# Question 29214: Exhaustive Associativity over All Ordered Triples [Return to catalogue page 147](https://ocme.evemisslab.com/en/questions/page/147/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261022-r05-finite-group-foundations/#q-20261013-advance-20261022-r05-finite-group-0039) 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. R0057-29214 · q-20261013-advance-20261022-r05-finite-group-0039 ## Original question and givens The complete finite Cayley table=[[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements. - G001 — The complete finite Cayley table=[[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements. ## 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-finite-group](https://ocme.evemisslab.com/en/mko/mko-finite-group/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-str22-group-associativity-audit, group-associativity-audit] After derived = [[n, table, bad_triples, associative]] Conditions: none Use this bank exact ordered tuple and hyphenated operation.Rational outputs reduce with positive denominators and OMIT/1 for integers,unlike the preceding legacy bank.Do not change original bilingual snapshots or meaning. Step identifiers and dependenciesRule: STR22_exact_interface_and_source_ORDERED_tuple_roles · Uses: G001 S002 · impliesBefore source roles = [5, 0, [4, 3, 1, 0, 2]] After derived = [[Yes, Yes, Yes, Yes, Yes]] Conditions: none Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms. Step identifiers and dependenciesRule: Group_EVERY_original_row_closure · Uses: G001, S001 S003 · impliesBefore source roles = [5, 1, [3, 2, 4, 1, 0]] After derived = [[Yes, Yes, Yes, Yes, Yes]] Conditions: none Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms. Step identifiers and dependenciesRule: Group_EVERY_original_row_closure · Uses: G001, S002 S004 · impliesBefore source roles = [5, 2, [1, 4, 0, 2, 3]] After derived = [[Yes, Yes, Yes, Yes, Yes]] Conditions: none Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms. Step identifiers and dependenciesRule: Group_EVERY_original_row_closure · Uses: G001, S003 S005 · impliesBefore source roles = [5, 3, [0, 1, 2, 3, 4]] After derived = [[Yes, Yes, Yes, Yes, Yes]] Conditions: none Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms. Step identifiers and dependenciesRule: Group_EVERY_original_row_closure · Uses: G001, S004 S006 · impliesBefore source roles = [5, 4, [2, 0, 3, 4, 1]] After derived = [[Yes, Yes, Yes, Yes, Yes]] Conditions: none Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms. Step identifiers and dependenciesRule: Group_EVERY_original_row_closure · Uses: G001, S005 S007 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 0, 0] After derived = [[2, 0, 3, 4, 1], [2, 0, 3, 4, 1], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S006 S008 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 0, 1] After derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S007 S009 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 0, 2] After derived = [[3, 2, 4, 1, 0], [3, 2, 4, 1, 0], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S008 S010 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 0, 3] After derived = [[4, 3, 1, 0, 2], [4, 3, 1, 0, 2], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S009 S011 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 0, 4] After derived = [[1, 4, 0, 2, 3], [1, 4, 0, 2, 3], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S010 S012 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 1, 0] After derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S011 S013 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 1, 1] After derived = [[1, 4, 0, 2, 3], [1, 4, 0, 2, 3], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S012 S014 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 1, 2] After derived = [[2, 0, 3, 4, 1], [2, 0, 3, 4, 1], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S013 S015 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 1, 3] After derived = [[3, 2, 4, 1, 0], [3, 2, 4, 1, 0], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S014 S016 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 1, 4] After derived = [[4, 3, 1, 0, 2], [4, 3, 1, 0, 2], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S015 S017 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 2, 0] After derived = [[3, 2, 4, 1, 0], [3, 2, 4, 1, 0], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S016 S018 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 2, 1] After derived = [[2, 0, 3, 4, 1], [2, 0, 3, 4, 1], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S017 S019 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 2, 2] After derived = [[4, 3, 1, 0, 2], [4, 3, 1, 0, 2], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S018 S020 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 2, 3] After derived = [[1, 4, 0, 2, 3], [1, 4, 0, 2, 3], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S019 S021 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 2, 4] After derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S020 S022 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 3, 0] After derived = [[4, 3, 1, 0, 2], [4, 3, 1, 0, 2], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S021 S023 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 3, 1] After derived = [[3, 2, 4, 1, 0], [3, 2, 4, 1, 0], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S022 S024 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 3, 2] After derived = [[1, 4, 0, 2, 3], [1, 4, 0, 2, 3], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S023 S025 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 3, 3] After derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S024 S026 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 3, 4] After derived = [[2, 0, 3, 4, 1], [2, 0, 3, 4, 1], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S025 S027 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 4, 0] After derived = [[1, 4, 0, 2, 3], [1, 4, 0, 2, 3], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S026 S028 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 4, 1] After derived = [[4, 3, 1, 0, 2], [4, 3, 1, 0, 2], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S027 S029 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 4, 2] After derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S028 S030 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 4, 3] After derived = [[2, 0, 3, 4, 1], [2, 0, 3, 4, 1], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S029 S031 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], 4, 4] After derived = [[3, 2, 4, 1, 0], [3, 2, 4, 1, 0], []] Conditions: none For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample. Step identifiers and dependenciesRule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S030 S032 · impliesBefore source roles = [5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]]] After derived = [[], Yes] Conditions: none The(a,b) loops and ALL c cover n³ ordered triples.Retain ALL bad triples in nested increasing order;only an empty list certifies associativity of THIS complete finite table,not an unknown structure. Step identifiers and dependenciesRule: Group_ALL_bad_triples_nested_source_order · Uses: G001, S031 ## Reverse checks - S032 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S031 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S030 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S029 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S028 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S027 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S026 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S025 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S024 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S023 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S022 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S021 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S020 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S019 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S018 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S017 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S016 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S015 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S014 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S013 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S012 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S011 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S010 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S009 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S008 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S007 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S006 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S005 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S004 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S003 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S002 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S001 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. ## Result (5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], [], 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: 123e5beee4fecedcdc4fc4772aa485bdd2b63765af712b818d22778010652188 - candidate record SHA-256: c89ffe5ca83dd02ed5c4b28e154aeab89d278d1b9d350f91b005d64cee775431 - semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6 - question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261022-r05-finite-group-0039.json)