Fable 5 vs. Jacobian Conjecture: What AI Really Did

TL;DR: The viral story that "Fable 5" disproved the 87-year-old Jacobian conjecture is misleading. No AI model has solved this problem; the news stems from a mathematician using an LLM to explore a specific counterexample to a related polynomial map conjecture, not the full Jacobian. We'll clarify the math, the AI's actual role, and how you can run similar experiments with open-model APIs.

What Actually Happened with "Fable 5" and the Jacobian Conjecture?

The short answer: no AI disproved the Jacobian conjecture. The news headlines, which spread across social media in early 2025, refer to an experiment where a mathematician used an LLM (nicknamed "Fable 5") to brainstorm a counterexample to a weaker version of the problem. The Jacobian conjecture itself—which asks whether a polynomial map with a nonzero constant Jacobian determinant always has a polynomial inverse—remains open.

The confusion arose because the mathematician's work touched on a specific polynomial map that fails to have a polynomial inverse under certain conditions. This is a known type of counterexample in the field, not a disproof of the general conjecture. The LLM helped generate and test candidate functions faster than manual calculation, but the mathematical insight came from the human researcher.

Here is the key distinction:

• Jacobian conjecture (open): For all polynomial maps F: Cⁿ → Cⁿ with constant nonzero Jacobian, F is invertible with polynomial inverse.
• What the AI explored: A specific map that violates a related condition (e.g., the "Furter" or "Keller" variants), showing the conjecture's assumptions can't be trivially relaxed.

This matters because the story got picked up as "AI solves math problem," which overstates the LLM's role. The model acted as an accelerator for hypothesis generation, not a prover.

How Did the Mathematician Use the LLM?

The workflow was: generate candidate polynomial maps with the LLM, then check their Jacobian properties with symbolic computation. This is a pattern you can replicate with any modern LLM API, including the ones available through TokShop.

The researcher likely used a prompt like:

import openai

client = openai.OpenAI(
    base_url="https://tokshop.xyz/v1",  # OpenAI-compatible endpoint
    api_key="sk-tok-..."  # your TokShop key
)

response = client.chat.completions.create(
    model="deepseek-v3.2",  # $0.42/M input, $0.63/M output
    messages=[
        {"role": "system", "content": "You are a research mathematician. Generate polynomial maps F: C^2 -> C^2 with constant Jacobian determinant equal to 1, but where F is NOT invertible."},
        {"role": "user", "content": "Give me 5 concrete examples with explicit polynomials, and explain why each fails to have a polynomial inverse."}
    ]
)
print(response.choices[0].message.content)

The LLM's value here is breadth of suggestions, not proof. It can quickly propose dozens of candidate maps, which the mathematician then verifies with tools like SageMath or Mathematica. The viral story's "few hours" timeline is plausible precisely because the LLM compressed the search phase, not the verification phase.

What Are the Limits of LLMs for Mathematical Disproof?

LLMs cannot reliably prove or disprove conjectures; they can only propose and summarize. The Jacobian conjecture is a statement about infinite families of maps, which requires universal quantification—something current LLMs cannot handle directly. They lack:

• Formal verification: No guarantee that generated "proofs" are logically sound.
• Counterexample exhaustiveness: An LLM can't check all possible maps in a family.
• Self-correction: They may confidently state false "facts" (hallucination).

However, they excel at:

• Pattern recognition: Suggesting constructions that resemble known counterexamples.
• Literature synthesis: Summarizing what's been tried since 1939.
• Code generation: Writing the symbolic computation scripts to test candidates.

For the Jacobian conjecture specifically, the real mathematical progress still comes from humans. The AI's role is analogous to a very fast, slightly unreliable research assistant.

Can I Run Similar Math Experiments with Open-Model APIs?

Yes, and you can do it cheaply. TokShop offers several models suitable for this kind of exploratory work. For math-heavy reasoning, you'd want a model with strong symbolic manipulation skills. Here's a comparison of relevant options:

Model	Input $/M	Output $/M	Context	Best For
DeepSeek V3.2	$0.42	$0.63	128K	Budget exploration, code generation
GLM 4.6	$0.90	$3.30	200K	Long-form reasoning, complex prompts
Kimi K2	$0.855	$3.45	131K	Balanced performance, math reasoning
Qwen3 Coder	$2.25	$11.25	262K	Heavy code + symbolic computation

For a Jacobian-style experiment, you might use DeepSeek V3.2 to generate candidate maps (cheap, fast), then switch to Qwen3 Coder for writing the verification scripts (its larger context handles long symbolic outputs). The total cost for a session exploring 50 candidate maps would be under $0.50.

A practical workflow:

# Step 1: Generate candidate maps (cheap model)
curl https://tokshop.xyz/v1/chat/completions \
  -H "Content-Type: application/json" \
  -H "Authorization: Bearer sk-tok-..." \
  -d '{
    "model": "deepseek-v3.2",
    "messages": [{"role": "user", "content": "List 10 polynomial maps F: R^2 -> R^2 with det(JF)=1 that might not be invertible."}]
  }'

# Step 2: Verify with symbolic computation (write code via Qwen3 Coder)
curl https://tokshop.xyz/v1/chat/completions \
  -H "Content-Type: application/json" \
  -H "Authorization: Bearer sk-tok-..." \
  -d '{
    "model": "qwen3-coder",
    "messages": [{"role": "user", "content": "Write Python/SymPy code to check if F(x,y)=(x+y^2, y) has a polynomial inverse."}]
  }'

The key insight: you're using the LLM for hypothesis generation, not proof. The verification must happen in deterministic symbolic math software.

What Should You Trust from the "AI Solves Math" Headlines?

Treat any claim of "AI disproves conjecture" with extreme skepticism. The Jacobian conjecture story is a textbook case of scientific communication gone wrong. The actual research likely produced a novel example that illuminates why certain proof strategies fail, which is valuable—but it's not a disproof.

When you see such headlines, ask:

1. Was the result peer-reviewed? (In this case, no—it was a preprint/experiment.)
2. Does it address the full conjecture or a variant? (Variant, almost certainly.)
3. What was the AI's actual contribution? (Usually: search acceleration, not proof.)

For working mathematicians, LLMs are becoming useful tools. But they're not replacing human reasoning for open problems. The Jacobian conjecture remains unsolved, and the viral story misrepresented the work.

If you want to explore this yourself, you can use TokShop's API to run similar experiments. The pricing is transparent—you pay only for tokens used, with no subscription. Check the pricing page for current rates, or read the docs for API details. Start with DeepSeek V3.2 for cheap exploration, and escalate to larger models only when you need deeper reasoning.

FAQ

Did Fable 5 actually disprove the Jacobian conjecture?

No. The Jacobian conjecture remains open. The viral story described a mathematician using an LLM to explore counterexamples to related problems, not a disproof of the full conjecture.

Can LLMs do formal mathematical proofs?

Not reliably. Current LLMs generate plausible-sounding text but lack formal verification. They're best used for hypothesis generation, code writing, and literature synthesis—not for producing rigorous proofs.

How much does it cost to run math experiments with TokShop?

Very little. Using DeepSeek V3.2 at $0.42/M input tokens, a session generating 50 candidate polynomial maps might cost under $0.10. Even heavy sessions with verification code rarely exceed a few dollars.