{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261025-r04-linear-independence-basis-0095", "pilot_number": "R0065-33245", "number": 33245, "family_id": "qf-mod25-basisdeep-r2-coordinate-uniqueness-audit", "title": { "zh-Hant": "第 33245 題：基底密度", "en": "Question 33245: Candidate Coordinate Lists and the Conditional Uniqueness Obligation" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261025-r04-linear-independence-basis-0095", "batch_id": "batch-20261013-advance-20261025-r04-linear-independence-basis-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-linear-independence-basis" ], "title_zh": "基底密度", "stem_zh": "只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,2],[0,1]],\"target\":[7,16],\"first_coordinates\":[7,2],\"second_coordinates\":[7,2]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。", "answer": { "kind": "expression", "value": "([[1, 2], [0, 1]], [7, 16], 1, [7, 2], [7, 16], 是, [7, 2], [7, 16], 是, 是)", "explanation_zh": "收束波核對=([[1, 2], [0, 1]], [7, 16], 1, [7, 2], [7, 16], 是, [7, 2], [7, 16], 是, 是)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；不新增維度或換基底定理；座標唯一性只在非零行列式有序R2基底後使用。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "基底密度" ] }, "variant": { "family_id": "qf-mod25-basisdeep-r2-coordinate-uniqueness-audit", "seed": 202610254095, "parameters": { "basis": [ [ 1, 2 ], [ 0, 1 ] ], "target": [ 7, 16 ], "first_coordinates": [ 7, 2 ], "second_coordinates": [ 7, 2 ], "domain": "R2_basis_density_without_dimension_extension", "operation": "basisdeep-r2-coordinate-uniqueness-audit", "semantic_template": "qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,2],[0,1]],\"first_coordinates\":[7,2],\"second_coordinates\":[7,2],\"target\":[7,16]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-13", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261013-advance-20261025-r04-linear-independence-basis-0095", "batch_id": "batch-20261013-advance-20261025-r04-linear-independence-basis-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "daf48ea6e352f03f6873cf0c9bda4518cdf0c2731b461b7d4b13be2e3dba8cdb", "family_id": "qf-mod25-basisdeep-r2-coordinate-uniqueness-audit", "seed": 202610254095, "semantic_template": "qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,2],[0,1]],\"first_coordinates\":[7,2],\"second_coordinates\":[7,2],\"target\":[7,16]}" }, "fields": { "title": "Candidate Coordinate Lists and the Conditional Uniqueness Obligation", "stem": "Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 2], [0, 1]], target=[7, 16], first integer coordinates=[7, 2], second=[7, 2]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).", "answer": { "kind": "expression", "value": "([[1, 2], [0, 1]], [7, 16], 1, [7, 2], [7, 16], Yes, [7, 2], [7, 16], Yes, Yes)", "explanation": "Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [7, 16], 1, [7, 2], [7, 16], Yes, [7, 2], [7, 16], Yes, Yes)." }, "skills": [ "Candidate Coordinate Lists and the Conditional Uniqueness Obligation" ], "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": "只深化既有有限R2基底語義。操作=basisdeep-r2-coordinate-uniqueness-audit；資料={\"basis\":[[1,2],[0,1]],\"target\":[7,16],\"first_coordinates\":[7,2],\"second_coordinates\":[7,2]}。請依 [basis,target,det,first,first_target,first_matches,second,second_target,second_matches,uniqueness_consistent] 欄序回答。", "en": "Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 2], [0, 1]], target=[7, 16], first integer coordinates=[7, 2], second=[7, 2]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-mod25-basisdeep-r2-coordinate-uniqueness-audit, basisdeep-r2-coordinate-uniqueness-audit, R2_basis_density_without_dimension_extension, []]", "after": "derived = [[basis, target, D, first, reconstruction1, match1, second, reconstruction2, match2, implication]]", "relation": "implies", "rule_id": "R65_SOURCE_bank_keys_operation_domain_ordered_output_roles", "conditions": [], "explanation": { "zh-Hant": "MOD25用LOCAL hyphen及具名domain，矩陣legacy用underscore/reals，BASElogic不補domain/op。