Step-by-step derivation
S001 · implies
Before
source operation=orbit-orbit-partition;original snapshot and mathematics retained
After
explicit ordered source output fields=[orbits, covered_points, pairwise_disjoint, partition_certificate]
Conditions: none
Retain the historical interface marker while listing ordered output fields without JavaScript.Fields and tags are not extra mathematical evidence;fixed No markers retain their claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL complete group-label carrier Fin(2);declared identity label=0;Cayley table row g,column h gives g*h=[[0, 1], [1, 0]]
After
ORIGINAL complete point carrier Fin(4);TOTAL action table row g,column x gives g*x=[[0, 1, 2, 3], [1, 0, 2, 3]];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], [1, 0]];ACTUAL declared group identity=0
After
ALL [g,e*g,g*e,both_equal_g] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes]];two-sided group identity=Yes
Conditions: none
Both e*g and g*e in the group table must equal g.This is not certified by an identity-action row on points.
Step identifiers and 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(2)
After
ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes]]
Conditions: none
Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.
Step identifiers and 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(2)
After
ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 0, 0, Yes]]
Conditions: none
Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.
Step identifiers and dependencies
Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S004
S006 · implies
Before
ORIGINAL group pair(g,h)=[1, 0];check ALL k in Fin(2)
After
ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 0, 0, Yes]]
Conditions: none
Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.
Step identifiers and dependencies
Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S005
S007 · implies
Before
ORIGINAL group pair(g,h)=[1, 1];check ALL k in Fin(2)
After
ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes]]
Conditions: none
Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.
Step identifiers and dependencies
Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S006
S008 · 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]]
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, S007
S009 · 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, 0, 0, Yes]]
After
first actual two-sided inverse witness=1;exists=Yes
Conditions: none
For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels.
Step identifiers and dependencies
Rule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S008
S010 · 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, S009
S011 · implies
Before
ACTUAL declared identity label=0;ORIGINAL identity action row=[0, 1, 2, 3];complete point carrier Fin(4)
After
ALL ordered [point,identity_image,equality] rows=[[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, 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, S010
S012 · 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]];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)=[0, 1];actual group product=1;action table roles unchanged
After
ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, Yes], [1, 0, 0, Yes], [2, 2, 2, Yes], [3, 3, 3, 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)=[1, 0];actual group product=1;action table roles unchanged
After
ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, Yes], [1, 0, 0, Yes], [2, 2, 2, Yes], [3, 3, 3, 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)=[1, 1];actual group product=0;action table roles unchanged
After
ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, 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 · calculation
Before
group action ordered group-pair count := mul(2, 2)
After
group action ordered group-pair count = 4
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S015
S017 · calculation
Before
group action complete triple-domain count := mul(4, 4)
After
group action complete triple-domain 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, S016
S018 · implies
Before
COMPLETE triple domain order^2*point_count=16;this derivation ACTUALLY checked all16 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, S017
S019 · implies
Before
ORIGINAL group-row label=0;ORIGINAL point-indexed action row=[0, 1, 2, 3]
After
sorted image MULTISET=[0, 1, 2, 3];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, S018
S020 · implies
Before
ORIGINAL group-row label=1;ORIGINAL point-indexed action row=[1, 0, 2, 3]
After
sorted image MULTISET=[0, 1, 2, 3];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, S019
S021 · implies
Before
CURRENT original group/action qualified:group=Yes,identity_action=Yes,compatibility=Yes,permutations=Yes;actual inverses=[0, 1]
After
only NOW use the action axioms;row permutations ALONE are insufficient;invalid group/action data are rejected,not reported as an orbit
Conditions: none
Qualify the actual group and both action axioms before using orbits.Permutation rows alone are insufficient.Reused frozen pure helpers receive this question's full tables and trace,not a link to an old derivation.
Step identifiers and dependencies
Rule: orbit_reject_unqualified_tables_before_orbit_claims · Uses: G001, S020
S022 · implies
Before
CURRENT identity=0;full inverses=[0, 1];action table=[[0, 1, 2, 3], [1, 0, 2, 3]]
After
reflexive:e*x=x;symmetric:y=g*x gives x=g_inverse*y;transitive:y=g*x,z=h*y gives z=(h*g)*x;therefore related iff the two full orbits are equal,and distinct orbits are disjoint
Conditions: none
Symmetry uses the actual inverse and action compatibility.Transitivity acts by g and then h,so the product is h*g,without assuming commutativity.If two orbits share a point,symmetry and transitivity give both inclusions;otherwise they are disjoint.
Step identifiers and dependencies
Rule: orbit_equivalence_reflexive_symmetric_transitive_hg_order · Uses: G001, S021
S023 · calculation
Before
orbit point0 distinct image count := sum(1, 1)
After
orbit point0 distinct image count = 2
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S022
S024 · implies
Before
ORIGINAL point=0;images in ALL group-index order=[0, 1];group order=2
After
full orbit sorted and deduplicated=[0, 1];orbit cardinality=2;group enumeration count=2;these counts need NOT agree
Conditions: none
Take images under every group element,not a single g,generator reachability or a stabilizer.Multiple elements may have the same image;keep group order distinct from orbit size.
Step identifiers and dependencies
Rule: orbit_all_original_group_rows_not_single_generator · Uses: G001, S023
S025 · calculation
Before
orbit point2 distinct image count := sum(1)
After
orbit point2 distinct image count = 1
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S024
S026 · implies
Before
ORIGINAL point=2;images in ALL group-index order=[2, 2];group order=2
After
full orbit sorted and deduplicated=[2];orbit cardinality=1;group enumeration count=2;these counts need NOT agree
Conditions: none
Take images under every group element,not a single g,generator reachability or a stabilizer.Multiple elements may have the same image;keep group order distinct from orbit size.
Step identifiers and dependencies
Rule: orbit_all_original_group_rows_not_single_generator · Uses: G001, S025
S027 · calculation
Before
orbit point3 distinct image count := sum(1)
After
orbit point3 distinct image count = 1
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S026
S028 · implies
Before
ORIGINAL point=3;images in ALL group-index order=[3, 3];group order=2
After
full orbit sorted and deduplicated=[3];orbit cardinality=1;group enumeration count=2;these counts need NOT agree
Conditions: none
Take images under every group element,not a single g,generator reachability or a stabilizer.Multiple elements may have the same image;keep group order distinct from orbit size.
Step identifiers and dependencies
Rule: orbit_all_original_group_rows_not_single_generator · Uses: G001, S027
S029 · implies
Before
ALL carrier points0..3;orbits ordered by first point=[[0, 1], [2], [3]]
After
sorted flattened coverage=[0, 1, 2, 3];pairwise disjoint=Yes;partition certificate=Yes
Conditions: none
The identity puts each point in its orbit.Start at the least unseen point each time,using the equal-or-disjoint result;also check full coverage and no duplicates rather than guessing a partition from its count.
Step identifiers and dependencies
Rule: orbit_partition_all_points_coverage_and_disjointness · Uses: G001, S028