{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-real-hilbert-space-01",
  "mko_id": "mko-real-hilbert-space",
  "title": {
    "zh-Hant": "同一矩陣誘導的有限維 Hilbert 證書",
    "en": "A finite-dimensional Hilbert certificate from one matrix"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-real-hilbert-space-01",
    "source": {
      "mko_id": "mko-real-hilbert-space",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "89b1eab58b853963348761bf140ffff91d2903053edbdf9baf157b32a5ac731e",
      "english_sha256": "3ac4455274e5bba35d48841ef7df59ae0ed9e3c9645b4035ebd5961f396f7052"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "algebra",
      "inner_norm_metric",
      "finite_dim_complete"
    ],
    "goal": {
      "zh-Hant": "驗 R^2 配 A=[[2,1],[1,2]] 的內積、範數、度量與完備性一致。",
      "en": "Verify the consistent inner product, norm, metric and completeness of R^2 with A=[[2,1],[1,2]]."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "V=R^2，A=[[2,1],[1,2]]；定義 <x,y>_A=x^T A y，範數及度量必由此同一內積誘導。",
          "en": "V=R^2, A=[[2,1],[1,2]]; define <x,y>_A=x^T A y and derive both norm and metric from this same inner product."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "x=(x_1,x_2),\\ y=(y_1,y_2)",
        "after_tex": "\\langle x,y\\rangle_A=2x_1y_1+x_1y_2+x_2y_1+2x_2y_2",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "展開矩陣乘積，固定所有結構共用 A。",
          "en": "Expand the matrix product, fixing A as the common source of every structure."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "交換 x,y 後式子相同，兩個交叉係數也相同。",
            "en": "Swapping x,y preserves the expression, including both cross coefficients."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S02",
        "uses": [
          "S01"
        ],
        "before_tex": "\\langle ax+bz,y\\rangle_A",
        "after_tex": "a\\langle x,y\\rangle_A+b\\langle z,y\\rangle_A",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐項分配乘法給實雙線性，對稱性也由展開式成立。",
          "en": "Distributivity gives real bilinearity termwise; the expanded expression also gives symmetry."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "S01"
          ],
          "explanation": {
            "zh-Hant": "a,b 可為任意實數，不是僅整數樣本。",
            "en": "a,b are arbitrary real scalars, not sampled integers."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S03",
        "uses": [
          "S01"
        ],
        "before_tex": "\\langle x,x\\rangle_A=2x_1^2+2x_1x_2+2x_2^2",
        "after_tex": "\\langle x,x\\rangle_A=(x_1+x_2)^2+x_1^2+x_2^2\\ge0",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "平方和給非負；等號迫使 x_1=x_2=0，所以正定。",
          "en": "The sum of squares is nonnegative, and equality forces both coordinates zero, proving positive definiteness."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "S01"
          ],
          "explanation": {
            "zh-Hant": "不是以行列式正或幾個數值樣本取代對所有 x 的證明。",
            "en": "No determinant shortcut or numerical sample replaces the all-x proof."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S02",
          "S03"
        ],
        "before_tex": "V=\\mathbb R^2,\\ \\langle\\ ,\\ \\rangle_A\\text{ symmetric bilinear positive definite}",
        "after_tex": "(V,\\langle\\ ,\\ \\rangle_A)\\text{ is a real inner-product space}",
        "relation": "implies",
        "rule": "inner_norm_metric",
        "conditions": [],
        "explanation": {
          "zh-Hant": "內積三責任全部具備，先確認空間資格。",
          "en": "All inner-product obligations are met, qualifying the space first."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "載體一直是 R^2，沒有換成有理向量子集。",
            "en": "The carrier remains R^2, not the subset of rational vectors."
          }
        },
        "leaf_basis": "inner_norm_metric"
      },
      {
        "id": "S05",
        "uses": [
          "S04"
        ],
        "before_tex": "\\|x\\|_A=\\sqrt{\\langle x,x\\rangle_A}",
        "after_tex": "\\|x\\|_A=\\sqrt{2x_1^2+2x_1x_2+2x_2^2}",
        "relation": "theorem_application",
        "rule": "inner_norm_metric",
        "conditions": [],
        "explanation": {
          "zh-Hant": "依內積誘導範數定理，這是真正範數；平方根取非負支。",
          "en": "By the inner-product induced-norm theorem this is a norm, using the nonnegative square root."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S04"
          ],
          "explanation": {
            "zh-Hant": "不能換成另一個矩陣或未綁定範數。",
            "en": "No different matrix or unbound norm may be substituted."
          }
        },
        "leaf_basis": "inner_norm_metric"
      },
      {
        "id": "S06",
        "uses": [
          "S05"
        ],
        "before_tex": "d_A(x,y)=\\|x-y\\|_A",
        "after_tex": "d_A(x,y)=\\sqrt{2(x_1-y_1)^2+2(x_1-y_1)(x_2-y_2)+2(x_2-y_2)^2}",
        "relation": "theorem_application",
        "rule": "inner_norm_metric",
        "conditions": [],
        "explanation": {
          "zh-Hant": "同一範數誘導度量；非負、分離、對稱與三角不等式由範數性得到。",
          "en": "The same norm induces the metric; its norm properties give nonnegativity, separation, symmetry and triangle inequality."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S05"
          ],
          "explanation": {
            "zh-Hant": "每個差向量仍在相同 V 內。",
            "en": "Every difference vector stays in the same V."
          }
        },
        "leaf_basis": "inner_norm_metric"
      },
      {
        "id": "S07",
        "uses": [
          "S03",
          "S05"
        ],
        "before_tex": "(x_1+x_2)^2\\le2(x_1^2+x_2^2)",
        "after_tex": "\\|x\\|_2^2\\le\\|x\\|_A^2\\le3\\|x\\|_2^2",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "上界由 (x_1-x_2)^2≥0 推得；下界由平方和直接得到，額外核對範數綁定。",
          "en": "The upper bound follows from (x_1-x_2)^2≥0 and the lower from the sum of squares, additionally checking the norm binding."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "界對所有實向量成立，但不是由此自動證無限維結論。",
            "en": "The bounds hold for all real vectors but imply no unprovided infinite-dimensional result."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S04",
          "S05",
          "S06"
        ],
        "before_tex": "\\dim_{\\mathbb R}V=2<\\infty",
        "after_tex": "(V,\\langle\\ ,\\ \\rangle_A,\\|\\ \\|_A,d_A)\\text{ is a real Hilbert space}",
        "relation": "theorem_application",
        "rule": "finite_dim_complete",
        "conditions": [
          "V=\\mathbb R^2",
          "A=A^T\\text{ positive definite}"
        ],
        "explanation": {
          "zh-Hant": "套命名有限維實內積完備定理，證每個 d_A-Cauchy 序列在同一 V 有 d_A 極限，這就是此處 Hilbert 責任。",
          "en": "Apply the named finite-dimensional real-inner-product completeness theorem: every d_A-Cauchy sequence has a d_A-limit in the same V, which supplies the Hilbert obligation."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "S05",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "證書作用於同一載體、內積、範數、度量；抽樣 Cauchy 序列無法取代它。",
            "en": "The certificate binds the same carrier, product, norm and metric; sampled Cauchy sequences cannot replace it."
          }
        },
        "leaf_basis": "finite_dim_complete"
      }
    ],
    "result": {
      "zh-Hant": "此 R^2 與同一 A 誘導的全部結構構成實 Hilbert 空間；證書是有限維完備定理。",
      "en": "This R^2, with all structures induced by the same A, is a real Hilbert space; completeness is supplied by the finite-dimensional completeness theorem."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "改用 Q^2 仍可照套有限維實內積定理。",
          "en": "Replacing R^2 by Q^2 permits the same real finite-dimensional theorem."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "Q^2 不是 R 向量空間，載體前提失效。",
          "en": "Q^2 is not a real vector space, so the carrier premise fails."
        }
      },
      {
        "claim": {
          "zh-Hant": "有限抽查 Cauchy 序列證完備。",
          "en": "Finitely checking Cauchy sequences proves completeness."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "完備量化所有 Cauchy 序列。",
          "en": "Completeness quantifies over every Cauchy sequence."
        }
      },
      {
        "claim": {
          "zh-Hant": "Hilbert 因此此空間緊緻。",
          "en": "Hilbert therefore makes this space compact."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "本稿沒有緊緻證書；R^2 本身亦無界。",
          "en": "No compactness certificate is supplied; R^2 itself is unbounded."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "carrier、inner product、induced norm、induced metric與completeness certificate必須描述同一空間。本輪使用有限維R^n標準或正定矩陣內積的命名完備定理證書；抽樣Cauchy序列不足。",
          "en": "The carrier, inner product, induced norm, induced metric and completeness certificate must describe the same space. This slice uses named completeness-theorem certificates for finite-dimensional R^n with a standard or positive-definite-matrix inner product. Sampled Cauchy sequences do not establish completeness."
        },
        "source": {
          "mko_id": "mko-real-hilbert-space",
          "sha256": "89b1eab58b853963348761bf140ffff91d2903053edbdf9baf157b32a5ac731e",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "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": "inner_norm_metric",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "正定對稱雙線性形式定義實內積；實內積的平方根自積是範數（含三角不等式），此範數的差向量值是度量。",
          "en": "A positive-definite symmetric bilinear form defines a real inner product; its square-root self-product is a norm, including the triangle inequality, and the difference norm is a metric."
        }
      },
      {
        "id": "finite_dim_complete",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "命名完備證書：每個有限維實內積空間在該內積誘導範數及度量下完備。",
          "en": "Named completeness certificate: every finite-dimensional real inner-product space is complete in the norm and metric induced by that inner product."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "quadratic-identity",
        "lhs": "2*x*x+2*x*y+2*y*y",
        "rhs": "(x+y)**2+x*x+y*y",
        "step": "S03",
        "vars": [
          "x",
          "y"
        ],
        "degree": 2
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "只證同一有限維實載體；不物化無限維、緊緻或可分結論。",
      "en": "Only this finite-dimensional real carrier; no infinite-dimensional, compactness or separability claim."
    },
    "md_symbols": [
      [
        "A",
        "固定正定矩陣",
        "fixed positive-definite matrix"
      ],
      [
        "\\|\\ \\|_A,d_A",
        "同源範數及度量",
        "matching norm and metric"
      ]
    ],
    "md_error_controls": [
      {
        "title": "五項同源 / Five-way binding",
        "zh": "每處固定 V、A、內積、範數、度量，不混搭。",
        "en": "Keep V, A, inner product, norm and metric consistently bound."
      },
      {
        "title": "命名完備證書 / Named completeness",
        "zh": "明示應用有限維實內積定理，而非幾個樣本。",
        "en": "Explicitly apply finite-dimensional real-inner-product completeness, not samples."
      }
    ],
    "md_open_items": []
  },
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    "original_source_sha256": "1dfb783bf683beff32490fed97cb7f8755ddeca7526828da95db41018e1e36fd",
    "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": [],
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    "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."
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}