{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261010-advance-20261014-r04-linear-equation-0065", "pilot_number": "R0060-30590", "number": 30590, "family_id": "qf-density14-linear-root-substitution-grid", "title": { "zh-Hant": "第 30590 題：根周圍的候選與回代", "en": "Question 30590: Exact Root Offsets and Original Substitution" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261010-advance-20261014-r04-linear-equation-0065", "batch_id": "batch-20261010-advance-20261014-r04-linear-equation-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-linear-equation" ], "title_zh": "根周圍的候選與回代", "stem_zh": "Q上ax+b=0；非零a才是一次；a=15≠0、b=2，offsets=[0/1, 15/2, -14/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。", "answer": { "kind": "expression", "value": "(-2/15, [-2/15, 221/30, -24/5], [0/1, 225/2, -70/1], [是, 否, 否])", "explanation_zh": "根周圍候選與回代殘差=(-2/15, [-2/15, 221/30, -24/5], [0/1, 225/2, -70/1], [是, 否, 否])。先求精確根，再逐一加偏移並回代原係數，不只看候選接近。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": [ "根周圍的候選與回代" ] }, "variant": { "family_id": "qf-density14-linear-root-substitution-grid", "seed": 202610144065, "parameters": { "a": 15, "b": 2, "offsets": [ [ 0, 1 ], [ 15, 2 ], [ -14, 3 ] ], "domain": "finite_Q_affine_row_protocol", "operation": "density14_linear_root_substitution_grid", "semantic_template": "qf-density14-linear-root-substitution-grid/[\"-2/15\",[\"-2/15\",\"221/30\",\"-24/5\"],[\"0/1\",\"225/2\",\"-70/1\"]]" } }, "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-0065", "batch_id": "batch-20261010-advance-20261014-r04-linear-equation-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "22cddaa975570aca2a314d3959c329217bac940fcfdf845d592b303ccd6c930a", "family_id": "qf-density14-linear-root-substitution-grid", "seed": 202610144065, "semantic_template": "qf-density14-linear-root-substitution-grid/[\"-2/15\",[\"-2/15\",\"221/30\",\"-24/5\"],[\"0/1\",\"225/2\",\"-70/1\"]]" }, "fields": { "title": "Exact Root Offsets and Original Substitution", "stem": "Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=15≠0, b=2, ordered offsets=[0/1, 15/2, -14/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers.", "answer": { "kind": "expression", "value": "(-2/15, [-2/15, 221/30, -24/5], [0/1, 225/2, -70/1], [Yes, No, No])", "explanation": "Find the exact root, add each offset, then substitute the original coefficients; numerical closeness is not exact solution eligibility. The exact result, retaining the requested order, is (-2/15, [-2/15, 221/30, -24/5], [0/1, 225/2, -70/1], [Yes, No, No])." }, "skills": [ "Exact Root Offsets and Original Substitution" ], "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；a=15≠0、b=2，offsets=[0/1, 15/2, -14/3]。依序答 (root, root+offset三值, 原式回代殘差三值, 成立資格三值)。", "en": "Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. a=15≠0, b=2, ordered offsets=[0/1, 15/2, -14/3]. Give (root, three root+offset candidates, original residuals, equation-eligibility mask). Use exact reduced fractions with positive denominators, retaining /1 for integers." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-density14-linear-root-substitution-grid, density14_linear_root_substitution_grid, finite_Q_affine_row_protocol]", "after": "derived = [[root, ALL_candidates, ALL_original_residuals, ALL_exact_solution_flags]]", "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 grid base ORIGINAL negated constant := sub(0, 2)", "after": "Linear grid base ORIGINAL negated constant = -2", "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 grid base ORIGINAL negated constant，不以小數近似替代。", "en": "Recompute Linear grid base ORIGINAL negated constant with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "2" ], "output": "-2", "label": "Linear grid base ORIGINAL negated constant" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Linear grid base ORIGINAL negative b over a := div(-2, 15)", "after": "Linear grid base ORIGINAL negative b over a = -2/15", "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 grid base ORIGINAL negative b over a，不以小數近似替代。", "en": "Recompute Linear grid base ORIGINAL negative b over a with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-2", "15" ], "output": "-2/15", "label": "Linear grid base ORIGINAL negative b over a" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Linear grid base reverse ORIGINAL a times root := mul(15, -2/15)", "after": "Linear grid base reverse ORIGINAL a times root = -2", "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 grid base reverse ORIGINAL a times root，不以小數近似替代。", "en": "Recompute Linear grid base reverse ORIGINAL a times root with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "15", "-2/15" ], "output": "-2", "label": "Linear grid base reverse ORIGINAL a times root" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Linear grid base reverse ORIGINAL residual plus b := sum(-2, 2)", "after": "Linear grid base 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 grid base reverse ORIGINAL residual plus b，不以小數近似替代。", "en": "Recompute Linear grid base reverse ORIGINAL residual plus b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "2" ], "output": "0", "label": "Linear grid base reverse ORIGINAL residual plus b" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [Linear grid base, 15, 2]", "after": "derived = [-2/15, 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": "source roles = [Linear grid original offset0, [0, 1]]", "after": "derived = [0/1]", "relation": "implies", "rule_id": "R60_RAW_SAFE_rational_pair_NONZERO_den_SIGN_normalized", "conditions": [], "explanation": { "zh-Hant": "raw兩component SAFE、den非0，可負den，先正規化sign/gcd；衍生fraction不限SAFE或1e6，整數保/1。", "en": "Both RAW pair components are SAFE and the denominator is nonzero.Negative denominators normalize sign/gcd;derived fractions are not SAFE/1e6-capped.Integer fractions retain/1." }, "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": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [Linear grid original offset1, [15, 2]]", "after": "derived = [15/2]", "relation": "implies", "rule_id": "R60_RAW_SAFE_rational_pair_NONZERO_den_SIGN_normalized", "conditions": [], "explanation": { "zh-Hant": "raw兩component SAFE、den非0，可負den，先正規化sign/gcd；衍生fraction不限SAFE或1e6，整數保/1。", "en": "Both RAW pair components are SAFE and the denominator is nonzero.Negative denominators normalize sign/gcd;derived fractions are not SAFE/1e6-capped.Integer fractions retain/1." }, "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": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [Linear grid original offset2, [-14, 3]]", "after": "derived = [-14/3]", "relation": "implies", "rule_id": "R60_RAW_SAFE_rational_pair_NONZERO_den_SIGN_normalized", "conditions": [], "explanation": { "zh-Hant": "raw兩component SAFE、den非0，可負den，先正規化sign/gcd；衍生fraction不限SAFE或1e6，整數保/1。", "en": "Both RAW pair components are SAFE and the denominator is nonzero.Negative denominators normalize sign/gcd;derived fractions are not SAFE/1e6-capped.Integer fractions retain/1." }, "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": "S010", "uses": [ "G001", "S009" ], "before": "Linear grid ORIGINAL root plus offset0 := sum(-2/15, 0)", "after": "Linear grid ORIGINAL root plus offset0 = -2/15", "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 grid ORIGINAL root plus offset0，不以小數近似替代。", "en": "Recompute Linear grid ORIGINAL root plus offset0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2/15", "0" ], "output": "-2/15", "label": "Linear grid ORIGINAL root plus offset0" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Linear grid ORIGINAL a times candidate0 := mul(15, -2/15)", "after": "Linear grid ORIGINAL a times candidate0 = -2", "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 grid ORIGINAL a times candidate0，不以小數近似替代。", "en": "Recompute Linear grid ORIGINAL a times candidate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "15", "-2/15" ], "output": "-2", "label": "Linear grid ORIGINAL a times candidate0" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Linear grid ORIGINAL candidate0 residual plus b := sum(-2, 2)", "after": "Linear grid ORIGINAL candidate0 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 grid ORIGINAL candidate0 residual plus b，不以小數近似替代。", "en": "Recompute Linear grid ORIGINAL candidate0 residual plus b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "2" ], "output": "0", "label": "Linear grid ORIGINAL candidate0 residual plus b" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Linear grid reverse a times offset0 := mul(15, 0)", "after": "Linear grid reverse a times offset0 = 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 grid reverse a times offset0，不以小數近似替代。", "en": "Recompute Linear grid reverse a times offset0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "15", "0" ], "output": "0", "label": "Linear grid reverse a times offset0" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "Linear grid ORIGINAL root plus offset1 := sum(-2/15, 15/2)", "after": "Linear grid ORIGINAL root plus offset1 = 221/30", "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 grid ORIGINAL root plus offset1，不以小數近似替代。", "en": "Recompute Linear grid ORIGINAL root plus offset1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2/15", "15/2" ], "output": "221/30", "label": "Linear grid ORIGINAL root plus offset1" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "Linear grid ORIGINAL a times candidate1 := mul(15, 221/30)", "after": "Linear grid ORIGINAL a times candidate1 = 221/2", "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 grid ORIGINAL a times