{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0001","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa042ce66d49587891943ba5d71009239b221592c1515019443d0d0307b3d7ea","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115001,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":0}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=0≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(0, [1], 1, C_zero_equals_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (0, [1], 1, C_zero_equals_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0002","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62e2ce818d6f7e3dd61901b850296373807afa55e1d0766542ae1376818f98ee","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115002,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":1}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=1≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(1, [1, 1], 1, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (1, [1, 1], 1, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0003","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3074ba56a16a96502e629bb0966b09d83b8ab599b731c4ac000c138e5b57bf32","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115003,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":2}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=2≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(2, [1, 1, 2], 2, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (2, [1, 1, 2], 2, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0004","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8de45a723afa432d166dbf74120624a53cbffe2a58c4eca2f240b31a0605e32f","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115004,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":3}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=3≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(3, [1, 1, 2, 5], 5, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (3, [1, 1, 2, 5], 5, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0005","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"219f0ba59786b8c3f7bd2ba1f6468f482784df731f51e3c3370984278459774b","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115005,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":4}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=4≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(4, [1, 1, 2, 5, 14], 14, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (4, [1, 1, 2, 5, 14], 14, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0006","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c40ed4146a09540a6985f89b27e13b311b97d05d64e391de82e8721a01d4ec78","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115006,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":5}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=5≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(5, [1, 1, 2, 5, 14, 42], 42, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (5, [1, 1, 2, 5, 14, 42], 42, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0007","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef4a1684a3f9e8d1dbf7c30eb5adaf89872d70ddc1fd1ed1fd2c134f5ae81059","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115007,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":6}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=6≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(6, [1, 1, 2, 5, 14, 42, 132], 132, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (6, [1, 1, 2, 5, 14, 42, 132], 132, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0008","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73936fe5fb400b27d252b520427ba81bd41d56395900782b328b83b692f0cd13","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115008,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":7}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=7≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(7, [1, 1, 2, 5, 14, 42, 132, 429], 429, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (7, [1, 1, 2, 5, 14, 42, 132, 429], 429, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0009","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd8452d2c21f6595c3d2bee2b4cb9f43d2453730d37953283699c927d1087568","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115009,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":8}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=8≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(8, [1, 1, 2, 5, 14, 42, 132, 429, 1430], 1430, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (8, [1, 1, 2, 5, 14, 42, 132, 429, 1430], 1430, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0010","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f90223c8e9328c032e8c4ac66d6cc5c92264e9b39f5b201f90761f5965b8626b","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115010,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":9}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=9≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(9, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862], 4862, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (9, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862], 4862, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0011","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef8ff8862bfa40f65171c6edc5c18895fad3d01aa95d52ee6d8921bce0f7f3a8","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115011,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":10}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=10≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(10, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796], 16796, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (10, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796], 16796, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0012","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fa6b4214b516b2b0184c95bb70bb5b89e28b1bef99f1354128bf3511f2baeba","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115012,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":11}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=11≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(11, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786], 58786, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (11, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786], 58786, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0013","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"170841cad5550668c542afcf23c434ca30db417ecced741fd96b2df54b2619b9","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115013,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":12}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=12≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(12, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012], 208012, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (12, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012], 208012, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0014","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e50277e95881ce03349a4be037fbd2220b54fe21e241d1b37d0815813acef3de","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115014,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":13}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=13≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(13, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900], 742900, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (13, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900], 742900, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0015","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e9568a90e9c060c32d9ec5c999bd83957646b7b8ed74d81c84fb8f7eaaea84f","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115015,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":14}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=14≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(14, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440], 2674440, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (14, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440], 2674440, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0016","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6832e8dcd3a0aee83280daa240dae38a5803e5850cebfc784c42315aef3802e","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115016,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":15}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=15≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(15, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845], 9694845, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (15, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845], 9694845, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0017","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d823e3cdd0b2a5a35b34826170d3d5fff3aacb59b00dec57013e7805793c491","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115017,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":16}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=16≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(16, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670], 35357670, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (16, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670], 35357670, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0018","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acaf30848d65f4242d8ab513abd6edc76954227ca7a024e91f3b6acd31eb4765","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115018,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":17}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=17≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(17, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790], 129644790, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (17, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790], 129644790, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0019","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1ed587da2a279a767d98e7e49d257ccfea40e91427ed795f4c2e8036938d122","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115019,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":18}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=18≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(18, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700], 477638700, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (18, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700], 477638700, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0020","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4da53987c0d6a2b4da7f1983c0f41943abae89f285d8adeb6d53b576fb69799d","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115020,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":19}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=19≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(19, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190], 1767263190, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (19, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190], 1767263190, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0021","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7fab0c2eb68c12a3d443216e7a8edb8e71fb3c5d12d01f25341984c17620ce1","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115021,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":20}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=20≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(20, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420], 6564120420, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (20, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420], 6564120420, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0022","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f90033fed8fbd54e71fdc6b83e25a772da052e90c687a1e8c07de0e67f563657","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115022,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":21}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=21≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(21, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020], 24466267020, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (21, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020], 24466267020, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0023","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b55b0e48644df7ef3a107a194bb931bf4011d97ad8867e0f5a92d8f484b733f2","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115023,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":22}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=22≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(22, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640], 91482563640, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (22, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640], 91482563640, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0024","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cbadab2bd2cdb312a74e78d4bd40ab2d0ac1679818b4ea0385aa96008b7943d","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115024,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":23}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(23, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650], 343059613650, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (23, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650], 343059613650, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0025","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7373adb021e3b70e6047df54789c08932953dd8c2fbef9bc7e99e465432940b3","family_id":"qf-mod42-catalan-recurrence-generation","seed":202611115025,"semantic_template":"qf-mod42-catalan-recurrence-generation/{\"n\":24}"},"fields":{"title":"Catalan Generation Starts with C Zero Equal to One","stem":"Use natural index n=24≥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, prefix, C_n, index_convention).","answer":{"kind":"expression","value":"(24, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324], 1289904147324, C_next_sum_i_zero_to_n_minus_one)","explanation":"Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (24, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324], 1289904147324, C_next_sum_i_zero_to_n_minus_one)."},"skills":["Catalan Generation Starts with C Zero Equal to One"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0026","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ebb76d672d28406e5e6998a2ceca7c9cc5717310fa25eadc4818387c7a1b9e7","family_id":"qf-mod42-catalan-closed-form","seed":202611115026,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":0}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=0≥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":"(0, 1, 1, 0, 1, 1, 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 (0, 1, 1, 0, 1, 1, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0027","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"757597fe79552b0b81d90aaa867658415d9b62ec54f915653d5872d33f3dc0e3","family_id":"qf-mod42-catalan-closed-form","seed":202611115027,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":1}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=1≥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":"(1, 2, 2, 0, 1, 1, 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 (1, 2, 2, 0, 1, 1, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0028","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc11b36f3e85a05d7c542f4301638a135e756306d779942b4977ffa38e849469","family_id":"qf-mod42-catalan-closed-form","seed":202611115028,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":2}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=2≥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":"(2, 6, 3, 0, 2, 2, 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 (2, 6, 3, 0, 2, 2, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0029","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f0dc6787fc721c3382bf0378eb32cd08c5b4f31fb06c0c786846e36ea514b84","family_id":"qf-mod42-catalan-closed-form","seed":202611115029,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":3}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=3≥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":"(3, 20, 4, 0, 5, 5, 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 (3, 20, 4, 0, 5, 5, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0030","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"448dff755c54a9367cf25bdabab4e754f721650039aa4f104b6ea2d13b0ca1df","family_id":"qf-mod42-catalan-closed-form","seed":202611115030,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":4}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=4≥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":"(4, 70, 5, 0, 14, 14, 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 (4, 70, 5, 0, 14, 14, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0031","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ef7915623279fc1d9d1f5ad594c9925164b5416316b1b11ec8263f440b15f7b","family_id":"qf-mod42-catalan-closed-form","seed":202611115031,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":5}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=5≥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":"(5, 252, 6, 0, 42, 42, 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 (5, 252, 6, 0, 42, 42, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0032","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e089d89eac6d772151cc6c789ef70e78945871d30d0828d083a1350f0e65ab5","family_id":"qf-mod42-catalan-closed-form","seed":202611115032,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":6}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=6≥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":"(6, 924, 7, 0, 132, 132, 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 (6, 924, 7, 0, 132, 132, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0033","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be79abeb7245c084ceac91775d9afed16ead3f67d52dbbef37ba3a618e116774","family_id":"qf-mod42-catalan-closed-form","seed":202611115033,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":7}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=7≥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":"(7, 3432, 8, 0, 429, 429, 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 (7, 3432, 8, 0, 429, 429, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0034","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"620e15ab0d95d24685232d50c928eb992089b6001e56a0a30e4849776cc86627","family_id":"qf-mod42-catalan-closed-form","seed":202611115034,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":8}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=8≥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":"(8, 12870, 9, 0, 1430, 1430, 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 (8, 12870, 9, 0, 1430, 1430, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0035","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"618d94f83f4ead825ae2d2e70b49ce580ce0de3c9fc86a9a99bbc81e26fd4cad","family_id":"qf-mod42-catalan-closed-form","seed":202611115035,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":9}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=9≥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":"(9, 48620, 10, 0, 4862, 4862, 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 (9, 48620, 10, 0, 4862, 4862, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0036","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c1ab14951329dbe2337945b1dd21f9f1ccea5fa3cffbddc7e9a3e1cbb381fa4","family_id":"qf-mod42-catalan-closed-form","seed":202611115036,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":10}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=10≥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":"(10, 184756, 11, 0, 16796, 16796, 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 (10, 184756, 11, 0, 16796, 16796, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0037","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4354d245e0b06c57edd145e5fd5afa098a16a6cc34d5f74171fe726ff56c185","family_id":"qf-mod42-catalan-closed-form","seed":202611115037,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":11}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=11≥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":"(11, 705432, 12, 0, 58786, 58786, 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 (11, 705432, 12, 0, 58786, 58786, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0038","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3807250e5dc1c9a2d24507eb35a7c5b8bf8b9be1ff774c4325f1bf06c5378c0","family_id":"qf-mod42-catalan-closed-form","seed":202611115038,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":12}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=12≥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":"(12, 2704156, 13, 0, 208012, 208012, 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 (12, 2704156, 13, 0, 208012, 208012, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0039","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f62905cf3c8ca98857660e5bc1b7c5678f5ae263eb1dba930c58c05965128e1b","family_id":"qf-mod42-catalan-closed-form","seed":202611115039,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":13}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=13≥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":"(13, 10400600, 14, 0, 742900, 742900, 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 (13, 10400600, 14, 0, 742900, 742900, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0040","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21a5046d79a895ea2dcdf401d67b25b244f7ecba9dcb3714d7069fef7c4915d5","family_id":"qf-mod42-catalan-closed-form","seed":202611115040,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":14}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=14≥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":"(14, 40116600, 15, 0, 2674440, 2674440, 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 (14, 40116600, 15, 0, 2674440, 2674440, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0041","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d65f31dce22fef179562b880eba5266412289317ba45ff871a4452c059f9c540","family_id":"qf-mod42-catalan-closed-form","seed":202611115041,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":15}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=15≥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":"(15, 155117520, 16, 0, 9694845, 9694845, 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 (15, 155117520, 16, 0, 9694845, 9694845, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0042","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f1c3325b1d40e1c35a3b49b63a3a4cff14ce21171e580f37b375b71c010f0db","family_id":"qf-mod42-catalan-closed-form","seed":202611115042,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":16}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=16≥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":"(16, 601080390, 17, 0, 35357670, 35357670, 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 (16, 601080390, 17, 0, 35357670, 35357670, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0043","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1d861f795b18470703ab108b3c9a265da20c1fafbee14788fa9c39e0ee2d2f6","family_id":"qf-mod42-catalan-closed-form","seed":202611115043,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":17}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=17≥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":"(17, 2333606220, 18, 0, 129644790, 129644790, 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 (17, 2333606220, 18, 0, 129644790, 129644790, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0044","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4129bbbec79858b4fb7ee571a34e12e1880d8ec4495e42ea0c8f028a8c415e26","family_id":"qf-mod42-catalan-closed-form","seed":202611115044,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":18}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=18≥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":"(18, 9075135300, 19, 0, 477638700, 477638700, 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 (18, 9075135300, 19, 0, 477638700, 477638700, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0045","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfc03a864fe406713a8d99c37503fbd65c5869c1fa5abaf1f5813863e13d29c9","family_id":"qf-mod42-catalan-closed-form","seed":202611115045,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":19}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=19≥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":"(19, 35345263800, 20, 0, 1767263190, 1767263190, 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 (19, 35345263800, 20, 0, 1767263190, 1767263190, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0046","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37c583c43bb19f931ebf92aaefe6dddea69aa82fcef42a7d4f3db3e73ab96ebb","family_id":"qf-mod42-catalan-closed-form","seed":202611115046,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":20}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=20≥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":"(20, 137846528820, 21, 0, 6564120420, 6564120420, 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 (20, 137846528820, 21, 0, 6564120420, 6564120420, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0047","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d0c41d69269e32fb1d568a63c6c3075d966402ca4fdafcdc2783635aead3926","family_id":"qf-mod42-catalan-closed-form","seed":202611115047,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":21}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=21≥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":"(21, 538257874440, 22, 0, 24466267020, 24466267020, 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 (21, 538257874440, 22, 0, 24466267020, 24466267020, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0048","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b86e2a6ff96925e0c1bb869ea501904deac6dbc627c972da410021458dc4942","family_id":"qf-mod42-catalan-closed-form","seed":202611115048,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":22}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=22≥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":"(22, 2104098963720, 23, 0, 91482563640, 91482563640, 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 (22, 2104098963720, 23, 0, 91482563640, 91482563640, 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."}}
{"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0050","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71686a487ded3b97e89c42fc9e670449e59c4fe867f67d9321f8a181a03633f8","family_id":"qf-mod42-catalan-closed-form","seed":202611115050,"semantic_template":"qf-mod42-catalan-closed-form/{\"n\":24}"},"fields":{"title":"Check Divisibility before Taking the Catalan Quotient","stem":"Use natural index n=24≥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":"(24, 32247603683100, 25, 0, 1289904147324, 1289904147324, 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 (24, 32247603683100, 25, 0, 1289904147324, 1289904147324, 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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0051","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6de38e28f1fb49bca03ab5c715bc23fcb295f7afae1c2878431c5b139d11caa4","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115051,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":0}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=0≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(0, 1, 1, 1, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (0, 1, 1, 1, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0052","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88eaa89bcd5d6bb60a980fcbb39551a4884bd315c56e3680a6cb9f0ea2615699","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115052,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":1}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=1≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(1, 1, 1, 1, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (1, 1, 1, 1, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0053","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cae3c5cd2efb3e4382b4f9e8bded5eb43a496e5816046fd91c05b581dc78229c","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115053,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":2}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=2≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(2, 2, 2, 2, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (2, 2, 2, 2, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0054","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"801d78d76f0eb72531f7eba1f9bc613242ef8f875f428c1b80751d8298429d09","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115054,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":3}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=3≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(3, 5, 5, 5, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (3, 5, 5, 5, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0055","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e89c8b638e17c77f3cad82e742b0429eab3cc3a1ff7e9e6ef828605a582984a4","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115055,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":4}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=4≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(4, 14, 14, 14, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (4, 14, 14, 14, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0056","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e616b4dee91bfac029f6d237e781e178ac7ff5016cfa154831bfbc00f87a8f2","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115056,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":5}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=5≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(5, 42, 42, 42, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (5, 42, 42, 42, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0057","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7736a9df3112187faead5bbf4c9d74b0e691565c06f11245f841ab229ddd17ae","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115057,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":6}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=6≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(6, 132, 132, 132, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (6, 132, 132, 132, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0058","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"baee5e4c1f6d5d8de8d4c3c22ade829a462a6edfe26afdcfed781d681f64f662","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115058,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":7}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=7≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(7, 429, 429, 429, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (7, 429, 429, 429, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0059","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4e71834d3884b105c147d32fcbb238f9e81fdd6a3b3bec71df2722e20e8711a","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115059,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":8}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=8≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(8, 1430, 1430, 1430, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (8, 1430, 1430, 1430, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0060","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d2c47d222acaba5ac03a0310abe3dbd7b511834ddda1245fe935f82c4305e4c","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115060,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":9}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=9≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(9, 4862, 4862, 4862, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (9, 4862, 4862, 4862, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0061","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64dbfa124e08488adce664b4553a2cb58fdb4baa12cd26d94b0d6667735c3816","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115061,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":10}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=10≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(10, 16796, 16796, 16796, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (10, 16796, 16796, 16796, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0062","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d032fdd82c11d5e568f67753a36b893d4fd63f29e99cfaf91fff08069625385","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115062,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":11}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=11≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(11, 58786, 58786, 58786, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (11, 58786, 58786, 58786, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0063","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bf98817f07ba27c209dff63ba484ae43d6a22583092d26901e060b2180cd9cd","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115063,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":12}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=12≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(12, 208012, 208012, 208012, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (12, 208012, 208012, 208012, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0064","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c71606dc043fa021b3f451e85b4b998eb55f65d17712b859e9279c04efa2ce19","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115064,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":13}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=13≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(13, 742900, 742900, 742900, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (13, 742900, 742900, 742900, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0065","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64449256c7b27fe0ced1d8025ab8ea1e06f0c5b0bb3888d75f1dc950d6a5d4ba","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115065,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":14}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=14≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(14, 2674440, 2674440, 2674440, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (14, 2674440, 2674440, 2674440, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0066","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c930488f60bc64d2a0e2065f844569c8c58d6f0ad1108e312b5e18005e59bd9f","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115066,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":15}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=15≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(15, 