{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261003-advance-20261004-r06-system-linear-equations-0041", "pilot_number": "R0064-32941", "number": 32941, "family_id": "qf-linear-system-solution-class", "title": { "zh-Hant": "第 32941 題：二元線性系統的解集合分類", "en": "Question 32941: Solution Classification of a Linear System" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261003-advance-20261004-r06-system-linear-equations-0041", "batch_id": "batch-20261003-advance-20261004-r06-system-linear-equations-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-system-linear-equations" ], "title_zh": "二元線性系統的解集合分類", "stem_zh": "實數論域中兩個限制 (1)x+(9)y=11 與 (2)x+(18)y=22 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。", "answer": { "kind": "short_text", "value": "無限多解", "explanation_zh": "D=0；解分類=無限多解。精確列消去同時檢查零列：0=非零 不一致；一致且缺少兩個主元時為無限多解，不在 D=0 時除法。" }, "difficulty": { "band": "upper_secondary", "level": 2, "skills": [ "二元線性系統的解集合分類" ] }, "variant": { "family_id": "qf-linear-system-solution-class", "seed": 202610046041, "parameters": { "rows": [ [ 1, 9, 11 ], [ 2, 18, 22 ] ], "domain": "reals", "operation": "solution_class", "semantic_template": "qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"9\",\"11\"],[\"1\",\"9\",\"11\"]]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication" ], "warning_zh": "僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "Codex independent peer (inherited model; exact runtime model not exposed)", "generated_at": "2026-10-03", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261003-advance-20261004-r06-system-linear-equations-0041", "batch_id": "batch-20261003-advance-20261004-r06-system-linear-equations-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "6ca1f42380dd3a99990e95ccf8741b93e3f861b9a5be831ad6ab76201bc97006", "family_id": "qf-linear-system-solution-class", "seed": 202610046041, "semantic_template": "qf-linear-system-solution-class/{\"domain\":\"reals\",\"operation\":\"solution_class\",\"rows\":[[\"1\",\"9\",\"11\"],[\"1\",\"9\",\"11\"]]}" }, "fields": { "title": "Solution Classification of a Linear System", "stem": "Over the reals, the constraints (1)x+(9)y=11 and (2)x+(18)y=22 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed.", "answer": { "kind": "short_text", "value": "infinitely many solutions", "explanation": "D=0 and the classification is infinitely many solutions. Check zero rows as well as augmented consistency: 0=nonzero is inconsistent; a consistent system with fewer than two pivots has infinitely many solutions. Do not divide by D=0." }, "skills": [ "Solution Classification of a Linear System" ], "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": "實數論域中兩個限制 (1)x+(9)y=11 與 (2)x+(18)y=22 必須同時成立。此系統是無解、唯一解還是無限多解？容許零係數列。", "en": "Over the reals, the constraints (1)x+(9)y=11 and (2)x+(18)y=22 must both hold. Classify the system as no solution, a unique solution, or infinitely many solutions. Zero-coefficient rows are allowed." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-linear-system-solution-class, solution_class, reals, []]", "after": "derived = [[SOURCE_solution_class_STRING]]", "relation": "implies", "rule_id": "R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles", "conditions": [], "explanation": { "zh-Hant": "legacy核來源精確欄位及operation/domain；BASE整數保metadata但不當oracle。先分NUMBER、字串、fractionpair或vector等式，不強套同一格式；原中英／AST／status不變。", "en": "Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,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": "source roles = [[[1, 9, 11], [2, 18, 22]]]", "after": "derived = [[coefficient_x, coefficient_y, right_side]]", "relation": "implies", "rule_id": "System_RAW_exact2_augmented_rows3_ALL_SAFEincl_unused_rhs", "conditions": [], "explanation": { "zh-Hant": "原兩row各[a,b,c]代表ax+by=c，第三欄是rhs，不是矩陣係數。全部6個raw都SAFE先驗，D題未用rhs也不能略過；原row次序不改。", "en": "Each ORIGINAL row[a,b,c]means ax+by=c;the THIRD column is RHS,not a coefficient.Type ALL SIX raw SAFE values,including unused RHS in a determinant task.Keep original row order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,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": "S003", "uses": [ "G001", "S002" ], "before": "System ORIGINAL a times e main product := mul(1, 18)", "after": "System ORIGINAL a times e main product = 18", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 System ORIGINAL a times e main product，不以小數近似替代。", "en": "Recompute System ORIGINAL a times e main product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "18" ], "output": "18", "label": "System ORIGINAL a times e main product" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "System ORIGINAL b times d other product := mul(9, 2)", "after": "System ORIGINAL b times d other product = 18", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 System ORIGINAL b times d other product，不以小數近似替代。", "en": "Recompute System ORIGINAL b times d other product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "9", "2" ], "output": "18", "label": "System ORIGINAL b times d other product" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "System ORIGINAL signed determinant ae minus bd := sub(18, 18)", "after": "System ORIGINAL signed determinant ae minus bd = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 System ORIGINAL signed determinant ae minus bd，不以小數近似替代。", "en": "Recompute System ORIGINAL signed determinant ae minus bd with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "18", "18" ], "output": "0", "label": "System ORIGINAL signed determinant ae minus bd" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [[[1, 9, 11], [2, 18, 22]]]", "after": "derived = [18, 18, 0]", "relation": "implies", "rule_id": "System_DETERMINANT_coefficients_ONLY_raw_order_sign", "conditions": [], "explanation": { "zh-Hant": "D=ae−bd只取兩row前兩欄，保原row順序與符號；rhs不算入，row交換會改D符號，不用semantic排序代替原資料。", "en": "D=ae−bd uses ONLY the first TWO columns,retaining ORIGINAL row order/sign.RHS is excluded;swapping rows changes the sign.Do not replace originals by semantic sorting." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,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": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [[[1, 9, 11], [2, 18, 22]], 0]", "after": "derived = [[]]", "relation": "implies", "rule_id": "System_DZERO_zero_row_contradiction_guard_BEFORE_rank_or_minors", "conditions": [], "explanation": { "zh-Hant": "D0先找0x+0y=非0的矛盾row；這種row不可能成立，不可因其他det minors也0就誤判相容，且不除D。", "en": "At D0,FIRST detect any0x+0y=nonzero contradiction.Such a row is impossible;zero other minors do not establish consistency,and D must not be divided." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,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": "S008", "uses": [ "G001", "S007" ], "before": "System rank1 ORIGINAL a times f augmented minor := mul(1, 22)", "after": "System rank1 ORIGINAL a times f augmented minor = 22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 System rank1 ORIGINAL a times f augmented minor，不以小數近似替代。", "en": "Recompute System rank1 ORIGINAL a times f augmented minor with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "22" ], "output": "22", "label": "System rank1 ORIGINAL a times f augmented minor" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "System rank1 ORIGINAL c times d augmented minor := mul(11, 2)", "after": "System rank1 ORIGINAL c times d augmented minor = 22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 System rank1 ORIGINAL c times d augmented minor，不以小數近似替代。", "en": "Recompute System rank1 ORIGINAL c times d augmented minor with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "11", "2" ], "output": "22", "label": "System rank1 ORIGINAL c times d augmented minor" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "System rank1 ORIGINAL augmented x minor := sub(22, 22)", "after": "System rank1 ORIGINAL augmented x minor = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 System rank1 ORIGINAL augmented x minor，不以小數近似替代。", "en": "Recompute System rank1 ORIGINAL augmented x minor with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "22", "22" ], "output": "0", "label": "System rank1 ORIGINAL augmented x minor" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "System rank1 ORIGINAL b times f augmented minor := mul(9, 22)", "after": "System rank1 ORIGINAL b times f augmented minor = 198", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 System rank1 ORIGINAL b times f augmented minor，不以小數近似替代。", "en": "Recompute System rank1 ORIGINAL b times f augmented minor with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "9", "22" ], "output": "198", "label": "System rank1 ORIGINAL b times f augmented minor" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "System rank1 ORIGINAL c times e augmented minor := mul(11, 18)", "after": "System rank1 ORIGINAL c times e augmented minor = 198", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 System rank1 ORIGINAL c times e augmented minor，不以小數近似替代。", "en": "Recompute System rank1 ORIGINAL c times e augmented minor with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "11", "18" ], "output": "198", "label": "System rank1 ORIGINAL c times e augmented minor" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "System rank1 ORIGINAL augmented y minor := sub(198, 198)", "after": "System rank1 ORIGINAL augmented y minor = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 System rank1 ORIGINAL augmented y minor，不以小數近似替代。", "en": "Recompute System rank1 ORIGINAL augmented y minor with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "198", "198" ], "output": "0", "label": "System rank1 ORIGINAL augmented y minor" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "source roles = [[[1, 9, 11], [2, 18, 22]]]", "after": "derived = [0, 0, 是]", "relation": "implies", "rule_id": "System_RANKONE_BOTH_augmented_minors_consistency", "conditions": [], "explanation": { "zh-Hant": "已排rank0及矛盾零row後，D0的係數rank1；兩個augmented minors都0才相容，任一非0是消去後常數矛盾。不把一個minor0當充分。", "en": "AFTER excluding rank0/zero-row contradictions,the D0 coefficient rank is1.BOTH augmented minors must vanish for consistency;any nonzero minor yields an elimination contradiction.One zero minor is insufficient." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,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": "S015", "uses": [ "G001", "S014" ], "before": "System free parameter ORIGINAL rhs over x pivot := div(11, 1)", "after": "System free parameter ORIGINAL rhs over x pivot = 11", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 System free parameter ORIGINAL rhs over x pivot，不以小數近似替代。", "en": "Recompute System free parameter ORIGINAL rhs over x pivot with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "11", "1" ], "output": "11", "label": "System free parameter ORIGINAL rhs over x pivot" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "System free parameter ORIGINAL negative y coefficient := sub(0, 9)", "after": "System free parameter ORIGINAL negative y coefficient = -9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 System free parameter ORIGINAL negative y coefficient，不以小數近似替代。", "en": "Recompute System free parameter ORIGINAL negative y coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "9" ], "output": "-9", "label": "System free parameter ORIGINAL negative y coefficient" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "System free parameter ORIGINAL x slope in free y := div(-9, 1)", "after": "System free parameter ORIGINAL x slope in free y = -9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 System free parameter ORIGINAL x slope in free y，不以小數近似替代。", "en": "Recompute System free parameter ORIGINAL x slope in free y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-9", "1" ], "output": "-9", "label": "System free parameter ORIGINAL x slope in free y" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "System free parameter reverse OTHER x coefficient times constant := mul(2, 11)", "after": "System free parameter reverse OTHER x coefficient times constant = 22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 System free parameter reverse OTHER x coefficient times constant，不以小數近似替代。", "en": "Recompute System free parameter reverse OTHER x coefficient times constant with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "11" ], "output": "22", "label": "System free parameter reverse OTHER x coefficient times constant" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "System free parameter reverse OTHER x coefficient times slope := mul(2, -9)", "after": "System free parameter reverse OTHER x coefficient times slope = -18", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 System free parameter reverse OTHER x coefficient times slope，不以小數近似替代。", "en": "Recompute System free parameter reverse OTHER x coefficient times slope with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "-9" ], "output": "-18", "label": "System free parameter reverse OTHER x coefficient times slope" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "System free parameter reverse OTHER add y coefficient := sum(-18, 18)", "after": "System free parameter reverse OTHER add y coefficient = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 System free parameter reverse OTHER add y coefficient，不以小數近似替代。", "en": "Recompute System free parameter reverse OTHER add y coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-18", "18" ], "output": "0", "label": "System free parameter reverse OTHER add y coefficient" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "source roles = [[[1, 9, 11], [2, 18, 22]], 0]", "after": "derived = [[x = 11/1 + (-9/1)*t, y = t], 22/1, 0/1, t arbitrary in R]", "relation": "implies", "rule_id": "System_RANKONE_explicit_free_REAL_parameter_and_ALL_row_identity", "conditions": [], "explanation": { "zh-Hant": "選實際非零係數作pivot，明列一個自由實參數t。代入另一row的常數正好rhs、t係數0，所以整個family都解；自由坐標取不同t產生無限多不同解。", "en": "Choose an ACTUAL nonzero pivot and exhibit a free REAL parameter t.Substitution into the OTHER row gives its exact RHS and zero t-coefficient,so the WHOLE family solves.Different free-coordinate t values give infinitely many distinct solutions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,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": "無限多解" }, "conclusion": { "canonical_value": "無限多解", "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-20261003-advance-20261004-r06-system-linear-equations-foundations", "batch_path": "public/data/questions/batches/batch-20261003-advance-20261004-r06-system-linear-equations-foundations/questions.jsonl", "batch_sha256": "1482e4d4f4926204693a79846ba7e7d847e4a3539335e7d8b21cb87ed8ec2ad8", "original_line": 41, "original_line_sha256": "4496f8daf5d3fc141925237093eba58c4c4c5ded05368b0bd98a831e8feb2774", "question_compact_sha256": "6ca1f42380dd3a99990e95ccf8741b93e3f861b9a5be831ad6ab76201bc97006", "english_row_compact_sha256": "3a774ab1d6ae49311175bbed5a92e344d9d9804ba1064b4d4fbd32870e662cbd", "english_manifest_path": "public/data/i18n/editions/batch-20261003-advance-20261004-r06-system-linear-equations-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261003-advance-20261004-r06-system-linear-equations-foundations.en.jsonl" }, "mko_bindings": { "mko-system-linear-equations": { "sha256": "62dc0d59fbb66d23a8ef47c7b2725a56713d925c7f2c8ef0a380c46552972403", "english_sha256": "7fc6d87498d9c2a7943b1715c26c7d3e2baf79d0f54d4beb9019b628da0010c4", "original_path": "public/data/mko/system-linear-equations.json", "english_path": "public/data/i18n/editions/mko-system-linear-equations.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "528186aa825c3054db15f4ce0a03ad58f72be7e7eb5bb236c4cc7ef34ff5a470", "original_candidate_sha256": "50b4cafaa832ada7edf7a460801c8a05dd1fb2479041d776028fd704c2790465", "original_candidate_line_sha256": "50b4cafaa832ada7edf7a460801c8a05dd1fb2479041d776028fd704c2790465" }, "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-p65", "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": "8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b", "question_first_witness_sha256": "066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 21, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "edbaf115a5728d07b806c493362c3cefef84e1c9b821f9d328fa187d6ce33fb1", "arithmetic_steps": 15 }, "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." } }