OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 25634: The Center Requires Commutation with Every Group Element

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

Return to catalogue page 129

← Question derivation index · Original question batch

R0050-25634 · q-20261015-advance-20261115-r02-finite-group-center-0009

Original question and givens

Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2], [13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3], [14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4], [15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5], [16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6], [17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7], [18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8], [19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9], [20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

  • G001 — Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2], [13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3], [14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4], [15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5], [16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6], [17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7], [18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8], [19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9], [20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Give (commutation_rows, center, certificate).

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 = [center-commutation-table]

After

derived = [[commutation_rows, center, certificate]]

Conditions: none

List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [Center ORIGINAL group, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2], [13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3], [14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4], [15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5], [16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6], [17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7], [18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8], [19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9], [20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10], [21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], 0]

After

derived = [22, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7], [8, 8, 8, 8], [9, 9, 9, 9], [10, 10, 10, 10], [11, 11, 11, 11], [12, 12, 12, 12], [13, 13, 13, 13], [14, 14, 14, 14], [15, 15, 15, 15], [16, 16, 16, 16], [17, 17, 17, 17], [18, 18, 18, 18], [19, 19, 19, 19], [20, 20, 20, 20], [21, 21, 21, 21]], Yes]

Conditions: none

Qualify the complete closed Cayley table and actual identity first,allowing order64 without assuming identity0.

Step identifiers and dependencies

Rule: Group_complete_table_actual_identity_bound64 · Uses: G001, S001

S003 · implies

Before

source roles = [Center ORIGINAL group, 22, 0]

After

derived = [[[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], 10648, [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

Conditions: none

Each pair-matrix entry checks every third element for associativity,covering all n³ triples,not samples.Inverses return the actual identity from both sides.

Step identifiers and dependencies

Rule: Group_ALL_triples_compact_pair_matrix_and_ALL_inverses · Uses: G001, S002

S004 · implies

Before

source roles = [0, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], Yes]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S003

S005 · implies

Before

source roles = [1, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S004

S006 · implies

Before

source roles = [2, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S005

S007 · implies

Before

source roles = [3, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S006

S008 · implies

Before

source roles = [4, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S007

S009 · implies

Before

source roles = [5, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S008

S010 · implies

Before

source roles = [6, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S009

S011 · implies

Before

source roles = [7, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S010

S012 · implies

Before

source roles = [8, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S011

S013 · implies

Before

source roles = [9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S012

S014 · implies

Before

source roles = [10, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S013

S015 · implies

Before

source roles = [11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S014

S016 · implies

Before

source roles = [12, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 13, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S015

S017 · implies

Before

source roles = [13, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 14, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3], [13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S016

S018 · implies

Before

source roles = [14, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 15, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4], [14, 15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S017

S019 · implies

Before

source roles = [15, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5], [15, 16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S018

S020 · implies

Before

source roles = [16, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6], [16, 17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S019

S021 · implies

Before

source roles = [17, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 18, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7], [17, 18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S020

S022 · implies

Before

source roles = [18, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 19, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8], [18, 19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S021

S023 · implies

Before

source roles = [19, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 20, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9], [19, 20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S022

S024 · implies

Before

source roles = [20, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 21, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10], [20, 21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S023

S025 · implies

Before

source roles = [21, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]]

After

derived = [[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [21, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes], No]

Conditions: none

For each z compare zg and gz across the full row,not the centralizer of one chosen g.

Step identifiers and dependencies

Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S024

S026 · implies

Before

source roles = [Center ACTUAL Z, [0], 0]

After

derived = [Yes, [[0, 0, 0, Yes]], Yes, [[0, 0, Yes]], Yes, Yes, [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes]], Yes]

Conditions: none

Check identity,all products and inverse closure separately before subgroup qualification,then every g for normality.These are actual properties that may individually hold even when a map fails;do not fill flags from a theorem name.

Step identifiers and dependencies

Rule: Subgroup_actual_identity_product_inverse_ALL_conjugates · Uses: G001, S025

S027 · implies

Before

source roles = [[0], [[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No], [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No], [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No], [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]]

After

derived = [Yes, Yes, central z: g*z*g^-1=z for EVERY g]

Conditions: none

Identity commutes with every g;products and inverses of all-g commuting elements do too.Right multiplying gz=zg gives gzg^-1=z,so central elements are fixed pointwise and the center is normal.No characteristic or class-equation claim.

Step identifiers and dependencies

Rule: Center_GENERAL_closure_and_pointwise_fixed_conjugation · Uses: G001, S026

Reverse checks

  • 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

([[0, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], [1, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [2, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [3, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [4, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [5, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [6, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [7, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [8, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [9, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [10, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, No, No, No, No, No, No, No, No, No]], [11, [Yes, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No, No]], [12, [Yes, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No, No]], [13, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No, No]], [14, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No, No]], [15, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No, No]], [16, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No, No]], [17, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No, No]], [18, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No, No]], [19, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No, No]], [20, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes, No]], [21, [Yes, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, No, Yes]]], [0], Z_G_contains_exactly_the_elements_commuting_with_every_g)

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: 1a4d06b291947b508896da2bbb54aec890300587567d070c6c650ee5ff79d75e
  • candidate record SHA-256: 03a4e7dc63090684ca833565657636de858985d6008465c606ff118355f51822
  • semantic review receipt SHA-256: 74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82
  • question-first witness SHA-256: 6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON