{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261116-r01-contraction-mapping-theorem-0060", "pilot_number": "R0051-26235", "number": 26235, "family_id": "qf-mod47-contract-fixed-point-bound", "title": { "zh-Hant": "第 26235 題：壓縮映射定理", "en": "Question 26235: The Geometric Tail Bound Controls the Exact Picard Error" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261116-r01-contraction-mapping-theorem-0060", "batch_id": "batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-contraction-mapping-theorem" ], "title_zh": "壓縮映射定理", "stem_zh": "完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,10,30],[10,0,20],[30,20,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/8, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)", "explanation_zh": "Wave6第十層核對=([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/8, 是, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)；依完整明示資料獨立重算；Picard orbit、unique fixed point與geometric tail bound使用exact arithmetic；q=1、非self-map、noncomplete或sampled-pairs不足；finite model不取代一般定理。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "壓縮映射定理" ] }, "variant": { "family_id": "qf-mod47-contract-fixed-point-bound", "seed": 202611161060, "parameters": { "metric": [ [ 0, 10, 30 ], [ 10, 0, 20 ], [ 30, 20, 0 ] ], "mapping": [ 0, 0, 1 ], "q": [ 1, 2 ], "start": 2, "steps": 6, "domain": "Exact_finite_contraction_mapping", "operation": "contract-fixed-point-bound", "semantic_template": "qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[10,1],[30,1]],[[10,1],[0,1],[20,1]],[[30,1],[20,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":6}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261116-r01-contraction-mapping-theorem-0060", "batch_id": "batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "8af47539ee1b36f93cda57a3cfca21155c75a7d304ff8259ad0c835892eca6de", "family_id": "qf-mod47-contract-fixed-point-bound", "seed": 202611161060, "semantic_template": "qf-mod47-contract-fixed-point-bound/{\"mapping\":[0,0,1],\"metric\":[[[0,1],[10,1],[30,1]],[[10,1],[0,1],[20,1]],[[30,1],[20,1],[0,1]]],\"q\":[1,2],\"start\":2,\"steps\":6}" }, "fields": { "title": "The Geometric Tail Bound Controls the Exact Picard Error", "stem": "Use the complete exact metric matrix=[[0, 10, 30], [10, 0, 20], [30, 20, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=6. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).", "answer": { "kind": "expression", "value": "([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/8, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)", "explanation": "Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/8, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)." }, "skills": [ "The Geometric Tail Bound Controls the Exact Picard Error" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "完整metric、total self-map與forall-pairs q<1 contraction責任。操作=contract-fixed-point-bound；資料={\"metric\":[[0,10,30],[10,0,20],[30,20,0]],\"mapping\":[0,0,1],\"q\":[1,2],\"start\":2,\"steps\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the complete exact metric matrix=[[0, 10, 30], [10, 0, 20], [30, 20, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Start=2, steps=6. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [contract-fixed-point-bound]", "after": "derived = [[orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "本頁直接列完整來源欄序，不需JavaScript；contract的6/4/6/6與IVT的8/7/8由原mode選，不補欄也不由答案形狀猜。", "en": "List the full source field order without JavaScript.Original modes select contract6/4/6/6 and IVT8/7/8 roles,without padding or answer-shape inference." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [[[0, 10, 30], [10, 0, 20], [30, 20, 0]]]", "after": "derived = [3, 是, 是, 是, [[0, 0, 0, 0, 0, 是], [0, 0, 1, 10, 10, 是], [0, 0, 2, 30, 30, 是], [0, 1, 0, 0, 20, 是], [0, 1, 1, 10, 10, 是], [0, 1, 2, 30, 30, 是], [0, 2, 0, 0, 60, 是], [0, 2, 1, 10, 50, 是], [0, 2, 2, 30, 30, 是], [1, 0, 0, 10, 10, 是], [1, 0, 1, 0, 20, 是], [1, 0, 2, 20, 40, 是], [1, 1, 0, 10, 10, 是], [1, 1, 1, 0, 0, 是], [1, 1, 2, 20, 20, 是], [1, 2, 0, 10, 50, 是], [1, 2, 1, 0, 40, 是], [1, 2, 2, 20, 20, 是], [2, 0, 0, 30, 30, 是], [2, 0, 1, 20, 40, 是], [2, 0, 2, 0, 60, 是], [2, 1, 0, 30, 30, 是], [2, 1, 1, 20, 20, 是], [2, 1, 2, 0, 40, 是], [2, 2, 0, 30, 30, 是], [2, 2, 1, 20, 20, 是], [2, 2, 2, 0, 0, 是]], 是]", "relation": "implies", "rule_id": "Contraction_COMPLETE_metric_axioms_ALL_triangles_finite_completeness", "conditions": [], "explanation": { "zh-Hant": "逐全部三元組核triangle，不由samples推metric。非空有限metric必complete：singleton顯然，否則取最小正距離delta，Cauchy尾段距離