{ "schema_version": "ocme-derivation-reader-v1", "guide_id": "derivation-mko-contraction-mapping-theorem-01", "mko_id": "mko-contraction-mapping-theorem", "title": { "zh-Hant": "有限度量的完整收縮證書", "en": "A complete contraction certificate on a finite metric" }, "contract": { "schema_version": "ocme-derivation-guide-v0.1", "guide_id": "derivation-mko-contraction-mapping-theorem-01", "source": { "mko_id": "mko-contraction-mapping-theorem", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "sha256": "ab6943f2789918c758734f9dd379c1ca82efe2111e3e7b2f6cd5c3be1eb4db5d", "english_sha256": "f40dc82c772b380f05cbd256c6019c0825fda576fd85a8614c86e43cdb7cb080" }, "mode": "worked_example", "audience_prerequisites": [ "algebra", "sets", "epsilon" ], "goal": { "zh-Hant": "在三點離散度量與常值 self-map 上核完備、唯一不動點及誤差界。", "en": "Audit completeness, a unique fixed point and the error bound for a constant self-map on a three-point discrete metric." }, "givens": [ { "id": "G1", "statement": { "zh-Hant": "X={a,b,c}，d(x,x)=0、異點 d=1；T(a)=T(b)=T(c)=a，q=1/2，x_0=c。", "en": "X={a,b,c}; d(x,x)=0 and d=1 on distinct points; T(a)=T(b)=T(c)=a, q=1/2, x_0=c." } } ], "steps": [ { "id": "S01", "uses": [ "G1" ], "before_tex": "D=\\begin{pmatrix}0&1&1\\\\1&0&1\\\\1&1&0\\end{pmatrix}", "after_tex": "d\\ge0,\\quad d(x,y)=0\\iff x=y,\\quad d(x,y)=d(y,x)", "relation": "implies", "rule": "sets", "conditions": [], "explanation": { "zh-Hant": "完整九項矩陣直接核非負、分離及對稱。", "en": "The full nine-entry matrix directly verifies nonnegativity, separation and symmetry." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1" ], "explanation": { "zh-Hant": "不是只給若干點對距離。", "en": "Distances are supplied for every pair, not sampled pairs." } }, "leaf_basis": "sets" }, { "id": "S02", "uses": [ "G1", "S01" ], "before_tex": "x=z\\text{ or }x\\ne z", "after_tex": "d(x,z)\\le d(x,y)+d(y,z)", "relation": "implies", "rule": "algebra", "conditions": [], "explanation": { "zh-Hant": "x=z 時左邊零；異點時右邊不可能兩項同為零，所以至少 1。這涵蓋全部 27 三元組。", "en": "If x=z the left side is zero; otherwise the two right terms cannot both vanish, so their sum is at least one. This covers all 27 triples." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01" ], "explanation": { "zh-Hant": "三角不等式有全三元組論證，非數值抽樣。", "en": "The triangle argument covers every triple, not numerical samples." } }, "leaf_basis": "algebra" }, { "id": "S03", "uses": [ "S01", "S02" ], "before_tex": "(z_n)\\text{ is }d\\text{-Cauchy},\\quad\\varepsilon=1/2", "after_tex": "\\exists N\\ \\forall m,n\\ge N:\\ d(z_m,z_n)<1/2\\Longrightarrow z_m=z_n", "relation": "implies", "rule": "epsilon", "conditions": [], "explanation": { "zh-Hant": "Cauchy 尾項只能全相同，故最終常值於 X 內並收斂，空間完備。", "en": "A Cauchy tail must be constant, hence eventually equals a point of X and converges; the space is complete." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "S01", "S02" ], "explanation": { "zh-Hant": "所有 Cauchy 序列由最小正距離控制，不是幾條樣本。", "en": "The minimum positive distance controls every Cauchy sequence, not selected samples." } }, "leaf_basis": "epsilon" }, { "id": "S04", "uses": [ "G1" ], "before_tex": "T(a)=T(b)=T(c)=a\\in X", "after_tex": "T:X\\to X\\text{ is a total self-map}", "relation": "implies", "rule": "sets", "conditions": [], "explanation": { "zh-Hant": "逐一核全部三個輸入且像仍在載體。", "en": "Every input is assigned and its image remains in the carrier." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1" ], "explanation": { "zh-Hant": "定理所需 self-map 不是任意去別空間的映射。", "en": "A self-map, not a map into another space, is required." } }, "leaf_basis": "sets" }, { "id": "S05", "uses": [ "G1", "S04" ], "before_tex": "x,y\\in X", "after_tex": "d(Tx,Ty)=0\\le\\tfrac12d(x,y),\\quad0\\le q<1", "relation": "implies", "rule": "algebra", "conditions": [], "explanation": { "zh-Hant": "所有九個有序點對像相等，所以完整收縮不等式成立。", "en": "Images agree for all nine ordered pairs, establishing the full contraction inequality." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S04" ], "explanation": { "zh-Hant": "q=1/2 嚴格小於 1；不是僅非擴張 q=1。", "en": "q=1/2 is strictly below one, not merely nonexpansive q=1." } }, "leaf_basis": "algebra" }, { "id": "S06", "uses": [ "G1", "S04" ], "before_tex": "Tx=x", "after_tex": "x=a\\text{ is the only fixed point}", "relation": "implies", "rule": "sets", "conditions": [], "explanation": { "zh-Hant": "a 映到自己；b,c 映到不同點 a，完整不動點枚舉只剩 a。", "en": "a maps to itself, while b,c map to the different point a; the exhaustive fixed-point list contains only a." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S04" ], "explanation": { "zh-Hant": "唯一性由全部點核對，亦可由 d≤q d 及 q<1 推出。", "en": "Uniqueness follows from all-point auditing and also from d≤q*d with q<1." } }, "leaf_basis": "sets" }, { "id": "S07", "uses": [ "G1", "S04", "S06" ], "before_tex": "x_{n+1}=Tx_n,\\quad x_0=c", "after_tex": "x_1=a,\\quad x_n=a\\ (n\\ge1),\\quad x_n\\to a", "relation": "implies", "rule": "sets", "conditions": [], "explanation": { "zh-Hant": "第一步到 a，之後用 T(a)=a 歸納整個無限 orbit。", "en": "The first step reaches a, and T(a)=a inductively fixes the entire remaining orbit." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S04", "S06" ], "explanation": { "zh-Hant": "不是只產生有限 Picard trace 就宣稱收斂。", "en": "Convergence is not claimed merely from a finite Picard trace." } }, "leaf_basis": "sets" }, { "id": "S08", "uses": [ "S06", "S07" ], "before_tex": "\\frac{q^n}{1-q}d(x_1,x_0)=2^{1-n}", "after_tex": "d(x_0,a)=1\\le2;\\quad d(x_n,a)=0\\le2^{1-n}\\ (n\\ge1)", "relation": "implies", "rule": "algebra", "conditions": [], "explanation": { "zh-Hant": "逐分支核全 n 的標準幾何誤差界，包括 n=0。", "en": "Check the standard geometric error bound for all n, including zero, by explicit branches." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "S06", "S07" ], "explanation": { "zh-Hant": "界支持本例，並不以此有限度量論證一般完備空間定理。", "en": "The bound certifies this instance without claiming a general complete-space proof." } }, "leaf_basis": "algebra" } ], "result": { "zh-Hant": "此有限度量完備，T 的唯一不動點 a；從 c 一步到達，全部誤差界成立。", "en": "This finite metric is complete, T has unique fixed point a, and the orbit from c reaches it in one step with every error bound satisfied." }, "negative_controls": [ { "claim": { "zh-Hant": "q=1 的恆等映射也必唯一不動點。", "en": "The identity with q=1 also has a unique fixed point." }, "expected": "reject", "reason": { "zh-Hant": "三個點皆固定，嚴格收縮前提失敗。", "en": "All three points are fixed; strict contraction fails." } }, { "claim": { "zh-Hant": "只核 (a,b) 就證所有點對收縮。", "en": "Checking only (a,b) proves contraction on every pair." }, "expected": "reject", "reason": { "zh-Hant": "全點對量詞不能由一對成功取代。", "en": "One success cannot replace the all-pair quantifier." } }, { "claim": { "zh-Hant": "映射離開 X 仍可使用 self-map 定理。", "en": "A map leaving X still qualifies for the self-map theorem." }, "expected": "reject", "reason": { "zh-Hant": "迭代及不動點責任須在同一 X。", "en": "Iteration and the fixed-point obligation require the same X." } } ], "status": "candidate", "gaps": [], "verification": { "structure": "checked_local_script", "mathematics": "candidate_argument", "formalization": "not_formalized" }, "rule_glossary": [ { "id": "mko_definition", "kind": "cited", "statement": { "zh-Hant": "題族以完整finite metric與total self-map逐對驗d(Tx,Ty)≤q d(x,y)，固定0≤q<1。從任意x0完整產生Picard orbit，枚舉fixed points並核唯一性；exact rational metric下驗d(x_n,x*)≤q^n/(1-q)d(x_1,x_0)。finite metric自動complete但只證指定模型；q=1、非self-map、非complete空間與sampled-pairs不足均不得套定理。", "en": "The exact finite families qualify every metric axiom and a complete total self-map, then check d(Tx,Ty)≤q*d(x,y) for every ordered pair with fixed0≤q<1. From a declared start generate the entire requested Picard orbit and enumerate fixed points to check uniqueness. Compare d(x_n,x*) with q^n/(1−q)*d(x_1,x_0) exactly. A finite metric space is complete, but its certificate proves only that specified model. q=1, failure of the self-map premise, a noncomplete space or sampled-pair checks do not license the theorem." }, "source": { "mko_id": "mko-contraction-mapping-theorem", "sha256": "ab6943f2789918c758734f9dd379c1ca82efe2111e3e7b2f6cd5c3be1eb4db5d", "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": "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": "epsilon", "kind": "prerequisite", "statement": { "zh-Hant": "收斂表示每個正 epsilon 有一個整數 N，使全部 n≥N 的誤差嚴格小於 epsilon；使用實數 Archimedean 性。", "en": "Convergence requires for every positive epsilon an integer N controlling every n≥N by a strict error bound; use the Archimedean property of the reals." } } ], "revision_log": [], "numeric_checks": [ { "id": "n-zero-bound", "lhs": "1", "rhs": "(1/2)**0/(1-1/2)", "step": "S08", "relation": "<=" } ] }, "presentation": { "md_scope": { "zh-Hant": "只證完整有限模型，不從此取代一般 Banach 定理。", "en": "Only the complete finite model, not a replacement for the general Banach theorem." }, "md_symbols": [ [ "D", "完整度量矩陣", "complete metric matrix" ], [ "T", "全 self-map", "total self-map" ], [ "q", "嚴格收縮常數", "strict contraction factor" ] ], "md_error_controls": [ { "title": "度量全表 / Complete metric", "zh": "先核全部公理及全域映射，再核收縮。", "en": "Qualify every metric axiom and the total map before contraction." }, { "title": "全 orbit / Whole orbit", "zh": "由固定點後歸納永遠固定，與有限 trace 分開。", "en": "Inductively prove it stays fixed forever, separately from a finite trace." } ], "md_open_items": [] }, "revision": { "id": "ocme-new142-reader-adoption-20261021", "collection": "coverage-20261021", "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "adopted_source_sha256": "9495e80b14cbe0e32e555d76b8ebed0ff8eb48f593477974ab06adb8325a82a6", "original_source_sha256": "9495e80b14cbe0e32e555d76b8ebed0ff8eb48f593477974ab06adb8325a82a6", "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": [], "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." } }