GPT-5.6 破解30年數學難題：AI真的會做研究了嗎？ | GPT-5.6 Solves 30-Year Convex Optimization Problem

7月 22, 2026

By Kit 小克 | AI Tool Observer | 2026-07-23

🇹🇼 GPT-5.6 破解30年數學難題：AI真的會做研究了嗎？

GPT-5.6 最近在 Hacker News 上引爆討論：一位研究者用單一 prompt，引導模型證明了「凸優化」（convex optimization）領域一個懸而未決 30 年的下界問題，且證明過程用 Lean 4 形式化驗證，完整編譯通過、沒有任何 sorry（未證明的跳過標記）。這不是一次性的巧合，而是 OpenAI 幾週前用類似手法證出 CDC（Complexity of Descent Conjecture）猜想之後的第二起案例。

GPT-5.6 到底做了什麼？

研究者證明的是：對一類標準凸函數做優化，任何演算法都至少需要 Omega(d^2) 次函數評估——也就是「下界」。而三十年前就已經有演算法達到這個上界，但沒人證明過「無法再更快」。證明下界比證明上界難得多：上界只要秀出一個方法能做到，下界卻要排除所有可能的方法，等於是在講整個問題的結構性極限。整段推理耗時約 148 分鐘，最後用 Lean 4 搭配 Mathlib 函式庫形式化，確保每一步都經得起機器檢查，而不是「聽起來合理」。

關鍵不是模型本身，是 prompt 工程

值得誠實講清楚的一點：這次成功高度依賴一個精心設計的 prompt——精確陳述待證問題、要求以 Boyd 與 Vandenberghe（凸優化教科書作者）的風格寫證明、並明確要求模型標記任何未被證明的步驟。換句話說，GPT-5.6 不是自己「發現」了這個問題並主動去解，而是在極度結構化的引導下完成推理。這說明目前 AI 在數學研究上的角色比較像「配備強大符號推理能力的助理」，而不是獨立研究者。

對開發者與研究者的實際意義

如果你的工作涉及形式化驗證（Lean、Coq 等），GPT-5.6 這類模型值得納入輔助證明流程，但仍需要專家設計 prompt 與審核每一步
「AI 解決數學難題」的新聞很吸睛，但實際門檻是：懂問題的人 + 懂怎麼問的人，缺一不可
形式化驗證（Lean 4 編譯通過）是目前判斷 AI 數學證明可信度的關鍵標準，比模型自己說「我證明了」可靠得多

好不好用，試了才知道。

🇺🇸 GPT-5.6 Solves 30-Year Convex Optimization Problem

GPT-5.6 became the top story on Hacker News this week after a researcher used a single carefully constructed prompt to guide the model through proving a 30-year-old open lower-bound problem in convex optimization, with the entire proof formalized in Lean 4 and compiling clean, with no sorry (the marker for unproven steps). This follows OpenAI's similar proof of the Complexity of Descent Conjecture (CDC) just weeks earlier, suggesting this isn't a one-off fluke.

What exactly did GPT-5.6 prove?

The result: optimizing over a standard class of convex functions requires at least Omega(d^2) function evaluations, a lower bound. An algorithm matching that upper bound has existed for thirty years, but nobody had proven you could not do better. Lower bounds are categorically harder than upper bounds: an upper bound just requires showing one method works, while a lower bound must rule out every possible approach, making a statement about the fundamental structure of the problem itself. The reasoning session took about 148 minutes, and the final proof was formalized using Lean 4 with the Mathlib library, so every step is machine-checked rather than merely sounding right.

The real story is prompt engineering, not the model alone

Worth being honest about: this result leaned heavily on a carefully designed prompt: a precise statement of the open problem, an instruction to write the proof in the style of Boyd and Vandenberghe's convex optimization textbook, and an explicit request to flag any unproven steps. GPT-5.6 did not discover this problem on its own initiative, it executed structured reasoning under tight expert guidance. That makes today's AI role closer to an assistant with strong symbolic reasoning than an independent researcher.

What this actually means for developers and researchers

If your work touches formal verification (Lean, Coq, etc.), GPT-5.6-class models are worth adding to your proof-assistance workflow, but you still need an expert to design the prompt and audit every step
"AI solves a math problem" headlines are eye-catching, but the real bar is: someone who understands the problem, plus someone who knows how to ask, neither is optional
Machine-verified formalization (a clean Lean 4 compile) is currently the most reliable way to judge AI-generated proofs, far more trustworthy than the model simply claiming it proved something

好不好用，試了才知道。
You won't know until you try it.

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