{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261011-advance-20261015-r07-coordinate-plane-0080", "pilot_number": "R0060-30730", "number": 30730, "family_id": "qf-density15-coordinate-first-mismatch-index", "title": { "zh-Hant": "第 30730 題：三對座標的首個不等", "en": "Question 30730: Three Ordered Coordinate Comparisons and First Mismatch" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261011-advance-20261015-r07-coordinate-plane-0080", "batch_id": "batch-20261011-advance-20261015-r07-coordinate-plane-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-coordinate-plane" ], "title_zh": "三對座標的首個不等", "stem_zh": "同一正交有向xy框架，座標有序對。原序三對[P,Q]=[[[-8, 1], [0, 0]], [[0, 4], [0, 0]], [[0, -1], [0, 0]]]。求 (每對P−Q分量殘差, 逐對相等資格, 首個不等0索引或無)。", "answer": { "kind": "expression", "value": "([[-8, 1], [0, 4], [0, -1]], [否, 否, 否], 0)", "explanation_zh": "三對分量差與首個不等=([[-8, 1], [0, 4], [0, -1]], [否, 否, 否], 0)。每對都核兩個分量，按原序取第一個失配；若全相同答無。軸點不屬象限原點同屬兩軸，逐分量差不當距離或圖形尺寸，不由示意圖猜座標。" }, "difficulty": { "band": "lower_secondary", "level": 1, "skills": [ "三對座標的首個不等" ] }, "variant": { "family_id": "qf-density15-coordinate-first-mismatch-index", "seed": 202610157080, "parameters": { "pairs": [ [ [ -8, 1 ], [ 0, 0 ] ], [ [ 0, 4 ], [ 0, 0 ] ], [ [ 0, -1 ], [ 0, 0 ] ] ], "frame": "same_orthogonal_xy_axes", "domain": "finite_ordered_coordinate_protocol", "operation": "density15_coordinate_first_mismatch_index", "semantic_template": "qf-density15-coordinate-first-mismatch-index/[[\"-8\",\"1\"],[\"0\",\"4\"],[\"0\",\"-1\"]]" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-11", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261011-advance-20261015-r07-coordinate-plane-0080", "batch_id": "batch-20261011-advance-20261015-r07-coordinate-plane-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "6e091f96608603f14bda823a7bfe51b081f6f3cb7ac4132a1235082c567d225b", "family_id": "qf-density15-coordinate-first-mismatch-index", "seed": 202610157080, "semantic_template": "qf-density15-coordinate-first-mismatch-index/[[\"-8\",\"1\"],[\"0\",\"4\"],[\"0\",\"-1\"]]" }, "fields": { "title": "Three Ordered Coordinate Comparisons and First Mismatch", "stem": "Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[-8, 1], [0, 0]], [[0, 4], [0, 0]], [[0, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).", "answer": { "kind": "expression", "value": "([[-8, 1], [0, 4], [0, -1]], [No, No, No], 0)", "explanation": "Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[-8, 1], [0, 4], [0, -1]], [No, No, No], 0)." }, "skills": [ "Three Ordered Coordinate Comparisons and First Mismatch" ], "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": "同一正交有向xy框架，座標有序對。原序三對[P,Q]=[[[-8, 1], [0, 0]], [[0, 4], [0, 0]], [[0, -1], [0, 0]]]。求 (每對P−Q分量殘差, 逐對相等資格, 首個不等0索引或無)。", "en": "Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[-8, 1], [0, 0]], [[0, 4], [0, 0]], [[0, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-density15-coordinate-first-mismatch-index, density15_coordinate_first_mismatch_index, finite_ordered_coordinate_protocol]", "after": "derived = [[ALL_P_minus_Q_pairs, ALL_equal_flags, FIRST_unequal_PAIR_zero_index_or_none]]", "relation": "implies", "rule_id": "R60_EXACT_perbank_operation_domain_FRAME_and_output_roles", "conditions": [], "explanation": { "zh-Hant": "density14/15用full-prefix underscores，density17用local hyphens。coordinate frame與circle frame不同，輸出欄序／型別依本family，不改原中英／JSON／status。", "en": "density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Coordinate original pair0 ORIGINAL P minus Q coordinate0 := sub(-8, 0)", "after": "Coordinate original pair0 ORIGINAL P minus Q coordinate0 = -8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Coordinate original pair0 ORIGINAL P minus Q coordinate0，不以小數近似替代。", "en": "Recompute Coordinate original pair0 ORIGINAL P minus Q coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-8", "0" ], "output": "-8", "label": "Coordinate original pair0 ORIGINAL P minus Q coordinate0" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Coordinate original pair0 ORIGINAL P minus Q coordinate1 := sub(1, 0)", "after": "Coordinate original pair0 ORIGINAL P minus Q coordinate1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Coordinate original pair0 ORIGINAL P minus Q coordinate1，不以小數近似替代。", "en": "Recompute Coordinate original pair0 ORIGINAL P minus Q coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "0" ], "output": "1", "label": "Coordinate original pair0 ORIGINAL P minus Q coordinate1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [Coordinate original pair0, [-8, 1], [0, 0]]", "after": "derived = [[-8, 1]]", "relation": "implies", "rule_id": "Coordinate_SOURCE_direction_PminusQ_not_length_QminusP", "conditions": [], "explanation": { "zh-Hant": "這個coordinate協議依原P-Q保兩個分量sign，和length bank的Q-P相反；差不是distance或圖形尺寸，derived可超SAFE。", "en": "THIS coordinate protocol retains both signs of ORIGINAL P-Q,OPPOSITE the length bank Q-P.A component difference is not distance or rendered size;derived values may exceed SAFE." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Coordinate original pair1 ORIGINAL P minus Q coordinate0 := sub(0, 0)", "after": "Coordinate original pair1 ORIGINAL P minus Q coordinate0 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Coordinate original pair1 ORIGINAL P minus Q coordinate0，不以小數近似替代。", "en": "Recompute Coordinate original pair1 ORIGINAL P minus Q coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "0" ], "output": "0", "label": "Coordinate original pair1 ORIGINAL P minus Q coordinate0" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Coordinate original pair1 ORIGINAL P minus Q coordinate1 := sub(4, 0)", "after": "Coordinate original pair1 ORIGINAL P minus Q coordinate1 = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Coordinate original pair1 ORIGINAL P minus Q coordinate1，不以小數近似替代。", "en": "Recompute Coordinate original pair1 ORIGINAL P minus Q coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "4", "0" ], "output": "4", "label": "Coordinate original pair1 ORIGINAL P minus Q coordinate1" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [Coordinate original pair1, [0, 4], [0, 0]]", "after": "derived = [[0, 4]]", "relation": "implies", "rule_id": "Coordinate_SOURCE_direction_PminusQ_not_length_QminusP", "conditions": [], "explanation": { "zh-Hant": "這個coordinate協議依原P-Q保兩個分量sign，和length bank的Q-P相反；差不是distance或圖形尺寸，derived可超SAFE。", "en": "THIS coordinate protocol retains both signs of ORIGINAL P-Q,OPPOSITE the length bank Q-P.A component difference is not distance or rendered size;derived values may exceed SAFE." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Coordinate original pair2 ORIGINAL P minus Q coordinate0 := sub(0, 0)", "after": "Coordinate original pair2 ORIGINAL P minus Q coordinate0 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Coordinate original pair2 ORIGINAL P minus Q coordinate0，不以小數近似替代。", "en": "Recompute Coordinate original pair2 ORIGINAL P minus Q coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "0" ], "output": "0", "label": "Coordinate original pair2 ORIGINAL P minus Q coordinate0" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Coordinate original pair2 ORIGINAL P minus Q coordinate1 := sub(-1, 0)", "after": "Coordinate original pair2 ORIGINAL P minus Q coordinate1 = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數做精確整數／有理計算；只有raw輸入驗SAFE或本family明定小界，衍生值不重套界、不作浮點近似。", "en": "Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Coordinate original pair2 ORIGINAL P minus Q coordinate1，不以小數近似替代。", "en": "Recompute Coordinate original pair2 ORIGINAL P minus Q coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-1", "0" ], "output": "-1", "label": "Coordinate original pair2 ORIGINAL P minus Q coordinate1" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "source roles = [Coordinate original pair2, [0, -1], [0, 0]]", "after": "derived = [[0, -1]]", "relation": "implies", "rule_id": "Coordinate_SOURCE_direction_PminusQ_not_length_QminusP", "conditions": [], "explanation": { "zh-Hant": "這個coordinate協議依原P-Q保兩個分量sign，和length bank的Q-P相反；差不是distance或圖形尺寸，derived可超SAFE。", "en": "THIS coordinate protocol retains both signs of ORIGINAL P-Q,OPPOSITE the length bank Q-P.A component difference is not distance or rendered size;derived values may exceed SAFE." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "source roles = [[[[-8, 1], [0, 0]], [[0, 4], [0, 0]], [[0, -1], [0, 0]]]]", "after": "derived = [[[-8, 1], [0, 4], [0, -1]], [否, 否, 否], 0]", "relation": "implies", "rule_id": "Coordinate_FIRST_unequal_PAIR_zero_based_not_coordinate_index", "conditions": [], "explanation": { "zh-Hant": "每個pair先驗兩分量相等，再按原三pair順序找第一個不相等的0-based index；index0不能被誤當none。全部相等回none，不是第一個分量1/2的索引。", "en": "First compare BOTH components within EACH pair,then find the FIRST unequal PAIR in original order using a ZERO-based index.Index0 is not none.All equal returns none;this is not a coordinate-component1/2 index." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "用原始資料、型別、索引與本步前提反向代回；一致性審查不等於可由結論唯一還原輸入。有限候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate." } }, "arithmetic": null } ], "math_model": { "result": [ [ [ "-8", "1" ], [ "0", "4" ], [ "0", "-1" ] ], [ false, false, false ], "0" ] }, "conclusion": { "canonical_value": "([[-8, 1], [0, 4], [0, -1]], [否, 否, 否], 0)", "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-20261011-advance-20261015-r07-coordinate-plane-foundations", "batch_path": "public/data/questions/batches/batch-20261011-advance-20261015-r07-coordinate-plane-foundations/questions.jsonl", "batch_sha256": "123b8ceaa888b690a0ae676ad07f8bab359ecadb66103e52a232095b913b44ff", "original_line": 80, "original_line_sha256": "7fd4c8ccfaec2fd81d359afa14b01dc0236dfa6e4c09476a55d138a8ea862a46", "question_compact_sha256": "6e091f96608603f14bda823a7bfe51b081f6f3cb7ac4132a1235082c567d225b", "english_row_compact_sha256": "1e5c811c7a9bbc54ee227029392b2c0b0275025817f7711e0ab041fb91fd2628", "english_manifest_path": "public/data/i18n/editions/batch-20261011-advance-20261015-r07-coordinate-plane-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261011-advance-20261015-r07-coordinate-plane-foundations.en.jsonl" }, "mko_bindings": { "mko-coordinate-plane": { "sha256": "eb010001cb85a8fe8eee5a876ee1ec40caddc67fe621f671741caacf48ab3c78", "english_sha256": "5da6135768256b641dcda707cae4d1c9b7a425ea489f9b1e850eaa6bf67fa0b3", "original_path": "public/data/mko/coordinate-plane.json", "english_path": "public/data/i18n/editions/mko-coordinate-plane.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "322533190c1eb10144c4c59b6c70cc2f8105108b5f4f3a39dd577712e7185c02", "original_candidate_sha256": "ccb0a502db9241d0ac774753a9860f822907d28ad4e4a68a64b856c9da2e3ae0", "original_candidate_line_sha256": "ccb0a502db9241d0ac774753a9860f822907d28ad4e4a68a64b856c9da2e3ae0" }, "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-20261026-011-p61", "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": "d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bf", "question_first_witness_sha256": "f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 11, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "d6992e42064fe47668769d435c419f3fbc2a7e41d1561ed87c9efadbddc09cd1", "arithmetic_steps": 6 }, "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." } }