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Question 10923: Operation Preservation for Every Source Pair

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0020-10923 · q-20261013-advance-20261023-r03-subgroup-homomorphism-0073

Original question and givens

Use complete qualified finite group Cayley tables with carrier labels 0 through size−1 and the given row/column order. Every group axiom is a premise already checked on the whole table. Results concern these finite groups; a homomorphism need not be injective or an isomorphism. source size=5, source table=[[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], target size=5, target table=[[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], total mapping=[3, 0, 2, 4, 1] in source-label order. Give (source size,target size,mapping,every bad [a,b] pair,homomorphism?). A bad pair violates f(a*b)=f(a)*f(b); use nested increasing source-label order.

  • G001 — Use complete qualified finite group Cayley tables with carrier labels 0 through size−1 and the given row/column order. Every group axiom is a premise already checked on the whole table. Results concern these finite groups; a homomorphism need not be injective or an isomorphism. source size=5, source table=[[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], target size=5, target table=[[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], total mapping=[3, 0, 2, 4, 1] in source-label order. Give (source size,target size,mapping,every bad [a,b] pair,homomorphism?). A bad pair violates f(a*b)=f(a)*f(b); use nested increasing source-label order.

Definition and scope

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.

Step-by-step derivation

S001 · implies

Before

group=source;carrier Fin(5);complete Cayley table=[[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]]

After

typed operation closure=Yes;all associativity failures=[]

Conditions: none

Type the complete operation table and check every associativity triple,not only cells used by a subset or map.These tasks require qualified finite groups.

Step identifiers and dependencies

Rule: complete_group_table_closure_associativity · Uses: G001

S002 · implies

Before

group=source;candidate identity=0;left products row=[3, 4, 0, 1, 2];right products column=[3, 4, 0, 1, 2]

After

both sides equal carrier labels=No

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S001

S003 · implies

Before

group=source;candidate identity=1;left products row=[4, 0, 1, 2, 3];right products column=[4, 0, 1, 2, 3]

After

both sides equal carrier labels=No

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S002

S004 · implies

Before

group=source;candidate identity=2;left products row=[0, 1, 2, 3, 4];right products column=[0, 1, 2, 3, 4]

After

both sides equal carrier labels=Yes

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S003

S005 · implies

Before

group=source;candidate identity=3;left products row=[1, 2, 3, 4, 0];right products column=[1, 2, 3, 4, 0]

After

both sides equal carrier labels=No

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S004

S006 · implies

Before

group=source;candidate identity=4;left products row=[2, 3, 4, 0, 1];right products column=[2, 3, 4, 0, 1]

After

both sides equal carrier labels=No

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S005

S007 · implies

Before

group=source;element=0;derived identity=2;two-sided inverse candidates=[4]

After

inverse=4

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S006

S008 · implies

Before

group=source;element=1;derived identity=2;two-sided inverse candidates=[3]

After

inverse=3

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S007

S009 · implies

Before

group=source;element=2;derived identity=2;two-sided inverse candidates=[2]

After

inverse=2

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S008

S010 · implies

Before

group=source;element=3;derived identity=2;two-sided inverse candidates=[1]

After

inverse=1

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S009

S011 · implies

Before

group=source;element=4;derived identity=2;two-sided inverse candidates=[0]

After

inverse=0

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S010

S012 · implies

Before

group=target;carrier Fin(5);complete Cayley table=[[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]]

After

typed operation closure=Yes;all associativity failures=[]

Conditions: none

Type the complete operation table and check every associativity triple,not only cells used by a subset or map.These tasks require qualified finite groups.

Step identifiers and dependencies

Rule: complete_group_table_closure_associativity · Uses: G001, S011

S013 · implies

Before

group=target;candidate identity=0;left products row=[3, 4, 0, 1, 2];right products column=[3, 4, 0, 1, 2]

After

both sides equal carrier labels=No

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S012

S014 · implies

Before

group=target;candidate identity=1;left products row=[4, 0, 1, 2, 3];right products column=[4, 0, 1, 2, 3]

After

both sides equal carrier labels=No

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S013

S015 · implies

Before

group=target;candidate identity=2;left products row=[0, 1, 2, 3, 4];right products column=[0, 1, 2, 3, 4]

After

both sides equal carrier labels=Yes

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S014

S016 · implies

Before

group=target;candidate identity=3;left products row=[1, 2, 3, 4, 0];right products column=[1, 2, 3, 4, 0]

After

both sides equal carrier labels=No

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S015

S017 · implies

Before

group=target;candidate identity=4;left products row=[2, 3, 4, 0, 1];right products column=[2, 3, 4, 0, 1]

After

both sides equal carrier labels=No

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S016

S018 · implies

Before

group=target;element=0;derived identity=2;two-sided inverse candidates=[4]

After

inverse=4

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S017

S019 · implies

Before

group=target;element=1;derived identity=2;two-sided inverse candidates=[3]

After

inverse=3

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S018

S020 · implies

Before

group=target;element=2;derived identity=2;two-sided inverse candidates=[2]

After

inverse=2

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S019

S021 · implies

Before

group=target;element=3;derived identity=2;two-sided inverse candidates=[1]

After

inverse=1

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S020

S022 · implies

Before

group=target;element=4;derived identity=2;two-sided inverse candidates=[0]

After

inverse=0

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S021

S023 · implies

Before

source size=5;target size=5;whole mapping=[3, 0, 2, 4, 1]

After

all original mapping cells typed before product preservation

Conditions: none

Type the entire total mapping in original source-index order before checking all products.Do not fill missing values from partial correct data.

