Next-Gen GPT Model Cracks 10 Open Math Problems for Just $2,000

A Single Tweet That Sent Shockwaves

Recently, a tweet from a team associated with OpenAI sparked widespread discussion across the AI and academic communities. It claims that an internal version of their next-generation flagship model produced 10 new results on long-standing open problems in mathematics and theoretical computer science—and the computational resources consumed to achieve this, when converted at GPT-5.6 Sol API pricing, amounted to roughly $2,000 in token costs.

If this claim is true, the signal it sends is far more significant than the numbers themselves: AI is no longer merely a tool that helps humans organize existing knowledge—it has begun producing original mathematical results on frontier problems that humans have yet to solve.

It should be noted that this remains a single-source claim from the vendor, with no accompanying full paper, proof details, or independent peer review. Therefore, while we should recognize its potentially groundbreaking significance, we must also maintain appropriate caution.

Why "Open Problems" Are Such Critical Words

The Watershed Between Recitation and Creation

Over the past few years, large language models' mathematical performance has primarily concentrated on two types of tasks: solving competition problems with known standard answers (such as AIME and IMO-level problems), and reproducing classic proofs from textbooks. These tasks are essentially retrieval, combination, and reasoning within the model's "knowledge boundary."

AIME (American Invitational Mathematics Examination) and IMO (International Mathematical Olympiad) are standard benchmarks for measuring AI mathematical reasoning capabilities. In 2024, DeepMind's AlphaProof and AlphaGeometry 2 demonstrated near-gold-medal performance on IMO-level problems, but these achievements are still fundamentally "closed-domain" reasoning—the problems have definite answers, and large numbers of similar problem types exist within the training distribution. A qualitative chasm separates "solving known problems" from "cracking unknown ones": the former can be accomplished through pattern recognition and search strategies, while the latter requires generating entirely new mathematical objects and constructing unprecedented lines of argument. This is the fundamental difference between research mathematics and competition mathematics.

"Long-standing open problems" represent an entirely different magnitude. These are problems that mathematicians have studied for years or even decades without resolution, and their answers simply do not exist in any training corpus. If a model can produce new results on such problems, it demonstrates not memory and pattern matching, but some degree of genuine reasoning and creative capability.

The Weight of Theoretical Computer Science

The tweet specifically mentions that results span both mathematics and theoretical computer science. Open problems in theoretical computer science—such as those in complexity theory, algorithmic lower bounds, and combinatorial optimization—typically require rigorous formal proofs. AI producing verifiable new results in these domains is valuable not just for the conclusions themselves, but for demonstrating that AI possesses the ability to construct rigorous logical chains.

The core open problem types in theoretical computer science (TCS) deserve further elaboration: The P vs NP problem asks whether "all problems that can be quickly verified can also be quickly solved"; circuit complexity lower bounds attempt to prove that certain functions cannot be computed by small circuits; additionally, there are communication complexity, space lower bounds for streaming algorithms, and approximation ratio limits in combinatorial optimization. The common characteristic of these problems is that they demand extremely rigorous mathematical proofs—often a single tiny logical flaw will cause an entire argument to collapse. Many TCS problems have remained unsolved for decades, not due to lack of computational power, but because humanity has not yet found the correct proof framework or combinatorial structure. If AI can achieve breakthroughs in this domain, it demonstrates not merely symbol manipulation, but the ability to "discover" new mathematical structures.

The Economics Behind $2,000

In this piece of information, the cost figure is perhaps the most easily overlooked yet most industry-disrupting element.

Background on GPT-5.6 Sol API pricing: In OpenAI's model naming system, the Sol series is believed to represent their highest-capability reasoning model line. API pricing is typically charged per input/output token, with high-end reasoning models priced far above general conversational models—for example, GPT-4o's output token price is approximately $15 per million tokens, while reasoning-enhanced models (such as the o1 series) can cost several times or even ten times more. $2,000 in reasoning costs roughly corresponds to tens of millions to hundreds of millions of generated tokens, meaning the model likely performed extensive internal "chain-of-thought" expansion, hypothesis verification, and backtracking search. Notably, this is only the inference-stage cost and does not include the sunk costs of model pre-training (potentially hundreds of millions of dollars) and post-training alignment.

$2,000 in token costs is virtually negligible in a corporate R&D budget. By comparison, the investment of a mathematics PhD researcher over months or even years costs orders of magnitude more. If AI can produce frontier research results at such low marginal costs, it will bring about a structural transformation in research productivity:

• Marginal costs of research plummet: Problems that previously required top-tier talent working long-term could potentially be explored in batch at low cost.
• The space for trial and error expands dramatically: Researchers can have models explore numerous conjecture paths in parallel, rapidly screening for promising directions.
• Human-AI division of labor is restructured: Human researchers may shift more toward problem definition, result verification, and theoretical integration.

Throughout the history of science, every major tool innovation has profoundly changed research efficiency and paradigms. Computer algebra systems (like Mathematica and Maple) automated symbolic computation from manual work; Monte Carlo simulations enabled statistical physics and financial engineering to explore complex systems beyond analytical reach; gene sequencing machines shifted biology from hypothesis-driven to data-driven. AI's intervention as a "cognitive tool" may represent yet another paradigm leap—not replacing a particular computational step, but replacing the "generation of insight," a core function previously considered the exclusive domain of human intellect. Its breadth and depth of impact may exceed any single tool revolution mentioned above.

Of course, $2,000 is merely the "hit cost" for producing these 10 results. Behind it likely lie numerous unsuccessful attempts, as well as the model's enormous training investment. Therefore, this figure more accurately reflects the marginal economics of the inference stage rather than full-pipeline costs.

Several Points Requiring Sober Assessment

Verifiability of Results

The value of mathematical results is highly dependent on proof correctness and reproducibility. "New results" produced by AI must undergo rigorous scrutiny from the mathematical community—either through formal verification tools (such as Lean or Coq) or peer review by human experts. History has seen AI produce "proofs" that appear correct but contain hidden flaws. Therefore, until independent verification is complete, the quality of these 10 results remains to be confirmed.

Lean and Coq are currently the most mainstream interactive theorem provers, based on Dependent Type Theory, allowing users to write mathematical proofs in a programming-language-like manner while the computer verifies each logical step's correctness. Lean 4 is widely adopted in the mathematical formalization community, and its core mathematics library Mathlib has formalized over 150,000 mathematical theorems. If AI-produced proofs can be translated into Lean code and pass compiler verification, their correctness receives machine-level guarantees—more certain than traditional peer review. However, converting natural language proofs into formalized code remains a challenge itself; many subtle mathematical intuitions and "obvious" steps need to be fully expanded into hundreds of lines of rigorous code.

Varying Weight of "New Results"

"New results" is a broad characterization. It could mean a complete proof of a famous conjecture, or it could be a boundary improvement on a problem, a special-case solution, or a counterexample construction. Results of different weight carry vastly different academic significance. Without specific details, we cannot judge what level these 10 results actually occupy.

Limitations of a Single Source

Currently, this information comes solely from the vendor's social media post, with no independent third-party cross-verification or complete technical materials in sight. For claims of this magnitude, academic convention dictates waiting for papers, data, and reproducible procedures to be made public before drawing conclusions.

What It Means If True

Setting aside caution, if this achievement is ultimately verified, it marks AI entering a new phase: evolving from a carrier of knowledge to a producer of knowledge.

The far-reaching implications of this transformation include:

1. Research paradigm shift: AI could become a standard collaborative tool for mathematicians and theoretical computer scientists, much like today's calculators or computer algebra systems.
2. Accelerating positive feedback: AI-assisted research results can in turn be used to improve next-generation models, creating a spiral of ascending capability.
3. Implications for AGI progress: The ability to genuinely solve open problems is viewed by many researchers as one of the key markers of general intelligence.

Within the AGI (Artificial General Intelligence) research community, there is ongoing debate about "what capabilities signify true general intelligence." Some researchers (such as François Chollet) emphasize abstract reasoning and generalization abilities, proposing the ARC benchmark as a measuring standard; others argue that producing verifiable new knowledge on problems humans have not yet solved is more convincing evidence of intelligence than any benchmark test. The logic behind this view is that solving open mathematical problems requires seamlessly integrating multiple cognitive abilities—search, abstraction, analogy, creative construction—making it the best embodiment of the word "general." If AI has indeed reached this level, it will redefine philosophical and scientific discussions about the nature of intelligence.

Conclusion

This brief tweet may be a milestone worth marking in AI's developmental history, or it may simply be a slightly exaggerated note in marketing materials. The truth depends on subsequent technical detail disclosures and academic verification.

Regardless of the outcome, it clearly points to a trend: AI's role in frontier scientific research is moving from the periphery to the core. For researchers, companies, and the entire technology industry, now may be precisely the moment to rethink "how humans and AI divide the labor of exploring the unknown." We look forward to more complete evidence surfacing, allowing this discussion to return from amazement to verifiable science.