{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261116-r01-contraction-mapping-theorem-0090", "pilot_number": "R0051-26215", "number": 26215, "family_id": "qf-mod47-contract-boundary", "title": { "zh-Hant": "第 26215 題：壓縮映射定理", "en": "Question 26215: Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261116-r01-contraction-mapping-theorem-0090", "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-boundary；資料={\"mode\":\"noncomplete_space\",\"scale\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(noncomplete_space, X=(0,15), T(x)=(x+15)/2, fixed_point=15_outside_X, [1, 2], 否)", "explanation_zh": "Wave6第十層核對=(noncomplete_space, X=(0,15), T(x)=(x+15)/2, fixed_point=15_outside_X, [1, 2], 否)；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-boundary", "seed": 202611161090, "parameters": { "mode": "noncomplete_space", "scale": 15, "domain": "Exact_finite_contraction_mapping", "operation": "contract-boundary", "semantic_template": "qf-mod47-contract-boundary/{\"mode\":\"noncomplete_space\",\"scale\":[15,1]}" } }, "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-0090", "batch_id": "batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "cc7dc1de2c11df5b42653b40407d2828ca1c05a3aa98a09ee6aaa79517b131b8", "family_id": "qf-mod47-contract-boundary", "seed": 202611161090, "semantic_template": "qf-mod47-contract-boundary/{\"mode\":\"noncomplete_space\",\"scale\":[15,1]}" }, "fields": { "title": "Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem", "stem": "Use boundary mode=noncomplete_space and positive exact scale=15. Give (mode, space_description, mapping_description, outside_fixed_point, q_as_fraction, theorem_application_claim).", "answer": { "kind": "expression", "value": "(noncomplete_space, X=(0,15), T(x)=(x+15)/2, fixed_point=15_outside_X, [1, 2], No)", "explanation": "For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (noncomplete_space, X=(0,15), T(x)=(x+15)/2, fixed_point=15_outside_X, [1, 2], No)." }, "skills": [ "Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem" ], "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-boundary；資料={\"mode\":\"noncomplete_space\",\"scale\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use boundary mode=noncomplete_space and positive exact scale=15. Give (mode, space_description, mapping_description, outside_fixed_point, q_as_fraction, theorem_application_claim)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [contract-boundary]", "after": "derived = [[mode, space_description, mapping_description, outside_fixed_point, q_as_fraction, theorem_application_claim]]", "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 = [noncomplete_space, 15]", "after": "derived = [noncomplete_space, X=(0,15), T(x)=(x+15)/2, fixed_point=15_outside_X, [1, 2], 否]", "relation": "implies", "rule_id": "Contraction_original_boundary_mode_actual_scope_not_invented_witness", "conditions": [], "explanation": { "zh-Hant": "q1的two-point identity有兩fixed points；非self-map不能迭代於同空間；(0,s)上(x+s)/2的fixed point s在外；sampled_pairs只保來源範圍敘述，沒提供可重算的完整metric/map，不冒稱已驗numeric反例。", "en": "The two-point identity at q1 has two fixed points;nonself maps do not iterate in the same space;on(0,s),(x+s)/2 has its fixed point s outside.sampled_pairs supplies only source scope,not a complete recomputable metric/map counterexample." }, "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 } ], "math_model": { "result": [ "noncomplete_space", "X=(0,15)", "T(x)=(x+15)/2", "fixed_point=15_outside_X", [ 1, 2 ], false ] }, "conclusion": { "canonical_value": "(noncomplete_space, X=(0,15), T(x)=(x+15)/2, fixed_point=15_outside_X, [1, 2], 否)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations/questions.jsonl", "batch_sha256": "268af34d2f0ede8f2948e060f19a97e857bedbcdc7359cad296d6741ee95280f", "original_line": 90, "original_line_sha256": "05d9a30627c7e1d4c96bacc78d913b5dd28100a9e1b90529142c2679f1f071da", "question_compact_sha256": "cc7dc1de2c11df5b42653b40407d2828ca1c05a3aa98a09ee6aaa79517b131b8", "english_row_compact_sha256": "2ea26f81d4e22a214edc9b5379c732a753f33f6229ea80d65b0599a5ebc03142", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations.en.jsonl" }, "mko_bindings": { "mko-contraction-mapping-theorem": { "sha256": "ab6943f2789918c758734f9dd379c1ca82efe2111e3e7b2f6cd5c3be1eb4db5d", "english_sha256": "f40dc82c772b380f05cbd256c6019c0825fda576fd85a8614c86e43cdb7cb080", "original_path": "public/data/mko/contraction-mapping-theorem.json", "english_path": "public/data/i18n/editions/mko-contraction-mapping-theorem.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "74f6f34e0a695e06f9d836b5f5c677f9f716d2291c75c8f2f5b809ff0935bcff", "original_candidate_sha256": "16607c52a331c6aaf4cd88837353a1bacf7d34b4e115b09d0e9b8e021a851237", "original_candidate_line_sha256": "16607c52a331c6aaf4cd88837353a1bacf7d34b4e115b09d0e9b8e021a851237" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-010-p52", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3", "question_first_witness_sha256": "ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 2, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "649bdc6c3e9e7d09dfbc9f19a661bf66d249bd0a2faa817a63fc0f20a1b267ff", "arithmetic_steps": 0 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }