{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261111-r05-catalan-numbers-0049", "pilot_number": "R0044-22724", "number": 22724, "family_id": "qf-mod42-catalan-closed-form", "title": { "zh-Hant": "第 22724 題：Catalan數", "en": "Question 22724: Check Divisibility before Taking the Catalan Quotient" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261111-r05-catalan-numbers-0049", "batch_id": "batch-20261014-advance-20261111-r05-catalan-numbers-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-catalan-numbers" ], "title_zh": "Catalan數", "stem_zh": "C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(23, 8233430727600, 24, 0, 343059613650, 343059613650, 是)", "explanation_zh": "Wave6第五層核對=(23, 8233430727600, 24, 0, 343059613650, 343059613650, 是)；依完整明示資料獨立重算；closed form division前必須驗(n+1) divides binomial(2n,n)，並與binomial difference cross-check；domain為n>=0；不可混用C_0/C_1 convention、negative index或integer truncation。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "Catalan數" ] }, "variant": { "family_id": "qf-mod42-catalan-closed-form", "seed": 202611115049, "parameters": { "n": 23, "domain": "Exact_Catalan_sequence", "operation": "catalan-closed-form", "semantic_template": "qf-mod42-catalan-closed-form/{\"n\":23}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261111-r05-catalan-numbers-0049", "batch_id": "batch-20261014-advance-20261111-r05-catalan-numbers-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "aa83fa2aa4784ee26fe59314b6dc35fba3981d15f7faa9810f44365df730f822", "family_id": "qf-mod42-catalan-closed-form", "seed": 202611115049, "semantic_template": "qf-mod42-catalan-closed-form/{\"n\":23}" }, "fields": { "title": "Check Divisibility before Taking the Catalan Quotient", "stem": "Use natural index n=23≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).", "answer": { "kind": "expression", "value": "(23, 8233430727600, 24, 0, 343059613650, 343059613650, Yes)", "explanation": "Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (23, 8233430727600, 24, 0, 343059613650, 343059613650, Yes)." }, "skills": [ "Check Divisibility before Taking the Catalan Quotient" ], "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": "C_0=1且C_(n+1)=sum_i C_i*C_(n-i)的zero-based indexing。操作=catalan-closed-form；資料={\"n\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use natural index n=23≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=catalan-closed-form;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[n, central_binomial, denominator, remainder, quotient, binomial_difference, equal]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "保留原題歷史介面標記，直接列來源欄序，不需JavaScript。tag不是額外證據，固定No依原聲稱範圍解讀；排序輔助只處理協定，不是數學oracle。", "en": "Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "ORIGINAL requested natural index n=23", "after": "initialize C_0=1;recurrence C_(j+1)=sum_i=0..j C_i*C_(j-i),including ALL splits", "relation": "implies", "rule_id": "catalan_base_case_empty_object_C0_one", "conditions": [], "explanation": { "zh-Hant": "空物件只有一種所以C0=1；第j+1項必含i0到j全部split，不採C1起算的位移慣例。", "en": "There is one empty object,so C0=1.Include every split i0..j for C_(j+1),without a C1-based index shift." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Catalan complete convolution for C_1 := sum(1)", "after": "Catalan complete convolution for C_1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_1，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1" ], "output": "1", "label": "Catalan complete convolution for C_1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "ORIGINAL index j=0;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 1, 1]]", "after": "C_1=1;ALL earlier values used from the established prefix=[1]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Catalan complete convolution for C_2 := sum(1, 1)", "after": "Catalan complete convolution for C_2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_2，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1" ], "output": "2", "label": "Catalan complete convolution for C_2" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "ORIGINAL index j=1;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 1, 1], [1, 1, 1, 1]]", "after": "C_2=2;ALL earlier values used from the established prefix=[1, 1]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Catalan complete convolution for C_3 := sum(2, 1, 2)", "after": "Catalan complete convolution for C_3 = 5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_3，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2", "1", "2" ], "output": "5", "label": "Catalan complete convolution for C_3" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "ORIGINAL index j=2;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 2, 2], [1, 1, 1, 1], [2, 2, 1, 2]]", "after": "C_3=5;ALL earlier values used from the established prefix=[1, 1, 2]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Catalan complete convolution for C_4 := sum(5, 2, 2, 5)", "after": "Catalan complete convolution for C_4 = 14", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_4，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "5", "2", "2", "5" ], "output": "14", "label": "Catalan complete convolution for C_4" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "ORIGINAL index j=3;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 5, 5], [1, 1, 2, 2], [2, 2, 1, 2], [3, 5, 1, 5]]", "after": "C_4=14;ALL earlier values used from the established prefix=[1, 1, 2, 5]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Catalan complete convolution for C_5 := sum(14, 5, 4, 5, 14)", "after": "Catalan complete convolution for C_5 = 42", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_5，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "14", "5", "4", "5", "14" ], "output": "42", "label": "Catalan complete convolution for C_5" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "ORIGINAL index j=4;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 14, 14], [1, 1, 5, 5], [2, 2, 2, 4], [3, 5, 1, 5], [4, 14, 1, 14]]", "after": "C_5=42;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Catalan complete convolution for C_6 := sum(42, 14, 10, 10, 14, 42)", "after": "Catalan complete convolution for C_6 = 132", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_6，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "42", "14", "10", "10", "14", "42" ], "output": "132", "label": "Catalan complete convolution for C_6" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "ORIGINAL index j=5;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 42, 42], [1, 1, 14, 14], [2, 2, 5, 10], [3, 5, 2, 10], [4, 14, 1, 14], [5, 42, 1, 42]]", "after": "C_6=132;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "Catalan complete convolution for C_7 := sum(132, 42, 28, 25, 28, 42, 132)", "after": "Catalan complete convolution for C_7 = 429", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_7，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_7 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "132", "42", "28", "25", "28", "42", "132" ], "output": "429", "label": "Catalan complete convolution for C_7" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "ORIGINAL index j=6;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 132, 132], [1, 1, 42, 42], [2, 2, 14, 28], [3, 5, 5, 25], [4, 14, 2, 28], [5, 42, 1, 42], [6, 132, 1, 132]]", "after": "C_7=429;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "Catalan complete convolution for C_8 := sum(429, 132, 84, 70, 70, 84, 132, 429)", "after": "Catalan complete convolution for C_8 = 1430", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_8，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_8 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "429", "132", "84", "70", "70", "84", "132", "429" ], "output": "1430", "label": "Catalan complete convolution for C_8" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "ORIGINAL index j=7;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 429, 429], [1, 1, 132, 132], [2, 2, 42, 84], [3, 5, 14, 70], [4, 14, 5, 70], [5, 42, 2, 84], [6, 132, 1, 132], [7, 429, 1, 429]]", "after": "C_8=1430;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "Catalan complete convolution for C_9 := sum(1430, 429, 264, 210, 196, 210, 264, 429, 1430)", "after": "Catalan complete convolution for C_9 = 4862", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_9，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_9 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1430", "429", "264", "210", "196", "210", "264", "429", "1430" ], "output": "4862", "label": "Catalan complete convolution for C_9" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "ORIGINAL index j=8;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 1430, 1430], [1, 1, 429, 429], [2, 2, 132, 264], [3, 5, 42, 210], [4, 14, 14, 196], [5, 42, 5, 210], [6, 132, 2, 264], [7, 429, 1, 429], [8, 1430, 1, 1430]]", "after": "C_9=4862;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "Catalan complete convolution for C_10 := sum(4862, 1430, 858, 660, 588, 588, 660, 858, 1430, 4862)", "after": "Catalan complete convolution for C_10 = 16796", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_10，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_10 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4862", "1430", "858", "660", "588", "588", "660", "858", "1430", "4862" ], "output": "16796", "label": "Catalan