{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261117-r06-path-connected-space-0069", "pilot_number": "R0052-26644", "number": 26644, "family_id": "qf-mod48-path-concatenation", "title": { "zh-Hant": "第 26644 題：道路連通空間", "en": "Question 26644: Two Rescaled Affine Branches Meet at the Exact Middle Point" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261117-r06-path-connected-space-0069", "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-concatenation；資料={\"left\":-21,\"right\":22,\"start\":-18,\"middle\":20,\"end\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(-18, 20, 18, 0<=t<=1/2:-18+2t(38), 1/2<=t<=1:20+(2t-1)(-2), both_branches_meet_at_middle, 是)", "explanation_zh": "Wave6第十一層核對=(-18, 20, 18, 0<=t<=1/2:-18+2t(38), 1/2<=t<=1:20+(2t-1)(-2), both_branches_meet_at_middle, 是)；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": 3, "skills": [ "道路連通空間" ] }, "variant": { "family_id": "qf-mod48-path-concatenation", "seed": 202611176069, "parameters": { "left": -21, "right": 22, "start": -18, "middle": 20, "end": 18, "domain": "Exact_explicit_path_connectedness", "operation": "path-concatenation", "semantic_template": "path-concatenation|{end:[18,1],left:[-21,1],middle:[20,1],right:[22,1],start:[-18,1]}" } }, "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-0069", "batch_id": "batch-20261015-advance-20261117-r06-path-connected-space-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "8d792e33691b26e201ad88264c620ca72c8e7a029a8d487f061bd06238bc4ba3", "family_id": "qf-mod48-path-concatenation", "seed": 202611176069, "semantic_template": "path-concatenation|{end:[18,1],left:[-21,1],middle:[20,1],right:[22,1],start:[-18,1]}" }, "fields": { "title": "Two Rescaled Affine Branches Meet at the Exact Middle Point", "stem": "Use real interval [-21,22] and contained start=-18,middle=20,end=18. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).", "answer": { "kind": "expression", "value": "(-18, 20, 18, 0<=t<=1/2:-18+2t(38), 1/2<=t<=1:20+(2t-1)(-2), both_branches_meet_at_middle, Yes)", "explanation": "Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-18, 20, 18, 0<=t<=1/2:-18+2t(38), 1/2<=t<=1:20+(2t-1)(-2), both_branches_meet_at_middle, Yes)." }, "skills": [ "Two Rescaled Affine Branches Meet at the Exact Middle Point" ], "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-concatenation；資料={\"left\":-21,\"right\":22,\"start\":-18,\"middle\":20,\"end\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use real interval [-21,22] and contained start=-18,middle=20,end=18. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [path-concatenation]", "after": "derived = [[start, middle, end, first_branch, second_branch, join, continuous_path]]", "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": "Path first ORIGINAL middle minus start := sub(20, -18)", "after": "Path first ORIGINAL middle minus start = 38", "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 first ORIGINAL middle minus start，不以小數近似替代。", "en": "Recompute Path first ORIGINAL middle minus start with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "20", "-18" ], "output": "38", "label": "Path first ORIGINAL middle minus start" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Path second ORIGINAL end minus middle := sub(18, 20)", "after": "Path second ORIGINAL end minus middle = -2", "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 second ORIGINAL end minus middle，不以小數近似替代。", "en": "Recompute Path second ORIGINAL end minus middle with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "18", "20" ], "output": "-2", "label": "Path second ORIGINAL end minus middle" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Path first branch JOIN at half := sum(-18, 38)", "after": "Path first branch JOIN at half = 20", "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 first branch JOIN at half，不以小數近似替代。", "en": "Recompute Path first branch JOIN at half with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-18", "38" ], "output": "20", "label": "Path first branch JOIN at half" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Path second branch JOIN at half := sum(20, 0)", "after": "Path second branch JOIN at half = 20", "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 second branch JOIN at half，不以小數近似替代。", "en": "Recompute Path second branch JOIN at half with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "20", "0" ], "output": "20", "label": "Path second branch JOIN at half" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [-21, 22, -18, 20, 18, 38, -2]", "after": "derived = [first weight=2t in[0,1], second weight=2t-1 in[0,1], 20, 20, 是]", "relation": "implies", "rule_id": "Path_TWO_whole_convex_branches_pasting_at_middle", "conditions": [], "explanation": { "zh-Hant": "前半以2t連start到middle，後半以2t-1連middle到end；各半區間全range由convexity保包含，各自連續，在t=1/2同取middle，pasting給全域連續。", "en": "On the first half use weight2t from start to middle;on the second use2t-1 from middle to end.Convexity controls each entire range.Both branches are continuous and agree at t=1/2,so the pasting lemma gives continuity on the whole interval." }, "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": [ "-18", "20", "18", "0<=t<=1/2:-18+2t(38)", "1/2<=t<=1:20+(2t-1)(-2)", "both_branches_meet_at_middle", true ] }, "conclusion": { "canonical_value": "(-18, 20, 18, 0<=t<=1/2:-18+2t(38), 1/2<=t<=1:20+(2t-1)(-2), both_branches_meet_at_middle, 是)", "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": 69, "original_line_sha256": "ce9350fb5a9b327b313544b94690adfcff2a9317965737c8b1118c821f0c6edc", "question_compact_sha256": "8d792e33691b26e201ad88264c620ca72c8e7a029a8d487f061bd06238bc4ba3", "english_row_compact_sha256": "7ddbe31eb03385333269ff4a20fb1d8aef6b09decad9a2393933fc2b955c2684", "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": "a9b5979711e45e59772d0f397fc3aba9289575022248ce124bc330e750b1b6fe", "original_candidate_line_sha256": "a9b5979711e45e59772d0f397fc3aba9289575022248ce124bc330e750b1b6fe", "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": "d1eff8ad64bd5524fe7c3415aadc73344f91435018e84c52ae0abdf4a30f0689", "arithmetic_steps": 4 }, "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." } }