# Question 21156: The Complete Orbits Partition the Whole Point Carrier [Return to catalogue page 106](https://ocme.evemisslab.com/en/questions/page/106/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261108-r01-finite-group-orbit-foundations/#q-20261014-advance-20261108-r01-finite-group-orbit-0031) 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. R0041-21156 · q-20261014-advance-20261108-r01-finite-group-orbit-0031 ## Original question and givens Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate). - G001 — Use complete group table=[[0, 1], [1, 0]], declared identity=0, point carrier Fin(3)={0,…,2} and total action_table=[[0, 1, 2], [1, 0, 2]]. Row g/column h gives g*h, and row g/column x of the action table gives g*x. The complete group and both action axioms must qualify before computing orbits; every group element is used. Give (orbits, covered_points, pairwise_disjoint, partition_certificate). ## 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-orbit](https://ocme.evemisslab.com/en/mko/mko-finite-group-orbit/) ## Step-by-step derivation S001 · impliesBefore source operation=orbit-orbit-partition;original snapshot and mathematics retained After explicit ordered source output fields=[orbits, covered_points, pairwise_disjoint, partition_certificate] Conditions: none Retain the historical interface marker while listing ordered output fields without JavaScript.Fields and tags are not extra mathematical evidence;fixed No markers retain their claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL complete group-label carrier Fin(2);declared identity label=0;Cayley table row g,column h gives g*h=[[0, 1], [1, 0]] After ORIGINAL complete point carrier Fin(3);TOTAL action table row g,column x gives g*x=[[0, 1, 2], [1, 0, 2]];all entries in range,so structural closure/totality checked;GROUP AXIOMS not assumed from shape Conditions: none The two tables have different column roles:group label h versus action point x.Complete in-range tables establish closure/totality only.A candidate group may fail its axioms;its table name does not certify them. Step identifiers and dependenciesRule: groupaction_two_table_roles_complete_candidate_not_group_assumption · Uses: G001, S001 S003 · impliesBefore ORIGINAL closed Cayley table=[[0, 1], [1, 0]];ACTUAL declared group identity=0 After ALL [g,e*g,g*e,both_equal_g] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes]];two-sided group identity=Yes Conditions: none Both e*g and g*e in the group table must equal g.This is not certified by an identity-action row on points. Step identifiers and dependenciesRule: groupaction_group_identity_both_sides_not_action_identity · Uses: G001, S002 S004 · impliesBefore ORIGINAL group pair(g,h)=[0, 0];check ALL k in Fin(2) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S003 S005 · impliesBefore ORIGINAL group pair(g,h)=[0, 1];check ALL k in Fin(2) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S004 S006 · impliesBefore ORIGINAL group pair(g,h)=[1, 0];check ALL k in Fin(2) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 0, 0, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S005 S007 · impliesBefore ORIGINAL group pair(g,h)=[1, 1];check ALL k in Fin(2) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes]] Conditions: none Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition. Step identifiers and dependenciesRule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S006 S008 · impliesBefore ORIGINAL group element g=0;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 0, 0, Yes], [1, 1, 1, No]] After first actual two-sided inverse witness=0;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S007 S009 · impliesBefore ORIGINAL group element g=1;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 1, 1, No], [1, 0, 0, Yes]] After first actual two-sided inverse witness=1;exists=Yes Conditions: none For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels. Step identifiers and dependenciesRule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S008 S010 · impliesBefore structurally closed COMPLETE group candidate;two-sided identity=Yes;ALL-triple associativity=Yes;ALL-element inverses=Yes After group_valid=Yes is the CONJUNCTION;False remains a reported candidate result,not a silent repair Conditions: none Group qualification is the conjunction of these complete obligations.A permutation-looking action table cannot repair a bad group table;do not replace the identity or modify the table. Step identifiers and dependenciesRule: groupaction_group_axiom_conjunction_candidate_may_fail · Uses: G001, S009 S011 · impliesBefore ACTUAL declared identity label=0;ORIGINAL identity action row=[0, 1, 2];complete point carrier Fin(3) After ALL ordered [point,identity_image,equality] rows=[[0, 0, Yes], [1, 1, Yes], [2, 2, Yes]];identity-action axiom=Yes Conditions: none Check e*x=x at every action point,using the actual declared identity label rather than assuming0.This action axiom is distinct from the two-sided identity law in the group table. Step identifiers and dependenciesRule: groupaction_every_point_identity_action_not_group_identity · Uses: G001, S010 S012 · impliesBefore ORIGINAL ordered group pair(g,h)=[0, 0];actual group product=0;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S011 S013 · impliesBefore ORIGINAL ordered group pair(g,h)=[0, 1];actual group product=1;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, Yes], [1, 0, 0, Yes], [2, 2, 2, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S012 S014 · impliesBefore ORIGINAL ordered group pair(g,h)=[1, 0];actual group product=1;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, Yes], [1, 0, 0, Yes], [2, 2, 2, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S013 S015 · impliesBefore ORIGINAL ordered group pair(g,h)=[1, 1];actual group product=0;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes]];earliest failure retained,not overwritten Conditions: none For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure. Step identifiers and dependenciesRule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S014 S016 · calculationBefore group action ordered group-pair count := mul(2, 2) After group action ordered group-pair count = 4 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore group action complete triple-domain count := mul(4, 3) After group action complete triple-domain count = 12 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore COMPLETE triple domain order^2*point_count=12;this derivation ACTUALLY checked all12 triples After compatibility=Yes;FIRST source-order failure[g,h,x,left,right]=null;null means NO failure anywhere Conditions: none The triple_count is the complete quantified domain size,not a measurement of an early-stopping prototype.This derivation actually checks every point for every g,h pair,so null certifies full compatibility and a first failure retains exact source data/order. Step identifiers and dependenciesRule: groupaction_full_domain_count_first_failure_not_early_stop_measurement · Uses: G001, S017 S019 · impliesBefore ORIGINAL group-row label=0;ORIGINAL point-indexed action row=[0, 1, 2] After sorted image MULTISET=[0, 1, 2];equals full point carrier=Yes;original row NOT reordered Conditions: none Sort only to test whether the image multiset covers the full carrier exactly once;do not reorder the original point-indexed row.Even all-permutation rows alone do not establish an action without the group and both action axioms. Step identifiers and dependenciesRule: groupaction_every_row_permutation_actual_images_not_validity_converse · Uses: G001, S018 S020 · impliesBefore ORIGINAL group-row label=1;ORIGINAL point-indexed action row=[1, 0, 2] After sorted image MULTISET=[0, 1, 2];equals full point carrier=Yes;original row NOT reordered Conditions: none Sort only to test whether the image multiset covers the full carrier exactly once;do not reorder the original point-indexed row.Even all-permutation rows alone do not establish an action without the group and both action axioms. Step identifiers and dependenciesRule: groupaction_every_row_permutation_actual_images_not_validity_converse · Uses: G001, S019 S021 · impliesBefore CURRENT original group/action qualified:group=Yes,identity_action=Yes,compatibility=Yes,permutations=Yes;actual inverses=[0, 1] After only NOW use the action axioms;row permutations ALONE are insufficient;invalid group/action data are rejected,not reported as an orbit Conditions: none Qualify the actual group and both action axioms before using orbits.Permutation rows alone are insufficient.Reused frozen pure helpers receive this question's full tables and trace,not a link to an old derivation. Step identifiers and dependenciesRule: orbit_reject_unqualified_tables_before_orbit_claims · Uses: G001, S020 S022 · impliesBefore CURRENT identity=0;full inverses=[0, 1];action table=[[0, 1, 2], [1, 0, 2]] After reflexive:e*x=x;symmetric:y=g*x gives x=g_inverse*y;transitive:y=g*x,z=h*y gives z=(h*g)*x;therefore related iff the two full orbits are equal,and distinct orbits are disjoint Conditions: none Symmetry uses the actual inverse and action compatibility.Transitivity acts by g and then h,so the product is h*g,without assuming commutativity.If two orbits share a point,symmetry and transitivity give both inclusions;otherwise they are disjoint. Step identifiers and dependenciesRule: orbit_equivalence_reflexive_symmetric_transitive_hg_order · Uses: G001, S021 S023 · calculationBefore orbit point0 distinct image count := sum(1, 1) After orbit point0 distinct image count = 2 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · impliesBefore ORIGINAL point=0;images in ALL group-index order=[0, 1];group order=2 After full orbit sorted and deduplicated=[0, 1];orbit cardinality=2;group enumeration count=2;these counts need NOT agree Conditions: none Take images under every group element,not a single g,generator reachability or a stabilizer.Multiple elements may have the same image;keep group order distinct from orbit size. Step identifiers and dependenciesRule: orbit_all_original_group_rows_not_single_generator · Uses: G001, S023 S025 · calculationBefore orbit point2 distinct image count := sum(1) After orbit point2 distinct image count = 1 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · impliesBefore ORIGINAL point=2;images in ALL group-index order=[2, 2];group order=2 After full orbit sorted and deduplicated=[2];orbit cardinality=1;group enumeration count=2;these counts need NOT agree Conditions: none Take images under every group element,not a single g,generator reachability or a stabilizer.Multiple elements may have the same image;keep group order distinct from orbit size. Step identifiers and dependenciesRule: orbit_all_original_group_rows_not_single_generator · Uses: G001, S025 S027 · impliesBefore ALL carrier points0..2;orbits ordered by first point=[[0, 1], [2]] After sorted flattened coverage=[0, 1, 2];pairwise disjoint=Yes;partition certificate=Yes Conditions: none The identity puts each point in its orbit.Start at the least unseen point each time,using the equal-or-disjoint result;also check full coverage and no duplicates rather than guessing a partition from its count. Step identifiers and dependenciesRule: orbit_partition_all_points_coverage_and_disjointness · Uses: G001, S026 ## Reverse checks - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 orbit point2 distinct image count 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 orbit point0 distinct image count 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 group action complete triple-domain count with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute group action ordered group-pair count with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 ([[0, 1], [2]], [0, 1, 2], Yes, 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: 96cf07991c31d1899cb400eb1a18043c70f4df7116df1758b5f9ab2b2d551452 - candidate record SHA-256: 5926055e9a2bbb5c9fd1a76b8dea8d9c26316a6b458636b3b0d0f030847f1938 - semantic review receipt SHA-256: 8f134403487694af4c70d2e00ec84bab542043ef4b3c54517c04f3c409c0262a - question-first witness SHA-256: ad772845088ed8e99f87f90d4264b05ea97be35775305a02011af5ee35f0f4ce - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261108-r01-finite-group-orbit-0031.json)