complete convolution for C_10" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "ORIGINAL index j=9;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 4862, 4862], [1, 1, 1430, 1430], [2, 2, 429, 858], [3, 5, 132, 660], [4, 14, 42, 588], [5, 42, 14, 588], [6, 132, 5, 660], [7, 429, 2, 858], [8, 1430, 1, 1430], [9, 4862, 1, 4862]]", "after": "C_10=16796;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "Catalan complete convolution for C_11 := sum(16796, 4862, 2860, 2145, 1848, 1764, 1848, 2145, 2860, 4862, 16796)", "after": "Catalan complete convolution for C_11 = 58786", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_11，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_11 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "16796", "4862", "2860", "2145", "1848", "1764", "1848", "2145", "2860", "4862", "16796" ], "output": "58786", "label": "Catalan complete convolution for C_11" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "ORIGINAL index j=10;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 16796, 16796], [1, 1, 4862, 4862], [2, 2, 1430, 2860], [3, 5, 429, 2145], [4, 14, 132, 1848], [5, 42, 42, 1764], [6, 132, 14, 1848], [7, 429, 5, 2145], [8, 1430, 2, 2860], [9, 4862, 1, 4862], [10, 16796, 1, 16796]]", "after": "C_11=58786;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "Catalan complete convolution for C_12 := sum(58786, 16796, 9724, 7150, 6006, 5544, 5544, 6006, 7150, 9724, 16796, 58786)", "after": "Catalan complete convolution for C_12 = 208012", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_12，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_12 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "58786", "16796", "9724", "7150", "6006", "5544", "5544", "6006", "7150", "9724", "16796", "58786" ], "output": "208012", "label": "Catalan complete convolution for C_12" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "ORIGINAL index j=11;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 58786, 58786], [1, 1, 16796, 16796], [2, 2, 4862, 9724], [3, 5, 1430, 7150], [4, 14, 429, 6006], [5, 42, 132, 5544], [6, 132, 42, 5544], [7, 429, 14, 6006], [8, 1430, 5, 7150], [9, 4862, 2, 9724], [10, 16796, 1, 16796], [11, 58786, 1, 58786]]", "after": "C_12=208012;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "Catalan complete convolution for C_13 := sum(208012, 58786, 33592, 24310, 20020, 18018, 17424, 18018, 20020, 24310, 33592, 58786, 208012)", "after": "Catalan complete convolution for C_13 = 742900", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_13，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_13 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "208012", "58786", "33592", "24310", "20020", "18018", "17424", "18018", "20020", "24310", "33592", "58786", "208012" ], "output": "742900", "label": "Catalan complete convolution for C_13" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "ORIGINAL index j=12;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 208012, 208012], [1, 1, 58786, 58786], [2, 2, 16796, 33592], [3, 5, 4862, 24310], [4, 14, 1430, 20020], [5, 42, 429, 18018], [6, 132, 132, 17424], [7, 429, 42, 18018], [8, 1430, 14, 20020], [9, 4862, 5, 24310], [10, 16796, 2, 33592], [11, 58786, 1, 58786], [12, 208012, 1, 208012]]", "after": "C_13=742900;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "Catalan complete convolution for C_14 := sum(742900, 208012, 117572, 83980, 68068, 60060, 56628, 56628, 60060, 68068, 83980, 117572, 208012, 742900)", "after": "Catalan complete convolution for C_14 = 2674440", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_14，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_14 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "742900", "208012", "117572", "83980", "68068", "60060", "56628", "56628", "60060", "68068", "83980", "117572", "208012", "742900" ], "output": "2674440", "label": "Catalan complete convolution for C_14" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "ORIGINAL index j=13;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 742900, 742900], [1, 1, 208012, 208012], [2, 2, 58786, 117572], [3, 5, 16796, 83980], [4, 14, 4862, 68068], [5, 42, 1430, 60060], [6, 132, 429, 56628], [7, 429, 132, 56628], [8, 1430, 42, 60060], [9, 4862, 14, 68068], [10, 16796, 5, 83980], [11, 58786, 2, 117572], [12, 208012, 1, 208012], [13, 742900, 1, 742900]]", "after": "C_14=2674440;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "Catalan complete convolution for C_15 := sum(2674440, 742900, 416024, 293930, 235144, 204204, 188760, 184041, 188760, 204204, 235144, 293930, 416024, 742900, 2674440)", "after": "Catalan complete convolution for C_15 = 9694845", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_15，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_15 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2674440", "742900", "416024", "293930", "235144", "204204", "188760", "184041", "188760", "204204", "235144", "293930", "416024", "742900", "2674440" ], "output": "9694845", "label": "Catalan complete convolution for C_15" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "ORIGINAL index j=14;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 2674440, 2674440], [1, 1, 742900, 742900], [2, 2, 208012, 416024], [3, 5, 58786, 293930], [4, 14, 16796, 235144], [5, 42, 4862, 204204], [6, 132, 1430, 188760], [7, 429, 429, 184041], [8, 1430, 132, 188760], [9, 4862, 42, 204204], [10, 16796, 14, 235144], [11, 58786, 5, 293930], [12, 208012, 2, 416024], [13, 742900, 1, 742900], [14, 2674440, 1, 2674440]]", "after": "C_15=9694845;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "Catalan complete convolution for C_16 := sum(9694845, 2674440, 1485800, 1040060, 823004, 705432, 641784, 613470, 613470, 641784, 705432, 823004, 1040060, 1485800, 2674440, 9694845)", "after": "Catalan complete convolution for C_16 = 35357670", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_16，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_16 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "9694845", "2674440", "1485800", "1040060", "823004", "705432", "641784", "613470", "613470", "641784", "705432", "823004", "1040060", "1485800", "2674440", "9694845" ], "output": "35357670", "label": "Catalan complete convolution for C_16" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "ORIGINAL index j=15;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 9694845, 9694845], [1, 1, 2674440, 2674440], [2, 2, 742900, 1485800], [3, 5, 208012, 1040060], [4, 14, 58786, 823004], [5, 42, 16796, 705432], [6, 132, 4862, 641784], [7, 429, 1430, 613470], [8, 1430, 429, 613470], [9, 4862, 132, 641784], [10, 16796, 42, 705432], [11, 58786, 14, 823004], [12, 208012, 5, 1040060], [13, 742900, 2, 1485800], [14, 2674440, 1, 2674440], [15, 9694845, 1, 9694845]]", "after": "C_16=35357670;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "Catalan complete convolution for C_17 := sum(35357670, 9694845, 5348880, 3714500, 2912168, 2469012, 2217072, 2085798, 2044900, 2085798, 2217072, 2469012, 2912168, 3714500, 5348880, 9694845, 35357670)", "after": "Catalan complete convolution for C_17 = 129644790", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_17，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_17 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "35357670", "9694845", "5348880", "3714500", "2912168", "2469012", "2217072", "2085798", "2044900", "2085798", "2217072", "2469012", "2912168", "3714500", "5348880", "9694845", "35357670" ], "output": "129644790", "label": "Catalan complete convolution for C_17" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "ORIGINAL index j=16;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 35357670, 35357670], [1, 1, 9694845, 9694845], [2, 2, 2674440, 5348880], [3, 5, 742900, 3714500], [4, 14, 208012, 2912168], [5, 42, 58786, 2469012], [6, 132, 16796, 2217072], [7, 429, 4862, 2085798], [8, 1430, 1430, 2044900], [9, 4862, 429, 2085798], [10, 16796, 132, 2217072], [11, 58786, 42, 2469012], [12, 208012, 14, 2912168], [13, 742900, 5, 3714500], [14, 2674440, 2, 5348880], [15, 9694845, 1, 9694845], [16, 35357670, 1, 35357670]]", "after": "C_17=129644790;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "Catalan complete convolution for C_18 := sum(129644790, 35357670, 19389690, 13372200, 10400600, 8736504, 7759752, 7205484, 6952660, 6952660, 7205484, 7759752, 8736504, 10400600, 13372200, 19389690, 35357670, 129644790)", "after": "Catalan complete convolution for C_18 = 477638700", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_18，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_18 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "129644790", "35357670", "19389690", "13372200", "10400600", "8736504", "7759752", "7205484", "6952660", "6952660", "7205484", "7759752", "8736504", "10400600", "13372200", "19389690", "35357670", "129644790" ], "output": "477638700", "label": "Catalan complete convolution for C_18" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "ORIGINAL index j=17;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 129644790, 129644790], [1, 1, 35357670, 35357670], [2, 2, 9694845, 19389690], [3, 5, 2674440, 13372200], [4, 14, 742900, 10400600], [5, 42, 208012, 8736504], [6, 132, 58786, 7759752], [7, 429, 16796, 7205484], [8, 1430, 4862, 6952660], [9, 4862, 1430, 6952660], [10, 16796, 429, 7205484], [11, 58786, 132, 7759752], [12, 208012, 