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R0023-12055 · q-20261013-advance-20261026-r01-finite-abelian-group-0030
Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,least positive order,returns to identity?).
G001 — Use the complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]] on Fin(6)={0,…,5}, in the given row/column label order. Closure, associativity, a unique two-sided identity and inverses must first qualify on the whole table. The identity need not be label 0. Do not assume commutativity from the family name or a few samples; these results concern only the specified finite group. element=3. Give (element,identity,least positive order,returns to identity?).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.
Before
carrier Fin(6);complete Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]]
After
all entries lie in carrier;do not assume identity0 or commutativity
Conditions: none
Fix the whole table and labels.Typed closure is only the first group premise;an Abelian family name does not replace the commutativity audit.
Rule: complete_finite_group_typed_table · Uses: G001
Before
a=0;b=0;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S001
Before
a=0;b=1;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[1, 2, 5, 4, 0, 3];a*(b*c) values=[1, 2, 5, 4, 0, 3];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S002
Before
a=0;b=2;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[2, 5, 3, 0, 1, 4];a*(b*c) values=[2, 5, 3, 0, 1, 4];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S003
Before
a=0;b=3;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[3, 4, 0, 2, 5, 1];a*(b*c) values=[3, 4, 0, 2, 5, 1];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S004
Before
a=0;b=4;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[4, 0, 1, 5, 3, 2];a*(b*c) values=[4, 0, 1, 5, 3, 2];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S005
Before
a=0;b=5;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[5, 3, 4, 1, 2, 0];a*(b*c) values=[5, 3, 4, 1, 2, 0];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S006
Before
a=1;b=0;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[1, 2, 5, 4, 0, 3];a*(b*c) values=[1, 2, 5, 4, 0, 3];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S007
Before
a=1;b=1;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[2, 5, 3, 0, 1, 4];a*(b*c) values=[2, 5, 3, 0, 1, 4];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S008
Before
a=1;b=2;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[5, 3, 4, 1, 2, 0];a*(b*c) values=[5, 3, 4, 1, 2, 0];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S009
Before
a=1;b=3;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[4, 0, 1, 5, 3, 2];a*(b*c) values=[4, 0, 1, 5, 3, 2];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S010
Before
a=1;b=4;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S011
Before
a=1;b=5;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[3, 4, 0, 2, 5, 1];a*(b*c) values=[3, 4, 0, 2, 5, 1];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S012
Before
a=2;b=0;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[2, 5, 3, 0, 1, 4];a*(b*c) values=[2, 5, 3, 0, 1, 4];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S013
Before
a=2;b=1;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[5, 3, 4, 1, 2, 0];a*(b*c) values=[5, 3, 4, 1, 2, 0];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S014
Before
a=2;b=2;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[3, 4, 0, 2, 5, 1];a*(b*c) values=[3, 4, 0, 2, 5, 1];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S015
Before
a=2;b=3;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S016
Before
a=2;b=4;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[1, 2, 5, 4, 0, 3];a*(b*c) values=[1, 2, 5, 4, 0, 3];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S017
Before
a=2;b=5;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[4, 0, 1, 5, 3, 2];a*(b*c) values=[4, 0, 1, 5, 3, 2];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S018
Before
a=3;b=0;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[3, 4, 0, 2, 5, 1];a*(b*c) values=[3, 4, 0, 2, 5, 1];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S019
Before
a=3;b=1;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[4, 0, 1, 5, 3, 2];a*(b*c) values=[4, 0, 1, 5, 3, 2];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S020
Before
a=3;b=2;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S021
Before
a=3;b=3;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[2, 5, 3, 0, 1, 4];a*(b*c) values=[2, 5, 3, 0, 1, 4];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S022
Before
a=3;b=4;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[5, 3, 4, 1, 2, 0];a*(b*c) values=[5, 3, 4, 1, 2, 0];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S023
Before
a=3;b=5;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[1, 2, 5, 4, 0, 3];a*(b*c) values=[1, 2, 5, 4, 0, 3];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S024
Before
a=4;b=0;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[4, 0, 1, 5, 3, 2];a*(b*c) values=[4, 0, 1, 5, 3, 2];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S025
Before
a=4;b=1;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S026
Before
a=4;b=2;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[1, 2, 5, 4, 0, 3];a*(b*c) values=[1, 2, 5, 4, 0, 3];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S027
Before
a=4;b=3;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[5, 3, 4, 1, 2, 0];a*(b*c) values=[5, 3, 4, 1, 2, 0];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S028
Before
a=4;b=4;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[3, 4, 0, 2, 5, 1];a*(b*c) values=[3, 4, 0, 2, 5, 1];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S029
Before
a=4;b=5;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[2, 5, 3, 0, 1, 4];a*(b*c) values=[2, 5, 3, 0, 1, 4];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S030
Before
a=5;b=0;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[5, 3, 4, 1, 2, 0];a*(b*c) values=[5, 3, 4, 1, 2, 0];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S031
Before
a=5;b=1;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[3, 4, 0, 2, 5, 1];a*(b*c) values=[3, 4, 0, 2, 5, 1];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S032
Before
a=5;b=2;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[4, 0, 1, 5, 3, 2];a*(b*c) values=[4, 0, 1, 5, 3, 2];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S033
Before
a=5;b=3;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[1, 2, 5, 4, 0, 3];a*(b*c) values=[1, 2, 5, 4, 0, 3];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S034
Before
a=5;b=4;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[2, 5, 3, 0, 1, 4];a*(b*c) values=[2, 5, 3, 0, 1, 4];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S035
Before
a=5;b=5;all c=[0, 1, 2, 3, 4, 5]
After
(a*b)*c values=[0, 1, 2, 3, 4, 5];a*(b*c) values=[0, 1, 2, 3, 4, 5];equal=Yes
Conditions: none
For each ordered(a,b),list both sides over every c.Only equality of every row qualifies the whole finite associativity obligation.