輸出tuple/矩陣/數字欄與fraction文字各依來源，不一律保/1；原中英/AST/status不變。", "en": "MOD25 uses LOCAL hyphens/named domains,legacy matrices underscores/reals;BASE logic acquires no domain/op.Output tuples,matrices,numeric fields and fraction texts follow THIS source,not a universal/1 rule.Retain originals/English/AST/status." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [ordered R2 basis COLUMNS vector0, [1, 2]]", "after": "derived = [[1, 2]]", "relation": "implies", "rule_id": "Basis_RAW_vector_1to6_components_bound1000", "conditions": [], "explanation": { "zh-Hant": "raw vector非空、1..6分量且每值±1000；只有R²座標題另驗2維，不對其他families憑domain名字強加2維。", "en": "RAW vectors are nonempty with1..6 components bounded±1000.ONLY the R² coordinate task adds dimension2;do not infer an extra2D gate elsewhere from a domain label." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [ordered R2 basis COLUMNS vector1, [0, 1]]", "after": "derived = [[0, 1]]", "relation": "implies", "rule_id": "Basis_RAW_vector_1to6_components_bound1000", "conditions": [], "explanation": { "zh-Hant": "raw vector非空、1..6分量且每值±1000；只有R²座標題另驗2維，不對其他families憑domain名字強加2維。", "en": "RAW vectors are nonempty with1..6 components bounded±1000.ONLY the R² coordinate task adds dimension2;do not infer an extra2D gate elsewhere from a domain label." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [R2 target, [7, 16]]", "after": "derived = [[7, 16]]", "relation": "implies", "rule_id": "Basis_RAW_vector_1to6_components_bound1000", "conditions": [], "explanation": { "zh-Hant": "raw vector非空、1..6分量且每值±1000；只有R²座標題另驗2維，不對其他families憑domain名字強加2維。", "en": "RAW vectors are nonempty with1..6 components bounded±1000.ONLY the R² coordinate task adds dimension2;do not infer an extra2D gate elsewhere from a domain label." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "R2 ORIGINAL b1x times b2y := mul(1, 1)", "after": "R2 ORIGINAL b1x times b2y = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 ORIGINAL b1x times b2y，不以小數近似替代。", "en": "Recompute R2 ORIGINAL b1x times b2y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "R2 ORIGINAL b1x times b2y" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "R2 ORIGINAL b2x times b1y := mul(0, 2)", "after": "R2 ORIGINAL b2x times b1y = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 ORIGINAL b2x times b1y，不以小數近似替代。", "en": "Recompute R2 ORIGINAL b2x times b1y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "2" ], "output": "0", "label": "R2 ORIGINAL b2x times b1y" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "R2 ORIGINAL signed column determinant := sub(1, 0)", "after": "R2 ORIGINAL signed column determinant = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 ORIGINAL signed column determinant，不以小數近似替代。", "en": "Recompute R2 ORIGINAL signed column determinant with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "0" ], "output": "1", "label": "R2 ORIGINAL signed column determinant" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "R2 true coordinate ORIGINAL targetx times b2y := mul(7, 1)", "after": "R2 true coordinate ORIGINAL targetx times b2y = 7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 true coordinate ORIGINAL targetx times b2y，不以小數近似替代。", "en": "Recompute R2 true coordinate ORIGINAL targetx times b2y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7", "1" ], "output": "7", "label": "R2 true coordinate ORIGINAL targetx times b2y" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "R2 true coordinate ORIGINAL b2x times targety := mul(0, 16)", "after": "R2 true coordinate ORIGINAL b2x times targety = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 true coordinate ORIGINAL b2x times targety，不以小數近似替代。", "en": "Recompute R2 true coordinate ORIGINAL b2x times targety with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "16" ], "output": "0", "label": "R2 true coordinate ORIGINAL b2x times targety" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "R2 true coordinate ORIGINAL first numerator := sub(7, 0)", "after": "R2 true coordinate ORIGINAL first numerator = 7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 true coordinate ORIGINAL first numerator，不以小數近似替代。", "en": "Recompute R2 true coordinate ORIGINAL first numerator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "7", "0" ], "output": "7", "label": "R2 true coordinate ORIGINAL first numerator" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "R2 true coordinate ORIGINAL b1x times targety := mul(1, 16)", "after": "R2 true coordinate ORIGINAL b1x times targety = 16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 true coordinate ORIGINAL b1x times targety，不以小數近似替代。", "en": "Recompute R2 true coordinate ORIGINAL b1x times targety with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "16" ], "output": "16", "label": "R2 true coordinate ORIGINAL b1x times targety" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "R2 true coordinate ORIGINAL targetx times b1y := mul(7, 2)", "after": "R2 true coordinate ORIGINAL targetx times b1y = 14", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 true coordinate ORIGINAL targetx times b1y，不以小數近似替代。", "en": "Recompute R2 true coordinate ORIGINAL targetx times b1y with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7", "2" ], "output": "14", "label": "R2 true coordinate ORIGINAL targetx times b1y" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "R2 true coordinate ORIGINAL second numerator := sub(16, 14)", "after": "R2 true coordinate ORIGINAL second numerator = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 true coordinate ORIGINAL second numerator，不以小數近似替代。", "en": "Recompute R2 true coordinate ORIGINAL second numerator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "16", "14" ], "output": "2", "label": "R2 true coordinate ORIGINAL second numerator" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "source roles = [R2 source helper TRUEcoordinate0, RAW-pair, [7, 1], [], 7, 1]", "after": "derived = [7, 1, 7]", "relation": "implies", "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations", "conditions": [], "explanation": { "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。", "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "R2 source helper TRUEcoordinate0 ORIGINAL raw numerator over nonzero denominator := div(7, 1)", "after": "R2 source helper TRUEcoordinate0 ORIGINAL raw numerator over nonzero denominator = 7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 source helper TRUEcoordinate0 ORIGINAL raw numerator over nonzero denominator，不以小數近似替代。", "en": "Recompute R2 source helper TRUEcoordinate0 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "7", "1" ], "output": "7", "label": "R2 source helper TRUEcoordinate0 ORIGINAL raw numerator over nonzero denominator" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "source roles = [R2 source helper TRUEcoordinate1, RAW-pair, [2, 1], [], 2, 1]", "after": "derived = [2, 1, 2]", "relation": "implies", "rule_id": "Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations", "conditions": [], "explanation": { "zh-Hant": "這是來源可接受運算切片：先cancel指定因子，再對傳入rational的num/den驗±100萬、den非0，之後才sign/gcd約分。超界是SOURCE切片不合，不表示數學式為假；其他衍生整數不普遍套此界。", "en": "This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "R2 source helper TRUEcoordinate1 ORIGINAL raw numerator over nonzero denominator := div(2, 1)", "after": "R2 source helper TRUEcoordinate1 ORIGINAL raw numerator over nonzero denominator = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 source helper TRUEcoordinate1 ORIGINAL raw numerator over nonzero denominator，不以小數近似替代。", "en": "Recompute R2 source helper TRUEcoordinate1 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "2", "1" ], "output": "2", "label": "R2 source helper TRUEcoordinate1 ORIGINAL raw numerator over nonzero denominator" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "R2 FIRST integer coordinate list ORIGINAL vector0 coordinate0 weighted term := mul(7, 1)", "after": "R2 FIRST integer coordinate list ORIGINAL vector0 coordinate0 weighted term = 7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 FIRST integer coordinate list ORIGINAL vector0 coordinate0 weighted term，不以小數近似替代。", "en": "Recompute R2 FIRST integer coordinate list ORIGINAL vector0 coordinate0 weighted term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7", "1" ], "output": "7", "label": "R2 