42, 8736504], [13, 742900, 14, 10400600], [14, 2674440, 5, 13372200], [15, 9694845, 2, 19389690], [16, 35357670, 1, 35357670], [17, 129644790, 1, 129644790]]", "after": "C_18=477638700;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "Catalan complete convolution for C_19 := sum(477638700, 129644790, 70715340, 48474225, 37442160, 31201800, 27457584, 25219194, 24018280, 23639044, 24018280, 25219194, 27457584, 31201800, 37442160, 48474225, 70715340, 129644790, 477638700)", "after": "Catalan complete convolution for C_19 = 1767263190", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_19，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_19 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "477638700", "129644790", "70715340", "48474225", "37442160", "31201800", "27457584", "25219194", "24018280", "23639044", "24018280", "25219194", "27457584", "31201800", "37442160", "48474225", "70715340", "129644790", "477638700" ], "output": "1767263190", "label": "Catalan complete convolution for C_19" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "ORIGINAL index j=18;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 477638700, 477638700], [1, 1, 129644790, 129644790], [2, 2, 35357670, 70715340], [3, 5, 9694845, 48474225], [4, 14, 2674440, 37442160], [5, 42, 742900, 31201800], [6, 132, 208012, 27457584], [7, 429, 58786, 25219194], [8, 1430, 16796, 24018280], [9, 4862, 4862, 23639044], [10, 16796, 1430, 24018280], [11, 58786, 429, 25219194], [12, 208012, 132, 27457584], [13, 742900, 42, 31201800], [14, 2674440, 14, 37442160], [15, 9694845, 5, 48474225], [16, 35357670, 2, 70715340], [17, 129644790, 1, 129644790], [18, 477638700, 1, 477638700]]", "after": "C_19=1767263190;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "Catalan complete convolution for C_20 := sum(1767263190, 477638700, 259289580, 176788350, 135727830, 112326480, 98062800, 89237148, 84063980, 81662152, 81662152, 84063980, 89237148, 98062800, 112326480, 135727830, 176788350, 259289580, 477638700, 1767263190)", "after": "Catalan complete convolution for C_20 = 6564120420", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_20，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_20 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1767263190", "477638700", "259289580", "176788350", "135727830", "112326480", "98062800", "89237148", "84063980", "81662152", "81662152", "84063980", "89237148", "98062800", "112326480", "135727830", "176788350", "259289580", "477638700", "1767263190" ], "output": "6564120420", "label": "Catalan complete convolution for C_20" } }, { "id": "S042", "uses": [ "G001", "S041" ], "before": "ORIGINAL index j=19;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 1767263190, 1767263190], [1, 1, 477638700, 477638700], [2, 2, 129644790, 259289580], [3, 5, 35357670, 176788350], [4, 14, 9694845, 135727830], [5, 42, 2674440, 112326480], [6, 132, 742900, 98062800], [7, 429, 208012, 89237148], [8, 1430, 58786, 84063980], [9, 4862, 16796, 81662152], [10, 16796, 4862, 81662152], [11, 58786, 1430, 84063980], [12, 208012, 429, 89237148], [13, 742900, 132, 98062800], [14, 2674440, 42, 112326480], [15, 9694845, 14, 135727830], [16, 35357670, 5, 176788350], [17, 129644790, 2, 259289580], [18, 477638700, 1, 477638700], [19, 1767263190, 1, 1767263190]]", "after": "C_20=6564120420;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S043", "uses": [ "G001", "S042" ], "before": "Catalan complete convolution for C_21 := sum(6564120420, 1767263190, 955277400, 648223950, 495007380, 407183490, 353026080, 318704100, 297457160, 285817532, 282105616, 285817532, 297457160, 318704100, 353026080, 407183490, 495007380, 648223950, 955277400, 1767263190, 6564120420)", "after": "Catalan complete convolution for C_21 = 24466267020", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_21，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_21 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "6564120420", "1767263190", "955277400", "648223950", "495007380", "407183490", "353026080", "318704100", "297457160", "285817532", "282105616", "285817532", "297457160", "318704100", "353026080", "407183490", "495007380", "648223950", "955277400", "1767263190", "6564120420" ], "output": "24466267020", "label": "Catalan complete convolution for C_21" } }, { "id": "S044", "uses": [ "G001", "S043" ], "before": "ORIGINAL index j=20;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 6564120420, 6564120420], [1, 1, 1767263190, 1767263190], [2, 2, 477638700, 955277400], [3, 5, 129644790, 648223950], [4, 14, 35357670, 495007380], [5, 42, 9694845, 407183490], [6, 132, 2674440, 353026080], [7, 429, 742900, 318704100], [8, 1430, 208012, 297457160], [9, 4862, 58786, 285817532], [10, 16796, 16796, 282105616], [11, 58786, 4862, 285817532], [12, 208012, 1430, 297457160], [13, 742900, 429, 318704100], [14, 2674440, 132, 353026080], [15, 9694845, 42, 407183490], [16, 35357670, 14, 495007380], [17, 129644790, 5, 648223950], [18, 477638700, 2, 955277400], [19, 1767263190, 1, 1767263190], [20, 6564120420, 1, 6564120420]]", "after": "C_21=24466267020;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S045", "uses": [ "G001", "S044" ], "before": "Catalan complete convolution for C_22 := sum(24466267020, 6564120420, 3534526380, 2388193500, 1815027060, 1485022140, 1279719540, 1147334760, 1062347000, 1011354344, 987369656, 987369656, 1011354344, 1062347000, 1147334760, 1279719540, 1485022140, 1815027060, 2388193500, 3534526380, 6564120420, 24466267020)", "after": "Catalan complete convolution for C_22 = 91482563640", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_22，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_22 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "24466267020", "6564120420", "3534526380", "2388193500", "1815027060", "1485022140", "1279719540", "1147334760", "1062347000", "1011354344", "987369656", "987369656", "1011354344", "1062347000", "1147334760", "1279719540", "1485022140", "1815027060", "2388193500", "3534526380", "6564120420", "24466267020" ], "output": "91482563640", "label": "Catalan complete convolution for C_22" } }, { "id": "S046", "uses": [ "G001", "S045" ], "before": "ORIGINAL index j=21;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 24466267020, 24466267020], [1, 1, 6564120420, 6564120420], [2, 2, 1767263190, 3534526380], [3, 5, 477638700, 2388193500], [4, 14, 129644790, 1815027060], [5, 42, 35357670, 1485022140], [6, 132, 9694845, 1279719540], [7, 429, 2674440, 1147334760], [8, 1430, 742900, 1062347000], [9, 4862, 208012, 1011354344], [10, 16796, 58786, 987369656], [11, 58786, 16796, 987369656], [12, 208012, 4862, 1011354344], [13, 742900, 1430, 1062347000], [14, 2674440, 429, 1147334760], [15, 9694845, 132, 1279719540], [16, 35357670, 42, 1485022140], [17, 129644790, 14, 1815027060], [18, 477638700, 5, 2388193500], [19, 1767263190, 2, 3534526380], [20, 6564120420, 1, 6564120420], [21, 24466267020, 1, 24466267020]]", "after": "C_22=91482563640;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S047", "uses": [ "G001", "S046" ], "before": "Catalan complete convolution for C_23 := sum(91482563640, 24466267020, 13128240840, 8836315950, 6686941800, 5445081180, 4667212440, 4159088505, 3824449200, 3611979800, 3493769552, 3455793796, 3493769552, 3611979800, 3824449200, 4159088505, 4667212440, 5445081180, 6686941800, 8836315950, 13128240840, 24466267020, 91482563640)", "after": "Catalan complete convolution for C_23 = 343059613650", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan complete convolution for C_23，不以小數近似替代。", "en": "Recompute Catalan complete convolution for C_23 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "91482563640", "24466267020", "13128240840", "8836315950", "6686941800", "5445081180", "4667212440", "4159088505", "3824449200", "3611979800", "3493769552", "3455793796", "3493769552", "3611979800", "3824449200", "4159088505", "4667212440", "5445081180", "6686941800", "8836315950", "13128240840", "24466267020", "91482563640" ], "output": "343059613650", "label": "Catalan complete convolution for C_23" } }, { "id": "S048", "uses": [ "G001", "S047" ], "before": "ORIGINAL index j=22;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 91482563640, 91482563640], [1, 1, 24466267020, 24466267020], [2, 2, 6564120420, 13128240840], [3, 5, 1767263190, 8836315950], [4, 14, 477638700, 6686941800], [5, 42, 129644790, 5445081180], [6, 132, 35357670, 4667212440], [7, 429, 9694845, 4159088505], [8, 1430, 2674440, 3824449200], [9, 4862, 742900, 3611979800], [10, 16796, 208012, 3493769552], [11, 58786, 58786, 3455793796], [12, 208012, 16796, 3493769552], [13, 742900, 4862, 3611979800], [14, 2674440, 1430, 3824449200], [15, 9694845, 429, 4159088505], [16, 35357670, 132, 4667212440], [17, 129644790, 42, 5445081180], [18, 477638700, 14, 6686941800], [19, 1767263190, 5, 8836315950], [20, 6564120420, 2, 13128240840], [21, 24466267020, 1, 24466267020], [22, 91482563640, 1, 91482563640]]", "after": "C_23=343059613650;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640]", "relation": "implies", "rule_id": "catalan_EVERY_convolution_split_product_and_prefix", "conditions": [], "explanation": { "zh-Hant": "逐全部split列原先已證的兩個C值與乘積，完整求和才是下一項；乘積即使超SAFE仍保exact BigInt語意，不截斷或轉float。", "en": "List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S049", "uses": [ "G001", "S048" ], "before": "Catalan central:ORIGINAL binomial N=46,K=23", "after": "use k=min(K,N-K)=23;initialize exact product1;empty product for k0 is1", "relation": "implies", "rule_id": "catalan_exact_binomial_product_initial_and_symmetry", "conditions": [], "explanation": { "zh-Hant": "以組合數對稱取較短乘積，保原N/K角色；每步先乘整數再做精確整除，不用浮點factorial近似。", "en": "Use binomial symmetry to shorten the product while retaining original N/K roles.Multiply before each exact division,without floating-point factorial approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S050", "uses": [ "G001", "S049" ], "before": "Catalan central binomial product at i=1 := mul(1, 24)", "after": "Catalan central binomial product at i=1 = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=1，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "24" ], "output": "24", "label": "Catalan central binomial product at i=1" } }, { "id": "S051", "uses": [ "G001", "S050" ], "before": "Catalan central exact binomial division by i=1 := div(24, 1)", "after": "Catalan central exact binomial division by i=1 = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=1，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "24", "1" ], "output": "24", "label": "Catalan central exact binomial division by i=1" } }, { "id": "S052", "uses": [ "G001", "S051" ], "before": "Catalan central binomial product at i=2 := mul(24, 25)", "after": "Catalan central binomial product at i=2 = 600", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=2，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "24", "25" ], "output": "600", "label": "Catalan central binomial product at i=2" } }, { "id": "S053", "uses": [ "G001", "S052" ], "before": "Catalan central exact binomial division by i=2 := div(600, 2)", "after": "Catalan central exact binomial division by i=2 = 300", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=2，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "600", "2" ], "output": "300", "label": "Catalan central exact binomial division by i=2" } }, { "id": "S054", "uses": [ "G001", "S053" ], "before": "Catalan central binomial product at i=3 := mul(300, 26)", "after": "Catalan central binomial product at i=3 = 7800", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=3，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "300", "26" ], "output": "7800", "label": "Catalan central binomial product at i=3" } }, { "id": "S055", "uses": [ "G001", "S054" ], "before": "Catalan central exact binomial division by i=3 := div(7800, 3)", "after": "Catalan central exact binomial division by i=3 = 2600", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=3，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "7800", "3" ], "output": "2600", "label": "Catalan central exact binomial division by i=3" } }, { "id": "S056", "uses": [ "G001", "S055" ], "before": "Catalan central binomial product at i=4 := mul(2600, 27)", "after": "Catalan central binomial product at i=4 = 70200", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=4，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2600", "27" ], "output": "70200", "label": "Catalan central binomial product at i=4" } }, { "id": "S057", "uses": [ "G001", "S056" ], "before": "Catalan central exact binomial division by i=4 := div(70200, 4)", "after": "Catalan central exact binomial division by i=4 = 17550", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=4，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "70200", "4" ], "output": "17550", "label": "Catalan central exact binomial division by i=4" } }, { "id": "S058", "uses": [ "G001", "S057" ], "before": "Catalan central binomial product at i=5 := mul(17550, 28)", "after": "Catalan central binomial product at i=5 = 491400", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=5，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "17550", "28" ], "output": "491400", "label": "Catalan central binomial product at i=5" } }, { "id": "S059", "uses": [ "G001", "S058" ], "before": "Catalan central exact binomial division by i=5 := div(491400, 5)", "after": "Catalan central exact binomial division by i=5 = 98280", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=5，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "491400", "5" ], "output": "98280", "label": "Catalan central exact binomial division by i=5" } }, { "id": "S060", "uses": [ "G001", "S059" ], "before": "Catalan central binomial product at i=6 := mul(98280, 29)", "after": "Catalan central binomial product at i=6 = 2850120", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=6，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "98280", "29" ], "output": "2850120", "label": "Catalan central binomial product at i=6" } }, { "id": "S061", "uses": [ "G001", "S060" ], "before": "Catalan central exact binomial division by i=6 := div(2850120, 6)", "after": "Catalan central exact binomial division by i=6 = 475020", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=6，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "2850120", "6" ], "output": "475020", "label": "Catalan central exact binomial division by i=6" } }, { "id": "S062", "uses": [ "G001", "S061" ], "before": "Catalan central binomial product at i=7 := mul(475020, 30)", "after": "Catalan central binomial product at i=7 = 14250600", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=7，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=7 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "475020", "30" ], "output": "14250600", "label": "Catalan central binomial product at i=7" } }, { "id": "S063", "uses": [ "G001", "S062" ], "before": "Catalan central exact binomial division by i=7 := div(14250600, 7)", "after": "Catalan central exact binomial division by i=7 = 2035800", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=7，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=7 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "14250600", "7" ], "output": "2035800", "label": "Catalan central exact binomial division by i=7" } }, { "id": "S064", "uses": [ "G001", "S063" ], "before": "Catalan central binomial product at i=8 := mul(2035800, 31)", "after": "Catalan central binomial product at i=8 = 63109800", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=8，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=8 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2035800", "31" ], "output": "63109800", "label": "Catalan central binomial product at i=8" } }, { "id": "S065", "uses": [ "G001", "S064" ], "before": "Catalan central exact binomial division by i=8 := div(63109800, 8)", "after": "Catalan central exact binomial division by i=8 = 7888725", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=8，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=8 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "63109800", "8" ], "output": "7888725", "label": "Catalan central exact binomial division by i=8" } }, { "id": "S066", "uses": [ "G001", "S065" ], "before": "Catalan central binomial product at i=9 := mul(7888725, 32)", "after": "Catalan central binomial product at i=9 = 252439200", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=9，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=9 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7888725", "32" ], "output": "252439200", "label": "Catalan central binomial product at i=9" } }, { "id": "S067", "uses": [ "G001", "S066" ], "before": "Catalan central exact binomial division by i=9 := div(252439200, 9)", "after": "Catalan central exact binomial division by i=9 = 28048800", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=9，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=9 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "252439200", "9" ], "output": "28048800", "label": "Catalan central exact binomial division by i=9" } }, { "id": "S068", "uses": [ "G001", "S067" ], "before": "Catalan central binomial product at i=10 := mul(28048800, 33)", "after": "Catalan central binomial product at i=10 = 925610400", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=10，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=10 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "28048800", "33" ], "output": "925610400", "label": "Catalan central binomial product at i=10" } }, { "id": "S069", "uses": [ "G001", "S068" ], "before": "Catalan central exact binomial division by i=10 := div(925610400, 10)", "after": "Catalan central exact binomial division by i=10 = 92561040", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=10，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=10 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "925610400", "10" ], "output": "92561040", "label": "Catalan central exact binomial division by i=10" } }, { "id": "S070", "uses": [ "G001", "S069" ], "before": "Catalan central binomial product at i=11 := mul(92561040, 34)", "after": "Catalan central binomial product at i=11 = 3147075360", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=11，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=11 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "92561040", "34" ], "output": "3147075360", "label": "Catalan central binomial product at i=11" } }, { "id": "S071", "uses": [ "G001", "S070" ], "before": "Catalan central exact binomial division by i=11 := div(3147075360, 11)", "after": "Catalan central exact binomial division by i=11 = 286097760", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=11，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=11 