Rule: associativity_every_third_label · Uses: G001, S036
Before
candidate identity=0;left products=[0, 1, 2, 3, 4, 5];right products=[0, 1, 2, 3, 4, 5]
After
both equal original carrier labels=Yes
Conditions: none
Require both e*x=x and x*e=x.Label0 has no special status;read both table directions.
Rule: two_sided_identity_derive_not_zero_label · Uses: G001, S037
Before
candidate identity=1;left products=[1, 2, 5, 4, 0, 3];right products=[1, 2, 5, 4, 0, 3]
After
both equal original carrier labels=No
Conditions: none
Require both e*x=x and x*e=x.Label0 has no special status;read both table directions.
Rule: two_sided_identity_derive_not_zero_label · Uses: G001, S038
Before
candidate identity=2;left products=[2, 5, 3, 0, 1, 4];right products=[2, 5, 3, 0, 1, 4]
After
both equal original carrier labels=No
Conditions: none
Require both e*x=x and x*e=x.Label0 has no special status;read both table directions.
Rule: two_sided_identity_derive_not_zero_label · Uses: G001, S039
Before
candidate identity=3;left products=[3, 4, 0, 2, 5, 1];right products=[3, 4, 0, 2, 5, 1]
After
both equal original carrier labels=No
Conditions: none
Require both e*x=x and x*e=x.Label0 has no special status;read both table directions.
Rule: two_sided_identity_derive_not_zero_label · Uses: G001, S040
Before
candidate identity=4;left products=[4, 0, 1, 5, 3, 2];right products=[4, 0, 1, 5, 3, 2]
After
both equal original carrier labels=No
Conditions: none
Require both e*x=x and x*e=x.Label0 has no special status;read both table directions.
Rule: two_sided_identity_derive_not_zero_label · Uses: G001, S041
Before
candidate identity=5;left products=[5, 3, 4, 1, 2, 0];right products=[5, 3, 4, 1, 2, 0]
After
both equal original carrier labels=No
Conditions: none
Require both e*x=x and x*e=x.Label0 has no special status;read both table directions.
Rule: two_sided_identity_derive_not_zero_label · Uses: G001, S042
Before
element=0;actual identity=0;two-sided inverse candidates=[0]
After
inverse=0
Conditions: none
Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning.
Rule: group_inverse_qualified_at_actual_identity · Uses: G001, S043
Before
element=1;actual identity=0;two-sided inverse candidates=[4]
After
inverse=4
Conditions: none
Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning.
Rule: group_inverse_qualified_at_actual_identity · Uses: G001, S044
Before
element=2;actual identity=0;two-sided inverse candidates=[3]
After
inverse=3
Conditions: none
Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning.
Rule: group_inverse_qualified_at_actual_identity · Uses: G001, S045
Before
element=3;actual identity=0;two-sided inverse candidates=[2]
After
inverse=2
Conditions: none
Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning.
Rule: group_inverse_qualified_at_actual_identity · Uses: G001, S046
Before
element=4;actual identity=0;two-sided inverse candidates=[1]
After
inverse=1
Conditions: none
Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning.
Rule: group_inverse_qualified_at_actual_identity · Uses: G001, S047
Before
element=5;actual identity=0;two-sided inverse candidates=[5]
After
inverse=5
Conditions: none
Find a two-sided inverse returning to the actual identity for every element.Missing any inverse prevents later group reasoning.
Rule: group_inverse_qualified_at_actual_identity · Uses: G001, S048
Before
element=3;actual identity=0
After
zero repetitions give=0
Conditions: none
Start powers at the actual identity.Zero repetitions return it;no extra commutativity premise has been assumed.
Rule: power_starts_at_actual_identity · Uses: G001, S049
Before
repetition=1;previous=0;element=3;table row=[0, 1, 2, 3, 4, 5]
After
new value=table[0][3]=3
Conditions: none
Look up the current accumulated value and element in the original table,retaining iteration order.Labels are not integer addition.
Rule: ordered_group_power_table_lookup · Uses: G001, S050
Before
repetition=2;previous=3;element=3;table row=[3, 4, 0, 2, 5, 1]
After
new value=table[3][3]=2
Conditions: none
Look up the current accumulated value and element in the original table,retaining iteration order.Labels are not integer addition.
Rule: ordered_group_power_table_lookup · Uses: G001, S051
Before
repetition=3;previous=2;element=3;table row=[2, 5, 3, 0, 1, 4]
After
new value=table[2][3]=0
Conditions: none
Look up the current accumulated value and element in the original table,retaining iteration order.Labels are not integer addition.
Rule: ordered_group_power_table_lookup · Uses: G001, S052
Before
successive distinct powers=[0, 3, 2];first repeat=0;actual identity=0
After
least positive return order=3;generated cyclic subgroup size=3
Conditions: none
If two powers agree,group inverses cancel the earlier power to give a positive return to identity.Thus the first repeat is the starting identity with no earlier positive return.The preceding powers form a cyclic subgroup,not necessarily the whole group.
Rule: first_positive_identity_return_and_cyclic_scope · Uses: G001, S053
S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.(3, 0, 3, Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 3294bacfa7ee49015d07326f6998b6df0b21388adde0df249aad704d9fb10090candidate record SHA-256: 25816c64f8b046f29ff454565221ecdd116bddb913ec23e58ac9dba28099becasemantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ffquestion-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18Lean: NOT_RUN · human review: NOT_PERFORMED