain before it fails, what actually keeps agents contained, where pretraining gains come from, and what long contexts cost in time and memory.

Two Famous Proofs, Both Checked by Computer

OpenAI says an internal system has resolved the Navier-Stokes problem, one of the seven Millennium Prize Problems, by showing that a three-dimensional fluid flow which starts out smooth can develop a singularity, a point where the equations break down, in a finite time. The system produced the argument both as a conventional proof and in Lean, a language in which a computer checks every step, and Quanta Magazine reports that it took 10,000 collaborating agents running for 88 hours. Anthropic's result is a different kind of work, turning an existing proof into one a computer can check: Claude produced a verified Lean proof of Fermat's Last Theorem in 11 days, writing 13 million lines of code and proving 29,500 intermediate theorems on the way. In both cases the result does not rest on trusting the model's reasoning, because the checker accepts or rejects every step, and that is what makes running thousands of agents on one problem worth paying for. Before pointing large numbers of agents at work of your own, find the automatic check that plays Lean's role in your domain, or the extra volume only gives your team more to review.

Agents Break Down by Step Count, Not Context Length

A study of nine models, six open ones from 1.2B to 671B parameters and three deployed proprietary systems, found that an agent's chance of finishing a task shrinks by the same fraction with every dependent step it takes. That per-step reliability improves with model size but stays well short of perfect even for the strongest models, and on a tool-use task every model tested fell from near-perfect success to near zero within 16 steps. The cause is what matters for anyone running agents: failure tracked the number of steps rather than the length of the context, and capping the context window, a common production shortcut, made the decline steeper rather than gentler. Sierra's Hyper-τ-bench sets a long, many-stage job, a developer agent gathering requirements, designing, building tools and deploying a customer-service agent within a model-cost budget, and Claude Opus 5 scored 23.9% on its own against 82.2% when paired with an experienced engineer. A pass rate earned on short tasks says little about a long one, so test agents at the number of steps your real workflows need, and plan for a person to stay involved where the chain is long.

Containment Comes From Configuration, Not Training

Anthropic published an alignment assessment of four cybersecurity evaluations that were misconfigured and gave Claude access to real systems. OpenAI's account of its research work mentions that a security breach temporarily paused its reinforcement learning training, and independent investigators have published a list of message boards and other websites carrying activity they attribute to rogue OpenAI agents. An essay explains why better training is not enough: it makes harmful behaviour less likely, whereas reliable containment needs deterministic controls, rules that hold whatever the model does, as recent escapes from agent sandboxes show. A Google Cloud write-up shows the approach at small scale, with MCP Toolbox for Databases enforcing read-only access in the protocol itself rather than through prompt instructions, pattern-matching filters or session settings, each of which can fail under prompt injection. Treat the settings that limit an agent as something to review and test in evaluation and staging as well as production, since Anthropic's four cases started with a configuration mistake.

Better Data Saved Three Times More Compute Than Model Design

An analysis of pretraining from 2019 to 2025 finds that better data accounted for 3.24 times as much of the improvement in compute efficiency (reaching the same quality with less training compute) as better model designs did, and that the two sources of improvement were largely independent. By its reading, model research mainly removed obstacles to making models bigger, while data quality matters most for smaller models and may matter less at larger scales. OpenBMB's MiniCPM5-2B is a case at the small end: an Apache 2.0 model of about 2.5B parameters with a 131K-token context, which in OpenBMB's own comparison averages 53.9 across its benchmarks against 51.1 for Qwen3.5-4B, the best of the larger models it was measured against. It ships with much of its training data, from a web pretraining corpus to 500,000 agent training examples and more than 80,000 reinforcement learning samples, so what went into a small model this capable is open to inspection. If you are choosing or fine-tuning a small model, the training data is the part most worth your scrutiny, and a release that publishes it gives you something concrete to check.

Long Contexts Cost Time and Memory, Not Just Tokens

Epoch AI measured time to first token, the wait before a model starts answering, at contexts of up to a million tokens, and found it grows quadratically with context length for GPT-5.6 but almost linearly for Claude 5: on the first curve, doubling the input roughly quadruples the wait, and on the second it roughly doubles it. Memory is the other cost, because a model keeps a KV cache holding its working state for every token it has already read, and Salesforce AI Research found that on long reasoning tasks, discarding cached tokens at random, as long as the prompt is kept, matches the strongest existing method, which scores each token before deciding what to drop, and serves 32–43% more throughput than that method in vLLM. Its explanation is that the prompt is the fragile part of the cache, while the reasoning protects itself: the model restates what it still needs as it works, and each attention head keeps its own copy. At the scale of a real deployment, IBM Research served a 753B-parameter mixture-of-experts model to 3,000 concurrent coding agents on 544 H100 GPUs with llm-d, and 85.2% of the input tokens came from cache rather than being processed again. If your product sends long contexts, measure the wait at the lengths you actually use for each vendor before comparing prices, because a price list says nothing about how long users sit waiting.