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "3147075360", "11" ], "output": "286097760", "label": "Catalan central exact binomial division by i=11" } }, { "id": "S072", "uses": [ "G001", "S071" ], "before": "Catalan central binomial product at i=12 := mul(286097760, 35)", "after": "Catalan central binomial product at i=12 = 10013421600", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=12，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=12 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "286097760", "35" ], "output": "10013421600", "label": "Catalan central binomial product at i=12" } }, { "id": "S073", "uses": [ "G001", "S072" ], "before": "Catalan central exact binomial division by i=12 := div(10013421600, 12)", "after": "Catalan central exact binomial division by i=12 = 834451800", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=12，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=12 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "10013421600", "12" ], "output": "834451800", "label": "Catalan central exact binomial division by i=12" } }, { "id": "S074", "uses": [ "G001", "S073" ], "before": "Catalan central binomial product at i=13 := mul(834451800, 36)", "after": "Catalan central binomial product at i=13 = 30040264800", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=13，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=13 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "834451800", "36" ], "output": "30040264800", "label": "Catalan central binomial product at i=13" } }, { "id": "S075", "uses": [ "G001", "S074" ], "before": "Catalan central exact binomial division by i=13 := div(30040264800, 13)", "after": "Catalan central exact binomial division by i=13 = 2310789600", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=13，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=13 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "30040264800", "13" ], "output": "2310789600", "label": "Catalan central exact binomial division by i=13" } }, { "id": "S076", "uses": [ "G001", "S075" ], "before": "Catalan central binomial product at i=14 := mul(2310789600, 37)", "after": "Catalan central binomial product at i=14 = 85499215200", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=14，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=14 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2310789600", "37" ], "output": "85499215200", "label": "Catalan central binomial product at i=14" } }, { "id": "S077", "uses": [ "G001", "S076" ], "before": "Catalan central exact binomial division by i=14 := div(85499215200, 14)", "after": "Catalan central exact binomial division by i=14 = 6107086800", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=14，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=14 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "85499215200", "14" ], "output": "6107086800", "label": "Catalan central exact binomial division by i=14" } }, { "id": "S078", "uses": [ "G001", "S077" ], "before": "Catalan central binomial product at i=15 := mul(6107086800, 38)", "after": "Catalan central binomial product at i=15 = 232069298400", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=15，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=15 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6107086800", "38" ], "output": "232069298400", "label": "Catalan central binomial product at i=15" } }, { "id": "S079", "uses": [ "G001", "S078" ], "before": "Catalan central exact binomial division by i=15 := div(232069298400, 15)", "after": "Catalan central exact binomial division by i=15 = 15471286560", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=15，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=15 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "232069298400", "15" ], "output": "15471286560", "label": "Catalan central exact binomial division by i=15" } }, { "id": "S080", "uses": [ "G001", "S079" ], "before": "Catalan central binomial product at i=16 := mul(15471286560, 39)", "after": "Catalan central binomial product at i=16 = 603380175840", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=16，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=16 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "15471286560", "39" ], "output": "603380175840", "label": "Catalan central binomial product at i=16" } }, { "id": "S081", "uses": [ "G001", "S080" ], "before": "Catalan central exact binomial division by i=16 := div(603380175840, 16)", "after": "Catalan central exact binomial division by i=16 = 37711260990", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=16，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=16 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "603380175840", "16" ], "output": "37711260990", "label": "Catalan central exact binomial division by i=16" } }, { "id": "S082", "uses": [ "G001", "S081" ], "before": "Catalan central binomial product at i=17 := mul(37711260990, 40)", "after": "Catalan central binomial product at i=17 = 1508450439600", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=17，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=17 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "37711260990", "40" ], "output": "1508450439600", "label": "Catalan central binomial product at i=17" } }, { "id": "S083", "uses": [ "G001", "S082" ], "before": "Catalan central exact binomial division by i=17 := div(1508450439600, 17)", "after": "Catalan central exact binomial division by i=17 = 88732378800", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=17，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=17 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1508450439600", "17" ], "output": "88732378800", "label": "Catalan central exact binomial division by i=17" } }, { "id": "S084", "uses": [ "G001", "S083" ], "before": "Catalan central binomial product at i=18 := mul(88732378800, 41)", "after": "Catalan central binomial product at i=18 = 3638027530800", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=18，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=18 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "88732378800", "41" ], "output": "3638027530800", "label": "Catalan central binomial product at i=18" } }, { "id": "S085", "uses": [ "G001", "S084" ], "before": "Catalan central exact binomial division by i=18 := div(3638027530800, 18)", "after": "Catalan central exact binomial division by i=18 = 202112640600", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=18，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=18 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "3638027530800", "18" ], "output": "202112640600", "label": "Catalan central exact binomial division by i=18" } }, { "id": "S086", "uses": [ "G001", "S085" ], "before": "Catalan central binomial product at i=19 := mul(202112640600, 42)", "after": "Catalan central binomial product at i=19 = 8488730905200", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=19，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=19 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "202112640600", "42" ], "output": "8488730905200", "label": "Catalan central binomial product at i=19" } }, { "id": "S087", "uses": [ "G001", "S086" ], "before": "Catalan central exact binomial division by i=19 := div(8488730905200, 19)", "after": "Catalan central exact binomial division by i=19 = 446775310800", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=19，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=19 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "8488730905200", "19" ], "output": "446775310800", "label": "Catalan central exact binomial division by i=19" } }, { "id": "S088", "uses": [ "G001", "S087" ], "before": "Catalan central binomial product at i=20 := mul(446775310800, 43)", "after": "Catalan central binomial product at i=20 = 19211338364400", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=20，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=20 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "446775310800", "43" ], "output": "19211338364400", "label": "Catalan central binomial product at i=20" } }, { "id": "S089", "uses": [ "G001", "S088" ], "before": "Catalan central exact binomial division by i=20 := div(19211338364400, 20)", "after": "Catalan central exact binomial division by i=20 = 960566918220", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=20，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=20 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "19211338364400", "20" ], "output": "960566918220", "label": "Catalan central exact binomial division by i=20" } }, { "id": "S090", "uses": [ "G001", "S089" ], "before": "Catalan central binomial product at i=21 := mul(960566918220, 44)", "after": "Catalan central binomial product at i=21 = 42264944401680", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=21，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=21 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "960566918220", "44" ], "output": "42264944401680", "label": "Catalan central binomial product at i=21" } }, { "id": "S091", "uses": [ "G001", "S090" ], "before": "Catalan central exact binomial division by i=21 := div(42264944401680, 21)", "after": "Catalan central exact binomial division by i=21 = 2012616400080", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=21，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=21 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "42264944401680", "21" ], "output": "2012616400080", "label": "Catalan central exact binomial division by i=21" } }, { "id": "S092", "uses": [ "G001", "S091" ], "before": "Catalan central binomial product at i=22 := mul(2012616400080, 45)", "after": "Catalan central binomial product at i=22 = 90567738003600", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=22，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=22 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2012616400080", "45" ], "output": "90567738003600", "label": "Catalan central binomial product at i=22" } }, { "id": "S093", "uses": [ "G001", "S092" ], "before": "Catalan central exact binomial division by i=22 := div(90567738003600, 22)", "after": "Catalan central exact binomial division by i=22 = 4116715363800", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=22，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=22 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "90567738003600", "22" ], "output": "4116715363800", "label": "Catalan central exact binomial division by i=22" } }, { "id": "S094", "uses": [ "G001", "S093" ], "before": "Catalan central binomial product at i=23 := mul(4116715363800, 46)", "after": "Catalan central binomial product at i=23 = 189368906734800", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central binomial product at i=23，不以小數近似替代。", "en": "Recompute Catalan central binomial product at i=23 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4116715363800", "46" ], "output": "189368906734800", "label": "Catalan central binomial product at i=23" } }, { "id": "S095", "uses": [ "G001", "S094" ], "before": "Catalan central exact binomial division by i=23 := div(189368906734800, 23)", "after": "Catalan central exact binomial division by i=23 = 8233430727600", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central exact binomial division by i=23，不以小數近似替代。", "en": "Recompute Catalan central exact binomial division by i=23 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "189368906734800", "23" ], "output": "8233430727600", "label": "Catalan central exact binomial division by i=23" } }, { "id": "S096", "uses": [ "G001", "S095" ], "before": "Catalan ORIGINAL denominator n plus1 := sum(23, 1)", "after": "Catalan ORIGINAL denominator n plus1 = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan ORIGINAL denominator n plus1，不以小數近似替代。", "en": "Recompute Catalan ORIGINAL denominator n plus1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "23", "1" ], "output": "24", "label": "Catalan ORIGINAL denominator n plus1" } }, { "id": "S097", "uses": [ "G001", "S096" ], "before": "ORIGINAL n=23;central binomial=8233430727600;positive denominator n+1=24", "after": "Euclidean quotient=343059613650,remainder=0;check remainder0 BEFORE treating quotient as Catalan integer", "relation": "implies", "rule_id": "catalan_remainder_BEFORE_exact_closed_division", "conditions": [], "explanation": { "zh-Hant": "先查餘數0才可作精確整數商，不以整數截斷掩蓋分母問題。n0時分母1與central1照樣保留。", "en": "Check remainder0 before taking an exact integer quotient.Do not hide nondivisibility by truncation;retain denominator1 and central1 at n0." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S098", "uses": [ "G001", "S097" ], "before": "Catalan exact closed quotient after remainder0 := div(8233430727600, 24)", "after": "Catalan exact closed quotient after remainder0 = 343059613650", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan exact closed quotient after remainder0，不以小數近似替代。", "en": "Recompute Catalan exact closed quotient after remainder0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "8233430727600", "24" ], "output": "343059613650", "label": "Catalan exact closed quotient after remainder0" } }, { "id": "S099", "uses": [ "G001", "S098" ], "before": "Catalan adjacent:ORIGINAL binomial N=46,K=24", "after": "use k=min(K,N-K)=22;initialize exact product1;empty product for k0 is1", "relation": "implies", "rule_id": "catalan_exact_binomial_product_initial_and_symmetry", "conditions": [], "explanation": { "zh-Hant": "以組合數對稱取較短乘積，保原N/K角色；每步先乘整數再做精確整除，不用浮點factorial近似。", "en": "Use binomial symmetry to shorten the product while retaining original N/K roles.Multiply before each exact division,without floating-point factorial approximations." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S100", "uses": [ "G001", "S099" ], "before": "Catalan adjacent binomial product at i=1 := mul(1, 25)", "after": "Catalan adjacent binomial product at i=1 = 25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=1，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "25" ], "output": "25", "label": "Catalan adjacent binomial product at i=1" } }, { "id": "S101", "uses": [ "G001", "S100" ], "before": "Catalan adjacent exact binomial division by i=1 := div(25, 1)", "after": "Catalan adjacent exact binomial division by i=1 = 25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=1，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "25", "1" ], "output": "25", "label": "Catalan adjacent exact binomial division by i=1" } }, { "id": "S102", "uses": [ "G001", "S101" ], "before": "Catalan adjacent binomial product at i=2 := mul(25, 26)", "after": "Catalan adjacent binomial product at i=2 = 650", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=2，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "25", "26" ], "output": "650", "label": "Catalan adjacent binomial product at i=2" } }, { "id": "S103", "uses": [ "G001", "S102" ], "before": "Catalan adjacent exact binomial division by i=2 := div(650, 2)", "after": "Catalan adjacent exact binomial division by i=2 = 325", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=2，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "650", "2" ], "output": "325", "label": "Catalan adjacent exact binomial division by i=2" } }, { "id": "S104", "uses": [ "G001", "S103" ], "before": "Catalan adjacent binomial product at i=3 := mul(325, 27)", "after": "Catalan adjacent binomial product at i=3 = 8775", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=3，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "325", "27" ], "output": "8775", "label": "Catalan adjacent binomial product at i=3" } }, { "id": "S105", "uses": [ "G001", "S104" ], "before": "Catalan adjacent exact binomial division by i=3 := div(8775, 3)", "after": "Catalan adjacent exact binomial division by i=3 = 2925", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=3，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "8775", "3" ], "output": "2925", "label": "Catalan adjacent exact binomial division by i=3" } }, { "id": "S106", "uses": [ "G001", "S105" ], "before": "Catalan adjacent binomial product at i=4 := mul(2925, 28)", "after": "Catalan adjacent binomial product at i=4 = 81900", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=4，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2925", "28" ], "output": "81900", "label": "Catalan adjacent binomial product at i=4" } }, { "id": "S107", "uses": [ "G001", "S106" ], "before": "Catalan adjacent exact binomial division by i=4 := div(81900, 4)", "after": "Catalan adjacent exact binomial division by i=4 = 20475", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=4，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=4 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "81900", "4" ], "output": "20475", "label": "Catalan adjacent exact binomial division by i=4" } }, { "id": "S108", "uses": [ "G001", "S107" ], "before": "Catalan adjacent binomial product at i=5 := mul(20475, 29)", "after": "Catalan adjacent binomial product at i=5 = 593775", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=5，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "20475", "29" ], "output": "593775", "label": "Catalan adjacent binomial product at i=5" } }, { "id": "S109", "uses": [ "G001", "S108" ], "before": "Catalan adjacent exact binomial division by i=5 := div(593775, 5)", "after": "Catalan adjacent exact binomial division by i=5 = 118755", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=5，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=5 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "593775", "5" ], "output": "118755", "label": "Catalan adjacent exact binomial division by i=5" } }, { "id": "S110", "uses": [ "G001", "S109" ], "before": "Catalan adjacent binomial product at i=6 := mul(118755, 30)", "after": "Catalan adjacent binomial product at i=6 = 3562650", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=6，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "118755", "30" ], "output": "3562650", "label": "Catalan adjacent binomial product at i=6" } }, { "id": "S111", "uses": [ "G001", "S110" ], "before": "Catalan adjacent exact binomial division by i=6 := div(3562650, 6)", "after": "Catalan adjacent exact binomial division by i=6 = 593775", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=6，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=6 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "3562650", "6" ], "output": "593775", "label": "Catalan adjacent exact binomial division by i=6" } }, { "id": "S112", "uses": [ "G001", "S111" ], "before": "Catalan adjacent binomial product at i=7 := mul(593775, 31)", "after": "Catalan adjacent binomial product at i=7 = 18407025", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=7，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=7 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "593775", "31" ], "output": "18407025", "label": "Catalan adjacent binomial product at i=7" } }, { "id": "S113", "uses": [ "G001", "S112" ], "before": "Catalan adjacent exact binomial division by i=7 := div(18407025, 7)", "after": "Catalan adjacent exact binomial division by i=7 = 2629575", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=7，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=7 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "18407025", "7" ], "output": "2629575", "label": "Catalan adjacent exact binomial division by i=7" } }, { "id": "S114", "uses": [ "G001", "S113" ], "before": "Catalan adjacent binomial product at i=8 := mul(2629575, 32)", "after": "Catalan adjacent binomial product at i=8 = 84146400", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=8，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=8 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2629575", "32" ], "output": "84146400", "label": "Catalan adjacent binomial product at i=8" } }, { "id": "S115", "uses": [ "G001", "S114" ], "before": "Catalan adjacent exact binomial division by i=8 := div(84146400, 8)", "after": "Catalan adjacent exact binomial division by i=8 = 10518300", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=8，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=8 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "84146400", "8" ], "output": "10518300", "label": "Catalan adjacent exact binomial division by i=8" } }, { "id": "S116", "uses": [ "G001", "S115" ], "before": "Catalan adjacent binomial product at i=9 := mul(10518300, 33)", "after": "Catalan adjacent binomial product at i=9 = 347103900", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=9，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=9 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "10518300", "33" ], "output": "347103900", "label": "Catalan adjacent binomial product at i=9" } }, { "id": "S117", "uses": [ "G001", "S116" ], "before": "Catalan adjacent exact binomial division by i=9 := div(347103900, 9)", "after": "Catalan adjacent exact binomial division by i=9 = 38567100", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=9，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=9 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "347103900", "9" ], "output": "38567100", "label": "Catalan adjacent exact binomial division by i=9" } }, { "id": "S118", "uses": [ "G001", "S117" ], "before": "Catalan adjacent binomial product at i=10 := mul(38567100, 34)", "after": "Catalan adjacent binomial product at i=10 = 1311281400", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=10，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=10 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "38567100", "34" ], "output": "1311281400", "label": "Catalan adjacent binomial product at i=10" } }, { "id": "S119", "uses": [ "G001", "S118" ], "before": "Catalan adjacent exact binomial division by i=10 := div(1311281400, 10)", "after": "Catalan adjacent exact binomial division by i=10 = 131128140", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=10，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=10 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1311281400", "10" ], "output": "131128140", "label": "Catalan adjacent exact binomial division by i=10" } }, { "id": "S120", "uses": [ "G001", "S119" ], "before": "Catalan adjacent binomial product at i=11 := mul(131128140, 35)", "after": "Catalan adjacent binomial product at i=11 = 4589484900", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=11，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=11 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "131128140", "35" ], "output": "4589484900", "label": "Catalan adjacent binomial product at i=11" } }, { "id": "S121", "uses": [ "G001", "S120" ], "before": "Catalan adjacent exact binomial division by i=11 := div(4589484900, 11)", "after": "Catalan adjacent exact binomial division by i=11 = 417225900", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=11，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=11 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "4589484900", "11" ], "output": "417225900", "label": "Catalan adjacent exact binomial division by i=11" } }, { "id": "S122", "uses": [ "G001", "S121" ], "before": "Catalan adjacent binomial product at i=12 := mul(417225900, 36)", "after": "Catalan adjacent binomial product at i=12 = 15020132400", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=12，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=12 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "417225900", "36" ], "output": "15020132400", "label": "Catalan adjacent binomial product at i=12" } }, { "id": "S123", "uses": [ "G001", "S122" ], "before": "Catalan adjacent exact binomial division by i=12 := div(15020132400, 12)", "after": "Catalan adjacent exact binomial division by i=12 = 1251677700", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=12，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=12 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "15020132400", "12" ], "output": "1251677700", "label": "Catalan adjacent exact binomial division by i=12" } }, { "id": "S124", "uses": [ "G001", "S123" ], "before": "Catalan adjacent binomial product at i=13 := mul(1251677700, 37)", "after": "Catalan adjacent binomial product at i=13 = 46312074900", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=13，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=13 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1251677700", "37" ], "output": "46312074900", "label": "Catalan adjacent binomial product at i=13" } }, { "id": "S125", "uses": [ "G001", "S124" ], "before": "Catalan adjacent exact binomial division by i=13 := div(46312074900, 13)", "after": "Catalan adjacent exact binomial division by i=13 = 3562467300", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=13，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=13 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "46312074900", "13" ], "output": "3562467300", "label": "Catalan adjacent exact binomial division by i=13" } }, { "id": "S126", "uses": [ "G001", "S125" ], "before": "Catalan adjacent binomial product at i=14 := mul(3562467300, 38)", "after": "Catalan adjacent binomial product at i=14 = 135373757400", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=14，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=14 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3562467300", "38" ], "output": "135373757400", "label": "Catalan adjacent binomial product at i=14" } }, { "id": "S127", "uses": [ "G001", "S126" ], "before": "Catalan adjacent exact binomial division by i=14 := div(135373757400, 14)", "after": "Catalan adjacent exact binomial division by i=14 = 9669554100", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=14，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=14 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "135373757400", "14" ], "output": "9669554100", "label": "Catalan adjacent exact binomial division by i=14" } }, { "id": "S128", "uses": [ "G001", "S127" ], "before": "Catalan adjacent binomial product at i=15 := mul(9669554100, 39)", "after": "Catalan adjacent binomial product at i=15 = 377112609900", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=15，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=15 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "9669554100", "39" ], "output": "377112609900", "label": "Catalan adjacent binomial product at i=15" } }, { "id": "S129", "uses": [ "G001", "S128" ], "before": "Catalan adjacent exact binomial division by i=15 := div(377112609900, 15)", "after": "Catalan adjacent exact binomial division by i=15 = 25140840660", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=15，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=15 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "377112609900", "15" ], "output": "25140840660", "label": "Catalan adjacent exact binomial division by i=15" } }, { "id": "S130", "uses": [ "G001", "S129" ], "before": "Catalan adjacent binomial product at i=16 := mul(25140840660, 40)", "after": "Catalan adjacent binomial product at i=16 = 1005633626400", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=16，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=16 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "25140840660", "40" ], "output": "1005633626400", "label": "Catalan adjacent binomial product at i=16" } }, { "id": "S131", "uses": [ "G001", "S130" ], "before": "Catalan adjacent exact binomial division by i=16 := div(1005633626400, 16)", "after": "Catalan adjacent exact binomial division by i=16 = 62852101650", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=16，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=16 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1005633626400", "16" ], "output": "62852101650", "label": "Catalan adjacent exact binomial division by i=16" } }, { "id": "S132", "uses": [ "G001", "S131" ], "before": "Catalan adjacent binomial product at i=17 := mul(62852101650, 41)", "after": "Catalan adjacent binomial product at i=17 = 2576936167650", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=17，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=17 