{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261007-advance-20261008-r07-definite-integral-0099", "pilot_number": "R0064-32624", "number": 32624, "family_id": "qf-integral-affine-left-sum", "title": { "zh-Hant": "第 32624 題：仿射式的左端有限和", "en": "Question 32624: Finite Left-Endpoint Sum of an Affine Function" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261007-advance-20261008-r07-definite-integral-0099", "batch_id": "batch-20261007-advance-20261008-r07-definite-integral-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-definite-integral" ], "title_zh": "仿射式的左端有限和", "stem_zh": "全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-24x+(11)，區間 [1,5] 分為 N=24 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。", "answer": { "kind": "expression", "value": "-236/1", "explanation_zh": "左端有限和=-236/1。產生端用等差級數閉式，答案是此有限和，不一般當成精確積分；常數式或零區間可恰與多項式矩積分相同。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "仿射式的左端有限和" ] }, "variant": { "family_id": "qf-integral-affine-left-sum", "seed": 202610087099, "parameters": { "m": -24, "c": 11, "a": 1, "b": 5, "N": 24, "operation": "affine_left_sum", "domain": "reals", "semantic_template": "qf-integral-affine-left-sum/{\"inputs\":[\"-24\",\"-13\",\"4\",\"24\"],\"operation\":\"affine_left_sum\"}" } }, "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-20261008-r07-definite-integral-0099", "batch_id": "batch-20261007-advance-20261008-r07-definite-integral-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "277bd8619d17823b71393f4770726c7b7b74536e60b6d6b02420b9d49c772df8", "family_id": "qf-integral-affine-left-sum", "seed": 202610087099, "semantic_template": "qf-integral-affine-left-sum/{\"inputs\":[\"-24\",\"-13\",\"4\",\"24\"],\"operation\":\"affine_left_sum\"}" }, "fields": { "title": "Finite Left-Endpoint Sum of an Affine Function", "stem": "On all R, f(x)=(-24)x+(11). For a=1≤b=5 and N=24 uniform cells, compute the finite left-endpoint sum with Δx=(b−a)/N and x_i=a+iΔx for i=0,…,N−1. Give a reduced fraction with positive denominator, retaining /1 for integers.", "answer": { "kind": "expression", "value": "-236/1", "explanation": "Exact termwise rational accumulation gives the finite sum -236/1. A finite left sum generally differs from the integral, even though constant or zero-length special cases can agree; this is not a general Riemann-integrability proof." }, "skills": [ "Finite Left-Endpoint Sum of an Affine Function" ], "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，以下函數／點均以明示實數模型解讀。f(x)=-24x+(11)，區間 [1,5] 分為 N=24 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。", "en": "On all R, f(x)=(-24)x+(11). For a=1≤b=5 and N=24 uniform cells, compute the finite left-endpoint sum with Δx=(b−a)/N and x_i=a+iΔx for i=0,…,N−1. Give a reduced fraction with positive denominator, retaining /1 for integers." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-integral-affine-left-sum, affine_left_sum, reals, []]", "after": "derived = [[finite_left_sum_fraction_RETAIN1]]", "relation": "implies", "rule_id": "R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles", "conditions": [], "explanation": { "zh-Hant": "legacy核來源精確欄位及operation/domain；BASE整數保metadata但不當oracle。先分NUMBER、字串、fractionpair或vector等式，不強套同一格式；原中英／AST／status不變。", "en": "Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Left sum ORIGINAL upper minus lower endpoint := sub(5, 1)", "after": "Left sum ORIGINAL upper minus lower endpoint = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL upper minus lower endpoint，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL upper minus lower endpoint with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "5", "1" ], "output": "4", "label": "Left sum ORIGINAL upper minus lower endpoint" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Left sum ORIGINAL width over positive cell count := div(4, 24)", "after": "Left sum ORIGINAL width over positive cell count = 1/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL width over positive cell count，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL width over positive cell count with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "4", "24" ], "output": "1/6", "label": "Left sum ORIGINAL width over positive cell count" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Left sum ORIGINAL sample0 index times width := mul(0, 1/6)", "after": "Left sum ORIGINAL sample0 index times width = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample0 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample0 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "1/6" ], "output": "0", "label": "Left sum ORIGINAL sample0 index times width" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Left sum ORIGINAL sample0 lower plus offset := sum(1, 0)", "after": "Left sum ORIGINAL sample0 lower plus offset = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample0 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample0 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "0" ], "output": "1", "label": "Left sum ORIGINAL sample0 lower plus offset" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Left sum ORIGINAL sample0 slope times x := mul(-24, 1)", "after": "Left sum ORIGINAL sample0 slope times x = -24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample0 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample0 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "1" ], "output": "-24", "label": "Left sum ORIGINAL sample0 slope times x" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Left sum ORIGINAL sample0 affine value plus c := sum(-24, 11)", "after": "Left sum ORIGINAL sample0 affine value plus c = -13", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample0 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample0 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-24", "11" ], "output": "-13", "label": "Left sum ORIGINAL sample0 affine value plus c" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Left sum ORIGINAL sample0 value times cell width := mul(-13, 1/6)", "after": "Left sum ORIGINAL sample0 value times cell width = -13/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample0 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample0 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-13", "1/6" ], "output": "-13/6", "label": "Left sum ORIGINAL sample0 value times cell width" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Left sum ORIGINAL accumulated term0 := sum(0, -13/6)", "after": "Left sum ORIGINAL accumulated term0 = -13/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term0，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "-13/6" ], "output": "-13/6", "label": "Left sum ORIGINAL accumulated term0" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "source roles = [0, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [1/1, -13/1, -13/6, -13/6]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Left sum ORIGINAL sample1 index times width := mul(1, 1/6)", "after": "Left sum ORIGINAL sample1 index times width = 1/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample1 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample1 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1/6" ], "output": "1/6", "label": "Left sum ORIGINAL sample1 index times width" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Left sum ORIGINAL sample1 lower plus offset := sum(1, 1/6)", "after": "Left sum ORIGINAL sample1 lower plus offset = 7/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample1 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample1 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1/6" ], "output": "7/6", "label": "Left sum ORIGINAL sample1 lower plus offset" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Left sum ORIGINAL sample1 slope times x := mul(-24, 7/6)", "after": "Left sum ORIGINAL sample1 slope times x = -28", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample1 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample1 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "7/6" ], "output": "-28", "label": "Left sum ORIGINAL sample1 slope times x" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "Left sum ORIGINAL sample1 affine value plus c := sum(-28, 11)", "after": "Left sum ORIGINAL sample1 affine value plus c = -17", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample1 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample1 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-28", "11" ], "output": "-17", "label": "Left sum ORIGINAL sample1 affine value plus c" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "Left sum ORIGINAL sample1 value times cell width := mul(-17, 1/6)", "after": "Left sum ORIGINAL sample1 value times cell width = -17/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample1 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample1 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-17", "1/6" ], "output": "-17/6", "label": "Left sum ORIGINAL sample1 value times cell width" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Left sum ORIGINAL accumulated term1 := sum(-13/6, -17/6)", "after": "Left sum ORIGINAL accumulated term1 = -5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term1，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-13/6", "-17/6" ], "output": "-5", "label": "Left