- Is it really worth worrying about a pure mathematics problem?
- What really happened: the race for mathematical reasoning
- What does "serious overhang" mean for your business?
- Why everyone is talking about it now: the Italian AI market in numbers
- From the Erdős problem to business processes: the concrete connection
- Three concrete scenarios: how does the situation change for an Italian company?
- Anthropic vs OpenAI vs Google: which platform to choose?
- What concrete risks do you need to manage before adopting these tools?
- How to get started: steps to begin without wasting budget?
- What mistakes do SMEs make when approaching advanced AI?
- How to tell if AI adoption is producing concrete results?
- Is it really worth it for your company to invest in advanced AI right now?
- How we see it at SHM
- Frequently Asked Questions
- Claude Mythos solved the 1946 Erdős conjecture in a single weekend; Google DeepMind solved 9 in the same timeframe with AlphaProof Nexus.
- The formal reasoning capabilities that enabled these results are the same ones powering logistics optimization, contract review, and dynamic pricing.
- 76% of Italian SMEs have not yet invested in AI: the gap with large enterprises widens every quarter.
- The right strategy isn't rushing to adopt every new thing, but experimenting in a controlled way and scaling only when the results are measurable.
Updated on October 5, 2026
A weekend to solve a problem open for 80 years. On May 26, 2026, Anthropic's Claude Mythos produced an elegant demonstration of the Erdős unit distance conjecture — and almost simultaneously, Google DeepMind solved nine others with AlphaProof Nexus. For Italian SMEs, the question isn't "how great is this result," but "what changes for me tomorrow?"
Is it really worth worrying about a pure mathematics problem?
Yes, because the capabilities that allowed solving the Erdős conjecture are the same ones AI models use to optimize complex business processes. It's not an academic exercise: it's a sign of what these tools can already do.
On May 26, 2026, an engineer at Anthropic announced that Claude Mythos had solved the unit distance conjecture formulated by mathematician Paul Erdős in 1946. The solution arrived "over the weekend" — during a weekend — and was described as an elegant proof, not a computational brute force.
Almost simultaneously, Google DeepMind published the AlphaProof Nexus paper on arXiv , which had independently solved 9 out of 353 open Erdős problems and proved 44 conjectures from the OEIS (the great online encyclopedia of mathematical sequences), using formal proofs verified in Lean 4 — an automated verification language — at a cost of a few hundred dollars per problem.
Two labs, two different approaches, similar results on the same problems. For the technical details of the original announcement, the original source on The Decoder reconstructs the whole story.
| Your situation | Is it worth looking into? | why |
|---|---|---|
| Tech SMEs with complex decision-making processes | Yes | Models with advanced reasoning can already automate analyses that currently require consultants. |
| Manufacturing company with complex logistics | Yes | Routing and storage optimization is one of the most mature use cases |
| Professional firm (legal, accounting) | Yes | Contract review and compliance are direct applications of formal reasoning |
| SMEs without structured data or documented processes | Not yet | First you need foundations: clean data and defined workflows |
| Company that has already adopted basic AI | Yes | It is time to consider making the jump to models with multi-step reasoning |
What really happened: the race for mathematical reasoning
In a matter of months, three of the world's leading AI labs solved math problems that had been open for decades. The progression is rapid and documented.
The news about Claude Mythos doesn't come out of nowhere. It's part of a clear trajectory that Quanta Magazine and others have reconstructed like the fastest acceleration ever seen in computational mathematics.
| Year | Model | Result |
|---|---|---|
| 2024 | AlphaProof (Google DeepMind) | Silver medal equivalent at the IMO (International Mathematical Olympiad) |
| 2025 | Gemini Deep Think (Google) | Gold medal at the IMO, solved autonomously within the standard time |
| May 2026 | Claude Mythos (Anthropic) + OpenAI | Erdős conjecture on unit distances independently solved |
| May 2026 | AlphaProof Nexus (Google DeepMind) | 9 open Erdős problems solved, 44 OEIS conjectures proven in Lean 4 |
| August 2026 | OpenAI (undisclosed model) | 249-page manuscript with 10 results in pure mathematics and theoretical computer science |
A critical point: the Claude Mythos demonstration is still based on an informal announcement by an Anthropic engineer. It has not yet passed formal peer review by the mathematical community. As mathematician Terence Tao pointed out on Quanta Magazine , AI without formal validation remains too unreliable for serious applications: the tool that makes results reliable is the Lean proof assistant, used precisely by AlphaProof Nexus. A promising sign, not yet a certified result.
What does "serious overhang" mean for your business?
The Anthropic engineer used this phrase to mean that the AI models available today have capabilities far superior to what companies are actually using. It's latent potential, not yet tapped.
In physics, overhang is a suspended mass ready to fall. In the AI context, the term indicates that already available models — not future ones — can tackle problems that until yesterday seemed unsolvable. However, most Italian SMEs have not yet structured internal processes capable of leveraging this level of reasoning.
The number of scientific publications on AI applied to formal mathematical reasoning almost doubled every year in both 2023 and 2024, according to the Communications of the ACM (March 2026) . Research is racing ahead. Business applications lag behind by 12-24 months. Whoever moves now takes up that space.
This data comes from 2025-2026 Research by the Digital Innovation in SMEs Observatory of the Polytechnic University of Milan , presented on May 21, 2026. The gap between large enterprises and SMEs isn't about access to tools: it's about priorities and internal structure.
Why everyone is talking about it now: the Italian AI market in numbers
The Italian AI market reached 1.8 billion euros in 2025, with 50% growth in a single year. The moment isn't in the future: it's already happening.
Generative AI — the models that generate text, code, and analysis — accounts for 46% of the total market, according to Artificial Intelligence Observatory of the Polytechnic University of Milan . It is not a niche: it is already the main component of the sector.
On the employment front, requests for AI skills in Italian job ads have grown by 93% in one year. 76% of offers for qualified profiles require at least one, according to the same Observatory. Those who do not train today will find the job market already reorganized around these skills tomorrow.
The ISTAT data is clear: 16.4% of Italian companies with at least 10 employees already use at least one AI technology, double the 8.2% in 2024. The gap between SMEs (15.7%) and large enterprises (53.1%) has risen to 37 percentage points. This data is available on ANSA/Teleborsa .
From the Erdős problem to business processes: the concrete connection
The capabilities that make it possible to solve a mathematical conjecture—constraint reasoning, hidden pattern search, generation of non-obvious solutions—are the exact same ones that AI models apply to real business problems.
The Erdős conjecture belongs to pure mathematics. But the skills a model demonstrates by solving that kind of problem don't stay in the lab. They transfer directly to industrial applications.
| Demonstrated capability in mathematics | Concrete business application | Who benefits |
|---|---|---|
| Formal proof for constraints | Contract analysis and automated compliance | Law firms, procurement departments, regulated companies |
| Searching for hidden patterns in complex structures | Logistics and supply chain optimization | Manufacturing, distribution, e-commerce |
| Generation of non-obvious solutions | Dynamic pricing with many interdependent variables | Retail, hospitality, SaaS |
| Verifiable multi-step reasoning | Code generation and review with fewer structural bugs | Software houses, internal IT teams |
| Summary of complex structures | Automatic analytical reports from raw data | Finance, marketing, operations |
Three concrete scenarios: how does the situation change for an Italian company?
The competitive advantage of advanced AI isn't the same for everyone. It depends on the company's digital maturity and the complexity of the processes to automate.
Three typical scenarios to understand where your company stands. The numbers are illustrative and serve to show the order of magnitude, not real client data.
| Scenario | Starting situation | Advanced AI application | Expected benefit |
|---|---|---|---|
| E-commerce with 50,000 SKUs | Manual pricing updated weekly | Multi-step reasoning AI model for daily dynamic pricing | Higher margins during peak demand hours |
| Consulting firm (20 people) | Contract review assigned to junior staff with senior supervision | AI for initial review of critical clauses and anomaly flagging | Reduced review time, seniors freed up for strategic work |
| Manufacturing company (components) | Logistics planning on an Excel sheet updated monthly | AI for routing and storage optimization on real-time data | Inventory reduction and transport costs on repetitive routes |
Anthropic vs OpenAI vs Google: which platform to choose?
The fact that three labs independently reached similar results is no coincidence: it is an intensifying tech race. For businesses, choosing a platform depends on practical criteria, not on who won the math competition.
Claude Mythos, OpenAI models, and Google DeepMind's AlphaProof Nexus are all investing in mathematical reasoning as a benchmark for general intelligence. For companies trying to choose which platform to adopt, this scenario raises some very real questions.
It is also worth reading the model comparison in the article on Intelligence Index 4.2: GPT-6 Astra vs Claude Fable 5.1 , which analyzes the operational differences between the main platforms.
| Choice criterion | What to evaluate | Practical note |
|---|---|---|
| Reliability of structured reasoning | Benchmarks on tasks similar to your use case | Test with your company's real data, not just public benchmarks |
| Integration with existing systems | Available APIs, native connectors, documentation | Check compatibility with your CRM, ERP, or management software |
| Sustainable pricing for SMBs | Cost per token or per API call, usage limits | Calculate the cost on a realistic volume, not on the basic plan |
| Data security and compliance | Where data is processed, DPA contracts | Essential for regulated sectors (healthcare, finance, legal) |
| Support and updates | Model update frequency, SLAs | A model that improves every 3 months changes your ROI expectations |
An often underestimated aspect concerns security. Claude Mythos has already shown significant capabilities on the vulnerability front as well: the article on Claude Mythos and cryptographic vulnerabilities explain what changes for those who manage digital infrastructure.
What concrete risks do you need to manage before adopting these tools?
The main risk isn't the technology: it's adopting it without a validation process. An AI model that reasons in an advanced way can also make advanced mistakes.
The speed at which these announcements follow one another makes objective evaluation difficult. The risk for companies is twofold: ignoring real developments out of excessive skepticism, or adopting immature tools based on unverified announcements.
The stance of Terence Tao — one of the world's most authoritative mathematicians — is clear: without formal validation, AI results remain unreliable for serious applications. The same principle applies in business: AI output not validated by a human review process is not ready for critical decisions.
| Risk | How it shows up | How to manage it |
|---|---|---|
| Premature adoption | Implementing an immature tool on critical processes | Pilot on circumscribed use cases, with metrics defined in advance |
| Lack of structured data | The model produces useless output because the input data is messy | First clean and document your data, then integrate AI |
| Vendor lock-in | The vendor changes pricing or policies and the company has no alternatives | Design the architecture so you can switch models without starting from scratch |
| Lack of human validation | Erroneous outputs are used in business decisions without control | Define who approves AI outputs before putting them into production |
| Underestimation of hidden costs | The real cost includes integration, training, and maintenance | Calculate the 12-month TCO (total cost of ownership), not just the license |
A related topic concerns regulatory restrictions on more advanced models: the article on Claude Mythos 5 and US export controls explains which regulatory constraints may limit access to certain models for European companies.
How to get started: steps to begin without wasting budget?
The most effective strategy isn't the fastest one: it's the one that starts with a precise audit, tests in a controlled way, and only scales when the results are measurable.
- 1 · Operations or IT manager Candidate process audit Map out the business processes that require structured reasoning: document analysis, optimization, reporting. Not all use cases need the same capabilities.
- 2 · Internal team + external consultant Data quality check Before integrating any AI model, check that your input data is clean, structured, and documented. An advanced model on messy data produces useless outputs.
- 3 · Project Manager Pilot on a limited use case Pick a non-critical process, define your success metrics ahead of time, and launch a 4-8 week test. Measure the time saved and the quality of the output.
- 4 · Management Evaluation and scaling decision If the pilot produces measurable results, plan to expand it to other processes. If it doesn't produce results, identify the bottleneck before investing further.
- 5 · Internal team Continuous training 7% of Italian SMEs have structured AI training programs. Those who invest in internal training today will have a structural advantage when next-generation models hit production.
What mistakes do SMEs make when approaching advanced AI?
The most common mistakes aren't about technology: they're about change management and the lack of a validation process.
- Starting with excitement, not the problem. Adopting a tool just because it is new, without identifying which problem it actually solves in your company.
- Skipping the data phase. Integrating an AI model without first checking the quality and structure of your input data. Garbage in, garbage out matters even more with advanced models.
- Not defining who validates the outputs. Putting AI outputs into production without a human approval process. Advanced reasoning can fail in advanced ways.
- Underestimating integration costs. The license cost is often the smallest part. Integration, training, and maintenance can triple the total cost.
- Failing to communicate change internally. Introducing AI tools without involving the teams that will use them leads to resistance and partial adoption.
How to tell if AI adoption is producing concrete results?
The KPIs to monitor depend on the use case, but there are cross-cutting metrics that show whether your investment is working or if there's an issue to fix.
| Indicator | What it measures | Positive signal |
|---|---|---|
| Average time to complete the process | Operational Efficiency | 20%+ reduction compared to the pre-AI baseline |
| Output error rate | Quality of the model's reasoning | Lower error rates than the previous manual process average |
| Percentage of outputs validated without edits | Model reliability for your use case | Above 70% after the settling-in period |
| Cost per output generated | Economic sustainability | Lower than the cost of the equivalent manual process |
| Internal adoption (% of teams using the tool) | Cultural shift | Above 60% 3 months after introduction |
Is it really worth it for your company to invest in advanced AI right now?
It is worth it if You have complex decision-making processes with many variables (logistics, pricing, document analysis), structured data, and a team willing to experiment. The time is now: the gap between those who adopt and those who wait widens every quarter.
A middle ground you are in a digital transition phase: first consolidate core data and processes, then evaluate AI integration. A limited pilot on a non-critical process is the best way to understand the real potential.
Not yet if you don't have structured data, documented processes, or an internal manager to handle experimentation yet. In this case, investing in advanced AI produces poor results and high hidden costs.
How we see it at SHM
For us, the news about Claude Mythos isn't just an academic curiosity. It's confirmation that AI models have already crossed the threshold of formal reasoning—and that the gap between the lab and the average SME is measured in months, not years.
We believe AI should be integrated to free companies from routine tasks, not to replace human ingenuity. The advanced reasoning of these models only makes sense if it serves to help people make better decisions, with more time and information at their disposal.
Let's work on this practically: from the Strategic AI integration in business processes, to business intelligence on structured data , right up to the AI training for corporate teams . If you want to figure out where to start in your company, the starting point is always an honest audit of what you already have.
To stay updated on these developments, follow the cluster AI and process automation : collects the most useful analyses for decision-makers, not for those who just want to follow the news.
It's also worth reading what's going on with AI agent costs: the article on Claude Fable 5.1 and cutting costs for agentic work shows how model pricing is changing fast to benefit SMEs.
Frequently Asked Questions
The announcement was made by an Anthropic engineer on May 26, 2026, and describes an elegant proof found over a weekend. At the time of publication, the proof had not yet passed formal review by the mathematical community. It is a promising sign, not yet a certified result.
The capabilities that allow an AI model to solve a mathematical conjecture — constraint reasoning, pattern finding, generating non-obvious solutions — are the same ones used to optimize logistics, analyze contracts, and manage dynamic pricing. Math is the proving ground; business processes are the application.
There is no one-size-fits-all answer. The choice depends on your specific use case, compatibility with existing systems, sustainable pricing for your volume, and data security requirements. The practical advice is to test with your company's real data, not just rely on public benchmarks.
The license cost is often the smallest part. Integration, team training, and maintenance can triple the total cost over 12 months. Before deciding, calculate the TCO (total cost of ownership) on a realistic volume of use, not on the basic plan.
It depends on your sector and your competitors. If you operate in a market where competitors are already adopting AI to optimize processes, delay translates into a measurable competitive disadvantage. If your sector is still in its early stages, you still have time to act in a structured rather than rushed manner.
Three practical questions: do you have structured and documented data? Do you have clear and measurable internal processes? Do you have an internal manager willing to handle the experiment? If the answer is yes to all three, you're ready for a pilot. If not, start from there.
The Erdős conjecture was open for 80 years: the real open problem for Italian SMEs is deciding whether to wait another 80 months or start right now.
Related articles
Discover more articles exploring similar topics, selected to offer you a more complete and stimulating perspective. Each piece of content is carefully chosen to enrich your experience.
