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Question 29215: Exhaustive Associativity over All Ordered Triples

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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← Question derivation index · Original question batch

R0057-29215 · q-20261013-advance-20261022-r05-finite-group-0040

Original question and givens

The complete finite Cayley table=[[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.

  • G001 — The complete finite Cayley table=[[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.

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

source roles = [qf-str22-group-associativity-audit, group-associativity-audit]

After

derived = [[n, table, bad_triples, associative]]

Conditions: none

Use this bank exact ordered tuple and hyphenated operation.Rational outputs reduce with positive denominators and OMIT/1 for integers,unlike the preceding legacy bank.Do not change original bilingual snapshots or meaning.

Step identifiers and dependencies

Rule: STR22_exact_interface_and_source_ORDERED_tuple_roles · Uses: G001

S002 · implies

Before

source roles = [5, 0, [3, 4, 1, 2, 0]]

After

derived = [[Yes, Yes, Yes, Yes, Yes]]

Conditions: none

Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms.

Step identifiers and dependencies

Rule: Group_EVERY_original_row_closure · Uses: G001, S001

S003 · implies

Before

source roles = [5, 1, [4, 2, 3, 0, 1]]

After

derived = [[Yes, Yes, Yes, Yes, Yes]]

Conditions: none

Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms.

Step identifiers and dependencies

Rule: Group_EVERY_original_row_closure · Uses: G001, S002

S004 · implies

Before

source roles = [5, 2, [1, 3, 0, 4, 2]]

After

derived = [[Yes, Yes, Yes, Yes, Yes]]

Conditions: none

Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms.

Step identifiers and dependencies

Rule: Group_EVERY_original_row_closure · Uses: G001, S003

S005 · implies

Before

source roles = [5, 3, [2, 0, 4, 1, 3]]

After

derived = [[Yes, Yes, Yes, Yes, Yes]]

Conditions: none

Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms.

Step identifiers and dependencies

Rule: Group_EVERY_original_row_closure · Uses: G001, S004

S006 · implies

Before

source roles = [5, 4, [0, 1, 2, 3, 4]]

After

derived = [[Yes, Yes, Yes, Yes, Yes]]

Conditions: none

Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms.

Step identifiers and dependencies

Rule: Group_EVERY_original_row_closure · Uses: G001, S005

S007 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 0, 0]

After

derived = [[2, 0, 4, 1, 3], [2, 0, 4, 1, 3], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S006

S008 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 0, 1]

After

derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S007

S009 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 0, 2]

After

derived = [[4, 2, 3, 0, 1], [4, 2, 3, 0, 1], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S008

S010 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 0, 3]

After

derived = [[1, 3, 0, 4, 2], [1, 3, 0, 4, 2], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S009

S011 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 0, 4]

After

derived = [[3, 4, 1, 2, 0], [3, 4, 1, 2, 0], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S010

S012 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 1, 0]

After

derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S011

S013 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 1, 1]

After

derived = [[1, 3, 0, 4, 2], [1, 3, 0, 4, 2], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S012

S014 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 1, 2]

After

derived = [[2, 0, 4, 1, 3], [2, 0, 4, 1, 3], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S013

S015 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 1, 3]

After

derived = [[3, 4, 1, 2, 0], [3, 4, 1, 2, 0], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S014

S016 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 1, 4]

After

derived = [[4, 2, 3, 0, 1], [4, 2, 3, 0, 1], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S015

S017 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 2, 0]

After

derived = [[4, 2, 3, 0, 1], [4, 2, 3, 0, 1], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S016

S018 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 2, 1]

After

derived = [[2, 0, 4, 1, 3], [2, 0, 4, 1, 3], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S017

S019 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 2, 2]

After

derived = [[3, 4, 1, 2, 0], [3, 4, 1, 2, 0], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S018

S020 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 2, 3]

After

derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S019

S021 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 2, 4]

After

derived = [[1, 3, 0, 4, 2], [1, 3, 0, 4, 2], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S020

S022 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 3, 0]

After

derived = [[1, 3, 0, 4, 2], [1, 3, 0, 4, 2], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S021

S023 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 3, 1]

After

derived = [[3, 4, 1, 2, 0], [3, 4, 1, 2, 0], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S022

S024 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 3, 2]

After

derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S023

S025 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 3, 3]

After

derived = [[4, 2, 3, 0, 1], [4, 2, 3, 0, 1], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S024

S026 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 3, 4]

After

derived = [[2, 0, 4, 1, 3], [2, 0, 4, 1, 3], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S025

S027 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 4, 0]

After

derived = [[3, 4, 1, 2, 0], [3, 4, 1, 2, 0], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S026

S028 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 4, 1]

After

derived = [[4, 2, 3, 0, 1], [4, 2, 3, 0, 1], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S027

S029 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 4, 2]

After

derived = [[1, 3, 0, 4, 2], [1, 3, 0, 4, 2], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S028

S030 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 4, 3]

After

derived = [[2, 0, 4, 1, 3], [2, 0, 4, 1, 3], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S029

S031 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], 4, 4]

After

derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S030

S032 · implies

Before

source roles = [5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]]]

After

derived = [[], Yes]

Conditions: none

The(a,b) loops and ALL c cover n³ ordered triples.Retain ALL bad triples in nested increasing order;only an empty list certifies associativity of THIS complete finite table,not an unknown structure.

Step identifiers and dependencies

Rule: Group_ALL_bad_triples_nested_source_order · Uses: G001, S031

Reverse checks

  • S032 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S031 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S030 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S029 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S028 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S027 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S026 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S025 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S024 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S023 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S022 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S021 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S020 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S019 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S018 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S017 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S016 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S015 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S014 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S013 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S012 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S011 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S010 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S009 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S008 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S007 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S006 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S005 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S004 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S003 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S002 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S001 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.

Result

(5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], [], 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: 4f52adf21a19d4e28e3be80afbaeef6b3001697502f793d6e1c4c0f19356e47c
  • candidate record SHA-256: 8727a0941e812d210ef13efb56ea3a6fb813d0dfd7c274c15c3f516dc11751d8
  • semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6
  • question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON