{
  "schema_version":"ocme-learning-path-registry-v0.1",
  "paths":[
    {
      "schema_version":"ocme-learning-path-v0.1",
      "id":"path-geometry-foundations",
      "title_zh":"幾何基礎到畢達哥拉斯定理",
      "audience":["general_learner","upper_secondary"],
      "goal_zh":"由點、線、面的基礎語言，經三角形、角、長度與直角三角形進入畢達哥拉斯定理。",
      "nodes":[
        {"object_id":"mko-point-line-plane","position":1,"requirement":"required","rationale_zh":"先區分幾何位置、延伸對象與圖示。"},
        {"object_id":"mko-triangle","position":2,"requirement":"required","rationale_zh":"由三個相異不共線點建立三角形、頂點與邊。"},
        {"object_id":"mko-angle","position":2,"requirement":"required","rationale_zh":"建立共同端點兩射線與角量的語言。"},
        {"object_id":"mko-euclidean-length","position":2,"requirement":"required","rationale_zh":"平行建立邊長與距離的意義。"},
        {"object_id":"mko-right-triangle","position":3,"requirement":"required","rationale_zh":"再以直角與邊確定定理適用的幾何對象。"},
        {"object_id":"mko-euclid-pythagorean-theorem","position":4,"requirement":"required","rationale_zh":"最後理解平方關係與證明。"}
      ],
      "edges":[
        {"from":"mko-point-line-plane","to":"mko-triangle","relation":"required_before"},
        {"from":"mko-point-line-plane","to":"mko-angle","relation":"required_before"},
        {"from":"mko-point-line-plane","to":"mko-euclidean-length","relation":"recommended_before"},
        {"from":"mko-angle","to":"mko-right-triangle","relation":"required_before"},
        {"from":"mko-triangle","to":"mko-right-triangle","relation":"required_before"},
        {"from":"mko-euclidean-length","to":"mko-right-triangle","relation":"recommended_before"},
        {"from":"mko-right-triangle","to":"mko-euclid-pythagorean-theorem","relation":"required_before"}
      ],
      "review":{"status":"candidate","reviewer":"Codex local AI","version":2}
    },
    {
      "schema_version":"ocme-learning-path-v0.1",
      "id":"path-function-to-limits",
      "title_zh":"集合、函數到趨近關係",
      "audience":["upper_secondary","undergraduate"],
      "goal_zh":"建立集合隸屬、總函數與趨近語義的基礎鏈。",
      "nodes":[
        {"object_id":"mko-set-membership","position":1,"requirement":"required","rationale_zh":"建立元素與集合之間的基本關係。"},
        {"object_id":"mko-function-mapping","position":2,"requirement":"required","rationale_zh":"理解定義域到陪域的總函數。"},
        {"object_id":"mko-tends-to-relation","position":3,"requirement":"required","rationale_zh":"在函數關係上引入趨近與濾子核心。"}
      ],
      "edges":[
        {"from":"mko-set-membership","to":"mko-function-mapping","relation":"recommended_before"},
        {"from":"mko-function-mapping","to":"mko-tends-to-relation","relation":"required_before"}
      ],
      "review":{"status":"reviewed","reviewer":"Neo.K / Aletheia","version":1}
    },
    {
      "schema_version":"ocme-learning-path-v0.1",
      "id":"path-modern-general-literacy",
      "title_zh":"現代通識數學樣本路徑",
      "audience":["general_learner"],
      "goal_zh":"以六個現有物件示範通識入口的多分支學習方式。",
      "nodes":[
        {"object_id":"mko-set-membership","position":1,"requirement":"required","rationale_zh":"先接觸現代數學常用的集合語言。"},
        {"object_id":"mko-function-mapping","position":2,"requirement":"required","rationale_zh":"建立函數作為映射的基本概念。"},
        {"object_id":"mko-euclidean-length","position":2,"requirement":"optional","rationale_zh":"平行進入幾何量與距離。"},
        {"object_id":"mko-right-triangle","position":3,"requirement":"optional","rationale_zh":"由長度進入直角幾何。"},
        {"object_id":"mko-euclid-pythagorean-theorem","position":4,"requirement":"optional","rationale_zh":"觀察幾何、代數與證明的連接。"},
        {"object_id":"mko-tends-to-relation","position":4,"requirement":"optional","rationale_zh":"由函數進入分析中的趨近語義。"}
      ],
      "edges":[
        {"from":"mko-set-membership","to":"mko-function-mapping","relation":"recommended_before"},
        {"from":"mko-euclidean-length","to":"mko-right-triangle","relation":"recommended_before"},
        {"from":"mko-right-triangle","to":"mko-euclid-pythagorean-theorem","relation":"recommended_before"},
        {"from":"mko-function-mapping","to":"mko-tends-to-relation","relation":"recommended_before"}
      ],
      "review":{"status":"reviewed","reviewer":"Neo.K / Aletheia","version":1}
    },
    {
      "schema_version":"ocme-learning-path-v0.1",
      "id":"path-formalized-mathematics-entry",
      "title_zh":"形式化數學入門樣本",
      "audience":["undergraduate","formalizer"],
      "goal_zh":"從集合與函數語義進入 Lean／Mathlib 的精確聲明與證據物件。",
      "nodes":[
        {"object_id":"mko-set-membership","position":1,"requirement":"required","rationale_zh":"對齊 Set 作為謂詞的形式語義。"},
        {"object_id":"mko-function-mapping","position":2,"requirement":"required","rationale_zh":"對齊總函數型別與唯一輸出。"},
        {"object_id":"mko-tends-to-relation","position":3,"requirement":"required","rationale_zh":"理解 Filter.Tendsto 的形式核心。"},
        {"object_id":"mko-euclid-pythagorean-theorem","position":4,"requirement":"optional","rationale_zh":"觀察百科語義到向量及邊長模型的形式橋。"}
      ],
      "edges":[
        {"from":"mko-set-membership","to":"mko-function-mapping","relation":"required_before"},
        {"from":"mko-function-mapping","to":"mko-tends-to-relation","relation":"required_before"}
      ],
      "review":{"status":"reviewed","reviewer":"Neo.K / Aletheia","version":1}
    },
    {
      "schema_version":"ocme-learning-path-v0.1",
      "id":"path-proof-and-evidence-literacy",
      "title_zh":"證明與證據素養",
      "audience":["general_learner","undergraduate"],
      "goal_zh":"先辨識證明義務，再區分人工證明、有限計算證據與精確形式證明。",
      "nodes":[
        {"object_id":"mko-proof","position":1,"requirement":"required","rationale_zh":"先建立前提、推理、結論與非證明證據的區分。"},
        {"object_id":"mko-euclid-pythagorean-theorem","position":2,"requirement":"required","rationale_zh":"同一物件同時具有人工、Python、FELRA 與 Lean 證據。"},
        {"object_id":"mko-right-triangle","position":3,"requirement":"optional","rationale_zh":"比較定義物件與定理物件的證據需求。"},
        {"object_id":"mko-tends-to-relation","position":3,"requirement":"optional","rationale_zh":"比較形式核心與完整分析語義的邊界。"}
      ],
      "edges":[
        {"from":"mko-proof","to":"mko-euclid-pythagorean-theorem","relation":"recommended_before"},
        {"from":"mko-euclid-pythagorean-theorem","to":"mko-right-triangle","relation":"extends"},
        {"from":"mko-euclid-pythagorean-theorem","to":"mko-tends-to-relation","relation":"alternative_entry"}
      ],
      "review":{"status":"reviewed","reviewer":"Neo.K / Aletheia","version":1}
    },
    {
      "schema_version":"ocme-learning-path-v0.1",
      "id":"path-foundational-language-candidate",
      "title_zh":"數學基礎語言候選路徑",
      "audience":["general_learner","upper_secondary"],
      "goal_zh":"從計數的具體語言前進到集合、命題與邏輯連接詞，建立閱讀後續數學物件所需的基礎語言。",
      "nodes":[
        {"object_id":"mko-natural-number","position":1,"requirement":"required","rationale_zh":"先用熟悉的計數對象建立數學概念與符號的區分。"},
        {"object_id":"mko-set","position":2,"requirement":"required","rationale_zh":"再以集合整理對象、元素與整體之間的語言。"},
        {"object_id":"mko-proposition","position":3,"requirement":"required","rationale_zh":"引入可被斷言、判定與證明的數學陳述。"},
        {"object_id":"mko-logical-connectives","position":4,"requirement":"required","rationale_zh":"用否定、合取、析取、蘊涵與雙條件組合命題。"},
        {"object_id":"mko-proof","position":5,"requirement":"required","rationale_zh":"以明示前提與合法推理建立結論。"}
      ],
      "edges":[
        {"from":"mko-natural-number","to":"mko-set","relation":"recommended_before"},
        {"from":"mko-set","to":"mko-proposition","relation":"recommended_before"},
        {"from":"mko-proposition","to":"mko-logical-connectives","relation":"required_before"},
        {"from":"mko-logical-connectives","to":"mko-proof","relation":"required_before"}
      ],
      "review":{"status":"candidate","reviewer":"Codex local AI","version":2}
    },
    {
      "schema_version":"ocme-learning-path-v0.1",
      "id":"path-domain-coverage-wave-one",
      "title_zh":"第一波領域覆蓋基礎",
      "audience":["general_learner","upper_secondary"],
      "goal_zh":"由自然數、集合與命題分支進入算術、方程、機率、歸納、階乘與基本組合計數。",
      "nodes":[
        {"object_id":"mko-natural-number","position":1,"requirement":"required","rationale_zh":"提供算術、歸納與有限計數的自然數基礎。"},
        {"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"提供樣本空間作為結果集合的基礎。"},
        {"object_id":"mko-proposition","position":1,"requirement":"required","rationale_zh":"提供歸納法所處理命題族的邏輯基礎。"},
        {"object_id":"mko-arithmetic-operations","position":2,"requirement":"required","rationale_zh":"解鎖代數表達與基本計數中的數量組合。"},
        {"object_id":"mko-sample-space","position":2,"requirement":"optional","rationale_zh":"從集合語言進入機率統計的結果論域。"},
        {"object_id":"mko-mathematical-induction","position":2,"requirement":"optional","rationale_zh":"從自然數與命題進入結構化證明方法。"},
        {"object_id":"mko-variable-expression","position":3,"requirement":"optional","rationale_zh":"從算術運算進入含變數的代數表達。"},
        {"object_id":"mko-equation","position":4,"requirement":"optional","rationale_zh":"由表達式進入等式、解集與代入驗證。"},
        {"object_id":"mko-linear-equation","position":5,"requirement":"optional","rationale_zh":"以非零一次係數收斂到唯一解與退化分類。"},
        {"object_id":"mko-counting-principle","position":3,"requirement":"optional","rationale_zh":"以自然數與算術運算組合有限選擇數量。"},
        {"object_id":"mko-factorial","position":3,"requirement":"optional","rationale_zh":"以自然數乘法與後繼遞迴建立有限乘積。"}
      ],
      "edges":[
        {"from":"mko-natural-number","to":"mko-arithmetic-operations","relation":"required_before"},
        {"from":"mko-arithmetic-operations","to":"mko-variable-expression","relation":"required_before"},
        {"from":"mko-variable-expression","to":"mko-equation","relation":"required_before"},
        {"from":"mko-equation","to":"mko-linear-equation","relation":"required_before"},
        {"from":"mko-set","to":"mko-sample-space","relation":"required_before"},
        {"from":"mko-natural-number","to":"mko-mathematical-induction","relation":"required_before"},
        {"from":"mko-proposition","to":"mko-mathematical-induction","relation":"required_before"},
        {"from":"mko-natural-number","to":"mko-counting-principle","relation":"required_before"},
        {"from":"mko-arithmetic-operations","to":"mko-counting-principle","relation":"required_before"},
        {"from":"mko-natural-number","to":"mko-factorial","relation":"required_before"},
        {"from":"mko-arithmetic-operations","to":"mko-factorial","relation":"required_before"}
      ],
      "review":{"status":"candidate","reviewer":"Codex local AI","version":1}
    },
    {
      "schema_version":"ocme-learning-path-v0.1",
      "id":"path-set-language-to-topology-candidate",
      "title_zh":"集合語言到拓撲前置候選路徑",
      "audience":["upper_secondary","undergraduate"],
      "goal_zh":"從集合、元素隸屬、子集合與基本集合運算建立閱讀拓撲空間定義所需的集合語言。",
      "nodes":[
        {"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"先確立集合是由隸屬條件刻畫的對象。"},
        {"object_id":"mko-set-membership","position":2,"requirement":"required","rationale_zh":"再以元素是否屬於集合表達運算定義。"},
        {"object_id":"mko-subset","position":3,"requirement":"required","rationale_zh":"由逐元素隸屬建立集合之間的包含關係。"},
        {"object_id":"mko-set-operations","position":3,"requirement":"required","rationale_zh":"平行建立聯集、交集、差集與相對補集。"}
      ],
      "edges":[
        {"from":"mko-set","to":"mko-set-membership","relation":"recommended_before"},
        {"from":"mko-set-membership","to":"mko-subset","relation":"required_before"},
        {"from":"mko-set-membership","to":"mko-set-operations","relation":"required_before"}
      ],
      "review":{"status":"candidate","reviewer":"Codex local AI","version":1}
    },
    {
      "schema_version":"ocme-learning-path-v0.1",
      "id":"path-probability-events-candidate",
      "title_zh":"樣本空間到事件候選路徑",
      "audience":["upper_secondary","undergraduate"],
      "goal_zh":"由集合語言與樣本空間建立事件作為結果子集合的精確邊界。",
      "nodes":[
        {"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"先建立集合語言。"},
        {"object_id":"mko-set-membership","position":2,"requirement":"required","rationale_zh":"區分結果元素與結果集合。"},
        {"object_id":"mko-subset","position":3,"requirement":"required","rationale_zh":"建立集合之間的包含關係。"},
        {"object_id":"mko-sample-space","position":3,"requirement":"required","rationale_zh":"指定試驗的全部可能結果。"},
        {"object_id":"mko-event","position":4,"requirement":"required","rationale_zh":"把事件定位為樣本空間中的結果集合。"}
      ],
      "edges":[
        {"from":"mko-set","to":"mko-set-membership","relation":"required_before"},
        {"from":"mko-set-membership","to":"mko-subset","relation":"required_before"},
        {"from":"mko-set","to":"mko-sample-space","relation":"required_before"},
        {"from":"mko-subset","to":"mko-event","relation":"required_before"},
        {"from":"mko-sample-space","to":"mko-event","relation":"required_before"}
      ],
      "review":{"status":"candidate","reviewer":"Codex local AI","version":1}
    },
    {
      "schema_version":"ocme-learning-path-v0.1",
      "id":"path-number-system-extension-candidate",
      "title_zh":"自然數、整數、有理數、實數到絕對值候選路徑",
      "audience":["general_learner","lower_secondary"],
      "goal_zh":"由計數用的自然數進入整數與有理數，以完備化語言建立實數入口，再用絕對值表達數線距離。",
      "nodes":[
        {"object_id":"mko-natural-number","position":1,"requirement":"required","rationale_zh":"先建立自然數與零的慣例。"},
        {"object_id":"mko-integer","position":2,"requirement":"required","rationale_zh":"加入負方向形成整數。"},
        {"object_id":"mko-rational-number","position":3,"requirement":"required","rationale_zh":"以兩個整數之比建立有理數，並排除零分母。"},
        {"object_id":"mko-real-number","position":4,"requirement":"required","rationale_zh":"由有理數嵌入與完備化進入實數系。"},
        {"object_id":"mko-absolute-value","position":5,"requirement":"optional","rationale_zh":"以分段函數表示實數到零的非負距離。"}
      ],
      "edges":[
        {"from":"mko-natural-number","to":"mko-integer","relation":"required_before"},
        {"from":"mko-integer","to":"mko-rational-number","relation":"required_before"},
        {"from":"mko-rational-number","to":"mko-real-number","relation":"required_before"},
        {"from":"mko-real-number","to":"mko-absolute-value","relation":"required_before"}
      ],
      "review":{"status":"candidate","reviewer":"Codex local AI","version":3}
    },
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20260930","title_zh":"七輪基礎擴充：數值、函數、計數與座標","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-rational-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-fraction","position":2,"requirement":"optional","rationale_zh":"分數 p/q 是有理數的一種表示；分子與分母為整數，分母必須非零。"},{"object_id":"mko-ratio-proportion","position":3,"requirement":"optional","rationale_zh":"本條目把數值比 a:b 解讀為 a/b，把比例定義為兩個合法比值相等。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-domain-codomain-range","position":4,"requirement":"optional","rationale_zh":"總函數 f:A→B 的定義域是 A，陪域是 B，值域是實際取得的輸出集合 f[A]。"},{"object_id":"mko-composition","position":5,"requirement":"optional","rationale_zh":"合成 g∘f 先求 f 的輸出，再把它送入 g；合成前須確認中間值可作為 g 的輸入。"},{"object_id":"mko-natural-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence","position":6,"requirement":"optional","rationale_zh":"本條目採從 0 起始的實數數列 a:N→R；數列項依索引排列，不能任意重排成集合。"},{"object_id":"mko-counting-principle","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-factorial","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-permutation-combination","position":7,"requirement":"optional","rationale_zh":"由 n 個相異物件不重複取 k 個，有序選取是排列，無序選取是組合。"},{"object_id":"mko-point-line-plane","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-coordinate-plane","position":8,"requirement":"optional","rationale_zh":"固定原點、互相垂直的有向實數軸與单位後，平面點用有序實數對 (x,y) 表示。"}],"edges":[{"from":"mko-rational-number","to":"mko-fraction","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-fraction","relation":"required_before"},{"from":"mko-fraction","to":"mko-ratio-proportion","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-domain-codomain-range","relation":"required_before"},{"from":"mko-set","to":"mko-domain-codomain-range","relation":"required_before"},{"from":"mko-domain-codomain-range","to":"mko-composition","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-sequence","relation":"required_before"},{"from":"mko-natural-number","to":"mko-sequence","relation":"required_before"},{"from":"mko-real-number","to":"mko-sequence","relation":"required_before"},{"from":"mko-counting-principle","to":"mko-permutation-combination","relation":"required_before"},{"from":"mko-factorial","to":"mko-permutation-combination","relation":"required_before"},{"from":"mko-real-number","to":"mko-coordinate-plane","relation":"required_before"},{"from":"mko-point-line-plane","to":"mko-coordinate-plane","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261003","title_zh":"七輪代數與機率基礎：冪、恆等式、函數與有限模型","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-power-root","position":2,"requirement":"optional","rationale_zh":"本輪限定實數底數的自然數冪，以及非負實數的主平方根，分開冪運算、根值與解集。"},{"object_id":"mko-variable-expression","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-rational-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-polynomial","position":3,"requirement":"optional","rationale_zh":"把多項式視為有限支撐的有理係數資料；合併同次項後再讀取係數與次數。"},{"object_id":"mko-factorization","position":4,"requirement":"optional","rationale_zh":"因式分解是在指定係數域中把多項式寫成因子乘積；驗收比較全部係數而不是若干代入值。"},{"object_id":"mko-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quadratic-equation","position":5,"requirement":"optional","rationale_zh":"對非零二次係數的實方程，用判別式區分實根數，將方程式、根集合與重根分開。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-function","position":6,"requirement":"optional","rationale_zh":"實函數 f(x)=mx+b 稱仿射函數；m≠0 時為一次函數，m=0 時為常數函數。"},{"object_id":"mko-coordinate-plane","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-graph","position":7,"requirement":"optional","rationale_zh":"固定直角座標系後，實值函數的圖形是全部合法輸入與其唯一輸出組成的有序對集合。"},{"object_id":"mko-event","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-ratio-proportion","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-probability","position":8,"requirement":"optional","rationale_zh":"在非空有限樣本空間上指定非負且總和為一的權重，事件機率為其成員權重之和。"}],"edges":[{"from":"mko-arithmetic-operations","to":"mko-power-root","relation":"required_before"},{"from":"mko-real-number","to":"mko-power-root","relation":"required_before"},{"from":"mko-variable-expression","to":"mko-polynomial","relation":"required_before"},{"from":"mko-power-root","to":"mko-polynomial","relation":"required_before"},{"from":"mko-rational-number","to":"mko-polynomial","relation":"required_before"},{"from":"mko-polynomial","to":"mko-factorization","relation":"required_before"},{"from":"mko-equation","to":"mko-quadratic-equation","relation":"required_before"},{"from":"mko-power-root","to":"mko-quadratic-equation","relation":"required_before"},{"from":"mko-polynomial","to":"mko-quadratic-equation","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-linear-function","relation":"required_before"},{"from":"mko-linear-equation","to":"mko-linear-function","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-function-graph","relation":"required_before"},{"from":"mko-coordinate-plane","to":"mko-function-graph","relation":"required_before"},{"from":"mko-event","to":"mko-probability","relation":"required_before"},{"from":"mko-ratio-proportion","to":"mko-probability","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261004","title_zh":"每日七輪第一期：量詞、離散結構與線性／条件模型","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-proposition","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":2,"requirement":"optional","rationale_zh":"全稱與存在量詞必須綁定論域；古典真值語義下，空論域的全稱為真、存在為假。"},{"object_id":"mko-proof","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-counterexample","position":3,"requirement":"optional","rationale_zh":"反例必須屬於被聲稱的論域且使全稱述詞失敗；域外值和找不到反例都不能自動證明原命題。"},{"object_id":"mko-set-membership","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-relation","position":4,"requirement":"optional","rationale_zh":"二元關係是指定 A×B 中有序對的集合，不要求每個輸入唯一或必有輸出。"},{"object_id":"mko-graph","position":5,"requirement":"optional","rationale_zh":"本輪限定有限簡單無向圖，邊是兩個相異頂點的無序集合，度數只計相鄰頂點。"},{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-coordinate-plane","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-vector","position":6,"requirement":"optional","rationale_zh":"本輪把平面向量限制為固定基底下的 R² 有序對，逐分量定義加法與純量乘法。"},{"object_id":"mko-linear-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-system-linear-equations","position":7,"requirement":"optional","rationale_zh":"同時滿足兩個實數線性限制才是解；非零行列式給唯一解，奇異與零列須另判一致性。"},{"object_id":"mko-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-conditional-probability","position":8,"requirement":"optional","rationale_zh":"條件事件必須有正機率；有限模型中 P(A|B)=P(A∩B)/P(B)，方向不能任意交換。"}],"edges":[{"from":"mko-proposition","to":"mko-quantifier","relation":"required_before"},{"from":"mko-set","to":"mko-quantifier","relation":"required_before"},{"from":"mko-quantifier","to":"mko-counterexample","relation":"required_before"},{"from":"mko-proof","to":"mko-counterexample","relation":"required_before"},{"from":"mko-set-membership","to":"mko-relation","relation":"required_before"},{"from":"mko-set","to":"mko-relation","relation":"required_before"},{"from":"mko-set","to":"mko-graph","relation":"required_before"},{"from":"mko-relation","to":"mko-graph","relation":"required_before"},{"from":"mko-real-number","to":"mko-vector","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-vector","relation":"required_before"},{"from":"mko-coordinate-plane","to":"mko-vector","relation":"required_before"},{"from":"mko-linear-equation","to":"mko-system-linear-equations","relation":"required_before"},{"from":"mko-vector","to":"mko-system-linear-equations","relation":"required_before"},{"from":"mko-probability","to":"mko-conditional-probability","relation":"required_before"},{"from":"mko-set-operations","to":"mko-conditional-probability","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261005","title_zh":"函數可逆性與有限機率矩數：每日七輪","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-injective","position":2,"requirement":"optional","rationale_zh":"單射要求不同合法輸入不能有同一輸出，不要求覆蓋整個陪域。"},{"object_id":"mko-surjective","position":3,"requirement":"optional","rationale_zh":"滿射要求陪域每個元素都有合法原像；相同輸出可以有多個原像。"},{"object_id":"mko-bijective","position":4,"requirement":"optional","rationale_zh":"雙射同時單射與滿射，等價於陪域每個元素恰有一個合法原像。"},{"object_id":"mko-composition","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-function","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-inverse-function","position":5,"requirement":"optional","rationale_zh":"雙射 f:A→B 的反函數 g:B→A 須驗證兩個方向的還原，不能把交換資料當成無條件反函數。"},{"object_id":"mko-sample-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-random-variable","position":6,"requirement":"optional","rationale_zh":"有限模型中的隨機變數是基本結果到實數的總函數；它不是基本結果，也不是權重本身。"},{"object_id":"mko-expectation","position":7,"requirement":"optional","rationale_zh":"有限期望是合法機率權重與隨機變數值的加權總和，線性性不要求獨立。"},{"object_id":"mko-power-root","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-variance","position":8,"requirement":"optional","rationale_zh":"本輪變異數為 E[(X−E[X])²]，不是 n−1 分母的樣本變異數；零權重離群值不影響它。"}],"edges":[{"from":"mko-function-mapping","to":"mko-injective","relation":"required_before"},{"from":"mko-quantifier","to":"mko-injective","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-surjective","relation":"required_before"},{"from":"mko-quantifier","to":"mko-surjective","relation":"required_before"},{"from":"mko-injective","to":"mko-bijective","relation":"required_before"},{"from":"mko-surjective","to":"mko-bijective","relation":"required_before"},{"from":"mko-bijective","to":"mko-inverse-function","relation":"required_before"},{"from":"mko-composition","to":"mko-inverse-function","relation":"required_before"},{"from":"mko-linear-function","to":"mko-inverse-function","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-random-variable","relation":"required_before"},{"from":"mko-sample-space","to":"mko-random-variable","relation":"required_before"},{"from":"mko-real-number","to":"mko-random-variable","relation":"required_before"},{"from":"mko-probability","to":"mko-random-variable","relation":"required_before"},{"from":"mko-random-variable","to":"mko-expectation","relation":"required_before"},{"from":"mko-probability","to":"mko-expectation","relation":"required_before"},{"from":"mko-expectation","to":"mko-variance","relation":"required_before"},{"from":"mko-power-root","to":"mko-variance","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261006","title_zh":"實數順序、變化率與座標幾何、資料摘要：七輪脈衝","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-rational-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":2,"requirement":"optional","rationale_zh":"在實數論域比較大小；精確有理數交叉乘積、有限極值及含端點區間是本輪切片。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quadratic-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quadratic-function","position":3,"requirement":"optional","rationale_zh":"實數全論域的 q(x)=a(x−h)²+k、a≠0；頂點、全域極值與對稱點的明示切片。"},{"object_id":"mko-ratio-proportion","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-graph","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-function","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-rate-of-change","position":4,"requirement":"optional","rationale_zh":"平均變化率是輸出差除以非零輸入差，不是瞬時導數。"},{"object_id":"mko-coordinate-plane","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-power-root","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-euclidean-length","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-distance-formula","position":5,"requirement":"optional","rationale_zh":"在同一等尺度正交座標系，用非負主平方根讀取二維歐氏距離。"},{"object_id":"mko-point-line-plane","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-circle","position":6,"requirement":"optional","rationale_zh":"圓是到固定中心距離恰等於半徑的點集，不是含內部的圓盤。"},{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-area","position":7,"requirement":"optional","rationale_zh":"先精確處理半開矩形及有限單位格聯集，含重疊、觸邊與空矩形；不冒充一般面積理論。"},{"object_id":"mko-descriptive-statistics","position":8,"requirement":"optional","rationale_zh":"保留重數的非空觀測列，精確計算均值、中位數、全距與所有最大頻率值。"}],"edges":[{"from":"mko-real-number","to":"mko-order-relation","relation":"required_before"},{"from":"mko-rational-number","to":"mko-order-relation","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-quadratic-function","relation":"required_before"},{"from":"mko-quadratic-equation","to":"mko-quadratic-function","relation":"required_before"},{"from":"mko-ratio-proportion","to":"mko-rate-of-change","relation":"required_before"},{"from":"mko-function-graph","to":"mko-rate-of-change","relation":"required_before"},{"from":"mko-linear-function","to":"mko-rate-of-change","relation":"required_before"},{"from":"mko-quadratic-function","to":"mko-rate-of-change","relation":"required_before"},{"from":"mko-coordinate-plane","to":"mko-distance-formula","relation":"required_before"},{"from":"mko-power-root","to":"mko-distance-formula","relation":"required_before"},{"from":"mko-euclidean-length","to":"mko-distance-formula","relation":"required_before"},{"from":"mko-euclidean-length","to":"mko-circle","relation":"required_before"},{"from":"mko-point-line-plane","to":"mko-circle","relation":"required_before"},{"from":"mko-distance-formula","to":"mko-circle","relation":"required_before"},{"from":"mko-euclidean-length","to":"mko-area","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-area","relation":"required_before"},{"from":"mko-coordinate-plane","to":"mko-area","relation":"required_before"},{"from":"mko-set-operations","to":"mko-area","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-descriptive-statistics","relation":"required_before"},{"from":"mko-rational-number","to":"mko-descriptive-statistics","relation":"required_before"},{"from":"mko-order-relation","to":"mko-descriptive-statistics","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261007","title_zh":"矩陣、實二階線性代數與標記三角形、座標變換：七輪脈衝","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-matrix","position":2,"requirement":"optional","rationale_zh":"矩陣以行、列組織元素；本輪使用非空有限實矩陣及精確整數陣列切片。"},{"object_id":"mko-vector","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-transformation","position":3,"requirement":"optional","rationale_zh":"先在 R²→R² 的矩陣切片驗證加法與純量相容，不把帶非零平移的仿射式叫線性映射。"},{"object_id":"mko-determinant","position":4,"requirement":"optional","rationale_zh":"2×2實矩陣的 ad−bc、非零行列式可逆性、乘積與純量縮放切片。"},{"object_id":"mko-quadratic-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-eigenvalue","position":5,"requirement":"optional","rationale_zh":"特徵向量須非零；候選特徵空間核含零向量，實二階矩陣可沒有實特徵值。"},{"object_id":"mko-triangle","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-distance-formula","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-congruence","position":6,"requirement":"optional","rationale_zh":"限定正交等單位平面中的標記非退化三角形，以三組對應邊長判全等，允許鏡射。"},{"object_id":"mko-ratio-proportion","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-power-root","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-similarity","position":7,"requirement":"optional","rationale_zh":"指定由A到B的正長度倍率 k，平方長度與指定面積以 k² 縮放，不把負座標係數當負長度倍率。"},{"object_id":"mko-coordinate-plane","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-geometric-transformations","position":8,"requirement":"optional","rationale_zh":"平移、原點逆時針四分之一圈的整數倍旋轉、指定軸鏡射是全平面等距；均勻縮放先分清零、±1與其他係數。"}],"edges":[{"from":"mko-arithmetic-operations","to":"mko-matrix","relation":"required_before"},{"from":"mko-real-number","to":"mko-matrix","relation":"required_before"},{"from":"mko-matrix","to":"mko-linear-transformation","relation":"required_before"},{"from":"mko-vector","to":"mko-linear-transformation","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-linear-transformation","relation":"required_before"},{"from":"mko-matrix","to":"mko-determinant","relation":"required_before"},{"from":"mko-real-number","to":"mko-determinant","relation":"required_before"},{"from":"mko-linear-transformation","to":"mko-eigenvalue","relation":"required_before"},{"from":"mko-determinant","to":"mko-eigenvalue","relation":"required_before"},{"from":"mko-quadratic-equation","to":"mko-eigenvalue","relation":"required_before"},{"from":"mko-triangle","to":"mko-congruence","relation":"required_before"},{"from":"mko-distance-formula","to":"mko-congruence","relation":"required_before"},{"from":"mko-determinant","to":"mko-congruence","relation":"required_before"},{"from":"mko-congruence","to":"mko-similarity","relation":"required_before"},{"from":"mko-ratio-proportion","to":"mko-similarity","relation":"required_before"},{"from":"mko-power-root","to":"mko-similarity","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-geometric-transformations","relation":"required_before"},{"from":"mko-coordinate-plane","to":"mko-geometric-transformations","relation":"required_before"},{"from":"mko-distance-formula","to":"mko-geometric-transformations","relation":"required_before"},{"from":"mko-similarity","to":"mko-geometric-transformations","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261008","title_zh":"實數極限、連續、導數與多項式積分：七輪脈衝","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-tends-to-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-rational-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence-limit","position":2,"requirement":"optional","rationale_zh":"從0起算的實數列以任意精度、某一自然尾界及整個尾段定義極限；本輪精確處理倒數尾項模型。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-absolute-value","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-factorization","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-limit","position":3,"requirement":"optional","rationale_zh":"在a附近排除a本身，以ε-δ控制函數值；點值不屬於去心極限條件。"},{"object_id":"mko-continuity","position":4,"requirement":"optional","rationale_zh":"在全實數論域連續需要點值存在且與兩側極限一致；補點與原函數連續不可混用。"},{"object_id":"mko-rate-of-change","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-polynomial","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-derivative","position":5,"requirement":"optional","rationale_zh":"導數是非零步差商的兩側極限，不是任何一個有限步差商；絕對值尖點另保留不存在。"},{"object_id":"mko-composition","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-derivative-rules","position":6,"requirement":"optional","rationale_zh":"四條微分規則先保留可微前提；本輪只在有限整數係數多項式上精確計算。"},{"object_id":"mko-antiderivative","position":7,"requirement":"optional","rationale_zh":"原函數須在宣告區間導回f；本輪全R多項式以常數項固定0的有理係數原函數作可重播代表。"},{"object_id":"mko-area","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-definite-integral","position":8,"requirement":"optional","rationale_zh":"連續多項式的有向定積分與精確矩公式；有限左端黎曼和只是有限和，不冒充一般積分或微積分基本定理完成。"}],"edges":[{"from":"mko-sequence","to":"mko-sequence-limit","relation":"required_before"},{"from":"mko-tends-to-relation","to":"mko-sequence-limit","relation":"required_before"},{"from":"mko-quantifier","to":"mko-sequence-limit","relation":"required_before"},{"from":"mko-rational-number","to":"mko-sequence-limit","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-function-limit","relation":"required_before"},{"from":"mko-tends-to-relation","to":"mko-function-limit","relation":"required_before"},{"from":"mko-absolute-value","to":"mko-function-limit","relation":"required_before"},{"from":"mko-factorization","to":"mko-function-limit","relation":"required_before"},{"from":"mko-function-limit","to":"mko-continuity","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-continuity","relation":"required_before"},{"from":"mko-function-limit","to":"mko-derivative","relation":"required_before"},{"from":"mko-rate-of-change","to":"mko-derivative","relation":"required_before"},{"from":"mko-polynomial","to":"mko-derivative","relation":"required_before"},{"from":"mko-absolute-value","to":"mko-derivative","relation":"required_before"},{"from":"mko-derivative","to":"mko-derivative-rules","relation":"required_before"},{"from":"mko-polynomial","to":"mko-derivative-rules","relation":"required_before"},{"from":"mko-composition","to":"mko-derivative-rules","relation":"required_before"},{"from":"mko-derivative","to":"mko-antiderivative","relation":"required_before"},{"from":"mko-derivative-rules","to":"mko-antiderivative","relation":"required_before"},{"from":"mko-rational-number","to":"mko-antiderivative","relation":"required_before"},{"from":"mko-function-limit","to":"mko-definite-integral","relation":"required_before"},{"from":"mko-polynomial","to":"mko-definite-integral","relation":"required_before"},{"from":"mko-antiderivative","to":"mko-definite-integral","relation":"required_before"},{"from":"mko-area","to":"mko-definite-integral","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261009","title_zh":"Core80最後種子與五輪既有題族密度","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-derivative","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-definite-integral","position":4,"requirement":"optional","rationale_zh":"原有多項式矩與有向有限區間切片：任意端點順序的三段積分；奇函數資格與對稱積分成對輸出；有序端點仿射左端有限和的精確誤差；非零常數函數積分的端點反解。"},{"object_id":"mko-continuity","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-antiderivative","position":3,"requirement":"optional","rationale_zh":"原有全R多項式與正分母有理係數切片：指定原函數係數、共享F0的兩常數差、由F(a)=V解C，以及候選F的導數係數。保持升冪、最簡分數與常數0規約。"},{"object_id":"mko-derivative-rules","position":2,"requirement":"optional","rationale_zh":"原有至多3次全R多項式切片：二階導數點值、乘積導數候選值的資格、兩個有序合成導數，以及和的導數完整升冪係數。合成可能升次，不截去高項。"},{"object_id":"mko-fundamental-theorem-calculus","position":5,"requirement":"optional","rationale_zh":"在全實數連續前提下，累積定積分的導數還原被積函數；合法原函數給有向積分端點差。"},{"object_id":"mko-graph","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-tree","position":2,"requirement":"optional","rationale_zh":"非空有限簡單無向圖同時連通且無環才是樹；保留單點、度1葉及加一條新邊的唯一環。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-tends-to-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-absolute-value","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-factorization","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-limit","position":2,"requirement":"optional","rationale_zh":"在既有去心實數模型內加厚：m≠0且|m|δ>ε時取h=ε/|m|，得到0<h<δ但誤差=ε；點值與極限成對輸出；左右極限保持順序；商只在x≠a求值。"},{"object_id":"mko-polynomial","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-composition","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-rational-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-area","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-transformation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-determinant","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quadratic-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-eigenvalue","position":2,"requirement":"optional","rationale_zh":"原有實二階方陣整數切片：降冪特徵多項式(1,−tr,det)、精確判別式、Mv−λv有序殘差、非零向量第一失配座標。沒有對角化、Jordan或複數譜擴充。"}],"edges":[{"from":"mko-derivative","to":"mko-fundamental-theorem-calculus","relation":"required_before"},{"from":"mko-definite-integral","to":"mko-fundamental-theorem-calculus","relation":"required_before"},{"from":"mko-continuity","to":"mko-fundamental-theorem-calculus","relation":"required_before"},{"from":"mko-antiderivative","to":"mko-fundamental-theorem-calculus","relation":"required_before"},{"from":"mko-derivative-rules","to":"mko-fundamental-theorem-calculus","relation":"required_before"},{"from":"mko-graph","to":"mko-tree","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-function-limit","relation":"required_before"},{"from":"mko-tends-to-relation","to":"mko-function-limit","relation":"required_before"},{"from":"mko-absolute-value","to":"mko-function-limit","relation":"required_before"},{"from":"mko-factorization","to":"mko-function-limit","relation":"required_before"},{"from":"mko-derivative","to":"mko-derivative-rules","relation":"required_before"},{"from":"mko-polynomial","to":"mko-derivative-rules","relation":"required_before"},{"from":"mko-composition","to":"mko-derivative-rules","relation":"required_before"},{"from":"mko-derivative","to":"mko-antiderivative","relation":"required_before"},{"from":"mko-derivative-rules","to":"mko-antiderivative","relation":"required_before"},{"from":"mko-rational-number","to":"mko-antiderivative","relation":"required_before"},{"from":"mko-function-limit","to":"mko-definite-integral","relation":"required_before"},{"from":"mko-polynomial","to":"mko-definite-integral","relation":"required_before"},{"from":"mko-antiderivative","to":"mko-definite-integral","relation":"required_before"},{"from":"mko-area","to":"mko-definite-integral","relation":"required_before"},{"from":"mko-linear-transformation","to":"mko-eigenvalue","relation":"required_before"},{"from":"mko-determinant","to":"mko-eigenvalue","relation":"required_before"},{"from":"mko-quadratic-equation","to":"mko-eigenvalue","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261010","title_zh":"七輪跨域密度：代數、集合、距離、有限概率與量詞","audience":["general_learner","upper_secondary"],"goal_zh":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","nodes":[{"object_id":"mko-variable-expression","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-power-root","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-rational-number","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-polynomial","position":2,"requirement":"optional","rationale_zh":"固定整數升冪係數至多3次的Q[x]切片；和與乘積输出完整正規化係數，乘積可升至6次；同一多項式雙點有序求值；p(x+c)完整係數以精確二項式展開核對。不把單點相同當多項式恆等。"},{"object_id":"mko-set-membership","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-set-operations","position":2,"requirement":"optional","rationale_zh":"有限明示整數集合正規化重複與排序；聯／交大小對、(A\\B)∪(B\\A)、U\\(A∪B)與(U\\A)∩(U\\B)的完整結果、三段有序分割。補集先要求A、B⊆明示U。只封閉此有限实例，不新增一般集合論形式證明。"},{"object_id":"mko-coordinate-plane","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-euclidean-length","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-distance-formula","position":2,"requirement":"optional","rationale_zh":"所有向量是同一等單位正交實平面從共同原點出發的整數座標差；精確平方對、候選非負資格與平方缺陷、有三個具索引距離的含同距分組排序，以及兩個完全平方距離的整數和。不是非正交／不等尺度或Prod預設metric；不拿兩段和當另一段距離。"},{"object_id":"mko-probability","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-conditional-probability","position":3,"requirement":"optional","rationale_zh":"S=0至n−1、全冪集、非負整數mass與明示正分母D，Σmass=D才為歸一化模型。B質量嚴格正，雙向另要求A正。枚舉交集／B概率對、雙向條件值、對全部singleton在B條件下的概率向量、候選值與真正條件值的有號缺陷。非空但mass0不能條件化；不宣稱Bayes或独立性。"},{"object_id":"mko-random-variable","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-expectation","position":2,"requirement":"optional","rationale_zh":"同一有限合法歸一化整數值模型：逐點wX向量、E[X−Y]、(E[XY],E[X]E[Y])有序對與給定均值下的正質量未知整數值反解。只有共同S與權重，不假設X/Y獨立；數值相等也不推出独立。未知slot明示null、正權重且解為整數；不做樣本估計或一般積分存在性。"},{"object_id":"mko-variance","position":3,"requirement":"optional","rationale_zh":"合法有限概率模型：逐項w(X−μ)²、(E[X²],μ²)精確對、先構造共模型X−Y再求其變異數，以及相對第一個正權重值的第一個正支持失配索引。不是n−1樣本估計；保留零權重異值；不把Var(X−Y)拆成兩個Var，也不聲稱協方差理論完成。"},{"object_id":"mko-proposition","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":2,"requirement":"optional","rationale_zh":"有限明示整數U及可判定大小／相等／非零整除述詞。完整真假元素分割、(¬∀P,¬∃P)、三個有序候選的域內反例资格mask，及有限U/V上(滿足∃y內條件的x數,滿足∀x內條件的y數)。空域全稱真、存在假；候選域外無资格；嵌套枚舉不交换量詞、不外推全部整數或一般一階逻輯證明。"}],"edges":[{"from":"mko-variable-expression","to":"mko-polynomial","relation":"required_before"},{"from":"mko-power-root","to":"mko-polynomial","relation":"required_before"},{"from":"mko-rational-number","to":"mko-polynomial","relation":"required_before"},{"from":"mko-set-membership","to":"mko-set-operations","relation":"required_before"},{"from":"mko-coordinate-plane","to":"mko-distance-formula","relation":"required_before"},{"from":"mko-power-root","to":"mko-distance-formula","relation":"required_before"},{"from":"mko-euclidean-length","to":"mko-distance-formula","relation":"required_before"},{"from":"mko-probability","to":"mko-conditional-probability","relation":"required_before"},{"from":"mko-set-operations","to":"mko-conditional-probability","relation":"required_before"},{"from":"mko-random-variable","to":"mko-expectation","relation":"required_before"},{"from":"mko-probability","to":"mko-expectation","relation":"required_before"},{"from":"mko-expectation","to":"mko-variance","relation":"required_before"},{"from":"mko-power-root","to":"mko-variance","relation":"required_before"},{"from":"mko-proposition","to":"mko-quantifier","relation":"required_before"},{"from":"mko-set","to":"mko-quantifier","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261011","title_zh":"七輪基礎補缺：自然數、表達式、集合與有限趨近模型","audience":["general_learner","upper_secondary"],"goal_zh":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","nodes":[{"object_id":"mko-natural-number","position":1,"requirement":"optional","rationale_zh":"固定N包含0。列有限後繼窗、雙點的部分自然前驅、有限閉區間大小與三個有理候選的N資格；0無自然前驅，不混用截斷pred、負數或分數。"},{"object_id":"mko-arithmetic-operations","position":2,"requirement":"optional","rationale_zh":"固定有理數或N輸入的Q結果切片。四則有序結果、括號雙式、逆元候選有號缺陷、自然數差商的資格；除零未定義，負差不截成0，商不是整除。"},{"object_id":"mko-variable-expression","position":3,"requirement":"optional","rationale_zh":"只指定Q中的四種小表達式，不建立通用AST。比較兩環境值、同時交換兩變數、內外層詞法遮蔽及a/(x-b)的原式定義域；零分子也不消除排除點，代數式不當方程式。"},{"object_id":"mko-set","position":1,"requirement":"optional","rationale_zh":"明示整數型別的有限集合。去重排序、相等及雙向差、候選元素的內外分割、至多4元素的全部子集合列舉；空集合的冪集只有空集合，不宣稱無限制概括或新集合論公理。"},{"object_id":"mko-set-membership","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-subset","position":2,"requirement":"optional","rationale_zh":"同一整數型別的有限集合。全部A不在B的元素、非嚴格／嚴格包含及雙向見證、三段包含鏈的阻擋元素、三個候選反例的域內資格；空集包含與相等不是嚴格包含，有限檢查不當普遍集合證明。"},{"object_id":"mko-sample-space","position":2,"requirement":"optional","rationale_zh":"有限且明示的可能結果協議。完整有序直積、記錄映射像及原結果纖維、候選清單的漏列／域外項、基本結果和事件隸屬的雙mask；不指定概率，不假設均勻、獨立或由未知試驗推得完整性。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-tends-to-relation","position":2,"requirement":"optional","rationale_zh":"來源及目標均明示為有限主濾子，映射在完整有限型別上為總函數。核對像生成集、f(S)不在T的阻擋點、反例候選資格與兩條合成像計算；允許空生成集的bottom濾子。只關閉明示有限模型，不把裸箭頭當一般數列／點極限，不新增形式Evidence。"}],"edges":[{"from":"mko-natural-number","to":"mko-arithmetic-operations","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-variable-expression","relation":"required_before"},{"from":"mko-set-membership","to":"mko-subset","relation":"required_before"},{"from":"mko-set","to":"mko-sample-space","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-tends-to-relation","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261012","title_zh":"七輪基礎推理與幾何：歸納、計數、命題、型別與長度","audience":["general_learner","upper_secondary"],"goal_zh":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","nodes":[{"object_id":"mko-natural-number","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-proposition","position":1,"requirement":"optional","rationale_zh":"固定古典整數比較陳述與有號算術見證。未代入的自由變數標未封閉；相等與否定分開；候選自報真假只作待核資料，與實際真假和正式證明項分離。不建立通用邏輯語言或新形式Evidence。"},{"object_id":"mko-mathematical-induction","position":2,"requirement":"optional","rationale_zh":"N從0起，分清歸納假設n與目標n+1。只核對明示整數遞推的有限前綴、仿射候選的基礎／步缺陷與有限檢查覆蓋；不把有限樣本當成任意n的歸納證明。"},{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-counting-principle","position":2,"requirement":"optional","rationale_zh":"有限選擇協議明示互斥標籤或完整有序直積，包含空分支與零選項；後續寬度可依前一步變化，候選等寬法核對資格和缺陷，不從總數巧合推得等寬或概率獨立。兩集合重疊由實際外延計數修正。"},{"object_id":"mko-set-membership","position":1,"requirement":"optional","rationale_zh":"有限同型別集合與完整特徵表。Fin(n)外候選只標不合型別，不盲讀謂詞；隸屬判定與插入／刪除結果、有限明示論域上的謂詞集合外延。保留Set α→Prop既有Evidence，不把程式容器當全部集合論。"},{"object_id":"mko-right-triangle","position":1,"requirement":"optional","rationale_zh":"候選整數角度先核正角與和180的協議，再檢90；兩個90加零角不能通過。合格直角資料標記頂點、對邊及銳角值；π有理倍數精確換算，不用容差、不取模、不把候選當已構造的非退化三角形。"},{"object_id":"mko-euclidean-length","position":1,"requirement":"optional","rationale_zh":"同一等單位正交L2實平面的整數座標切片。保留有向座標差及長度平方；完全平方才給整數主距離；中點為精確有理數，兩段平方分別核對。軸上路徑與端點直線長分開，零距離不排除；不重播舊浮點參考Evidence或聲稱全部度量公理完成。"},{"object_id":"mko-function-mapping","position":2,"requirement":"optional","rationale_zh":"完整有限型別上的總映射，允許空定義域及唯一空表，非空源到空陪域不可宣稱總函數。圖形、實際值域與未用陪域分開；型別外查詢標不合型別；兩張同型別總表逐點核差異。保留既有總函數core Evidence，不混作部分、有副作用或會失敗的程式函數。"}],"edges":[{"from":"mko-natural-number","to":"mko-mathematical-induction","relation":"required_before"},{"from":"mko-proposition","to":"mko-mathematical-induction","relation":"required_before"},{"from":"mko-natural-number","to":"mko-counting-principle","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-counting-principle","relation":"required_before"},{"from":"mko-set-membership","to":"mko-function-mapping","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261013","title_zh":"七輪有限結構：畢氏、數系、推理、事件與標記三角","audience":["general_learner","upper_secondary"],"goal_zh":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","nodes":[{"object_id":"mko-right-triangle","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-euclidean-length","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-euclid-pythagorean-theorem","position":2,"requirement":"optional","rationale_zh":"正整數兩股與明示歐氏直角邊長協議。平方欄位有序，完全平方才取正整數斜邊或缺股；候選負c即使平方殘差0也不合非退化長度。保留既有FELRA、向量核心及宣告side model證據，不宣稱任意仿射橋已完成。"},{"object_id":"mko-natural-number","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-integer","position":2,"requirement":"optional","rationale_zh":"明示Z及有理候選。歐式整除要求正b及0≤r<b，負a不用截斷餘數；淨位移與步量分開；相反數及雙向差有序；負整數、零與非整數分數的資格和實值分組，有限切片不當環公理。"},{"object_id":"mko-rational-number","position":3,"requirement":"optional","rationale_zh":"分子與非零整數分母，包含負分母表示，答案正分母精確規約。和差、次序及精確差、除法資格與回乘、缺失加數回代，不用浮點或近似，不聲稱完整Rat構造或域公理。"},{"object_id":"mko-logical-connectives","position":2,"requirement":"optional","rationale_zh":"兩份完整八行0/1古典檔表逐行做固定合取／析取、材料蘊涵／雙條件、兩路德摩根與含括／排他或。材料蘊涵不作因果，逐行真假不當所有推理規則證明或通用邏輯語言。"},{"object_id":"mko-proof","position":3,"requirement":"optional","rationale_zh":"有限Pi標籤、給定前提與單前提MP邊。最短推導深度、型別資格、候選序列逐行審查與刪規則影響；無種子循環不自證，錯行不加入可用集。只是明示玩具推理協議，不當通用邏輯完備性、人審或形式Evidence。"},{"object_id":"mko-proposition","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-sample-space","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-subset","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-event","position":2,"requirement":"optional","rationale_zh":"明示非空有限S，區分F=P(S)與分塊聯集族。補事件／聯交外補、分塊的部分相交見證及最小閉包、候選觀測資格與首發生；有限S不自動讓全部子集可測，沒有概率或獨立假設。"},{"object_id":"mko-point-line-plane","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-triangle","position":2,"requirement":"optional","rationale_zh":"整數標記A/B/C的歐氏頂點協議。相異／共線見證、合格三角形的對邊平方、循環及交換標記的座標行列式、對邊中點與三段平方；候選退化資料不冒稱三角形，有限座標核對不作一般幾何構造證明。"}],"edges":[{"from":"mko-right-triangle","to":"mko-euclid-pythagorean-theorem","relation":"required_before"},{"from":"mko-euclidean-length","to":"mko-euclid-pythagorean-theorem","relation":"required_before"},{"from":"mko-natural-number","to":"mko-integer","relation":"required_before"},{"from":"mko-integer","to":"mko-rational-number","relation":"required_before"},{"from":"mko-logical-connectives","to":"mko-proof","relation":"required_before"},{"from":"mko-proposition","to":"mko-logical-connectives","relation":"required_before"},{"from":"mko-sample-space","to":"mko-event","relation":"required_before"},{"from":"mko-subset","to":"mko-event","relation":"required_before"},{"from":"mko-point-line-plane","to":"mko-triangle","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261014","title_zh":"七輪基礎深化：角度、數線、方程、階乘與圓邊界","audience":["general_learner","upper_secondary"],"goal_zh":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","nodes":[{"object_id":"mko-point-line-plane","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-angle","position":2,"requirement":"optional","rationale_zh":"0到180度非定向角的精確度數協議。拆分先核資格，互餘與互補分开，雙門檻殘差和三段累加只作明示算術；超180累加量不冒充本條目角，不由示意圖、射線長或定向旋轉推測量值。"},{"object_id":"mko-rational-number","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-real-number","position":2,"requirement":"optional","rationale_zh":"只核有理點在R的嵌入切片：有序中點、有限唯一點列及閉包區間、含端點的閉區間交集與分離距離、四個點的有向偏移及距離。接觸是點區間而非空；不是Cauchy完備性、無理數分類或一般連續性證明。"},{"object_id":"mko-variable-expression","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-equation","position":2,"requirement":"optional","rationale_zh":"給定Q上的原式雙側值、有限候選與可逆條件。乘零能改變解集；含x-h分母時先排除x=h，交乘殘差零不恢復未定義點。同量平移核兩側值及殘差，不建立通用AST或一般方程求解完備性。"},{"object_id":"mko-linear-equation","position":3,"requirement":"optional","rationale_zh":"a非零才屬Q上一元一次方程；規一列、參數切片的退化常數等式、根周有理偏移回代與兩個一次方程的根／共同解。a=0可能全Q或無解但不是一次，有限核對不作一般線性系統或唯一性形式Evidence。"},{"object_id":"mko-natural-number","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-factorial","position":2,"requirement":"optional","rationale_zh":"N從0起且0!=1，用BigInt核相鄰階乘精確比、含空乘積的窗口、候選自報的真值與遞迴缺陷、省去一次已存在因子後回乘。報告用有界正規十進整數字串，不以浮點或遞迴自洽冒充正確階乘；不聲稱排列、整除性或漸近普遍證明。"},{"object_id":"mko-absolute-value","position":3,"requirement":"optional","rationale_zh":"只核有理嵌入的分段絕對值：負分母表示先正規化，三點距離與路徑差分開，閉半徑四點代入與有理目標的根／三候選核對。負目標無根，零根去重；有限切片非全部實數性質、三角不等式或一般不等式求解證明。"},{"object_id":"mko-euclidean-length","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-distance-formula","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-circle","position":2,"requirement":"optional","rationale_zh":"等單位正交歐氏框架與非負整數半徑。三點平方殘差分類但圓只取邊界；軸向兩個對徑端點、半徑非負尺度乘算、水平截線平方餘量與整數候選。r=0只有中心且軸向表示折疊，負餘量不取根，不聲稱一般圓API或相似定理的形式橋。"}],"edges":[{"from":"mko-point-line-plane","to":"mko-angle","relation":"required_before"},{"from":"mko-rational-number","to":"mko-real-number","relation":"required_before"},{"from":"mko-variable-expression","to":"mko-equation","relation":"required_before"},{"from":"mko-equation","to":"mko-linear-equation","relation":"required_before"},{"from":"mko-natural-number","to":"mko-factorial","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-factorial","relation":"required_before"},{"from":"mko-real-number","to":"mko-absolute-value","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-absolute-value","relation":"required_before"},{"from":"mko-euclidean-length","to":"mko-circle","relation":"required_before"},{"from":"mko-point-line-plane","to":"mko-circle","relation":"required_before"},{"from":"mko-distance-formula","to":"mko-circle","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261015","title_zh":"七輪型別資料與選取：分數、比例、映射、合成與座標","audience":["general_learner","upper_secondary"],"goal_zh":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","nodes":[{"object_id":"mko-rational-number","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-fraction","position":2,"requirement":"optional","rationale_zh":"合法整數分數表示，原分母非零且負分母可用。真加法對照分子分母分別加的候選、四值相鄰差鏈、倒數先核零、兩次原始縮放表示與零縮放不合法；表示與數值分開，不把有限算術當商集或良定性形式證明。"},{"object_id":"mko-ratio-proportion","position":3,"requirement":"optional","rationale_zh":"Q數值比先核後項非零：正反向比、同k的完整三點律、同單位非負總量的正權分配、候選比例雙後項與交乘殘差。零總量的0:0仍非法，交乘零不恢復非法比；有限模型不外推因果、所有輸入或幾何相似。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-domain-codomain-range","position":2,"requirement":"optional","rationale_zh":"Fin(n)到Fin(m)完整表的域／陪域／值域、兩個候選陪域值的原像、單格替換與值域增刪、一列協議候選表稽核。未使用格也須合型；重複同值資料列違反一列協議但不代表多值函數，不混同資料表示稽核與一般函數證明。"},{"object_id":"mko-composition","position":3,"requirement":"optional","rationale_zh":"先f後g且先核中間域；完整表路徑、g在明示子域的總表限制、兩序Q仿射合成與常數差、三總表的兩種括號路徑。未入g域不求值，未使用的g/h格仍須合型；合成不當乘法，不作全部函數等式形式證明。"},{"object_id":"mko-natural-number","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-sequence","position":2,"requirement":"optional","rationale_zh":"N從0起的明示有限索引、給定仿射規則的窗口、同四項前綴的兩個有限延伸、五項前綴和與有序差。次序與重複保留；空窗口不灌水、相同候選不算不同延伸，有限前綴不推定唯一無限規則、收斂或無窮級數。"},{"object_id":"mko-counting-principle","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-factorial","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-permutation-combination","position":2,"requirement":"optional","rationale_zh":"相異標記Fin(n)物件無放回。有序／無序計數及k!與空選取、三個候選取樣資格、含／排物件0的組合分割、固定前r個物件順序後餘量與完成計數比。空選取只有一種，不套重複、不可區分、環形模型或擅稱概率。"},{"object_id":"mko-point-line-plane","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-coordinate-plane","position":2,"requirement":"optional","rationale_zh":"同一正交有向xy框架中的整數有序對：四點欄位／軸／象限、逐分量差與相等、交換欄位候選、三對的首個不等。原點同屬兩軸，軸點不屬象限；交換欄位不作幾何反射形式橋，分量差不當距離或图形尺寸。"}],"edges":[{"from":"mko-rational-number","to":"mko-fraction","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-fraction","relation":"required_before"},{"from":"mko-fraction","to":"mko-ratio-proportion","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-domain-codomain-range","relation":"required_before"},{"from":"mko-set","to":"mko-domain-codomain-range","relation":"required_before"},{"from":"mko-domain-codomain-range","to":"mko-composition","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-sequence","relation":"required_before"},{"from":"mko-natural-number","to":"mko-sequence","relation":"required_before"},{"from":"mko-real-number","to":"mko-sequence","relation":"required_before"},{"from":"mko-counting-principle","to":"mko-permutation-combination","relation":"required_before"},{"from":"mko-factorial","to":"mko-permutation-combination","relation":"required_before"},{"from":"mko-real-number","to":"mko-coordinate-plane","relation":"required_before"},{"from":"mko-point-line-plane","to":"mko-coordinate-plane","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261016","title_zh":"七輪線性與有限映射：向量、矩陣、方程及單滿射","audience":["general_learner","upper_secondary"],"goal_zh":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","nodes":[{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-coordinate-plane","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-vector","position":2,"requirement":"optional","rationale_zh":"固定標準基底下的整數向量：兩純量加權、相反差、純量和兩側與候選分量殘差。零純量与零向量合法，逐分量不當內積或長度。"},{"object_id":"mko-matrix","position":2,"requirement":"optional","rationale_zh":"非空1至5行列整數矩陣：轉置再轉置、同尺寸加法的兩側轉置、三個1起算索引資格與候選轉置逐項殘差。形狀不合先拒絕求值，不當求逆或反序。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-linear-transformation","position":3,"requirement":"optional","rationale_zh":"明示全R²到R²列向量模型Mv：基底列重建、加法兩側、純量兩側與Mv+b的零點障礙。零矩陣可以線性，有限測值不替任意函數證明線性。"},{"object_id":"mko-determinant","position":3,"requirement":"optional","rationale_zh":"R上2×2有號行列式：交换兩行的負號、t²縮放、通常行乘列的兩序乘積、絕對值候選誤判。實數域非零可逆不搬至整數環，負行列式不取絕對值。"},{"object_id":"mko-linear-equation","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-system-linear-equations","position":3,"requirement":"optional","rationale_zh":"R上整數係數的兩元兩式：有理候選逐式殘差、係數與增廣秩區分無解／唯一／無限多解、加倍列消元與第一列縮放。t=0不是可逆列操作，不能只看行列式零便稱無解。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-injective","position":2,"requirement":"optional","rationale_zh":"完整Fin總表單射：列出全部碰撞、移去一個輸入後保留原標籤、附加輸出的陪域資格與限制到明示子集合。空源单射真，碰撞必有兩個不同輸入；重複表示不當多值函數。"},{"object_id":"mko-surjective","position":2,"requirement":"optional","rationale_zh":"完整Fin總表滿射：全纖維與漏項、更換陪域先容納全部輸出、刪輸入保原標籤後漏項、兩個候選目標的資格與首原像。空到空滿射真，空到非空不滿；值域不自動當陪域。"}],"edges":[{"from":"mko-real-number","to":"mko-vector","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-vector","relation":"required_before"},{"from":"mko-coordinate-plane","to":"mko-vector","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-matrix","relation":"required_before"},{"from":"mko-real-number","to":"mko-matrix","relation":"required_before"},{"from":"mko-matrix","to":"mko-linear-transformation","relation":"required_before"},{"from":"mko-vector","to":"mko-linear-transformation","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-linear-transformation","relation":"required_before"},{"from":"mko-matrix","to":"mko-determinant","relation":"required_before"},{"from":"mko-real-number","to":"mko-determinant","relation":"required_before"},{"from":"mko-linear-equation","to":"mko-system-linear-equations","relation":"required_before"},{"from":"mko-vector","to":"mko-system-linear-equations","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-injective","relation":"required_before"},{"from":"mko-quantifier","to":"mko-injective","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-surjective","relation":"required_before"},{"from":"mko-quantifier","to":"mko-surjective","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261017","title_zh":"七輪離散結構：雙射、反函數、關係、圖樹與反例","audience":["general_learner","upper_secondary"],"goal_zh":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","nodes":[{"object_id":"mko-injective","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-surjective","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-bijective","position":2,"requirement":"optional","rationale_zh":"完整Fin總表把單射、滿射與雙射分開；全纖維／碰撞／漏項、交換關係資格、交換兩個輸出標籤、基數存在性與給定表分開。相同基數不審定給定f；空到空雙射真。"},{"object_id":"mko-composition","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-linear-function","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-inverse-function","position":3,"requirement":"optional","rationale_zh":"完整Fin總表先雙射再建反表；候選g逐格合型後才核兩側identity；查詢先陪域資格。不把反函數當函數值倒數。R到R仿射恢復限非零斜率，兩個合法有理輸入以最簡正分母回代。"},{"object_id":"mko-set-membership","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-relation","position":2,"requirement":"optional","rationale_zh":"R是明示Fin來源×目標的有序對集合：完整投影／漏源、子集合像與全纖維、逆關係再逆、單值與總表分開。關係可多值或缺源；重複對與集合列項正規化，逆關係不自動給反函數。"},{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-rational-number","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":2,"requirement":"optional","rationale_zh":"R通常全序的正原分母有理切片：有序三筆穩定排序、兩閉區間交會與端點、三個會員與有號餘量、純量正向／反向／零合併。要求實際新數值，不只換平移參數重出相同比較；不外推偏序或複數。"},{"object_id":"mko-graph","position":3,"requirement":"optional","rationale_zh":"簡單無向Fin圖：完整鄰接矩陣／度數與握手、全頂點鄰居／非鄰居分割、有效邊toggle的度數改變、明示子集合誘導圖且保原標籤。反向或重複邊不增加邊，無自環；本graph輪不作連通或樹判定。"},{"object_id":"mko-tree","position":4,"requirement":"optional","rationale_zh":"非空樹資格與完整簡單無向圖資料：components／有環與空域互達、原樹與端點資格後唯一道路、度1葉刪除保原標籤、原樹上新增不存在邊的唯一環。空圖不是樹；單點無邊是樹但度0非葉，邊數不單獨判樹。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-proof","position":1,"requirement":"required","rationale_zh":"本密度脈衝所需的既有 Canonical 前置。"},{"object_id":"mko-counterexample","position":2,"requirement":"optional","rationale_zh":"明示有限Fin論域與完整布林述詞表：全失敗集及最小反例、三個候選的域資格、限制子域後見證是否仍有效、明示加入一個新述詞值。域外不查P也不當反例；空域全稱量化真，有限枚舉不證無限域。"}],"edges":[{"from":"mko-injective","to":"mko-bijective","relation":"required_before"},{"from":"mko-surjective","to":"mko-bijective","relation":"required_before"},{"from":"mko-bijective","to":"mko-inverse-function","relation":"required_before"},{"from":"mko-composition","to":"mko-inverse-function","relation":"required_before"},{"from":"mko-linear-function","to":"mko-inverse-function","relation":"required_before"},{"from":"mko-set-membership","to":"mko-relation","relation":"required_before"},{"from":"mko-set","to":"mko-relation","relation":"required_before"},{"from":"mko-real-number","to":"mko-order-relation","relation":"required_before"},{"from":"mko-rational-number","to":"mko-order-relation","relation":"required_before"},{"from":"mko-set","to":"mko-graph","relation":"required_before"},{"from":"mko-relation","to":"mko-graph","relation":"required_before"},{"from":"mko-graph","to":"mko-tree","relation":"required_before"},{"from":"mko-quantifier","to":"mko-counterexample","relation":"required_before"},{"from":"mko-proof","to":"mko-counterexample","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261018","title_zh":"擴展數學地圖第一批：拓撲、數論、決定問題與形式陳述","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-topological-space","position":2,"requirement":"optional","rationale_zh":"拓撲空間由底集合與一族開集組成；開集族包含空集與全空間，並對任意聯集與有限交封閉。"},{"object_id":"mko-natural-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-natural-divisibility","position":3,"requirement":"optional","rationale_zh":"自然數 a 整除 b，表示存在自然數乘數 k 使 b=a·k；零的兩側行為必須分開。"},{"object_id":"mko-gcd-coprime","position":4,"requirement":"optional","rationale_zh":"gcd(a,b) 是同時整除 a、b 且被每個公因數整除的自然數；gcd=1 時稱互質。"},{"object_id":"mko-prime-number","position":5,"requirement":"optional","rationale_zh":"自然數質數是大於或等於 2，且正因數只有 1 與自身的自然數。"},{"object_id":"mko-modular-congruence","position":6,"requirement":"optional","rationale_zh":"固定正模數 n 時，a 與 b 模 n 同餘表示兩者除以 n 的餘數相同。"},{"object_id":"mko-proposition","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-decision-problem","position":7,"requirement":"optional","rationale_zh":"決定問題由輸入集合 X 與 yes-instance 子集合 Y⊆X 指定；decider 必須對每個合法輸入終止並回傳正確布林答案。"},{"object_id":"mko-lean-formal-statement","position":8,"requirement":"optional","rationale_zh":"形式陳述是在固定版本、imports、環境與 local context 中成功 elaboration，且型別為 Prop 的核心項。"}],"edges":[{"from":"mko-set","to":"mko-topological-space","relation":"required_before"},{"from":"mko-set-operations","to":"mko-topological-space","relation":"required_before"},{"from":"mko-natural-number","to":"mko-natural-divisibility","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-natural-divisibility","relation":"required_before"},{"from":"mko-natural-divisibility","to":"mko-gcd-coprime","relation":"required_before"},{"from":"mko-natural-divisibility","to":"mko-prime-number","relation":"required_before"},{"from":"mko-natural-number","to":"mko-modular-congruence","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-modular-congruence","relation":"required_before"},{"from":"mko-set","to":"mko-decision-problem","relation":"required_before"},{"from":"mko-proposition","to":"mko-decision-problem","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-decision-problem","relation":"required_before"},{"from":"mko-proposition","to":"mko-lean-formal-statement","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261019","title_zh":"擴展數學地圖第二批：拓撲語言、計算正確性、漸近界與證明項","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-topological-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subset","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-open-closed-set","position":2,"requirement":"optional","rationale_zh":"在固定拓撲空間中，開集屬於拓撲；閉集的補集為開集，同一集合可以同時開且閉。"},{"object_id":"mko-set-membership","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-neighborhood","position":3,"requirement":"optional","rationale_zh":"集合 N 是點 x 的鄰域，當且僅當 N 包含某個含 x 的開集；N 本身不必開。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-topological-continuity","position":4,"requirement":"optional","rationale_zh":"拓撲映射連續，表示目標空間每個開集的逆像在來源空間仍是開集。"},{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-natural-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-computational-model","position":5,"requirement":"optional","rationale_zh":"有限狀態計算模型以狀態集、初始狀態集、步驟關係與停止狀態集描述可能執行。"},{"object_id":"mko-proposition","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-proof","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-algorithm-correctness","position":6,"requirement":"optional","rationale_zh":"算法正確性分開部分正確性與終止性；總正確性需要兩者同時成立。"},{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-asymptotic-resource-bound","position":7,"requirement":"optional","rationale_zh":"漸近上界、下界與緊界必須相對輸入大小、成本模型及全尾段量化定義。"},{"object_id":"mko-lean-formal-statement","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-lean-proof-term","position":8,"requirement":"optional","rationale_zh":"證明項是在固定形式環境與 context 中具有命題 P 型別，並可由 kernel 檢查的項 t。"}],"edges":[{"from":"mko-topological-space","to":"mko-open-closed-set","relation":"required_before"},{"from":"mko-subset","to":"mko-open-closed-set","relation":"required_before"},{"from":"mko-set-operations","to":"mko-open-closed-set","relation":"required_before"},{"from":"mko-topological-space","to":"mko-neighborhood","relation":"required_before"},{"from":"mko-open-closed-set","to":"mko-neighborhood","relation":"required_before"},{"from":"mko-set-membership","to":"mko-neighborhood","relation":"required_before"},{"from":"mko-topological-space","to":"mko-topological-continuity","relation":"required_before"},{"from":"mko-open-closed-set","to":"mko-topological-continuity","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-topological-continuity","relation":"required_before"},{"from":"mko-set","to":"mko-computational-model","relation":"required_before"},{"from":"mko-relation","to":"mko-computational-model","relation":"required_before"},{"from":"mko-natural-number","to":"mko-computational-model","relation":"required_before"},{"from":"mko-computational-model","to":"mko-algorithm-correctness","relation":"required_before"},{"from":"mko-proposition","to":"mko-algorithm-correctness","relation":"required_before"},{"from":"mko-proof","to":"mko-algorithm-correctness","relation":"required_before"},{"from":"mko-natural-number","to":"mko-asymptotic-resource-bound","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-asymptotic-resource-bound","relation":"required_before"},{"from":"mko-order-relation","to":"mko-asymptotic-resource-bound","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-asymptotic-resource-bound","relation":"required_before"},{"from":"mko-lean-formal-statement","to":"mko-lean-proof-term","relation":"required_before"},{"from":"mko-proof","to":"mko-lean-proof-term","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261020","title_zh":"應用數學基礎第一批：動力、微分方程、數值、最佳化、物理、生物與金融","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-natural-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-discrete-dynamical-system","position":2,"requirement":"optional","rationale_zh":"離散時間動力系統以狀態空間、自映射與自然數時間描述逐步演化；有限軌道核對不自動給出一般穩定性。"},{"object_id":"mko-derivative","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-ordinary-differential-equation","position":3,"requirement":"optional","rationale_zh":"常微分方程關聯未知單變數函數及其有限階導數；候選函數的殘差核對不等於存在唯一性定理。"},{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-absolute-value","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-ratio-proportion","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-numerical-approximation-error","position":4,"requirement":"optional","rationale_zh":"絕對誤差是近似值與參照值之差的絕對值；相對誤差另要求參照值非零並除以其絕對值。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-optimization-problem","position":5,"requirement":"optional","rationale_zh":"最佳化問題在明示可行集合上尋找使目標函數最小或最大的點；最優值與最優解集合需分開。"},{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-dimensional-analysis","position":6,"requirement":"optional","rationale_zh":"量綱分析用基本量綱指數向量追蹤乘、除與冪，並核對方程兩側量綱是否一致。"},{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-power-root","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-discrete-exponential-growth","position":7,"requirement":"optional","rationale_zh":"固定時間步與固定非負因子 r 的離散模型滿足 x_(n+1)=r x_n，因此 x_n=x_0 r^n。"},{"object_id":"mko-compound-interest","position":8,"requirement":"optional","rationale_zh":"固定每期利率 i 且無額外現金流時，餘額滿足 B_(n+1)=(1+i)B_n，故 B_n=B_0(1+i)^n。"}],"edges":[{"from":"mko-set","to":"mko-discrete-dynamical-system","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-discrete-dynamical-system","relation":"required_before"},{"from":"mko-natural-number","to":"mko-discrete-dynamical-system","relation":"required_before"},{"from":"mko-derivative","to":"mko-ordinary-differential-equation","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-ordinary-differential-equation","relation":"required_before"},{"from":"mko-equation","to":"mko-ordinary-differential-equation","relation":"required_before"},{"from":"mko-real-number","to":"mko-numerical-approximation-error","relation":"required_before"},{"from":"mko-absolute-value","to":"mko-numerical-approximation-error","relation":"required_before"},{"from":"mko-ratio-proportion","to":"mko-numerical-approximation-error","relation":"required_before"},{"from":"mko-set","to":"mko-optimization-problem","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-optimization-problem","relation":"required_before"},{"from":"mko-order-relation","to":"mko-optimization-problem","relation":"required_before"},{"from":"mko-ratio-proportion","to":"mko-dimensional-analysis","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-dimensional-analysis","relation":"required_before"},{"from":"mko-equation","to":"mko-dimensional-analysis","relation":"required_before"},{"from":"mko-sequence","to":"mko-discrete-exponential-growth","relation":"required_before"},{"from":"mko-power-root","to":"mko-discrete-exponential-growth","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-discrete-exponential-growth","relation":"required_before"},{"from":"mko-sequence","to":"mko-compound-interest","relation":"required_before"},{"from":"mko-power-root","to":"mko-compound-interest","relation":"required_before"},{"from":"mko-ratio-proportion","to":"mko-compound-interest","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261021","title_zh":"應用數學基礎第二批：表徵、診斷、週期、初值、二分、線性規劃與受限成長","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-variable-expression","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-coordinate-plane","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-mathematical-representation","position":2,"requirement":"optional","rationale_zh":"數學表徵把同一對象的文字規則、符號式、有限表格、序對與座標資料分開記錄，並以明示語義橋核對一致性。"},{"object_id":"mko-proposition","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-counterexample","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-diagnostic-mathematical-assessment","position":3,"requirement":"optional","rationale_zh":"診斷性數學評量記錄把題目、期望、作答與明示判定規則綁在一起，只產生可追溯的規則符合狀態。"},{"object_id":"mko-discrete-dynamical-system","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-orbit-fixed-periodic","position":4,"requirement":"optional","rationale_zh":"反覆迭代自映射形成軌道；固定點週期為1，週期點的最小正週期須和任意返回時間分開。"},{"object_id":"mko-ordinary-differential-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-domain-codomain-range","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-initial-value-problem","position":5,"requirement":"optional","rationale_zh":"初值問題在微分方程外指定某點的函數值或導數值；候選核對與存在唯一性是不同責任。"},{"object_id":"mko-numerical-approximation-error","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-continuity","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-bisection-method","position":6,"requirement":"optional","rationale_zh":"二分法在已提供的端點與中點符號資料上取中點並保留符號相反或零值的半區，有限步寬度減半。"},{"object_id":"mko-optimization-problem","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-system-linear-equations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-vector","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-programming","position":7,"requirement":"optional","rationale_zh":"線性規劃在線性等式或不等式限制的可行域上最佳化線性目標函數；本輪題庫只驗完整有限候選網格。"},{"object_id":"mko-discrete-exponential-growth","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-discrete-logistic-growth","position":8,"requirement":"optional","rationale_zh":"固定容量K與0≤r≤1時，本條目以x_(n+1)=x_n+r x_n(1-x_n/K)描述每步受限增量。"}],"edges":[{"from":"mko-variable-expression","to":"mko-mathematical-representation","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-mathematical-representation","relation":"required_before"},{"from":"mko-coordinate-plane","to":"mko-mathematical-representation","relation":"required_before"},{"from":"mko-proposition","to":"mko-diagnostic-mathematical-assessment","relation":"required_before"},{"from":"mko-counterexample","to":"mko-diagnostic-mathematical-assessment","relation":"required_before"},{"from":"mko-mathematical-representation","to":"mko-diagnostic-mathematical-assessment","relation":"required_before"},{"from":"mko-discrete-dynamical-system","to":"mko-orbit-fixed-periodic","relation":"required_before"},{"from":"mko-sequence","to":"mko-orbit-fixed-periodic","relation":"required_before"},{"from":"mko-equation","to":"mko-orbit-fixed-periodic","relation":"required_before"},{"from":"mko-ordinary-differential-equation","to":"mko-initial-value-problem","relation":"required_before"},{"from":"mko-domain-codomain-range","to":"mko-initial-value-problem","relation":"required_before"},{"from":"mko-numerical-approximation-error","to":"mko-bisection-method","relation":"required_before"},{"from":"mko-continuity","to":"mko-bisection-method","relation":"required_before"},{"from":"mko-order-relation","to":"mko-bisection-method","relation":"required_before"},{"from":"mko-equation","to":"mko-bisection-method","relation":"required_before"},{"from":"mko-optimization-problem","to":"mko-linear-programming","relation":"required_before"},{"from":"mko-linear-equation","to":"mko-linear-programming","relation":"required_before"},{"from":"mko-system-linear-equations","to":"mko-linear-programming","relation":"required_before"},{"from":"mko-vector","to":"mko-linear-programming","relation":"required_before"},{"from":"mko-discrete-exponential-growth","to":"mko-discrete-logistic-growth","relation":"required_before"},{"from":"mko-sequence","to":"mko-discrete-logistic-growth","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-discrete-logistic-growth","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261022","title_zh":"現代結構第一批：振子、折現、複數、向量空間、有限群、度量與分布","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-ordinary-differential-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-derivative-rules","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-power-root","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-harmonic-oscillator-model","position":2,"requirement":"optional","rationale_zh":"無阻尼線性簡諧振子以m x''+k x=0及m,k>0定義；本輪不宣稱三角函數閉式解。"},{"object_id":"mko-compound-interest","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-fraction","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-present-value-discounting","position":3,"requirement":"optional","rationale_zh":"固定正累積因子與期數時，未來金額除以累積因子冪得到現值。"},{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-coordinate-plane","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-complex-number","position":4,"requirement":"optional","rationale_zh":"複數以實數有序對(a,b)表示a+bi，並以i²=-1定義加減乘法。"},{"object_id":"mko-vector","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-vector-space","position":5,"requirement":"optional","rationale_zh":"實向量空間配備向量加法與實數純量乘法，滿足加法交換群及相容的純量作用公理。"},{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-proof","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-group","position":6,"requirement":"optional","rationale_zh":"有限群由有限集合與完整封閉Cayley表構成，並滿足結合律、唯一單位元及每元素逆元。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-metric-space","position":7,"requirement":"optional","rationale_zh":"度量空間由集合與距離函數組成，距離非負、只在相同點為零、對稱且滿足三角不等式。"},{"object_id":"mko-random-variable","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-discrete-probability-distribution","position":8,"requirement":"optional","rationale_zh":"有限離散分布是在有限結果值上指定非負機率質量，全部質量精確和為一。"}],"edges":[{"from":"mko-ordinary-differential-equation","to":"mko-harmonic-oscillator-model","relation":"required_before"},{"from":"mko-derivative-rules","to":"mko-harmonic-oscillator-model","relation":"required_before"},{"from":"mko-power-root","to":"mko-harmonic-oscillator-model","relation":"required_before"},{"from":"mko-compound-interest","to":"mko-present-value-discounting","relation":"required_before"},{"from":"mko-fraction","to":"mko-present-value-discounting","relation":"required_before"},{"from":"mko-equation","to":"mko-present-value-discounting","relation":"required_before"},{"from":"mko-real-number","to":"mko-complex-number","relation":"required_before"},{"from":"mko-coordinate-plane","to":"mko-complex-number","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-complex-number","relation":"required_before"},{"from":"mko-vector","to":"mko-real-vector-space","relation":"required_before"},{"from":"mko-real-number","to":"mko-real-vector-space","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-real-vector-space","relation":"required_before"},{"from":"mko-set","to":"mko-finite-group","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-finite-group","relation":"required_before"},{"from":"mko-relation","to":"mko-finite-group","relation":"required_before"},{"from":"mko-proof","to":"mko-finite-group","relation":"required_before"},{"from":"mko-set","to":"mko-metric-space","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-metric-space","relation":"required_before"},{"from":"mko-relation","to":"mko-metric-space","relation":"required_before"},{"from":"mko-real-number","to":"mko-metric-space","relation":"required_before"},{"from":"mko-order-relation","to":"mko-metric-space","relation":"required_before"},{"from":"mko-random-variable","to":"mko-finite-discrete-probability-distribution","relation":"required_before"},{"from":"mko-probability","to":"mko-finite-discrete-probability-distribution","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-finite-discrete-probability-distribution","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261023","title_zh":"現代結構第二批：複數不變量、生成、群同態、收斂、測度、過程與多變數函數","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-complex-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-absolute-value","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-power-root","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-complex-modulus-conjugate","position":2,"requirement":"optional","rationale_zh":"複數共軛保留實部並反轉虛部；模平方為實部與虛部平方和。"},{"object_id":"mko-real-vector-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-system-linear-equations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-combination-span","position":3,"requirement":"optional","rationale_zh":"線性組合是有限個向量的純量加權和；生成空間收集全部有限線性組合。"},{"object_id":"mko-finite-group","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subset","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subgroup-homomorphism","position":4,"requirement":"optional","rationale_zh":"子群在繼承運算下自身成群；群同態保持乘法，kernel與image由完整映射表導出。"},{"object_id":"mko-metric-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-metric-convergence","position":5,"requirement":"optional","rationale_zh":"度量空間序列收斂到x，表示每個正epsilon終有所有n≥N落入x的epsilon球。"},{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sigma-algebra-measure-space","position":6,"requirement":"optional","rationale_zh":"σ-代數含空集並對補集與可數聯集封閉；測度非負、空集為零且可數可加。"},{"object_id":"mko-finite-discrete-probability-distribution","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-random-variable","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-discrete-stochastic-process","position":7,"requirement":"optional","rationale_zh":"離散隨機過程是共同機率空間上的離散時間隨機變數族；邊際分布不決定聯合律。"},{"object_id":"mko-vector","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-coordinate-plane","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-multivariable-real-function","position":8,"requirement":"optional","rationale_zh":"多變數實函數把明示實向量域中的輸入映到實數；座標角色與定義域是函數資料。"}],"edges":[{"from":"mko-complex-number","to":"mko-complex-modulus-conjugate","relation":"required_before"},{"from":"mko-absolute-value","to":"mko-complex-modulus-conjugate","relation":"required_before"},{"from":"mko-power-root","to":"mko-complex-modulus-conjugate","relation":"required_before"},{"from":"mko-real-vector-space","to":"mko-linear-combination-span","relation":"required_before"},{"from":"mko-system-linear-equations","to":"mko-linear-combination-span","relation":"required_before"},{"from":"mko-finite-group","to":"mko-subgroup-homomorphism","relation":"required_before"},{"from":"mko-subset","to":"mko-subgroup-homomorphism","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-subgroup-homomorphism","relation":"required_before"},{"from":"mko-metric-space","to":"mko-metric-convergence","relation":"required_before"},{"from":"mko-sequence","to":"mko-metric-convergence","relation":"required_before"},{"from":"mko-quantifier","to":"mko-metric-convergence","relation":"required_before"},{"from":"mko-set","to":"mko-sigma-algebra-measure-space","relation":"required_before"},{"from":"mko-set-operations","to":"mko-sigma-algebra-measure-space","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-sigma-algebra-measure-space","relation":"required_before"},{"from":"mko-sequence","to":"mko-sigma-algebra-measure-space","relation":"required_before"},{"from":"mko-quantifier","to":"mko-sigma-algebra-measure-space","relation":"required_before"},{"from":"mko-real-number","to":"mko-sigma-algebra-measure-space","relation":"required_before"},{"from":"mko-finite-discrete-probability-distribution","to":"mko-discrete-stochastic-process","relation":"required_before"},{"from":"mko-sequence","to":"mko-discrete-stochastic-process","relation":"required_before"},{"from":"mko-random-variable","to":"mko-discrete-stochastic-process","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-multivariable-real-function","relation":"required_before"},{"from":"mko-vector","to":"mko-multivariable-real-function","relation":"required_before"},{"from":"mko-coordinate-plane","to":"mko-multivariable-real-function","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261024","title_zh":"現代結構第三批：基底、可測性、Markov、偏導、PDE與凸幾何","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-linear-combination-span","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-system-linear-equations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-determinant","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-independence-basis","position":2,"requirement":"optional","rationale_zh":"線性獨立排除非平凡零和關係；基底同時線性獨立並生成已宣告空間。"},{"object_id":"mko-sigma-algebra-measure-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-measurable-function","position":3,"requirement":"optional","rationale_zh":"來源與目標各固定sigma代數時，函數可測表示每個目標可測集的逆像皆在來源sigma代數。"},{"object_id":"mko-discrete-stochastic-process","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-matrix","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-conditional-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-markov-chain","position":4,"requirement":"optional","rationale_zh":"有限時間齊次Markov鏈以row-stochastic轉移核與需要時的初始律指定有限路徑機率。"},{"object_id":"mko-multivariable-real-function","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-derivative","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-rate-of-change","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-partial-derivative","position":5,"requirement":"optional","rationale_zh":"偏導數固定其他座標，只沿一個座標切片取h趨近零的單變數導數。"},{"object_id":"mko-real-vector-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-ratio-proportion","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-convex-combination","position":6,"requirement":"optional","rationale_zh":"有限凸組合用非負精確權重加權有限非空向量列，且權重總和恰為一。"},{"object_id":"mko-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-derivative-rules","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-partial-differential-equation","position":7,"requirement":"optional","rationale_zh":"偏微分方程關聯多變數未知函數與一個或多個偏導數；殘差與主部分類需固定算子convention。"},{"object_id":"mko-subset","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-convex-set","position":8,"requirement":"optional","rationale_zh":"集合凸表示任兩點及每個lambda∈[0,1]形成的整條線段仍在集合內。"}],"edges":[{"from":"mko-linear-combination-span","to":"mko-linear-independence-basis","relation":"required_before"},{"from":"mko-system-linear-equations","to":"mko-linear-independence-basis","relation":"required_before"},{"from":"mko-determinant","to":"mko-linear-independence-basis","relation":"required_before"},{"from":"mko-sigma-algebra-measure-space","to":"mko-measurable-function","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-measurable-function","relation":"required_before"},{"from":"mko-set-operations","to":"mko-measurable-function","relation":"required_before"},{"from":"mko-discrete-stochastic-process","to":"mko-finite-markov-chain","relation":"required_before"},{"from":"mko-matrix","to":"mko-finite-markov-chain","relation":"required_before"},{"from":"mko-conditional-probability","to":"mko-finite-markov-chain","relation":"required_before"},{"from":"mko-multivariable-real-function","to":"mko-partial-derivative","relation":"required_before"},{"from":"mko-derivative","to":"mko-partial-derivative","relation":"required_before"},{"from":"mko-rate-of-change","to":"mko-partial-derivative","relation":"required_before"},{"from":"mko-real-vector-space","to":"mko-finite-convex-combination","relation":"required_before"},{"from":"mko-ratio-proportion","to":"mko-finite-convex-combination","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-finite-convex-combination","relation":"required_before"},{"from":"mko-partial-derivative","to":"mko-partial-differential-equation","relation":"required_before"},{"from":"mko-equation","to":"mko-partial-differential-equation","relation":"required_before"},{"from":"mko-derivative-rules","to":"mko-partial-differential-equation","relation":"required_before"},{"from":"mko-finite-convex-combination","to":"mko-convex-set","relation":"required_before"},{"from":"mko-subset","to":"mko-convex-set","relation":"required_before"},{"from":"mko-real-vector-space","to":"mko-convex-set","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261025","title_zh":"現代結構 Wave3 收束：邊界值、凸函數、凸最佳化與四項密度深化","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-partial-differential-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-domain-codomain-range","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-boundary-value-problem","position":2,"requirement":"optional","rationale_zh":"邊界值問題把定義域內的微分方程責任與邊界上的值或導數資料同時指定；候選通過不等於適定性。"},{"object_id":"mko-convex-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-convex-function","position":2,"requirement":"optional","rationale_zh":"凸域上的函數為凸，若每個兩點凸組合處的函數值不超過相同權重的端點函數值。"},{"object_id":"mko-optimization-problem","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-vector-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-convex-optimization-problem","position":3,"requirement":"optional","rationale_zh":"凸最佳化問題在凸可行域上最小化凸目標；問題資格、可行性、求值與全域最優是不同責任。"},{"object_id":"mko-linear-combination-span","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-system-linear-equations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-determinant","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-independence-basis","position":2,"requirement":"optional","rationale_zh":"深化既有R²基底語義，以顯式重建與非平凡零和證書區分span、dependence與座標唯一性。"},{"object_id":"mko-sigma-algebra-measure-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-measurable-function","position":2,"requirement":"optional","rationale_zh":"深化完整有限可測空間上的逆像補集／聯集律、常值映射及來源sigma代數細化。"},{"object_id":"mko-discrete-stochastic-process","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-matrix","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-conditional-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-markov-chain","position":2,"requirement":"optional","rationale_zh":"深化精確有限時間齊次kernel的n步矩陣、端點joint與給定初態的條件路徑機率。"},{"object_id":"mko-multivariable-real-function","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-derivative","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-rate-of-change","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-partial-derivative","position":2,"requirement":"optional","rationale_zh":"深化二次多項式的x／y偏導多項式、固定x的y切片及符號差商消去後極限。"}],"edges":[{"from":"mko-partial-differential-equation","to":"mko-boundary-value-problem","relation":"required_before"},{"from":"mko-domain-codomain-range","to":"mko-boundary-value-problem","relation":"required_before"},{"from":"mko-equation","to":"mko-boundary-value-problem","relation":"required_before"},{"from":"mko-convex-set","to":"mko-convex-function","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-convex-function","relation":"required_before"},{"from":"mko-order-relation","to":"mko-convex-function","relation":"required_before"},{"from":"mko-convex-function","to":"mko-convex-optimization-problem","relation":"required_before"},{"from":"mko-optimization-problem","to":"mko-convex-optimization-problem","relation":"required_before"},{"from":"mko-real-vector-space","to":"mko-convex-optimization-problem","relation":"required_before"},{"from":"mko-linear-combination-span","to":"mko-linear-independence-basis","relation":"required_before"},{"from":"mko-system-linear-equations","to":"mko-linear-independence-basis","relation":"required_before"},{"from":"mko-determinant","to":"mko-linear-independence-basis","relation":"required_before"},{"from":"mko-sigma-algebra-measure-space","to":"mko-measurable-function","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-measurable-function","relation":"required_before"},{"from":"mko-set-operations","to":"mko-measurable-function","relation":"required_before"},{"from":"mko-discrete-stochastic-process","to":"mko-finite-markov-chain","relation":"required_before"},{"from":"mko-matrix","to":"mko-finite-markov-chain","relation":"required_before"},{"from":"mko-conditional-probability","to":"mko-finite-markov-chain","relation":"required_before"},{"from":"mko-multivariable-real-function","to":"mko-partial-derivative","relation":"required_before"},{"from":"mko-derivative","to":"mko-partial-derivative","relation":"required_before"},{"from":"mko-rate-of-change","to":"mko-partial-derivative","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261026","title_zh":"結構深度Wave4第一批：阿貝爾、範數、Cauchy、joint law、協方差、圖同構與等價關係","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-finite-group","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-abelian-group","position":2,"requirement":"optional","rationale_zh":"有限群若完整Cayley表對所有元素對都滿足交換律，則為有限阿貝爾群。"},{"object_id":"mko-real-vector-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-absolute-value","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-normed-vector-space","position":3,"requirement":"optional","rationale_zh":"實向量空間配備正定、絕對齊次且滿足三角不等式的實值範數。"},{"object_id":"mko-metric-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-cauchy-sequence","position":4,"requirement":"optional","rationale_zh":"度量空間序列為Cauchy，若每個正epsilon最終所有兩項彼此距離皆小於epsilon。"},{"object_id":"mko-finite-discrete-probability-distribution","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-random-variable","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sample-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-joint-distribution","position":5,"requirement":"optional","rationale_zh":"共同有限機率空間上多個隨機變數的值組與精確質量共同指定有限聯合分布。"},{"object_id":"mko-expectation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-variance","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-covariance","position":6,"requirement":"optional","rationale_zh":"同一有限聯合分布下協方差為E[XY]-E[X]E[Y]；零協方差通常不推出獨立。"},{"object_id":"mko-graph","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-bijective","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-graph-isomorphism","position":7,"requirement":"optional","rationale_zh":"有限簡單圖同構是頂點雙射，對所有頂點對完整保持鄰接與非鄰接。"},{"object_id":"mko-logical-connectives","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-equivalence-relation","position":8,"requirement":"optional","rationale_zh":"固定載體上的關係若完整滿足反身、對稱與傳遞，則為等價關係。"}],"edges":[{"from":"mko-finite-group","to":"mko-finite-abelian-group","relation":"required_before"},{"from":"mko-quantifier","to":"mko-finite-abelian-group","relation":"required_before"},{"from":"mko-real-vector-space","to":"mko-real-normed-vector-space","relation":"required_before"},{"from":"mko-absolute-value","to":"mko-real-normed-vector-space","relation":"required_before"},{"from":"mko-order-relation","to":"mko-real-normed-vector-space","relation":"required_before"},{"from":"mko-metric-space","to":"mko-cauchy-sequence","relation":"required_before"},{"from":"mko-sequence","to":"mko-cauchy-sequence","relation":"required_before"},{"from":"mko-quantifier","to":"mko-cauchy-sequence","relation":"required_before"},{"from":"mko-finite-discrete-probability-distribution","to":"mko-finite-joint-distribution","relation":"required_before"},{"from":"mko-random-variable","to":"mko-finite-joint-distribution","relation":"required_before"},{"from":"mko-probability","to":"mko-finite-joint-distribution","relation":"required_before"},{"from":"mko-sample-space","to":"mko-finite-joint-distribution","relation":"required_before"},{"from":"mko-finite-joint-distribution","to":"mko-covariance","relation":"required_before"},{"from":"mko-expectation","to":"mko-covariance","relation":"required_before"},{"from":"mko-variance","to":"mko-covariance","relation":"required_before"},{"from":"mko-random-variable","to":"mko-covariance","relation":"required_before"},{"from":"mko-graph","to":"mko-graph-isomorphism","relation":"required_before"},{"from":"mko-bijective","to":"mko-graph-isomorphism","relation":"required_before"},{"from":"mko-relation","to":"mko-graph-isomorphism","relation":"required_before"},{"from":"mko-relation","to":"mko-equivalence-relation","relation":"required_before"},{"from":"mko-logical-connectives","to":"mko-equivalence-relation","relation":"required_before"},{"from":"mko-quantifier","to":"mko-equivalence-relation","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261027","title_zh":"結構深度Wave4根節點：環、度量完備性、線性結構、獨立與圖路徑","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-finite-abelian-group","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-unital-ring","position":2,"requirement":"optional","rationale_zh":"有限含一環在同一完整有限載體上具加法阿貝爾群、乘法結合律、左右分配律及與零不同的乘法單位元。"},{"object_id":"mko-real-normed-vector-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-metric-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-vector-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-norm-induced-metric","position":3,"requirement":"optional","rationale_zh":"同一實賦範向量空間上的範數以向量差的範數定義度量，度量公理由範數公理全稱推出。"},{"object_id":"mko-cauchy-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-metric-convergence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-complete-metric-space","position":4,"requirement":"optional","rationale_zh":"每個Cauchy序列都收斂到空間內某點的度量空間稱為完備。"},{"object_id":"mko-linear-combination-span","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-bilinear-form","position":5,"requirement":"optional","rationale_zh":"實向量空間上的雙線性形式分別對兩個輸入保持向量加法與實純量乘法。"},{"object_id":"mko-general-linear-map","position":6,"requirement":"optional","rationale_zh":"一般實向量空間間映射若對所有向量與實純量保持線性組合，則為線性映射。"},{"object_id":"mko-event","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-conditional-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-event-independence","position":7,"requirement":"optional","rationale_zh":"兩事件獨立以交集機率等於邊際機率乘積定義；此定義不依賴零機率條件式。"},{"object_id":"mko-graph","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-graph-walk-simple-path","position":8,"requirement":"optional","rationale_zh":"有限無向簡單圖的walk要求每對連續頂點相鄰；simple path另要求所有頂點互異。"}],"edges":[{"from":"mko-finite-abelian-group","to":"mko-finite-unital-ring","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-finite-unital-ring","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-finite-unital-ring","relation":"required_before"},{"from":"mko-real-normed-vector-space","to":"mko-norm-induced-metric","relation":"required_before"},{"from":"mko-metric-space","to":"mko-norm-induced-metric","relation":"required_before"},{"from":"mko-real-vector-space","to":"mko-norm-induced-metric","relation":"required_before"},{"from":"mko-cauchy-sequence","to":"mko-complete-metric-space","relation":"required_before"},{"from":"mko-metric-convergence","to":"mko-complete-metric-space","relation":"required_before"},{"from":"mko-metric-space","to":"mko-complete-metric-space","relation":"required_before"},{"from":"mko-real-vector-space","to":"mko-real-bilinear-form","relation":"required_before"},{"from":"mko-linear-combination-span","to":"mko-real-bilinear-form","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-real-bilinear-form","relation":"required_before"},{"from":"mko-real-vector-space","to":"mko-general-linear-map","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-general-linear-map","relation":"required_before"},{"from":"mko-linear-combination-span","to":"mko-general-linear-map","relation":"required_before"},{"from":"mko-event","to":"mko-event-independence","relation":"required_before"},{"from":"mko-probability","to":"mko-event-independence","relation":"required_before"},{"from":"mko-conditional-probability","to":"mko-event-independence","relation":"required_before"},{"from":"mko-graph","to":"mko-graph-walk-simple-path","relation":"required_before"},{"from":"mko-sequence","to":"mko-graph-walk-simple-path","relation":"required_before"},{"from":"mko-relation","to":"mko-graph-walk-simple-path","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261028","title_zh":"結構深度Wave4第二層：交換環、理想、Banach、內積、算子、獨立與連通分量","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-finite-unital-ring","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-commutative-ring","position":2,"requirement":"optional","rationale_zh":"有限含一環若乘法對全部元素對交換，則為有限交換環；零因子仍可能存在。"},{"object_id":"mko-subset","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-abelian-group","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-two-sided-ideal","position":3,"requirement":"optional","rationale_zh":"有限環子集若為加法子群且吸收父環的左右乘法，則為雙邊理想。"},{"object_id":"mko-real-normed-vector-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-norm-induced-metric","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-complete-metric-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-vector-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-banach-space","position":4,"requirement":"optional","rationale_zh":"在自身範數誘導度量下完備的實賦範向量空間稱為實Banach空間。"},{"object_id":"mko-real-bilinear-form","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-inner-product-space","position":5,"requirement":"optional","rationale_zh":"對稱且正定的實雙線性形式構成實內積；半正定或抽樣為正不足。"},{"object_id":"mko-general-linear-map","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-bounded-linear-operator","position":6,"requirement":"optional","rationale_zh":"賦範空間間線性映射若有單一非負常數統一控制所有輸出範數，則為有界線性算子。"},{"object_id":"mko-event-independence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-joint-distribution","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-random-variable","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-measurable-function","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-random-variable-independence","position":7,"requirement":"optional","rationale_zh":"同一有限joint law下，隨機變數獨立要求全部值對的joint mass分解為邊際質量乘積。"},{"object_id":"mko-graph-walk-simple-path","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-equivalence-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-graph","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-connected-component","position":8,"requirement":"optional","rationale_zh":"有限無向簡單圖中，由存在路徑所定義的頂點等價類恰為連通分量。"}],"edges":[{"from":"mko-finite-unital-ring","to":"mko-finite-commutative-ring","relation":"required_before"},{"from":"mko-quantifier","to":"mko-finite-commutative-ring","relation":"required_before"},{"from":"mko-finite-unital-ring","to":"mko-finite-two-sided-ideal","relation":"required_before"},{"from":"mko-subset","to":"mko-finite-two-sided-ideal","relation":"required_before"},{"from":"mko-finite-abelian-group","to":"mko-finite-two-sided-ideal","relation":"required_before"},{"from":"mko-real-normed-vector-space","to":"mko-real-banach-space","relation":"required_before"},{"from":"mko-norm-induced-metric","to":"mko-real-banach-space","relation":"required_before"},{"from":"mko-complete-metric-space","to":"mko-real-banach-space","relation":"required_before"},{"from":"mko-real-vector-space","to":"mko-real-banach-space","relation":"required_before"},{"from":"mko-real-bilinear-form","to":"mko-real-inner-product-space","relation":"required_before"},{"from":"mko-order-relation","to":"mko-real-inner-product-space","relation":"required_before"},{"from":"mko-real-vector-space","to":"mko-real-inner-product-space","relation":"required_before"},{"from":"mko-general-linear-map","to":"mko-bounded-linear-operator","relation":"required_before"},{"from":"mko-real-normed-vector-space","to":"mko-bounded-linear-operator","relation":"required_before"},{"from":"mko-order-relation","to":"mko-bounded-linear-operator","relation":"required_before"},{"from":"mko-event-independence","to":"mko-random-variable-independence","relation":"required_before"},{"from":"mko-finite-joint-distribution","to":"mko-random-variable-independence","relation":"required_before"},{"from":"mko-random-variable","to":"mko-random-variable-independence","relation":"required_before"},{"from":"mko-measurable-function","to":"mko-random-variable-independence","relation":"required_before"},{"from":"mko-graph-walk-simple-path","to":"mko-connected-component","relation":"required_before"},{"from":"mko-equivalence-relation","to":"mko-connected-component","relation":"required_before"},{"from":"mko-graph","to":"mko-connected-component","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261029","title_zh":"結構深度Wave4第三層：整環、多項式、內積範數、條件期望、filtration、IID與等價類","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-finite-commutative-ring","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-proposition","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-integral-domain","position":2,"requirement":"optional","rationale_zh":"有限交換含一環若1不等於0且沒有非零零因子，則為有限整環。"},{"object_id":"mko-polynomial","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-polynomial-ring-over-finite-commutative-ring","position":3,"requirement":"optional","rationale_zh":"係數取自有限交換環的普通多項式以有限支撐係數序列表示，但所有任意有限次多項式合成的載體仍是無限。"},{"object_id":"mko-real-inner-product-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-power-root","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-inner-product-induced-norm","position":4,"requirement":"optional","rationale_zh":"同一實內積以向量自內積的非負平方根定義範數；三角不等式依Cauchy–Schwarz全稱證書。"},{"object_id":"mko-finite-joint-distribution","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-measurable-function","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-expectation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-conditional-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-random-variable","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-conditional-expectation","position":5,"requirement":"optional","rationale_zh":"有限機率空間中，相對分割生成sigma代數的條件期望在每個正質量區塊取加權平均並保持區塊積分。"},{"object_id":"mko-sigma-algebra-measure-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-discrete-stochastic-process","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-filtration","position":6,"requirement":"optional","rationale_zh":"有限時間filtration是在同一機率空間上隨時間資訊遞增的sigma代數序列。"},{"object_id":"mko-random-variable-independence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-iid-sequence","position":7,"requirement":"optional","rationale_zh":"IID序列同時要求每個變數有相同分布，且每個有限子族滿足mutual independence。"},{"object_id":"mko-equivalence-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set-membership","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subset","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-equivalence-class","position":8,"requirement":"optional","rationale_zh":"固定完整等價關係下，元素a的等價類收集全部與a相關的載體元素。"}],"edges":[{"from":"mko-finite-commutative-ring","to":"mko-finite-integral-domain","relation":"required_before"},{"from":"mko-proposition","to":"mko-finite-integral-domain","relation":"required_before"},{"from":"mko-quantifier","to":"mko-finite-integral-domain","relation":"required_before"},{"from":"mko-finite-commutative-ring","to":"mko-polynomial-ring-over-finite-commutative-ring","relation":"required_before"},{"from":"mko-polynomial","to":"mko-polynomial-ring-over-finite-commutative-ring","relation":"required_before"},{"from":"mko-sequence","to":"mko-polynomial-ring-over-finite-commutative-ring","relation":"required_before"},{"from":"mko-real-inner-product-space","to":"mko-inner-product-induced-norm","relation":"required_before"},{"from":"mko-power-root","to":"mko-inner-product-induced-norm","relation":"required_before"},{"from":"mko-order-relation","to":"mko-inner-product-induced-norm","relation":"required_before"},{"from":"mko-finite-joint-distribution","to":"mko-conditional-expectation","relation":"required_before"},{"from":"mko-measurable-function","to":"mko-conditional-expectation","relation":"required_before"},{"from":"mko-expectation","to":"mko-conditional-expectation","relation":"required_before"},{"from":"mko-conditional-probability","to":"mko-conditional-expectation","relation":"required_before"},{"from":"mko-random-variable","to":"mko-conditional-expectation","relation":"required_before"},{"from":"mko-sigma-algebra-measure-space","to":"mko-finite-filtration","relation":"required_before"},{"from":"mko-discrete-stochastic-process","to":"mko-finite-filtration","relation":"required_before"},{"from":"mko-sequence","to":"mko-finite-filtration","relation":"required_before"},{"from":"mko-measurable-function","to":"mko-finite-filtration","relation":"required_before"},{"from":"mko-random-variable-independence","to":"mko-iid-sequence","relation":"required_before"},{"from":"mko-discrete-stochastic-process","to":"mko-iid-sequence","relation":"required_before"},{"from":"mko-sequence","to":"mko-iid-sequence","relation":"required_before"},{"from":"mko-finite-joint-distribution","to":"mko-iid-sequence","relation":"required_before"},{"from":"mko-expectation","to":"mko-iid-sequence","relation":"required_before"},{"from":"mko-equivalence-relation","to":"mko-equivalence-class","relation":"required_before"},{"from":"mko-set-membership","to":"mko-equivalence-class","relation":"required_before"},{"from":"mko-subset","to":"mko-equivalence-class","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261030","title_zh":"結構深度Wave4第四層：有限域、Hilbert、Martingale、商集、基數、CDF與依機率收斂","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-finite-integral-domain","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-unital-ring","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-abelian-group","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-field","position":2,"requirement":"optional","rationale_zh":"有限交換含一環中每個非零元素皆有乘法逆元時構成有限域。"},{"object_id":"mko-real-inner-product-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-inner-product-induced-norm","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-norm-induced-metric","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-complete-metric-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-hilbert-space","position":3,"requirement":"optional","rationale_zh":"在同一內積誘導範數與度量下完備的實內積空間為實Hilbert空間。"},{"object_id":"mko-finite-filtration","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-conditional-expectation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-discrete-stochastic-process","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-expectation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-measurable-function","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-martingale","position":4,"requirement":"optional","rationale_zh":"相對同一filtration適應且可積的有限過程，若每一步下一值條件期望等於現在值a.s.，則為Martingale。"},{"object_id":"mko-equivalence-class","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-equivalence-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quotient-set","position":5,"requirement":"optional","rationale_zh":"完整等價關係的全部互異等價類組成商集，並分割原載體。"},{"object_id":"mko-injective","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-surjective","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-bijective","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-cardinality-comparison","position":6,"requirement":"optional","rationale_zh":"明示source與target間的單射、滿射、雙射分別提供有限集合基數比較證書。"},{"object_id":"mko-finite-discrete-probability-distribution","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-cumulative-distribution-function","position":7,"requirement":"optional","rationale_zh":"實值隨機變數的CDF以F(t)=P(X小於等於t)定義，為單調右連續且有限離散跳躍等於點質量。"},{"object_id":"mko-finite-joint-distribution","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-random-variable","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-metric-convergence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-convergence-in-probability","position":8,"requirement":"optional","rationale_zh":"X_n依機率收斂到X，若對每個epsilon>0，偏差超過epsilon的機率趨近0。"}],"edges":[{"from":"mko-finite-integral-domain","to":"mko-finite-field","relation":"required_before"},{"from":"mko-finite-unital-ring","to":"mko-finite-field","relation":"required_before"},{"from":"mko-finite-abelian-group","to":"mko-finite-field","relation":"required_before"},{"from":"mko-real-inner-product-space","to":"mko-real-hilbert-space","relation":"required_before"},{"from":"mko-inner-product-induced-norm","to":"mko-real-hilbert-space","relation":"required_before"},{"from":"mko-norm-induced-metric","to":"mko-real-hilbert-space","relation":"required_before"},{"from":"mko-complete-metric-space","to":"mko-real-hilbert-space","relation":"required_before"},{"from":"mko-finite-filtration","to":"mko-martingale","relation":"required_before"},{"from":"mko-conditional-expectation","to":"mko-martingale","relation":"required_before"},{"from":"mko-discrete-stochastic-process","to":"mko-martingale","relation":"required_before"},{"from":"mko-expectation","to":"mko-martingale","relation":"required_before"},{"from":"mko-measurable-function","to":"mko-martingale","relation":"required_before"},{"from":"mko-equivalence-class","to":"mko-quotient-set","relation":"required_before"},{"from":"mko-set","to":"mko-quotient-set","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-quotient-set","relation":"required_before"},{"from":"mko-equivalence-relation","to":"mko-quotient-set","relation":"required_before"},{"from":"mko-injective","to":"mko-cardinality-comparison","relation":"required_before"},{"from":"mko-surjective","to":"mko-cardinality-comparison","relation":"required_before"},{"from":"mko-bijective","to":"mko-cardinality-comparison","relation":"required_before"},{"from":"mko-set","to":"mko-cardinality-comparison","relation":"required_before"},{"from":"mko-finite-discrete-probability-distribution","to":"mko-cumulative-distribution-function","relation":"required_before"},{"from":"mko-order-relation","to":"mko-cumulative-distribution-function","relation":"required_before"},{"from":"mko-probability","to":"mko-cumulative-distribution-function","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-cumulative-distribution-function","relation":"required_before"},{"from":"mko-finite-joint-distribution","to":"mko-convergence-in-probability","relation":"required_before"},{"from":"mko-random-variable","to":"mko-convergence-in-probability","relation":"required_before"},{"from":"mko-probability","to":"mko-convergence-in-probability","relation":"required_before"},{"from":"mko-sequence","to":"mko-convergence-in-probability","relation":"required_before"},{"from":"mko-metric-convergence","to":"mko-convergence-in-probability","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261031","title_zh":"Wave4收束：可數、二分圖、分布收斂、商環、著色與兩項深度密度","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-cardinality-comparison","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-natural-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-injective","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-surjective","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-at-most-countable-set","position":2,"requirement":"optional","rationale_zh":"集合若有限或存在到自然數的明示單射，則為至多可數；不等同可數無限。"},{"object_id":"mko-graph","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subset","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-bipartite-graph","position":2,"requirement":"optional","rationale_zh":"有限無向簡單圖若全體頂點可分為兩個互斥部分且每條邊跨部，則為二分圖。"},{"object_id":"mko-cumulative-distribution-function","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-limit","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-joint-distribution","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-convergence-in-distribution","position":2,"requirement":"optional","rationale_zh":"極限CDF每個連續點上F_n逐點收斂到F時，稱依分布收斂。"},{"object_id":"mko-finite-unital-ring","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-two-sided-ideal","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-equivalence-class","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quotient-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-quotient-ring","position":2,"requirement":"optional","rationale_zh":"有限環依proper雙邊理想的加法cosets形成商環，運算須與代表元選擇無關。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-graph-coloring","position":2,"requirement":"optional","rationale_zh":"proper k-coloring使用至多k種顏色並使每條邊兩端顏色不同。"},{"object_id":"mko-finite-integral-domain","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-abelian-group","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-field","position":2,"requirement":"optional","rationale_zh":"深化完整有限域表上的characteristic、非零乘法群、仿射方程與Frobenius恆等。"},{"object_id":"mko-finite-filtration","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-conditional-expectation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-discrete-stochastic-process","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-expectation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-measurable-function","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-martingale","position":2,"requirement":"optional","rationale_zh":"深化有限filtration上martingale difference、affine stability、公平partial sum與常期望反例。"}],"edges":[{"from":"mko-cardinality-comparison","to":"mko-at-most-countable-set","relation":"required_before"},{"from":"mko-natural-number","to":"mko-at-most-countable-set","relation":"required_before"},{"from":"mko-injective","to":"mko-at-most-countable-set","relation":"required_before"},{"from":"mko-surjective","to":"mko-at-most-countable-set","relation":"required_before"},{"from":"mko-graph","to":"mko-bipartite-graph","relation":"required_before"},{"from":"mko-subset","to":"mko-bipartite-graph","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-bipartite-graph","relation":"required_before"},{"from":"mko-cumulative-distribution-function","to":"mko-convergence-in-distribution","relation":"required_before"},{"from":"mko-function-limit","to":"mko-convergence-in-distribution","relation":"required_before"},{"from":"mko-sequence","to":"mko-convergence-in-distribution","relation":"required_before"},{"from":"mko-finite-joint-distribution","to":"mko-convergence-in-distribution","relation":"required_before"},{"from":"mko-finite-unital-ring","to":"mko-finite-quotient-ring","relation":"required_before"},{"from":"mko-finite-two-sided-ideal","to":"mko-finite-quotient-ring","relation":"required_before"},{"from":"mko-equivalence-class","to":"mko-finite-quotient-ring","relation":"required_before"},{"from":"mko-quotient-set","to":"mko-finite-quotient-ring","relation":"required_before"},{"from":"mko-graph","to":"mko-graph-coloring","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-graph-coloring","relation":"required_before"},{"from":"mko-order-relation","to":"mko-graph-coloring","relation":"required_before"},{"from":"mko-finite-integral-domain","to":"mko-finite-field","relation":"required_before"},{"from":"mko-finite-unital-ring","to":"mko-finite-field","relation":"required_before"},{"from":"mko-finite-abelian-group","to":"mko-finite-field","relation":"required_before"},{"from":"mko-finite-filtration","to":"mko-martingale","relation":"required_before"},{"from":"mko-conditional-expectation","to":"mko-martingale","relation":"required_before"},{"from":"mko-discrete-stochastic-process","to":"mko-martingale","relation":"required_before"},{"from":"mko-expectation","to":"mko-martingale","relation":"required_before"},{"from":"mko-measurable-function","to":"mko-martingale","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261101","title_zh":"Wave5根節點：子空間、拓撲覆蓋、幾乎必然收斂、匹配、φ函數與有限自動機","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-real-vector-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subset","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-vector-subspace","position":2,"requirement":"optional","rationale_zh":"實向量空間的非空子集若含零向量並對向量加法與每個實純量乘法封閉，則為向量子空間。"},{"object_id":"mko-open-closed-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-topological-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-closure-and-interior","position":3,"requirement":"optional","rationale_zh":"集合閉包是所有閉超集的交，內部是所有開子集的聯；兩者相對於指定拓撲。"},{"object_id":"mko-open-cover-and-subcover","position":4,"requirement":"optional","rationale_zh":"開覆蓋由指定拓撲的開集組成且聯集包含目標集合；子覆蓋須取自原覆蓋並仍覆蓋同一目標。"},{"object_id":"mko-random-variable","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sigma-algebra-measure-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-convergence-in-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-almost-sure-convergence","position":5,"requirement":"optional","rationale_zh":"同一機率空間上的隨機變數序列若在零機率例外集外逐點收斂，則幾乎必然收斂。"},{"object_id":"mko-graph","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-graph-matching","position":6,"requirement":"optional","rationale_zh":"有限圖的matching是彼此不共享端點的邊子集；空matching合法。"},{"object_id":"mko-prime-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-gcd-coprime","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-counting-principle","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-euler-totient-function","position":7,"requirement":"optional","rationale_zh":"Euler φ(n)計數一個完整剩餘代表系中與n互質的代表；約定φ(1)=1。"},{"object_id":"mko-computational-model","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-discrete-dynamical-system","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-automaton","position":8,"requirement":"optional","rationale_zh":"DFA由有限狀態集、有限字母表、總轉移函數、唯一初始狀態與接受狀態子集組成。"}],"edges":[{"from":"mko-real-vector-space","to":"mko-vector-subspace","relation":"required_before"},{"from":"mko-subset","to":"mko-vector-subspace","relation":"required_before"},{"from":"mko-quantifier","to":"mko-vector-subspace","relation":"required_before"},{"from":"mko-open-closed-set","to":"mko-closure-and-interior","relation":"required_before"},{"from":"mko-topological-space","to":"mko-closure-and-interior","relation":"required_before"},{"from":"mko-set-operations","to":"mko-closure-and-interior","relation":"required_before"},{"from":"mko-topological-space","to":"mko-open-cover-and-subcover","relation":"required_before"},{"from":"mko-open-closed-set","to":"mko-open-cover-and-subcover","relation":"required_before"},{"from":"mko-subset","to":"mko-open-cover-and-subcover","relation":"required_before"},{"from":"mko-set-operations","to":"mko-open-cover-and-subcover","relation":"required_before"},{"from":"mko-random-variable","to":"mko-almost-sure-convergence","relation":"required_before"},{"from":"mko-sequence","to":"mko-almost-sure-convergence","relation":"required_before"},{"from":"mko-probability","to":"mko-almost-sure-convergence","relation":"required_before"},{"from":"mko-sigma-algebra-measure-space","to":"mko-almost-sure-convergence","relation":"required_before"},{"from":"mko-convergence-in-probability","to":"mko-almost-sure-convergence","relation":"required_before"},{"from":"mko-graph","to":"mko-graph-matching","relation":"required_before"},{"from":"mko-subset","to":"mko-graph-matching","relation":"required_before"},{"from":"mko-prime-number","to":"mko-euler-totient-function","relation":"required_before"},{"from":"mko-gcd-coprime","to":"mko-euler-totient-function","relation":"required_before"},{"from":"mko-counting-principle","to":"mko-euler-totient-function","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-euler-totient-function","relation":"required_before"},{"from":"mko-computational-model","to":"mko-finite-automaton","relation":"required_before"},{"from":"mko-discrete-dynamical-system","to":"mko-finite-automaton","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-finite-automaton","relation":"required_before"},{"from":"mko-set","to":"mko-finite-automaton","relation":"required_before"},{"from":"mko-sequence","to":"mko-finite-automaton","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261102","title_zh":"Wave5第二層：維度、核像、拓撲邊界、緊緻、Bayes與正規語言","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-real-vector-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-independence-basis","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-cardinality-comparison","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-vector-space-dimension","position":2,"requirement":"optional","rationale_zh":"有限維向量空間的維度是任一有限基底的元素數；基底不同但基數不變。"},{"object_id":"mko-general-linear-map","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-vector-subspace","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-map-kernel","position":3,"requirement":"optional","rationale_zh":"線性映射T的核是全部滿足T(x)=0的來源向量，並形成來源子空間。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-map-image","position":4,"requirement":"optional","rationale_zh":"線性映射的像是所有輸出T(x)的集合；矩陣映射的像等於column span。"},{"object_id":"mko-closure-and-interior","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-topological-boundary","position":5,"requirement":"optional","rationale_zh":"集合A的拓撲邊界為closure(A)減去interior(A)，亦等於closure(A)交closure(complement A)。"},{"object_id":"mko-topological-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-open-cover-and-subcover","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-compact-topological-space","position":6,"requirement":"optional","rationale_zh":"拓撲空間緊緻，若每個開覆蓋都有有限子覆蓋；每個有限拓撲空間皆緊緻。"},{"object_id":"mko-conditional-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-event","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-bayes-theorem","position":7,"requirement":"optional","rationale_zh":"正分母條件下，Bayes定理以共同intersection與priors/likelihoods精確反轉條件機率。"},{"object_id":"mko-finite-automaton","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-decision-problem","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-regular-language","position":8,"requirement":"optional","rationale_zh":"正規語言是存在一個完整DFA可辨識其全部字串的語言。"}],"edges":[{"from":"mko-real-vector-space","to":"mko-vector-space-dimension","relation":"required_before"},{"from":"mko-linear-independence-basis","to":"mko-vector-space-dimension","relation":"required_before"},{"from":"mko-cardinality-comparison","to":"mko-vector-space-dimension","relation":"required_before"},{"from":"mko-general-linear-map","to":"mko-linear-map-kernel","relation":"required_before"},{"from":"mko-vector-subspace","to":"mko-linear-map-kernel","relation":"required_before"},{"from":"mko-equation","to":"mko-linear-map-kernel","relation":"required_before"},{"from":"mko-general-linear-map","to":"mko-linear-map-image","relation":"required_before"},{"from":"mko-vector-subspace","to":"mko-linear-map-image","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-linear-map-image","relation":"required_before"},{"from":"mko-closure-and-interior","to":"mko-topological-boundary","relation":"required_before"},{"from":"mko-set-operations","to":"mko-topological-boundary","relation":"required_before"},{"from":"mko-topological-space","to":"mko-compact-topological-space","relation":"required_before"},{"from":"mko-open-cover-and-subcover","to":"mko-compact-topological-space","relation":"required_before"},{"from":"mko-quantifier","to":"mko-compact-topological-space","relation":"required_before"},{"from":"mko-conditional-probability","to":"mko-bayes-theorem","relation":"required_before"},{"from":"mko-event","to":"mko-bayes-theorem","relation":"required_before"},{"from":"mko-probability","to":"mko-bayes-theorem","relation":"required_before"},{"from":"mko-finite-automaton","to":"mko-regular-language","relation":"required_before"},{"from":"mko-set","to":"mko-regular-language","relation":"required_before"},{"from":"mko-sequence","to":"mko-regular-language","relation":"required_before"},{"from":"mko-decision-problem","to":"mko-regular-language","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261103","title_zh":"Wave5第三層：矩陣秩、函數列逐點收斂、級數、統計模型、連通、最大匹配與模反元素","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-matrix","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-combination-span","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-vector-space-dimension","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-matrix-rank","position":2,"requirement":"optional","rationale_zh":"矩陣秩是row/column space維度，可由精確rank-preserving消去的pivot數計算。"},{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence-limit","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-pointwise-convergence-of-functions","position":3,"requirement":"optional","rationale_zh":"同一domain上的函數列逐點收斂，若每個固定x的值序列收斂至f(x)，且N可依x與epsilon改變。"},{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-infinite-series","position":4,"requirement":"optional","rationale_zh":"無窮級數以partial-sum sequence的極限定義；項趨零必要但不充分。"},{"object_id":"mko-finite-joint-distribution","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-parametric-statistical-model","position":5,"requirement":"optional","rationale_zh":"參數統計模型為每個theta指定同一有限可測sample space上的完整機率law P_theta。"},{"object_id":"mko-topological-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-open-closed-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-connected-topological-space","position":6,"requirement":"optional","rationale_zh":"拓撲空間連通，若不存在由兩個不交非空開集組成的separation；等價於無非平凡clopen subset。"},{"object_id":"mko-graph-matching","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-optimization-problem","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-maximum-matching","position":7,"requirement":"optional","rationale_zh":"最大匹配是在完整有限圖全部合法matching中邊數最大者；可有多個且不必perfect。"},{"object_id":"mko-gcd-coprime","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-modular-congruence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-modular-inverse","position":8,"requirement":"optional","rationale_zh":"n≥2時，a模n有乘法反元素恰當gcd(a,n)=1；反元素在模n意義唯一。"}],"edges":[{"from":"mko-matrix","to":"mko-matrix-rank","relation":"required_before"},{"from":"mko-linear-combination-span","to":"mko-matrix-rank","relation":"required_before"},{"from":"mko-vector-space-dimension","to":"mko-matrix-rank","relation":"required_before"},{"from":"mko-sequence","to":"mko-pointwise-convergence-of-functions","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-pointwise-convergence-of-functions","relation":"required_before"},{"from":"mko-sequence-limit","to":"mko-pointwise-convergence-of-functions","relation":"required_before"},{"from":"mko-quantifier","to":"mko-pointwise-convergence-of-functions","relation":"required_before"},{"from":"mko-sequence","to":"mko-infinite-series","relation":"required_before"},{"from":"mko-sequence-limit","to":"mko-infinite-series","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-infinite-series","relation":"required_before"},{"from":"mko-finite-joint-distribution","to":"mko-parametric-statistical-model","relation":"required_before"},{"from":"mko-probability","to":"mko-parametric-statistical-model","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-parametric-statistical-model","relation":"required_before"},{"from":"mko-set","to":"mko-parametric-statistical-model","relation":"required_before"},{"from":"mko-topological-space","to":"mko-connected-topological-space","relation":"required_before"},{"from":"mko-open-closed-set","to":"mko-connected-topological-space","relation":"required_before"},{"from":"mko-set-operations","to":"mko-connected-topological-space","relation":"required_before"},{"from":"mko-graph-matching","to":"mko-maximum-matching","relation":"required_before"},{"from":"mko-optimization-problem","to":"mko-maximum-matching","relation":"required_before"},{"from":"mko-gcd-coprime","to":"mko-modular-inverse","relation":"required_before"},{"from":"mko-modular-congruence","to":"mko-modular-inverse","relation":"required_before"},{"from":"mko-equation","to":"mko-modular-inverse","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261104","title_zh":"Wave5第四層：rank-nullity、一致收斂、絕對收斂、估計量、CRT、生成樹與Lp收斂","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-linear-map-kernel","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-map-image","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-matrix-rank","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-vector-space-dimension","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-rank-nullity-theorem","position":2,"requirement":"optional","rationale_zh":"有限維線性映射T:V→W滿足dim V=dim ker T+rank T。"},{"object_id":"mko-pointwise-convergence-of-functions","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-metric-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-absolute-value","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-uniform-convergence-of-functions","position":3,"requirement":"optional","rationale_zh":"一致收斂要求同一N(epsilon)同時控制domain全部x。"},{"object_id":"mko-infinite-series","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence-limit","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-absolute-convergence-of-series","position":4,"requirement":"optional","rationale_zh":"sum |a_n|收斂時sum a_n絕對收斂；絕對收斂推出收斂，反向不一般成立。"},{"object_id":"mko-parametric-statistical-model","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-random-variable","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sample-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-statistical-estimator","position":5,"requirement":"optional","rationale_zh":"估計量是固定且不依未知theta的可測total statistic；觀察後數值是estimate。"},{"object_id":"mko-integer","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-modular-congruence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-gcd-coprime","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-chinese-remainder-theorem","position":6,"requirement":"optional","rationale_zh":"正兩兩互質模數的同餘系統在模乘積意義下存在唯一解。"},{"object_id":"mko-tree","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-graph","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-connected-component","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-spanning-tree","position":7,"requirement":"optional","rationale_zh":"生成樹是包含父圖全部頂點的連通無圈子圖。"},{"object_id":"mko-expectation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-power-root","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sigma-algebra-measure-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-lp-convergence","position":8,"requirement":"optional","rationale_zh":"共同機率空間上固定整數p≥1時，E|X_n−X|^p→0稱Lp收斂。"}],"edges":[{"from":"mko-linear-map-kernel","to":"mko-rank-nullity-theorem","relation":"required_before"},{"from":"mko-linear-map-image","to":"mko-rank-nullity-theorem","relation":"required_before"},{"from":"mko-matrix-rank","to":"mko-rank-nullity-theorem","relation":"required_before"},{"from":"mko-vector-space-dimension","to":"mko-rank-nullity-theorem","relation":"required_before"},{"from":"mko-pointwise-convergence-of-functions","to":"mko-uniform-convergence-of-functions","relation":"required_before"},{"from":"mko-metric-space","to":"mko-uniform-convergence-of-functions","relation":"required_before"},{"from":"mko-absolute-value","to":"mko-uniform-convergence-of-functions","relation":"required_before"},{"from":"mko-quantifier","to":"mko-uniform-convergence-of-functions","relation":"required_before"},{"from":"mko-infinite-series","to":"mko-absolute-convergence-of-series","relation":"required_before"},{"from":"mko-absolute-value","to":"mko-absolute-convergence-of-series","relation":"required_before"},{"from":"mko-sequence-limit","to":"mko-absolute-convergence-of-series","relation":"required_before"},{"from":"mko-parametric-statistical-model","to":"mko-statistical-estimator","relation":"required_before"},{"from":"mko-random-variable","to":"mko-statistical-estimator","relation":"required_before"},{"from":"mko-sample-space","to":"mko-statistical-estimator","relation":"required_before"},{"from":"mko-integer","to":"mko-chinese-remainder-theorem","relation":"required_before"},{"from":"mko-modular-congruence","to":"mko-chinese-remainder-theorem","relation":"required_before"},{"from":"mko-gcd-coprime","to":"mko-chinese-remainder-theorem","relation":"required_before"},{"from":"mko-quantifier","to":"mko-chinese-remainder-theorem","relation":"required_before"},{"from":"mko-tree","to":"mko-spanning-tree","relation":"required_before"},{"from":"mko-graph","to":"mko-spanning-tree","relation":"required_before"},{"from":"mko-connected-component","to":"mko-spanning-tree","relation":"required_before"},{"from":"mko-random-variable","to":"mko-lp-convergence","relation":"required_before"},{"from":"mko-expectation","to":"mko-lp-convergence","relation":"required_before"},{"from":"mko-absolute-value","to":"mko-lp-convergence","relation":"required_before"},{"from":"mko-sequence-limit","to":"mko-lp-convergence","relation":"required_before"},{"from":"mko-power-root","to":"mko-lp-convergence","relation":"required_before"},{"from":"mko-sigma-algebra-measure-space","to":"mko-lp-convergence","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261105","title_zh":"Wave5第五層：直和、對偶、Euclidean演算法、Euler trail、同胚、抽樣分布與弱大數律","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-vector-subspace","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-independence-basis","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-direct-sum","position":2,"requirement":"optional","rationale_zh":"二個子空間之和若每個向量有唯一分解便是內直和；要宣稱覆蓋母空間還須另證生成。"},{"object_id":"mko-general-linear-map","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-vector-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-dual-vector-space","position":3,"requirement":"optional","rationale_zh":"實向量空間的對偶空間由所有實值線性泛函組成；座標covector表示依賴基底。"},{"object_id":"mko-gcd-coprime","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-natural-divisibility","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-algorithm-correctness","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-euclidean-algorithm","position":4,"requirement":"optional","rationale_zh":"Euclidean演算法以精確帶餘除法、gcd invariant與嚴格餘數下降求最大公因數。"},{"object_id":"mko-graph-walk-simple-path","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-graph","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-connected-component","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-eulerian-trail","position":5,"requirement":"optional","rationale_zh":"Euler trail恰用每條邊一次；非零邊支撐連通且奇度數為0或2時分別有circuit或open trail。"},{"object_id":"mko-topological-continuity","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-bijective","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-inverse-function","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-topological-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-homeomorphism","position":6,"requirement":"optional","rationale_zh":"同胚是雙射、連續且逆映射亦連續的映射；一般continuous bijection仍可能不是同胚。"},{"object_id":"mko-parametric-statistical-model","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-statistical-estimator","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-discrete-probability-distribution","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sampling-distribution","position":7,"requirement":"optional","rationale_zh":"固定theta時，估計量把樣本分布pushforward成統計量的抽樣分布；全部theta形成一族laws。"},{"object_id":"mko-iid-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-expectation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-variance","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-convergence-in-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-weak-law-large-numbers","position":8,"requirement":"optional","rationale_zh":"IID且具有有限variance時，Chebyshev界給樣本平均依機率收斂到共同mean。"}],"edges":[{"from":"mko-vector-subspace","to":"mko-direct-sum","relation":"required_before"},{"from":"mko-set-operations","to":"mko-direct-sum","relation":"required_before"},{"from":"mko-linear-independence-basis","to":"mko-direct-sum","relation":"required_before"},{"from":"mko-general-linear-map","to":"mko-dual-vector-space","relation":"required_before"},{"from":"mko-real-vector-space","to":"mko-dual-vector-space","relation":"required_before"},{"from":"mko-real-number","to":"mko-dual-vector-space","relation":"required_before"},{"from":"mko-gcd-coprime","to":"mko-euclidean-algorithm","relation":"required_before"},{"from":"mko-natural-divisibility","to":"mko-euclidean-algorithm","relation":"required_before"},{"from":"mko-algorithm-correctness","to":"mko-euclidean-algorithm","relation":"required_before"},{"from":"mko-graph-walk-simple-path","to":"mko-eulerian-trail","relation":"required_before"},{"from":"mko-graph","to":"mko-eulerian-trail","relation":"required_before"},{"from":"mko-connected-component","to":"mko-eulerian-trail","relation":"required_before"},{"from":"mko-topological-continuity","to":"mko-homeomorphism","relation":"required_before"},{"from":"mko-bijective","to":"mko-homeomorphism","relation":"required_before"},{"from":"mko-inverse-function","to":"mko-homeomorphism","relation":"required_before"},{"from":"mko-topological-space","to":"mko-homeomorphism","relation":"required_before"},{"from":"mko-parametric-statistical-model","to":"mko-sampling-distribution","relation":"required_before"},{"from":"mko-statistical-estimator","to":"mko-sampling-distribution","relation":"required_before"},{"from":"mko-finite-discrete-probability-distribution","to":"mko-sampling-distribution","relation":"required_before"},{"from":"mko-iid-sequence","to":"mko-weak-law-large-numbers","relation":"required_before"},{"from":"mko-expectation","to":"mko-weak-law-large-numbers","relation":"required_before"},{"from":"mko-variance","to":"mko-weak-law-large-numbers","relation":"required_before"},{"from":"mko-convergence-in-probability","to":"mko-weak-law-large-numbers","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261106","title_zh":"Wave5封閉層：色數、Clique與獨立集、估計量偏誤與變異數、Fermat小定理、Diophantine方程與商空間","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-graph-coloring","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-natural-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-optimization-problem","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-chromatic-number","position":2,"requirement":"optional","rationale_zh":"有限簡單圖的色數是proper頂點著色所需顏色數的最小值，需同時具備上界與下界證書。"},{"object_id":"mko-graph","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subset","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-clique-and-independent-set","position":3,"requirement":"optional","rationale_zh":"Clique內每對不同頂點相鄰，獨立集內每對不同頂點不相鄰；最大與極大不可混同。"},{"object_id":"mko-sampling-distribution","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-expectation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-parametric-statistical-model","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-estimator-bias","position":4,"requirement":"optional","rationale_zh":"估計量在theta的偏誤是其theta抽樣分布期望減去目標參數；無偏要求對所有theta成立。"},{"object_id":"mko-variance","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-estimator-variance","position":5,"requirement":"optional","rationale_zh":"估計量變異數逐theta衡量sampling law相對自身期望的二次離差，不等同偏誤或MSE。"},{"object_id":"mko-prime-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-modular-congruence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-power-root","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-fermat-little-theorem","position":6,"requirement":"optional","rationale_zh":"p為質數時a^p同餘a；若p不整除a則可化為a^(p-1)同餘1，逆命題不成立。"},{"object_id":"mko-integer","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-gcd-coprime","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-euclidean-algorithm","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-diophantine-equation","position":7,"requirement":"optional","rationale_zh":"非退化整數方程ax+by=c有解恰當gcd(a,b)整除c，全部解由一個整數參數給出。"},{"object_id":"mko-vector-subspace","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quotient-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-vector-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-equivalence-class","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-vector-space-dimension","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quotient-vector-space","position":8,"requirement":"optional","rationale_zh":"V依子空間W的cosets形成商向量空間，運算須與代表元選擇無關。"}],"edges":[{"from":"mko-graph-coloring","to":"mko-chromatic-number","relation":"required_before"},{"from":"mko-natural-number","to":"mko-chromatic-number","relation":"required_before"},{"from":"mko-optimization-problem","to":"mko-chromatic-number","relation":"required_before"},{"from":"mko-graph","to":"mko-clique-and-independent-set","relation":"required_before"},{"from":"mko-subset","to":"mko-clique-and-independent-set","relation":"required_before"},{"from":"mko-sampling-distribution","to":"mko-estimator-bias","relation":"required_before"},{"from":"mko-expectation","to":"mko-estimator-bias","relation":"required_before"},{"from":"mko-parametric-statistical-model","to":"mko-estimator-bias","relation":"required_before"},{"from":"mko-sampling-distribution","to":"mko-estimator-variance","relation":"required_before"},{"from":"mko-variance","to":"mko-estimator-variance","relation":"required_before"},{"from":"mko-prime-number","to":"mko-fermat-little-theorem","relation":"required_before"},{"from":"mko-modular-congruence","to":"mko-fermat-little-theorem","relation":"required_before"},{"from":"mko-power-root","to":"mko-fermat-little-theorem","relation":"required_before"},{"from":"mko-integer","to":"mko-linear-diophantine-equation","relation":"required_before"},{"from":"mko-gcd-coprime","to":"mko-linear-diophantine-equation","relation":"required_before"},{"from":"mko-linear-equation","to":"mko-linear-diophantine-equation","relation":"required_before"},{"from":"mko-euclidean-algorithm","to":"mko-linear-diophantine-equation","relation":"required_before"},{"from":"mko-vector-subspace","to":"mko-quotient-vector-space","relation":"required_before"},{"from":"mko-quotient-set","to":"mko-quotient-vector-space","relation":"required_before"},{"from":"mko-real-vector-space","to":"mko-quotient-vector-space","relation":"required_before"},{"from":"mko-equivalence-class","to":"mko-quotient-vector-space","relation":"required_before"},{"from":"mko-vector-space-dimension","to":"mko-quotient-vector-space","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261107","title_zh":"Wave6根層：有限群作用、上下確界、拓撲基底、Markov不等式、鴿籠原理、Euler定理與子數列","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-finite-group","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-group-action","position":2,"requirement":"optional","rationale_zh":"有限群在有限集合上的作用是滿足單位與乘法相容律的total action table。"},{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-supremum-and-infimum","position":3,"requirement":"optional","rationale_zh":"非空且有界實數集合的supremum是最小上界，infimum是最大下界，兩者未必屬於集合。"},{"object_id":"mko-topological-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-open-closed-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-topological-basis","position":4,"requirement":"optional","rationale_zh":"拓撲基底以覆蓋carrier與逐點局部交細化兩條公理生成拓撲。"},{"object_id":"mko-random-variable","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-expectation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-markov-inequality","position":5,"requirement":"optional","rationale_zh":"非負且有限期望的隨機變數在正閾值上的尾機率不超過E[X]/a。"},{"object_id":"mko-counting-principle","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-cardinality-comparison","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-pigeonhole-principle","position":6,"requirement":"optional","rationale_zh":"有限total map的domain比codomain大時必有兩個不同元素共享像。"},{"object_id":"mko-euler-totient-function","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-gcd-coprime","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-modular-congruence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-power-root","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-euler-theorem-number-theory","position":7,"requirement":"optional","rationale_zh":"n>=2且gcd(a,n)=1時，a的phi(n)次方模n同餘於1。"},{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-injective","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subsequence","position":8,"requirement":"optional","rationale_zh":"子數列由嚴格遞增的自然數索引映射選取原數列項並保留次序。"}],"edges":[{"from":"mko-finite-group","to":"mko-finite-group-action","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-finite-group-action","relation":"required_before"},{"from":"mko-set","to":"mko-finite-group-action","relation":"required_before"},{"from":"mko-real-number","to":"mko-supremum-and-infimum","relation":"required_before"},{"from":"mko-order-relation","to":"mko-supremum-and-infimum","relation":"required_before"},{"from":"mko-set","to":"mko-supremum-and-infimum","relation":"required_before"},{"from":"mko-quantifier","to":"mko-supremum-and-infimum","relation":"required_before"},{"from":"mko-topological-space","to":"mko-topological-basis","relation":"required_before"},{"from":"mko-open-closed-set","to":"mko-topological-basis","relation":"required_before"},{"from":"mko-set-operations","to":"mko-topological-basis","relation":"required_before"},{"from":"mko-random-variable","to":"mko-markov-inequality","relation":"required_before"},{"from":"mko-expectation","to":"mko-markov-inequality","relation":"required_before"},{"from":"mko-probability","to":"mko-markov-inequality","relation":"required_before"},{"from":"mko-order-relation","to":"mko-markov-inequality","relation":"required_before"},{"from":"mko-counting-principle","to":"mko-pigeonhole-principle","relation":"required_before"},{"from":"mko-cardinality-comparison","to":"mko-pigeonhole-principle","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-pigeonhole-principle","relation":"required_before"},{"from":"mko-euler-totient-function","to":"mko-euler-theorem-number-theory","relation":"required_before"},{"from":"mko-gcd-coprime","to":"mko-euler-theorem-number-theory","relation":"required_before"},{"from":"mko-modular-congruence","to":"mko-euler-theorem-number-theory","relation":"required_before"},{"from":"mko-power-root","to":"mko-euler-theorem-number-theory","relation":"required_before"},{"from":"mko-sequence","to":"mko-subsequence","relation":"required_before"},{"from":"mko-injective","to":"mko-subsequence","relation":"required_before"},{"from":"mko-order-relation","to":"mko-subsequence","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261108","title_zh":"Wave6第二層：群軌道、單調收斂、子空間拓撲、Chebyshev、容斥、唯一質因數分解與有向圖","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-finite-group-action","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-equivalence-class","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-group-orbit","position":2,"requirement":"optional","rationale_zh":"有限群作用下，一點被全部群元素送到的像集合形成軌道，全部軌道分割carrier。"},{"object_id":"mko-supremum-and-infimum","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence-limit","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-monotone-sequence-convergence-theorem","position":3,"requirement":"optional","rationale_zh":"非遞減且有上界的實數列收斂至其項集合supremum；非遞增版本對偶收斂至infimum。"},{"object_id":"mko-topological-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subset","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-topological-basis","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subspace-topology","position":4,"requirement":"optional","rationale_zh":"Y⊆X的相對拓撲由全部母空間open sets與Y的交集組成。"},{"object_id":"mko-markov-inequality","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-variance","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-absolute-value","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-chebyshev-inequality","position":5,"requirement":"optional","rationale_zh":"有限均值與變異數的隨機變數偏離均值至少epsilon的機率不超過variance/epsilon²。"},{"object_id":"mko-set-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-counting-principle","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-cardinality-comparison","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-mathematical-induction","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-inclusion-exclusion-principle","position":6,"requirement":"optional","rationale_zh":"有限indexed set family的union cardinality是全部非空index subsets交集大小的交替和。"},{"object_id":"mko-prime-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-natural-divisibility","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-euclidean-algorithm","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-gcd-coprime","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-unique-prime-factorization","position":7,"requirement":"optional","rationale_zh":"每個n>1有prime factorization且prime multiset唯一，只允許因數次序差異。"},{"object_id":"mko-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-directed-graph","position":8,"requirement":"optional","rationale_zh":"有限loop-free simple digraph由有限vertices與不含loop、無平行同向重邊的ordered-pair edge set組成。"}],"edges":[{"from":"mko-finite-group-action","to":"mko-finite-group-orbit","relation":"required_before"},{"from":"mko-equivalence-class","to":"mko-finite-group-orbit","relation":"required_before"},{"from":"mko-supremum-and-infimum","to":"mko-monotone-sequence-convergence-theorem","relation":"required_before"},{"from":"mko-sequence-limit","to":"mko-monotone-sequence-convergence-theorem","relation":"required_before"},{"from":"mko-order-relation","to":"mko-monotone-sequence-convergence-theorem","relation":"required_before"},{"from":"mko-topological-space","to":"mko-subspace-topology","relation":"required_before"},{"from":"mko-subset","to":"mko-subspace-topology","relation":"required_before"},{"from":"mko-topological-basis","to":"mko-subspace-topology","relation":"required_before"},{"from":"mko-markov-inequality","to":"mko-chebyshev-inequality","relation":"required_before"},{"from":"mko-variance","to":"mko-chebyshev-inequality","relation":"required_before"},{"from":"mko-absolute-value","to":"mko-chebyshev-inequality","relation":"required_before"},{"from":"mko-set-operations","to":"mko-inclusion-exclusion-principle","relation":"required_before"},{"from":"mko-counting-principle","to":"mko-inclusion-exclusion-principle","relation":"required_before"},{"from":"mko-cardinality-comparison","to":"mko-inclusion-exclusion-principle","relation":"required_before"},{"from":"mko-mathematical-induction","to":"mko-inclusion-exclusion-principle","relation":"required_before"},{"from":"mko-prime-number","to":"mko-unique-prime-factorization","relation":"required_before"},{"from":"mko-natural-divisibility","to":"mko-unique-prime-factorization","relation":"required_before"},{"from":"mko-mathematical-induction","to":"mko-unique-prime-factorization","relation":"required_before"},{"from":"mko-euclidean-algorithm","to":"mko-unique-prime-factorization","relation":"required_before"},{"from":"mko-gcd-coprime","to":"mko-unique-prime-factorization","relation":"required_before"},{"from":"mko-relation","to":"mko-finite-directed-graph","relation":"required_before"},{"from":"mko-set","to":"mko-finite-directed-graph","relation":"required_before"},{"from":"mko-cardinality-comparison","to":"mko-finite-directed-graph","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261109","title_zh":"Wave6第三層：Stabilizer、Bolzano-Weierstrass、緊緻子集、全期望、二項式、Möbius與CFG","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-finite-group-action","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subgroup-homomorphism","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-group-stabilizer","position":2,"requirement":"optional","rationale_zh":"指定點的stabilizer是有限群作用中固定該點的全部群元素，並形成子群。"},{"object_id":"mko-subsequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence-limit","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-supremum-and-infimum","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-bolzano-weierstrass-theorem-real","position":3,"requirement":"optional","rationale_zh":"每個整體有界的實數列都有一個以嚴格遞增無限索引取得的收斂子數列。"},{"object_id":"mko-compact-topological-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subspace-topology","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-compact-subset","position":4,"requirement":"optional","rationale_zh":"K⊆X在subspace topology下compact，等價地每個relative-open cover有有限subcover。"},{"object_id":"mko-conditional-expectation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-expectation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-event","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-law-of-total-expectation","position":5,"requirement":"optional","rationale_zh":"同一有限機率空間上，完整partition各cell的conditional mean按cell probability加權後等於E[X]。"},{"object_id":"mko-permutation-combination","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-mathematical-induction","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-polynomial","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-binomial-theorem","position":6,"requirement":"optional","rationale_zh":"交換乘法環境中，natural exponent n的二項式冪以binomial coefficients完整展開。"},{"object_id":"mko-prime-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-unique-prime-factorization","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-mobius-function","position":7,"requirement":"optional","rationale_zh":"Möbius函數在positive integers上由完整prime multiplicities決定：mu(1)=1、含平方質因數為0、squarefree取parity sign。"},{"object_id":"mko-computational-model","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-context-free-grammar","position":8,"requirement":"optional","rationale_zh":"CFG由有限且不交的nonterminals、terminals、start與單一nonterminal左側的finite productions生成語言。"}],"edges":[{"from":"mko-finite-group-action","to":"mko-finite-group-stabilizer","relation":"required_before"},{"from":"mko-subgroup-homomorphism","to":"mko-finite-group-stabilizer","relation":"required_before"},{"from":"mko-subsequence","to":"mko-bolzano-weierstrass-theorem-real","relation":"required_before"},{"from":"mko-sequence-limit","to":"mko-bolzano-weierstrass-theorem-real","relation":"required_before"},{"from":"mko-real-number","to":"mko-bolzano-weierstrass-theorem-real","relation":"required_before"},{"from":"mko-supremum-and-infimum","to":"mko-bolzano-weierstrass-theorem-real","relation":"required_before"},{"from":"mko-compact-topological-space","to":"mko-compact-subset","relation":"required_before"},{"from":"mko-subspace-topology","to":"mko-compact-subset","relation":"required_before"},{"from":"mko-conditional-expectation","to":"mko-law-of-total-expectation","relation":"required_before"},{"from":"mko-expectation","to":"mko-law-of-total-expectation","relation":"required_before"},{"from":"mko-event","to":"mko-law-of-total-expectation","relation":"required_before"},{"from":"mko-permutation-combination","to":"mko-binomial-theorem","relation":"required_before"},{"from":"mko-mathematical-induction","to":"mko-binomial-theorem","relation":"required_before"},{"from":"mko-polynomial","to":"mko-binomial-theorem","relation":"required_before"},{"from":"mko-prime-number","to":"mko-mobius-function","relation":"required_before"},{"from":"mko-unique-prime-factorization","to":"mko-mobius-function","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-mobius-function","relation":"required_before"},{"from":"mko-computational-model","to":"mko-context-free-grammar","relation":"required_before"},{"from":"mko-set","to":"mko-context-free-grammar","relation":"required_before"},{"from":"mko-sequence","to":"mko-context-free-grammar","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261110","title_zh":"Wave6第四層：Orbit-Stabilizer、MVT、緊緻像、全變異數、遞迴、Möbius反演與PDA","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-finite-group-orbit","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-group-stabilizer","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-cardinality-comparison","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-orbit-stabilizer-theorem","position":2,"requirement":"optional","rationale_zh":"完整有限群作用中，同一點的orbit大小乘以stabilizer大小等於群階。"},{"object_id":"mko-derivative","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-continuity","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-rate-of-change","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-mean-value-theorem","position":3,"requirement":"optional","rationale_zh":"若a<b，f在[a,b]連續且在(a,b)可微，則某個內點c的導數等於端點割線斜率。"},{"object_id":"mko-compact-subset","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-topological-continuity","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-continuous-image-of-compact","position":4,"requirement":"optional","rationale_zh":"若K在X中compact且f:X→Y連續，則image f(K)在Y中compact。"},{"object_id":"mko-law-of-total-expectation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-variance","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-conditional-expectation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-law-of-total-variance","position":5,"requirement":"optional","rationale_zh":"同一有限law與完整partition生成的G下，總變異數等於cell內變異數期望加cell均值變異數。"},{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-recurrence-relation","position":6,"requirement":"optional","rationale_zh":"order-r recurrence以明示initial values與total single-valued update rule逐項決定sequence。"},{"object_id":"mko-mobius-function","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-natural-divisibility","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-counting-principle","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-mobius-inversion-finite-divisors","position":7,"requirement":"optional","rationale_zh":"若F(n)是所有正因數d上的f(d)總和，則以Möbius權重可精確反演f(n)。"},{"object_id":"mko-computational-model","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-automaton","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-context-free-grammar","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-pushdown-automaton","position":8,"requirement":"optional","rationale_zh":"nondeterministic PDA以有限控制、input與stack transitions運算，本輪固定讀完input且到final state接受。"}],"edges":[{"from":"mko-finite-group-orbit","to":"mko-orbit-stabilizer-theorem","relation":"required_before"},{"from":"mko-finite-group-stabilizer","to":"mko-orbit-stabilizer-theorem","relation":"required_before"},{"from":"mko-cardinality-comparison","to":"mko-orbit-stabilizer-theorem","relation":"required_before"},{"from":"mko-derivative","to":"mko-mean-value-theorem","relation":"required_before"},{"from":"mko-continuity","to":"mko-mean-value-theorem","relation":"required_before"},{"from":"mko-rate-of-change","to":"mko-mean-value-theorem","relation":"required_before"},{"from":"mko-order-relation","to":"mko-mean-value-theorem","relation":"required_before"},{"from":"mko-compact-subset","to":"mko-continuous-image-of-compact","relation":"required_before"},{"from":"mko-topological-continuity","to":"mko-continuous-image-of-compact","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-continuous-image-of-compact","relation":"required_before"},{"from":"mko-law-of-total-expectation","to":"mko-law-of-total-variance","relation":"required_before"},{"from":"mko-variance","to":"mko-law-of-total-variance","relation":"required_before"},{"from":"mko-conditional-expectation","to":"mko-law-of-total-variance","relation":"required_before"},{"from":"mko-sequence","to":"mko-recurrence-relation","relation":"required_before"},{"from":"mko-equation","to":"mko-recurrence-relation","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-recurrence-relation","relation":"required_before"},{"from":"mko-mobius-function","to":"mko-mobius-inversion-finite-divisors","relation":"required_before"},{"from":"mko-natural-divisibility","to":"mko-mobius-inversion-finite-divisors","relation":"required_before"},{"from":"mko-counting-principle","to":"mko-mobius-inversion-finite-divisors","relation":"required_before"},{"from":"mko-computational-model","to":"mko-pushdown-automaton","relation":"required_before"},{"from":"mko-finite-automaton","to":"mko-pushdown-automaton","relation":"required_before"},{"from":"mko-sequence","to":"mko-pushdown-automaton","relation":"required_before"},{"from":"mko-context-free-grammar","to":"mko-pushdown-automaton","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261111","title_zh":"Wave6第五層：Lagrange、高階導數、Hausdorff、連續CDF、Catalan、二次剩餘與NFA","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-finite-group","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subgroup-homomorphism","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-natural-divisibility","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-cardinality-comparison","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quotient-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-lagrange-theorem-finite-groups","position":2,"requirement":"optional","rationale_zh":"有限群G的子群H以等大左陪集分割G，因此|G|=[G:H]|H|且|H|整除|G|。"},{"object_id":"mko-derivative","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-derivative-rules","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-mathematical-induction","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-higher-order-derivative","position":3,"requirement":"optional","rationale_zh":"以f^(0)=f及f^(n+1)=(f^(n))'遞迴定義每個存在的有限階導數。"},{"object_id":"mko-topological-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-neighborhood","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-hausdorff-space","position":4,"requirement":"optional","rationale_zh":"拓撲空間X若每對不同點都有不交open neighborhoods即為Hausdorff。"},{"object_id":"mko-cumulative-distribution-function","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-continuity","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-limit","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sigma-algebra-measure-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-continuous-real-distribution-function","position":5,"requirement":"optional","rationale_zh":"本節點以CDF本身連續定義continuous distribution；CDF仍須nondecreasing、right-continuous且兩端極限0與1。"},{"object_id":"mko-recurrence-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-binomial-theorem","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-counting-principle","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-catalan-numbers","position":6,"requirement":"optional","rationale_zh":"Catalan數採C_0=1，並以convolution recurrence或兩個exact closed forms定義同一zero-based sequence。"},{"object_id":"mko-modular-congruence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-prime-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-power-root","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quadratic-residue","position":7,"requirement":"optional","rationale_zh":"本節點採odd prime p上的nonzero convention：a mod p非零且等於某個nonzero square時為quadratic residue。"},{"object_id":"mko-finite-automaton","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subset","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-nondeterministic-finite-automaton","position":8,"requirement":"optional","rationale_zh":"epsilon-NFA以狀態、input symbol或epsilon對應可能後繼集合，存在一條full-input final path即接受。"}],"edges":[{"from":"mko-finite-group","to":"mko-lagrange-theorem-finite-groups","relation":"required_before"},{"from":"mko-subgroup-homomorphism","to":"mko-lagrange-theorem-finite-groups","relation":"required_before"},{"from":"mko-natural-divisibility","to":"mko-lagrange-theorem-finite-groups","relation":"required_before"},{"from":"mko-cardinality-comparison","to":"mko-lagrange-theorem-finite-groups","relation":"required_before"},{"from":"mko-quotient-set","to":"mko-lagrange-theorem-finite-groups","relation":"required_before"},{"from":"mko-derivative","to":"mko-higher-order-derivative","relation":"required_before"},{"from":"mko-derivative-rules","to":"mko-higher-order-derivative","relation":"required_before"},{"from":"mko-mathematical-induction","to":"mko-higher-order-derivative","relation":"required_before"},{"from":"mko-topological-space","to":"mko-hausdorff-space","relation":"required_before"},{"from":"mko-neighborhood","to":"mko-hausdorff-space","relation":"required_before"},{"from":"mko-quantifier","to":"mko-hausdorff-space","relation":"required_before"},{"from":"mko-cumulative-distribution-function","to":"mko-continuous-real-distribution-function","relation":"required_before"},{"from":"mko-continuity","to":"mko-continuous-real-distribution-function","relation":"required_before"},{"from":"mko-function-limit","to":"mko-continuous-real-distribution-function","relation":"required_before"},{"from":"mko-order-relation","to":"mko-continuous-real-distribution-function","relation":"required_before"},{"from":"mko-sigma-algebra-measure-space","to":"mko-continuous-real-distribution-function","relation":"required_before"},{"from":"mko-recurrence-relation","to":"mko-catalan-numbers","relation":"required_before"},{"from":"mko-binomial-theorem","to":"mko-catalan-numbers","relation":"required_before"},{"from":"mko-counting-principle","to":"mko-catalan-numbers","relation":"required_before"},{"from":"mko-natural-divisibility","to":"mko-catalan-numbers","relation":"required_before"},{"from":"mko-modular-congruence","to":"mko-quadratic-residue","relation":"required_before"},{"from":"mko-prime-number","to":"mko-quadratic-residue","relation":"required_before"},{"from":"mko-power-root","to":"mko-quadratic-residue","relation":"required_before"},{"from":"mko-finite-automaton","to":"mko-nondeterministic-finite-automaton","relation":"required_before"},{"from":"mko-relation","to":"mko-nondeterministic-finite-automaton","relation":"required_before"},{"from":"mko-subset","to":"mko-nondeterministic-finite-automaton","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261112","title_zh":"Wave6第六層：Cauchy、Taylor、緊緻閉集、瑕積分、收斂半徑、Legendre與子集合構造","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-lagrange-theorem-finite-groups","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-prime-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-group","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-group-action","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-cauchy-theorem-finite-groups","position":2,"requirement":"optional","rationale_zh":"若質數p整除有限群G的階，則G含有一個exact order為p的元素。"},{"object_id":"mko-derivative-rules","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-polynomial","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-factorial","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-higher-order-derivative","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-taylor-polynomial","position":3,"requirement":"optional","rationale_zh":"在中心a由f的零階到n階導數值依factorial縮放組成n階Taylor多項式。"},{"object_id":"mko-hausdorff-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-compact-subset","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-open-closed-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-compact-subset-closed-in-hausdorff","position":4,"requirement":"optional","rationale_zh":"若K是Hausdorff空間X的緊緻子集，則K在X中為閉集。"},{"object_id":"mko-definite-integral","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-limit","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence-limit","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-absolute-convergence-of-series","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-improper-integral","position":5,"requirement":"optional","rationale_zh":"無界區間或含奇點的積分以適當有限區間積分的one-sided limits定義，且每個必要分段須獨立收斂。"},{"object_id":"mko-infinite-series","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-supremum-and-infimum","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-absolute-value","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-power-root","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-power-series-radius-of-convergence","position":6,"requirement":"optional","rationale_zh":"冪級數在中心c周圍具有內部絕對收斂、外部發散的半徑R，邊界點須逐案判定。"},{"object_id":"mko-quadratic-residue","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-legendre-symbol","position":7,"requirement":"optional","rationale_zh":"對奇質數p，Legendre符號以0、1、-1分別編碼zero class、nonzero quadratic residue與nonresidue。"},{"object_id":"mko-nondeterministic-finite-automaton","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-automaton","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subset","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subset-construction-nfa-to-dfa","position":8,"requirement":"optional","rationale_zh":"以epsilon-NFA狀態子集合為DFA狀態，start、transition與acceptance按epsilon closure定義即可保持語言。"}],"edges":[{"from":"mko-lagrange-theorem-finite-groups","to":"mko-cauchy-theorem-finite-groups","relation":"required_before"},{"from":"mko-prime-number","to":"mko-cauchy-theorem-finite-groups","relation":"required_before"},{"from":"mko-finite-group","to":"mko-cauchy-theorem-finite-groups","relation":"required_before"},{"from":"mko-finite-group-action","to":"mko-cauchy-theorem-finite-groups","relation":"required_before"},{"from":"mko-derivative-rules","to":"mko-taylor-polynomial","relation":"required_before"},{"from":"mko-polynomial","to":"mko-taylor-polynomial","relation":"required_before"},{"from":"mko-factorial","to":"mko-taylor-polynomial","relation":"required_before"},{"from":"mko-higher-order-derivative","to":"mko-taylor-polynomial","relation":"required_before"},{"from":"mko-hausdorff-space","to":"mko-compact-subset-closed-in-hausdorff","relation":"required_before"},{"from":"mko-compact-subset","to":"mko-compact-subset-closed-in-hausdorff","relation":"required_before"},{"from":"mko-open-closed-set","to":"mko-compact-subset-closed-in-hausdorff","relation":"required_before"},{"from":"mko-definite-integral","to":"mko-improper-integral","relation":"required_before"},{"from":"mko-function-limit","to":"mko-improper-integral","relation":"required_before"},{"from":"mko-sequence-limit","to":"mko-improper-integral","relation":"required_before"},{"from":"mko-absolute-convergence-of-series","to":"mko-improper-integral","relation":"required_before"},{"from":"mko-infinite-series","to":"mko-power-series-radius-of-convergence","relation":"required_before"},{"from":"mko-supremum-and-infimum","to":"mko-power-series-radius-of-convergence","relation":"required_before"},{"from":"mko-absolute-value","to":"mko-power-series-radius-of-convergence","relation":"required_before"},{"from":"mko-power-root","to":"mko-power-series-radius-of-convergence","relation":"required_before"},{"from":"mko-quadratic-residue","to":"mko-legendre-symbol","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-legendre-symbol","relation":"required_before"},{"from":"mko-nondeterministic-finite-automaton","to":"mko-subset-construction-nfa-to-dfa","relation":"required_before"},{"from":"mko-finite-automaton","to":"mko-subset-construction-nfa-to-dfa","relation":"required_before"},{"from":"mko-subset","to":"mko-subset-construction-nfa-to-dfa","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261113","title_zh":"Wave6第七層：正規子群、實指數、解析OGF、Euler判準、Turing判定、梯度與流網路","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-finite-group","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subgroup-homomorphism","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-normal-subgroup","position":2,"requirement":"optional","rationale_zh":"有限群G的子群H若對每個g皆滿足gHg^{-1}=H，則H為正規子群；等價地每個左陪集等於相應右陪集。"},{"object_id":"mko-infinite-series","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence-limit","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-power-series-radius-of-convergence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-factorial","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-absolute-convergence-of-series","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-binomial-theorem","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-exponential-function","position":3,"requirement":"optional","rationale_zh":"實指數函數以對每個實數絕對收斂的冪級數E(x)=sum_{n≥0}x^n/n!定義，並滿足E(x+y)=E(x)E(y)與E(x)>0。"},{"object_id":"mko-recurrence-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-analytic-ordinary-generating-function","position":4,"requirement":"optional","rationale_zh":"數列(a_n)在明示收斂域內以ordinary power series A(x)=sum_{n≥0}a_nx^n解析編碼。"},{"object_id":"mko-euler-theorem-number-theory","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quadratic-residue","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-legendre-symbol","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-euler-criterion-quadratic-residue","position":5,"requirement":"optional","rationale_zh":"若p為奇質數且p不整除a，則a^((p-1)/2)模p等於Legendre符號(a/p)的模p代表。"},{"object_id":"mko-computational-model","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-decision-problem","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-automaton","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-algorithm-correctness","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-turing-machine-decision-model","position":6,"requirement":"optional","rationale_zh":"固定deterministic single-tape Turing機語義；decider必須對每個合法輸入停機並給正確accept/reject答案。"},{"object_id":"mko-partial-derivative","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-vector","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-multivariable-real-function","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-gradient-vector","position":7,"requirement":"optional","rationale_zh":"在固定standard Euclidean coordinates中，可微實值函數的一階偏導依座標排列成梯度向量。"},{"object_id":"mko-finite-directed-graph","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-programming","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-flow-network","position":8,"requirement":"optional","rationale_zh":"有限有向網路中的非負流須逐邊不超過capacity，並在source與sink以外每個頂點滿足流量守恆。"}],"edges":[{"from":"mko-finite-group","to":"mko-finite-normal-subgroup","relation":"required_before"},{"from":"mko-subgroup-homomorphism","to":"mko-finite-normal-subgroup","relation":"required_before"},{"from":"mko-quantifier","to":"mko-finite-normal-subgroup","relation":"required_before"},{"from":"mko-infinite-series","to":"mko-real-exponential-function","relation":"required_before"},{"from":"mko-real-number","to":"mko-real-exponential-function","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-real-exponential-function","relation":"required_before"},{"from":"mko-sequence-limit","to":"mko-real-exponential-function","relation":"required_before"},{"from":"mko-power-series-radius-of-convergence","to":"mko-real-exponential-function","relation":"required_before"},{"from":"mko-factorial","to":"mko-real-exponential-function","relation":"required_before"},{"from":"mko-absolute-convergence-of-series","to":"mko-real-exponential-function","relation":"required_before"},{"from":"mko-binomial-theorem","to":"mko-real-exponential-function","relation":"required_before"},{"from":"mko-recurrence-relation","to":"mko-analytic-ordinary-generating-function","relation":"required_before"},{"from":"mko-power-series-radius-of-convergence","to":"mko-analytic-ordinary-generating-function","relation":"required_before"},{"from":"mko-sequence","to":"mko-analytic-ordinary-generating-function","relation":"required_before"},{"from":"mko-euler-theorem-number-theory","to":"mko-euler-criterion-quadratic-residue","relation":"required_before"},{"from":"mko-quadratic-residue","to":"mko-euler-criterion-quadratic-residue","relation":"required_before"},{"from":"mko-legendre-symbol","to":"mko-euler-criterion-quadratic-residue","relation":"required_before"},{"from":"mko-computational-model","to":"mko-turing-machine-decision-model","relation":"required_before"},{"from":"mko-decision-problem","to":"mko-turing-machine-decision-model","relation":"required_before"},{"from":"mko-finite-automaton","to":"mko-turing-machine-decision-model","relation":"required_before"},{"from":"mko-algorithm-correctness","to":"mko-turing-machine-decision-model","relation":"required_before"},{"from":"mko-partial-derivative","to":"mko-gradient-vector","relation":"required_before"},{"from":"mko-vector","to":"mko-gradient-vector","relation":"required_before"},{"from":"mko-multivariable-real-function","to":"mko-gradient-vector","relation":"required_before"},{"from":"mko-finite-directed-graph","to":"mko-finite-flow-network","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-finite-flow-network","relation":"required_before"},{"from":"mko-linear-programming","to":"mko-finite-flow-network","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261114","title_zh":"Wave6第八層：kernel、商群、共軛類、標準常態、MSE、LP標準形與頂點覆蓋","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-subgroup-homomorphism","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-normal-subgroup","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-group-homomorphism-kernel","position":2,"requirement":"optional","rationale_zh":"有限群同態f:G→Q映到目標identity的完整原像是G的正規子群，且同態fibers恰為kernel的陪集。"},{"object_id":"mko-quotient-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-group","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-quotient-group","position":3,"requirement":"optional","rationale_zh":"有限群G依正規子群N的陪集形成商群，元素是cosets且乘法(gN)(hN)=(gh)N必須與代表元無關。"},{"object_id":"mko-finite-group-action","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-group-orbit","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-group-conjugacy-class","position":4,"requirement":"optional","rationale_zh":"有限群G以conjugation g·x=gxg^{-1}作用於自身，其orbits形成共軛類並分割G。"},{"object_id":"mko-continuous-real-distribution-function","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-improper-integral","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-exponential-function","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-power-root","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-expectation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-variance","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-standard-normal-distribution","position":5,"requirement":"optional","rationale_zh":"標準常態Z以Gaussian density phi(x)=(2pi)^(-1/2)exp(-x^2/2)與其連續CDF Phi定義，並具有mean 0、variance 1。"},{"object_id":"mko-estimator-bias","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-estimator-variance","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sampling-distribution","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-parametric-statistical-model","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-estimator-mean-squared-error","position":6,"requirement":"optional","rationale_zh":"scalar parameter theta下，估計量T的MSE_theta(T)=E_theta[(T-theta)^2]由完整sampling distribution逐theta定義。"},{"object_id":"mko-linear-programming","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-matrix","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-system-linear-equations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-program-standard-form","position":7,"requirement":"optional","rationale_zh":"本節點固定唯一primal convention：maximize c^T x，subject to Ax≤b與x≥0。"},{"object_id":"mko-graph","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subset","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-vertex-cover","position":8,"requirement":"optional","rationale_zh":"有限simple undirected graph的頂點子集C若與每條edge至少一端相交，則C為vertex cover。"}],"edges":[{"from":"mko-subgroup-homomorphism","to":"mko-finite-group-homomorphism-kernel","relation":"required_before"},{"from":"mko-finite-normal-subgroup","to":"mko-finite-group-homomorphism-kernel","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-finite-group-homomorphism-kernel","relation":"required_before"},{"from":"mko-finite-normal-subgroup","to":"mko-finite-quotient-group","relation":"required_before"},{"from":"mko-quotient-set","to":"mko-finite-quotient-group","relation":"required_before"},{"from":"mko-finite-group","to":"mko-finite-quotient-group","relation":"required_before"},{"from":"mko-finite-group-action","to":"mko-finite-group-conjugacy-class","relation":"required_before"},{"from":"mko-finite-group-orbit","to":"mko-finite-group-conjugacy-class","relation":"required_before"},{"from":"mko-finite-group","to":"mko-finite-group-conjugacy-class","relation":"required_before"},{"from":"mko-continuous-real-distribution-function","to":"mko-standard-normal-distribution","relation":"required_before"},{"from":"mko-improper-integral","to":"mko-standard-normal-distribution","relation":"required_before"},{"from":"mko-real-exponential-function","to":"mko-standard-normal-distribution","relation":"required_before"},{"from":"mko-power-root","to":"mko-standard-normal-distribution","relation":"required_before"},{"from":"mko-expectation","to":"mko-standard-normal-distribution","relation":"required_before"},{"from":"mko-variance","to":"mko-standard-normal-distribution","relation":"required_before"},{"from":"mko-estimator-bias","to":"mko-estimator-mean-squared-error","relation":"required_before"},{"from":"mko-estimator-variance","to":"mko-estimator-mean-squared-error","relation":"required_before"},{"from":"mko-expectation","to":"mko-estimator-mean-squared-error","relation":"required_before"},{"from":"mko-sampling-distribution","to":"mko-estimator-mean-squared-error","relation":"required_before"},{"from":"mko-parametric-statistical-model","to":"mko-estimator-mean-squared-error","relation":"required_before"},{"from":"mko-linear-programming","to":"mko-linear-program-standard-form","relation":"required_before"},{"from":"mko-matrix","to":"mko-linear-program-standard-form","relation":"required_before"},{"from":"mko-system-linear-equations","to":"mko-linear-program-standard-form","relation":"required_before"},{"from":"mko-order-relation","to":"mko-linear-program-standard-form","relation":"required_before"},{"from":"mko-graph","to":"mko-vertex-cover","relation":"required_before"},{"from":"mko-subset","to":"mko-vertex-cover","relation":"required_before"},{"from":"mko-quantifier","to":"mko-vertex-cover","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261115","title_zh":"Wave6第九層：第一同構、群中心、bias–variance、LP對偶、max-flow/min-cut、最小頂點覆蓋與many-one reduction","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-finite-group-homomorphism-kernel","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-quotient-group","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subgroup-homomorphism","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-bijective","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-first-isomorphism-theorem-finite-groups","position":2,"requirement":"optional","rationale_zh":"有限群同態f:G→Q由kernel cosets誘導出G/ker f到im f的群同構，而非自動同構到整個codomain。"},{"object_id":"mko-finite-group","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-group-conjugacy-class","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-normal-subgroup","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-group-center","position":3,"requirement":"optional","rationale_zh":"有限群G的中心Z(G)由與每個群元素交換的全部元素構成，且是G的正規子群。"},{"object_id":"mko-estimator-mean-squared-error","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-estimator-bias","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-estimator-variance","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-bias-variance-decomposition","position":4,"requirement":"optional","rationale_zh":"逐參數theta，scalar estimator的MSE等於其variance加上bias平方。"},{"object_id":"mko-linear-program-standard-form","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-optimization-problem","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-program-duality","position":5,"requirement":"optional","rationale_zh":"對固定primal max c^Tx、Ax≤b、x≥0，其dual為min b^Ty、A^Ty≥c、y≥0；feasible pair滿足弱對偶。"},{"object_id":"mko-finite-flow-network","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-programming","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-max-flow-min-cut-theorem","position":6,"requirement":"optional","rationale_zh":"有限容量網路的最大feasible flow value等於全部s–t cuts中的最小capacity。"},{"object_id":"mko-vertex-cover","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-cardinality-comparison","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-minimum-vertex-cover","position":7,"requirement":"optional","rationale_zh":"有限simple graph中，cardinality在全部vertex covers裡最小的候選是minimum vertex cover。"},{"object_id":"mko-decision-problem","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-composition","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-turing-machine-decision-model","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-many-one-reduction","position":8,"requirement":"optional","rationale_zh":"A≤mB表示存在total computable f，使每個x皆滿足x∈A恰當f(x)∈B。"