Question 21823: Fixing This Point Is Exactly Membership in Its Stabilizer

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0042-21823 · q-20261014-advance-20261109-r01-finite-group-stabilizer-0073

Original question and givens

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Test group_element=2. Give (group_element, point, image, fixes_point, in_stabilizer).

G001 — Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, point carrier Fin(8)={0,…,7} and total action_table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 5, 6, 7]]. Group row g/column h gives g*h; action row g/column x gives g*x. Every group and action axiom must qualify before the stabilizer is computed. Specified point=6. Test group_element=2. Give (group_element, point, image, fixes_point, in_stabilizer).

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-stabilizer

Step-by-step derivation

S001 · implies

Before

source operation=stabilizer-membership-audit;original snapshot and mathematics retained

After

explicit ordered source output fields=[group_element, point, image, fixes_point, in_stabilizer]

Conditions: none

Keep the original historical interface marker while listing ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their source claim scope.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

ORIGINAL complete group-label carrier Fin(4);declared identity label=0;Cayley table row g,column h gives g*h=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]]

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, 0, 4, 5, 6, 7], [2, 3, 0, 1, 4, 5, 6, 7], [3, 0, 1, 2, 4, 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 dependencies

Rule: groupaction_two_table_roles_complete_candidate_not_group_assumption · Uses: G001, S001

S003 · implies

Before

ORIGINAL closed Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]];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]];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 dependencies

Rule: groupaction_group_identity_both_sides_not_action_identity · Uses: G001, S002

S004 · implies

Before

ORIGINAL group pair(g,h)=[0, 0];check ALL k in Fin(4)

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]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S003

S005 · implies

Before

ORIGINAL group pair(g,h)=[0, 1];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S004

S006 · implies

Before

ORIGINAL group pair(g,h)=[0, 2];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S005

S007 · implies

Before

ORIGINAL group pair(g,h)=[0, 3];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S006

S008 · implies

Before

ORIGINAL group pair(g,h)=[1, 0];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S007

S009 · implies

Before

ORIGINAL group pair(g,h)=[1, 1];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S008

S010 · implies

Before

ORIGINAL group pair(g,h)=[1, 2];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S009

S011 · implies

Before

ORIGINAL group pair(g,h)=[1, 3];check ALL k in Fin(4)

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]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S010

S012 · implies

Before

ORIGINAL group pair(g,h)=[2, 0];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S011

S013 · implies

Before

ORIGINAL group pair(g,h)=[2, 1];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S012

S014 · implies

Before

ORIGINAL group pair(g,h)=[2, 2];check ALL k in Fin(4)

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]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S013

S015 · implies

Before

ORIGINAL group pair(g,h)=[2, 3];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S014

S016 · implies

Before

ORIGINAL group pair(g,h)=[3, 0];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S015

S017 · implies

Before

ORIGINAL group pair(g,h)=[3, 1];check ALL k in Fin(4)

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]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S016

S018 · implies

Before

ORIGINAL group pair(g,h)=[3, 2];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S017

S019 · implies

Before

ORIGINAL group pair(g,h)=[3, 3];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 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 dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S018

S020 · implies

Before

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]]

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 dependencies

Rule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S019

S021 · implies

Before

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, 0, 0, Yes]]

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 dependencies

Rule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S020

S022 · implies

Before

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, 0, 0, Yes], [3, 1, 1, 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 dependencies

Rule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S021

S023 · implies

Before

ORIGINAL group element g=3;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 3, 3, No], [1, 0, 0, Yes], [2, 1, 1, No], [3, 2, 2, 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 dependencies

Rule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S022

S024 · implies

Before

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 dependencies

Rule: groupaction_group_axiom_conjunction_candidate_may_fail · Uses: G001, S023

S025 · implies

Before

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 dependencies

Rule: groupaction_every_point_identity_action_not_group_identity · Uses: G001, S024

S026 · implies

Before

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 dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S025

S027 · implies

Before

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, 0, 0, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S026

S028 · implies

Before

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, 0, 0, Yes], [3, 1, 1, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S027

S029 · implies

Before

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, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S028

S030 · implies

Before

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, 0, 0, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S029

S031 · implies

Before

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, 0, 0, Yes], [3, 1, 1, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S030

S032 · implies

Before

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, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S031

S033 · implies

Before

ORIGINAL ordered group pair(g,h)=[1, 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

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S032

S034 · implies

Before

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, 0, 0, Yes], [3, 1, 1, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S033

S035 · implies

Before

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, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S034

S036 · implies

Before

ORIGINAL ordered group pair(g,h)=[2, 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

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S035

S037 · implies

Before

ORIGINAL ordered group pair(g,h)=[2, 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, 0, 0, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S036

S038 · implies

Before

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, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S037

S039 · implies

Before

ORIGINAL ordered group pair(g,h)=[3, 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

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S038

S040 · implies

Before

ORIGINAL ordered group pair(g,h)=[3, 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, 0, 0, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S039

S041 · implies

Before

ORIGINAL ordered group pair(g,h)=[3, 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, 0, 0, Yes], [3, 1, 1, Yes], [4, 4, 4, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S040

S042 · calculation

Before

group action ordered group-pair count := mul(4, 4)

After

group action ordered group-pair count = 16

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

group action complete triple-domain count := mul(16, 8)

After

group action complete triple-domain count = 128

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · implies

Before

COMPLETE triple domain order^2*point_count=128;this derivation ACTUALLY checked all128 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 dependencies

Rule: groupaction_full_domain_count_first_failure_not_early_stop_measurement · Uses: G001, S043

S045 · implies

Before

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 dependencies

Rule: groupaction_every_row_permutation_actual_images_not_validity_converse · Uses: G001, S044

S046 · implies

Before

ORIGINAL group-row label=1;ORIGINAL point-indexed action row=[1, 2, 3, 0, 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

Step identifiers and dependencies

Rule: groupaction_every_row_permutation_actual_images_not_validity_converse · Uses: G001, S045

S047 · implies

Before

ORIGINAL group-row label=2;ORIGINAL point-indexed action row=[2, 3, 0, 1, 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

Step identifiers and dependencies

Rule: groupaction_every_row_permutation_actual_images_not_validity_converse · Uses: G001, S046

S048 · implies

Before

ORIGINAL group-row label=3;ORIGINAL point-indexed action row=[3, 0, 1, 2, 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

Step identifiers and dependencies

Rule: groupaction_every_row_permutation_actual_images_not_validity_converse · Uses: G001, S047

S049 · implies

Before

CURRENT original group/action:group=Yes,identity_action=Yes,compatibility=Yes,permutations=Yes;actual identity=0

After

qualified full action;actual inverse list=[0, 3, 2, 1];permutation rows alone would NOT suffice

Conditions: none

Qualify the actual group and both action axioms first.Permutation rows alone do not suffice.Use the original identity label,not assumed0;frozen pure helpers receive this question's tables and trace.

Step identifiers and dependencies

Rule: stabilizer_qualified_actual_group_action_before_fixing · Uses: G001, S048

S050 · calculation

Before

Stabilizer point6 complete fixing count := sum(1, 1, 1, 1)

After

Stabilizer point6 complete fixing count = 4

Conditions: none

Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · implies

Before

ORIGINAL point=6;ALL [group_element,image,fixes] rows=[[0, 6, Yes], [1, 6, Yes], [2, 6, Yes], [3, 6, Yes]]

After

full stabilizer in ORIGINAL group order=[0, 1, 2, 3];stabilizer order=4

Conditions: none

The stabilizer is the set of all group elements fixing this point,not its orbit or one fixed-element witness.A tested g may fail to fix the point while the identity remains in the full stabilizer.

Step identifiers and dependencies

Rule: stabilizer_ALL_group_elements_fixing_not_orbit · Uses: G001, S050

S052 · implies

Before

ORIGINAL group element=2,point=6,actual image=6;complete stabilizer=[0, 1, 2, 3]

After

fixes_point=Yes;in_stabilizer=Yes;flags agree by the full fixing-set definition

Conditions: none

Read the actual image of the specified g,compare with the point,and separately check the full member list.The existence of the identity does not make every g fix the point.

Step identifiers and dependencies

Rule: stabilizer_tested_element_image_iff_membership · Uses: G001, S051

Reverse checks

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 — Recompute Stabilizer point6 complete fixing count with exact rational arithmetic, not rounded decimals.
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 — Recompute group action complete triple-domain count with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Recompute group action ordered group-pair count with exact rational arithmetic, not rounded decimals.
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

(2, 6, 6, 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: 4161968203b7c3827b4fec75e88a05c66eceeee2d3a2434e5dba00b53f14f8fd
candidate record SHA-256: dcfe5ca85cdd04fa878602fae1bc66814009ac21a85d45710ba597061ab7bf9b
semantic review receipt SHA-256: dd44555041dfbb6fe9936aa00b9a5f8c5982c2784ef61dd0cbd5d1a303a49c1a
question-first witness SHA-256: 9a6011263f6ac53b9a1fd57addf2f3e2f6088b54802c539f332d689480dc2d17
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON