Step-by-step derivation
S001 · implies
Before
source operation=groupaction-compatibility-audit;original snapshot and mathematics retained
After
explicit ordered source output fields=[triple_count, first_failure, compatible, certificate]
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(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(9);TOTAL action table row g,column x gives g*x=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 0, 5, 6, 7, 8], [2, 3, 4, 0, 1, 5, 6, 7, 8], [3, 4, 0, 1, 2, 5, 6, 7, 8], [4, 0, 1, 2, 3, 5, 6, 7, 8]];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 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], [8, 8, 8, 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, S002
S004 · 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, 4, 4, Yes], [4, 0, 0, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, 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, S003
S005 · 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, 4, 4, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, 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, S004
S006 · 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, 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], [8, 8, 8, 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, S005
S007 · implies
Before
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], [8, 8, 8, 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, S006
S008 · 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, 4, 4, Yes], [4, 0, 0, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, 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, S007
S009 · 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, 4, 4, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, 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, S008
S010 · 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, 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], [8, 8, 8, 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, S009
S011 · implies
Before
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], [8, 8, 8, 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, S010
S012 · implies
Before
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], [8, 8, 8, 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, S011
S013 · 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, 4, 4, Yes], [3, 0, 0, Yes], [4, 1, 1, Yes], [5, 5, 5, Yes], [6, 6, 6, Yes], [7, 7, 7, Yes], [8, 8, 8, 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, S012
S014 · 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, 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], [8, 8, 8, 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, S013
S015 · implies
Before
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], [8, 8, 8, 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, S014
S016 · implies
Before
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], [8, 8, 8, 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, S015
S017 · implies
Before
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], [8, 8, 8, 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, S016
S018 · 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, 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], [8, 8, 8, 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, S017
S019 · implies
Before
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], [8, 8, 8, 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, S018
S020 · implies
Before
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], [8, 8, 8, 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, S019
S021 · implies
Before
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], [8, 8, 8, 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, S020
S022 · implies
Before
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], [8, 8, 8, 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, S021
S023 · implies
Before
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], [8, 8, 8, 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, S022
S024 · implies
Before
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], [8, 8, 8, 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, S023
S025 · implies
Before
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], [8, 8, 8, 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, S024
S026 · implies
Before
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], [8, 8, 8, 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)=[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], [8, 8, 8, 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 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S027
S029 · calculation
Before
group action complete triple-domain count := mul(25, 9)
After
group action complete triple-domain count = 225
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, S028
S030 · implies
Before
COMPLETE triple domain order^2*point_count=225;this derivation ACTUALLY checked all225 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, S029
S031 · implies
Before
this component audited ALL group-group-point compatibility triples
After
compatibility flag alone does NOT establish identity-action or group qualification
Conditions: none
Complete compatibility is one obligation,not a replacement for e*x=x or group qualification.Even passing this component does not additionally claim a valid action.
Step identifiers and dependencies
Rule: groupaction_compatibility_component_not_full_action_claim · Uses: G001, S030