Step identifiers and dependencies

Rule: whole_group_mapping_type_guard · Uses: G001, S022

S024 · implies

Before

source pair=[0, 0];source product=3;mapped operands=[3, 3]

After

f(source product)=4;target product=4;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S023

S025 · implies

Before

source pair=[0, 1];source product=4;mapped operands=[3, 0]

After

f(source product)=1;target product=1;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S024

S026 · implies

Before

source pair=[0, 2];source product=0;mapped operands=[3, 2]

After

f(source product)=3;target product=3;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S025

S027 · implies

Before

source pair=[0, 3];source product=1;mapped operands=[3, 4]

After

f(source product)=0;target product=0;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S026

S028 · implies

Before

source pair=[0, 4];source product=2;mapped operands=[3, 1]

After

f(source product)=2;target product=2;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S027

S029 · implies

Before

source pair=[1, 0];source product=4;mapped operands=[0, 3]

After

f(source product)=1;target product=1;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S028

S030 · implies

Before

source pair=[1, 1];source product=0;mapped operands=[0, 0]

After

f(source product)=3;target product=3;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S029

S031 · implies

Before

source pair=[1, 2];source product=1;mapped operands=[0, 2]

After

f(source product)=0;target product=0;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S030

S032 · implies

Before

source pair=[1, 3];source product=2;mapped operands=[0, 4]

After

f(source product)=2;target product=2;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S031

S033 · implies

Before

source pair=[1, 4];source product=3;mapped operands=[0, 1]

After

f(source product)=4;target product=4;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S032

S034 · implies

Before

source pair=[2, 0];source product=0;mapped operands=[2, 3]

After

f(source product)=3;target product=3;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S033

S035 · implies

Before

source pair=[2, 1];source product=1;mapped operands=[2, 0]

After

f(source product)=0;target product=0;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S034

S036 · implies

Before

source pair=[2, 2];source product=2;mapped operands=[2, 2]

After

f(source product)=2;target product=2;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S035

S037 · implies

Before

source pair=[2, 3];source product=3;mapped operands=[2, 4]

After

f(source product)=4;target product=4;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S036

S038 · implies

Before

source pair=[2, 4];source product=4;mapped operands=[2, 1]

After

f(source product)=1;target product=1;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S037

S039 · implies

Before

source pair=[3, 0];source product=1;mapped operands=[4, 3]

After

f(source product)=0;target product=0;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S038

S040 · implies

Before

source pair=[3, 1];source product=2;mapped operands=[4, 0]

After

f(source product)=2;target product=2;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S039

S041 · implies

Before

source pair=[3, 2];source product=3;mapped operands=[4, 2]

After

f(source product)=4;target product=4;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S040

S042 · implies

Before

source pair=[3, 3];source product=4;mapped operands=[4, 4]

After

f(source product)=1;target product=1;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S041

S043 · implies

Before

source pair=[3, 4];source product=0;mapped operands=[4, 1]

After

f(source product)=3;target product=3;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S042

S044 · implies

Before

source pair=[4, 0];source product=2;mapped operands=[1, 3]

After

f(source product)=2;target product=2;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S043

S045 · implies

Before

source pair=[4, 1];source product=3;mapped operands=[1, 0]

After

f(source product)=4;target product=4;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S044

S046 · implies

Before

source pair=[4, 2];source product=4;mapped operands=[1, 2]

After

f(source product)=1;target product=1;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S045

S047 · implies

Before

source pair=[4, 3];source product=0;mapped operands=[1, 4]

After

f(source product)=3;target product=3;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S046

S048 · implies

Before

source pair=[4, 4];source product=1;mapped operands=[1, 1]

After

f(source product)=0;target product=0;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S047

S049 · implies

Before

all bad source ordered pairs=[]

After

homomorphism=Yes

Conditions: none

Preservation for all ordered source pairs is required.One failure supplies a counterexample;one success does not establish the universal condition.

Step identifiers and dependencies

Rule: whole_homomorphism_audit · Uses: G001, S048

Reverse checks

  • 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.

Result

(5, 5, [3, 0, 2, 4, 1], [], Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 50e87ae6b6897304abd87bbe5f2046c7bc477eec9e5112e428c47c17cb113c81
  • candidate record SHA-256: c269ed0214240687e55b67a9e324be6dd876be8053c6d50b791e765bc86f3315
  • semantic review receipt SHA-256: b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598f
  • question-first witness SHA-256: 3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON