{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-path-connected-space-01",
  "mko_id": "mko-path-connected-space",
  "title": {
    "zh-Hant": "凸盒中的全稱路徑與拼接",
    "en": "Universal paths and concatenation in a convex box"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-path-connected-space-01",
    "source": {
      "mko_id": "mko-path-connected-space",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "6df47a43396e20219abd3fe5ab3d088f1aaf4a1022bc9628ef0fe164b6a45e17",
      "english_sha256": "bb59736c31f281af474355507a3b5172c4fd2b948847c2332ec737e88490c538"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "sets",
      "algebra",
      "polynomial",
      "pasting",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "證 [0,1]^2 路徑連通，並核兩段路徑的接點。",
      "en": "Prove [0,1]^2 path connected and audit a two-piece concatenation."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "B=[0,1]^2；任意 x=(x_1,x_2),y=(y_1,y_2)∈B，路徑定義域為通常拓撲的 [0,1]。",
          "en": "B=[0,1]^2; x=(x_1,x_2), y=(y_1,y_2) are arbitrary in B, and the path domain is [0,1] with its usual topology."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "0\\le t\\le1",
        "after_tex": "\\gamma(t)=(1-t)x+ty",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "對任意端點明示一條完整函數，而非有限點列。",
          "en": "Specify a whole function for arbitrary endpoints, not a finite point list."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "每個 t 都有同型兩座標值。",
            "en": "Every t has a two-coordinate value of the required type."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "0\\le x_i,y_i,t\\le1",
        "after_tex": "0\\le(1-t)x_i+ty_i\\le(1-t)+t=1\\quad(i\\in\\{1,2\\})",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [
          "t\\in[0,1]"
        ],
        "explanation": {
          "zh-Hant": "非負權重和為一，兩個座標都留在盒中。",
          "en": "The nonnegative weights sum to one, keeping both coordinates in the box."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "這是全部 t 的範圍證書，不是抽樣。",
            "en": "This certifies the range for every t, not sampled values."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S03",
        "uses": [
          "S01"
        ],
        "before_tex": "\\gamma(0),\\gamma(1)",
        "after_tex": "\\gamma(0)=x,\\quad\\gamma(1)=y",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "代入端點得指定起終點。",
          "en": "Endpoint substitution gives the prescribed start and end."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "S01"
          ],
          "explanation": {
            "zh-Hant": "端點正確不取代連續與全範圍責任。",
            "en": "Correct endpoints do not replace continuity and full-range containment."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S04",
        "uses": [
          "S01",
          "S02"
        ],
        "before_tex": "\\gamma_i(t)=(1-t)x_i+ty_i",
        "after_tex": "\\gamma:[0,1]\\to B\\text{ continuous}",
        "relation": "implies",
        "rule": "polynomial",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每座標為仿射多項式，連續；Euclidean 距離下可由差值範數 |t-s|‖y-x‖直接核。",
          "en": "Each coordinate is affine and continuous; in Euclidean distance the difference norm is |t-s| times the norm of y-x."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "若 x=y 是常值映射，連續仍成立。",
            "en": "If x=y the map is constant and remains continuous."
          }
        },
        "leaf_basis": "polynomial"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S02",
          "S03",
          "S04"
        ],
        "before_tex": "x,y\\in B\\text{ arbitrary}",
        "after_tex": "B\\text{ is path connected}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每對點均有已核端點、連續性和全範圍的路徑，完成全稱定義。",
          "en": "Every pair has a path with certified endpoints, continuity and whole range, meeting the universal definition."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S03",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "任意 x,y 的前提不能由一對成功取代。",
            "en": "Arbitrary x,y cannot be replaced by one successful pair."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S06",
        "uses": [
          "G1"
        ],
        "before_tex": "\\alpha(1)=\\beta(0),\\quad\\eta(t)=\\begin{cases}\\alpha(2t)&t\\le1/2\\\\\\beta(2t-1)&t\\ge1/2\\end{cases}",
        "after_tex": "\\eta(1/2)=\\alpha(1)=\\beta(0)",
        "relation": "implies",
        "rule": "pasting",
        "conditions": [
          "\\alpha,\\beta:[0,1]\\to B\\text{ continuous},\\ \\alpha(1)=\\beta(0)"
        ],
        "explanation": {
          "zh-Hant": "兩條在 B 內的連續路徑若接點相同，可用重參數化拼接。",
          "en": "Continuous B-valued paths with a common join can be concatenated by reparametrization."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "此為另加條件的拼接，必須核共有接點。",
            "en": "This is a conditional concatenation and requires the shared join."
          }
        },
        "leaf_basis": "pasting"
      },
      {
        "id": "S07",
        "uses": [
          "S06"
        ],
        "before_tex": "[0,1]=[0,1/2]\\cup[1/2,1]",
        "after_tex": "\\eta\\text{ continuous},\\quad\\eta(0)=\\alpha(0),\\ \\eta(1)=\\beta(1)",
        "relation": "theorem_application",
        "rule": "pasting",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩閉段覆蓋定義域，接點相合，套有限閉拼接引理，範圍仍在 B。",
          "en": "The two closed pieces cover the domain and agree at their join, so finite closed pasting gives continuity and B-valued range."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S06"
          ],
          "explanation": {
            "zh-Hant": "沒有忽略 t=1/2 的一致性。",
            "en": "Agreement at t=1/2 has not been omitted."
          }
        },
        "leaf_basis": "pasting"
      },
      {
        "id": "S08",
        "uses": [
          "S05"
        ],
        "before_tex": "B=U\\cup V\\text{ a hypothetical nonempty open separation}",
        "after_tex": "\\gamma^{-1}(U),\\gamma^{-1}(V)\\text{ would separate }[0,1]\\text{, contradiction}",
        "relation": "implies",
        "rule": "pasting",
        "conditions": [],
        "explanation": {
          "zh-Hant": "取分離兩側各一點，路徑逆像給非空不交開覆蓋，與區間連通矛盾，所以 B 亦連通。",
          "en": "Choose one endpoint in each side; path preimages give a nonempty disjoint open cover, contradicting connectedness of the interval. Thus B is connected too."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S05"
          ],
          "explanation": {
            "zh-Hant": "只證路徑連通推出連通，未聲稱一般逆命題。",
            "en": "This proves only path connected implies connected, not the general converse."
          }
        },
        "leaf_basis": "pasting"
      }
    ],
    "result": {
      "zh-Hant": "盒 B 路徑連通且連通；拼接需接點相合。",
      "en": "The box B is path connected and connected; concatenation requires matching joins."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "端點符合指定起終點，就自動是連續路徑。",
          "en": "Correct endpoints automatically give a continuous path."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "跳躍函數可有正確端點但不連續。",
          "en": "A jump can have the correct endpoints without continuity."
        }
      },
      {
        "claim": {
          "zh-Hant": "R\\{0} 可由 -1 路徑走到 1。",
          "en": "A path in R minus {0} can join -1 to 1."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "IVT 迫使路徑取值 0，違反範圍。",
          "en": "IVT forces the path to take 0, violating its range."
        }
      },
      {
        "claim": {
          "zh-Hant": "任意兩條路徑不核接點即可拼接。",
          "en": "Any two paths concatenate without a join check."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "左右極限若不同，拼接在中點不連續。",
          "en": "Different one-sided values make the concatenation discontinuous at the midpoint."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "path是continuous gamma:[0,1]→X且gamma(0)=x、gamma(1)=y。exact families以實interval與finite-dimensional box的linear path、兩條path的piecewise concatenation核endpoint、range與接點；disconnected gap與兩點discrete space用connected-image責任排除。只給有限sample points、只對某一對端點有path或忽略continuity都不證path connected。path connected推出connected，但一般converse不成立。",
          "en": "A path is a continuous gamma:[0,1]→X with gamma(0)=x and gamma(1)=y. Exact families check linear paths in real intervals and finite-dimensional boxes, together with piecewise concatenation, including endpoints, full range containment and the shared join. Connected-image or IVT reasoning excludes a path across a disconnected gap or between the two points of a discrete space. Finite samples, continuity-free endpoint data or a path for one pair do not establish the universal property. Path connectedness implies connectedness, while the general converse fails."
        },
        "source": {
          "mko_id": "mko-path-connected-space",
          "sha256": "6df47a43396e20219abd3fe5ab3d088f1aaf4a1022bc9628ef0fe164b6a45e17",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "sets",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "集合的聯、交、補集、包含與函數逆像以逐元素定義計算。",
          "en": "Compute unions, intersections, complements, inclusion and preimages element by element."
        }
      },
      {
        "id": "algebra",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實數域的等式運算；除法前分母必非零，不等式乘正數保持方向。",
          "en": "Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities."
        }
      },
      {
        "id": "polynomial",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "多項式在實數上連續且可逐項微分；d(x^k)/dx=k x^(k-1)，常數導數為零。",
          "en": "Real polynomials are continuous and differentiate termwise: the derivative of x^k is k*x^(k-1), and constants differentiate to zero."
        }
      },
      {
        "id": "pasting",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限閉覆蓋上連續且在交集相合的映射可拼接為連續映射；[0,1] 在通常拓撲連通。",
          "en": "Continuous maps agreeing on overlaps of a finite closed cover paste continuously; [0,1] is connected in its usual topology."
        }
      },
      {
        "id": "definition_here",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "路徑是 [0,1] 上連續、端點正確且全範圍在空間內的映射；每對點皆有路徑才路徑連通。",
          "en": "A path is a continuous map from [0,1] with the specified endpoints and entire range in the space; path connectedness requires one for every pair."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "通常實拓撲與明示線性路徑；不把連通一般反推路徑連通。",
      "en": "Usual real topologies and explicit affine paths; no general connectedness converse."
    },
    "md_symbols": [
      [
        "\\gamma",
        "仿射路徑",
        "affine path"
      ],
      [
        "\\eta",
        "拼接路徑",
        "concatenated path"
      ]
    ],
    "md_error_controls": [
      {
        "title": "全參數範圍 / All-parameter range",
        "zh": "用凸組合不等式核全部 t。",
        "en": "Use convex-combination inequalities for every t."
      },
      {
        "title": "三責任 / Three obligations",
        "zh": "端點、連續性與範圍須分別成立。",
        "en": "Endpoints, continuity and range must each hold."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "collection": "coverage-20261021",
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    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
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    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite.",
    "english_binding_revision": {
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      "path": "content/edition-symbol-bindings-20261101/revision.json",
      "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c"
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    "source_quotation_revision": {
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}