Question 29255: Two-Sided Identity and Two-Sided Inverse Inventories

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0057-29255 · q-20261013-advance-20261022-r05-finite-group-0055

Original question and givens

The complete finite Cayley table=[[1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 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 closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.

G001 — The complete finite Cayley table=[[1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 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 closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.

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.

mko-finite-group

Step-by-step derivation

S001 · implies

Before

source roles = [qf-str22-group-identity-inverse-inventory, group-identity-inverse-inventory]

After

derived = [[n, table, identities, inverse_rows, all_have_inverse]]

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

After

derived = [[1, 2, 3, 4, 0], [1, 2, 3, 4, 0], No]

Conditions: none

The SAME e must satisfy e*x=x AND x*e=x for EVERY x.A one-sided condition or one row alone is insufficient.

Step identifiers and dependencies

Rule: Group_EACH_candidate_identity_BOTH_sides_ALL_elements · Uses: G001, S006

S008 · implies

Before

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

After

derived = [[2, 3, 4, 0, 1], [2, 3, 4, 0, 1], No]

Conditions: none

The SAME e must satisfy e*x=x AND x*e=x for EVERY x.A one-sided condition or one row alone is insufficient.

Step identifiers and dependencies

Rule: Group_EACH_candidate_identity_BOTH_sides_ALL_elements · Uses: G001, S007

S009 · implies

Before

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

After

derived = [[3, 4, 0, 1, 2], [3, 4, 0, 1, 2], No]

Conditions: none

The SAME e must satisfy e*x=x AND x*e=x for EVERY x.A one-sided condition or one row alone is insufficient.

Step identifiers and dependencies

Rule: Group_EACH_candidate_identity_BOTH_sides_ALL_elements · Uses: G001, S008

S010 · implies

Before

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

After

derived = [[4, 0, 1, 2, 3], [4, 0, 1, 2, 3], No]

Conditions: none

The SAME e must satisfy e*x=x AND x*e=x for EVERY x.A one-sided condition or one row alone is insufficient.

Step identifiers and dependencies

Rule: Group_EACH_candidate_identity_BOTH_sides_ALL_elements · Uses: G001, S009

S011 · implies

Before

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

After

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

Conditions: none

The SAME e must satisfy e*x=x AND x*e=x for EVERY x.A one-sided condition or one row alone is insufficient.

Step identifiers and dependencies

Rule: Group_EACH_candidate_identity_BOTH_sides_ALL_elements · Uses: G001, S010

S012 · implies

Before

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

After

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

Conditions: none

Enumerate EVERY y for this x,and require BOTH products equal the SAME e.Retain all candidates;without established associativity,do not assume inverse uniqueness.

Step identifiers and dependencies

Rule: Group_EVERY_element_ALL_two_sided_inverse_candidates · Uses: G001, S011

S013 · implies

Before

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

After

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

Conditions: none

Enumerate EVERY y for this x,and require BOTH products equal the SAME e.Retain all candidates;without established associativity,do not assume inverse uniqueness.

Step identifiers and dependencies

Rule: Group_EVERY_element_ALL_two_sided_inverse_candidates · Uses: G001, S012

S014 · implies

Before

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

After

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

Conditions: none

Enumerate EVERY y for this x,and require BOTH products equal the SAME e.Retain all candidates;without established associativity,do not assume inverse uniqueness.

Step identifiers and dependencies

Rule: Group_EVERY_element_ALL_two_sided_inverse_candidates · Uses: G001, S013

S015 · implies

Before

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

After

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

Conditions: none

Enumerate EVERY y for this x,and require BOTH products equal the SAME e.Retain all candidates;without established associativity,do not assume inverse uniqueness.

Step identifiers and dependencies

Rule: Group_EVERY_element_ALL_two_sided_inverse_candidates · Uses: G001, S014

S016 · implies

Before

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

After

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

Conditions: none

Enumerate EVERY y for this x,and require BOTH products equal the SAME e.Retain all candidates;without established associativity,do not assume inverse uniqueness.

Step identifiers and dependencies

Rule: Group_EVERY_element_ALL_two_sided_inverse_candidates · Uses: G001, S015

S017 · implies

Before

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

After

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

Conditions: none

Without a unique two-sided e,the inverse-row list is empty and all-have-inverse is No,not vacuous True.An identity/inverse inventory still does not include associativity.

Step identifiers and dependencies

Rule: Group_unique_identity_required_NO_vacuous_inverse_claim · Uses: G001, S016

Reverse checks

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, [[1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4]], [4], [[0, [3]], [1, [2]], [2, [1]], [3, [0]], [4, [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: b0b14bd07aafb70917952fd9b693408058c631bdbba7842652badb594e8c1320
candidate record SHA-256: 396813d84a8ce81b1743671bc8603da9236da4e3eddc844e7cce61f0d96d8ca0
semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6
question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON