# Question 22715: Check Divisibility before Taking the Catalan Quotient [Return to catalogue page 114](https://ocme.evemisslab.com/en/questions/page/114/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261111-r05-catalan-numbers-foundations/#q-20261014-advance-20261111-r05-catalan-numbers-0040) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0044-22715 · q-20261014-advance-20261111-r05-catalan-numbers-0040 ## Original question and givens 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). - G001 — 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). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-catalan-numbers](https://ocme.evemisslab.com/en/mko/mko-catalan-numbers/) ## Step-by-step derivation S001 · impliesBefore source operation=catalan-closed-form;original snapshot and mathematics retained After explicit ordered source output fields=[n, central_binomial, denominator, remainder, quotient, binomial_difference, equal] Conditions: none Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL requested natural index n=14 After initialize C_0=1;recurrence C_(j+1)=sum_i=0..j C_i*C_(j-i),including ALL splits Conditions: none There is one empty object,so C0=1.Include every split i0..j for C_(j+1),without a C1-based index shift. Step identifiers and dependenciesRule: catalan_base_case_empty_object_C0_one · Uses: G001, S001 S003 · calculationBefore Catalan complete convolution for C_1 := sum(1) After Catalan complete convolution for C_1 = 1 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore ORIGINAL index j=0;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 1, 1]] After C_1=1;ALL earlier values used from the established prefix=[1] Conditions: none List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats. Step identifiers and dependenciesRule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S003 S005 · calculationBefore Catalan complete convolution for C_2 := sum(1, 1) After Catalan complete convolution for C_2 = 2 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore ORIGINAL index j=1;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 1, 1], [1, 1, 1, 1]] After C_2=2;ALL earlier values used from the established prefix=[1, 1] Conditions: none List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats. Step identifiers and dependenciesRule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S005 S007 · calculationBefore Catalan complete convolution for C_3 := sum(2, 1, 2) After Catalan complete convolution for C_3 = 5 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore ORIGINAL index j=2;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 2, 2], [1, 1, 1, 1], [2, 2, 1, 2]] After C_3=5;ALL earlier values used from the established prefix=[1, 1, 2] Conditions: none List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats. Step identifiers and dependenciesRule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S007 S009 · calculationBefore Catalan complete convolution for C_4 := sum(5, 2, 2, 5) After Catalan complete convolution for C_4 = 14 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore ORIGINAL index j=3;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 5, 5], [1, 1, 2, 2], [2, 2, 1, 2], [3, 5, 1, 5]] After C_4=14;ALL earlier values used from the established prefix=[1, 1, 2, 5] Conditions: none List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats. Step identifiers and dependenciesRule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S009 S011 · calculationBefore Catalan complete convolution for C_5 := sum(14, 5, 4, 5, 14) After Catalan complete convolution for C_5 = 42 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore ORIGINAL index j=4;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 14, 14], [1, 1, 5, 5], [2, 2, 2, 4], [3, 5, 1, 5], [4, 14, 1, 14]] After C_5=42;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14] Conditions: none List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats. Step identifiers and dependenciesRule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S011 S013 · calculationBefore Catalan complete convolution for C_6 := sum(42, 14, 10, 10, 14, 42) After Catalan complete convolution for C_6 = 132 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore ORIGINAL index j=5;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 42, 42], [1, 1, 14, 14], [2, 2, 5, 10], [3, 5, 2, 10], [4, 14, 1, 14], [5, 42, 1, 42]] After C_6=132;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42] Conditions: none List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats. Step identifiers and dependenciesRule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S013 S015 · calculationBefore Catalan complete convolution for C_7 := sum(132, 42, 28, 25, 28, 42, 132) After Catalan complete convolution for C_7 = 429 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore ORIGINAL index j=6;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 132, 132], [1, 1, 42, 42], [2, 2, 14, 28], [3, 5, 5, 25], [4, 14, 2, 28], [5, 42, 1, 42], [6, 132, 1, 132]] After C_7=429;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132] Conditions: none List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats. Step identifiers and dependenciesRule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S015 S017 · calculationBefore Catalan complete convolution for C_8 := sum(429, 132, 84, 70, 70, 84, 132, 429) After Catalan complete convolution for C_8 = 1430 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore ORIGINAL index j=7;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 429, 429], [1, 1, 132, 132], [2, 2, 42, 84], [3, 5, 14, 70], [4, 14, 5, 70], [5, 42, 2, 84], [6, 132, 1, 132], [7, 429, 1, 429]] After C_8=1430;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429] Conditions: none List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats. Step identifiers and dependenciesRule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S017 S019 · calculationBefore Catalan complete convolution for C_9 := sum(1430, 429, 264, 210, 196, 210, 264, 429, 1430) After Catalan complete convolution for C_9 = 4862 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · impliesBefore ORIGINAL index j=8;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 1430, 1430], [1, 1, 429, 429], [2, 2, 132, 264], [3, 5, 42, 210], [4, 14, 14, 196], [5, 42, 5, 210], [6, 132, 2, 264], [7, 429, 1, 429], [8, 1430, 1, 1430]] After C_9=4862;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430] Conditions: none List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats. Step identifiers and dependenciesRule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S019 S021 · calculationBefore Catalan complete convolution for C_10 := sum(4862, 1430, 858, 660, 588, 588, 660, 858, 1430, 4862) After Catalan complete convolution for C_10 = 16796 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · impliesBefore ORIGINAL index j=9;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 4862, 4862], [1, 1, 1430, 1430], [2, 2, 429, 858], [3, 5, 132, 660], [4, 14, 42, 588], [5, 42, 14, 588], [6, 132, 5, 660], [7, 429, 2, 858], [8, 1430, 1, 1430], [9, 4862, 1, 4862]] After C_10=16796;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862] Conditions: none List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats. Step identifiers and dependenciesRule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S021 S023 · calculationBefore Catalan complete convolution for C_11 := sum(16796, 4862, 2860, 2145, 1848, 1764, 1848, 2145, 2860, 4862, 16796) After Catalan complete convolution for C_11 = 58786 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · impliesBefore ORIGINAL index j=10;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 16796, 16796], [1, 1, 4862, 4862], [2, 2, 1430, 2860], [3, 5, 429, 2145], [4, 14, 132, 1848], [5, 42, 42, 1764], [6, 132, 14, 1848], [7, 429, 5, 2145], [8, 1430, 2, 2860], [9, 4862, 1, 4862], [10, 16796, 1, 16796]] After C_11=58786;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796] Conditions: none List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats. Step identifiers and dependenciesRule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S023 S025 · calculationBefore Catalan complete convolution for C_12 := sum(58786, 16796, 9724, 7150, 6006, 5544, 5544, 6006, 7150, 9724, 16796, 58786) After Catalan complete convolution for C_12 = 208012 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · impliesBefore ORIGINAL index j=11;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 58786, 58786], [1, 1, 16796, 16796], [2, 2, 4862, 9724], [3, 5, 1430, 7150], [4, 14, 429, 6006], [5, 42, 132, 5544], [6, 132, 42, 5544], [7, 429, 14, 6006], [8, 1430, 5, 7150], [9, 4862, 2, 9724], [10, 16796, 1, 16796], [11, 58786, 1, 58786]] After C_12=208012;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786] Conditions: none List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats. Step identifiers and dependenciesRule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S025 S027 · calculationBefore Catalan complete convolution for C_13 := sum(208012, 58786, 33592, 24310, 20020, 18018, 17424, 18018, 20020, 24310, 33592, 58786, 208012) After Catalan complete convolution for C_13 = 742900 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · impliesBefore ORIGINAL index j=12;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 208012, 208012], [1, 1, 58786, 58786], [2, 2, 16796, 33592], [3, 5, 4862, 24310], [4, 14, 1430, 20020], [5, 42, 429, 18018], [6, 132, 132, 17424], [7, 429, 42, 18018], [8, 1430, 14, 20020], [9, 4862, 5, 24310], [10, 16796, 2, 33592], [11, 58786, 1, 58786], [12, 208012, 1, 208012]] After C_13=742900;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012] Conditions: none List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats. Step identifiers and dependenciesRule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S027 S029 · calculationBefore Catalan complete convolution for C_14 := sum(742900, 208012, 117572, 83980, 68068, 60060, 56628, 56628, 60060, 68068, 83980, 117572, 208012, 742900) After Catalan complete convolution for C_14 = 2674440 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · impliesBefore ORIGINAL index j=13;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 742900, 742900], [1, 1, 208012, 208012], [2, 2, 58786, 117572], [3, 5, 16796, 83980], [4, 14, 4862, 68068], [5, 42, 1430, 60060], [6, 132, 429, 56628], [7, 429, 132, 56628], [8, 1430, 42, 60060], [9, 4862, 14, 68068], [10, 16796, 5, 83980], [11, 58786, 2, 117572], [12, 208012, 1, 208012], [13, 742900, 1, 742900]] After C_14=2674440;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900] Conditions: none List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats. Step identifiers and dependenciesRule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S029 S031 · impliesBefore Catalan central:ORIGINAL binomial N=28,K=14 After use k=min(K,N-K)=14;initialize exact product1;empty product for k0 is1 Conditions: none Use binomial symmetry to shorten the product while retaining original N/K roles.Multiply before each exact division,without floating-point factorial approximations. Step identifiers and dependenciesRule: catalan_exact_binomial_product_initial_and_symmetry · Uses: G001, S030 S032 · calculationBefore Catalan central binomial product at i=1 := mul(1, 15) After Catalan central binomial product at i=1 = 15 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore Catalan central exact binomial division by i=1 := div(15, 1) After Catalan central exact binomial division by i=1 = 15 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore Catalan central binomial product at i=2 := mul(15, 16) After Catalan central binomial product at i=2 = 240 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore Catalan central exact binomial division by i=2 := div(240, 2) After Catalan central exact binomial division by i=2 = 120 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore Catalan central binomial product at i=3 := mul(120, 17) After Catalan central binomial product at i=3 = 2040 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore Catalan central exact binomial division by i=3 := div(2040, 3) After Catalan central exact binomial division by i=3 = 680 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore Catalan central binomial product at i=4 := mul(680, 18) After Catalan central binomial product at i=4 = 12240 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore Catalan central exact binomial division by i=4 := div(12240, 4) After Catalan central exact binomial division by i=4 = 3060 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore Catalan central binomial product at i=5 := mul(3060, 19) After Catalan central binomial product at i=5 = 58140 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore Catalan central exact binomial division by i=5 := div(58140, 5) After Catalan central exact binomial division by i=5 = 11628 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore Catalan central binomial product at i=6 := mul(11628, 20) After Catalan central binomial product at i=6 = 232560 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore Catalan central exact binomial division by i=6 := div(232560, 6) After Catalan central exact binomial division by i=6 = 38760 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore Catalan central binomial product at i=7 := mul(38760, 21) After Catalan central binomial product at i=7 = 813960 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore Catalan central exact binomial division by i=7 := div(813960, 7) After Catalan central exact binomial division by i=7 = 116280 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore Catalan central binomial product at i=8 := mul(116280, 22) After Catalan central binomial product at i=8 = 2558160 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore Catalan central exact binomial division by i=8 := div(2558160, 8) After Catalan central exact binomial division by i=8 = 319770 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore Catalan central binomial product at i=9 := mul(319770, 23) After Catalan central binomial product at i=9 = 7354710 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · calculationBefore Catalan central exact binomial division by i=9 := div(7354710, 9) After Catalan central exact binomial division by i=9 = 817190 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · calculationBefore Catalan central binomial product at i=10 := mul(817190, 24) After Catalan central binomial product at i=10 = 19612560 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · calculationBefore Catalan central exact binomial division by i=10 := div(19612560, 10) After Catalan central exact binomial division by i=10 = 1961256 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · calculationBefore Catalan central binomial product at i=11 := mul(1961256, 25) After Catalan central binomial product at i=11 = 49031400 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S051 S053 · calculationBefore Catalan central exact binomial division by i=11 := div(49031400, 11) After Catalan central exact binomial division by i=11 = 4457400 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · calculationBefore Catalan central binomial product at i=12 := mul(4457400, 26) After Catalan central binomial product at i=12 = 115892400 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S053 S055 · calculationBefore Catalan central exact binomial division by i=12 := div(115892400, 12) After Catalan central exact binomial division by i=12 = 9657700 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054 S056 · calculationBefore Catalan central binomial product at i=13 := mul(9657700, 27) After Catalan central binomial product at i=13 = 260757900 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055 S057 · calculationBefore Catalan central exact binomial division by i=13 := div(260757900, 13) After Catalan central exact binomial division by i=13 = 20058300 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056 S058 · calculationBefore Catalan central binomial product at i=14 := mul(20058300, 28) After Catalan central binomial product at i=14 = 561632400 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057 S059 · calculationBefore Catalan central exact binomial division by i=14 := div(561632400, 14) After Catalan central exact binomial division by i=14 = 40116600 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058 S060 · calculationBefore Catalan ORIGINAL denominator n plus1 := sum(14, 1) After Catalan ORIGINAL denominator n plus1 = 15 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · impliesBefore ORIGINAL n=14;central binomial=40116600;positive denominator n+1=15 After Euclidean quotient=2674440,remainder=0;check remainder0 BEFORE treating quotient as Catalan integer Conditions: none Check remainder0 before taking an exact integer quotient.Do not hide nondivisibility by truncation;retain denominator1 and central1 at n0. Step identifiers and dependenciesRule: catalan_remainder_BEFORE_exact_closed_division · Uses: G001, S060 S062 · calculationBefore Catalan exact closed quotient after remainder0 := div(40116600, 15) After Catalan exact closed quotient after remainder0 = 2674440 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S061 S063 · impliesBefore Catalan adjacent:ORIGINAL binomial N=28,K=15 After use k=min(K,N-K)=13;initialize exact product1;empty product for k0 is1 Conditions: none Use binomial symmetry to shorten the product while retaining original N/K roles.Multiply before each exact division,without floating-point factorial approximations. Step identifiers and dependenciesRule: catalan_exact_binomial_product_initial_and_symmetry · Uses: G001, S062 S064 · calculationBefore Catalan adjacent binomial product at i=1 := mul(1, 16) After Catalan adjacent binomial product at i=1 = 16 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S063 S065 · calculationBefore Catalan adjacent exact binomial division by i=1 := div(16, 1) After Catalan adjacent exact binomial division by i=1 = 16 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S064 S066 · calculationBefore Catalan adjacent binomial product at i=2 := mul(16, 17) After Catalan adjacent binomial product at i=2 = 272 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S065 S067 · calculationBefore Catalan adjacent exact binomial division by i=2 := div(272, 2) After Catalan adjacent exact binomial division by i=2 = 136 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066 S068 · calculationBefore Catalan adjacent binomial product at i=3 := mul(136, 18) After Catalan adjacent binomial product at i=3 = 2448 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · calculationBefore Catalan adjacent exact binomial division by i=3 := div(2448, 3) After Catalan adjacent exact binomial division by i=3 = 816 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S068 S070 · calculationBefore Catalan adjacent binomial product at i=4 := mul(816, 19) After Catalan adjacent binomial product at i=4 = 15504 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069 S071 · calculationBefore Catalan adjacent exact binomial division by i=4 := div(15504, 4) After Catalan adjacent exact binomial division by i=4 = 3876 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S070 S072 · calculationBefore Catalan adjacent binomial product at i=5 := mul(3876, 20) After Catalan adjacent binomial product at i=5 = 77520 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · calculationBefore Catalan adjacent exact binomial division by i=5 := div(77520, 5) After Catalan adjacent exact binomial division by i=5 = 15504 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S072 S074 · calculationBefore Catalan adjacent binomial product at i=6 := mul(15504, 21) After Catalan adjacent binomial product at i=6 = 325584 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S073 S075 · calculationBefore Catalan adjacent exact binomial division by i=6 := div(325584, 6) After Catalan adjacent exact binomial division by i=6 = 54264 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S074 S076 · calculationBefore Catalan adjacent binomial product at i=7 := mul(54264, 22) After Catalan adjacent binomial product at i=7 = 1193808 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S075 S077 · calculationBefore Catalan adjacent exact binomial division by i=7 := div(1193808, 7) After Catalan adjacent exact binomial division by i=7 = 170544 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S076 S078 · calculationBefore Catalan adjacent binomial product at i=8 := mul(170544, 23) After Catalan adjacent binomial product at i=8 = 3922512 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S077 S079 · calculationBefore Catalan adjacent exact binomial division by i=8 := div(3922512, 8) After Catalan adjacent exact binomial division by i=8 = 490314 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S078 S080 · calculationBefore Catalan adjacent binomial product at i=9 := mul(490314, 24) After Catalan adjacent binomial product at i=9 = 11767536 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S079 S081 · calculationBefore Catalan adjacent exact binomial division by i=9 := div(11767536, 9) After Catalan adjacent exact binomial division by i=9 = 1307504 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S080 S082 · calculationBefore Catalan adjacent binomial product at i=10 := mul(1307504, 25) After Catalan adjacent binomial product at i=10 = 32687600 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S081 S083 · calculationBefore Catalan adjacent exact binomial division by i=10 := div(32687600, 10) After Catalan adjacent exact binomial division by i=10 = 3268760 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S082 S084 · calculationBefore Catalan adjacent binomial product at i=11 := mul(3268760, 26) After Catalan adjacent binomial product at i=11 = 84987760 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S083 S085 · calculationBefore Catalan adjacent exact binomial division by i=11 := div(84987760, 11) After Catalan adjacent exact binomial division by i=11 = 7726160 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S084 S086 · calculationBefore Catalan adjacent binomial product at i=12 := mul(7726160, 27) After Catalan adjacent binomial product at i=12 = 208606320 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S085 S087 · calculationBefore Catalan adjacent exact binomial division by i=12 := div(208606320, 12) After Catalan adjacent exact binomial division by i=12 = 17383860 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S086 S088 · calculationBefore Catalan adjacent binomial product at i=13 := mul(17383860, 28) After Catalan adjacent binomial product at i=13 = 486748080 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S087 S089 · calculationBefore Catalan adjacent exact binomial division by i=13 := div(486748080, 13) After Catalan adjacent exact binomial division by i=13 = 37442160 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S088 S090 · calculationBefore Catalan central minus adjacent binomial := sub(40116600, 37442160) After Catalan central minus adjacent binomial = 2674440 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S089 S091 · impliesBefore ORIGINAL n=14;recurrence value=2674440;central=40116600,adjacent=37442160,denominator=15 After closed quotient=2674440;binomial difference=2674440;ALL THREE agree;balanced paths counted by choose(2n,n),bad paths first-crossing reflection counted by choose(2n,n+1) Conditions: none Reflect a bad balanced path through its first crossing of−1 to obtain a path with n+1 ups and n−1 downs,so good paths are the binomial difference.First-return decomposition U P D Q gives the Catalan convolution with empty-path base1,uniquely determining the same count.The adjacent-binomial ratio yields central/(n+1).This is a general argument,not extrapolation from finitely many equalities. Step identifiers and dependenciesRule: catalan_general_Dyck_reflection_and_first_return_recurrence · Uses: G001, S090 ## Reverse checks - S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S090 · premise_and_result_audit — Recompute Catalan central minus adjacent binomial with exact rational arithmetic, not rounded decimals. - S089 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=13 with exact rational arithmetic, not rounded decimals. - S088 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=13 with exact rational arithmetic, not rounded decimals. - S087 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=12 with exact rational arithmetic, not rounded decimals. - S086 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=12 with exact rational arithmetic, not rounded decimals. - S085 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=11 with exact rational arithmetic, not rounded decimals. - S084 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=11 with exact rational arithmetic, not