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "62852101650", "41" ], "output": "2576936167650", "label": "Catalan adjacent binomial product at i=17" } }, { "id": "S133", "uses": [ "G001", "S132" ], "before": "Catalan adjacent exact binomial division by i=17 := div(2576936167650, 17)", "after": "Catalan adjacent exact binomial division by i=17 = 151584480450", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=17，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=17 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "2576936167650", "17" ], "output": "151584480450", "label": "Catalan adjacent exact binomial division by i=17" } }, { "id": "S134", "uses": [ "G001", "S133" ], "before": "Catalan adjacent binomial product at i=18 := mul(151584480450, 42)", "after": "Catalan adjacent binomial product at i=18 = 6366548178900", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=18，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=18 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "151584480450", "42" ], "output": "6366548178900", "label": "Catalan adjacent binomial product at i=18" } }, { "id": "S135", "uses": [ "G001", "S134" ], "before": "Catalan adjacent exact binomial division by i=18 := div(6366548178900, 18)", "after": "Catalan adjacent exact binomial division by i=18 = 353697121050", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=18，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=18 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "6366548178900", "18" ], "output": "353697121050", "label": "Catalan adjacent exact binomial division by i=18" } }, { "id": "S136", "uses": [ "G001", "S135" ], "before": "Catalan adjacent binomial product at i=19 := mul(353697121050, 43)", "after": "Catalan adjacent binomial product at i=19 = 15208976205150", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=19，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=19 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "353697121050", "43" ], "output": "15208976205150", "label": "Catalan adjacent binomial product at i=19" } }, { "id": "S137", "uses": [ "G001", "S136" ], "before": "Catalan adjacent exact binomial division by i=19 := div(15208976205150, 19)", "after": "Catalan adjacent exact binomial division by i=19 = 800472431850", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=19，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=19 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "15208976205150", "19" ], "output": "800472431850", "label": "Catalan adjacent exact binomial division by i=19" } }, { "id": "S138", "uses": [ "G001", "S137" ], "before": "Catalan adjacent binomial product at i=20 := mul(800472431850, 44)", "after": "Catalan adjacent binomial product at i=20 = 35220787001400", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=20，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=20 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "800472431850", "44" ], "output": "35220787001400", "label": "Catalan adjacent binomial product at i=20" } }, { "id": "S139", "uses": [ "G001", "S138" ], "before": "Catalan adjacent exact binomial division by i=20 := div(35220787001400, 20)", "after": "Catalan adjacent exact binomial division by i=20 = 1761039350070", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=20，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=20 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "35220787001400", "20" ], "output": "1761039350070", "label": "Catalan adjacent exact binomial division by i=20" } }, { "id": "S140", "uses": [ "G001", "S139" ], "before": "Catalan adjacent binomial product at i=21 := mul(1761039350070, 45)", "after": "Catalan adjacent binomial product at i=21 = 79246770753150", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=21，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=21 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1761039350070", "45" ], "output": "79246770753150", "label": "Catalan adjacent binomial product at i=21" } }, { "id": "S141", "uses": [ "G001", "S140" ], "before": "Catalan adjacent exact binomial division by i=21 := div(79246770753150, 21)", "after": "Catalan adjacent exact binomial division by i=21 = 3773655750150", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=21，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=21 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "79246770753150", "21" ], "output": "3773655750150", "label": "Catalan adjacent exact binomial division by i=21" } }, { "id": "S142", "uses": [ "G001", "S141" ], "before": "Catalan adjacent binomial product at i=22 := mul(3773655750150, 46)", "after": "Catalan adjacent binomial product at i=22 = 173588164506900", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent binomial product at i=22，不以小數近似替代。", "en": "Recompute Catalan adjacent binomial product at i=22 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3773655750150", "46" ], "output": "173588164506900", "label": "Catalan adjacent binomial product at i=22" } }, { "id": "S143", "uses": [ "G001", "S142" ], "before": "Catalan adjacent exact binomial division by i=22 := div(173588164506900, 22)", "after": "Catalan adjacent exact binomial division by i=22 = 7890371113950", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan adjacent exact binomial division by i=22，不以小數近似替代。", "en": "Recompute Catalan adjacent exact binomial division by i=22 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "173588164506900", "22" ], "output": "7890371113950", "label": "Catalan adjacent exact binomial division by i=22" } }, { "id": "S144", "uses": [ "G001", "S143" ], "before": "Catalan central minus adjacent binomial := sub(8233430727600, 7890371113950)", "after": "Catalan central minus adjacent binomial = 343059613650", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色、原索引與已證中間量精確代值，不改為浮點近似或其他題目的資料。", "en": "Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Catalan central minus adjacent binomial，不以小數近似替代。", "en": "Recompute Catalan central minus adjacent binomial with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "8233430727600", "7890371113950" ], "output": "343059613650", "label": "Catalan central minus adjacent binomial" } }, { "id": "S145", "uses": [ "G001", "S144" ], "before": "ORIGINAL n=23;recurrence value=343059613650;central=8233430727600,adjacent=7890371113950,denominator=24", "after": "closed quotient=343059613650;binomial difference=343059613650;ALL THREE agree;balanced paths counted by choose(2n,n),bad paths first-crossing reflection counted by choose(2n,n+1)", "relation": "implies", "rule_id": "catalan_general_Dyck_reflection_and_first_return_recurrence", "conditions": [], "explanation": { "zh-Hant": "Dyck path首次跌到−1的prefix反射，與n+1個up／n−1個down路徑雙射；good數是兩binomial之差。首次返回分解U P D Q給C_(j+1)=ΣC_i C_(j−i)，空path給C0=1，所以同一遞迴唯一決定此數；adjacent比率再化差為central/(n+1)。不是只由有限數值猜一般恆等。", "en": "Reflect a bad balanced path through its first crossing of−1 to obtain a path with n+1 ups and n−1 downs,so good paths are the binomial difference.First-return decomposition U P D Q gives the Catalan convolution with empty-path base1,uniquely determining the same count.The adjacent-binomial ratio yields central/(n+1).This is a general argument,not extrapolation from finitely many equalities." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ 23, "8233430727600", "24", "0", "343059613650", "343059613650", true ] }, "conclusion": { "canonical_value": "(23, 8233430727600, 24, 0, 343059613650, 343059613650, 是)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261111-r05-catalan-numbers-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261111-r05-catalan-numbers-foundations/questions.jsonl", "batch_sha256": "ceaecd210e147af2eb1f9c63d2057f658f4e7d1ec07cc95c12ee88d2eb3cb856", "original_line": 49, "original_line_sha256": "9c1a1cbd478456c58364178088e8fb8866881ca07258374a64e9f88df4eed769", "question_compact_sha256": "aa83fa2aa4784ee26fe59314b6dc35fba3981d15f7faa9810f44365df730f822", "english_row_compact_sha256": "22443e5d7f8e094fc993ab529713f50dad2b8d15b4565a3c0706f669dab234a4", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261111-r05-catalan-numbers-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261111-r05-catalan-numbers-foundations.en.jsonl" }, "mko_bindings": { "mko-catalan-numbers": { "sha256": "2d847df82bf2c534d59aedb78304a0f59d94a87bf4454e0eee1886df489e21b5", "english_sha256": "328d37ca081c47ad3e92abe05ddd5028dfeb23321136b79cc8b3d101c89e814d", "original_path": "public/data/mko/catalan-numbers.json", "english_path": "public/data/i18n/editions/mko-catalan-numbers.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "4db6f09f48d29a15f89787c4aac3ae0abe7981d9f906e3fbecc88b60dd686db5", "original_candidate_sha256": "c132762c5d03028df8e611ce88a60277ef3e31e2d9d37368a36bfd17f654a39c", "original_candidate_line_sha256": "c132762c5d03028df8e611ce88a60277ef3e31e2d9d37368a36bfd17f654a39c" }, "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-20261026-009-p45", "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": "89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81", "question_first_witness_sha256": "b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 145, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "72780748a0b49a7b982355c4b64230575e0c2db7217da35ffa0427c92476cd5f", "arithmetic_steps": 116 }, "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." } }