FIRST integer coordinate list ORIGINAL vector0 coordinate0 weighted term" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "R2 FIRST integer coordinate list ORIGINAL vector1 coordinate0 weighted term := mul(2, 0)", "after": "R2 FIRST integer coordinate list ORIGINAL vector1 coordinate0 weighted term = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 FIRST integer coordinate list ORIGINAL vector1 coordinate0 weighted term，不以小數近似替代。", "en": "Recompute R2 FIRST integer coordinate list ORIGINAL vector1 coordinate0 weighted term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "0" ], "output": "0", "label": "R2 FIRST integer coordinate list ORIGINAL vector1 coordinate0 weighted term" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "R2 FIRST integer coordinate list ORIGINAL coordinate0 all terms sum := sum(7, 0)", "after": "R2 FIRST integer coordinate list ORIGINAL coordinate0 all terms sum = 7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 FIRST integer coordinate list ORIGINAL coordinate0 all terms sum，不以小數近似替代。", "en": "Recompute R2 FIRST integer coordinate list ORIGINAL coordinate0 all terms sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "7", "0" ], "output": "7", "label": "R2 FIRST integer coordinate list ORIGINAL coordinate0 all terms sum" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "source roles = [R2 FIRST integer coordinate list, 0, [[1, 2], [0, 1]], [7, 2]]", "after": "derived = [[7, 0], 7]", "relation": "implies", "rule_id": "Basis_EACH_coordinate_complete_linear_combination_no_derived_raw_cap", "conditions": [], "explanation": { "zh-Hant": "每個coordinate保全部coeff*vector分量，再加總；不對衍生重構值重套raw vector/係數界。", "en": "For EACH coordinate retain EVERY coefficient*vector component before summing.Do not recap derived reconstructions as RAW vector/coefficient inputs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "R2 FIRST integer coordinate list ORIGINAL vector0 coordinate1 weighted term := mul(7, 2)", "after": "R2 FIRST integer coordinate list ORIGINAL vector0 coordinate1 weighted term = 14", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 FIRST integer coordinate list ORIGINAL vector0 coordinate1 weighted term，不以小數近似替代。", "en": "Recompute R2 FIRST integer coordinate list ORIGINAL vector0 coordinate1 weighted term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7", "2" ], "output": "14", "label": "R2 FIRST integer coordinate list ORIGINAL vector0 coordinate1 weighted term" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "R2 FIRST integer coordinate list ORIGINAL vector1 coordinate1 weighted term := mul(2, 1)", "after": "R2 FIRST integer coordinate list ORIGINAL vector1 coordinate1 weighted term = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 FIRST integer coordinate list ORIGINAL vector1 coordinate1 weighted term，不以小數近似替代。", "en": "Recompute R2 FIRST integer coordinate list ORIGINAL vector1 coordinate1 weighted term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "R2 FIRST integer coordinate list ORIGINAL vector1 coordinate1 weighted term" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "R2 FIRST integer coordinate list ORIGINAL coordinate1 all terms sum := sum(14, 2)", "after": "R2 FIRST integer coordinate list ORIGINAL coordinate1 all terms sum = 16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 FIRST integer coordinate list ORIGINAL coordinate1 all terms sum，不以小數近似替代。", "en": "Recompute R2 FIRST integer coordinate list ORIGINAL coordinate1 all terms sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "14", "2" ], "output": "16", "label": "R2 FIRST integer coordinate list ORIGINAL coordinate1 all terms sum" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "source roles = [R2 FIRST integer coordinate list, 1, [[1, 2], [0, 1]], [7, 2]]", "after": "derived = [[14, 2], 16]", "relation": "implies", "rule_id": "Basis_EACH_coordinate_complete_linear_combination_no_derived_raw_cap", "conditions": [], "explanation": { "zh-Hant": "每個coordinate保全部coeff*vector分量，再加總；不對衍生重構值重套raw vector/係數界。", "en": "For EACH coordinate retain EVERY coefficient*vector component before summing.Do not recap derived reconstructions as RAW vector/coefficient inputs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "R2 SECOND integer coordinate list ORIGINAL vector0 coordinate0 weighted term := mul(7, 1)", "after": "R2 SECOND integer coordinate list ORIGINAL vector0 coordinate0 weighted term = 7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 SECOND integer coordinate list ORIGINAL vector0 coordinate0 weighted term，不以小數近似替代。", "en": "Recompute R2 SECOND integer coordinate list ORIGINAL vector0 coordinate0 weighted term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7", "1" ], "output": "7", "label": "R2 SECOND integer coordinate list ORIGINAL vector0 coordinate0 weighted term" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "R2 SECOND integer coordinate list ORIGINAL vector1 coordinate0 weighted term := mul(2, 0)", "after": "R2 SECOND integer coordinate list ORIGINAL vector1 coordinate0 weighted term = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 SECOND integer coordinate list ORIGINAL vector1 coordinate0 weighted term，不以小數近似替代。", "en": "Recompute R2 SECOND integer coordinate list ORIGINAL vector1 coordinate0 weighted term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "0" ], "output": "0", "label": "R2 SECOND integer coordinate list ORIGINAL vector1 coordinate0 weighted term" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "R2 SECOND integer coordinate list ORIGINAL coordinate0 all terms sum := sum(7, 0)", "after": "R2 SECOND integer coordinate list ORIGINAL coordinate0 all terms sum = 7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 SECOND integer coordinate list ORIGINAL coordinate0 all terms sum，不以小數近似替代。", "en": "Recompute R2 SECOND integer coordinate list ORIGINAL coordinate0 all terms sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "7", "0" ], "output": "7", "label": "R2 SECOND integer coordinate list ORIGINAL coordinate0 all terms sum" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "source roles = [R2 SECOND integer coordinate list, 0, [[1, 2], [0, 1]], [7, 2]]", "after": "derived = [[7, 0], 7]", "relation": "implies", "rule_id": "Basis_EACH_coordinate_complete_linear_combination_no_derived_raw_cap", "conditions": [], "explanation": { "zh-Hant": "每個coordinate保全部coeff*vector分量，再加總；不對衍生重構值重套raw vector/係數界。", "en": "For EACH coordinate retain EVERY coefficient*vector component before summing.Do not recap derived reconstructions as RAW vector/coefficient inputs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "R2 SECOND integer coordinate list ORIGINAL vector0 coordinate1 weighted term := mul(7, 2)", "after": "R2 SECOND integer coordinate list ORIGINAL vector0 coordinate1 weighted term = 14", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 SECOND integer coordinate list ORIGINAL vector0 coordinate1 weighted term，不以小數近似替代。", "en": "Recompute R2 SECOND integer coordinate list ORIGINAL vector0 coordinate1 weighted term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7", "2" ], "output": "14", "label": "R2 SECOND integer coordinate list ORIGINAL vector0 coordinate1 weighted term" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "R2 SECOND integer coordinate list ORIGINAL vector1 coordinate1 weighted term := mul(2, 1)", "after": "R2 SECOND integer coordinate list ORIGINAL vector1 coordinate1 weighted term = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 SECOND integer coordinate list ORIGINAL vector1 coordinate1 weighted term，不以小數近似替代。", "en": "Recompute R2 SECOND integer coordinate list ORIGINAL vector1 coordinate1 weighted term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "R2 SECOND integer coordinate list ORIGINAL vector1 coordinate1 weighted term" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "R2 SECOND integer coordinate list ORIGINAL coordinate1 all terms sum := sum(14, 2)", "after": "R2 SECOND integer coordinate list ORIGINAL coordinate1 all terms sum = 16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保原操作數與次序作精確運算；有理計算另驗此MOD25來源的pre-reduction界。不是把所有衍生整數都截成100萬，也不用浮點近似。", "en": "Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 R2 SECOND integer coordinate list ORIGINAL coordinate1 all terms sum，不以小數近似替代。", "en": "Recompute R2 SECOND integer coordinate list ORIGINAL coordinate1 all terms sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "14", "2" ], "output": "16", "label": "R2 SECOND integer coordinate list ORIGINAL coordinate1 all terms sum" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "source roles = [R2 SECOND integer coordinate list, 1, [[1, 2], [0, 1]], [7, 2]]", "after": "derived = [[14, 2], 16]", "relation": "implies", "rule_id": "Basis_EACH_coordinate_complete_linear_combination_no_derived_raw_cap", "conditions": [], "explanation": { "zh-Hant": "每個coordinate保全部coeff*vector分量，再加總；不對衍生重構值重套raw vector/係數界。", "en": "For EACH coordinate retain EVERY coefficient*vector component before summing.Do not recap derived reconstructions as RAW vector/coefficient inputs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "source roles = [[[1, 2], [0, 1]], [7, 16], 1]", "after": "derived = [[7, 2], [7, 2], [7, 16], 是, [7, 2], [7, 16], 是, 是]", "relation": "implies", "rule_id": "Basis_R2_nonzero_column_basis_source_TRUEcoordinate_gate_and_conditional_uniqueness", "conditions": [], "explanation": { "zh-Hant": "basis是有序column basis，D非0才有唯一座標；來源helper即使不輸出真座標也必須過rational運算界。兩組提交是integer coords，逐組重構；最後是「兩組都對⇒相等」，若有一組不對，True不認證兩組皆有效。", "en": "The ordered vectors are COLUMNS;D≠0 qualifies unique coordinates.The SOURCE true-coordinate helper must pass its rational envelope EVEN when not in the output.Submitted lists are INTEGER coordinates;reconstruct each.The final BOTH-valid⇒equal implication may be true with an invalid list and does not certify both." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、型別、欄序、論域與明示前提反向核對；不表示結論唯一還原前提。有限來源候選、獨立審查、形式證明及發布各自分開。", "en": "Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate." } }, "arithmetic": null } ], "math_model": { "result": [ [ [ 1, 2 ], [ 0, 1 ] ], [ 7, 16 ], 1, [ 7, 2 ], [ 7, 16 ], true, [ 7, 2 ], [ 7, 16 ], true, true ] }, "conclusion": { "canonical_value": "([[1, 2], [0, 1]], [7, 16], 1, [7, 2], [7, 16], 是, [7, 2], [7, 16], 是, 是)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261013-advance-20261025-r04-linear-independence-basis-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261025-r04-linear-independence-basis-foundations/questions.jsonl", "batch_sha256": "27f05761dc76f45a861163c0fea8c42129f9765370396ea0909c413aa85b68e4", "original_line": 95, "original_line_sha256": "78404bd63c4f3f1c18e6210b2553a82a67caa6413de9cf7165e8a4a13a915a90", "question_compact_sha256": "daf48ea6e352f03f6873cf0c9bda4518cdf0c2731b461b7d4b13be2e3dba8cdb", "english_row_compact_sha256": "f1ea62098aaef5abd24d89b7f300d4f9c2e60b10edde4dd3d2e21b8d88374a74", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261025-r04-linear-independence-basis-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261025-r04-linear-independence-basis-foundations.en.jsonl" }, "mko_bindings": { "mko-linear-independence-basis": { "sha256": "78c45988a718f6a7ca4acf2aba00ea4df794c8f141cefe822bf72206fa2b2f62", "english_sha256": "48c43771b8f6734ece79626851e05a5dca9bb984732eb3f69cffe2c6e364a912", "original_path": "public/data/mko/linear-independence-basis.json", "english_path": "public/data/i18n/editions/mko-linear-independence-basis.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "871e2c05026567a1deae741d6b90b0588100c6482ee167c7f9f45af5b72d8606", "original_candidate_sha256": "cdad7cf78e6f91fb2ca2a1809ac573d6ecd195cfee8324091f3aa27b57df8d44", "original_candidate_line_sha256": "cdad7cf78e6f91fb2ca2a1809ac573d6ecd195cfee8324091f3aa27b57df8d44" }, "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-20261026-012-p66", "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": "92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6", "question_first_witness_sha256": "012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 34, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "7205774ee6f8c2d53a31564c531bbd5672071869c5f1824206629316e65b0602", "arithmetic_steps": 23 }, "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." } }