{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261010-advance-20261014-r04-linear-equation-0086", "pilot_number": "R0060-30611", "number": 30611, "family_id": "qf-density14-linear-two-row-coincidence", "title": { "zh-Hant": "第 30611 題：兩行一次式的共同根", "en": "Question 30611: Two Affine Roots and a Common Root" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261010-advance-20261014-r04-linear-equation-0086", "batch_id": "batch-20261010-advance-20261014-r04-linear-equation-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-linear-equation" ], "title_zh": "兩行一次式的共同根", "stem_zh": "Q上ax+b=0；非零a才是一次；有序rows=[[2, -10], [-3, -9]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。", "answer": { "kind": "expression", "value": "(5/1, -3/1, 否, 無)", "explanation_zh": "兩個一次方程的根與共同解=(5/1, -3/1, 否, 無)。各行獨立求根，用精確相等判共同解，不能以係數外觀判相同。退化常數等式不是一次方程，有限回代非一般線性系統證明。" }, "difficulty": { "band": "lower_secondary", "level": 2, "skills": [ "兩行一次式的共同根" ] }, "variant": { "family_id": "qf-density14-linear-two-row-coincidence", "seed": 202610144086, "parameters": { "rows": [ [ 2, -10 ], [ -3, -9 ] ], "domain": "finite_Q_affine_row_protocol", "operation": "density14_linear_two_row_coincidence", "semantic_template": "qf-density14-linear-two-row-coincidence/[\"5/1\",\"-3/1\"]" } }, "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": "2026-10-10", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261010-advance-20261014-r04-linear-equation-0086", "batch_id": "batch-20261010-advance-20261014-r04-linear-equation-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "dfc5575d78aca0c0978ab8ae134ca079142fc6ebddf0523a6df7c83c49862db9", "family_id": "qf-density14-linear-two-row-coincidence", "seed": 202610144086, "semantic_template": "qf-density14-linear-two-row-coincidence/[\"5/1\",\"-3/1\"]" }, "fields": { "title": "Two Affine Roots and a Common Root", "stem": "Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -10], [-3, -9]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.", "answer": { "kind": "expression", "value": "(5/1, -3/1, No, none)", "explanation": "Solve each row independently and compare roots exactly; coefficient appearance alone does not decide common solutions. The exact result, retaining the requested order, is (5/1, -3/1, No, none)." }, "skills": [ "Two Affine Roots and a Common Root" ], "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": "Q上ax+b=0；非零a才是一次；有序rows=[[2, -10], [-3, -9]]，每行[a,b]且a≠0。依序答 (root1, root2, 共同根存在是否, 共同root或無)。", "en": "Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -10], [-3, -9]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-density14-linear-two-row-coincidence, density14_linear_two_row_coincidence, finite_Q_affine_row_protocol]", "after": "derived = [[root1, root2, equal, common_root_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": "Linear original row0 ORIGINAL negated constant := sub(0, -10)", "after": "Linear original row0 ORIGINAL negated constant = 10", "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 Linear original row0 ORIGINAL negated constant，不以小數近似替代。", "en": "Recompute Linear original row0 ORIGINAL negated constant with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "-10" ], "output": "10", "label": "Linear original row0 ORIGINAL negated constant" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Linear original row0 ORIGINAL negative b over a := div(10, 2)", "after": "Linear original row0 ORIGINAL negative b over a = 5", "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 Linear original row0 ORIGINAL negative b over a，不以小數近似替代。", "en": "Recompute Linear original row0 ORIGINAL negative b over a with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "10", "2" ], "output": "5", "label": "Linear original row0 ORIGINAL negative b over a" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Linear original row0 reverse ORIGINAL a times root := mul(2, 5)", "after": "Linear original row0 reverse ORIGINAL a times root = 10", "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 Linear original row0 reverse ORIGINAL a times root，不以小數近似替代。", "en": "Recompute Linear original row0 reverse ORIGINAL a times root with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "5" ], "output": "10", "label": "Linear original row0 reverse ORIGINAL a times root" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Linear original row0 reverse ORIGINAL residual plus b := sum(10, -10)", "after": "Linear original row0 reverse ORIGINAL residual plus b = 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 Linear original row0 reverse ORIGINAL residual plus b，不以小數近似替代。", "en": "Recompute Linear original row0 reverse ORIGINAL residual plus b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "10", "-10" ], "output": "0", "label": "Linear original row0 reverse ORIGINAL residual plus b" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [Linear original row0, 2, -10]", "after": "derived = [5/1, 0/1]", "relation": "implies", "rule_id": "Linear_NONZERO_a_exact_Q_root_AND_original_substitution", "conditions": [], "explanation": { "zh-Hant": "a非0才可除，ax+b=0 iff x=-b/a；以原係數回代殘差0確認。負a也約成正den，derived root不截SAFE，不用近似。", "en": "ONLY nonzero a permits division:ax+b=0 IFF x=-b/a.Substitute into the ORIGINAL coefficients to verify zero residual.Normalize a negative a to a positive denominator;derived roots are not SAFE-capped or approximated." }, "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": "S007", "uses": [ "G001", "S006" ], "before": "Linear original row1 ORIGINAL negated constant := sub(0, -9)", "after": "Linear original row1 ORIGINAL negated constant = 9", "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 Linear original row1 ORIGINAL negated constant，不以小數近似替代。", "en": "Recompute Linear original row1 ORIGINAL negated constant with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "-9" ], "output": "9", "label": "Linear original row1 ORIGINAL negated constant" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Linear original row1 ORIGINAL negative b over a := div(9, -3)", "after": "Linear original row1 ORIGINAL negative b over a = -3", "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 Linear original row1 ORIGINAL negative b over a，不以小數近似替代。", "en": "Recompute Linear original row1 ORIGINAL negative b over a with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "9", "-3" ], "output": "-3", "label": "Linear original row1 ORIGINAL negative b over a" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Linear original row1 reverse ORIGINAL a times root := mul(-3, -3)", "after": "Linear original row1 reverse ORIGINAL a times root = 9", "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 Linear original row1 reverse ORIGINAL a times root，不以小數近似替代。", "en": "Recompute Linear original row1 reverse ORIGINAL a times root with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-3", "-3" ], "output": "9", "label": "Linear original row1 reverse ORIGINAL a times root" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Linear original row1 reverse ORIGINAL residual plus b := sum(9, -9)", "after": "Linear original row1 reverse ORIGINAL residual plus b = 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 Linear original row1 reverse ORIGINAL residual plus b，不以小數近似替代。", "en": "Recompute Linear original row1 reverse ORIGINAL residual plus b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "9", "-9" ], "output": "0", "label": "Linear original row1 reverse ORIGINAL residual plus b" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "source roles = [Linear original row1, -3, -9]", "after": "derived = [-3/1, 0/1]", "relation": "implies", "rule_id": "Linear_NONZERO_a_exact_Q_root_AND_original_substitution", "conditions": [], "explanation": { "zh-Hant": "a非0才可除，ax+b=0 iff x=-b/a；以原係數回代殘差0確認。負a也約成正den，derived root不截SAFE，不用近似。", "en": "ONLY nonzero a permits division:ax+b=0 IFF x=-b/a.Substitute into the ORIGINAL coefficients to verify zero residual.Normalize a negative a to a positive denominator;derived roots are not SAFE-capped or approximated." }, "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": "S012", "uses": [ "G001", "S011" ], "before": "source roles = [[[2, -10], [-3, -9]]]", "after": "derived = [[5/1, -3/1], 否, 無]", "relation": "implies", "rule_id": "Linear_TWO_independent_nonzero_rows_root_coincidence_NOT_coefficients", "conditions": [], "explanation": { "zh-Hant": "兩row都先驗a非0，各自解根再精確比較。係數不同仍可同根；無共同根回none，不造數值0或以近似判相等。", "en": "Require nonzero a in BOTH rows,solve each root independently,then compare exactly.Different coefficients can share a root.Without a common root return none,not numerical0 or approximate equality." }, "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": [ "5/1", "-3/1", false, "無" ] }, "conclusion": { "canonical_value": "(5/1, -3/1, 否, 無)", "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-20261010-advance-20261014-r04-linear-equation-foundations", "batch_path": "public/data/questions/batches/batch-20261010-advance-20261014-r04-linear-equation-foundations/questions.jsonl", "batch_sha256": "2fbe5515808bcc208c053f531c3c0e90d3c29d02bad07d36e1d0e64738410d2e", "original_line": 86, "original_line_sha256": "f2313a16fb423d838f125dc4e39b3ce2706e4a39d62f8816c19215867677c65f", "question_compact_sha256": "dfc5575d78aca0c0978ab8ae134ca079142fc6ebddf0523a6df7c83c49862db9", "english_row_compact_sha256": "c02d96f051bd46e9953b757184e611c7542133f2a8585c344f8e7c5933523752", "english_manifest_path": "public/data/i18n/editions/batch-20261010-advance-20261014-r04-linear-equation-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261010-advance-20261014-r04-linear-equation-foundations.en.jsonl" }, "mko_bindings": { "mko-linear-equation": { "sha256": "70cc85ab6a989077a196caad95d1800fef7d3e5988f0a9cdc5c0c700600ef109", "english_sha256": "02c486fa9699473d8637ad7a935af439f6824eb0d4d00498a72e210457f394d9", "original_path": "public/data/mko/linear-equation.json", "english_path": "public/data/i18n/editions/mko-linear-equation.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "322533190c1eb10144c4c59b6c70cc2f8105108b5f4f3a39dd577712e7185c02", "original_candidate_sha256": "8072904ad1693c63c76a4878417460280c1c0b350672dae6674302a474ca54c7", "original_candidate_line_sha256": "8072904ad1693c63c76a4878417460280c1c0b350672dae6674302a474ca54c7" }, "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": 12, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "5a5424a48d2d26b0cb6dcfd4e482896d54db718895676c3f8907a5bb24f7324c", "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." } }