Can AI Really Solve Hard Math Problems? A Rational Look at LLM Research Breakthroughs

Another "Unsolved Problem" Cracked by AI? Don't Believe It Just Yet

Recently, a piece of news sparked heated debate in the Reddit community: "GPT-5.6 Solves Yet Another Unsolved Problem." Headlines like this tend to quickly ignite the public's imagination—has the large language model already acquired the ability to push beyond the boundaries of human knowledge?

However, as practitioners who have long observed the AI field, we need to keep a clear head amid the excitement. Such claims appear frequently, but the cases that truly withstand academic scrutiny are few and far between. This article uses this topic as a springboard to take a deep dive into the real capability boundaries of large models when it comes to solving mathematical and scientific problems.

What Does It Really Mean for a Large Model to "Solve" a Problem

The Mathematical Community Has Strict Standards for "Solving an Unsolved Problem"

First, we need to clarify a key concept: the so-called "solving an unsolved problem" has an extremely strict standard in the mathematical community. A genuine breakthrough must simultaneously satisfy three conditions: the problem itself is a recognized open problem in the field, the proof provided is logically rigorous and error-free, and it can pass the test of peer review.

Many widely circulated cases of "AI cracking hard problems" on social media actually fall into one of the following categories: the model re-derived a known but relatively obscure result, produced a "proof" that appears reasonable but actually contains logical gaps, or completed part of the reasoning steps under heavy guidance from human experts. The value of these achievements cannot be denied, but they are fundamentally different from "independently solving a world-class problem."

From AlphaProof to LLM Reasoning: How Big Is the Gap?

AI has indeed made substantial progress in the field of mathematics. DeepMind's AlphaProof reached a silver-medal level at the International Mathematical Olympiad (IMO), and models from OpenAI and Google have also demonstrated potential in formal proofs. These systems typically combine reinforcement learning, formal verification tools (such as Lean), and large-scale search strategies, rather than relying solely on the "intuition" of a language model.

It is worth understanding in depth that the working principle of formal proof systems (such as Lean, Coq, and Isabelle) is fundamentally different from that of large language models. Formal systems require every reasoning step to conform to a predefined axiom system, with a computer mechanically verifying each derivation step—there is no room for semantic ambiguity or "skipping steps." Any invalid derivation is directly rejected by the system. AlphaProof precisely uses Lean as its backend verification tool, combining the neural network's "conjecturing ability" with the formal system's "verification ability" to achieve trustworthy mathematical reasoning. By contrast, a piece of "proof text" output by a conversational large model is natural language generated based on statistical patterns, and there is no guarantee that every step holds logically rigorous. The credibility of the two is simply not in the same league.

Why Such Breakthrough Claims Warrant Extra Caution

The Hallucination Problem Is Especially Fatal in Mathematical Contexts

The most widely known flaw of large language models is "hallucination"—outputting incorrect content in an extremely confident tone. To understand the root of this problem: the training objective of a large language model is to maximize the prediction probability of the next token (text fragment), which is essentially an extremely complex "text continuation" mechanism. During training, the model learns what kinds of mathematical statements "sound reasonable," but it cannot truly distinguish between "a fluently phrased proposition" and "a proposition that holds logically rigorous." This means the model can entirely produce a "proof" that is structurally complete, uses standard terminology, and carries a confident tone, yet in which some key step actually swaps a premise or skips a necessary condition. In a context like mathematical proof, where the demand for rigor is extremely high, a tiny logical leap or an unverified assumption is enough to completely invalidate the entire "proof."

What's even more tricky is that the erroneous proofs generated by models are often highly convincing on the surface—even ordinary readers, and sometimes even some professionals, find it hard to spot the problem right away. This is precisely why screenshot claims from communities like Reddit can only serve as a clue awaiting investigation, rather than a credible conclusion, before they have been authoritatively verified.

The "GPT-5.6" Version Number Itself Is Questionable

Version naming like "GPT-5.6" also deserves attention. OpenAI's official product naming system often diverges from the version numbers spread by the community, and the internet is full of content based on speculation or even mockery. In the absence of official release notes, technical reports, or reproducible evidence, assertions about the capabilities of a specific version should be approached with caution.

A Rational Understanding of AI's Value in Assisting Research

A Research Accelerator, Not an Independent Researcher

At the current stage, it is more accurate to position large models as "research accelerators" or "intelligent assistance tools." They can quickly search literature, propose candidate ideas, assist in verifying derivations, and even offer inspiring approaches to structured problems. Some mathematicians have publicly stated that AI has provided real help in exploring mathematical conjectures and checking computational processes.

But from "assisting human thinking" to "independently achieving original breakthroughs," there is still a considerable distance to go. A genuine research breakthrough requires a complete closed loop of problem definition, methodological innovation, and rigorous verification—and this is precisely the weakest link for current large models.

A Practical Framework for Judging the Authenticity of AI Breakthrough News

Faced with an endless stream of "AI cracks hard problem" news, you can make independent judgments from the following dimensions:

• Is there reproducible evidence: papers, code, and formal proof files are far more credible than a single screenshot;
• Has it passed peer review: recognition by authoritative journals or academic conferences is an important credibility threshold;
• Is there independent third-party verification: cross-confirmation from multiple sources is far more reliable than a single claim;
• The authority of the publishing channel: an official technical report carries entirely different weight from an anonymous community post.

Conclusion: Watching AI Rationally Is the Real Tech Literacy

Headlines like "GPT-5.6 Solves Yet Another Unsolved Problem" reflect the public's high expectations of AI's capabilities, while also exposing the widespread tendency toward exaggeration in information dissemination. We should neither blindly dismiss AI's genuine progress in the research field, nor let unverified sensational headlines sway our judgment.

What truly deserves ongoing attention is the sustainable path of AI and humans collaborating to solve complex problems, and the reliability improvements brought by tools such as formal verification. In this rapidly evolving field, maintaining rationality and critical thinking may be the very literacy we should possess most when facing every "major breakthrough."

Key Takeaways