rounded decimals. - S083 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=10 with exact rational arithmetic, not rounded decimals. - S082 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=10 with exact rational arithmetic, not rounded decimals. - S081 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=9 with exact rational arithmetic, not rounded decimals. - S080 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=9 with exact rational arithmetic, not rounded decimals. - S079 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=8 with exact rational arithmetic, not rounded decimals. - S078 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=8 with exact rational arithmetic, not rounded decimals. - S077 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=7 with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=7 with exact rational arithmetic, not rounded decimals. - S075 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=6 with exact rational arithmetic, not rounded decimals. - S074 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=6 with exact rational arithmetic, not rounded decimals. - S073 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=5 with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=5 with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=4 with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=4 with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=3 with exact rational arithmetic, not rounded decimals. - S068 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=3 with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=2 with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=2 with exact rational arithmetic, not rounded decimals. - S065 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=1 with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=1 with exact rational arithmetic, not rounded decimals. - S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S062 · premise_and_result_audit — Recompute Catalan exact closed quotient after remainder0 with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S060 · premise_and_result_audit — Recompute Catalan ORIGINAL denominator n plus1 with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=14 with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Recompute Catalan central binomial product at i=14 with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=13 with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute Catalan central binomial product at i=13 with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=12 with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute Catalan central binomial product at i=12 with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=11 with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute Catalan central binomial product at i=11 with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=10 with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute Catalan central binomial product at i=10 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=9 with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute Catalan central binomial product at i=9 with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=8 with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute Catalan central binomial product at i=8 with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=7 with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute Catalan central binomial product at i=7 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=6 with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute Catalan central binomial product at i=6 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=5 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute Catalan central binomial product at i=5 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=4 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute Catalan central binomial product at i=4 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=3 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute Catalan central binomial product at i=3 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=2 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute Catalan central binomial product at i=2 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=1 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Catalan central binomial product at i=1 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S029 · premise_and_result_audit — Recompute Catalan complete convolution for C_14 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Recompute Catalan complete convolution for C_13 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Recompute Catalan complete convolution for C_12 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Recompute Catalan complete convolution for C_11 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Recompute Catalan complete convolution for C_10 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Recompute Catalan complete convolution for C_9 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Recompute Catalan complete convolution for C_8 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Recompute Catalan complete convolution for C_7 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Recompute Catalan complete convolution for C_6 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute Catalan complete convolution for C_5 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute Catalan complete convolution for C_4 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute Catalan complete convolution for C_3 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute Catalan complete convolution for C_2 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Recompute Catalan complete convolution for C_1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (14, 40116600, 15, 0, 2674440, 2674440, Yes) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 21a5046d79a895ea2dcdf401d67b25b244f7ecba9dcb3714d7069fef7c4915d5 - candidate record SHA-256: 399f59af8a9230583cbeea764a06044570e4277df76b7d2254f214c25b1b76aa - semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81 - question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0040.json)