{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261005-advance-20261006-r03-rate-of-change-0067", "pilot_number": "R0056-28617", "number": 28617, "family_id": "qf-rate-affine-average", "title": { "zh-Hant": "第 28617 題：實仿射函數的平均變化率", "en": "Question 28617: Average Rate of a Real Affine Function" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261005-advance-20261006-r03-rate-of-change-0067", "batch_id": "batch-20261005-advance-20261006-r03-rate-of-change-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-rate-of-change" ], "title_zh": "實仿射函數的平均變化率", "stem_zh": "在實數論域指定 f(x)=4x+(41)，求 x0=0 到 x1=17 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。", "answer": { "kind": "expression", "value": "4/1", "explanation_zh": "平均變化率=4/1。仿射函數的輸出差為 m 倍輸入差，非零差相消後為 m，常數式 m=0 也合法；這不是導數題。" }, "difficulty": { "band": "upper_secondary", "level": 2, "skills": [ "實仿射函數的平均變化率" ] }, "variant": { "family_id": "qf-rate-affine-average", "seed": 202610063067, "parameters": { "m": 4, "b": 41, "x0": 0, "x1": 17, "domain": "reals", "operation": "affine_average", "semantic_template": "qf-rate-affine-average/{\"inputs\":[\"4\"],\"operation\":\"affine_average\"}" } }, "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-r03-rate-of-change-0067", "batch_id": "batch-20261005-advance-20261006-r03-rate-of-change-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "adbd7bc45004547f86f80952001b5cdcb0a5e5cd382b21d9a07c279cdf90931f", "family_id": "qf-rate-affine-average", "seed": 202610063067, "semantic_template": "qf-rate-affine-average/{\"inputs\":[\"4\"],\"operation\":\"affine_average\"}" }, "fields": { "title": "Average Rate of a Real Affine Function", "stem": "On the full real domain, f(x)=(4)x+(41). For distinct inputs 0 and 17, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers.", "answer": { "kind": "expression", "value": "4/1", "explanation": "The common constant cancels in the endpoint difference, leaving slope m=4. The exact average rate is 4/1, including zero slope. This is an endpoint average rate, not a derivative." }, "skills": [ "Average Rate of a Real Affine Function" ], "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": "在實數論域指定 f(x)=4x+(41)，求 x0=0 到 x1=17 的平均變化率，兩個輸入不同；答案採最簡正分母 p/q。", "en": "On the full real domain, f(x)=(4)x+(41). For distinct inputs 0 and 17, find the average rate of change. Give a reduced fraction with positive denominator, retaining /1 for integers." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-rate-affine-average, affine_average]", "after": "derived = [exact_average_rate_fraction]", "relation": "implies", "rule_id": "R56_source_answer_role_and_EXACT_family_interface", "conditions": [], "explanation": { "zh-Hant": "本題輸出依原scalar／set／是非／fraction角色；有理數整數保留/1。不加未宣告domain，不改中英來源、公式或題目。", "en": "Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。", "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Rate ORIGINAL x1 minus x0 := sub(17, 0)", "after": "Rate ORIGINAL x1 minus x0 = 17", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。", "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Rate ORIGINAL x1 minus x0，不以小數近似替代。", "en": "Recompute Rate ORIGINAL x1 minus x0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "17", "0" ], "output": "17", "label": "Rate ORIGINAL x1 minus x0" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Rate affine ORIGINAL m times x0 := mul(4, 0)", "after": "Rate affine ORIGINAL m times x0 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。", "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Rate affine ORIGINAL m times x0，不以小數近似替代。", "en": "Recompute Rate affine ORIGINAL m times x0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "0" ], "output": "0", "label": "Rate affine ORIGINAL m times x0" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Rate affine ORIGINAL endpoint0 plus b := sum(0, 41)", "after": "Rate affine ORIGINAL endpoint0 plus b = 41", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。", "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Rate affine ORIGINAL endpoint0 plus b，不以小數近似替代。", "en": "Recompute Rate affine ORIGINAL endpoint0 plus b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "41" ], "output": "41", "label": "Rate affine ORIGINAL endpoint0 plus b" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Rate affine ORIGINAL m times x1 := mul(4, 17)", "after": "Rate affine ORIGINAL m times x1 = 68", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。", "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Rate affine ORIGINAL m times x1，不以小數近似替代。", "en": "Recompute Rate affine ORIGINAL m times x1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "17" ], "output": "68", "label": "Rate affine ORIGINAL m times x1" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Rate affine ORIGINAL endpoint1 plus b := sum(68, 41)", "after": "Rate affine ORIGINAL endpoint1 plus b = 109", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。", "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Rate affine ORIGINAL endpoint1 plus b，不以小數近似替代。", "en": "Recompute Rate affine ORIGINAL endpoint1 plus b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "68", "41" ], "output": "109", "label": "Rate affine ORIGINAL endpoint1 plus b" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [4, 41, 0, 17]", "after": "derived = [[41, 109], 4]", "relation": "implies", "rule_id": "Rate_WHOLE_affine_endpoint_subtraction", "conditions": [], "explanation": { "zh-Hant": "由完整f(x)=mx+b代兩端，b在差中消去，dy=m*dx。m=0也合法；平均割線斜率不是另取導數作答案。", "en": "Substitute both endpoints in the WHOLE law f(x)=mx+b.The intercept cancels,so dy=m*dx;m=0 is allowed.This is an endpoint secant computation,not a substituted derivative." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。", "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Rate ORIGINAL y1 minus y0 := sub(109, 41)", "after": "Rate ORIGINAL y1 minus y0 = 68", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。", "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Rate ORIGINAL y1 minus y0，不以小數近似替代。", "en": "Recompute Rate ORIGINAL y1 minus y0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "109", "41" ], "output": "68", "label": "Rate ORIGINAL y1 minus y0" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Rate ORIGINAL dy over dx := div(68, 17)", "after": "Rate ORIGINAL dy over dx = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。", "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Rate ORIGINAL dy over dx，不以小數近似替代。", "en": "Recompute Rate ORIGINAL dy over dx with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "68", "17" ], "output": "4", "label": "Rate ORIGINAL dy over dx" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "source roles = [0, 17, 41, 109, 17, 68]", "after": "derived = [4/1]", "relation": "implies", "rule_id": "Rate_oriented_quotient_reverse_BOTH_differences", "conditions": [], "explanation": { "zh-Hant": "保持同一端點次序，平均率=(y1-y0)/(x1-x0)。可同時逆轉兩個差，不可只逆一個；可負、可0，分母約正後整數也保/1。這不由有限樣本證一般導數。", "en": "Keep the SAME endpoint orientation:(y1-y0)/(x1-x0).Both differences may reverse together,not just one.The result may be negative or0;normalize a positive denominator and retain/1.This finite endpoint rate is not a general derivative certificate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。", "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance." } }, "arithmetic": null } ], "math_model": { "result": "4/1" }, "conclusion": { "canonical_value": "4/1", "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-r03-rate-of-change-foundations", "batch_path": "public/data/questions/batches/batch-20261005-advance-20261006-r03-rate-of-change-foundations/questions.jsonl", "batch_sha256": "f7f2bbd6cbef58468a160367d229b21789cc36fdb8bdd1e96573aa6af31dfae1", "original_line": 67, "original_line_sha256": "cb4e9c7fed9de508f8d3cce9b587eedc6d394ff6c2e6c595c65f6d44a003932a", "question_compact_sha256": "adbd7bc45004547f86f80952001b5cdcb0a5e5cd382b21d9a07c279cdf90931f", "english_row_compact_sha256": "0fb74930ef589afc1e30327e930934cd48ea1b286abf1a3f35d6f38064c88077", "english_manifest_path": "public/data/i18n/editions/batch-20261005-advance-20261006-r03-rate-of-change-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261005-advance-20261006-r03-rate-of-change-foundations.en.jsonl" }, "mko_bindings": { "mko-rate-of-change": { "sha256": "8c12a896998e745bbd3743c80ae005d1ffbc28d69ea33072b8e08d03c3837416", "english_sha256": "da425c268629ac1b5344a831974cfbf2e90a380deb76eb9cf2eab84d01f97ec2", "original_path": "public/data/mko/rate-of-change.json", "english_path": "public/data/i18n/editions/mko-rate-of-change.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "e340023047850507ee3fe2ae1466d606cfcbd26849aa3bb168d53248f38097e5", "original_candidate_sha256": "48b204d96b1928ee467343750c652e1a82f63a4750bd3a13bf026f92593b4771", "original_candidate_line_sha256": "48b204d96b1928ee467343750c652e1a82f63a4750bd3a13bf026f92593b4771" }, "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-p57", "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": "8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8", "question_first_witness_sha256": "3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759f" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 10, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "187eb6927f0a5442859ef166e82eed762c0be9b4a0f5df5802eaa415ad593dae", "arithmetic_steps": 7 }, "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." } }