}],"edges":[{"from":"mko-finite-group-homomorphism-kernel","to":"mko-first-isomorphism-theorem-finite-groups","relation":"required_before"},{"from":"mko-finite-quotient-group","to":"mko-first-isomorphism-theorem-finite-groups","relation":"required_before"},{"from":"mko-subgroup-homomorphism","to":"mko-first-isomorphism-theorem-finite-groups","relation":"required_before"},{"from":"mko-bijective","to":"mko-first-isomorphism-theorem-finite-groups","relation":"required_before"},{"from":"mko-finite-group","to":"mko-finite-group-center","relation":"required_before"},{"from":"mko-finite-group-conjugacy-class","to":"mko-finite-group-center","relation":"required_before"},{"from":"mko-finite-normal-subgroup","to":"mko-finite-group-center","relation":"required_before"},{"from":"mko-quantifier","to":"mko-finite-group-center","relation":"required_before"},{"from":"mko-estimator-mean-squared-error","to":"mko-bias-variance-decomposition","relation":"required_before"},{"from":"mko-estimator-bias","to":"mko-bias-variance-decomposition","relation":"required_before"},{"from":"mko-estimator-variance","to":"mko-bias-variance-decomposition","relation":"required_before"},{"from":"mko-linear-program-standard-form","to":"mko-linear-program-duality","relation":"required_before"},{"from":"mko-optimization-problem","to":"mko-linear-program-duality","relation":"required_before"},{"from":"mko-order-relation","to":"mko-linear-program-duality","relation":"required_before"},{"from":"mko-finite-flow-network","to":"mko-max-flow-min-cut-theorem","relation":"required_before"},{"from":"mko-optimization-problem","to":"mko-max-flow-min-cut-theorem","relation":"required_before"},{"from":"mko-linear-programming","to":"mko-max-flow-min-cut-theorem","relation":"required_before"},{"from":"mko-vertex-cover","to":"mko-minimum-vertex-cover","relation":"required_before"},{"from":"mko-optimization-problem","to":"mko-minimum-vertex-cover","relation":"required_before"},{"from":"mko-cardinality-comparison","to":"mko-minimum-vertex-cover","relation":"required_before"},{"from":"mko-decision-problem","to":"mko-many-one-reduction","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-many-one-reduction","relation":"required_before"},{"from":"mko-composition","to":"mko-many-one-reduction","relation":"required_before"},{"from":"mko-turing-machine-decision-model","to":"mko-many-one-reduction","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261116","title_zh":"Wave6第十層：contraction、IVT、Taylor remainder、finite MLE、sufficiency、toposort與multiplicative order","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-orbit-fixed-periodic","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-complete-metric-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-metric-convergence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-infinite-series","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-contraction-mapping-theorem","position":2,"requirement":"optional","rationale_zh":"非空完備度量空間上的self-map若以固定q<1壓縮所有距離，則有唯一fixed point且Picard orbit收斂。"},{"object_id":"mko-continuity","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-supremum-and-infimum","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-intermediate-value-theorem","position":3,"requirement":"optional","rationale_zh":"連續實函數在閉區間端點值之間取到每個中間值。"},{"object_id":"mko-taylor-polynomial","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-higher-order-derivative","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-mean-value-theorem","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-absolute-value","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-taylor-remainder-theorem","position":4,"requirement":"optional","rationale_zh":"足夠高階可微函數的Taylor approximation error由Lagrange remainder與derivative bound控制。"},{"object_id":"mko-parametric-statistical-model","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-statistical-estimator","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-optimization-problem","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-maximum-likelihood-estimator","position":5,"requirement":"optional","rationale_zh":"有限參數模型中，MLE選取使觀察likelihood最大的參數，tie需固定且與theta無關。"},{"object_id":"mko-sampling-distribution","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-conditional-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-sufficient-statistic","position":6,"requirement":"optional","rationale_zh":"finite model中，給定statistic value後sample conditional law不依theta時，T對theta充分。"},{"object_id":"mko-finite-directed-graph","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-algorithm-correctness","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-mathematical-induction","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-topological-ordering-dag","position":7,"requirement":"optional","rationale_zh":"有限有向圖存在topological ordering恰當其無directed cycle。"},{"object_id":"mko-modular-inverse","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-gcd-coprime","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-natural-divisibility","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-euler-theorem-number-theory","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-multiplicative-order-mod-n","position":8,"requirement":"optional","rationale_zh":"gcd(a,n)=1時，ord_n(a)是a^k≡1 mod n的最小正整數k。"}],"edges":[{"from":"mko-orbit-fixed-periodic","to":"mko-contraction-mapping-theorem","relation":"required_before"},{"from":"mko-complete-metric-space","to":"mko-contraction-mapping-theorem","relation":"required_before"},{"from":"mko-metric-convergence","to":"mko-contraction-mapping-theorem","relation":"required_before"},{"from":"mko-infinite-series","to":"mko-contraction-mapping-theorem","relation":"required_before"},{"from":"mko-continuity","to":"mko-intermediate-value-theorem","relation":"required_before"},{"from":"mko-real-number","to":"mko-intermediate-value-theorem","relation":"required_before"},{"from":"mko-order-relation","to":"mko-intermediate-value-theorem","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-intermediate-value-theorem","relation":"required_before"},{"from":"mko-supremum-and-infimum","to":"mko-intermediate-value-theorem","relation":"required_before"},{"from":"mko-taylor-polynomial","to":"mko-taylor-remainder-theorem","relation":"required_before"},{"from":"mko-higher-order-derivative","to":"mko-taylor-remainder-theorem","relation":"required_before"},{"from":"mko-mean-value-theorem","to":"mko-taylor-remainder-theorem","relation":"required_before"},{"from":"mko-absolute-value","to":"mko-taylor-remainder-theorem","relation":"required_before"},{"from":"mko-parametric-statistical-model","to":"mko-finite-maximum-likelihood-estimator","relation":"required_before"},{"from":"mko-statistical-estimator","to":"mko-finite-maximum-likelihood-estimator","relation":"required_before"},{"from":"mko-optimization-problem","to":"mko-finite-maximum-likelihood-estimator","relation":"required_before"},{"from":"mko-parametric-statistical-model","to":"mko-finite-sufficient-statistic","relation":"required_before"},{"from":"mko-statistical-estimator","to":"mko-finite-sufficient-statistic","relation":"required_before"},{"from":"mko-sampling-distribution","to":"mko-finite-sufficient-statistic","relation":"required_before"},{"from":"mko-conditional-probability","to":"mko-finite-sufficient-statistic","relation":"required_before"},{"from":"mko-finite-directed-graph","to":"mko-topological-ordering-dag","relation":"required_before"},{"from":"mko-order-relation","to":"mko-topological-ordering-dag","relation":"required_before"},{"from":"mko-algorithm-correctness","to":"mko-topological-ordering-dag","relation":"required_before"},{"from":"mko-mathematical-induction","to":"mko-topological-ordering-dag","relation":"required_before"},{"from":"mko-modular-inverse","to":"mko-multiplicative-order-mod-n","relation":"required_before"},{"from":"mko-gcd-coprime","to":"mko-multiplicative-order-mod-n","relation":"required_before"},{"from":"mko-natural-divisibility","to":"mko-multiplicative-order-mod-n","relation":"required_before"},{"from":"mko-euler-theorem-number-theory","to":"mko-multiplicative-order-mod-n","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261117","title_zh":"Wave6第十一層：series tests、uniform limits、product／quotient topology、path connectivity與components","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-infinite-series","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence-limit","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-monotone-sequence-convergence-theorem","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-series-comparison-test","position":2,"requirement":"optional","rationale_zh":"非負項級數可由逐項或最終逐項上界與下界傳遞收斂或發散。"},{"object_id":"mko-absolute-convergence-of-series","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-absolute-value","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-ratio-test-series","position":3,"requirement":"optional","rationale_zh":"非零項絕對值相鄰比極限小於一時絕對收斂，大於一時發散，等於一時不判定。"},{"object_id":"mko-uniform-convergence-of-functions","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-continuity","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-supremum-and-infimum","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-uniform-limit-continuity-theorem","position":4,"requirement":"optional","rationale_zh":"同一domain上的連續函數列若一致收斂至f，則極限函數f連續。"},{"object_id":"mko-topological-basis","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-topological-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-topological-continuity","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-composition","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-product-topology","position":5,"requirement":"optional","rationale_zh":"有限多個拓撲空間的Cartesian product由open rectangles生成乘積拓撲。"},{"object_id":"mko-quotient-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-surjective","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quotient-topology","position":6,"requirement":"optional","rationale_zh":"滿射q:X→Y使U⊆Y恰在q逆像開時為開，從而賦予Y商拓撲。"},{"object_id":"mko-connected-topological-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-intermediate-value-theorem","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-path-connected-space","position":7,"requirement":"optional","rationale_zh":"空間中任意兩點若可由單位區間的連續映射連接，則空間道路連通。"},{"object_id":"mko-equivalence-class","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subset","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-closure-and-interior","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-connected-component","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-subspace-topology","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-topological-connected-component","position":8,"requirement":"optional","rationale_zh":"指定點的連通分支是包含該點的極大連通子集；所有分支分割空間且各自閉。"}],"edges":[{"from":"mko-infinite-series","to":"mko-series-comparison-test","relation":"required_before"},{"from":"mko-order-relation","to":"mko-series-comparison-test","relation":"required_before"},{"from":"mko-sequence-limit","to":"mko-series-comparison-test","relation":"required_before"},{"from":"mko-monotone-sequence-convergence-theorem","to":"mko-series-comparison-test","relation":"required_before"},{"from":"mko-absolute-convergence-of-series","to":"mko-ratio-test-series","relation":"required_before"},{"from":"mko-sequence-limit","to":"mko-ratio-test-series","relation":"required_before"},{"from":"mko-absolute-value","to":"mko-ratio-test-series","relation":"required_before"},{"from":"mko-infinite-series","to":"mko-ratio-test-series","relation":"required_before"},{"from":"mko-series-comparison-test","to":"mko-ratio-test-series","relation":"required_before"},{"from":"mko-uniform-convergence-of-functions","to":"mko-uniform-limit-continuity-theorem","relation":"required_before"},{"from":"mko-continuity","to":"mko-uniform-limit-continuity-theorem","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-uniform-limit-continuity-theorem","relation":"required_before"},{"from":"mko-supremum-and-infimum","to":"mko-uniform-limit-continuity-theorem","relation":"required_before"},{"from":"mko-topological-basis","to":"mko-finite-product-topology","relation":"required_before"},{"from":"mko-topological-space","to":"mko-finite-product-topology","relation":"required_before"},{"from":"mko-set-operations","to":"mko-finite-product-topology","relation":"required_before"},{"from":"mko-relation","to":"mko-finite-product-topology","relation":"required_before"},{"from":"mko-topological-continuity","to":"mko-finite-product-topology","relation":"required_before"},{"from":"mko-composition","to":"mko-finite-product-topology","relation":"required_before"},{"from":"mko-topological-space","to":"mko-quotient-topology","relation":"required_before"},{"from":"mko-quotient-set","to":"mko-quotient-topology","relation":"required_before"},{"from":"mko-surjective","to":"mko-quotient-topology","relation":"required_before"},{"from":"mko-topological-continuity","to":"mko-quotient-topology","relation":"required_before"},{"from":"mko-composition","to":"mko-quotient-topology","relation":"required_before"},{"from":"mko-topological-continuity","to":"mko-path-connected-space","relation":"required_before"},{"from":"mko-connected-topological-space","to":"mko-path-connected-space","relation":"required_before"},{"from":"mko-real-number","to":"mko-path-connected-space","relation":"required_before"},{"from":"mko-order-relation","to":"mko-path-connected-space","relation":"required_before"},{"from":"mko-finite-product-topology","to":"mko-path-connected-space","relation":"required_before"},{"from":"mko-intermediate-value-theorem","to":"mko-path-connected-space","relation":"required_before"},{"from":"mko-composition","to":"mko-path-connected-space","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-path-connected-space","relation":"required_before"},{"from":"mko-connected-topological-space","to":"mko-topological-connected-component","relation":"required_before"},{"from":"mko-equivalence-class","to":"mko-topological-connected-component","relation":"required_before"},{"from":"mko-subset","to":"mko-topological-connected-component","relation":"required_before"},{"from":"mko-closure-and-interior","to":"mko-topological-connected-component","relation":"required_before"},{"from":"mko-connected-component","to":"mko-topological-connected-component","relation":"required_before"},{"from":"mko-subspace-topology","to":"mko-topological-connected-component","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261118","title_zh":"Wave6第十二層：CLT、SLLN、confidence coverage、Wilson、p-adic valuation與regular languages","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-iid-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-convergence-in-distribution","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-expectation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-variance","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-standard-normal-distribution","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sigma-algebra-measure-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-binomial-theorem","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-central-limit-theorem-finite-variance-iid","position":2,"requirement":"optional","rationale_zh":"共同law且0<σ²<∞的無限IID序列，其正確中心化與sqrt(n)標準化和依分布收斂至標準常態。"},{"object_id":"mko-almost-sure-convergence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-strong-law-large-numbers-finite-valued-iid","position":3,"requirement":"optional","rationale_zh":"有限值無限IID序列的樣本平均幾乎必然收斂至共同期望。"},{"object_id":"mko-sampling-distribution","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-statistical-estimator","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-parametric-statistical-model","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-probability","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-finite-confidence-interval-coverage","position":4,"requirement":"optional","rationale_zh":"有限模型中預先固定的interval-valued procedure逐參數覆蓋真值的機率，定義其frequentist coverage。"},{"object_id":"mko-prime-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-modular-congruence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-factorial","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-modular-inverse","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-wilson-theorem","position":5,"requirement":"optional","rationale_zh":"整數n>1為質數，恰當(n-1)!模n同餘於-1。"},{"object_id":"mko-natural-divisibility","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-unique-prime-factorization","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-integer","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-absolute-value","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-p-adic-valuation-integers","position":6,"requirement":"optional","rationale_zh":"對質數p與非零整數a，v_p(a)是p整除|a|的最高次數；本節固定v_p(0)=infinity。"},{"object_id":"mko-regular-language","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-mathematical-induction","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-regular-expression","position":7,"requirement":"optional","rationale_zh":"正規表示式由empty language、epsilon、alphabet symbols、union、concatenation與Kleene star遞迴建構語言。"},{"object_id":"mko-finite-automaton","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-proof","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-pigeonhole-principle","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-quantifier","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-pumping-lemma-regular-languages","position":8,"requirement":"optional","rationale_zh":"每個正規語言都有p，使每個長度至少p的成員word存在可反覆pumping且仍留在語言的分解。"}],"edges":[{"from":"mko-iid-sequence","to":"mko-central-limit-theorem-finite-variance-iid","relation":"required_before"},{"from":"mko-convergence-in-distribution","to":"mko-central-limit-theorem-finite-variance-iid","relation":"required_before"},{"from":"mko-expectation","to":"mko-central-limit-theorem-finite-variance-iid","relation":"required_before"},{"from":"mko-variance","to":"mko-central-limit-theorem-finite-variance-iid","relation":"required_before"},{"from":"mko-standard-normal-distribution","to":"mko-central-limit-theorem-finite-variance-iid","relation":"required_before"},{"from":"mko-sigma-algebra-measure-space","to":"mko-central-limit-theorem-finite-variance-iid","relation":"required_before"},{"from":"mko-binomial-theorem","to":"mko-central-limit-theorem-finite-variance-iid","relation":"required_before"},{"from":"mko-iid-sequence","to":"mko-strong-law-large-numbers-finite-valued-iid","relation":"required_before"},{"from":"mko-almost-sure-convergence","to":"mko-strong-law-large-numbers-finite-valued-iid","relation":"required_before"},{"from":"mko-expectation","to":"mko-strong-law-large-numbers-finite-valued-iid","relation":"required_before"},{"from":"mko-sigma-algebra-measure-space","to":"mko-strong-law-large-numbers-finite-valued-iid","relation":"required_before"},{"from":"mko-variance","to":"mko-strong-law-large-numbers-finite-valued-iid","relation":"required_before"},{"from":"mko-sampling-distribution","to":"mko-finite-confidence-interval-coverage","relation":"required_before"},{"from":"mko-statistical-estimator","to":"mko-finite-confidence-interval-coverage","relation":"required_before"},{"from":"mko-parametric-statistical-model","to":"mko-finite-confidence-interval-coverage","relation":"required_before"},{"from":"mko-probability","to":"mko-finite-confidence-interval-coverage","relation":"required_before"},{"from":"mko-order-relation","to":"mko-finite-confidence-interval-coverage","relation":"required_before"},{"from":"mko-prime-number","to":"mko-wilson-theorem","relation":"required_before"},{"from":"mko-modular-congruence","to":"mko-wilson-theorem","relation":"required_before"},{"from":"mko-factorial","to":"mko-wilson-theorem","relation":"required_before"},{"from":"mko-modular-inverse","to":"mko-wilson-theorem","relation":"required_before"},{"from":"mko-prime-number","to":"mko-p-adic-valuation-integers","relation":"required_before"},{"from":"mko-natural-divisibility","to":"mko-p-adic-valuation-integers","relation":"required_before"},{"from":"mko-unique-prime-factorization","to":"mko-p-adic-valuation-integers","relation":"required_before"},{"from":"mko-integer","to":"mko-p-adic-valuation-integers","relation":"required_before"},{"from":"mko-absolute-value","to":"mko-p-adic-valuation-integers","relation":"required_before"},{"from":"mko-regular-language","to":"mko-regular-expression","relation":"required_before"},{"from":"mko-set","to":"mko-regular-expression","relation":"required_before"},{"from":"mko-sequence","to":"mko-regular-expression","relation":"required_before"},{"from":"mko-mathematical-induction","to":"mko-regular-expression","relation":"required_before"},{"from":"mko-regular-language","to":"mko-pumping-lemma-regular-languages","relation":"required_before"},{"from":"mko-finite-automaton","to":"mko-pumping-lemma-regular-languages","relation":"required_before"},{"from":"mko-proof","to":"mko-pumping-lemma-regular-languages","relation":"required_before"},{"from":"mko-pigeonhole-principle","to":"mko-pumping-lemma-regular-languages","relation":"required_before"},{"from":"mko-quantifier","to":"mko-pumping-lemma-regular-languages","relation":"required_before"},{"from":"mko-mathematical-induction","to":"mko-pumping-lemma-regular-languages","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}},
    {"schema_version":"ocme-learning-path-v0.1","id":"path-pulse-foundations-20261119","title_zh":"Wave6第十三層閉合：Newton、梯形積分、梯度下降、Lagrange與三輪核心密度","audience":["general_learner","upper_secondary"],"goal_zh":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","nodes":[{"object_id":"mko-derivative","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-numerical-approximation-error","position":2,"requirement":"optional","rationale_zh":"在既有誤差MKO上加厚affine equation的residual／forward error、誤差傳播、conditioning witness與tolerance邊界。"},{"object_id":"mko-algorithm-correctness","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-equation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-sequence","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-derivative-rules","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-newton-method","position":3,"requirement":"optional","rationale_zh":"在導數非零的目前點，以切線零點更新x至x-f(x)/f'(x)；收斂是另需局部假設的責任。"},{"object_id":"mko-definite-integral","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-order-relation","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-trapezoidal-quadrature","position":3,"requirement":"optional","rationale_zh":"在嚴格遞增partition上逐段以端點平均乘各自寬度，得到定積分的複合梯形近似。"},{"object_id":"mko-gradient-vector","position":2,"requirement":"optional","rationale_zh":"在既有standard Euclidean coordinate語義上加厚gradient線性、負梯度內積、symmetric quadratic差分與metric邊界。"},{"object_id":"mko-optimization-problem","position":2,"requirement":"optional","rationale_zh":"在完整有限可行域與鄰接關係上分開local/global最小、strict improvement、lower-bound certificate及空域邊界。"},{"object_id":"mko-real-vector-space","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-gradient-descent","position":3,"requirement":"optional","rationale_zh":"在固定Euclidean coordinates中以正步長沿負梯度更新；下降與收斂仍需目標及步長假設。"},{"object_id":"mko-system-linear-equations","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-matrix-rank","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-linear-independence-basis","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-lagrange-multiplier-condition","position":3,"requirement":"optional","rationale_zh":"正則等式約束局部極值點的目標梯度，必為constraint gradients的線性組合。"},{"object_id":"mko-real-number","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-absolute-value","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-ratio-proportion","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-partial-derivative","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-vector","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-multivariable-real-function","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-set","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"},{"object_id":"mko-function-mapping","position":1,"requirement":"required","rationale_zh":"本脈衝新物件的既有 Canonical 前置。"}],"edges":[{"from":"mko-derivative","to":"mko-newton-method","relation":"required_before"},{"from":"mko-numerical-approximation-error","to":"mko-newton-method","relation":"required_before"},{"from":"mko-algorithm-correctness","to":"mko-newton-method","relation":"required_before"},{"from":"mko-equation","to":"mko-newton-method","relation":"required_before"},{"from":"mko-sequence","to":"mko-newton-method","relation":"required_before"},{"from":"mko-derivative-rules","to":"mko-newton-method","relation":"required_before"},{"from":"mko-definite-integral","to":"mko-trapezoidal-quadrature","relation":"required_before"},{"from":"mko-numerical-approximation-error","to":"mko-trapezoidal-quadrature","relation":"required_before"},{"from":"mko-arithmetic-operations","to":"mko-trapezoidal-quadrature","relation":"required_before"},{"from":"mko-order-relation","to":"mko-trapezoidal-quadrature","relation":"required_before"},{"from":"mko-sequence","to":"mko-trapezoidal-quadrature","relation":"required_before"},{"from":"mko-gradient-vector","to":"mko-gradient-descent","relation":"required_before"},{"from":"mko-optimization-problem","to":"mko-gradient-descent","relation":"required_before"},{"from":"mko-sequence","to":"mko-gradient-descent","relation":"required_before"},{"from":"mko-real-vector-space","to":"mko-gradient-descent","relation":"required_before"},{"from":"mko-gradient-vector","to":"mko-lagrange-multiplier-condition","relation":"required_before"},{"from":"mko-optimization-problem","to":"mko-lagrange-multiplier-condition","relation":"required_before"},{"from":"mko-system-linear-equations","to":"mko-lagrange-multiplier-condition","relation":"required_before"},{"from":"mko-matrix-rank","to":"mko-lagrange-multiplier-condition","relation":"required_before"},{"from":"mko-linear-independence-basis","to":"mko-lagrange-multiplier-condition","relation":"required_before"},{"from":"mko-real-number","to":"mko-numerical-approximation-error","relation":"required_before"},{"from":"mko-absolute-value","to":"mko-numerical-approximation-error","relation":"required_before"},{"from":"mko-ratio-proportion","to":"mko-numerical-approximation-error","relation":"required_before"},{"from":"mko-partial-derivative","to":"mko-gradient-vector","relation":"required_before"},{"from":"mko-vector","to":"mko-gradient-vector","relation":"required_before"},{"from":"mko-multivariable-real-function","to":"mko-gradient-vector","relation":"required_before"},{"from":"mko-set","to":"mko-optimization-problem","relation":"required_before"},{"from":"mko-function-mapping","to":"mko-optimization-problem","relation":"required_before"},{"from":"mko-order-relation","to":"mko-optimization-problem","relation":"required_before"}],"review":{"status":"candidate","reviewer":"Codex local AI","version":1}}
  ]
}