# Question 22394: The Same Point Binds Both Factors in Orbit-Stabilizer Counting [Return to catalogue page 112](https://ocme.evemisslab.com/en/questions/page/112/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261110-r01-orbit-stabilizer-theorem-foundations/#q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0019) 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. R0043-22394 · q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0019 ## Original question and givens Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, point carrier Fin(8)={0,…,7}, total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]] and specified point=2. Row g/column h gives g*h, and action row g/column x gives g*x. Certify every group/action axiom, then enumerate all group elements with the same action and point binding. Free, faithful or transitive action is not required. Give (group_order, point, orbit, orbit_size, stabilizer, stabilizer_size, product, equal). - G001 — Use complete group table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, point carrier Fin(8)={0,…,7}, total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]] and specified point=2. Row g/column h gives g*h, and action row g/column x gives g*x. Certify every group/action axiom, then enumerate all group elements with the same action and point binding. Free, faithful or transitive action is not required. Give (group_order, point, orbit, orbit_size, stabilizer, stabilizer_size, product, 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-orbit-stabilizer-theorem](https://ocme.evemisslab.com/en/mko/mko-orbit-stabilizer-theorem/) ## Step-by-step derivation S001 · impliesBefore source operation=orbitstab-counting-identity;original snapshot and mathematics retained After explicit ordered source output fields=[group_order, point, orbit, orbit_size, stabilizer, stabilizer_size, product, equal] Conditions: none Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and 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(5);declared identity label=0;Cayley table row g,column h gives g*h=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] After ORIGINAL complete point carrier Fin(8);TOTAL action table row g,column x gives g*x=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 0, 5, 6, 7], [2, 3, 4, 0, 1, 5, 6, 7], [3, 4, 0, 1, 2, 5, 6, 7], [4, 0, 1, 2, 3, 5, 6, 7]];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, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]];ACTUAL declared group identity=0 After ALL [g,e*g,g*e,both_equal_g] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, 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(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, 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(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 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)=[0, 2];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 0, 0, Yes], [4, 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, S005 S007 · impliesBefore ORIGINAL group pair(g,h)=[0, 3];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, 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 pair(g,h)=[0, 4];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, 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, S007 S009 · impliesBefore ORIGINAL group pair(g,h)=[1, 0];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 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, S008 S010 · impliesBefore ORIGINAL group pair(g,h)=[1, 1];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 0, 0, Yes], [4, 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, S009 S011 · impliesBefore ORIGINAL group pair(g,h)=[1, 2];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, 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, S010 S012 · impliesBefore ORIGINAL group pair(g,h)=[1, 3];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, 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, S011 S013 · impliesBefore ORIGINAL group pair(g,h)=[1, 4];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, 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, S012 S014 · impliesBefore ORIGINAL group pair(g,h)=[2, 0];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 0, 0, Yes], [4, 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, S013 S015 · impliesBefore ORIGINAL group pair(g,h)=[2, 1];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, 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, S014 S016 · impliesBefore ORIGINAL group pair(g,h)=[2, 2];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, 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, S015 S017 · impliesBefore ORIGINAL group pair(g,h)=[2, 3];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, 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, S016 S018 · impliesBefore ORIGINAL group pair(g,h)=[2, 4];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 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, S017 S019 · impliesBefore ORIGINAL group pair(g,h)=[3, 0];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, 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, S018 S020 · impliesBefore ORIGINAL group pair(g,h)=[3, 1];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, 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, S019 S021 · impliesBefore ORIGINAL group pair(g,h)=[3, 2];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, 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, S020 S022 · impliesBefore ORIGINAL group pair(g,h)=[3, 3];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 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, S021 S023 · impliesBefore ORIGINAL group pair(g,h)=[3, 4];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 0, 0, Yes], [4, 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, S022 S024 · impliesBefore ORIGINAL group pair(g,h)=[4, 0];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 4, 4, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, 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, S023 S025 · impliesBefore ORIGINAL group pair(g,h)=[4, 1];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes], [4, 4, 4, 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, S024 S026 · impliesBefore ORIGINAL group pair(g,h)=[4, 2];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 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, S025 S027 · impliesBefore ORIGINAL group pair(g,h)=[4, 3];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 0, 0, Yes], [4, 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, S026 S028 · impliesBefore ORIGINAL group pair(g,h)=[4, 4];check ALL k in Fin(5) After ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, 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, S027 S029 · 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], [2, 2, 2, No], [3, 3, 3, No], [4, 4, 4, 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, S028 S030 · impliesBefore ORIGINAL group element g=1;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 1, 1, No], [1, 2, 2, No], [2, 3, 3, No], [3, 4, 4, No], [4, 0, 0, Yes]] After first actual two-sided inverse witness=4;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, S029 S031 · impliesBefore ORIGINAL group element g=2;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 2, 2, No], [1, 3, 3, No], [2, 4, 4, No], [3, 0, 0, Yes], [4, 1, 1, No]] After first actual two-sided inverse witness=3;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, S030 S032 · impliesBefore ORIGINAL group element g=3;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 3, 3, No], [1, 4, 4, No], [2, 0, 0, Yes], [3, 1, 1, No], [4, 2, 2, No]] After first actual two-sided inverse witness=2;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, S031 S033 · impliesBefore ORIGINAL group element g=4;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 4, 4, No], [1, 0, 0, Yes], [2, 1, 1, No], [3, 2, 2, No], [4, 3, 3, No]] 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, S032 S034 · 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, S033 S035 · impliesBefore ACTUAL declared identity label=0;ORIGINAL identity action row=[0, 1, 2, 3, 4, 5, 6, 7];complete point carrier Fin(8) After ALL ordered [point,identity_image,equality] rows=[[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes], [7, 7, 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, S034 S036 · 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], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S035 S037 · 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, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 0, 0, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S036 S038 · impliesBefore ORIGINAL ordered group pair(g,h)=[0, 2];actual group product=2;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S037 S039 · impliesBefore ORIGINAL ordered group pair(g,h)=[0, 3];actual group product=3;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S038 S040 · impliesBefore ORIGINAL ordered group pair(g,h)=[0, 4];actual group product=4;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 4, 4, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S039 S041 · 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, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 0, 0, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S040 S042 · impliesBefore ORIGINAL ordered group pair(g,h)=[1, 1];actual group product=2;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S041 S043 · impliesBefore ORIGINAL ordered group pair(g,h)=[1, 2];actual group product=3;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S042 S044 · impliesBefore ORIGINAL ordered group pair(g,h)=[1, 3];actual group product=4;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 4, 4, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S043 S045 · impliesBefore ORIGINAL ordered group pair(g,h)=[1, 4];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], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S044 S046 · impliesBefore ORIGINAL ordered group pair(g,h)=[2, 0];actual group product=2;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S045 S047 · impliesBefore ORIGINAL ordered group pair(g,h)=[2, 1];actual group product=3;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S046 S048 · impliesBefore ORIGINAL ordered group pair(g,h)=[2, 2];actual group product=4;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 4, 4, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S047 S049 · impliesBefore ORIGINAL ordered group pair(g,h)=[2, 3];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], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S048 S050 · impliesBefore ORIGINAL ordered group pair(g,h)=[2, 4];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, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 0, 0, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S049 S051 · impliesBefore ORIGINAL ordered group pair(g,h)=[3, 0];actual group product=3;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S050 S052 · impliesBefore ORIGINAL ordered group pair(g,h)=[3, 1];actual group product=4;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 4, 4, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S051 S053 · impliesBefore ORIGINAL ordered group pair(g,h)=[3, 2];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], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S052 S054 · impliesBefore ORIGINAL ordered group pair(g,h)=[3, 3];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, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 0, 0, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S053 S055 · impliesBefore ORIGINAL ordered group pair(g,h)=[3, 4];actual group product=2;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S054 S056 · impliesBefore ORIGINAL ordered group pair(g,h)=[4, 0];actual group product=4;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 4, 4, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 3, 3, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S055 S057 · impliesBefore ORIGINAL ordered group pair(g,h)=[4, 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], [3, 3, 3, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S056 S058 · impliesBefore ORIGINAL ordered group pair(g,h)=[4, 2];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, 2, 2, Yes], [2, 3, 3, Yes], [3, 4, 4, Yes], [4, 0, 0, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S057 S059 · impliesBefore ORIGINAL ordered group pair(g,h)=[4, 3];actual group product=2;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 4, 4, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S058 S060 · impliesBefore ORIGINAL ordered group pair(g,h)=[4, 4];actual group product=3;action table roles unchanged After ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, Yes], [1, 4, 4, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes], [4, 2, 2, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, 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, S059 S061 · calculationBefore group action ordered group-pair count := mul(5, 5) After group action ordered group-pair count = 25 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, S060 S062 · calculationBefore group action complete triple-domain count := mul(25, 8) After group action complete triple-domain count = 200 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, S061 S063 · impliesBefore COMPLETE triple domain order^2*point_count=200;this derivation ACTUALLY checked all200 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, S062 S064 · impliesBefore ORIGINAL group-row label=0;ORIGINAL point-indexed action row=[0, 1, 2, 3, 4, 5, 6, 7] After sorted image MULTISET=[0, 1, 2, 3, 4, 5, 6, 7];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, S063 S065 · impliesBefore ORIGINAL group-row label=1;ORIGINAL point-indexed action row=[1, 2, 3, 4, 0, 5, 6, 7] After sorted image MULTISET=[0, 1, 2, 3, 4, 5, 6, 7];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, S064 S066 · impliesBefore ORIGINAL group-row label=2;ORIGINAL point-indexed action row=[2, 3, 4, 0, 1, 5, 6, 7] After sorted image MULTISET=[0, 1, 2, 3, 4, 5, 6, 7];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, S065 S067 · impliesBefore ORIGINAL group-row label=3;ORIGINAL point-indexed action row=[3, 4, 0, 1, 2, 5, 6, 7] After sorted image MULTISET=[0, 1, 2, 3, 4, 5, 6, 7];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, S066 S068 · impliesBefore ORIGINAL group-row label=4;ORIGINAL point-indexed action row=[4, 0, 1, 2, 3, 5, 6, 7] After sorted image MULTISET=[0, 1, 2, 3, 4, 5, 6, 7];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, S067 S069 · impliesBefore CURRENT qualified group/action T=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]],identity=0,inverse list=[0, 4, 3, 2, 1];ORIGINAL SAME point=2 After ALL [g,g*point]=[[0, 2], [1, 3], [2, 4], [3, 0], [4, 1]];full orbit=[0, 1, 2, 3, 4];SAME-point stabilizer=[0];ALL [target,fiber] rows=[[0, [3]], [1, [4]], [2, [0]], [3, [1]], [4, [2]]];action map onto orbit,NOT necessarily whole carrier Conditions: none Qualify the actual group and both action axioms,then enumerate all g at the same point.Another point's stabilizer or a proper sample cannot replace these bindings.No free,faithful or transitive premise is needed. Step identifiers and dependenciesRule: orbitstab_qualified_SAME_action_point_full_fibers · Uses: G001, S068 S070 · impliesBefore ACTUAL SAME-point stabilizer=[0];identity=0 After ALL [g,h,product]=[[0, 0, 0]];ALL [g,inverse]=[[0, 0]];identity/product/inverse stay in SAME stabilizer Conditions: none The identity fixes the point;products of fixers fix it,and acting back by an inverse also fixes it.Check every member in the actual table,so the left cosets use this genuine subgroup,not any equally sized list. Step identifiers and dependenciesRule: orbitstab_SAME_point_stabilizer_actual_subgroup · Uses: G001, S069 S071 · calculationBefore Orbit-stabilizer fiber target0 complete size := sum(1) After Orbit-stabilizer fiber target0 complete size = 1 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S070 S072 · impliesBefore ORIGINAL orbit target=0;LEAST group-index representative g0=3;SAME stabilizer=[0];full fiber=[3] After ALL [h,g0*h,(g0*h)*point]=[[0, 3, 0]];ALL [g,g0_inverse*g,g0*(g0_inverse*g)]=[[3, 0, 3]];LEFT coset sorted=[3] equals fiber;fiber size=1 Conditions: none The map h->g0*h sends this point's stabilizer into the fibre;g->g0_inverse*g lands back in it by action compatibility.The compositions are identities,so the maps are inverse bijections.Do not substitute a right coset;the multiplication order works for nonabelian groups too. Step identifiers and dependenciesRule: orbitstab_ACTUAL_left_coset_fiber_two_inverse_maps · Uses: G001, S071 S073 · calculationBefore Orbit-stabilizer fiber target1 complete size := sum(1) After Orbit-stabilizer fiber target1 complete size = 1 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S072 S074 · impliesBefore ORIGINAL orbit target=1;LEAST group-index representative g0=4;SAME stabilizer=[0];full fiber=[4] After ALL [h,g0*h,(g0*h)*point]=[[0, 4, 1]];ALL [g,g0_inverse*g,g0*(g0_inverse*g)]=[[4, 0, 4]];LEFT coset sorted=[4] equals fiber;fiber size=1 Conditions: none The map h->g0*h sends this point's stabilizer into the fibre;g->g0_inverse*g lands back in it by action compatibility.The compositions are identities,so the maps are inverse bijections.Do not substitute a right coset;the multiplication order works for nonabelian groups too. Step identifiers and dependenciesRule: orbitstab_ACTUAL_left_coset_fiber_two_inverse_maps · Uses: G001, S073 S075 · calculationBefore Orbit-stabilizer fiber target2 complete size := sum(1) After Orbit-stabilizer fiber target2 complete size = 1 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S074 S076 · impliesBefore ORIGINAL orbit target=2;LEAST group-index representative g0=0;SAME stabilizer=[0];full fiber=[0] After ALL [h,g0*h,(g0*h)*point]=[[0, 0, 2]];ALL [g,g0_inverse*g,g0*(g0_inverse*g)]=[[0, 0, 0]];LEFT coset sorted=[0] equals fiber;fiber size=1 Conditions: none The map h->g0*h sends this point's stabilizer into the fibre;g->g0_inverse*g lands back in it by action compatibility.The compositions are identities,so the maps are inverse bijections.Do not substitute a right coset;the multiplication order works for nonabelian groups too. Step identifiers and dependenciesRule: orbitstab_ACTUAL_left_coset_fiber_two_inverse_maps · Uses: G001, S075 S077 · calculationBefore Orbit-stabilizer fiber target3 complete size := sum(1) After Orbit-stabilizer fiber target3 complete size = 1 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S076 S078 · impliesBefore ORIGINAL orbit target=3;LEAST group-index representative g0=1;SAME stabilizer=[0];full fiber=[1] After ALL [h,g0*h,(g0*h)*point]=[[0, 1, 3]];ALL [g,g0_inverse*g,g0*(g0_inverse*g)]=[[1, 0, 1]];LEFT coset sorted=[1] equals fiber;fiber size=1 Conditions: none The map h->g0*h sends this point's stabilizer into the fibre;g->g0_inverse*g lands back in it by action compatibility.The compositions are identities,so the maps are inverse bijections.Do not substitute a right coset;the multiplication order works for nonabelian groups too. Step identifiers and dependenciesRule: orbitstab_ACTUAL_left_coset_fiber_two_inverse_maps · Uses: G001, S077 S079 · calculationBefore Orbit-stabilizer fiber target4 complete size := sum(1) After Orbit-stabilizer fiber target4 complete size = 1 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S078 S080 · impliesBefore ORIGINAL orbit target=4;LEAST group-index representative g0=2;SAME stabilizer=[0];full fiber=[2] After ALL [h,g0*h,(g0*h)*point]=[[0, 2, 4]];ALL [g,g0_inverse*g,g0*(g0_inverse*g)]=[[2, 0, 2]];LEFT coset sorted=[2] equals fiber;fiber size=1 Conditions: none The map h->g0*h sends this point's stabilizer into the fibre;g->g0_inverse*g lands back in it by action compatibility.The compositions are identities,so the maps are inverse bijections.Do not substitute a right coset;the multiplication order works for nonabelian groups too. Step identifiers and dependenciesRule: orbitstab_ACTUAL_left_coset_fiber_two_inverse_maps · Uses: G001, S079 S081 · calculationBefore Orbit-stabilizer complete orbit count := sum(1, 1, 1, 1, 1) After Orbit-stabilizer complete orbit count = 5 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S080 S082 · calculationBefore Orbit-stabilizer SAME-point stabilizer count := sum(1) After Orbit-stabilizer SAME-point stabilizer count = 1 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S081 S083 · calculationBefore Orbit-stabilizer SAME-bound orbit times stabilizer sizes := mul(5, 1) After Orbit-stabilizer SAME-bound orbit times stabilizer sizes = 5 Conditions: none Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S082 S084 · impliesBefore ALL original group elements partitioned by actual action-map image;fibers=[[0, [3]], [1, [4]], [2, [0]], [3, [1]], [4, [2]]] After each fiber bijective with SAME stabilizer of size1;orbit size=5;group order=5=5*1=5 Conditions: none Each g has exactly one image,so these fibres are disjoint and cover G.The explicit bijections establish equal sizes;this is not merely a coincidental numerical product.Surjectivity onto the orbit does not imply surjectivity onto the entire point carrier. Step identifiers and dependenciesRule: orbitstab_partition_group_into_equal_size_fibers_count · Uses: G001, S083 ## Reverse checks - S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S083 · premise_and_result_audit — Recompute Orbit-stabilizer SAME-bound orbit times stabilizer sizes with exact rational arithmetic, not rounded decimals. - S082 · premise_and_result_audit — Recompute Orbit-stabilizer SAME-point stabilizer count with exact rational arithmetic, not rounded decimals. - S081 · premise_and_result_audit — Recompute Orbit-stabilizer complete orbit count with exact rational arithmetic, not rounded decimals. - S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S079 · premise_and_result_audit — Recompute Orbit-stabilizer fiber target4 complete size with exact rational arithmetic, not rounded decimals. - S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S077 · premise_and_result_audit — Recompute Orbit-stabilizer fiber target3 complete size with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S075 · premise_and_result_audit — Recompute Orbit-stabilizer fiber target2 complete size with exact rational arithmetic, not rounded decimals. - S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S073 · premise_and_result_audit — Recompute Orbit-stabilizer fiber target1 complete size with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S071 · premise_and_result_audit — Recompute Orbit-stabilizer fiber target0 complete size with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 group action complete triple-domain count with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Recompute group action ordered group-pair count with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — 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 (5, 2, [0, 1, 2, 3, 4], 5, [0], 1, 5, 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: aa2907e90d747544aae4ed649cf80610e01a5379c640256f4c9737eec2ca07da - candidate record SHA-256: 03a4468669eb5e60ce8cac767295443a119cf0cfbdf26d2dc335881de89f0c82 - semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5 - question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0019.json)