sum ORIGINAL accumulated term1" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "source roles = [1, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [7/6, -17/1, -17/6, -5/1]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "Left sum ORIGINAL sample2 index times width := mul(2, 1/6)", "after": "Left sum ORIGINAL sample2 index times width = 1/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample2 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample2 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1/6" ], "output": "1/3", "label": "Left sum ORIGINAL sample2 index times width" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "Left sum ORIGINAL sample2 lower plus offset := sum(1, 1/3)", "after": "Left sum ORIGINAL sample2 lower plus offset = 4/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample2 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample2 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1/3" ], "output": "4/3", "label": "Left sum ORIGINAL sample2 lower plus offset" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "Left sum ORIGINAL sample2 slope times x := mul(-24, 4/3)", "after": "Left sum ORIGINAL sample2 slope times x = -32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample2 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample2 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "4/3" ], "output": "-32", "label": "Left sum ORIGINAL sample2 slope times x" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "Left sum ORIGINAL sample2 affine value plus c := sum(-32, 11)", "after": "Left sum ORIGINAL sample2 affine value plus c = -21", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample2 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample2 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-32", "11" ], "output": "-21", "label": "Left sum ORIGINAL sample2 affine value plus c" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "Left sum ORIGINAL sample2 value times cell width := mul(-21, 1/6)", "after": "Left sum ORIGINAL sample2 value times cell width = -7/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample2 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample2 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-21", "1/6" ], "output": "-7/2", "label": "Left sum ORIGINAL sample2 value times cell width" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "Left sum ORIGINAL accumulated term2 := sum(-5, -7/2)", "after": "Left sum ORIGINAL accumulated term2 = -17/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term2，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-5", "-7/2" ], "output": "-17/2", "label": "Left sum ORIGINAL accumulated term2" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "source roles = [2, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [4/3, -21/1, -7/2, -17/2]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "Left sum ORIGINAL sample3 index times width := mul(3, 1/6)", "after": "Left sum ORIGINAL sample3 index times width = 1/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample3 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample3 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "1/6" ], "output": "1/2", "label": "Left sum ORIGINAL sample3 index times width" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "Left sum ORIGINAL sample3 lower plus offset := sum(1, 1/2)", "after": "Left sum ORIGINAL sample3 lower plus offset = 3/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample3 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample3 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1/2" ], "output": "3/2", "label": "Left sum ORIGINAL sample3 lower plus offset" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "Left sum ORIGINAL sample3 slope times x := mul(-24, 3/2)", "after": "Left sum ORIGINAL sample3 slope times x = -36", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample3 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample3 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "3/2" ], "output": "-36", "label": "Left sum ORIGINAL sample3 slope times x" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "Left sum ORIGINAL sample3 affine value plus c := sum(-36, 11)", "after": "Left sum ORIGINAL sample3 affine value plus c = -25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample3 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample3 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-36", "11" ], "output": "-25", "label": "Left sum ORIGINAL sample3 affine value plus c" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "Left sum ORIGINAL sample3 value times cell width := mul(-25, 1/6)", "after": "Left sum ORIGINAL sample3 value times cell width = -25/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample3 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample3 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-25", "1/6" ], "output": "-25/6", "label": "Left sum ORIGINAL sample3 value times cell width" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "Left sum ORIGINAL accumulated term3 := sum(-17/2, -25/6)", "after": "Left sum ORIGINAL accumulated term3 = -38/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term3，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-17/2", "-25/6" ], "output": "-38/3", "label": "Left sum ORIGINAL accumulated term3" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "source roles = [3, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [3/2, -25/1, -25/6, -38/3]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "Left sum ORIGINAL sample4 index times width := mul(4, 1/6)", "after": "Left sum ORIGINAL sample4 index times width = 2/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample4 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample4 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "1/6" ], "output": "2/3", "label": "Left sum ORIGINAL sample4 index times width" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "Left sum ORIGINAL sample4 lower plus offset := sum(1, 2/3)", "after": "Left sum ORIGINAL sample4 lower plus offset = 5/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample4 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample4 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "2/3" ], "output": "5/3", "label": "Left sum ORIGINAL sample4 lower plus offset" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "Left sum ORIGINAL sample4 slope times x := mul(-24, 5/3)", "after": "Left sum ORIGINAL sample4 slope times x = -40", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample4 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample4 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "5/3" ], "output": "-40", "label": "Left sum ORIGINAL sample4 slope times x" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "Left sum ORIGINAL sample4 affine value plus c := sum(-40, 11)", "after": "Left sum ORIGINAL sample4 affine value plus c = -29", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample4 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample4 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-40", "11" ], "output": "-29", "label": "Left sum ORIGINAL sample4 affine value plus c" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "Left sum ORIGINAL sample4 value times cell width := mul(-29, 1/6)", "after": "Left sum ORIGINAL sample4 value times cell width = -29/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample4 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample4 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-29", "1/6" ], "output": "-29/6", "label": "Left sum ORIGINAL sample4 value times cell width" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "Left sum ORIGINAL accumulated term4 := sum(-38/3, -29/6)", "after": "Left sum ORIGINAL accumulated term4 = -35/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term4，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-38/3", "-29/6" ], "output": "-35/2", "label": "Left sum ORIGINAL accumulated term4" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "source roles = [4, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [5/3, -29/1, -29/6, -35/2]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "Left sum ORIGINAL sample5 index times width := mul(5, 1/6)", "after": "Left sum ORIGINAL sample5 index times width = 5/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample5 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample5 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "5", "1/6" ], "output": "5/6", "label": "Left sum ORIGINAL sample5 index times width" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "Left sum ORIGINAL sample5 lower plus offset := sum(1, 5/6)", "after": "Left sum ORIGINAL sample5 lower plus offset = 11/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample5 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample5 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "5/6" ], "output": "11/6", "label": "Left sum ORIGINAL sample5 lower plus offset" } }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "Left sum ORIGINAL sample5 slope times x := mul(-24, 11/6)", "after": "Left sum ORIGINAL sample5 slope times x = -44", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample5 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample5 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "11/6" ], "output": "-44", "label": "Left sum ORIGINAL sample5 slope times x" } }, { "id": "S042", "uses": [ "G001", "S041" ], "before": "Left sum ORIGINAL sample5 affine value plus c := sum(-44, 11)", "after": "Left sum ORIGINAL sample5 affine value plus c = -33", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample5 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample5 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-44", "11" ], "output": "-33", "label": "Left sum ORIGINAL sample5 affine value plus c" } }, { "id": "S043", "uses": [ "G001", "S042" ], "before": "Left sum ORIGINAL sample5 value times cell width := mul(-33, 1/6)", "after": "Left sum ORIGINAL sample5 value times cell width = -11/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample5 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample5 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-33", "1/6" ], "output": "-11/2", "label": "Left sum ORIGINAL sample5 value times cell width" } }, { "id": "S044", "uses": [ "G001", "S043" ], "before": "Left sum ORIGINAL accumulated term5 := sum(-35/2, -11/2)", "after": "Left sum ORIGINAL accumulated term5 = -23", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term5，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-35/2", "-11/2" ], "output": "-23", "label": "Left sum ORIGINAL accumulated term5" } }, { "id": "S045", "uses": [ "G001", "S044" ], "before": "source roles = [5, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [11/6, -33/1, -11/2, -23/1]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S046", "uses": [ "G001", "S045" ], "before": "Left sum ORIGINAL sample6 index times width := mul(6, 1/6)", "after": "Left sum ORIGINAL sample6 index times width = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample6 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample6 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "1/6" ], "output": "1", "label": "Left sum ORIGINAL sample6 index times width" } }, { "id": "S047", "uses": [ "G001", "S046" ], "before": "Left sum ORIGINAL sample6 lower plus offset := sum(1, 1)", "after": "Left sum ORIGINAL sample6 lower plus offset = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample6 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample6 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1" ], "output": "2", "label": "Left sum ORIGINAL sample6 lower plus offset" } }, { "id": "S048", "uses": [ "G001", "S047" ], "before": "Left sum ORIGINAL sample6 slope times x := mul(-24, 2)", "after": "Left sum ORIGINAL sample6 slope times x = -48", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample6 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample6 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "2" ], "output": "-48", "label": "Left sum ORIGINAL sample6 slope times x" } }, { "id": "S049", "uses": [ "G001", "S048" ], "before": "Left sum ORIGINAL sample6 affine value plus c := sum(-48, 11)", "after": "Left sum ORIGINAL sample6 affine value plus c = -37", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample6 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample6 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-48", "11" ], "output": "-37", "label": "Left sum ORIGINAL sample6 affine value plus c" } }, { "id": "S050", "uses": [ "G001", "S049" ], "before": "Left sum ORIGINAL sample6 value times cell width := mul(-37, 1/6)", "after": "Left sum ORIGINAL sample6 value times cell width = -37/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample6 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample6 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-37", "1/6" ], "output": "-37/6", "label": "Left sum ORIGINAL sample6 value times cell width" } }, { "id": "S051", "uses": [ "G001", "S050" ], "before": "Left sum ORIGINAL accumulated term6 := sum(-23, -37/6)", "after": "Left sum ORIGINAL accumulated term6 = -175/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term6，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-23", "-37/6" ], "output": "-175/6", "label": "Left sum ORIGINAL accumulated term6" } }, { "id": "S052", "uses": [ "G001", "S051" ], "before": "source roles = [6, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [2/1, -37/1, -37/6, -175/6]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S053", "uses": [ "G001", "S052" ], "before": "Left sum ORIGINAL sample7 index times width := mul(7, 1/6)", "after": "Left sum ORIGINAL sample7 index times width = 7/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample7 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample7 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7", "1/6" ], "output": "7/6", "label": "Left sum ORIGINAL sample7 index times width" } }, { "id": "S054", "uses": [ "G001", "S053" ], "before": "Left sum ORIGINAL sample7 lower plus offset := sum(1, 7/6)", "after": "Left sum ORIGINAL sample7 lower plus offset = 13/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample7 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample7 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "7/6" ], "output": "13/6", "label": "Left sum ORIGINAL sample7 lower plus offset" } }, { "id": "S055", "uses": [ "G001", "S054" ], "before": "Left sum ORIGINAL sample7 slope times x := mul(-24, 13/6)", "after": "Left sum ORIGINAL sample7 slope times x = -52", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample7 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample7 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "13/6" ], "output": "-52", "label": "Left sum ORIGINAL sample7 slope times x" } }, { "id": "S056", "uses": [ "G001", "S055" ], "before": "Left sum ORIGINAL sample7 affine value plus c := sum(-52, 11)", "after": "Left sum ORIGINAL sample7 affine value plus c = -41", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample7 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample7 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-52", "11" ], "output": "-41", "label": "Left sum ORIGINAL sample7 affine value plus c" } }, { "id": "S057", "uses": [ "G001", "S056" ], "before": "Left sum ORIGINAL sample7 value times cell width := mul(-41, 1/6)", "after": "Left sum ORIGINAL sample7 value times cell width = -41/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample7 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample7 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-41", "1/6" ], "output": "-41/6", "label": "Left sum ORIGINAL sample7 value times cell width" } }, { "id": "S058", "uses": [ "G001", "S057" ], "before": "Left sum ORIGINAL accumulated term7 := sum(-175/6, -41/6)", "after": "Left sum ORIGINAL accumulated term7 = -36", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term7，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term7 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-175/6", "-41/6" ], "output": "-36", "label": "Left sum ORIGINAL accumulated term7" } }, { "id": "S059", "uses": [ "G001", "S058" ], "before": "source roles = [7, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [13/6, -41/1, -41/6, -36/1]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S060", "uses": [ "G001", "S059" ], "before": "Left sum ORIGINAL sample8 index times width := mul(8, 1/6)", "after": "Left sum ORIGINAL sample8 index times width = 4/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample8 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample8 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "8", "1/6" ], "output": "4/3", "label": "Left sum ORIGINAL sample8 index times width" } }, { "id": "S061", "uses": [ "G001", "S060" ], "before": "Left sum ORIGINAL sample8 lower plus offset := sum(1, 4/3)", "after": "Left sum ORIGINAL sample8 lower plus offset = 7/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample8 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample8 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "4/3" ], "output": "7/3", "label": "Left sum ORIGINAL sample8 lower plus offset" } }, { "id": "S062", "uses": [ "G001", "S061" ], "before": "Left sum ORIGINAL sample8 slope times x := mul(-24, 7/3)", "after": "Left sum ORIGINAL sample8 slope times x = -56", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample8 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample8 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "7/3" ], "output": "-56", "label": "Left sum ORIGINAL sample8 slope times x" } }, { "id": "S063", "uses": [ "G001", "S062" ], "before": "Left sum ORIGINAL sample8 affine value plus c := sum(-56, 11)", "after": "Left sum ORIGINAL sample8 affine value plus c = -45", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample8 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample8 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-56", "11" ], "output": "-45", "label": "Left sum ORIGINAL sample8 affine value plus c" } }, { "id": "S064", "uses": [ "G001", "S063" ], "before": "Left sum ORIGINAL sample8 value times cell width := mul(-45, 1/6)", "after": "Left sum ORIGINAL sample8 value times cell width = -15/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample8 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample8 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-45", "1/6" ], "output": "-15/2", "label": "Left sum ORIGINAL sample8 value times cell width" } }, { "id": "S065", "uses": [ "G001", "S064" ], "before": "Left sum ORIGINAL accumulated term8 := sum(-36, -15/2)", "after": "Left sum ORIGINAL accumulated term8 = -87/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term8，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term8 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-36", "-15/2" ], "output": "-87/2", "label": "Left sum ORIGINAL accumulated term8" } }, { "id": "S066", "uses": [ "G001", "S065" ], "before": "source roles = [8, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [7/3, -45/1, -15/2, -87/2]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S067", "uses": [ "G001", "S066" ], "before": "Left sum ORIGINAL sample9 index times width := mul(9, 1/6)", "after": "Left sum ORIGINAL sample9 index times width = 3/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample9 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample9 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "9", "1/6" ], "output": "3/2", "label": "Left sum ORIGINAL sample9 index times width" } }, { "id": "S068", "uses": [ "G001", "S067" ], "before": "Left sum ORIGINAL sample9 lower plus offset := sum(1, 3/2)", "after": "Left sum ORIGINAL sample9 lower plus offset = 5/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample9 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample9 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "3/2" ], "output": "5/2", "label": "Left sum ORIGINAL sample9 lower plus offset" } }, { "id": "S069", "uses": [ "G001", "S068" ], "before": "Left sum ORIGINAL sample9 slope times x := mul(-24, 5/2)", "after": "Left sum ORIGINAL sample9 slope times x = -60", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample9 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample9 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "5/2" ], "output": "-60", "label": "Left sum ORIGINAL sample9 slope times x" } }, { "id": "S070", "uses": [ "G001", "S069" ], "before": "Left sum ORIGINAL sample9 affine value plus c := sum(-60, 11)", "after": "Left sum ORIGINAL sample9 affine value plus c = -49", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample9 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample9 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-60", "11" ], "output": "-49", "label": "Left sum ORIGINAL sample9 affine value plus c" } }, { "id": "S071", "uses": [ "G001", "S070" ], "before": "Left sum ORIGINAL sample9 value times cell width := mul(-49, 1/6)", "after": "Left sum ORIGINAL sample9 value times cell width = -49/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample9 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample9 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-49", "1/6" ], "output": "-49/6", "label": "Left sum ORIGINAL sample9 value times cell width" } }, { "id": "S072", "uses": [ "G001", "S071" ], "before": "Left sum ORIGINAL accumulated term9 := sum(-87/2, -49/6)", "after": "Left sum ORIGINAL accumulated term9 = -155/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term9，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term9 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-87/2", "-49/6" ], "output": "-155/3", "label": "Left sum ORIGINAL accumulated term9" } }, { "id": "S073", "uses": [ "G001", "S072" ], "before": "source roles = [9, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [5/2, -49/1, -49/6, -155/3]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S074", "uses": [ "G001", "S073" ], "before": "Left sum ORIGINAL sample10 index times width := mul(10, 1/6)", "after": "Left sum ORIGINAL sample10 index times width = 5/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample10 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample10 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "10", "1/6" ], "output": "5/3", "label": "Left sum ORIGINAL sample10 index times width" } }, { "id": "S075", "uses": [ "G001", "S074" ], "before": "Left sum ORIGINAL sample10 lower plus offset := sum(1, 5/3)", "after": "Left sum ORIGINAL sample10 lower plus offset = 8/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample10 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample10 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "5/3" ], "output": "8/3", "label": "Left sum ORIGINAL sample10 lower plus offset" } }, { "id": "S076", "uses": [ "G001", "S075" ], "before": "Left sum ORIGINAL sample10 slope times x := mul(-24, 8/3)", "after": "Left sum ORIGINAL sample10 slope times x = -64", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample10 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample10 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "8/3" ], "output": "-64", "label": "Left sum ORIGINAL sample10 slope times x" } }, { "id": "S077", "uses": [ "G001", "S076" ], "before": "Left sum ORIGINAL sample10 affine value plus c := sum(-64, 11)", "after": "Left sum ORIGINAL sample10 affine value plus c = -53", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample10 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample10 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-64", "11" ], "output": "-53", "label": "Left sum ORIGINAL sample10 affine value plus c" } }, { "id": "S078", "uses": [ "G001", "S077" ], "before": "Left sum ORIGINAL sample10 value times cell width := mul(-53, 1/6)", "after": "Left sum ORIGINAL sample10 value times cell width = -53/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample10 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample10 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-53", "1/6" ], "output": "-53/6", "label": "Left sum ORIGINAL sample10 value times cell width" } }, { "id": "S079", "uses": [ "G001", "S078" ], "before": "Left sum ORIGINAL accumulated term10 := sum(-155/3, -53/6)", "after": "Left sum ORIGINAL accumulated term10 = -121/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term10，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term10 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-155/3", "-53/6" ], "output": "-121/2", "label": "Left sum ORIGINAL accumulated term10" } }, { "id": "S080", "uses": [ "G001", "S079" ], "before": "source roles = [10, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [8/3, -53/1, -53/6, -121/2]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S081", "uses": [ "G001", "S080" ], "before": "Left sum ORIGINAL sample11 index times width := mul(11, 1/6)", "after": "Left sum ORIGINAL sample11 index times width = 11/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample11 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample11 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "11", "1/6" ], "output": "11/6", "label": "Left sum ORIGINAL sample11 index times width" } }, { "id": "S082", "uses": [ "G001", "S081" ], "before": "Left sum ORIGINAL sample11 lower plus offset := sum(1, 11/6)", "after": "Left sum ORIGINAL sample11 lower plus offset = 17/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample11 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample11 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "11/6" ], "output": "17/6", "label": "Left sum ORIGINAL sample11 lower plus offset" } }, { "id": "S083", "uses": [ "G001", "S082" ], "before": "Left sum ORIGINAL sample11 slope times x := mul(-24, 17/6)", "after": "Left sum ORIGINAL sample11 slope times x = -68", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample11 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample11 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "17/6" ], "output": "-68", "label": "Left sum ORIGINAL sample11 slope times x" } }, { "id": "S084", "uses": [ "G001", "S083" ], "before": "Left sum ORIGINAL sample11 affine value plus c := sum(-68, 11)", "after": "Left sum ORIGINAL sample11 affine value plus c = -57", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample11 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample11 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-68", "11" ], "output": "-57", "label": "Left sum ORIGINAL sample11 affine value plus c" } }, { "id": "S085", "uses": [ "G001", "S084" ], "before": "Left sum ORIGINAL sample11 value times cell width := mul(-57, 1/6)", "after": "Left sum ORIGINAL sample11 value times cell width = -19/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample11 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample11 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-57", "1/6" ], "output": "-19/2", "label": "Left sum ORIGINAL sample11 value times cell width" } }, { "id": "S086", "uses": [ "G001", "S085" ], "before": "Left sum ORIGINAL accumulated term11 := sum(-121/2, -19/2)", "after": "Left sum ORIGINAL accumulated term11 = -70", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term11，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term11 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-121/2", "-19/2" ], "output": "-70", "label": "Left sum ORIGINAL accumulated term11" } }, { "id": "S087", "uses": [ "G001", "S086" ], "before": "source roles = [11, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [17/6, -57/1, -19/2, -70/1]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S088", "uses": [ "G001", "S087" ], "before": "Left sum ORIGINAL sample12 index times width := mul(12, 1/6)", "after": "Left sum ORIGINAL sample12 index times width = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample12 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample12 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "12", "1/6" ], "output": "2", "label": "Left sum ORIGINAL sample12 index times width" } }, { "id": "S089", "uses": [ "G001", "S088" ], "before": "Left sum ORIGINAL sample12 lower plus offset := sum(1, 2)", "after": "Left sum ORIGINAL sample12 lower plus offset = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample12 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample12 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "2" ], "output": "3", "label": "Left sum ORIGINAL sample12 lower plus offset" } }, { "id": "S090", "uses": [ "G001", "S089" ], "before": "Left sum ORIGINAL sample12 slope times x := mul(-24, 3)", "after": "Left sum ORIGINAL sample12 slope times x = -72", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample12 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample12 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "3" ], "output": "-72", "label": "Left sum ORIGINAL sample12 slope times x" } }, { "id": "S091", "uses": [ "G001", "S090" ], "before": "Left sum ORIGINAL sample12 affine value plus c := sum(-72, 11)", "after": "Left sum ORIGINAL sample12 affine value plus c = -61", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample12 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample12 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-72", "11" ], "output": "-61", "label": "Left sum ORIGINAL sample12 affine value plus c" } }, { "id": "S092", "uses": [ "G001", "S091" ], "before": "Left sum ORIGINAL sample12 value times cell width := mul(-61, 1/6)", "after": "Left sum ORIGINAL sample12 value times cell width = -61/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample12 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample12 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-61", "1/6" ], "output": "-61/6", "label": "Left sum ORIGINAL sample12 value times cell width" } }, { "id": "S093", "uses": [ "G001", "S092" ], "before": "Left sum ORIGINAL accumulated term12 := sum(-70, -61/6)", "after": "Left sum ORIGINAL accumulated term12 = -481/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term12，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term12 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-70", "-61/6" ], "output": "-481/6", "label": "Left sum ORIGINAL accumulated term12" } }, { "id": "S094", "uses": [ "G001", "S093" ], "before": "source roles = [12, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [3/1, -61/1, -61/6, -481/6]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S095", "uses": [ "G001", "S094" ], "before": "Left sum ORIGINAL sample13 index times width := mul(13, 1/6)", "after": "Left sum ORIGINAL sample13 index times width = 13/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample13 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample13 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "13", "1/6" ], "output": "13/6", "label": "Left sum ORIGINAL sample13 index times width" } }, { "id": "S096", "uses": [ "G001", "S095" ], "before": "Left sum ORIGINAL sample13 lower plus offset := sum(1, 13/6)", "after": "Left sum ORIGINAL sample13 lower plus offset = 19/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample13 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample13 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "13/6" ], "output": "19/6", "label": "Left sum ORIGINAL sample13 lower plus offset" } }, { "id": "S097", "uses": [ "G001", "S096" ], "before": "Left sum ORIGINAL sample13 slope times x := mul(-24, 19/6)", "after": "Left sum ORIGINAL sample13 slope times x = -76", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample13 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample13 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "19/6" ], "output": "-76", "label": "Left sum ORIGINAL sample13 slope times x" } }, { "id": "S098", "uses": [ "G001", "S097" ], "before": "Left sum ORIGINAL sample13 affine value plus c := sum(-76, 11)", "after": "Left sum ORIGINAL sample13 affine value plus c = -65", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample13 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample13 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-76", "11" ], "output": "-65", "label": "Left sum ORIGINAL sample13 affine value plus c" } }, { "id": "S099", "uses": [ "G001", "S098" ], "before": "Left sum ORIGINAL sample13 value times cell width := mul(-65, 1/6)", "after": "Left sum ORIGINAL sample13 value times cell width = -65/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample13 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample13 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-65", "1/6" ], "output": "-65/6", "label": "Left sum ORIGINAL sample13 value times cell width" } }, { "id": "S100", "uses": [ "G001", "S099" ], "before": "Left sum ORIGINAL accumulated term13 := sum(-481/6, -65/6)", "after": "Left sum ORIGINAL accumulated term13 = -91", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term13，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term13 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-481/6", "-65/6" ], "output": "-91", "label": "Left sum ORIGINAL accumulated term13" } }, { "id": "S101", "uses": [ "G001", "S100" ], "before": "source roles = [13, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [19/6, -65/1, -65/6, -91/1]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S102", "uses": [ "G001", "S101" ], "before": "Left sum ORIGINAL sample14 index times width := mul(14, 1/6)", "after": "Left sum ORIGINAL sample14 index times width = 7/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample14 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample14 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "14", "1/6" ], "output": "7/3", "label": "Left sum ORIGINAL sample14 index times width" } }, { "id": "S103", "uses": [ "G001", "S102" ], "before": "Left sum ORIGINAL sample14 lower plus offset := sum(1, 7/3)", "after": "Left sum ORIGINAL sample14 lower plus offset = 10/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample14 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample14 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "7/3" ], "output": "10/3", "label": "Left sum ORIGINAL sample14 lower plus offset" } }, { "id": "S104", "uses": [ "G001", "S103" ], "before": "Left sum ORIGINAL sample14 slope times x := mul(-24, 10/3)", "after": "Left sum ORIGINAL sample14 slope times x = -80", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample14 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample14 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "10/3" ], "output": "-80", "label": "Left sum ORIGINAL sample14 slope times x" } }, { "id": "S105", "uses": [ "G001", "S104" ], "before": "Left sum ORIGINAL sample14 affine value plus c := sum(-80, 11)", "after": "Left sum ORIGINAL sample14 affine value plus c = -69", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample14 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample14 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-80", "11" ], "output": "-69", "label": "Left sum ORIGINAL sample14 affine value plus c" } }, { "id": "S106", "uses": [ "G001", "S105" ], "before": "Left sum ORIGINAL sample14 value times cell width := mul(-69, 1/6)", "after": "Left sum ORIGINAL sample14 value times cell width = -23/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample14 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample14 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-69", "1/6" ], "output": "-23/2", "label": "Left sum ORIGINAL sample14 value times cell width" } }, { "id": "S107", "uses": [ "G001", "S106" ], "before": "Left sum ORIGINAL accumulated term14 := sum(-91, -23/2)", "after": "Left sum ORIGINAL accumulated term14 = -205/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term14，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term14 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-91", "-23/2" ], "output": "-205/2", "label": "Left sum ORIGINAL accumulated term14" } }, { "id": "S108", "uses": [ "G001", "S107" ], "before": "source roles = [14, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [10/3, -69/1, -23/2, -205/2]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S109", "uses": [ "G001", "S108" ], "before": "Left sum ORIGINAL sample15 index times width := mul(15, 1/6)", "after": "Left sum ORIGINAL sample15 index times width = 5/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample15 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample15 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "15", "1/6" ], "output": "5/2", "label": "Left sum ORIGINAL sample15 index times width" } }, { "id": "S110", "uses": [ "G001", "S109" ], "before": "Left sum ORIGINAL sample15 lower plus offset := sum(1, 5/2)", "after": "Left sum ORIGINAL sample15 lower plus offset = 7/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample15 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample15 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "5/2" ], "output": "7/2", "label": "Left