The mathematicians’ problem is becoming OpenAI’s problem

On 11 September, 25 Fields medallists published a declaration, A Severe Misalignment of AI in Mathematics, three days after OpenAI announced a claimed resolution of the Navier–Stokes problem.  Their objection was not that machines might solve difficult problems. They were concerned that results could arrive without the explanation, scrutiny or attribution through which mathematics advances, and that solving a benchmark problem was being confused with developing a deeper understanding of it. The declaration says that “the goals of the AI companies and the goals of the mathematical community are severely misaligned”.

For the past year, my Growth Under Hard Limits advisory work with industry and government has examined a related problem in commercial AI. OpenAI and Anthropic are spending billions developing capabilities that other companies can increasingly reproduce or put to work at a fraction of the original development cost. The distinction between creating an advance and capturing its economic value matters well beyond mathematics. It matters to investors financing the next generation of models, and to operators deciding whether they need to pay frontier prices to run their businesses.

OpenAI’s release on 6 October of more than 700 AI-generated manuscripts, later listed as 719, gave mathematicians a body of work to assess, with Lean formalisations for about 42% of the top-line results when checked on 12 October. On 15 October OpenAI pledged $5 million to a new Simons Foundation programme supporting evaluation and understanding of AI-assisted mathematics, part of a $15 million joint initiative with Anthropic and the foundation. That is a constructive response, although funding the assessment of a discovery and bearing the cost of producing it remain very different propositions.

The cost of being first

DeepSeek’s R1 release in January 2025 gave a practical indication of how far model capability can travel. The company trained smaller open-weight models, built on Qwen and Llama bases, using examples from its own larger reasoning system. Its 32-billion-parameter Qwen-based model scored 72.6% against R1’s 79.8% on AIME 2024, 94.3% against 97.3% on MATH-500, 62.1% against 71.5% on GPQA Diamond and 57.2% against 65.9% on LiveCodeBench. Across these four tests it retained between 87% and 97% of the teacher’s score, with weights that other developers could download and run.

Those percentages do not mean that a smaller model can complete 90% of a business’s work. A long sequence of decisions can fail when a few small errors accumulate, and reliability, data security and integration may count for more than a benchmark result. But for a defined workload there is little commercial reason to pay for capability beyond the accuracy the task requires, particularly if a smaller model can meet that requirement on equipment the business already owns.

Figure 1. Published DeepSeek-R1 and DeepSeek-R1-Distill-Qwen-32B benchmark results. Retention means the student’s score divided by the teacher’s on each test, not general equivalence.

This is familiar territory in telecoms and utilities. An operator can analyse gateway telemetry with a compact model to identify recurring Wi-Fi faults, distinguish coverage problems from failing equipment and prioritise support before a customer calls. A utility has millions of meter and network events to classify, but relatively few decisions that need expensive multi-step reasoning. In both cases, the business already has the devices, data and customer relationship. The economic return comes from fewer truck rolls, lower contact-centre costs, longer equipment life and better service. Keeping appropriate inference on local infrastructure can also reduce the movement of sensitive data outside the operating environment.

The economics of frontier research are moving in the opposite direction. Epoch AI estimates that the training cost of frontier language models has risen 3.5 times annually since 2020, against about 2.4 times annually over a longer period beginning in 2016. Its 2024 analysis suggested that the largest training runs could exceed $1 billion by 2027 if that trend continued. Neither growth rate can persist indefinitely, and training is only part of the expenditure. Failed experiments, research teams, data, evaluation and the capacity to serve customers must all be funded before a new release generates revenue.

The Wall Street Journal reported around $750 billion of OpenAI computing expenditure planned through 2030, while Reuters described at least $518 billion of Anthropic infrastructure commitments over the coming decade, about 80% of them payable regardless of usage. These are differently defined commitments, not training budgets, but they show the financial exposure involved in maintaining a lead that competitors may narrow within months. Some obligations last much longer than the pricing advantage of the models they support.

What happens to the premium

Meta’s 12 October Muse Wonder release offers another form of competition. Meta presented the model as frontier class and priced its API at $4 per million input tokens and $20 per million output tokens, although its claimed standing against rival models is based on Meta’s own benchmark selection. Meta can fund this work from an advertising business and a vast consumer distribution network. Google can draw on its cloud business, TPU infrastructure and existing products. The model does not have to recover every research dollar on its own.

OpenAI and Anthropic have strong distribution partners and substantial revenues, but neither has an established advertising or device business generating cash on that scale. They depend much more directly on monetising intelligence and the products built around it. For enterprises, self-hosting does not make inference free: depreciation, electricity, engineering, cybersecurity and utilisation still determine the bill. Nor will a small model always win on the cost of a completed task once failures and retries are counted. For high-volume, bounded tasks on an installed equipment estate, however, local inference can be materially cheaper than buying premium API access for every decision.

Figure 2. Illustrative inference bill when selected tasks move to a model priced at 10% of a frontier tariff. Constant token mix; excludes orchestration, tests and retries.

Consider a business able to route 80% of its token-weighted workload to a model charging one-tenth of the frontier tariff. Keeping the remaining 20% on the frontier reduces its blended model bill by 72%; routing 90% cuts it by 81%. Those are illustrative savings before additional engineering and validation, not a forecast of how many tasks can safely move. They indicate why organisations will keep testing cheaper models as performance improves, and why a frontier developer may find itself competing on price for work that its earlier research made possible.

Cheaper inference can also expand the market. Tasks that were once uneconomic become possible, while agents may consume more tokens as they take on longer jobs. Frontier laboratories could therefore grow revenue strongly even while losing share of routine inference. The harder question for investors is whether that growth earns an adequate return on research, infrastructure and financing, rather than whether the token market gets larger.

The transfer of proprietary capability also has limits. In February Anthropic alleged that DeepSeek, Moonshot and MiniMax had used more than 16 million Claude exchanges in unauthorised distillation campaigns. These are Anthropic’s allegations, not established findings, and distilling a developer’s own model or one used under a suitable licence is ordinary practice. Mathematical research and proprietary model output are governed by different legal arrangements; the parallel here concerns who pays for advances and who later benefits from them, not a claim that they confer identical ownership rights.

Terms of service can restrict unauthorised extraction, but they cannot stop competitors from learning to build comparable systems through independent research, open weights and permitted methods. OpenAI and Anthropic can themselves offer cheaper models, while retaining a premium for dependable performance on difficult multi-stage tasks. They must still fund further research while competing for the large volume of work where being a little better makes no financial difference.

Where the pressure is coming from

Anthropic’s figures reportedly presented to investors on 14 October demonstrate both the opportunity and the pressure. Third-quarter revenue reached $16.8 billion from $11.6 billion in Q2, with an 85% adjusted gross margin and an adjusted operating profit. Quarterly growth nevertheless slowed to about 45%, against roughly 145% in the preceding quarter. The larger revenue base explains some of the change, and reports also point to reduced distillation-related revenue and growing competition from established customers developing models themselves. I suspect lower-cost Chinese open-weight alternatives are contributing as well, although no public breakdown establishes how much of the slowdown they explain.

Moonshot released Kimi K3 on 16 July as a 2.8-trillion-parameter open-weight model. On the published GDPval-AA v2 comparison of professional work it ranked behind only Claude Fable 5 and GPT-5.6 Sol, but benchmark position is not a measure of cost per successful task. The listed API prices nevertheless matter: K3 at $3 per million input tokens and $15 per million output tokens, against $10 and $50 for Claude Fable 5, a 70% reduction on both rates. Chinese models handled between 57% and 67% of OpenRouter token traffic in mid-September, up from 6% to 13% in February. That is platform usage, not revenue, but the speed of the change is difficult to dismiss. Vercel’s separate AI Gateway data make the pricing point even more clearly: open-weight models handled 56% of August tokens but only 14% of spending, up from 29% of tokens and less than 4% of spending in June.

Figure 3. On OpenRouter, Chinese models’ reported share of token traffic rose from 6–13% in February to 57–67% in September 2026. The adjacent Vercel panel concerns all open-weight models, not Chinese models alone; the two platforms are not directly comparable.

China is also narrowing some of the hardware and operating-cost gaps. Huawei has targeted around 750,000 Ascend 950PR accelerators in 2026, though reported domestic demand exceeds available supply, and memory maker ChangXin Memory Technologies (CXMT) is making progress in advanced DRAM. Provinces including Gansu, Guizhou and Inner Mongolia have offered electricity discounts of up to 50% to eligible data centres using domestic chips. Some reported power prices were about 5.6 US cents per kilowatt-hour, compared with an average US industrial rate around 9.1 cents. These are location-specific subsidies, not a national electricity advantage, but they lower the hurdle for domestic AI infrastructure.

At a Singapore conference on 12 October, Broadcom chief executive Hock Tan said many AI investors “have no clue what they are investing in” and are “almost like they are in a casino”. He also identified capital as a growing constraint on the build-out. The warning carries weight because Broadcom sells equipment into that expansion and has itself agreed substantial financing for Anthropic-related infrastructure. The problem for investors is not whether AI has commercial demand, which it plainly does, but whether the income from successive model generations can cover the capital committed to producing and serving them.

The difference between owning infrastructure and earning attractive returns on it has become more visible in the credit market. On 15 October, the spread on default protection for CoreWeave debt rose above 1,100 basis points as its debt-funded expansion came under scrutiny. GPU capacity built with expensive borrowing, heavy depreciation and uncertain utilisation is not automatically a valuable asset, however strong the demand looks today. An established telecoms or utility estate is a different proposition: its network, devices and customer access were financed for an operating business, and AI can be another source of return on assets already in use.

My Growth Under Hard Limits work keeps bringing me back to that distinction. For operators, the most valuable AI system may not be the one scoring highest on a public leaderboard. It may be a specialist model that cuts avoidable engineer visits across millions of broadband lines, helps detect failing connectivity before a meter loses service, or reduces the cost of a routine decision by a few pence at very large scale. The financial benefit belongs first to the company that owns the service relationship and can put the model to work reliably, rather than necessarily to the laboratory that first demonstrated the capability.

Frontier research will continue to command a premium when the task is difficult enough, and cheaper intelligence will allow more work to be automated than businesses can justify today. I am less convinced that a lead in benchmark performance, however impressive, will be enough to secure the largest share of those returns. The mathematicians are asking how new results become lasting knowledge when discovery outpaces assessment and attribution. The corresponding question for frontier AI investors is how long a costly advance can retain its price once capable alternatives become available, and who is best placed to turn that intelligence into an enduring business.

Note on method. Benchmark retention is the student’s published score divided by its teacher’s score on each named test, not general capability. Model-routing calculations use a fixed token mix and a 10:1 price ratio without engineering, retry or quality-control costs. API list prices are not cost per completed task. Infrastructure commitments are as reported by the named outlets and are not directly comparable. OpenRouter measures token traffic, not spending or all industry demand. Anthropic’s quarterly results were reported from its investor meeting and have not been independently audited.

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