Step-by-step derivation
S001 · implies
Before
source operation=groupaction-qualification;original snapshot and mathematics retained
After
explicit ordered source output fields=[group_order, point_count, group_valid, identity_action, compatible, rows_are_permutations, valid_action]
Conditions: none
Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the 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(6);TOTAL action table row g,column x gives g*x=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 0, 4, 5], [2, 3, 0, 1, 4, 5], [3, 0, 1, 2, 4, 5]];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];complete point carrier Fin(6)
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]];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]];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]];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, 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]];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, 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]];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, 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]];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, 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]];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, 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]];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, 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]];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, 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]];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, 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]];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, 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]];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, 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]];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, 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]];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, 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]];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, 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]];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, 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]];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, 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, 6)
After
group action complete triple-domain count = 96
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=96;this derivation ACTUALLY checked all96 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]
After
sorted image MULTISET=[0, 1, 2, 3, 4, 5];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]
After
sorted image MULTISET=[0, 1, 2, 3, 4, 5];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, S045
S047 · implies
Before
ORIGINAL group-row label=2;ORIGINAL point-indexed action row=[2, 3, 0, 1, 4, 5]
After
sorted image MULTISET=[0, 1, 2, 3, 4, 5];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, S046
S048 · implies
Before
ORIGINAL group-row label=3;ORIGINAL point-indexed action row=[3, 0, 1, 2, 4, 5]
After
sorted image MULTISET=[0, 1, 2, 3, 4, 5];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, S047
S049 · implies
Before
ACTUAL qualified group and BOTH action axioms hold;for each g actual group inverse h=[0, 3, 2, 1]
After
A_g composed A_h=A_(g*h)=A_e=identity and reverse composition likewise;therefore EVERY group element induces a permutation;actual row flags agree=Yes
Conditions: none
Use each actual group inverse and complete compatibility to make both action compositions the identity,hence every action row is a bijection.This direction derives permutations from the axioms,not validity from permutation rows alone.
Step identifiers and dependencies
Rule: groupaction_valid_axioms_imply_permutations_via_inverses · Uses: G001, S048