{"version":"2026-10-19.boundary.1","asOf":"2026-10-19","month":"2026-10","domain":{"id":"math","title":"数学 · 推导与证明","summary":"数值答案、几何构造、机器可检证明和自然语言研究证明需要不同验收；竞赛突破不能代表整个数学职业。","frontierId":"F07","sourceIds":["BS02","BS03"],"terms":[{"term":"形式化 / Lean","definition":"把问题与条件写为机器可检查的逻辑命题；Lean检查所写命题的证明，人检查它是不是原问题。"},{"term":"CoT / pass@k","definition":"CoT是先生成推理步骤；Pass@K表示在指定评测标准下生成K个候选，至少一个通过的题目比例。M01的答案Pass@K核对最终答案，CoT-Pass@K另由模型裁判检查过程；两者都不等于形式证明内核通过。"},{"term":"自然语言核对器","definition":"用模型检查候选论证的语言流程，可能漏错；与严格规则的证明内核不同。"},{"term":"复数标量乘法","definition":"一次复数与复数相乘的算术操作计数；不能直接当成实数乘法次数或程序运行时间。"}]},"focusTask":"M01","caseStudy":{"id":"M01","explanationId":"answer-and-proof","title":"数值推导：先确定你要验收哪种正确","lead":"得到对的数字，不一定给出了成立的推导。先区分答案、推理步骤和形式证明，再选择核验方法。","fit":"本案例是多步数值推导。形式证明只作为关联M03独立阅读，不把答案评测当证明。","editorialAt":"2026-10-16","sourceIds":["BM01","BM02","BF06","BM11","BM12","BM13"],"steps":[{"title":"明确题意与产物","actor":"人决定","action":"说明变量、单位和条件；明确要数值答案还是成立推导。","output":"题意、条件与验收目标","gate":"条件不能偷偷替换","judgment":"M01-step-1","editorialReason":"本步骤是本站为交付和接收提出的组织安排；没有本地逐步成功率。","evidenceRelations":[]},{"title":"列出推导与表达式","actor":"AI草拟","action":"逐步把条件变成表达式，列出每一步依据。","output":"带条件的推导草稿","gate":"每一步运算前先核对语义","judgment":"M01-step-2","editorialReason":"本步骤是本站为交付和接收提出的组织安排；没有本地逐步成功率。","evidenceRelations":[]},{"title":"复算与反例检查","actor":"程序执行","action":"复算数值；检查单位、边界条件和错误表达式。","output":"复算记录与反例","gate":"结果相同不代表过程等价","judgment":"M01-step-3","editorialReason":"本步骤是本站为交付和接收提出的组织安排；没有本地逐步成功率。","evidenceRelations":[{"sourceId":"BF06","role":"support","reason":"限定数值问答的程序执行方法；未测本资源约束题","locator":"v4 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floor((40-5y)/2))的非负值，可独立核对最优41。\n失败候选x=20/3,y=16/3虽给连续上界124/3，却违反整数条件。若设备上限改为4，则应重算为x=8,y=4、收益40；不能沿用旧答案。\n交付需包括条件表、单位、整数性、枚举/求解输入、可行性核验、边界与改题结果。","note":"小例子仅解释验收差异；未形成研究级数学证明或模型成绩。"},"failures":[{"trigger":"数字正确，却用不成立的中间等式","why":"答案命中可能掩盖过程错误","action":"逐步检查依据；不能修正则拒绝推导"},{"trigger":"多次尝试找到答案，只保留成功一条","why":"隐藏尝试预算、挑选成本和失败","action":"记录全部尝试与选取规则；不把Pass@K当单次保证"},{"trigger":"连续解很好看，但工作包不可拆分","why":"题意到模型时遗漏整数约束，答案不具可执行性","action":"返回条件表，按整数重新求解并核对资源上限"}],"trial":{"scope":"一组条件明确、可独立复算的数值题；不扩大成开放证明或教学效果","baseline":"保存同一题意下人的答案/过程核验方式；分别计时","acceptance":["答案与单位可复算","每个推导步骤有依据、原条件未改","重复尝试/裁判配置/人的核验独立记录"],"timeFields":["活动人时：题意确认","活动人时：人工过程审查","活动人时：复算","活动人时：反例与边界检查","活动人时：返工","活动人时：提示、交互与监督","活动人时：接管与适用的维护","另记：机器运行、等待及端到端历时；重叠活动不重复累计；若等待占用本人，注明计量约定"],"stop":"题意被替换、过程不成立或裁判无法确认时，拒绝推导并由人接管"},"comparisons":[{"title":"程序执行接口","sourceId":"BF06","relationship":"部分覆盖","task":"GSM8K 1,318题","people":"固定小学多步题","setup":"同Codex；少样本贪心CoT/PoT","outcome":"63.1% → 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