{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261007-advance-20261009-r06-definite-integral-0063", "pilot_number": "R0006-03863", "number": 3863, "family_id": "qf-density-integral-left-error", "title": { "zh-Hant": "第 3863 題：精確矩積分減左端有限和", "en": "Question 3863: Exact Moment Integral Minus a Finite Left Sum" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261007-advance-20261009-r06-definite-integral-0063", "batch_id": "batch-20261007-advance-20261009-r06-definite-integral-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-definite-integral" ], "title_zh": "精確矩積分減左端有限和", "stem_zh": "全實數論域；升冪係數的受限多項式模型。f(x)=13x+(0)，其升冪係數=[0, 13]；a=-2、b=-1，N=13等份左端和S_N。求 I(a,b)−S_N。", "answer": { "kind": "expression", "value": "1/2", "explanation_zh": "多項式矩減左端有限和=1/2。產生端用等差級數導出的精確誤差m(b−a)²/(2N)；S_N仍是有限和，負斜率的差可負，不當作近似證據。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "精確矩積分減左端有限和" ] }, "variant": { "family_id": "qf-density-integral-left-error", "seed": 202610096063, "parameters": { "m": 13, "c": 0, "a": -2, "b": -1, "N": 13, "operation": "density_integral_left_error", "domain": "reals", "semantic_template": "qf-density-integral-left-error/[13,\"1\",13]" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-07", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261007-advance-20261009-r06-definite-integral-0063", "batch_id": "batch-20261007-advance-20261009-r06-definite-integral-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "9cf4030df9616b1539b5fc66b2f2a8db18c2c991601e8bbf79032ec471f24a45", "family_id": "qf-density-integral-left-error", "seed": 202610096063, "semantic_template": "qf-density-integral-left-error/[13,\"1\",13]" }, "fields": { "title": "Exact Moment Integral Minus a Finite Left Sum", "stem": "On all R, f(x)=(13)x+(0). On a=-2≤b=-1, use N=13 uniform left cells. Find the signed error I−S_N, exact moment integral minus the finite left sum. Use reduced positive-denominator fractions, retaining /1 for integers.", "answer": { "kind": "expression", "value": "1/2", "explanation": "Compute the polynomial moment and finite left sum separately, giving I−S_N=1/2. The error may be negative; equality in a constant or zero-width case does not make every finite sum an integral." }, "skills": [ "Exact Moment Integral Minus a Finite Left Sum" ], "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": "全實數論域；升冪係數的受限多項式模型。f(x)=13x+(0)，其升冪係數=[0, 13]；a=-2、b=-1，N=13等份左端和S_N。求 I(a,b)−S_N。", "en": "On all R, f(x)=(13)x+(0). On a=-2≤b=-1, use N=13 uniform left cells. Find the signed error I−S_N, exact moment integral minus the finite left sum. Use reduced positive-denominator fractions, retaining /1 for integers." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "explicit real polynomial moment or constant integrand model", "after": "oriented I(a,b)=H(b)−H(a);do not sort endpoints", "relation": "implies", "rule_id": "polynomial_moment_scope", "conditions": [], "explanation": { "zh-Hant": "受限多項式在實數上連續可積；用逐項原函數端點差求精確有向積分，不把絕對面積或有限和當積分。", "en": "Real polynomials are continuous and integrable;use termwise primitive endpoint differences for exact oriented integrals,not absolute area or a finite sum." }, "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": "S002", "uses": [ "G001", "S001" ], "before": "interval width := sub(-1, -2)", "after": "interval width = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先求非負區間寬度，允許零寬。", "en": "Compute the nonnegative interval width,allowing zero width." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 interval width，不以小數近似替代。", "en": "Recompute interval width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-1", "-2" ], "output": "1", "label": "interval width" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "partition step := div(1, 13)", "after": "partition step = 1/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "除以正整數N得到精確步長。", "en": "Divide by positive integer N to get the exact step." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 partition step，不以小數近似替代。", "en": "Recompute partition step with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "13" ], "output": "1/13", "label": "partition step" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "left sample0 offset := mul(0, 1/13)", "after": "left sample0 offset = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。", "en": "Left samples use i=0,...,N−1,not the final right endpoint." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample0 offset，不以小數近似替代。", "en": "Recompute left sample0 offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "1/13" ], "output": "0", "label": "left sample0 offset" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "left sample0 x := sum(-2, 0)", "after": "left sample0 x = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "起點加上第i個偏移得到採樣位置。", "en": "Add the indexed offset to the start." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample0 x，不以小數近似替代。", "en": "Recompute left sample0 x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "0" ], "output": "-2", "label": "left sample0 x" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "left sample0 mx := mul(13, -2)", "after": "left sample0 mx = -26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "代入仿射函數乘法部分。", "en": "Evaluate the product term of the affine integrand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample0 mx，不以小數近似替代。", "en": "Recompute left sample0 mx with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13", "-2" ], "output": "-26", "label": "left sample0 mx" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "left sample0 value := sum(-26, 0)", "after": "left sample0 value = -26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加常數c得到本採樣值。", "en": "Add c to obtain this sample value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample0 value，不以小數近似替代。", "en": "Recompute left sample0 value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-26", "0" ], "output": "-26", "label": "left sample0 value" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "left sample0 rectangle := mul(-26, 1/13)", "after": "left sample0 rectangle = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。", "en": "Multiply by the step to get the signed rectangle contribution,without absolute values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample0 rectangle，不以小數近似替代。", "en": "Recompute left sample0 rectangle with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-26", "1/13" ], "output": "-2", "label": "left sample0 rectangle" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "[-2, -26, -2]", "after": "finite left-sum term index=0", "relation": "implies", "rule_id": "left_partition_term", "conditions": [], "explanation": { "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。", "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof." }, "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": "S010", "uses": [ "G001", "S009" ], "before": "left sample1 offset := mul(1, 1/13)", "after": "left sample1 offset = 1/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。", "en": "Left samples use i=0,...,N−1,not the final right endpoint." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample1 offset，不以小數近似替代。", "en": "Recompute left sample1 offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1/13" ], "output": "1/13", "label": "left sample1 offset" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "left sample1 x := sum(-2, 1/13)", "after": "left sample1 x = -25/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "起點加上第i個偏移得到採樣位置。", "en": "Add the indexed offset to the start." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample1 x，不以小數近似替代。", "en": "Recompute left sample1 x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "1/13" ], "output": "-25/13", "label": "left sample1 x" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "left sample1 mx := mul(13, -25/13)", "after": "left sample1 mx = -25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "代入仿射函數乘法部分。", "en": "Evaluate the product term of the affine integrand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample1 mx，不以小數近似替代。", "en": "Recompute left sample1 mx with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13", "-25/13" ], "output": "-25", "label": "left sample1 mx" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "left sample1 value := sum(-25, 0)", "after": "left sample1 value = -25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加常數c得到本採樣值。", "en": "Add c to obtain this sample value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample1 value，不以小數近似替代。", "en": "Recompute left sample1 value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-25", "0" ], "output": "-25", "label": "left sample1 value" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "left sample1 rectangle := mul(-25, 1/13)", "after": "left sample1 rectangle = -25/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。", "en": "Multiply by the step to get the signed rectangle contribution,without absolute values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample1 rectangle，不以小數近似替代。", "en": "Recompute left sample1 rectangle with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-25", "1/13" ], "output": "-25/13", "label": "left sample1 rectangle" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "[-25/13, -25, -25/13]", "after": "finite left-sum term index=1", "relation": "implies", "rule_id": "left_partition_term", "conditions": [], "explanation": { "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。", "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof." }, "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": "S016", "uses": [ "G001", "S015" ], "before": "left sample2 offset := mul(2, 1/13)", "after": "left sample2 offset = 2/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。", "en": "Left samples use i=0,...,N−1,not the final right endpoint." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample2 offset，不以小數近似替代。", "en": "Recompute left sample2 offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1/13" ], "output": "2/13", "label": "left sample2 offset" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "left sample2 x := sum(-2, 2/13)", "after": "left sample2 x = -24/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "起點加上第i個偏移得到採樣位置。", "en": "Add the indexed offset to the start." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample2 x，不以小數近似替代。", "en": "Recompute left sample2 x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "2/13" ], "output": "-24/13", "label": "left sample2 x" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "left sample2 mx := mul(13, -24/13)", "after": "left sample2 mx = -24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "代入仿射函數乘法部分。", "en": "Evaluate the product term of the affine integrand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample2 mx，不以小數近似替代。", "en": "Recompute left sample2 mx with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13", "-24/13" ], "output": "-24", "label": "left sample2 mx" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "left sample2 value := sum(-24, 0)", "after": "left sample2 value = -24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加常數c得到本採樣值。", "en": "Add c to obtain this sample value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample2 value，不以小數近似替代。", "en": "Recompute left sample2 value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-24", "0" ], "output": "-24", "label": "left sample2 value" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "left sample2 rectangle := mul(-24, 1/13)", "after": "left sample2 rectangle = -24/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。", "en": "Multiply by the step to get the signed rectangle contribution,without absolute values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample2 rectangle，不以小數近似替代。", "en": "Recompute left sample2 rectangle with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "1/13" ], "output": "-24/13", "label": "left sample2 rectangle" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "[-24/13, -24, -24/13]", "after": "finite left-sum term index=2", "relation": "implies", "rule_id": "left_partition_term", "conditions": [], "explanation": { "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。", "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof." }, "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": "S022", "uses": [ "G001", "S021" ], "before": "left sample3 offset := mul(3, 1/13)", "after": "left sample3 offset = 3/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。", "en": "Left samples use i=0,...,N−1,not the final right endpoint." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample3 offset，不以小數近似替代。", "en": "Recompute left sample3 offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "1/13" ], "output": "3/13", "label": "left sample3 offset" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "left sample3 x := sum(-2, 3/13)", "after": "left sample3 x = -23/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "起點加上第i個偏移得到採樣位置。", "en": "Add the indexed offset to the start." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample3 x，不以小數近似替代。", "en": "Recompute left sample3 x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "3/13" ], "output": "-23/13", "label": "left sample3 x" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "left sample3 mx := mul(13, -23/13)", "after": "left sample3 mx = -23", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "代入仿射函數乘法部分。", "en": "Evaluate the product term of the affine integrand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample3 mx，不以小數近似替代。", "en": "Recompute left sample3 mx with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13", "-23/13" ], "output": "-23", "label": "left sample3 mx" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "left sample3 value := sum(-23, 0)", "after": "left sample3 value = -23", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加常數c得到本採樣值。", "en": "Add c to obtain this sample value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample3 value，不以小數近似替代。", "en": "Recompute left sample3 value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-23", "0" ], "output": "-23", "label": "left sample3 value" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "left sample3 rectangle := mul(-23, 1/13)", "after": "left sample3 rectangle = -23/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。", "en": "Multiply by the step to get the signed rectangle contribution,without absolute values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample3 rectangle，不以小數近似替代。", "en": "Recompute left sample3 rectangle with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-23", "1/13" ], "output": "-23/13", "label": "left sample3 rectangle" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "[-23/13, -23, -23/13]", "after": "finite left-sum term index=3", "relation": "implies", "rule_id": "left_partition_term", "conditions": [], "explanation": { "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。", "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof." }, "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": "left sample4 offset := mul(4, 1/13)", "after": "left sample4 offset = 4/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。", "en": "Left samples use i=0,...,N−1,not the final right endpoint." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample4 offset，不以小數近似替代。", "en": "Recompute left sample4 offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "1/13" ], "output": "4/13", "label": "left sample4 offset" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "left sample4 x := sum(-2, 4/13)", "after": "left sample4 x = -22/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "起點加上第i個偏移得到採樣位置。", "en": "Add the indexed offset to the start." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample4 x，不以小數近似替代。", "en": "Recompute left sample4 x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "4/13" ], "output": "-22/13", "label": "left sample4 x" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "left sample4 mx := mul(13, -22/13)", "after": "left sample4 mx = -22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "代入仿射函數乘法部分。", "en": "Evaluate the product term of the affine integrand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample4 mx，不以小數近似替代。", "en": "Recompute left sample4 mx with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13", "-22/13" ], "output": "-22", "label": "left sample4 mx" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "left sample4 value := sum(-22, 0)", "after": "left sample4 value = -22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加常數c得到本採樣值。", "en": "Add c to obtain this sample value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample4 value，不以小數近似替代。", "en": "Recompute left sample4 value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-22", "0" ], "output": "-22", "label": "left sample4 value" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "left sample4 rectangle := mul(-22, 1/13)", "after": "left sample4 rectangle = -22/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。", "en": "Multiply by the step to get the signed rectangle contribution,without absolute values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample4 rectangle，不以小數近似替代。", "en": "Recompute left sample4 rectangle with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-22", "1/13" ], "output": "-22/13", "label": "left sample4 rectangle" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "[-22/13, -22, -22/13]", "after": "finite left-sum term index=4", "relation": "implies", "rule_id": "left_partition_term", "conditions": [], "explanation": { "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。", "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof." }, "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": "left sample5 offset := mul(5, 1/13)", "after": "left sample5 offset = 5/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。", "en": "Left samples use i=0,...,N−1,not the final right endpoint." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample5 offset，不以小數近似替代。", "en": "Recompute left sample5 offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "5", "1/13" ], "output": "5/13", "label": "left sample5 offset" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "left sample5 x := sum(-2, 5/13)", "after": "left sample5 x = -21/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "起點加上第i個偏移得到採樣位置。", "en": "Add the indexed offset to the start." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample5 x，不以小數近似替代。", "en": "Recompute left sample5 x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "5/13" ], "output": "-21/13", "label": "left sample5 x" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "left sample5 mx := mul(13, -21/13)", "after": "left sample5 mx = -21", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "代入仿射函數乘法部分。", "en": "Evaluate the product term of the affine integrand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample5 mx，不以小數近似替代。", "en": "Recompute left sample5 mx with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13", "-21/13" ], "output": "-21", "label": "left sample5 mx" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "left sample5 value := sum(-21, 0)", "after": "left sample5 value = -21", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加常數c得到本採樣值。", "en": "Add c to obtain this sample value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample5 value，不以小數近似替代。", "en": "Recompute left sample5 value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-21", "0" ], "output": "-21", "label": "left sample5 value" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "left sample5 rectangle := mul(-21, 1/13)", "after": "left sample5 rectangle = -21/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。", "en": "Multiply by the step to get the signed rectangle contribution,without absolute values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample5 rectangle，不以小數近似替代。", "en": "Recompute left sample5 rectangle with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-21", "1/13" ], "output": "-21/13", "label": "left sample5 rectangle" } }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "[-21/13, -21, -21/13]", "after": "finite left-sum term index=5", "relation": "implies", "rule_id": "left_partition_term", "conditions": [], "explanation": { "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。", "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof." }, "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": "S040", "uses": [ "G001", "S039" ], "before": "left sample6 offset := mul(6, 1/13)", "after": "left sample6 offset = 6/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。", "en": "Left samples use i=0,...,N−1,not the final right endpoint." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample6 offset，不以小數近似替代。", "en": "Recompute left sample6 offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "1/13" ], "output": "6/13", "label": "left sample6 offset" } }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "left sample6 x := sum(-2, 6/13)", "after": "left sample6 x = -20/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "起點加上第i個偏移得到採樣位置。", "en": "Add the indexed offset to the start." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample6 x，不以小數近似替代。", "en": "Recompute left sample6 x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "6/13" ], "output": "-20/13", "label": "left sample6 x" } }, { "id": "S042", "uses": [ "G001", "S041" ], "before": "left sample6 mx := mul(13, -20/13)", "after": "left sample6 mx = -20", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "代入仿射函數乘法部分。", "en": "Evaluate the product term of the affine integrand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample6 mx，不以小數近似替代。", "en": "Recompute left sample6 mx with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13", "-20/13" ], "output": "-20", "label": "left sample6 mx" } }, { "id": "S043", "uses": [ "G001", "S042" ], "before": "left sample6 value := sum(-20, 0)", "after": "left sample6 value = -20", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加常數c得到本採樣值。", "en": "Add c to obtain this sample value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample6 value，不以小數近似替代。", "en": "Recompute left sample6 value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-20", "0" ], "output": "-20", "label": "left sample6 value" } }, { "id": "S044", "uses": [ "G001", "S043" ], "before": "left sample6 rectangle := mul(-20, 1/13)", "after": "left sample6 rectangle = -20/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。", "en": "Multiply by the step to get the signed rectangle contribution,without absolute values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample6 rectangle，不以小數近似替代。", "en": "Recompute left sample6 rectangle with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-20", "1/13" ], "output": "-20/13", "label": "left sample6 rectangle" } }, { "id": "S045", "uses": [ "G001", "S044" ], "before": "[-20/13, -20, -20/13]", "after": "finite left-sum term index=6", "relation": "implies", "rule_id": "left_partition_term", "conditions": [], "explanation": { "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。", "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof." }, "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": "S046", "uses": [ "G001", "S045" ], "before": "left sample7 offset := mul(7, 1/13)", "after": "left sample7 offset = 7/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。", "en": "Left samples use i=0,...,N−1,not the final right endpoint." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample7 offset，不以小數近似替代。", "en": "Recompute left sample7 offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7", "1/13" ], "output": "7/13", "label": "left sample7 offset" } }, { "id": "S047", "uses": [ "G001", "S046" ], "before": "left sample7 x := sum(-2, 7/13)", "after": "left sample7 x = -19/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "起點加上第i個偏移得到採樣位置。", "en": "Add the indexed offset to the start." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample7 x，不以小數近似替代。", "en": "Recompute left sample7 x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "7/13" ], "output": "-19/13", "label": "left sample7 x" } }, { "id": "S048", "uses": [ "G001", "S047" ], "before": "left sample7 mx := mul(13, -19/13)", "after": "left sample7 mx = -19", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "代入仿射函數乘法部分。", "en": "Evaluate the product term of the affine integrand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample7 mx，不以小數近似替代。", "en": "Recompute left sample7 mx with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13", "-19/13" ], "output": "-19", "label": "left sample7 mx" } }, { "id": "S049", "uses": [ "G001", "S048" ], "before": "left sample7 value := sum(-19, 0)", "after": "left sample7 value = -19", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加常數c得到本採樣值。", "en": "Add c to obtain this sample value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample7 value，不以小數近似替代。", "en": "Recompute left sample7 value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-19", "0" ], "output": "-19", "label": "left sample7 value" } }, { "id": "S050", "uses": [ "G001", "S049" ], "before": "left sample7 rectangle := mul(-19, 1/13)", "after": "left sample7 rectangle = -19/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。", "en": "Multiply by the step to get the signed rectangle contribution,without absolute values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample7 rectangle，不以小數近似替代。", "en": "Recompute left sample7 rectangle with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-19", "1/13" ], "output": "-19/13", "label": "left sample7 rectangle" } }, { "id": "S051", "uses": [ "G001", "S050" ], "before": "[-19/13, -19, -19/13]", "after": "finite left-sum term index=7", "relation": "implies", "rule_id": "left_partition_term", "conditions": [], "explanation": { "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。", "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof." }, "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": "S052", "uses": [ "G001", "S051" ], "before": "left sample8 offset := mul(8, 1/13)", "after": "left sample8 offset = 8/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。", "en": "Left samples use i=0,...,N−1,not the final right endpoint." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample8 offset，不以小數近似替代。", "en": "Recompute left sample8 offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "8", "1/13" ], "output": "8/13", "label": "left sample8 offset" } }, { "id": "S053", "uses": [ "G001", "S052" ], "before": "left sample8 x := sum(-2, 8/13)", "after": "left sample8 x = -18/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "起點加上第i個偏移得到採樣位置。", "en": "Add the indexed offset to the start." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample8 x，不以小數近似替代。", "en": "Recompute left sample8 x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "8/13" ], "output": "-18/13", "label": "left sample8 x" } }, { "id": "S054", "uses": [ "G001", "S053" ], "before": "left sample8 mx := mul(13, -18/13)", "after": "left sample8 mx = -18", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "代入仿射函數乘法部分。", "en": "Evaluate the product term of the affine integrand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample8 mx，不以小數近似替代。", "en": "Recompute left sample8 mx with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13", "-18/13" ], "output": "-18", "label": "left sample8 mx" } }, { "id": "S055", "uses": [ "G001", "S054" ], "before": "left sample8 value := sum(-18, 0)", "after": "left sample8 value = -18", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加常數c得到本採樣值。", "en": "Add c to obtain this sample value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample8 value，不以小數近似替代。", "en": "Recompute left sample8 value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-18", "0" ], "output": "-18", "label": "left sample8 value" } }, { "id": "S056", "uses": [ "G001", "S055" ], "before": "left sample8 rectangle := mul(-18, 1/13)", "after": "left sample8 rectangle = -18/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。", "en": "Multiply by the step to get the signed rectangle contribution,without absolute values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample8 rectangle，不以小數近似替代。", "en": "Recompute left sample8 rectangle with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-18", "1/13" ], "output": "-18/13", "label": "left sample8 rectangle" } }, { "id": "S057", "uses": [ "G001", "S056" ], "before": "[-18/13, -18, -18/13]", "after": "finite left-sum term index=8", "relation": "implies", "rule_id": "left_partition_term", "conditions": [], "explanation": { "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。", "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof." }, "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": "S058", "uses": [ "G001", "S057" ], "before": "left sample9 offset := mul(9, 1/13)", "after": "left sample9 offset = 9/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。", "en": "Left samples use i=0,...,N−1,not the final right endpoint." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample9 offset，不以小數近似替代。", "en": "Recompute left sample9 offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "9", "1/13" ], "output": "9/13", "label": "left sample9 offset" } }, { "id": "S059", "uses": [ "G001", "S058" ], "before": "left sample9 x := sum(-2, 9/13)", "after": "left sample9 x = -17/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "起點加上第i個偏移得到採樣位置。", "en": "Add the indexed offset to the start." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample9 x，不以小數近似替代。", "en": "Recompute left sample9 x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "9/13" ], "output": "-17/13", "label": "left sample9 x" } }, { "id": "S060", "uses": [ "G001", "S059" ], "before": "left sample9 mx := mul(13, -17/13)", "after": "left sample9 mx = -17", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "代入仿射函數乘法部分。", "en": "Evaluate the product term of the affine integrand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample9 mx，不以小數近似替代。", "en": "Recompute left sample9 mx with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13", "-17/13" ], "output": "-17", "label": "left sample9 mx" } }, { "id": "S061", "uses": [ "G001", "S060" ], "before": "left sample9 value := sum(-17, 0)", "after": "left sample9 value = -17", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加常數c得到本採樣值。", "en": "Add c to obtain this sample value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample9 value，不以小數近似替代。", "en": "Recompute left sample9 value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-17", "0" ], "output": "-17", "label": "left sample9 value" } }, { "id": "S062", "uses": [ "G001", "S061" ], "before": "left sample9 rectangle := mul(-17, 1/13)", "after": "left sample9 rectangle = -17/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。", "en": "Multiply by the step to get the signed rectangle contribution,without absolute values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample9 rectangle，不以小數近似替代。", "en": "Recompute left sample9 rectangle with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-17", "1/13" ], "output": "-17/13", "label": "left sample9 rectangle" } }, { "id": "S063", "uses": [ "G001", "S062" ], "before": "[-17/13, -17, -17/13]", "after": "finite left-sum term index=9", "relation": "implies", "rule_id": "left_partition_term", "conditions": [], "explanation": { "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。", "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof." }, "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": "S064", "uses": [ "G001", "S063" ], "before": "left sample10 offset := mul(10, 1/13)", "after": "left sample10 offset = 10/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。", "en": "Left samples use i=0,...,N−1,not the final right endpoint." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample10 offset，不以小數近似替代。", "en": "Recompute left sample10 offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "10", "1/13" ], "output": "10/13", "label": "left sample10 offset" } }, { "id": "S065", "uses": [ "G001", "S064" ], "before": "left sample10 x := sum(-2, 10/13)", "after": "left sample10 x = -16/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "起點加上第i個偏移得到採樣位置。", "en": "Add the indexed offset to the start." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample10 x，不以小數近似替代。", "en": "Recompute left sample10 x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "10/13" ], "output": "-16/13", "label": "left sample10 x" } }, { "id": "S066", "uses": [ "G001", "S065" ], "before": "left sample10 mx := mul(13, -16/13)", "after": "left sample10 mx = -16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "代入仿射函數乘法部分。", "en": "Evaluate the product term of the affine integrand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample10 mx，不以小數近似替代。", "en": "Recompute left sample10 mx with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13", "-16/13" ], "output": "-16", "label": "left sample10 mx" } }, { "id": "S067", "uses": [ "G001", "S066" ], "before": "left sample10 value := sum(-16, 0)", "after": "left sample10 value = -16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加常數c得到本採樣值。", "en": "Add c to obtain this sample value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample10 value，不以小數近似替代。", "en": "Recompute left sample10 value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-16", "0" ], "output": "-16", "label": "left sample10 value" } }, { "id": "S068", "uses": [ "G001", "S067" ], "before": "left sample10 rectangle := mul(-16, 1/13)", "after": "left sample10 rectangle = -16/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。", "en": "Multiply by the step to get the signed rectangle contribution,without absolute values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample10 rectangle，不以小數近似替代。", "en": "Recompute left sample10 rectangle with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-16", "1/13" ], "output": "-16/13", "label": "left sample10 rectangle" } }, { "id": "S069", "uses": [ "G001", "S068" ], "before": "[-16/13, -16, -16/13]", "after": "finite left-sum term index=10", "relation": "implies", "rule_id": "left_partition_term", "conditions": [], "explanation": { "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。", "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof." }, "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": "S070", "uses": [ "G001", "S069" ], "before": "left sample11 offset := mul(11, 1/13)", "after": "left sample11 offset = 11/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。", "en": "Left samples use i=0,...,N−1,not the final right endpoint." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample11 offset，不以小數近似替代。", "en": "Recompute left sample11 offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "11", "1/13" ], "output": "11/13", "label": "left sample11 offset" } }, { "id": "S071", "uses": [ "G001", "S070" ], "before": "left sample11 x := sum(-2, 11/13)", "after": "left sample11 x = -15/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "起點加上第i個偏移得到採樣位置。", "en": "Add the indexed offset to the start." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample11 x，不以小數近似替代。", "en": "Recompute left sample11 x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "11/13" ], "output": "-15/13", "label": "left sample11 x" } }, { "id": "S072", "uses": [ "G001", "S071" ], "before": "left sample11 mx := mul(13, -15/13)", "after": "left sample11 mx = -15", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "代入仿射函數乘法部分。", "en": "Evaluate the product term of the affine integrand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample11 mx，不以小數近似替代。", "en": "Recompute left sample11 mx with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13", "-15/13" ], "output": "-15", "label": "left sample11 mx" } }, { "id": "S073", "uses": [ "G001", "S072" ], "before": "left sample11 value := sum(-15, 0)", "after": "left sample11 value = -15", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加常數c得到本採樣值。", "en": "Add c to obtain this sample value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample11 value，不以小數近似替代。", "en": "Recompute left sample11 value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-15", "0" ], "output": "-15", "label": "left sample11 value" } }, { "id": "S074", "uses": [ "G001", "S073" ], "before": "left sample11 rectangle := mul(-15, 1/13)", "after": "left sample11 rectangle = -15/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。", "en": "Multiply by the step to get the signed rectangle contribution,without absolute values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample11 rectangle，不以小數近似替代。", "en": "Recompute left sample11 rectangle with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-15", "1/13" ], "output": "-15/13", "label": "left sample11 rectangle" } }, { "id": "S075", "uses": [ "G001", "S074" ], "before": "[-15/13, -15, -15/13]", "after": "finite left-sum term index=11", "relation": "implies", "rule_id": "left_partition_term", "conditions": [], "explanation": { "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。", "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof." }, "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": "S076", "uses": [ "G001", "S075" ], "before": "left sample12 offset := mul(12, 1/13)", "after": "left sample12 offset = 12/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "左端採樣用i=0,...,N−1，不含最後右端點。", "en": "Left samples use i=0,...,N−1,not the final right endpoint." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample12 offset，不以小數近似替代。", "en": "Recompute left sample12 offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "12", "1/13" ], "output": "12/13", "label": "left sample12 offset" } }, { "id": "S077", "uses": [ "G001", "S076" ], "before": "left sample12 x := sum(-2, 12/13)", "after": "left sample12 x = -14/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "起點加上第i個偏移得到採樣位置。", "en": "Add the indexed offset to the start." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample12 x，不以小數近似替代。", "en": "Recompute left sample12 x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "12/13" ], "output": "-14/13", "label": "left sample12 x" } }, { "id": "S078", "uses": [ "G001", "S077" ], "before": "left sample12 mx := mul(13, -14/13)", "after": "left sample12 mx = -14", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "代入仿射函數乘法部分。", "en": "Evaluate the product term of the affine integrand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample12 mx，不以小數近似替代。", "en": "Recompute left sample12 mx with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13", "-14/13" ], "output": "-14", "label": "left sample12 mx" } }, { "id": "S079", "uses": [ "G001", "S078" ], "before": "left sample12 value := sum(-14, 0)", "after": "left sample12 value = -14", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加常數c得到本採樣值。", "en": "Add c to obtain this sample value." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample12 value，不以小數近似替代。", "en": "Recompute left sample12 value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-14", "0" ], "output": "-14", "label": "left sample12 value" } }, { "id": "S080", "uses": [ "G001", "S079" ], "before": "left sample12 rectangle := mul(-14, 1/13)", "after": "left sample12 rectangle = -14/13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "樣本值乘步長是有符號矩形項，不取絕對值。", "en": "Multiply by the step to get the signed rectangle contribution,without absolute values." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 left sample12 rectangle，不以小數近似替代。", "en": "Recompute left sample12 rectangle with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-14", "1/13" ], "output": "-14/13", "label": "left sample12 rectangle" } }, { "id": "S081", "uses": [ "G001", "S080" ], "before": "[-14/13, -14, -14/13]", "after": "finite left-sum term index=12", "relation": "implies", "rule_id": "left_partition_term", "conditions": [], "explanation": { "zh-Hant": "每個分割恰列一個左端項，這只是有限和資料，不冒稱已做極限證明。", "en": "List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof." }, "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": "S082", "uses": [ "G001", "S081" ], "before": "finite left sum := sum(-2, -25/13, -24/13, -23/13, -22/13, -21/13, -20/13, -19/13, -18/13, -17/13, -16/13, -15/13, -14/13)", "after": "finite left sum = -20", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "把N項完整相加。", "en": "Add all N contributions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 finite left sum，不以小數近似替代。", "en": "Recompute finite left sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "-25/13", "-24/13", "-23/13", "-22/13", "-21/13", "-20/13", "-19/13", "-18/13", "-17/13", "-16/13", "-15/13", "-14/13" ], "output": "-20", "label": "finite left sum" } }, { "id": "S083", "uses": [ "G001", "S082" ], "before": "affine integral primitive coefficient1 := div(0, 1)", "after": "affine integral primitive coefficient1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "x^k的原函數係數是c/(k+1)，正整數k+1非零。", "en": "The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine integral primitive coefficient1，不以小數近似替代。", "en": "Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "1" ], "output": "0", "label": "affine integral primitive coefficient1" } }, { "id": "S084", "uses": [ "G001", "S083" ], "before": "affine integral primitive coefficient2 := div(13, 2)", "after": "affine integral primitive coefficient2 = 13/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "x^k的原函數係數是c/(k+1)，正整數k+1非零。", "en": "The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine integral primitive coefficient2，不以小數近似替代。", "en": "Recompute affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "13", "2" ], "output": "13/2", "label": "affine integral primitive coefficient2" } }, { "id": "S085", "uses": [ "G001", "S084" ], "before": "[0, 0, 13/2]", "after": "H′=the supplied polynomial;integrate from -2 to -1", "relation": "implies", "rule_id": "polynomial_primitive_identity", "conditions": [], "explanation": { "zh-Hant": "逐項微分回原係數確定原函數；積分常數在端點差相消。", "en": "Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference." }, "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": "S086", "uses": [ "G001", "S085" ], "before": "affine integral H(b) ascending coefficients=[0, 0, 13/2]", "after": "evaluate at x=-1", "relation": "implies", "rule_id": "polynomial_evaluation_order", "conditions": [], "explanation": { "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。", "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list." }, "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": "S087", "uses": [ "G001", "S086" ], "before": "affine integral H(b) Horner multiply before degree1 := mul(13/2, -1)", "after": "affine integral H(b) Horner multiply before degree1 = -13/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先乘同一代入點。", "en": "Multiply by the same evaluation point first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine integral H(b) Horner multiply before degree1，不以小數近似替代。", "en": "Recompute affine integral H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13/2", "-1" ], "output": "-13/2", "label": "affine integral H(b) Horner multiply before degree1" } }, { "id": "S088", "uses": [ "G001", "S087" ], "before": "affine integral H(b) Horner add degree1 := sum(-13/2, 0)", "after": "affine integral H(b) Horner add degree1 = -13/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加此位置的完整係數。", "en": "Then add the full coefficient at this position." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine integral H(b) Horner add degree1，不以小數近似替代。", "en": "Recompute affine integral H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-13/2", "0" ], "output": "-13/2", "label": "affine integral H(b) Horner add degree1" } }, { "id": "S089", "uses": [ "G001", "S088" ], "before": "affine integral H(b) Horner multiply before degree0 := mul(-13/2, -1)", "after": "affine integral H(b) Horner multiply before degree0 = 13/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先乘同一代入點。", "en": "Multiply by the same evaluation point first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine integral H(b) Horner multiply before degree0，不以小數近似替代。", "en": "Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-13/2", "-1" ], "output": "13/2", "label": "affine integral H(b) Horner multiply before degree0" } }, { "id": "S090", "uses": [ "G001", "S089" ], "before": "affine integral H(b) Horner add degree0 := sum(13/2, 0)", "after": "affine integral H(b) Horner add degree0 = 13/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加此位置的完整係數。", "en": "Then add the full coefficient at this position." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine integral H(b) Horner add degree0，不以小數近似替代。", "en": "Recompute affine integral H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "13/2", "0" ], "output": "13/2", "label": "affine integral H(b) Horner add degree0" } }, { "id": "S091", "uses": [ "G001", "S090" ], "before": "affine integral H(a) ascending coefficients=[0, 0, 13/2]", "after": "evaluate at x=-2", "relation": "implies", "rule_id": "polynomial_evaluation_order", "conditions": [], "explanation": { "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。", "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list." }, "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": "S092", "uses": [ "G001", "S091" ], "before": "affine integral H(a) Horner multiply before degree1 := mul(13/2, -2)", "after": "affine integral H(a) Horner multiply before degree1 = -13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先乘同一代入點。", "en": "Multiply by the same evaluation point first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine integral H(a) Horner multiply before degree1，不以小數近似替代。", "en": "Recompute affine integral H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13/2", "-2" ], "output": "-13", "label": "affine integral H(a) Horner multiply before degree1" } }, { "id": "S093", "uses": [ "G001", "S092" ], "before": "affine integral H(a) Horner add degree1 := sum(-13, 0)", "after": "affine integral H(a) Horner add degree1 = -13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加此位置的完整係數。", "en": "Then add the full coefficient at this position." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine integral H(a) Horner add degree1，不以小數近似替代。", "en": "Recompute affine integral H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-13", "0" ], "output": "-13", "label": "affine integral H(a) Horner add degree1" } }, { "id": "S094", "uses": [ "G001", "S093" ], "before": "affine integral H(a) Horner multiply before degree0 := mul(-13, -2)", "after": "affine integral H(a) Horner multiply before degree0 = 26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先乘同一代入點。", "en": "Multiply by the same evaluation point first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine integral H(a) Horner multiply before degree0，不以小數近似替代。", "en": "Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-13", "-2" ], "output": "26", "label": "affine integral H(a) Horner multiply before degree0" } }, { "id": "S095", "uses": [ "G001", "S094" ], "before": "affine integral H(a) Horner add degree0 := sum(26, 0)", "after": "affine integral H(a) Horner add degree0 = 26", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加此位置的完整係數。", "en": "Then add the full coefficient at this position." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine integral H(a) Horner add degree0，不以小數近似替代。", "en": "Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "26", "0" ], "output": "26", "label": "affine integral H(a) Horner add degree0" } }, { "id": "S096", "uses": [ "G001", "S095" ], "before": "affine integral oriented moment := sub(13/2, 26)", "after": "affine integral oriented moment = -39/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "終點值減起點值，端點逆序時自然變號。", "en": "Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 affine integral oriented moment，不以小數近似替代。", "en": "Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "13/2", "26" ], "output": "-39/2", "label": "affine integral oriented moment" } }, { "id": "S097", "uses": [ "G001", "S096" ], "before": "integral minus left sum := sub(-39/2, -20)", "after": "integral minus left sum = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "方向固定為I−S_N；負斜率可產生負差，不是絕對誤差。", "en": "The requested direction is I−S_N;a negative slope may give a negative difference,not absolute error." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 integral minus left sum，不以小數近似替代。", "en": "Recompute integral minus left sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-39/2", "-20" ], "output": "1/2", "label": "integral minus left sum" } }, { "id": "S098", "uses": [ "G001", "S097" ], "before": "I=-39/2;S_N=-20;N=13", "after": "signed finite-sum difference=1/2", "relation": "implies", "rule_id": "finite_sum_vs_integral", "conditions": [], "explanation": { "zh-Hant": "這是精確有限N的差；不把一次數值符合當作一般收斂速度或任意函數的積分證明。", "en": "This is the exact difference for finite N,not evidence of a general convergence rate or integrability of arbitrary functions." }, "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": "1/2", "integral": "-39/2", "left_sum": "-20", "step": "1/13", "partition_values": [ [ "-2", "-26", "-2" ], [ "-25/13", "-25", "-25/13" ], [ "-24/13", "-24", "-24/13" ], [ "-23/13", "-23", "-23/13" ], [ "-22/13", "-22", "-22/13" ], [ "-21/13", "-21", "-21/13" ], [ "-20/13", "-20", "-20/13" ], [ "-19/13", "-19", "-19/13" ], [ "-18/13", "-18", "-18/13" ], [ "-17/13", "-17", "-17/13" ], [ "-16/13", "-16", "-16/13" ], [ "-15/13", "-15", "-15/13" ], [ "-14/13", "-14", "-14/13" ] ] }, "conclusion": { "canonical_value": "1/2", "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-20261007-advance-20261009-r06-definite-integral-foundations", "batch_path": "public/data/questions/batches/batch-20261007-advance-20261009-r06-definite-integral-foundations/questions.jsonl", "batch_sha256": "00d0f47b298d3ba323a3ab2d1343f4bc4c7badcb593d0c1c108a6245ce665cfa", "original_line": 63, "original_line_sha256": "8055b770d702bf3d06153bb9f26233f2390f60f716a6b5b67e1edbb6d79de56c", "question_compact_sha256": "9cf4030df9616b1539b5fc66b2f2a8db18c2c991601e8bbf79032ec471f24a45", "english_row_compact_sha256": "36f1d8560a4347795a935faebe44ed3fbf57ca60a419ffb3767a54ff4d5eb4a1", "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261009-r06-definite-integral-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261009-r06-definite-integral-foundations.en.jsonl" }, "mko_bindings": { "mko-definite-integral": { "sha256": "f1fc56fc2ac48870dcaa53e6269bb2e365c1529c5540683ba6b6cd103ddcc5f7", "english_sha256": "cd470242748859a6d2bf765f69c47fa6a116eca1f2e64fb1a5edea6fe686d6a4", "original_path": "public/data/mko/definite-integral.json", "english_path": "public/data/i18n/editions/mko-definite-integral.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "6bac5f83f28cdd9c992a469306bfdf5c5415de17d11847cb9b17b4d13934c7e6", "original_candidate_sha256": "54199de6155ee8cdabe1480a5c22ddb558cd4e9bd7166d52b17433d5bfe06b15", "original_candidate_line_sha256": "54199de6155ee8cdabe1480a5c22ddb558cd4e9bd7166d52b17433d5bfe06b15" }, "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-20261026-004-p7", "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": "568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d", "question_first_witness_sha256": "d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 98, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "7722dd7af695504f939c71d0458fc1f9e5e44cefe7a32bc6906dff0ae725221f", "arithmetic_steps": 80 }, "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." } }