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Question 20147: Every Point Must Be Fixed by the Declared Identity Row

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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R0039-20147 · q-20261014-advance-20261107-r01-finite-group-action-0047

Original question and givens

Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).

  • G001 — Use complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]] with row g/column h giving g*h, declared identity=0, point carrier Fin(7)={0,…,6}, and total action_table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]] with row g/column x giving g*x. Check the full group axioms separately from e*x=x and (g*h)*x=g*(h*x). Permutation rows alone are not an action certificate. Give (declared_identity, point_checks, identity_action).

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 operation=groupaction-identity-audit;original snapshot and mathematics retained

After

explicit ordered source output fields=[declared_identity, point_checks, identity_action]

Conditions: none

Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the source claim scope.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

ORIGINAL complete group-label carrier Fin(3);declared identity label=0;Cayley table row g,column h gives g*h=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]

After

ORIGINAL complete point carrier Fin(7);TOTAL action table row g,column x gives g*x=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 0, 3, 4, 5, 6], [2, 0, 1, 3, 4, 5, 6]];all entries in range,so structural closure/totality checked;GROUP AXIOMS not assumed from shape

Conditions: none

The two tables have different column roles:group label h versus action point x.Complete in-range tables establish closure/totality only.A candidate group may fail its axioms;its table name does not certify them.

Step identifiers and dependencies

Rule: groupaction_two_table_roles_complete_candidate_not_group_assumption · Uses: G001, S001

S003 · implies

Before

ACTUAL declared identity label=0;ORIGINAL identity action row=[0, 1, 2, 3, 4, 5, 6];complete point carrier Fin(7)

After

ALL ordered [point,identity_image,equality] rows=[[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]];identity-action axiom=Yes

Conditions: none

Check e*x=x at every action point,using the actual declared identity label rather than assuming0.This action axiom is distinct from the two-sided identity law in the group table.

Step identifiers and dependencies

Rule: groupaction_every_point_identity_action_not_group_identity · Uses: G001, S002

S004 · implies

Before

this component audited ONLY the complete identity action row

After

identity-action flag alone does NOT establish group qualification or action compatibility

Conditions: none

This output answers the identity-row obligation only;the flag does not replace the remaining axioms or promote one component to valid_action.

Step identifiers and dependencies

Rule: groupaction_identity_component_not_full_action_claim · Uses: G001, S003

Reverse checks

  • 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

(0, [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes], [4, 4, Yes], [5, 5, Yes], [6, 6, Yes]], 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: ce92d186d6875f8101ed716b69c1811eb3caff975206ff726ccd16dacdfb8dd1
  • candidate record SHA-256: d21cb1644b8673da97f17f4c2db98de667adae15038be86a8a74d1aace821d31
  • semantic review receipt SHA-256: 6e11e35fafe32f16900ce9b522d01acb7b2ee6b463ab179392076cfbe1af8034
  • question-first witness SHA-256: cc0fa361fc25151729e70264340d2eae94f179f6f2a2edfe80ba5704fbbd0682
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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