sum ORIGINAL sample15 lower plus offset" } }, { "id": "S111", "uses": [ "G001", "S110" ], "before": "Left sum ORIGINAL sample15 slope times x := mul(-24, 7/2)", "after": "Left sum ORIGINAL sample15 slope times x = -84", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample15 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample15 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "7/2" ], "output": "-84", "label": "Left sum ORIGINAL sample15 slope times x" } }, { "id": "S112", "uses": [ "G001", "S111" ], "before": "Left sum ORIGINAL sample15 affine value plus c := sum(-84, 11)", "after": "Left sum ORIGINAL sample15 affine value plus c = -73", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample15 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample15 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-84", "11" ], "output": "-73", "label": "Left sum ORIGINAL sample15 affine value plus c" } }, { "id": "S113", "uses": [ "G001", "S112" ], "before": "Left sum ORIGINAL sample15 value times cell width := mul(-73, 1/6)", "after": "Left sum ORIGINAL sample15 value times cell width = -73/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample15 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample15 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-73", "1/6" ], "output": "-73/6", "label": "Left sum ORIGINAL sample15 value times cell width" } }, { "id": "S114", "uses": [ "G001", "S113" ], "before": "Left sum ORIGINAL accumulated term15 := sum(-205/2, -73/6)", "after": "Left sum ORIGINAL accumulated term15 = -344/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term15，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term15 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-205/2", "-73/6" ], "output": "-344/3", "label": "Left sum ORIGINAL accumulated term15" } }, { "id": "S115", "uses": [ "G001", "S114" ], "before": "source roles = [15, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [7/2, -73/1, -73/6, -344/3]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S116", "uses": [ "G001", "S115" ], "before": "Left sum ORIGINAL sample16 index times width := mul(16, 1/6)", "after": "Left sum ORIGINAL sample16 index times width = 8/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample16 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample16 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "16", "1/6" ], "output": "8/3", "label": "Left sum ORIGINAL sample16 index times width" } }, { "id": "S117", "uses": [ "G001", "S116" ], "before": "Left sum ORIGINAL sample16 lower plus offset := sum(1, 8/3)", "after": "Left sum ORIGINAL sample16 lower plus offset = 11/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample16 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample16 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "8/3" ], "output": "11/3", "label": "Left sum ORIGINAL sample16 lower plus offset" } }, { "id": "S118", "uses": [ "G001", "S117" ], "before": "Left sum ORIGINAL sample16 slope times x := mul(-24, 11/3)", "after": "Left sum ORIGINAL sample16 slope times x = -88", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample16 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample16 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "11/3" ], "output": "-88", "label": "Left sum ORIGINAL sample16 slope times x" } }, { "id": "S119", "uses": [ "G001", "S118" ], "before": "Left sum ORIGINAL sample16 affine value plus c := sum(-88, 11)", "after": "Left sum ORIGINAL sample16 affine value plus c = -77", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample16 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample16 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-88", "11" ], "output": "-77", "label": "Left sum ORIGINAL sample16 affine value plus c" } }, { "id": "S120", "uses": [ "G001", "S119" ], "before": "Left sum ORIGINAL sample16 value times cell width := mul(-77, 1/6)", "after": "Left sum ORIGINAL sample16 value times cell width = -77/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample16 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample16 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-77", "1/6" ], "output": "-77/6", "label": "Left sum ORIGINAL sample16 value times cell width" } }, { "id": "S121", "uses": [ "G001", "S120" ], "before": "Left sum ORIGINAL accumulated term16 := sum(-344/3, -77/6)", "after": "Left sum ORIGINAL accumulated term16 = -255/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term16，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term16 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-344/3", "-77/6" ], "output": "-255/2", "label": "Left sum ORIGINAL accumulated term16" } }, { "id": "S122", "uses": [ "G001", "S121" ], "before": "source roles = [16, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [11/3, -77/1, -77/6, -255/2]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S123", "uses": [ "G001", "S122" ], "before": "Left sum ORIGINAL sample17 index times width := mul(17, 1/6)", "after": "Left sum ORIGINAL sample17 index times width = 17/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample17 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample17 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "17", "1/6" ], "output": "17/6", "label": "Left sum ORIGINAL sample17 index times width" } }, { "id": "S124", "uses": [ "G001", "S123" ], "before": "Left sum ORIGINAL sample17 lower plus offset := sum(1, 17/6)", "after": "Left sum ORIGINAL sample17 lower plus offset = 23/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample17 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample17 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "17/6" ], "output": "23/6", "label": "Left sum ORIGINAL sample17 lower plus offset" } }, { "id": "S125", "uses": [ "G001", "S124" ], "before": "Left sum ORIGINAL sample17 slope times x := mul(-24, 23/6)", "after": "Left sum ORIGINAL sample17 slope times x = -92", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample17 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample17 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "23/6" ], "output": "-92", "label": "Left sum ORIGINAL sample17 slope times x" } }, { "id": "S126", "uses": [ "G001", "S125" ], "before": "Left sum ORIGINAL sample17 affine value plus c := sum(-92, 11)", "after": "Left sum ORIGINAL sample17 affine value plus c = -81", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample17 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample17 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-92", "11" ], "output": "-81", "label": "Left sum ORIGINAL sample17 affine value plus c" } }, { "id": "S127", "uses": [ "G001", "S126" ], "before": "Left sum ORIGINAL sample17 value times cell width := mul(-81, 1/6)", "after": "Left sum ORIGINAL sample17 value times cell width = -27/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample17 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample17 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-81", "1/6" ], "output": "-27/2", "label": "Left sum ORIGINAL sample17 value times cell width" } }, { "id": "S128", "uses": [ "G001", "S127" ], "before": "Left sum ORIGINAL accumulated term17 := sum(-255/2, -27/2)", "after": "Left sum ORIGINAL accumulated term17 = -141", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term17，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term17 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-255/2", "-27/2" ], "output": "-141", "label": "Left sum ORIGINAL accumulated term17" } }, { "id": "S129", "uses": [ "G001", "S128" ], "before": "source roles = [17, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [23/6, -81/1, -27/2, -141/1]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S130", "uses": [ "G001", "S129" ], "before": "Left sum ORIGINAL sample18 index times width := mul(18, 1/6)", "after": "Left sum ORIGINAL sample18 index times width = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample18 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample18 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "18", "1/6" ], "output": "3", "label": "Left sum ORIGINAL sample18 index times width" } }, { "id": "S131", "uses": [ "G001", "S130" ], "before": "Left sum ORIGINAL sample18 lower plus offset := sum(1, 3)", "after": "Left sum ORIGINAL sample18 lower plus offset = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample18 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample18 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "3" ], "output": "4", "label": "Left sum ORIGINAL sample18 lower plus offset" } }, { "id": "S132", "uses": [ "G001", "S131" ], "before": "Left sum ORIGINAL sample18 slope times x := mul(-24, 4)", "after": "Left sum ORIGINAL sample18 slope times x = -96", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample18 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample18 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "4" ], "output": "-96", "label": "Left sum ORIGINAL sample18 slope times x" } }, { "id": "S133", "uses": [ "G001", "S132" ], "before": "Left sum ORIGINAL sample18 affine value plus c := sum(-96, 11)", "after": "Left sum ORIGINAL sample18 affine value plus c = -85", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample18 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample18 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-96", "11" ], "output": "-85", "label": "Left sum ORIGINAL sample18 affine value plus c" } }, { "id": "S134", "uses": [ "G001", "S133" ], "before": "Left sum ORIGINAL sample18 value times cell width := mul(-85, 1/6)", "after": "Left sum ORIGINAL sample18 value times cell width = -85/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample18 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample18 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-85", "1/6" ], "output": "-85/6", "label": "Left sum ORIGINAL sample18 value times cell width" } }, { "id": "S135", "uses": [ "G001", "S134" ], "before": "Left sum ORIGINAL accumulated term18 := sum(-141, -85/6)", "after": "Left sum ORIGINAL accumulated term18 = -931/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term18，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term18 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-141", "-85/6" ], "output": "-931/6", "label": "Left sum ORIGINAL accumulated term18" } }, { "id": "S136", "uses": [ "G001", "S135" ], "before": "source roles = [18, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [4/1, -85/1, -85/6, -931/6]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S137", "uses": [ "G001", "S136" ], "before": "Left sum ORIGINAL sample19 index times width := mul(19, 1/6)", "after": "Left sum ORIGINAL sample19 index times width = 19/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample19 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample19 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "19", "1/6" ], "output": "19/6", "label": "Left sum ORIGINAL sample19 index times width" } }, { "id": "S138", "uses": [ "G001", "S137" ], "before": "Left sum ORIGINAL sample19 lower plus offset := sum(1, 19/6)", "after": "Left sum ORIGINAL sample19 lower plus offset = 25/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample19 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample19 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "19/6" ], "output": "25/6", "label": "Left sum ORIGINAL sample19 lower plus offset" } }, { "id": "S139", "uses": [ "G001", "S138" ], "before": "Left sum ORIGINAL sample19 slope times x := mul(-24, 25/6)", "after": "Left sum ORIGINAL sample19 slope times x = -100", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample19 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample19 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "25/6" ], "output": "-100", "label": "Left sum ORIGINAL sample19 slope times x" } }, { "id": "S140", "uses": [ "G001", "S139" ], "before": "Left sum ORIGINAL sample19 affine value plus c := sum(-100, 11)", "after": "Left sum ORIGINAL sample19 affine value plus c = -89", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample19 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample19 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-100", "11" ], "output": "-89", "label": "Left sum ORIGINAL sample19 affine value plus c" } }, { "id": "S141", "uses": [ "G001", "S140" ], "before": "Left sum ORIGINAL sample19 value times cell width := mul(-89, 1/6)", "after": "Left sum ORIGINAL sample19 value times cell width = -89/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample19 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample19 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-89", "1/6" ], "output": "-89/6", "label": "Left sum ORIGINAL sample19 value times cell width" } }, { "id": "S142", "uses": [ "G001", "S141" ], "before": "Left sum ORIGINAL accumulated term19 := sum(-931/6, -89/6)", "after": "Left sum ORIGINAL accumulated term19 = -170", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term19，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term19 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-931/6", "-89/6" ], "output": "-170", "label": "Left sum ORIGINAL accumulated term19" } }, { "id": "S143", "uses": [ "G001", "S142" ], "before": "source roles = [19, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [25/6, -89/1, -89/6, -170/1]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S144", "uses": [ "G001", "S143" ], "before": "Left sum ORIGINAL sample20 index times width := mul(20, 1/6)", "after": "Left sum ORIGINAL sample20 index times width = 10/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample20 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample20 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "20", "1/6" ], "output": "10/3", "label": "Left sum ORIGINAL sample20 index times width" } }, { "id": "S145", "uses": [ "G001", "S144" ], "before": "Left sum ORIGINAL sample20 lower plus offset := sum(1, 10/3)", "after": "Left sum ORIGINAL sample20 lower plus offset = 13/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample20 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample20 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "10/3" ], "output": "13/3", "label": "Left sum ORIGINAL sample20 lower plus offset" } }, { "id": "S146", "uses": [ "G001", "S145" ], "before": "Left sum ORIGINAL sample20 slope times x := mul(-24, 13/3)", "after": "Left sum ORIGINAL sample20 slope times x = -104", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample20 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample20 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "13/3" ], "output": "-104", "label": "Left sum ORIGINAL sample20 slope times x" } }, { "id": "S147", "uses": [ "G001", "S146" ], "before": "Left sum ORIGINAL sample20 affine value plus c := sum(-104, 11)", "after": "Left sum ORIGINAL sample20 affine value plus c = -93", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample20 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample20 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-104", "11" ], "output": "-93", "label": "Left sum ORIGINAL sample20 affine value plus c" } }, { "id": "S148", "uses": [ "G001", "S147" ], "before": "Left sum ORIGINAL sample20 value times cell width := mul(-93, 1/6)", "after": "Left sum ORIGINAL sample20 value times cell width = -31/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample20 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample20 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-93", "1/6" ], "output": "-31/2", "label": "Left sum ORIGINAL sample20 value times cell width" } }, { "id": "S149", "uses": [ "G001", "S148" ], "before": "Left sum ORIGINAL accumulated term20 := sum(-170, -31/2)", "after": "Left sum ORIGINAL accumulated term20 = -371/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term20，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term20 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-170", "-31/2" ], "output": "-371/2", "label": "Left sum ORIGINAL accumulated term20" } }, { "id": "S150", "uses": [ "G001", "S149" ], "before": "source roles = [20, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [13/3, -93/1, -31/2, -371/2]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S151", "uses": [ "G001", "S150" ], "before": "Left sum ORIGINAL sample21 index times width := mul(21, 1/6)", "after": "Left sum ORIGINAL sample21 index times width = 7/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample21 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample21 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "21", "1/6" ], "output": "7/2", "label": "Left sum ORIGINAL sample21 index times width" } }, { "id": "S152", "uses": [ "G001", "S151" ], "before": "Left sum ORIGINAL sample21 lower plus offset := sum(1, 7/2)", "after": "Left sum ORIGINAL sample21 lower plus offset = 9/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample21 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample21 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "7/2" ], "output": "9/2", "label": "Left sum ORIGINAL sample21 lower plus offset" } }, { "id": "S153", "uses": [ "G001", "S152" ], "before": "Left sum ORIGINAL sample21 slope times x := mul(-24, 9/2)", "after": "Left sum ORIGINAL sample21 slope times x = -108", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample21 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample21 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "9/2" ], "output": "-108", "label": "Left sum ORIGINAL sample21 slope times x" } }, { "id": "S154", "uses": [ "G001", "S153" ], "before": "Left sum ORIGINAL sample21 affine value plus c := sum(-108, 11)", "after": "Left sum ORIGINAL sample21 affine value plus c = -97", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample21 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample21 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-108", "11" ], "output": "-97", "label": "Left sum ORIGINAL sample21 affine value plus c" } }, { "id": "S155", "uses": [ "G001", "S154" ], "before": "Left sum ORIGINAL sample21 value times cell width := mul(-97, 1/6)", "after": "Left sum ORIGINAL sample21 value times cell width = -97/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample21 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample21 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-97", "1/6" ], "output": "-97/6", "label": "Left sum ORIGINAL sample21 value times cell width" } }, { "id": "S156", "uses": [ "G001", "S155" ], "before": "Left sum ORIGINAL accumulated term21 := sum(-371/2, -97/6)", "after": "Left sum ORIGINAL accumulated term21 = -605/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term21，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term21 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-371/2", "-97/6" ], "output": "-605/3", "label": "Left sum ORIGINAL accumulated term21" } }, { "id": "S157", "uses": [ "G001", "S156" ], "before": "source roles = [21, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [9/2, -97/1, -97/6, -605/3]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S158", "uses": [ "G001", "S157" ], "before": "Left sum ORIGINAL sample22 index times width := mul(22, 1/6)", "after": "Left sum ORIGINAL sample22 index times width = 11/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample22 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample22 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "22", "1/6" ], "output": "11/3", "label": "Left sum ORIGINAL sample22 index times width" } }, { "id": "S159", "uses": [ "G001", "S158" ], "before": "Left sum ORIGINAL sample22 lower plus offset := sum(1, 11/3)", "after": "Left sum ORIGINAL sample22 lower plus offset = 14/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample22 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample22 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "11/3" ], "output": "14/3", "label": "Left sum ORIGINAL sample22 lower plus offset" } }, { "id": "S160", "uses": [ "G001", "S159" ], "before": "Left sum ORIGINAL sample22 slope times x := mul(-24, 14/3)", "after": "Left sum ORIGINAL sample22 slope times x = -112", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample22 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample22 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "14/3" ], "output": "-112", "label": "Left sum ORIGINAL sample22 slope times x" } }, { "id": "S161", "uses": [ "G001", "S160" ], "before": "Left sum ORIGINAL sample22 affine value plus c := sum(-112, 11)", "after": "Left sum ORIGINAL sample22 affine value plus c = -101", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample22 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample22 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-112", "11" ], "output": "-101", "label": "Left sum ORIGINAL sample22 affine value plus c" } }, { "id": "S162", "uses": [ "G001", "S161" ], "before": "Left sum ORIGINAL sample22 value times cell width := mul(-101, 1/6)", "after": "Left sum ORIGINAL sample22 value times cell width = -101/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample22 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample22 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-101", "1/6" ], "output": "-101/6", "label": "Left sum ORIGINAL sample22 value times cell width" } }, { "id": "S163", "uses": [ "G001", "S162" ], "before": "Left sum ORIGINAL accumulated term22 := sum(-605/3, -101/6)", "after": "Left sum ORIGINAL accumulated term22 = -437/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term22，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term22 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-605/3", "-101/6" ], "output": "-437/2", "label": "Left sum ORIGINAL accumulated term22" } }, { "id": "S164", "uses": [ "G001", "S163" ], "before": "source roles = [22, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [14/3, -101/1, -101/6, -437/2]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S165", "uses": [ "G001", "S164" ], "before": "Left sum ORIGINAL sample23 index times width := mul(23, 1/6)", "after": "Left sum ORIGINAL sample23 index times width = 23/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample23 index times width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample23 index times width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "23", "1/6" ], "output": "23/6", "label": "Left sum ORIGINAL sample23 index times width" } }, { "id": "S166", "uses": [ "G001", "S165" ], "before": "Left sum ORIGINAL sample23 lower plus offset := sum(1, 23/6)", "after": "Left sum ORIGINAL sample23 lower plus offset = 29/6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample23 lower plus offset，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample23 lower plus offset with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "23/6" ], "output": "29/6", "label": "Left sum ORIGINAL sample23 lower plus offset" } }, { "id": "S167", "uses": [ "G001", "S166" ], "before": "Left sum ORIGINAL sample23 slope times x := mul(-24, 29/6)", "after": "Left sum ORIGINAL sample23 slope times x = -116", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample23 slope times x，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample23 slope times x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-24", "29/6" ], "output": "-116", "label": "Left sum ORIGINAL sample23 slope times x" } }, { "id": "S168", "uses": [ "G001", "S167" ], "before": "Left sum ORIGINAL sample23 affine value plus c := sum(-116, 11)", "after": "Left sum ORIGINAL sample23 affine value plus c = -105", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample23 affine value plus c，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample23 affine value plus c with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-116", "11" ], "output": "-105", "label": "Left sum ORIGINAL sample23 affine value plus c" } }, { "id": "S169", "uses": [ "G001", "S168" ], "before": "Left sum ORIGINAL sample23 value times cell width := mul(-105, 1/6)", "after": "Left sum ORIGINAL sample23 value times cell width = -35/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL sample23 value times cell width，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL sample23 value times cell width with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-105", "1/6" ], "output": "-35/2", "label": "Left sum ORIGINAL sample23 value times cell width" } }, { "id": "S170", "uses": [ "G001", "S169" ], "before": "Left sum ORIGINAL accumulated term23 := sum(-437/2, -35/2)", "after": "Left sum ORIGINAL accumulated term23 = -236", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。", "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Left sum ORIGINAL accumulated term23，不以小數近似替代。", "en": "Recompute Left sum ORIGINAL accumulated term23 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-437/2", "-35/2" ], "output": "-236", "label": "Left sum ORIGINAL accumulated term23" } }, { "id": "S171", "uses": [ "G001", "S170" ], "before": "source roles = [23, 24, -24, 11, 1, 5, 1/6]", "after": "derived = [29/6, -105/1, -35/2, -236/1]", "relation": "implies", "rule_id": "LeftSum_EVERY_fixed_LEFT_sample_value_weight_and_running_sum", "conditions": [], "explanation": { "zh-Hant": "逐i=0..N−1取左端點a+iΔ，先求affine值再乘Δ及累加；i=N右端點不在此和。每個分數精確，零寬／常數也先驗N與所有raw。", "en": "For EACH i=0..N−1,use the LEFT point a+iΔ,evaluate the affine law,multiply by Δ and accumulate.The right endpoint i=N is EXCLUDED.All fractions are exact;zero width/constants still validate N and EVERY raw field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S172", "uses": [ "G001", "S171" ], "before": "source roles = [24, -24, 11, 1, 5]", "after": "derived = [1/6, -236/1]", "relation": "implies", "rule_id": "LeftSum_FIXED_finite_partition_NOT_integral_or_limit_claim", "conditions": [], "explanation": { "zh-Hant": "這是固定N的有限左端和，一般不等於積分；常數／零寬特例可相等，不代表任意函數可積性或N→∞證明。", "en": "This is a FIXED-N finite left sum,generally not the integral.Constants/zero-width cases may agree,without proving arbitrary integrability or a limit as N→∞." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。", "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate." } }, "arithmetic": null } ], "math_model": { "result": "-236/1" }, "conclusion": { "canonical_value": "-236/1", "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-20261008-r07-definite-integral-foundations", "batch_path": "public/data/questions/batches/batch-20261007-advance-20261008-r07-definite-integral-foundations/questions.jsonl", "batch_sha256": "f08b46f392acfe2749b49fcea9773a4b95227d50a72cafa6f06b32903b00deef", "original_line": 99, "original_line_sha256": "1cfe5a755e256685efaa9b552565109fc84144e9920e14c407dbb41be58d244e", "question_compact_sha256": "277bd8619d17823b71393f4770726c7b7b74536e60b6d6b02420b9d49c772df8", "english_row_compact_sha256": "55d026f0d42d61f8e7e8d086c6d1bbcf8d0540d3cf2e5b53bb9c79a65292885e", "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261008-r07-definite-integral-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261008-r07-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": "528186aa825c3054db15f4ce0a03ad58f72be7e7eb5bb236c4cc7ef34ff5a470", "original_candidate_sha256": "166b756025214e8fffaf49801cd4ad74ba3cdca7dc8e72c86ed871df21d107f7", "original_candidate_line_sha256": "166b756025214e8fffaf49801cd4ad74ba3cdca7dc8e72c86ed871df21d107f7" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-011-p65", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b", "question_first_witness_sha256": "066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 172, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "9de2b0e261cda401653cbc3856628ee2e2c4edb6a4c405a22841890cb757fa7c", "arithmetic_steps": 146 }, "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." } }