{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261117-r06-path-connected-space-0038", "pilot_number": "R0052-26613", "number": 26613, "family_id": "qf-mod48-path-box-linear", "title": { "zh-Hant": "第 26613 題：道路連通空間", "en": "Question 26613: Coordinatewise Convex Weights Keep the Entire Path in the Box" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261117-r06-path-connected-space-0038", "batch_id": "batch-20261015-advance-20261117-r06-path-connected-space-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-path-connected-space" ], "title_zh": "道路連通空間", "stem_zh": "unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-14,-15],\"upper\":[15,16],\"start\":[-13,-14],\"end\":[14,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([-14, -15], [15, 16], [-13, -14], [14, 15], [-13+(27)t, -14+(29)t], 是, coordinatewise_convex_combination_stays_in_box)", "explanation_zh": "Wave6第十一層核對=([-14, -15], [15, 16], [-13, -14], [14, 15], [-13+(27)t, -14+(29)t], 是, coordinatewise_convex_combination_stays_in_box)；依完整明示資料獨立重算；linear／box／concatenated paths逐段exact核對；有限samples、graph walk或單一端點對不足；connected一般不推出path connected。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "道路連通空間" ] }, "variant": { "family_id": "qf-mod48-path-box-linear", "seed": 202611176038, "parameters": { "lower": [ -14, -15 ], "upper": [ 15, 16 ], "start": [ -13, -14 ], "end": [ 14, 15 ], "domain": "Exact_explicit_path_connectedness", "operation": "path-box-linear", "semantic_template": "path-box-linear|{end:[[14,1],[15,1]],lower:[[-14,1],[-15,1]],start:[[-13,1],[-14,1]],upper:[[15,1],[16,1]]}" } }, "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；不構成一般series、uniform-limit或topology定理證明、人工審定、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-20261117-r06-path-connected-space-0038", "batch_id": "batch-20261015-advance-20261117-r06-path-connected-space-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "760d7838a5decbb251eb70b21e4b8145ddda5ff9022bb31764142876e63fd023", "family_id": "qf-mod48-path-box-linear", "seed": 202611176038, "semantic_template": "path-box-linear|{end:[[14,1],[15,1]],lower:[[-14,1],[-15,1]],start:[[-13,1],[-14,1]],upper:[[15,1],[16,1]]}" }, "fields": { "title": "Coordinatewise Convex Weights Keep the Entire Path in the Box", "stem": "Use coordinate bounds lower=[-14, -15],upper=[15, 16], start=[-13, -14],end=[14, 15]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).", "answer": { "kind": "expression", "value": "([-14, -15], [15, 16], [-13, -14], [14, 15], [-13+(27)t, -14+(29)t], Yes, coordinatewise_convex_combination_stays_in_box)", "explanation": "Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-14, -15], [15, 16], [-13, -14], [14, 15], [-13+(27)t, -14+(29)t], Yes, coordinatewise_convex_combination_stays_in_box)." }, "skills": [ "Coordinatewise Convex Weights Keep the Entire Path in the Box" ], "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": "unit-interval path的continuity、兩端點與全range containment責任。操作=path-box-linear；資料={\"lower\":[-14,-15],\"upper\":[15,16],\"start\":[-13,-14],\"end\":[14,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use coordinate bounds lower=[-14, -15],upper=[15, 16], start=[-13, -14],end=[14, 15]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [path-box-linear]", "after": "derived = [[lower, upper, start, end, formulas, continuous_path, certificate]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "直接列出此操作及mode的完整來源欄序，不需JavaScript；布林、字串與null角色保持區別，不補欄或以答案形狀猜介面。", "en": "List this operation/mode exact source field order without JavaScript.Keep Boolean,string and null roles distinct;do not pad fields or infer the interface from answer shape." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。", "en": "Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [[-14, -15], [15, 16], [-13, -14], [14, 15]]", "after": "derived = [2, 是]", "relation": "implies", "rule_id": "Path_ALL_coordinate_endpoint_premises", "conditions": [], "explanation": { "zh-Hant": "逐每座標驗上下界與兩endpoint包含，維度須完全相同；允許退化座標與反向走。", "en": "Check both endpoints against each coordinate interval with matching dimensions.Degenerate coordinates and reverse orientation are allowed." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。", "en": "Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Path coordinate0 ORIGINAL end minus start := sub(14, -13)", "after": "Path coordinate0 ORIGINAL end minus start = 27", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色精確代值並保留符號運算；不用浮點近似，也不因額外教學運算縮小原題域。", "en": "Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Path coordinate0 ORIGINAL end minus start，不以小數近似替代。", "en": "Recompute Path coordinate0 ORIGINAL end minus start with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "14", "-13" ], "output": "27", "label": "Path coordinate0 ORIGINAL end minus start" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [0, -14, 15, -13, 14, 27]", "after": "derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -14, 15, -13, 14]", "relation": "implies", "rule_id": "Path_WHOLE_coordinate_convex_range_and_continuity", "conditions": [], "explanation": { "zh-Hant": "對所有t∈[0,1]，(1-t),t非負且和為1，故此座標夾在兩endpoint間且仍在[lower,upper]；affine函數連續，不是只驗endpoint。", "en": "For every t in[0,1],weights(1-t),t are nonnegative and sum to1.Thus this coordinate lies between its endpoints and within its bounds;the affine coordinate is continuous,not merely endpoint-checked." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。", "en": "Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Path coordinate1 ORIGINAL end minus start := sub(15, -14)", "after": "Path coordinate1 ORIGINAL end minus start = 29", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "以原始角色精確代值並保留符號運算；不用浮點近似，也不因額外教學運算縮小原題域。", "en": "Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Path coordinate1 ORIGINAL end minus start，不以小數近似替代。", "en": "Recompute Path coordinate1 ORIGINAL end minus start with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "15", "-14" ], "output": "29", "label": "Path coordinate1 ORIGINAL end minus start" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [1, -15, 16, -14, 15, 29]", "after": "derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -15, 16, -14, 15]", "relation": "implies", "rule_id": "Path_WHOLE_coordinate_convex_range_and_continuity", "conditions": [], "explanation": { "zh-Hant": "對所有t∈[0,1]，(1-t),t非負且和為1，故此座標夾在兩endpoint間且仍在[lower,upper]；affine函數連續，不是只驗endpoint。", "en": "For every t in[0,1],weights(1-t),t are nonnegative and sum to1.Thus this coordinate lies between its endpoints and within its bounds;the affine coordinate is continuous,not merely endpoint-checked." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查原題的完整資料、型別、欄序與本步條件；有限核對不升格為無限定理的形式證明。", "en": "Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems." } }, "arithmetic": null } ], "math_model": { "result": [ [ "-14", "-15" ], [ "15", "16" ], [ "-13", "-14" ], [ "14", "15" ], [ "-13+(27)t", "-14+(29)t" ], true, "coordinatewise_convex_combination_stays_in_box" ] }, "conclusion": { "canonical_value": "([-14, -15], [15, 16], [-13, -14], [14, 15], [-13+(27)t, -14+(29)t], 是, coordinatewise_convex_combination_stays_in_box)", "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-20261117-r06-path-connected-space-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261117-r06-path-connected-space-foundations/questions.jsonl", "batch_sha256": "2455669d9f59c769c41b1462899255c75387a7fdce752cd36e1fad7467bb3f32", "original_line": 38, "original_line_sha256": "9c0964ceebe0e7a564535be0e777d6466e5654782a713b4012aeadeec93e876d", "question_compact_sha256": "760d7838a5decbb251eb70b21e4b8145ddda5ff9022bb31764142876e63fd023", "english_row_compact_sha256": "63f5475b42c0d14a0d3a55d64c10bc0c6323a3464a2d87a8dd746f6d54ff8dc6", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261117-r06-path-connected-space-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261117-r06-path-connected-space-foundations.en.jsonl" }, "mko_bindings": { "mko-path-connected-space": { "sha256": "6df47a43396e20219abd3fe5ab3d088f1aaf4a1022bc9628ef0fe164b6a45e17", "english_sha256": "bb59736c31f281af474355507a3b5172c4fd2b948847c2332ec737e88490c538", "original_path": "public/data/mko/path-connected-space.json", "english_path": "public/data/i18n/editions/mko-path-connected-space.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "a3def7368821a4d3dc7757fefd3c4a099713f8de8fb8e6f3434025e9d7b5a6d0", "original_candidate_sha256": "ed6a0f196557013eea5bc3057af803b125b5154afd4d32ad6ccca9b8661a0276", "original_candidate_line_sha256": "ed6a0f196557013eea5bc3057af803b125b5154afd4d32ad6ccca9b8661a0276", "mko_binding_revision": { "id": "edition-symbol-bindings-20261101", "path": "content/edition-symbol-bindings-20261101/revision.json", "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c", "previous_english_sha256": { "mko-path-connected-space": "b17c14e650f00c01de1e7a49d95f7fbc2a4d9dcdac715c039f9e61fb4f9de607" } } }, "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-p53", "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": "729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252", "question_first_witness_sha256": "d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 6, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "a3ad93d7f8e97586f1681a3e2d4cd811a9636224fe5089e27ec1add6ce9572ca", "arithmetic_steps": 2 }, "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." } }