candidate1，不以小數近似替代。", "en": "Recompute Linear grid ORIGINAL a times candidate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "15", "221/30" ], "output": "221/2", "label": "Linear grid ORIGINAL a times candidate1" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Linear grid ORIGINAL candidate1 residual plus b := sum(221/2, 2)", "after": "Linear grid ORIGINAL candidate1 residual plus b = 225/2", "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 grid ORIGINAL candidate1 residual plus b，不以小數近似替代。", "en": "Recompute Linear grid ORIGINAL candidate1 residual plus b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "221/2", "2" ], "output": "225/2", "label": "Linear grid ORIGINAL candidate1 residual plus b" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "Linear grid reverse a times offset1 := mul(15, 15/2)", "after": "Linear grid reverse a times offset1 = 225/2", "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 grid reverse a times offset1，不以小數近似替代。", "en": "Recompute Linear grid reverse a times offset1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "15", "15/2" ], "output": "225/2", "label": "Linear grid reverse a times offset1" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "Linear grid ORIGINAL root plus offset2 := sum(-2/15, -14/3)", "after": "Linear grid ORIGINAL root plus offset2 = -24/5", "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 grid ORIGINAL root plus offset2，不以小數近似替代。", "en": "Recompute Linear grid ORIGINAL root plus offset2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2/15", "-14/3" ], "output": "-24/5", "label": "Linear grid ORIGINAL root plus offset2" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "Linear grid ORIGINAL a times candidate2 := mul(15, -24/5)", "after": "Linear grid ORIGINAL a times candidate2 = -72", "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 grid ORIGINAL a times candidate2，不以小數近似替代。", "en": "Recompute Linear grid ORIGINAL a times candidate2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "15", "-24/5" ], "output": "-72", "label": "Linear grid ORIGINAL a times candidate2" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "Linear grid ORIGINAL candidate2 residual plus b := sum(-72, 2)", "after": "Linear grid ORIGINAL candidate2 residual plus b = -70", "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 grid ORIGINAL candidate2 residual plus b，不以小數近似替代。", "en": "Recompute Linear grid ORIGINAL candidate2 residual plus b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-72", "2" ], "output": "-70", "label": "Linear grid ORIGINAL candidate2 residual plus b" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "Linear grid reverse a times offset2 := mul(15, -14/3)", "after": "Linear grid reverse a times offset2 = -70", "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 grid reverse a times offset2，不以小數近似替代。", "en": "Recompute Linear grid reverse a times offset2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "15", "-14/3" ], "output": "-70", "label": "Linear grid reverse a times offset2" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "source roles = [15, 2, -2/15, [0/1, 15/2, -14/3]]", "after": "derived = [[-2/15, 221/30, -24/5], [0/1, 225/2, -70/1], [是, 否, 否]]", "relation": "implies", "rule_id": "Linear_grid_EVERY_original_equation_residual_exact_zero_NOT_nearness", "conditions": [], "explanation": { "zh-Hant": "每個candidate以原a*x+b算，不用距根接近或offset的外觀。反算residual=a*offset；a非0，所以offset0才真解，負den先normalize。", "en": "For EVERY candidate evaluate the ORIGINAL a*x+b,not nearness to a root or raw offset appearance.The reverse check is residual=a*offset;with a!=0,ONLY offset0 gives a solution.Normalize negative denominators first." }, "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": [ "-2/15", [ "-2/15", "221/30", "-24/5" ], [ "0/1", "225/2", "-70/1" ], [ true, false, false ] ] }, "conclusion": { "canonical_value": "(-2/15, [-2/15, 221/30, -24/5], [0/1, 225/2, -70/1], [是, 否, 否])", "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": 65, "original_line_sha256": "8b240f4fd349fef342667673accf219c7801ec78d99eee3edb7e393e52efec0a", "question_compact_sha256": "22cddaa975570aca2a314d3959c329217bac940fcfdf845d592b303ccd6c930a", "english_row_compact_sha256": "2fa521c42aedb1a3f1f9368c15f7b8e71d1ea1962e843b4a580fb1fb636a9f08", "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": "173c8ab8f1efcd4ef797972b6bf508ccfe14dea96bea879560352d4cc50ba012", "original_candidate_line_sha256": "173c8ab8f1efcd4ef797972b6bf508ccfe14dea96bea879560352d4cc50ba012" }, "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": 22, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "86014211672a2aacd0f56c7cc1492c3d81787e0b98bd55cf1764016b092f6201", "arithmetic_steps": 16 }, "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." } }