9694845, 9694845, 9694845, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (15, 9694845, 9694845, 9694845, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0067","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58d4ba1451040446a333a524b2bc7bc6c1206a89a1364b5670bf023ca4915316","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115067,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":16}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=16≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(16, 35357670, 35357670, 35357670, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (16, 35357670, 35357670, 35357670, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0068","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b07dda5e5d7149981bef53a2140d3fdb53fc3dced0324747e2e9d8a3f7539d3","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115068,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":17}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=17≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(17, 129644790, 129644790, 129644790, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (17, 129644790, 129644790, 129644790, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0069","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8445922d4ac5c1bf9018eb46b86572eef156471d87ab0e897adca5419168cfcf","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115069,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":18}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=18≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(18, 477638700, 477638700, 477638700, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (18, 477638700, 477638700, 477638700, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0070","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1200497042b411799a6ecd3d24be968118f8b67826295a316a7b0a1960cf5146","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115070,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":19}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=19≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(19, 1767263190, 1767263190, 1767263190, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (19, 1767263190, 1767263190, 1767263190, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0071","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe01780c7850b8138b6398d6d2d695d1f3909f61ebd01e9aeadfc68a8ee41909","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115071,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":20}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=20≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(20, 6564120420, 6564120420, 6564120420, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (20, 6564120420, 6564120420, 6564120420, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0072","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69447479a2e23ce3ec90f4024ec3598dc35d8e5b99b25d736373aa146e98cc0b","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115072,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":21}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=21≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(21, 24466267020, 24466267020, 24466267020, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (21, 24466267020, 24466267020, 24466267020, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0073","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"452a9793c7e0e6bbffdea3ae19639e3f6506f09e2133bd846d5aaa365b0e99bc","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115073,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":22}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=22≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(22, 91482563640, 91482563640, 91482563640, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (22, 91482563640, 91482563640, 91482563640, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0074","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68cd55a236b6e1d60a69cbb6bb2f10eb2714ff27934047e4fdb10818b87d97cd","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115074,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":23}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(23, 343059613650, 343059613650, 343059613650, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (23, 343059613650, 343059613650, 343059613650, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0075","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bfc1e981bc6ac34113e21ded8e261e78ffb061287215ee0f5175b3efd428671","family_id":"qf-mod42-catalan-recurrence-closed-cross-check","seed":202611115075,"semantic_template":"qf-mod42-catalan-recurrence-closed-cross-check/{\"n\":24}"},"fields":{"title":"The Recurrence and Both Closed Forms Use the Same Zero-Based Index","stem":"Use natural index n=24≥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, recurrence_value, closed_value, difference_value, all_equal, certificate).","answer":{"kind":"expression","value":"(24, 1289904147324, 1289904147324, 1289904147324, Yes, C_zero_index_convention_preserved)","explanation":"Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (24, 1289904147324, 1289904147324, 1289904147324, Yes, C_zero_index_convention_preserved)."},"skills":["The Recurrence and Both Closed Forms Use the Same Zero-Based Index"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0076","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73a07846ee0c3382724e231346fb55827fcea99f121422a05cfe3155d126a32b","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115076,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-1}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-1. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-1, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-1, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0077","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da03eda4af4c71adcf0c4a77671f69f7592f910c77be1667a926f3c248571b05","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115077,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-2}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-2. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-2, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-2, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0078","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b7a39b801816d22676319e28c24ce84696e45a9c37238e19fce88744a27ac20","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115078,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-3}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-3. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-3, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-3, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0079","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c85ff3797ad2fbfc84a61275eca3f2608b887774de8f978b57f924e6bf329c6d","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115079,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-4}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-4. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-4, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-4, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0080","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdda37d9e3ef729efe65a3e670eb6f12bbe3f767cc01ec234e95c8e71f76d261","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115080,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-5}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-5. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-5, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-5, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0081","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c0a7ad157d402bb5525372b61688de201927ca76566956179220dcd1411add7","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115081,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-6}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-6. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-6, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-6, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0082","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bf054bc2e84af819d46b2eec1552ed2d1a80f76a35b81bcd54992eba4b1f79f","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115082,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-7}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-7. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-7, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-7, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0083","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26f092ce884fd47dc2e7ab6eb982002c677bc32f05240ae76baf5994e1d812a7","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115083,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-8}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-8. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-8, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-8, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0084","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e72335df4b3b5a0aeeacd7b5340e5e1ed6c02e76a55f48521620fc3e3aa155a","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115084,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-9}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-9. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-9, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-9, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0085","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6ef8392cc83b2456d33bcceb892d00da97b5ff27e923f4bd86646e48f2ceed9","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115085,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-10}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-10. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-10, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-10, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0086","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"687a7ddcf1a440f1e172225556f5739b08fa5e43c49ccaafef3698dfe5cf845c","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115086,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-11}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-11. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-11, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-11, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0087","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a320d87b9406f6f6cfff6f65394018ef74e29ea5b440a12bd08b2cf9d01020e","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115087,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-12}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-12. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-12, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-12, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0088","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3beb8b67e9065c31997fe74d0b3d4fedd533300dce065e91dec939a1300b52a0","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115088,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-13}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-13. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-13, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-13, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0089","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4eda80640c7bef738e5d5cee7e8a962b4373d1650a524655720a242f4871a58","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115089,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-14}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-14. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-14, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-14, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0090","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5426ceb9c9a99faa4d2e27a4dc9f8c623bcdacecba51d36274aa13bbec86b119","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115090,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-15}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-15. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-15, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-15, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0091","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38c1c87e529612fea7f680f8c87fbbf6df81c760c7357ce77a7248d5145bc9cf","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115091,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-16}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-16. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-16, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-16, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0092","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5178d115ba560c7d3cb2774f091b1b2ea94ac14575814453211fd2f132ad6bc","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115092,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-17}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-17. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-17, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-17, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0093","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3931ec65b8d8e87e0681689665ea2c4041e75ef81656f793ca1a6bf8f1ff97c","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115093,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-18}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-18. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-18, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-18, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0094","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c5e6ca22935cc8e4b2850d8a7638f82bb6048784f7e8c0fe9c894a44d97a998","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115094,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-19}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-19. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-19, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-19, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0095","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9d5df0760109d46de53db3baa9924a587a958a9a181103bfd56f6366566093e","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115095,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-20}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-20. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-20, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-20, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0096","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbe6ee037b027705be5bd8ea56dff0c45856ec9638fd85e9f8908c5ee32415be","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115096,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-21}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-21. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-21, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-21, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0097","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"813ee061974167407ad29f0f4a7624a7a0b3a3622e38ad08bd1196b1d0fed031","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115097,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-22}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-22. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-22, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-22, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0098","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4f7db91b42fbca3835f67a77c646ebcc420bb34d61e33ed598f87a717715460","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115098,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-23}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-23. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-23, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-23, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0099","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"193324899016478650ce5437114308928558fd0fd02459b7359281052d962361","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115099,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-24}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-24. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-24, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-24, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r05-catalan-numbers-0100","batch_id":"batch-20261014-advance-20261111-r05-catalan-numbers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"904148908237ba77a47745825eb689e9077f639fdf608cc2bdfc5229203fcc1f","family_id":"qf-mod42-catalan-negative-index-boundary","seed":202611115100,"semantic_template":"qf-mod42-catalan-negative-index-boundary/{\"n\":-25}"},"fields":{"title":"Negative Indices Are Outside This Catalan Domain","stem":"Use negative integer n=-25. Give (n, in_domain, value_or_null, boundary).","answer":{"kind":"expression","value":"(-25, No, null, Catalan_domain_is_natural_numbers_including_zero)","explanation":"Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-25, No, null, Catalan_domain_is_natural_numbers_including_zero)."},"skills":["Negative Indices Are Outside This Catalan Domain"],"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."}}