{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-binomial-theorem-01",
  "mko_id": "mko-binomial-theorem",
  "title": {
    "zh-Hant": "選取y因子的完整二項展開",
    "en": "A full binomial expansion by choosing y factors"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-binomial-theorem-01",
    "source": {
      "mko_id": "mko-binomial-theorem",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "c8fefe3783fc4f7633b4b22b8cf149d96a683900f181e3b42c690409ce90e5b7",
      "english_sha256": "1086dfe7c31f4e6bf8934681a0a080b50c14cb828f1a8b17e59143ec2cf06c83"
    },
    "mode": "theorem_proof",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "證一般係數責任並完整展開n=4。",
      "en": "Establish coefficient counting and expand n=4 completely."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "n∈N0，x、y位於commutative real polynomial ring；實例n=4。",
          "en": "n belongs to N0 and x,y are in a commutative real polynomial ring; the instance is n=4."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "(x+y)^n=\\prod_{i=1}^{n}(x+y)",
        "after_tex": "\\sum_{S\\subseteq\\{i\\in\\mathbb Z:1\\le i\\le n\\}}x^{n-|S|}y^{|S|}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每個factor選x或y，以選y的位置集合S索引完整分配展開。",
          "en": "Choose x or y in each factor, indexing the full distributive expansion by positions S choosing y."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "index set是全部1≤i≤n整數；每個factor選擇都保留。",
            "en": "The index set contains every integer1≤i≤n; no factor choices are omitted."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "|S|=k",
        "after_tex": "\\#\\{S:|S|=k\\}=\\binom nk,\\quad x^{n-|S|}y^{|S|}=x^{n-k}y^k",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "按y因子數k分組，每組有comb(n,k)項。",
          "en": "Group by the number k of y-factors; each group has comb(n,k) terms."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "commutativity讓不同位置選擇合併為同一monomial。",
            "en": "Commutativity permits choices at different positions to merge into one monomial."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "k\\in\\{j\\in\\mathbb Z:0\\le j\\le n\\}",
        "after_tex": "(x+y)^n=\\sum_{k=0}^{n}\\binom nk x^{n-k}y^k",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "所有子集依大小partition，包含k=0與k=n兩端。",
          "en": "Subset sizes partition every choice, including endpoints k=0 and k=n."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "指數由選y的k決定，不交換角色。",
            "en": "The y-exponent is the selected count k; roles are not exchanged."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "n=4,\\quad k=0,1,2,3,4",
        "after_tex": "\\binom40,\\binom41,\\binom42,\\binom43,\\binom44=1,4,6,4,1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "以factorial組合公式完整求五個係數。",
          "en": "Use the factorial combination formula for all five coefficients."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "中間係數6不是4。",
            "en": "The middle coefficient is 6, not 4."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "(x+y)^4",
        "after_tex": "x^4+4x^3y+6x^2y^2+4xy^3+y^4",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "將全部係數與正確指數配對。",
          "en": "Pair every coefficient with the correct exponents."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "每monomial總degree皆4。",
            "en": "Each monomial has total degree4."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "x=2,\\quad y=-1",
        "after_tex": "(2-1)^4=16-32+24-8+1=1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "有負y的exact evaluation核符號與端點。",
          "en": "An exact evaluation at negative y checks signs and endpoint terms."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "奇次y項負、偶次項正。",
            "en": "Odd y-powers are negative and even powers positive."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "n=0",
        "after_tex": "(x+y)^0=1=\\binom00x^0y^0",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "空乘積與唯一空子集給constant1。",
          "en": "The empty product and sole empty subset give constant 1."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "這是polynomial identity，包含x+y=0時的常數多項式約定。",
            "en": "This is a polynomial identity with the constant-polynomial convention even at x+y=0."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "xy\\ne yx\\quad\\text{(outside scope)}",
        "after_tex": "(x+y)^2=x^2+xy+yx+y^2\\text{ need not equal }x^2+2xy+y^2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "列出noncommuting邊界，定位分組所需假設。",
          "en": "Display the noncommuting boundary, identifying the assumption needed for grouping."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不把commutative證明外推矩陣。",
            "en": "Do not extend the commutative proof to arbitrary matrices."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "按子集完整證二項公式；n=4係數為1、4、6、4、1。",
      "en": "Grouping all subsets proves the binomial formula; at n=4 coefficients are 1,4,6,4,1."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "(x+y)^4只有x4+y4。",
          "en": "(x+y)^4 is only x4+y4."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "x=y=1時左16、右2。",
          "en": "At x=y=1 the left side is 16 and the right2."
        }
      },
      {
        "claim": {
          "zh-Hant": "n=-1仍是同一有限展開。",
          "en": "n=-1 has the same finite expansion."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "n屬N0是必要scope，負指數不是此polynomial。",
          "en": "The scope requires n in N0; a negative power is not this polynomial."
        }
      },
      {
        "claim": {
          "zh-Hant": "任意矩陣x、y也可合併xy與yx。",
          "en": "Arbitrary matrices permit merging xy and yx."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "不交換時xy與yx可不同。",
          "en": "For noncommuting matrices they can differ."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "n∈N_0，(x+y)^n=sum_{k=0}^n C(n,k)x^(n-k)y^k，coefficients exact且indices/exponents不可錯置。n=0給constant1。combinatorial witness選k個y factors；negative/noninteger exponent與noncommuting variables不在本輪。",
          "en": "For n∈N_0, (x+y)^n=sum over k=0..n of C(n,k)*x^(n−k)*y^k. Coefficients are exact, all indices and both endpoint terms are retained, and the exponents must not be interchanged. At n=0 the polynomial is constant 1. A combinatorial witness chooses k y-factors from n factors. Negative/noninteger exponents and noncommuting variables are outside this finite-polynomial slice."
        },
        "source": {
          "mko_id": "mko-binomial-theorem",
          "sha256": "c8fefe3783fc4f7633b4b22b8cf149d96a683900f181e3b42c690409ce90e5b7",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限分配律、可交換乘法、大小k子集數comb(n,k)=n!/[k!(n-k)!]；空乘積為1。",
          "en": "Finite distributivity, commutative multiplication and the count comb(n,k)=n!/[k!(n-k)!] of size-k subsets; the empty product is 1."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "comb(4,2)",
        "rhs": "6"
      },
      {
        "id": "N02",
        "lhs": "(x+y)**4",
        "rhs": "x**4+4*x**3*y+6*x**2*y**2+4*x*y**3+y**4",
        "vars": [
          "x",
          "y"
        ],
        "degree": 4
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "n非負整數，x、y可交換；不含非整數指數。",
      "en": "Nonnegative integer n and commuting x,y; no noninteger exponents."
    },
    "md_symbols": [
      [
        "k",
        "y因子數",
        "number of y-factors"
      ],
      [
        "S",
        "選y的位置子集",
        "positions choosing y"
      ]
    ],
    "md_error_controls": [
      {
        "title": "指數 / Exponents",
        "zh": "k個y對應x^(n-k)y^k。",
        "en": "k y-choices correspond to x^(n-k)y^k."
      },
      {
        "title": "完整係數 / All coefficients",
        "zh": "五項包括兩個端點；exact polynomial grid僅核n=4 identity。",
        "en": "Keep all five terms including endpoints; the exact polynomial grid checks the n=4 identity."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "ee47015e5d6cac87bd69650c5b71ea2b0a20f9578bdc2c3e785bdccb17535ada",
    "original_source_sha256": "a2fc26decd1525ce395cef607d62fc498d1cac913a33f9d6de68ffd0ffe794d8",
    "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": [
      "mko_definition",
      "work"
    ],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
      "BR142-S155",
      "BR142-S154",
      "BR142-S150",
      "BR142-S151",
      "BR142-S152",
      "BR142-S153"
    ],
    "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."
  }
}