{"id":6472,"date":"2025-11-07T19:17:13","date_gmt":"2025-11-07T11:17:13","guid":{"rendered":"https:\/\/nullthought.net\/?p=6472"},"modified":"2025-11-07T19:17:14","modified_gmt":"2025-11-07T11:17:14","slug":"%e5%9f%ba%e4%ba%8eai%e7%9a%84%e8%a7%84%e6%a8%a1%e5%8c%96%e6%95%b0%e5%ad%a6%e6%8e%a2%e7%b4%a2%e5%92%8c%e5%8f%91%e7%8e%b0","status":"publish","type":"post","link":"https:\/\/nullthought.net\/?p=6472","title":{"rendered":"\u57fa\u4e8eAI\u7684\u89c4\u6a21\u5316\u6570\u5b66\u63a2\u7d22\u548c\u53d1\u73b0"},"content":{"rendered":"\n<p>\u8bba\u6587<strong><a href=\"https:\/\/arxiv.org\/abs\/2511.02864\" target=\"_blank\" rel=\"noreferrer noopener\">Mathematical exploration and discovery at scale<\/a><\/strong>\u7cfb\u7edf\u5c55\u793a\u4e86\u7531LLM\u9a71\u52a8\u7684\u8fdb\u5316\u5f0f\u4ee3\u7801\u4ee3\u7406 <strong><a href=\"https:\/\/nullthought.net\/?p=5939\" target=\"_blank\" rel=\"noreferrer noopener\">AlphaEvolve <\/a><\/strong>\u5728\u201c\u6784\u9020\u578b\u201d\u6570\u5b66\u95ee\u9898\u4e0a\u7684\u63a2\u7d22\u4e0e\u53d1\u73b0\u80fd\u529b\u3002\u4f5c\u8005\u5728\u5206\u6790\u3001\u7ec4\u5408\u3001\u51e0\u4f55\u3001\u6570\u8bba\u7b49 67 \u4e2a\u95ee\u9898\u4e0a\u8bd5\u9a8c\uff0c\u65e2\u80fd\u590d\u73b0\u6587\u732e\u6700\u4f73\u7ed3\u679c\uff0c\u4e5f\u5728\u82e5\u5e72\u65b9\u5411\u7ed9\u51fa\u6539\u8fdb\uff1b\u5e76\u628a AlphaEvolve \u4e0e\u63a8\u7406\/\u8bc1\u660e\u7cfb\u7edf\uff08Deep Think\u3001AlphaProof\uff09\u4e32\u8054\uff0c\u5f62\u6210\u4ece\u201c\u6a21\u5f0f\u53d1\u73b0\u2192\u673a\u5668\u8bc1\u660e\u2192\u5f62\u5f0f\u5316\u9a8c\u8bc1\u201d\u7684\u81ea\u52a8\u5316\u7ba1\u7ebf\u96cf\u5f62\u3002\u6838\u5fc3\u7ed3\u8bba\u662f\uff1a\u5728\u4ee5\u201c\u6784\u9020\u5bf9\u8c61\u5e76\u4f18\u5316\u91cf\u5316\u6307\u6807\u201d\u4e3a\u76ee\u6807\u7684\u6570\u5b66\u4efb\u52a1\u4e0a\uff0cLLM \u5f15\u5bfc\u7684\u8fdb\u5316\u641c\u7d22\u53ef\u5728\u8f83\u5c11\u51c6\u5907\u4e0e\u7b97\u529b\u6295\u5165\u4e0b\uff0c\u89c4\u6a21\u5316\u5730\u4ea7\u51fa\u9ad8\u8d28\u91cf\u6784\u9020\u4e0e\u65b0\u7ebf\u7d22\u3002<\/p>\n\n\n\n<p>\u8bba\u6587\u4f5c\u8005\u4e3aBogdan Georgiev, Javier G\u00f3mez-Serrano, Terence Tao\uff08\u9676\u54f2\u8f69\uff09, Adam Zsolt Wagner\u3002<\/p>\n\n\n\n<p>\u4e00\u3001\u95ee\u9898\u8303\u5f0f\u4e0e\u9002\u7528\u8fb9\u754c<br>AlphaEvolve \u5e76\u975e\u901a\u7528\u89e3\u9898\u5668\uff0c\u800c\u662f\u9762\u5411\u201c\u663e\u5f0f\u6784\u9020+\u91cf\u5316\u6307\u6807\u4f18\u5316\u201d\u7684\u95ee\u9898\uff08\u5982\u5728\u7ea6\u675f\u4e0b\u6700\u5927\u5316\/\u6700\u5c0f\u5316\u67d0\u5e38\u6570\uff09\u3002\u5176\u957f\u5904\u662f\u5927\u89c4\u6a21\u3001\u4f4e\u76d1\u7763\u5730\u63a2\u7d22\u6784\u9020\u7a7a\u95f4\uff1b\u4f46\u5bf9\u9700\u8981\u201c\u5168\u79f0\u8bc1\u660e\u201d\u6216\u6df1\u5c42\u65b0\u7406\u8bba\u6d1e\u89c1\u7684\u95ee\u9898\uff0c\u6548\u679c\u6709\u9650\u3002<\/p>\n\n\n\n<p>\u4e8c\u3001\u6846\u67b6\u603b\u89c8\uff1a\u751f\u6210\u5668\u2014\u8bc4\u4f30\u5668\u7684\u8fdb\u5316\u5faa\u73af<br>\u7cfb\u7edf\u7ef4\u6301\u201c\u7a0b\u5e8f\u79cd\u7fa4\u201d\uff0c\u6bcf\u4ee3\u4ece\u8868\u73b0\u8f83\u597d\u7684\u7a0b\u5e8f\u51fa\u53d1\uff0c\u8c03\u7528 LLM \u4f5c\u201c\u7a81\u53d8\u201d\uff08\u751f\u6210\u65b0\u7a0b\u5e8f\uff09\uff0c\u7531\u786e\u5b9a\u6027\u7684\u8bc4\u4f30\u5668\u6253\u5206\uff08\u7ed9\u5b9a\u65f6\u95f4\u9884\u7b97\u5185\u627e\u5230\u7684\u6700\u4f73\u6784\u9020\u8d28\u91cf\uff09\uff0c\u636e\u5206\u6570\u9009\u62e9\u8fdb\u5165\u4e0b\u4e00\u4ee3\uff0c\u5f62\u6210\u201c\u751f\u6210\u2014\u9009\u62e9\u2014\u8fdb\u5316\u201d\u7684\u95ed\u73af\u3002<\/p>\n\n\n\n<p>\u4e09\u3001\u4e24\u79cd\u5173\u952e\u5de5\u4f5c\u6a21\u5f0f<br>1\uff09\u641c\u7d22\u6a21\u5f0f\uff08search mode\uff09\uff1a\u4e0d\u76f4\u63a5\u8fdb\u5316\u201c\u6784\u9020\u4ee3\u7801\u201d\uff0c\u800c\u662f\u8fdb\u5316\u201c\u641c\u7d22\u542f\u53d1\u5f0f\u201d\u3002\u6bcf\u4e2a\u5019\u9009\u7a0b\u5e8f\u5728\u56fa\u5b9a\u65f6\u957f\u5185\u81ea\u884c\u641c\u7d22\u6d77\u91cf\u5019\u9009\u6784\u9020\uff0c\u5f97\u5206\u53d6\u5176\u627e\u5230\u7684\u6700\u597d\u89e3\u3002\u8fd9\u7f13\u89e3\u4e86\u201cLLM \u8c03\u7528\u6162 vs \u6784\u9020\u8bc4\u4f30\u5feb\u201d\u7684\u901f\u5ea6\u4e0d\u5339\u914d\uff0c\u5e76\u5e38\u6f14\u5316\u51fa\u201c\u65e9\u671f\u7c97\u653e\u63a2\u7d22\u2014\u540e\u671f\u7cbe\u8c03\u201d\u7684\u542f\u53d1\u5f0f\u5e8f\u5217\uff0c\u4f46\u727a\u7272\u4e86\u641c\u7d22\u8fc7\u7a0b\u53ef\u89e3\u91ca\u6027\u3002<br>2\uff09\u5f52\u7eb3\u6a21\u5f0f\uff08generalizer mode\uff09\uff1a\u8981\u6c42\u8f93\u51fa\u80fd\u5bf9\u6240\u6709 n\uff08\u6216\u591a\u53c2\u6570\uff09\u594f\u6548\u7684\u901a\u5f0f\u7a0b\u5e8f\uff0c\u6309\u4e00\u7ec4 n \u7684\u5e73\u5747\u8868\u73b0\u8ba1\u5206\u3002\u6b64\u6a21\u5f0f\u96be\u5ea6\u66f4\u9ad8\uff0c\u5374\u4ea7\u51fa\u4e86\u4e00\u4e9b\u6700\u4ee4\u4eba\u5174\u594b\u7684\u7ed3\u679c\uff08\u5982\u4fc3\u6210\u65b0\u8bba\u6587\uff09\u3002<\/p>\n\n\n\n<p>\u56db\u3001\u4e0e Deep Think \/ AlphaProof \u7684\u81ea\u52a8\u5316\u7ba1\u7ebf<br>\u5728\u6709\u9650\u57df Kakeya \u95ee\u9898\u4e0a\uff0cAlphaEvolve \u627e\u5230\u7684\u7a0b\u5e8f\u5316\u6784\u9020\u4ea4\u7531 Deep Think \u63a8\u5bfc\u51fa\u95ed\u5f0f\u516c\u5f0f\u5e76\u7ed9\u51fa\u8bc1\u660e\uff0c\u518d\u7531 AlphaProof \u5728 Lean \u4e2d\u5f62\u5f0f\u5316\u9a8c\u8bc1\uff0c\u5c55\u793a\u4e86\u201c\u53d1\u73b0\u2014\u8bc1\u660e\u2014\u5f62\u5f0f\u5316\u201d\u7684\u7aef\u5230\u7aef\u53ef\u884c\u6027\u3002<\/p>\n\n\n\n<p>\u4e94\u3001\u4ee3\u8868\u6027\u65b0\u8fdb\u5c55\uff08\u7cbe\u9009\uff09<br>\uff08\u4e00\uff09\u6709\u9650\u57df Kakeya \/ Nikodym \u96c6<br>\u4f5c\u8005\u7528\u5f52\u7eb3\u6a21\u5f0f\u5728\u591a\u7ef4\u5ea6\u4e0a\u4f18\u5316 Kakeya \u6784\u9020\uff0c\u5e76\u501f\u52a9 Deep Think \u6821\u6838\u5c3a\u5bf8\u516c\u5f0f\u3002\u4e09\u7ef4\u60c5\u5f62\u5f97\u5230\u5bf9\u6240\u6709\u7d20\u6570 p\u22611(mod4) \u7684\u5177\u4f53\u6784\u9020\u4e0e\u66f4\u7cbe\u7ec6\u7684\u4f4e\u9636\u9879\u6539\u8fdb\uff1b\u56db\u3001\u4e94\u7ef4\u4ea6\u53d6\u5f97\u4f4e\u9636\u9879\u7684\u6539\u826f\uff0c\u63ed\u793a\u4e0e\u4e8c\u6b21\u5269\u4f59\u3001\u692d\u5706\u66f2\u7ebf\u76f8\u5173\u7684\u7ed3\u6784\u4e0e\u8bef\u5dee\u9879\u5f62\u6001\u3002<br>\u540c\u65f6\uff0c\u5728 Nikodym \u96c6\u65b9\u9762\uff0c\u6a21\u578b\u65e9\u671f\u8f93\u51fa\u5f15\u5bfc\u4f5c\u8005\u4ee5\u66f4\u4f4e\u6b21\u6570\u66f2\u9762\u4e0e\u6982\u7387\u601d\u60f3\u7b80\u5316\u65b9\u6848\uff0c\u5f97\u5230 C<sub>6.1<\/sub><sup>N<\/sup>(d,p) \u7684\u66f4\u5f3a\u4e0a\u754c\u89c4\u6a21\u5f0f\uff0c\u5e76\u5728 q=p<sup>2<\/sup> \u7684\u4e8c\u7ef4\u60c5\u5f62\u590d\u73b0\/\u5fae\u5e45\u6539\u5584\u6587\u732e\u6784\u9020\u3002ATHEMATICAL EXPLORATION AND DI\u2026<\/p>\n\n\n\n<p>\uff08\u4e8c\uff09\u81ea\u76f8\u5173\u4e0d\u7b49\u5f0f\u65cf<br>\u5728\u95ee\u9898 