{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20260924-equation-linear-solution-class-0010", "pilot_number": "R0062-31560", "number": 31560, "family_id": "qf-equation-linear-solution-class", "title": { "zh-Hant": "第 31560 題：一般一次方程的解分類", "en": "Question 31560: Classifying a General Linear Equation" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20260924-equation-linear-solution-class-0010", "batch_id": "batch-20260924-equation-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-equation" ], "title_zh": "一般一次方程的解分類", "stem_zh": "在有理數論域分類 11x+(9)=14x+(3) 的解集。", "answer": { "kind": "short_text", "value": "唯一解 x=2", "explanation_zh": "(11−14)x=-6，故 唯一解 x=2。" }, "difficulty": { "band": "lower_secondary", "level": 2, "skills": [ "一次方程", "解分類" ] }, "variant": { "family_id": "qf-equation-linear-solution-class", "seed": 20261234, "parameters": { "form": "a*x+b=c*x+d", "a": 11, "b": 9, "c": 14, "d": 3, "domain": "rationals", "semantic_template": "qf-equation-linear-solution-class/{\"a\":11,\"b\":9,\"c\":14,\"d\":3,\"domain\":\"rationals\",\"form\":\"a*x+b=c*x+d\"}" } }, "verification": { "status": "candidate", "checks": [ "schema_pending", "independent_recomputation_pending" ], "warning_zh": "本題為 candidate 練習；有限精確重算不是形式證明或人工審查。" }, "provenance": { "generator_role": "question_producer", "generator_model": "GPT-5.6 Terra", "generated_at": "2026-09-24", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20260924-equation-linear-solution-class-0010", "batch_id": "batch-20260924-equation-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "d8de12d14101cc68b4fb409579838b1f82acb49f6275a200b1ec9c1da9204d03", "family_id": "qf-equation-linear-solution-class", "seed": 20261234, "semantic_template": "qf-equation-linear-solution-class/{\"a\":11,\"b\":9,\"c\":14,\"d\":3,\"domain\":\"rationals\",\"form\":\"a*x+b=c*x+d\"}" }, "fields": { "title": "Classifying a General Linear Equation", "stem": "Over the rational numbers, classify the solution set of 11x+(9)=14x+(3).", "answer": { "kind": "short_text", "value": "unique solution x=2", "explanation": "(11−14)x=-6, so the equation has unique solution x=2." }, "skills": [ "linear equations", "solution classification" ], "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": "在有理數論域分類 11x+(9)=14x+(3) 的解集。", "en": "Over the rational numbers, classify the solution set of 11x+(9)=14x+(3)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-equation-linear-solution-class, rationals, no-operation-field, a*x+b=c*x+d]", "after": "derived = [[all_Q_no_solution_or_unique_compact_rational_STRING]]", "relation": "implies", "rule_id": "R62_EXACT_source_keys_domain_operation_form_AND_output_types", "conditions": [], "explanation": { "zh-Hant": "按原bank核欄位：BASE方程用form及論域，BASE事件／補集無operation，階乘無domain；ext18用local-hyphen，其他legacy用underscore。輸出型別不統一成tuple或fraction。", "en": "Check THIS bank interface:BASE equations use form/domain;BASE events/complement have no operation;factorials have no domain.ext18 uses local hyphens,other legacy operations underscores.Do not force every answer into a tuple or fraction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。", "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [11, 9, a*x+b=c*x+d, rationals]", "after": "derived = [是]", "relation": "implies", "rule_id": "Equation_BASE_integer_coefficients_declared_Z_OR_Q_NOT_blanket_SAFE", "conditions": [], "explanation": { "zh-Hant": "原BASE bank驗整數而非另訂SAFE界；本候選核對凍結題的精確整數資料，不宣稱任意unsafe JavaScript數值行為已認證。Z解與Q解分開。", "en": "The BASE bank checks integers,not an added SAFE limit.This candidate checks the frozen exact integer slice,not arbitrary unsafe JavaScript-number behavior.Distinguish solutions over Z and Q." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。", "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Equation class ORIGINAL a minus c := sub(11, 14)", "after": "Equation class ORIGINAL a minus c = -3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Equation class ORIGINAL a minus c，不以小數近似替代。", "en": "Recompute Equation class ORIGINAL a minus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "11", "14" ], "output": "-3", "label": "Equation class ORIGINAL a minus c" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Equation class ORIGINAL d minus b := sub(3, 9)", "after": "Equation class ORIGINAL d minus b = -6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Equation class ORIGINAL d minus b，不以小數近似替代。", "en": "Recompute Equation class ORIGINAL d minus b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "3", "9" ], "output": "-6", "label": "Equation class ORIGINAL d minus b" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [11, 9, 14, 3]", "after": "derived = [-3, -6]", "relation": "implies", "rule_id": "Equation_CLASS_both_sides_move_terms_equivalent_Q", "conditions": [], "explanation": { "zh-Hant": "兩邊同減cx及b，原式等價於(a−c)x=d−b。此時還沒除以a−c，必須先分類退化情形。", "en": "Subtract cx and b from BOTH sides to obtain the equivalent(a−c)x=d−b.IMPORTANT:do not divide by a−c until degeneracy is classified." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。", "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Equation class ORIGINAL constant over nonzero coefficient := div(-6, -3)", "after": "Equation class ORIGINAL constant over nonzero coefficient = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Equation class ORIGINAL constant over nonzero coefficient，不以小數近似替代。", "en": "Recompute Equation class ORIGINAL constant over nonzero coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-6", "-3" ], "output": "2", "label": "Equation class ORIGINAL constant over nonzero coefficient" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Equation class reverse ORIGINAL a times x := mul(11, 2)", "after": "Equation class reverse ORIGINAL a times x = 22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Equation class reverse ORIGINAL a times x，不以小數近似替代。", "en": "Recompute Equation class reverse ORIGINAL a times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "11", "2" ], "output": "22", "label": "Equation class reverse ORIGINAL a times x" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Equation class reverse ORIGINAL ax plus b := sum(22, 9)", "after": "Equation class reverse ORIGINAL ax plus b = 31", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Equation class reverse ORIGINAL ax plus b，不以小數近似替代。", "en": "Recompute Equation class reverse ORIGINAL ax plus b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "22", "9" ], "output": "31", "label": "Equation class reverse ORIGINAL ax plus b" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Equation class reverse ORIGINAL c times x := mul(14, 2)", "after": "Equation class reverse ORIGINAL c times x = 28", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Equation class reverse ORIGINAL c times x，不以小數近似替代。", "en": "Recompute Equation class reverse ORIGINAL c times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "14", "2" ], "output": "28", "label": "Equation class reverse ORIGINAL c times x" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Equation class reverse ORIGINAL cx plus d := sum(28, 3)", "after": "Equation class reverse ORIGINAL cx plus d = 31", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Equation class reverse ORIGINAL cx plus d，不以小數近似替代。", "en": "Recompute Equation class reverse ORIGINAL cx plus d with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "28", "3" ], "output": "31", "label": "Equation class reverse ORIGINAL cx plus d" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Equation class reverse ORIGINAL sides difference := sub(31, 31)", "after": "Equation class reverse ORIGINAL sides difference = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Equation class reverse ORIGINAL sides difference，不以小數近似替代。", "en": "Recompute Equation class reverse ORIGINAL sides difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "31", "31" ], "output": "0", "label": "Equation class reverse ORIGINAL sides difference" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "source roles = [11, 9, 14, 3, -3, -6]", "after": "derived = [2/1, 31, 31, 0, 唯一解 x=2]", "relation": "implies", "rule_id": "Equation_CLASS_NONZERO_unique_Q_root_positive_den_OMIT1", "conditions": [], "explanation": { "zh-Hant": "非零係數才除，得到唯一Q解。約分及正分母後，整數解的顯示省/1；用原兩側逐項回代，不只驗整理過的式子。", "en": "ONLY the nonzero coefficient permits division,giving a UNIQUE Q solution.Reduce to a positive denominator and OMIT/1 for integer display.Substitute termwise into BOTH ORIGINAL sides,not only the rearranged equation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。", "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate." } }, "arithmetic": null } ], "math_model": { "result": "唯一解 x=2" }, "conclusion": { "canonical_value": "唯一解 x=2", "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-20260924-equation-foundations", "batch_path": "public/data/questions/batches/batch-20260924-equation-foundations/questions.jsonl", "batch_sha256": "6d31f2307850ca273164b6b9024cb43f5b2102f0834c5fef9c5dd2285ffb256a", "original_line": 85, "original_line_sha256": "7840c22625735fceb171ec5c6c9c93ff5bf62f84d0edeee979846f5430cddb8c", "question_compact_sha256": "d8de12d14101cc68b4fb409579838b1f82acb49f6275a200b1ec9c1da9204d03", "english_row_compact_sha256": "7a95bdd765c5d52fc62c146415e1ed36c71692f88124a548b000e14e2c6feae5", "english_manifest_path": "public/data/i18n/editions/batch-20260924-equation-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20260924-equation-foundations.en.jsonl" }, "mko_bindings": { "mko-equation": { "sha256": "6f8d6ca379da35727c14f320d99d0bc8c53c47ad478d0f462d30a4127aea0a78", "english_sha256": "de50265f38fde7fafcc45aa22db603721664273dbafb01a18c77a7d51b393bdd", "original_path": "public/data/mko/equation.json", "english_path": "public/data/i18n/editions/mko-equation.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "59d384ca3d958b27165f4295ff96efb155e7b12d3fb70bcdd0ed68148513c05a", "original_candidate_sha256": "90b955429d3ffbd66791b711fb4a7a862708d3d5b8b785167a226fba36f060dc", "original_candidate_line_sha256": "90b955429d3ffbd66791b711fb4a7a862708d3d5b8b785167a226fba36f060dc" }, "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-p63", "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": "963099ffc9f754069fdce6792949a85ed350472465eaceb3038fa8011cf672bd", "question_first_witness_sha256": "d7655324c5d11528e1af90b757384cebd3313e19dc5f6f5b9b8ea7624fb06f6f" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 12, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "fed22f61e1fb2d63f493ab4356a4d297bd0ce8860bd8b040048ec0717f059ed3", "arithmetic_steps": 8 }, "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." } }