{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261007-advance-20261008-r02-function-limit-0067", "pilot_number": "R0063-32292", "number": 32292, "family_id": "qf-function-limit-affine-delta", "title": { "zh-Hant": "第 32292 題：仿射 δ 見證的嚴格內部", "en": "Question 32292: Affine δ Witness with Strict Punctured Interior" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261007-advance-20261008-r02-function-limit-0067", "batch_id": "batch-20261007-advance-20261008-r02-function-limit-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-function-limit" ], "title_zh": "仿射 δ 見證的嚴格內部", "stem_zh": "去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(16)(x−A)，斜率 m=16。ε=2/1>0、δ=2/16>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？", "answer": { "kind": "short_text", "value": "是", "explanation_zh": "仿射界交叉乘積=(32, 32)；非嚴格比較=是。去心內部是嚴格不等式，因此 |m|δ=ε 仍合格；斜率零時任何正 δ 都合格，真值旗標不偽寫失敗不等式。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "仿射 δ 見證的嚴格內部" ] }, "variant": { "family_id": "qf-function-limit-affine-delta", "seed": 202610082067, "parameters": { "m": 16, "eps_n": 2, "eps_d": 1, "delta_n": 2, "delta_d": 16, "operation": "affine_delta_validity", "domain": "punctured_reals", "semantic_template": "qf-function-limit-affine-delta/{\"inputs\":[\"16\",\"2/1\",\"1/8\"],\"operation\":\"affine_delta_validity\"}" } }, "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": "2026-10-07", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261007-advance-20261008-r02-function-limit-0067", "batch_id": "batch-20261007-advance-20261008-r02-function-limit-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "bf7051e7ba3904d30ba63f051958a1dd0a8920748c63bb932145a5900eaec618", "family_id": "qf-function-limit-affine-delta", "seed": 202610082067, "semantic_template": "qf-function-limit-affine-delta/{\"inputs\":[\"16\",\"2/1\",\"1/8\"],\"operation\":\"affine_delta_validity\"}" }, "fields": { "title": "Affine δ Witness with Strict Punctured Interior", "stem": "For arbitrary real A,L, define f(x)=L+(16)(x−A) for x≠A on R∖{A}. With ε=2/1>0 and δ=2/16>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?", "answer": { "kind": "short_text", "value": "Yes", "explanation": "Check |m|δ≤ε by exact cross products, giving Yes. Equality is permitted because the punctured input distance is strictly below δ. If m=0 every positive δ works. The target is L, not an undefined f(A)." }, "skills": [ "Affine δ Witness with Strict Punctured Interior" ], "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": "去心實數論域，採完整兩側實數鄰域。固定任意實數 A、L，對 x≠A 定義 f(x)=L+(16)(x−A)，斜率 m=16。ε=2/1>0、δ=2/16>0。是否保證 0<|x−A|<δ 時 |f(x)−L|<ε？", "en": "For arbitrary real A,L, define f(x)=L+(16)(x−A) for x≠A on R∖{A}. With ε=2/1>0 and δ=2/16>0, does 0<|x−A|<δ guarantee |f(x)−L|<ε?" } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-function-limit-affine-delta, affine_delta_validity, punctured_reals, []]", "after": "derived = [[nonstrict_budget_YesNo_STRING]]", "relation": "implies", "rule_id": "R63_SOURCE_bank_keys_operation_domain_output_roles_metadata_not_oracle", "conditions": [], "explanation": { "zh-Hant": "BASE來源沒有本輪legacy的通用operation/domain/額外欄位禁令；保metadata而不當答案。PULSE與legacy則核來源精確欄位、operation及domain。不要把tuple、/1或輸入界強套全部題族。", "en": "BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。", "en": "Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Delta ORIGINAL absolute slope times delta numerator := mul(16, 2)", "after": "Delta ORIGINAL absolute slope times delta numerator = 32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Delta ORIGINAL absolute slope times delta numerator，不以小數近似替代。", "en": "Recompute Delta ORIGINAL absolute slope times delta numerator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "16", "2" ], "output": "32", "label": "Delta ORIGINAL absolute slope times delta numerator" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Delta ORIGINAL left cross product times