{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261102-r02-linear-map-kernel-0076", "pilot_number": "R0033-17126", "number": 17126, "family_id": "qf-mod33-kernel-bounded-search-boundary", "title": { "zh-Hant": "第 17126 題：線性映射的核", "en": "Question 17126: A Bounded Integer Box Is Not the Full Real Kernel" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261102-r02-linear-map-kernel-0076", "batch_id": "batch-20261014-advance-20261102-r02-linear-map-kernel-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-linear-map-kernel" ], "title_zh": "線性映射的核", "stem_zh": "同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,1],[2,2]],\"bound\":1}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。", "answer": { "kind": "expression", "value": "(1, [[-1, 1], [0, 0], [1, -1]], [[1, -1]], 否, bounded_integer_search_not_full_real_kernel)", "explanation_zh": "Wave5第二層核對=(1, [[-1, 1], [0, 0], [1, -1]], [[1, -1]], 否, bounded_integer_search_not_full_real_kernel)；依完整明示資料重算；kernel只解T(x)=0且使用symbolic full-real certificate；bounded integer search只回答bounded問題；非零fiber不是kernel。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "線性映射的核" ] }, "variant": { "family_id": "qf-mod33-kernel-bounded-search-boundary", "seed": 202611022076, "parameters": { "matrix": [ [ 1, 1 ], [ 2, 2 ] ], "bound": 1, "domain": "Exact_R2_matrix_kernel", "operation": "kernel-bounded-search-boundary", "semantic_template": "qf-mod33-kernel-bounded-search-boundary/{\"bound\":1,\"matrix\":[[1,1],[2,2]]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261102-r02-linear-map-kernel-0076", "batch_id": "batch-20261014-advance-20261102-r02-linear-map-kernel-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "8267a67fc972a1b10e9d7e1d0cb4da5ba2a1745a90679779fc3bb30978d114a2", "family_id": "qf-mod33-kernel-bounded-search-boundary", "seed": 202611022076, "semantic_template": "qf-mod33-kernel-bounded-search-boundary/{\"bound\":1,\"matrix\":[[1,1],[2,2]]}" }, "fields": { "title": "A Bounded Integer Box Is Not the Full Real Kernel", "stem": "Use this same exact integer matrix A=[[1, 1], [2, 2]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=1. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).", "answer": { "kind": "expression", "value": "(1, [[-1, 1], [0, 0], [1, -1]], [[1, -1]], No, bounded_integer_search_not_full_real_kernel)", "explanation": "List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (1, [[-1, 1], [0, 0], [1, -1]], [[1, -1]], No, bounded_integer_search_not_full_real_kernel)." }, "skills": [ "A Bounded Integer Box Is Not the Full Real Kernel" ], "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": "同一精確2×2線性映射T:R²到R²。操作=kernel-bounded-search-boundary；資料={\"matrix\":[[1,1],[2,2]],\"bound\":1}。請依 [bound,integer_kernel_points,global_kernel_basis,complete,boundary] 欄序回答。", "en": "Use this same exact integer matrix A=[[1, 1], [2, 2]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=1. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "real map determinant diagonal product := mul(1, 2)", "after": "real map determinant diagonal product = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 real map determinant diagonal product，不以小數近似替代。", "en": "Recompute real map determinant diagonal product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "real map determinant diagonal product" } }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "real map determinant off-diagonal product := mul(1, 2)", "after": "real map determinant off-diagonal product = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 real map determinant off-diagonal product，不以小數近似替代。", "en": "Recompute real map determinant off-diagonal product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "real map determinant off-diagonal product" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "real map exact determinant := sub(2, 2)", "after": "real map exact determinant = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 real map exact determinant，不以小數近似替代。", "en": "Recompute real map exact determinant with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "2", "2" ], "output": "0", "label": "real map exact determinant" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "same ordered integer matrix=[[1, 1], [2, 2]];usual real map T:R2 to R2;original columns=[[1, 2], [1, 2]]", "after": "exact determinant=0;rank=1", "relation": "implies", "rule_id": "real_map_same_matrix_det_and_nonzero_rank_cases", "conditions": [], "explanation": { "zh-Hant": "非零det給秩2；det=0時還要看是否全零，才分秩0或1。固定同一矩陣與通常實向量空間，不把列當欄或自動把像當整個codomain。", "en": "A nonzero determinant gives rank2.With determinant0,distinguish the zero matrix from a nonzero singular matrix to get rank0 or1.Keep the same matrix and real vector spaces;do not replace columns by rows or assume the image is the entire codomain." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "primitive kernel first coordinate := div(1, 1)", "after": "primitive kernel first coordinate = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 primitive kernel first coordinate，不以小數近似替代。", "en": "Recompute primitive kernel first coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "primitive kernel first coordinate" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "primitive kernel second coordinate := div(-1, 1)", "after": "primitive kernel second coordinate = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 primitive kernel second coordinate，不以小數近似替代。", "en": "Recompute primitive kernel second coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-1", "1" ], "output": "-1", "label": "primitive kernel second coordinate" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "primitive kernel canonical sign coordinate0 := mul(1, 1)", "after": "primitive kernel canonical sign coordinate0 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 primitive kernel canonical sign coordinate0，不以小數近似替代。", "en": "Recompute primitive kernel canonical sign coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "primitive kernel canonical sign coordinate0" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "primitive kernel canonical sign coordinate1 := mul(1, -1)", "after": "primitive kernel canonical sign coordinate1 = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 primitive kernel canonical sign coordinate1，不以小數近似替代。", "en": "Recompute primitive kernel canonical sign coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "-1" ], "output": "-1", "label": "primitive kernel canonical sign coordinate1" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "other row dependence factor := div(2, 1)", "after": "other row dependence factor = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 other row dependence factor，不以小數近似替代。", "en": "Recompute other row dependence factor with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "2", "1" ], "output": "2", "label": "other row dependence factor" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "other row dependence coordinate0 := mul(2, 1)", "after": "other row dependence coordinate0 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 other row dependence coordinate0，不以小數近似替代。", "en": "Recompute other row dependence coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "other row dependence coordinate0" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "other row dependence coordinate1 := mul(2, 1)", "after": "other row dependence coordinate1 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 other row dependence coordinate1，不以小數近似替代。", "en": "Recompute other row dependence coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "other row dependence coordinate1" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "FIRST nonzero original row index=0;row=[1, 1];gcd=1;raw integer kernel direction=[1, -1];canonical sign=1", "after": "primitive kernel basis=[[1, -1]];other original row=[2, 2]=(2)*selected row", "relation": "implies", "rule_id": "real_map_rank1_kernel_canonical_row_and_full_equation", "conditions": [], "explanation": { "zh-Hant": "用首個非零原列解u·x+v·y=0，除gcd得到primitive整數方向，再讓第一個非零座標為正。已核另一列為它的倍數，所以兩列方程等價於這一個，不只憑det零猜解集。", "en": "Use the first nonzero original row to solve u*x+v*y=0.Divide by the gcd for a primitive integer direction and make its first nonzero coordinate positive.The other row is verified to be a multiple,so both equations reduce to this one rather than a guess from determinant0." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "canonical kernel basis vector row0 first term := mul(1, 1)", "after": "canonical kernel basis vector row0 first term = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 canonical kernel basis vector row0 first term，不以小數近似替代。", "en": "Recompute canonical kernel basis vector row0 first term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "canonical kernel basis vector row0 first term" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "canonical kernel basis vector row0 second term := mul(1, -1)", "after": "canonical kernel basis vector row0 second term = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 canonical kernel basis vector row0 second term，不以小數近似替代。", "en": "Recompute canonical kernel basis vector row0 second term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "-1" ], "output": "-1", "label": "canonical kernel basis vector row0 second term" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "canonical kernel basis vector image coordinate0 := sum(1, -1)", "after": "canonical kernel basis vector image coordinate0 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 canonical kernel basis vector image coordinate0，不以小數近似替代。", "en": "Recompute canonical kernel basis vector image coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "-1" ], "output": "0", "label": "canonical kernel basis vector image coordinate0" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "canonical kernel basis vector row1 first term := mul(2, 1)", "after": "canonical kernel basis vector row1 first term = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 canonical kernel basis vector row1 first term，不以小數近似替代。", "en": "Recompute canonical kernel basis vector row1 first term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "canonical kernel basis vector row1 first term" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "canonical kernel basis vector row1 second term := mul(2, -1)", "after": "canonical kernel basis vector row1 second term = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 canonical kernel basis vector row1 second term，不以小數近似替代。", "en": "Recompute canonical kernel basis vector row1 second term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "-1" ], "output": "-2", "label": "canonical kernel basis vector row1 second term" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "canonical kernel basis vector image coordinate1 := sum(2, -2)", "after": "canonical kernel basis vector image coordinate1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 canonical kernel basis vector image coordinate1，不以小數近似替代。", "en": "Recompute canonical kernel basis vector image coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2", "-2" ], "output": "0", "label": "canonical kernel basis vector image coordinate1" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "one nonzero row equation(1)*x+(1)*y=0;nonzero verified kernel vector=[1, -1];chosen nonzero vector coordinate=0", "after": "EVERY real solution is lambda*[1, -1];choose lambda=solution_coordinate0/(1);conversely every real lambda multiple maps to0", "relation": "implies", "rule_id": "real_map_rank1_kernel_all_real_solutions_not_one_test", "conditions": [], "explanation": { "zh-Hant": "若w0非零，v亦非零，取λ=x/w0後由兩個零方程相減得v(y−λw1)=0，故y=λw1。若w0=0，則v=0、u非零，方程強制x=0而y任意。反向由線性性得每個實倍都在kernel，故不是整數格點清單。", "en": "If w0 is nonzero,then v is nonzero.Set λ=x/w0;subtract the two zero equations to get v(y−λw1)=0,hence y=λw1.If w0=0,then v=0 and nonzero u forces x=0 while y is arbitrary.Conversely linearity sends every real multiple to zero.This describes the full real kernel,not an integer grid." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "other column dependence factor := div(1, 1)", "after": "other column dependence factor = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 other column dependence factor，不以小數近似替代。", "en": "Recompute other column dependence factor with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "other column dependence factor" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "other column dependence coordinate0 := mul(1, 1)", "after": "other column dependence coordinate0 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 other column dependence coordinate0，不以小數近似替代。", "en": "Recompute other column dependence coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "other column dependence coordinate0" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "other column dependence coordinate1 := mul(1, 2)", "after": "other column dependence coordinate1 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 other column dependence coordinate1，不以小數近似替代。", "en": "Recompute other column dependence coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "other column dependence coordinate1" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "FIRST nonzero ORIGINAL column index=0;basis vector kept unnormalized=[1, 2];other original column=[1, 2]=(1)*basis", "after": "for ANY real(s,t),M(s,t)=((1)*s+(1)*t)*basis;EVERY lambda*basis obtained from input lambda*e0", "relation": "implies", "rule_id": "real_map_rank1_image_both_inclusions_original_column", "conditions": [], "explanation": { "zh-Hant": "每個輸出是原兩欄的線性組合，已核它們同在此一欄的span；反向把λ放在該原欄的輸入座標即得到λ倍basis。故像恰為完整實span，不能把原欄改成primitive或拿列代替。", "en": "Every output is a linear combination of the original columns,now verified to lie in this one column's span.Conversely place λ in that original column's input coordinate to obtain λ times the basis.So the image is the full real span.Do not normalize the original column or substitute a row." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "real kernel source dimension minus rank := sub(2, 1)", "after": "real kernel source dimension minus rank = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一原始角色與已核中間量精確計算，保留分母、正負號、座標與索引。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 real kernel source dimension minus rank，不以小數近似替代。", "en": "Recompute real kernel source dimension minus rank with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "2", "1" ], "output": "1", "label": "real kernel source dimension minus rank" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "same real map rank=1;full kernel basis=[[1, -1]];full image basis=[[1, 2]]", "after": "nullity=1;image basis cardinality=1;rank plus nullity=2", "relation": "implies", "rule_id": "real_map_full_space_basis_counts_not_grid_counts", "conditions": [], "explanation": { "zh-Hant": "這些是已證完整實空間的basis個數，不是搜尋格點數；rank、nullity與像basis數量分開保留，同一來源維度為2。", "en": "Count bases of the established full real spaces,not searched grid points.Keep rank,nullity and image-basis count distinct;the same source dimension is2." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "kernel bounded search:integer point in x-outer/y-inner order=[-1, -1];same matrix=[[1, 1], [2, 2]];target=[0, 0]", "after": "exact row-dot image=[-2, -4];equals target=否", "relation": "implies", "rule_id": "real_map_every_bounded_integer_point_exact_image", "conditions": [], "explanation": { "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。", "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "kernel bounded search:integer point in x-outer/y-inner order=[-1, 0];same matrix=[[1, 1], [2, 2]];target=[0, 0]", "after": "exact row-dot image=[-1, -2];equals target=否", "relation": "implies", "rule_id": "real_map_every_bounded_integer_point_exact_image", "conditions": [], "explanation": { "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。", "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "kernel bounded search:integer point in x-outer/y-inner order=[-1, 1];same matrix=[[1, 1], [2, 2]];target=[0, 0]", "after": "exact row-dot image=[0, 0];equals target=是", "relation": "implies", "rule_id": "real_map_every_bounded_integer_point_exact_image", "conditions": [], "explanation": { "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。", "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "kernel bounded search:integer point in x-outer/y-inner order=[0, -1];same matrix=[[1, 1], [2, 2]];target=[0, 0]", "after": "exact row-dot image=[-1, -2];equals target=否", "relation": "implies", "rule_id": "real_map_every_bounded_integer_point_exact_image", "conditions": [], "explanation": { "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。", "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "kernel bounded search:integer point in x-outer/y-inner order=[0, 0];same matrix=[[1, 1], [2, 2]];target=[0, 0]", "after": "exact row-dot image=[0, 0];equals target=是", "relation": "implies", "rule_id": "real_map_every_bounded_integer_point_exact_image", "conditions": [], "explanation": { "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。", "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "kernel bounded search:integer point in x-outer/y-inner order=[0, 1];same matrix=[[1, 1], [2, 2]];target=[0, 0]", "after": "exact row-dot image=[1, 2];equals target=否", "relation": "implies", "rule_id": "real_map_every_bounded_integer_point_exact_image", "conditions": [], "explanation": { "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。", "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "kernel bounded search:integer point in x-outer/y-inner order=[1, -1];same matrix=[[1, 1], [2, 2]];target=[0, 0]", "after": "exact row-dot image=[0, 0];equals target=是", "relation": "implies", "rule_id": "real_map_every_bounded_integer_point_exact_image", "conditions": [], "explanation": { "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。", "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "kernel bounded search:integer point in x-outer/y-inner order=[1, 0];same matrix=[[1, 1], [2, 2]];target=[0, 0]", "after": "exact row-dot image=[1, 2];equals target=否", "relation": "implies", "rule_id": "real_map_every_bounded_integer_point_exact_image", "conditions": [], "explanation": { "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。", "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "kernel bounded search:integer point in x-outer/y-inner order=[1, 1];same matrix=[[1, 1], [2, 2]];target=[0, 0]", "after": "exact row-dot image=[2, 4];equals target=否", "relation": "implies", "rule_id": "real_map_every_bounded_integer_point_exact_image", "conditions": [], "explanation": { "zh-Hant": "只掃指定整數盒，逐點用原矩陣兩列點積核目標相等；保留x外層、y內層遞增順序，這個有限枚舉不承擔全部實向量的責任。", "en": "Scan only the specified integer box and test both exact row-dot coordinates against the target.Keep x-outer/y-inner increasing order.This finite enumeration does not quantify over all real inputs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "kernel bounded search:bound=1;complete integer box pair count=9", "after": "all hits in original scan order=[[-1, 1], [0, 0], [1, -1]]", "relation": "implies", "rule_id": "real_map_complete_bounded_list_not_unbounded_search", "conditions": [], "explanation": { "zh-Hant": "上下界含端點，兩座標共有(2b+1)²組；每組已核，清單只對這個盒完整。", "en": "Both endpoints are included,giving(2b+1)² pairs.Every pair was checked;completeness of the list is relative to this box only." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "bounded integer kernel points=[[-1, 1], [0, 0], [1, -1]];SEPARATE symbolic full-real basis=[[1, -1]]", "after": "general completeness inference from box alone=否;bounded_integer_search_not_full_real_kernel", "relation": "implies", "rule_id": "kernel_box_scope_not_denial_of_symbolic_or_special_equality", "conditions": [], "explanation": { "zh-Hant": "No只限制從整數盒推全部實kernel的推論，不否認另給的symbolic全域basis。某些特例盒內清單可能碰巧等於kernel，也不能把一般方法界線誤說成每例集合必不相等。", "en": "The No limits inference from an integer box alone to the full real kernel.It does not deny the separate symbolic basis.In special cases the list may even equal the kernel;the general evidence boundary is not a claim of unequal sets in every instance." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ 1, [ [ -1, 1 ], [ 0, 0 ], [ 1, -1 ] ], [ [ 1, -1 ] ], false, "bounded_integer_search_not_full_real_kernel" ] }, "conclusion": { "canonical_value": "(1, [[-1, 1], [0, 0], [1, -1]], [[1, -1]], 否, bounded_integer_search_not_full_real_kernel)", "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-20261014-advance-20261102-r02-linear-map-kernel-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261102-r02-linear-map-kernel-foundations/questions.jsonl", "batch_sha256": "8d4468971875ea86c054b2480a40fc13968893d248cef1004921db212a4abffe", "original_line": 76, "original_line_sha256": "344130084ed374f05a68a9a455701acd905c09a322d49d63889faf7d420da9ab", "question_compact_sha256": "8267a67fc972a1b10e9d7e1d0cb4da5ba2a1745a90679779fc3bb30978d114a2", "english_row_compact_sha256": "45403a9b946c8917010aca9ad583730b8f2689aa29ef062518caae5e6196f938", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261102-r02-linear-map-kernel-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261102-r02-linear-map-kernel-foundations.en.jsonl" }, "mko_bindings": { "mko-linear-map-kernel": { "sha256": "01a060bc8fa8fe1e302b8dcb559e8b1a9d5f6607cde6d5973ab75c1e4d0d3abc", "english_sha256": "52a1b6b1a8e08eee92c49d6980e6b664033cf0b4e9173c76496b428aa1db8a5c", "original_path": "public/data/mko/linear-map-kernel.json", "english_path": "public/data/i18n/editions/mko-linear-map-kernel.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "c6f506c0e91ce68861b828c110bfee6002681c5a6e691f1ba5ae296376b30ad5", "original_candidate_sha256": "98575fe5a7faf34e9c5e84eb2a2f8c3b82bc41fb3d766172c63b27475b8ff912", "original_candidate_line_sha256": "98575fe5a7faf34e9c5e84eb2a2f8c3b82bc41fb3d766172c63b27475b8ff912" }, "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-20261026-008-p34", "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": "4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3", "question_first_witness_sha256": "22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 36, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "bac6d9ccd040b0db0868214eb9f38e4bae0126d75fc0cd169f8dfc7acd283e0b", "arithmetic_steps": 20 }, "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." } }