{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261005-advance-20261006-r01-order-relation-0024", "pilot_number": "R0067-34124", "number": 34124, "family_id": "qf-order-rational-comparison", "title": { "zh-Hant": "第 34124 題：正分母有理數的精確順序", "en": "Question 34124: Exact Order of Positive-Denominator Rationals" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261005-advance-20261006-r01-order-relation-0024", "batch_id": "batch-20261005-advance-20261006-r01-order-relation-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-order-relation" ], "title_zh": "正分母有理數的精確順序", "stem_zh": "依實數順序，比較 -24/5 與 13/4，分母明示為正整數；回答 <、= 或 >。", "answer": { "kind": "short_text", "value": "<", "explanation_zh": "交叉乘積=-96,65。正分母交叉相乘保留順序，以精確整數乘積比較，不靠小數外觀。" }, "difficulty": { "band": "lower_secondary", "level": 3, "skills": [ "正分母有理數的精確順序" ] }, "variant": { "family_id": "qf-order-rational-comparison", "seed": 202610061024, "parameters": { "p": -24, "q": 5, "r": 13, "s": 4, "operation": "rational_comparison", "semantic_template": "qf-order-rational-comparison/{\"inputs\":[\"-24/5\",\"13/4\"],\"operation\":\"rational_comparison\"}" } }, "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": "2026-10-05", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261005-advance-20261006-r01-order-relation-0024", "batch_id": "batch-20261005-advance-20261006-r01-order-relation-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "7896de0057ad784c916e2e734eeb74bb709c93a8b72529ebe5b17feb74157499", "family_id": "qf-order-rational-comparison", "seed": 202610061024, "semantic_template": "qf-order-rational-comparison/{\"inputs\":[\"-24/5\",\"13/4\"],\"operation\":\"rational_comparison\"}" }, "fields": { "title": "Exact Order of Positive-Denominator Rationals", "stem": "Under the usual real order, compare (-24)/5 and (13)/4. Both denominators are positive. Give <, =, or > in that order.", "answer": { "kind": "short_text", "value": "<", "explanation": "The exact cross products are -96 and 65, giving <. Positive denominators preserve the comparison direction." }, "skills": [ "Exact Order of Positive-Denominator Rationals" ], "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": "依實數順序，比較 -24/5 與 13/4，分母明示為正整數；回答 <、= 或 >。", "en": "Under the usual real order, compare (-24)/5 and (13)/4. Both denominators are positive. Give <, =, or > in that order." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-order-rational-comparison, [operation, p, q, r, s]]", "after": "derived = [rational_comparison, no-invented-domain]", "relation": "implies", "rule_id": "R67_SOURCE_specific_keys_operation_domain_and_numeric_bank", "conditions": [], "explanation": { "zh-Hant": "按本bank核keys與operation/domain：legacy為各表underscore，MOD50用local hyphen；BASE參數較寬不加未知欄限制。Number.isInteger與SAFE兩種來源界分開，原JSON數字表示仍完整保存。", "en": "Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Order ORIGINAL first numerator times second positive denominator := mul(-24, 4)", "after": "Order ORIGINAL first numerator times second positive denominator = -96", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。", "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Order ORIGINAL first numerator times second positive denominator，不以小數近似替代。", "en": "Recompute Order ORIGINAL first numerator times second positive denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "4" ], "output": "-96", "label": "Order ORIGINAL first numerator times second positive denominator" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Order ORIGINAL second numerator times first positive denominator := mul(13, 5)", "after": "Order ORIGINAL second numerator times first positive denominator = 65", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。", "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Order ORIGINAL second numerator times first positive denominator，不以小數近似替代。", "en": "Recompute Order ORIGINAL second numerator times first positive denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13", "5" ], "output": "65", "label": "Order ORIGINAL second numerator times first positive denominator" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [-24, 5, 13, 4]", "after": "derived = [-96, 65, <]", "relation": "implies", "rule_id": "Order_POSITIVE_denominators_cross_products_preserve_direction", "conditions": [], "explanation": { "zh-Hant": "兩分母皆正，乘共同正數q*s不反轉順序，所以比較p*s與r*q。相等包含未約分的等值分數；衍生交叉乘積不重套RAW SAFE界。", "en": "Both denominators are POSITIVE,so multiplying by q*s preserves direction.Compare p*s with r*q.Equality includes unreduced equivalent fractions.Derived cross products are not recapped at RAW SAFE." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。", "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof." } }, "arithmetic": null } ], "math_model": { "result": "<" }, "conclusion": { "canonical_value": "<", "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-20261005-advance-20261006-r01-order-relation-foundations", "batch_path": "public/data/questions/batches/batch-20261005-advance-20261006-r01-order-relation-foundations/questions.jsonl", "batch_sha256": "7445d3cd6c8f4cc77448d51656238ce18229a8b5084f76112e419933e63d33a8", "original_line": 24, "original_line_sha256": "a82233c51fdfa1eb48c3227171a22b5e2fe5c80405e0363848d33eff207e1322", "question_compact_sha256": "7896de0057ad784c916e2e734eeb74bb709c93a8b72529ebe5b17feb74157499", "english_row_compact_sha256": "fae66b78821d7dff0cd73d28970817e4280a083bd253d83cffd33885fe24a413", "english_manifest_path": "public/data/i18n/editions/batch-20261005-advance-20261006-r01-order-relation-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261005-advance-20261006-r01-order-relation-foundations.en.jsonl" }, "mko_bindings": { "mko-order-relation": { "sha256": "95554189da3a451f8a69edd139e154fd9254524119837ea1643487a5646a2dc3", "english_sha256": "3bb53f0ce66755124a9e09f8d185cdebfbb4e6f8cc93a7733a5c08ceba6bdcd6", "original_path": "public/data/mko/order-relation.json", "english_path": "public/data/i18n/editions/mko-order-relation.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "6bb156cec7bdb6baf2497762702fc01f94fd1db55a5926bc575b12ff9237a4de", "original_candidate_sha256": "14678522bc9098ab225a2756f1e3bb7443f6a6c91e8953a2497704ab42e5e829", "original_candidate_line_sha256": "14678522bc9098ab225a2756f1e3bb7443f6a6c91e8953a2497704ab42e5e829" }, "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-20261026-012-p68", "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": "c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c", "question_first_witness_sha256": "12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 4, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "e4ccd04cc36b379397633fe2a8699ea9d86e50c9db35c38b44369160199280b0", "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." } }