epsilon denominator := mul(32, 1)", "after": "Delta ORIGINAL left cross product times epsilon denominator = 32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Delta ORIGINAL left cross product times epsilon denominator，不以小數近似替代。", "en": "Recompute Delta ORIGINAL left cross product times epsilon denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "32", "1" ], "output": "32", "label": "Delta ORIGINAL left cross product times epsilon denominator" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Delta ORIGINAL epsilon numerator times delta denominator := mul(2, 16)", "after": "Delta ORIGINAL epsilon numerator times delta denominator = 32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。", "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Delta ORIGINAL epsilon numerator times delta denominator，不以小數近似替代。", "en": "Recompute Delta ORIGINAL epsilon numerator times delta denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "16" ], "output": "32", "label": "Delta ORIGINAL epsilon numerator times delta denominator" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [16, 2, 1, 2, 16]", "after": "derived = [2/1, 1/8, 32, 32, 是]", "relation": "implies", "rule_id": "Limit_DELTA_ALL_positive_components_NONSTRICT_budget_STRICT_punctured_input", "conditions": [], "explanation": { "zh-Hant": "先驗四個raw分數分量全正，包括m0情形。因輸入距離嚴格小於δ，保證條件是|m|δ≤ε而非<；等號合法，m0誤差0。目標是L，不是未定義f(A)，A/L為任意固定實數而非新增參數。", "en": "FIRST validate ALL FOUR positive raw fraction components,EVEN at m0.Because the input distance is STRICTLY below δ,the budget criterion is|m|δ≤ε,NOT<.Equality is legal;m0 gives error0.The target is L,not undefined f(A);A/L are arbitrary fixed reals,not invented parameters." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。", "en": "Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null } ], "math_model": { "result": "是" }, "conclusion": { "canonical_value": "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-20261007-advance-20261008-r02-function-limit-foundations", "batch_path": "public/data/questions/batches/batch-20261007-advance-20261008-r02-function-limit-foundations/questions.jsonl", "batch_sha256": "1176f30642e50d59427c3026d10f3097c800f4e5fa9dd6f7e3796b1c401da318", "original_line": 67, "original_line_sha256": "268eca202fee57e5bb503eac6d58ca095adef7ba2c0c0b54405e86b48ad298cd", "question_compact_sha256": "bf7051e7ba3904d30ba63f051958a1dd0a8920748c63bb932145a5900eaec618", "english_row_compact_sha256": "7afed8381ee2c690bde6253f5dd44f78a48b605419754f1ec7d89bd0cc2b8529", "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261008-r02-function-limit-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261008-r02-function-limit-foundations.en.jsonl" }, "mko_bindings": { "mko-function-limit": { "sha256": "2dce49b3fcec9f9cf28bc85107ed2884a57b36ccc297c60e4ef5836b4a3c5ca3", "english_sha256": "608f319ceb9afeb2eb284c43c4a1d6d8d19c4e40dce81f0ac4df60cbb55f4c44", "original_path": "public/data/mko/function-limit.json", "english_path": "public/data/i18n/editions/mko-function-limit.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "f3271d6047df957dbf3d61120bd3dd374784c828b17849887e14caf59d93f745", "original_candidate_sha256": "0484f5a14296e3e8e19f8ef916c635a232558d2ce1cdeb0dab08d32f61bd3dc3", "original_candidate_line_sha256": "0484f5a14296e3e8e19f8ef916c635a232558d2ce1cdeb0dab08d32f61bd3dc3" }, "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-p64", "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": "acf03b776f4855e3869810bcbecf1116e3a2b11dfb1e2a5dfbcf8135ad24a7d0", "question_first_witness_sha256": "eadbbac3220988b52754f32450fb536d627217306d1ce28a49134f56c21bfa88" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 5, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "04d3f7b1b53a2175e0fd8707acf3c0dbdc2b84c660bca89bf8f716e25f5ad050", "arithmetic_steps": 3 }, "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." } }