Step-by-step derivation
S001 · implies
Before
source operation=cauchy-composite-divisor-boundary;original snapshot and mathematics retained
After
explicit ordered source output fields=[group_order, composite_divisor, element_order_rows, divisor_divides_order, exists_element_of_requested_order, boundary]
Conditions: none
Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL closed Cayley table=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 0, 1], [3, 2, 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], [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, S001
S003 · 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, S002
S004 · 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, 0, 0, Yes], [2, 3, 3, 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, S003
S005 · 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, S004
S006 · 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, 2, 2, Yes], [2, 1, 1, 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, S005
S007 · 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, 0, 0, Yes], [2, 3, 3, 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, 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, S007
S009 · 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, 2, 2, Yes], [2, 1, 1, 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, S008
S010 · 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, 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, S009
S011 · 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, S010
S012 · 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, 2, 2, Yes], [2, 1, 1, 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, S011
S013 · 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, S012
S014 · 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, 0, 0, Yes], [2, 3, 3, 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, S013
S015 · 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, 2, 2, Yes], [2, 1, 1, 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, 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, S015
S017 · 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, 0, 0, Yes], [2, 3, 3, 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, S016
S018 · 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, 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, S017
S019 · 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, S018
S020 · 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], [2, 3, 3, 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, S019
S021 · 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, S020
S022 · 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, 2, 2, No], [2, 1, 1, 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, S021
S023 · 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, S022
S024 · implies
Before
CURRENT ORIGINAL complete Cayley table=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 0, 1], [3, 2, 1, 0]];actual identity=0;order=4
After
ALL group axioms qualified;actual inverse list=[0, 1, 2, 3];no point carrier or group-action assumption is used
Conditions: none
Certify closure,identity,associativity and two-sided inverses in the group table itself.Do not import an old8-element action-table limit or assume cyclicity/commutativity.The frozen pure helper receives only this question's table and trace.
Step identifiers and dependencies
Rule: group_CURRENT_complete_certified_Cayley_NOT_action_wrapper · Uses: G001, S023
S025 · implies
Before
Cauchy ORIGINAL element0:ORIGINAL element=0;positive exponent=1;previous power=0;right multiply ORIGINAL g via table
After
g^1=0;identity=0;is_identity=Yes;all earlier POSITIVE powers are nonidentity
Conditions: none
Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e.
Step identifiers and dependencies
Rule: cauchy_EVERY_positive_power_until_FIRST_identity_return · Uses: G001, S024
S026 · implies
Before
Cauchy ORIGINAL element0:CURRENT certified group order=4,element=0;ALL powers through first return=[[1, 0]]
After
exact positive order=1;cyclic powers before repetition form a subgroup of this size;its equal-size left cosets partition G,so this order divides group order
Conditions: none
Among e,g,...,g^|G|,a repetition cancels to a positive return at most|G|.Powers before the minimal return are distinct and close under multiplication/inverse modulo that order.Their left cosets prove divisibility;one power equation alone does not.
Step identifiers and dependencies
Rule: cauchy_finite_order_exists_and_cyclic_subgroup_divisibility · Uses: G001, S025
S027 · implies
Before
Cauchy ORIGINAL element1:ORIGINAL element=1;positive exponent=1;previous power=0;right multiply ORIGINAL g via table
After
g^1=1;identity=0;is_identity=No;all earlier POSITIVE powers are nonidentity
Conditions: none
Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e.
Step identifiers and dependencies
Rule: cauchy_EVERY_positive_power_until_FIRST_identity_return · Uses: G001, S026
S028 · implies
Before
Cauchy ORIGINAL element1:ORIGINAL element=1;positive exponent=2;previous power=1;right multiply ORIGINAL g via table
After
g^2=0;identity=0;is_identity=Yes;all earlier POSITIVE powers are nonidentity
Conditions: none
Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e.
Step identifiers and dependencies
Rule: cauchy_EVERY_positive_power_until_FIRST_identity_return · Uses: G001, S027
S029 · implies
Before
Cauchy ORIGINAL element1:CURRENT certified group order=4,element=1;ALL powers through first return=[[1, 1], [2, 0]]
After
exact positive order=2;cyclic powers before repetition form a subgroup of this size;its equal-size left cosets partition G,so this order divides group order
Conditions: none
Among e,g,...,g^|G|,a repetition cancels to a positive return at most|G|.Powers before the minimal return are distinct and close under multiplication/inverse modulo that order.Their left cosets prove divisibility;one power equation alone does not.
Step identifiers and dependencies
Rule: cauchy_finite_order_exists_and_cyclic_subgroup_divisibility · Uses: G001, S028
S030 · implies
Before
Cauchy ORIGINAL element2:ORIGINAL element=2;positive exponent=1;previous power=0;right multiply ORIGINAL g via table
After
g^1=2;identity=0;is_identity=No;all earlier POSITIVE powers are nonidentity
Conditions: none
Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e.
Step identifiers and dependencies
Rule: cauchy_EVERY_positive_power_until_FIRST_identity_return · Uses: G001, S029
S031 · implies
Before
Cauchy ORIGINAL element2:ORIGINAL element=2;positive exponent=2;previous power=2;right multiply ORIGINAL g via table
After
g^2=0;identity=0;is_identity=Yes;all earlier POSITIVE powers are nonidentity
Conditions: none
Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e.
Step identifiers and dependencies
Rule: cauchy_EVERY_positive_power_until_FIRST_identity_return · Uses: G001, S030
S032 · implies
Before
Cauchy ORIGINAL element2:CURRENT certified group order=4,element=2;ALL powers through first return=[[1, 2], [2, 0]]
After
exact positive order=2;cyclic powers before repetition form a subgroup of this size;its equal-size left cosets partition G,so this order divides group order
Conditions: none
Among e,g,...,g^|G|,a repetition cancels to a positive return at most|G|.Powers before the minimal return are distinct and close under multiplication/inverse modulo that order.Their left cosets prove divisibility;one power equation alone does not.
Step identifiers and dependencies
Rule: cauchy_finite_order_exists_and_cyclic_subgroup_divisibility · Uses: G001, S031
S033 · implies
Before
Cauchy ORIGINAL element3:ORIGINAL element=3;positive exponent=1;previous power=0;right multiply ORIGINAL g via table
After
g^1=3;identity=0;is_identity=No;all earlier POSITIVE powers are nonidentity
Conditions: none
Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e.
Step identifiers and dependencies
Rule: cauchy_EVERY_positive_power_until_FIRST_identity_return · Uses: G001, S032
S034 · implies
Before
Cauchy ORIGINAL element3:ORIGINAL element=3;positive exponent=2;previous power=3;right multiply ORIGINAL g via table
After
g^2=0;identity=0;is_identity=Yes;all earlier POSITIVE powers are nonidentity
Conditions: none
Start at e but test only positive exponents1,2,...The first identity return is the exact order.Do not use exponent0 or infer exact order merely from g^p=e.
Step identifiers and dependencies
Rule: cauchy_EVERY_positive_power_until_FIRST_identity_return · Uses: G001, S033
S035 · implies
Before
Cauchy ORIGINAL element3:CURRENT certified group order=4,element=3;ALL powers through first return=[[1, 3], [2, 0]]
After
exact positive order=2;cyclic powers before repetition form a subgroup of this size;its equal-size left cosets partition G,so this order divides group order
Conditions: none
Among e,g,...,g^|G|,a repetition cancels to a positive return at most|G|.Powers before the minimal return are distinct and close under multiplication/inverse modulo that order.Their left cosets prove divisibility;one power equation alone does not.
Step identifiers and dependencies
Rule: cauchy_finite_order_exists_and_cyclic_subgroup_divisibility · Uses: G001, S034
S036 · implies
Before
Cauchy ORIGINAL composite divisor:ORIGINAL integer=4;complete possible trial divisors=[2]
After
ALL [divisor,remainder] rows=[[2, 0]];prime=No;a composite>1 has a factor at most its square root
Conditions: none
Check the entire square-root divisor range.The empty range qualifies prime2,but not0 or1.Samples,square tables or an Euler criterion do not replace primality qualification.
Step identifiers and dependencies
Rule: finite_prime_ALL_trial_divisors_through_square_root · Uses: G001, S035
S037 · implies
Before
CURRENT group order=4;ORIGINAL composite divisor=4;COMPLETE element-order rows=[[0, 1], [1, 2], [2, 2], [3, 2]]
After
divisor divides order=Yes;exists element of requested order=No in THIS complete group;not a claim composite element orders never occur
Conditions: none
Require actual divisibility and exhaustive absence of that order in this complete group.Other groups may have composite-order elements;this model does not deny them or settle existence of subgroups of that size.
Step identifiers and dependencies
Rule: cauchy_composite_counterexample_ACTUAL_absence_not_general_negation · Uses: G001, S036