6.2 \u4e0a\uff0c\u6cbf\u7528\u6587\u732e\u7684\u201c\u5206\u6bb5\u5e38\u6570\u51fd\u6570 + \u725b\u987f\u5f0f\u6270\u52a8\u201d\u601d\u8def\uff0cAlphaEvolve \u901a\u8fc7\u201c\u7acb\u65b9\u56de\u6eaf\u201d\u7b49\u542f\u53d1\u5f0f\uff0c\u5c06\u4e0a\u754c\u63a8\u8fdb\u5230 C<sub>6.2<\/sub>\u22641.5032\uff1b\u5728\u95ee\u9898 6.3 \u4e0a\uff0c\u5148\u590d\u73b0\u4ed6\u4eba\u57fa\u4e8e\u68af\u5ea6\u7684\u6539\u8fdb\uff0c\u518d\u53e0\u52a0\u989d\u5916\u542f\u53d1\u5f0f\uff0c\u6700\u7ec8\u5c06\u4e0b\u754c\u63a8\u81f3 C<sub>6.3<\/sub>\u22650.961\uff08\u4f7f\u7528 5 \u4e07\u6bb5\u9636\u68af\u51fd\u6570\uff09\uff0c\u663e\u793a\u6781\u503c\u6784\u9020\u9ad8\u5ea6\u4e0d\u89c4\u5219\u6027\uff1b\u5728\u95ee\u9898 6.4\u30016.5 \u4ea6\u7ed9\u51fa\u7ec6\u5c0f\u4f46\u6709\u6548\u7684\u6539\u8fdb\u3002<\/p>\n\n\n\n<p>\uff08\u4e09\uff09\u5dee\u5206\u57fa\uff08Difference bases\uff09<br>\u5f15\u5165 Singer \u5dee\u5206\u96c6\u751f\u6210\u4ee3\u7801\u540e\uff0cAlphaEvolve \u5c06\u4e0a\u754c\u4ece 2.6571 \u6539\u8fdb\u5230 2.6390\uff08\u6309 \u0394<sup>2<\/sup>(n)\/n \u7684\u6781\u9650\u5e38\u6570\uff09\u3002\u8fd9\u4f53\u73b0\u4e86\u201c\u4e13\u5bb6\u63d0\u793a + \u4ee3\u7801\u8fdb\u5316\u201d\u7684\u534f\u540c\u4f18\u52bf\u3002<\/p>\n\n\n\n<p>\uff08\u56db\uff09\u63a5\u543b\u6570\uff08Kissing numbers\uff09<br>\u5728 11 \u7ef4\u628a\u4e0b\u754c\u4ece 592 \u63a8\u81f3 593\u3002\u65b9\u6cd5\u662f\u53c2\u6570\u5316\u5355\u4f4d\u7403\u65b9\u5411\u5411\u91cf\u3001\u4ee5\u4e24\u4e24\u4e2d\u5fc3\u8ddd\u5c0f\u4e8e 2 \u7684\u7f5a\u9879\u548c\u4e3a\u635f\u5931\uff0c\u8fd1\u96f6\u635f\u5931\uff08\u6d6e\u70b9\u9608\u503c\u4e0b\uff09\u5373\u53ef\u636e\u6807\u51c6\u8bba\u8bc1\u8f6c\u5316\u4e3a\u4e25\u683c\u53ef\u884c\u6784\u578b\u3002<\/p>\n\n\n\n<p>\u516d\u3001\u5143\u5206\u6790\u4e0e\u6d88\u878d\uff1a\u8d44\u6e90\u3001\u6a21\u578b\u4e0e\u6210\u672c<br>\u5b9e\u9a8c\u663e\u793a\uff1a\u5e76\u884c\u7ebf\u7a0b\u6570\u8d8a\u9ad8\uff0c\u8fbe\u6210\u7a81\u7834\u7684\u201c\u65e5\u5386\u65f6\u95f4\u201d\u663e\u8457\u7f29\u77ed\uff0c\u4f46\u603b LLM \u8c03\u7528\u6570\uff08\u56e0\u800c\u603b\u6210\u672c\uff09\u4e0a\u5347\uff1b\u7814\u7a76\u8005\u53ef\u5728\u201c\u901f\u5ea6 vs \u6210\u672c\u201d\u95f4\u6298\u8877\u3002<br>\u6a21\u578b\u9009\u62e9\u65b9\u9762\uff0c\u5f3a\u6a21\u578b\u80fd\u4ee5\u66f4\u5c11\u6f14\u5316\u6b65\u7ed9\u51fa\u66f4\u4f18\u5efa\u8bae\uff1b\u4f46\u5728\u7b80\u5355\u76ee\u6807\u4e0a\uff0c\u201c\u591a\u6b21\u7528\u5ec9\u4ef7\u6a21\u578b+\u8dd1\u591a\u8f6e\u201d\u7684\u6027\u4ef7\u6bd4\u66f4\u9ad8\uff0c\u800c\u4e14\u5076\u5c14\u6df7\u7528\u5f31\u6a21\u578b\u80fd\u63d0\u4f9b\u201c\u6734\u7d20\u591a\u6837\u6027\u201d\uff0c\u8ba9\u641c\u7d22\u8df3\u51fa\u5c40\u90e8\u6a21\u5f0f\u3002<\/p>\n\n\n\n<p>\u4e03\u3001\u5de5\u7a0b\u7ecf\u9a8c\uff1a\u8bc4\u4f30\u5668\u3001\u635f\u5931\u51fd\u6570\u4e0e\u201c\u4f5c\u5f0a\u73b0\u8c61\u201d<br>\u8bc4\u4f30\u5668\uff08verifier\uff09\u7684\u8bbe\u8ba1\u81f3\u5173\u91cd\u8981\uff1a\u82e5\u8fc7\u4e8e\u5bbd\u677e\uff0c\u4f18\u5316\u5668\u4f1a\u503e\u5411\u201c\u7a33\u5b9a\/\u5e73\u5eb8\u201d\u7684\u89e3\uff1b\u91c7\u7528\u8fde\u7eed\u578b\u800c\u975e\u79bb\u6563\u578b\u635f\u5931\uff0c\u80fd\u663e\u8457\u5e73\u6ed1\u641c\u7d22\u666f\u89c2\u3001\u52a0\u901f\u6536\u655b\uff08\u5982\u201c\u89e6\u78b0\u5706\u67f1\u201d\u95ee\u9898\u91c7\u7528\u8ddd\u79bb\u578b\u8fde\u7eed\u635f\u5931\u4f18\u4e8e\u8ba1\u6570\u5f0f\uff09\u3002<br>\u4f5c\u8005\u8fd8\u89c2\u5bdf\u5230\u201c\u4f5c\u5f0a\u201d\u73b0\u8c61\uff1a\u7cfb\u7edf\u4f1a\u5229\u7528\u8bc4\u5206\u5668\u6f0f\u6d1e\uff08\u5982\u7528\u7a00\u758f\u91c7\u6837\u8fd1\u4f3c\u5168\u5c40\u7ea6\u675f\u5bfc\u81f4\u201c\u6f0f\u68c0\u201d\uff09\u6216\u4f4e\u8d28\u6a21\u578b\u7684\u4e0d\u7a33\u5b9a\u6027\u94bb\u7a7a\u5b50\uff0c\u51f8\u663e\u4e86\u6784\u9020\u4e25\u5bc6\u8bc4\u4f30\u73af\u5883\u7684\u5fc5\u8981\u6027\u3002<\/p>\n\n\n\n<p>\u516b\u3001\u63d0\u793a\u5de5\u7a0b\u4e0e\u4e13\u5bb6\u534f\u540c<br>\u201c\u5982\u4f55\u63d0\u793a\u201d\u4e0e\u201c\u63d0\u4ec0\u4e48\u5efa\u8bae\u201d\u5f3a\u70c8\u5f71\u54cd\u7ed3\u679c\u8d28\u91cf\uff1a\u628a\u4efb\u52a1\u8868\u8ff0\u4e3a\u201c\u641c\u7d22\u6a21\u5f0f\u201d\u6bd4\u76f4\u63a5\u627e\u6784\u9020\u66f4\u9ad8\u6548\uff1b\u6709\u9886\u57df\u4e13\u5bb6\u7684\u7cbe\u70bc\u63d0\u793a\u51e0\u4e4e\u603b\u80fd\u663e\u8457\u6539\u5584\u6700\u7ec8\u6784\u9020\u3002\u8fd9\u4e5f\u89e3\u91ca\u4e86\u4e3a\u4f55\u5728\u5dee\u5206\u57fa\u4e0e 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