Question 25331: Explicit Representative Dependence Blocks This Quotient Multiplication

Original question and givens

Use complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 0, 9, 10, 11, 12, 13, 14, 15, 8], [2, 3, 4, 5, 6, 7, 0, 1, 10, 11, 12, 13, 14, 15, 8, 9], [3, 4, 5, 6, 7, 0, 1, 2, 11, 12, 13, 14, 15, 8, 9, 10], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 6, 7, 0, 1, 2, 3, 4, 13, 14, 15, 8, 9, 10, 11, 12], [6, 7, 0, 1, 2, 3, 4, 5, 14, 15, 8, 9, 10, 11, 12, 13], [7, 0, 1, 2, 3, 4, 5, 6, 15, 8, 9, 10, 11, 12, 13, 14], [8, 15, 14, 13, 12, 11, 10, 9, 0, 7, 6, 5, 4, 3, 2, 1], [9, 8, 15, 14, 13, 12, 11, 10, 1, 0, 7, 6, 5, 4, 3, 2], [10, 9, 8, 15, 14, 13, 12, 11, 2, 1, 0, 7, 6, 5, 4, 3], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [12, 11, 10, 9, 8, 15, 14, 13, 4, 3, 2, 1, 0, 7, 6, 5], [13, 12, 11, 10, 9, 8, 15, 14, 5, 4, 3, 2, 1, 0, 7, 6], [14, 13, 12, 11, 10, 9, 8, 15, 6, 5, 4, 3, 2, 1, 0, 7], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0, explicit subset=[0, 8] and the entire left-coset partition. Subgroup qualification precedes normality. Quotient elements are coset sets, not representatives; coset indices follow the source canonical element-list order. Give (subgroup, normal, cosets, representative_counterexample, well_defined, quotient_group_certificate, boundary).

• G001 — Use complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 0, 9, 10, 11, 12, 13, 14, 15, 8], [2, 3, 4, 5, 6, 7, 0, 1, 10, 11, 12, 13, 14, 15, 8, 9], [3, 4, 5, 6, 7, 0, 1, 2, 11, 12, 13, 14, 15, 8, 9, 10], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 6, 7, 0, 1, 2, 3, 4, 13, 14, 15, 8, 9, 10, 11, 12], [6, 7, 0, 1, 2, 3, 4, 5, 14, 15, 8, 9, 10, 11, 12, 13], [7, 0, 1, 2, 3, 4, 5, 6, 15, 8, 9, 10, 11, 12, 13, 14], [8, 15, 14, 13, 12, 11, 10, 9, 0, 7, 6, 5, 4, 3, 2, 1], [9, 8, 15, 14, 13, 12, 11, 10, 1, 0, 7, 6, 5, 4, 3, 2], [10, 9, 8, 15, 14, 13, 12, 11, 2, 1, 0, 7, 6, 5, 4, 3], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [12, 11, 10, 9, 8, 15, 14, 13, 4, 3, 2, 1, 0, 7, 6, 5], [13, 12, 11, 10, 9, 8, 15, 14, 5, 4, 3, 2, 1, 0, 7, 6], [14, 13, 12, 11, 10, 9, 8, 15, 6, 5, 4, 3, 2, 1, 0, 7], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0, explicit subset=[0, 8] and the entire left-coset partition. Subgroup qualification precedes normality. Quotient elements are coset sets, not representatives; coset indices follow the source canonical element-list order. Give (subgroup, normal, cosets, representative_counterexample, well_defined, quotient_group_certificate, boundary).

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-quotient-group

Step-by-step derivation

source roles = [quotient-nonnormal-boundary]

derived = [[subgroup, normal, cosets, representative_counterexample, well_defined, quotient_group_certificate, boundary]]

source roles = [[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 0, 9, 10, 11, 12, 13, 14, 15, 8], [2, 3, 4, 5, 6, 7, 0, 1, 10, 11, 12, 13, 14, 15, 8, 9], [3, 4, 5, 6, 7, 0, 1, 2, 11, 12, 13, 14, 15, 8, 9, 10], [4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11], [5, 6, 7, 0, 1, 2, 3, 4, 13, 14, 15, 8, 9, 10, 11, 12], [6, 7, 0, 1, 2, 3, 4, 5, 14, 15, 8, 9, 10, 11, 12, 13], [7, 0, 1, 2, 3, 4, 5, 6, 15, 8, 9, 10, 11, 12, 13, 14], [8, 15, 14, 13, 12, 11, 10, 9, 0, 7, 6, 5, 4, 3, 2, 1], [9, 8, 15, 14, 13, 12, 11, 10, 1, 0, 7, 6, 5, 4, 3, 2], [10, 9, 8, 15, 14, 13, 12, 11, 2, 1, 0, 7, 6, 5, 4, 3], [11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4], [12, 11, 10, 9, 8, 15, 14, 13, 4, 3, 2, 1, 0, 7, 6, 5], [13, 12, 11, 10, 9, 8, 15, 14, 5, 4, 3, 2, 1, 0, 7, 6], [14, 13, 12, 11, 10, 9, 8, 15, 6, 5, 4, 3, 2, 1, 0, 7], [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], 0]

derived = [16, [[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]], Yes]

source roles = [16, 0]

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]], 4096, [0, 7, 6, 5, 4, 3, 2, 1, 8, 9, 10, 11, 12, 13, 14, 15]]

source roles = [0, [0, 8]]

derived = [[[0, 0, 0, Yes], [0, 8, 8, Yes], [8, 0, 8, Yes], [8, 8, 0, Yes]], [[0, 0, Yes], [8, 8, Yes]], Yes, [[0, [0, 8]], [1, [0, 10]], [2, [0, 12]], [3, [0, 14]], [4, [0, 8]], [5, [0, 10]], [6, [0, 12]], [7, [0, 14]], [8, [0, 8]], [9, [0, 10]], [10, [0, 12]], [11, [0, 14]], [12, [0, 8]], [13, [0, 10]], [14, [0, 12]], [15, [0, 14]]], No]

source roles = [[0, 8], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]]

derived = [[[0, 8], [1, 9], [2, 10], [3, 11], [4, 12], [5, 13], [6, 14], [7, 15]], [0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7], Yes]

source roles = [0, 0, [0, 8], [0, 8]]

derived = [[[0, 0, 0, 0], [0, 8, 8, 0], [8, 0, 8, 0], [8, 8, 0, 0]], [0], Yes]

source roles = [0, 1, [0, 8], [1, 9]]

derived = [[[0, 1, 1, 1], [0, 9, 9, 1], [8, 1, 15, 7], [8, 9, 7, 7]], [1, 7], No]

source roles = [0, 2, [0, 8], [2, 10]]

derived = [[[0, 2, 2, 2], [0, 10, 10, 2], [8, 2, 14, 6], [8, 10, 6, 6]], [2, 6], No]

source roles = [0, 3, [0, 8], [3, 11]]

derived = [[[0, 3, 3, 3], [0, 11, 11, 3], [8, 3, 13, 5], [8, 11, 5, 5]], [3, 5], No]

source roles = [0, 4, [0, 8], [4, 12]]

derived = [[[0, 4, 4, 4], [0, 12, 12, 4], [8, 4, 12, 4], [8, 12, 4, 4]], [4], Yes]

source roles = [0, 5, [0, 8], [5, 13]]

derived = [[[0, 5, 5, 5], [0, 13, 13, 5], [8, 5, 11, 3], [8, 13, 3, 3]], [3, 5], No]

source roles = [0, 6, [0, 8], [6, 14]]

derived = [[[0, 6, 6, 6], [0, 14, 14, 6], [8, 6, 10, 2], [8, 14, 2, 2]], [2, 6], No]

source roles = [0, 7, [0, 8], [7, 15]]

derived = [[[0, 7, 7, 7], [0, 15, 15, 7], [8, 7, 9, 1], [8, 15, 1, 1]], [1, 7], No]

source roles = [1, 0, [1, 9], [0, 8]]

derived = [[[1, 0, 1, 1], [1, 8, 9, 1], [9, 0, 9, 1], [9, 8, 1, 1]], [1], Yes]

source roles = [1, 1, [1, 9], [1, 9]]

derived = [[[1, 1, 2, 2], [1, 9, 10, 2], [9, 1, 8, 0], [9, 9, 0, 0]], [0, 2], No]

source roles = [1, 2, [1, 9], [2, 10]]

derived = [[[1, 2, 3, 3], [1, 10, 11, 3], [9, 2, 15, 7], [9, 10, 7, 7]], [3, 7], No]

source roles = [1, 3, [1, 9], [3, 11]]

derived = [[[1, 3, 4, 4], [1, 11, 12, 4], [9, 3, 14, 6], [9, 11, 6, 6]], [4, 6], No]

source roles = [1, 4, [1, 9], [4, 12]]

derived = [[[1, 4, 5, 5], [1, 12, 13, 5], [9, 4, 13, 5], [9, 12, 5, 5]], [5], Yes]

source roles = [1, 5, [1, 9], [5, 13]]

derived = [[[1, 5, 6, 6], [1, 13, 14, 6], [9, 5, 12, 4], [9, 13, 4, 4]], [4, 6], No]

source roles = [1, 6, [1, 9], [6, 14]]

derived = [[[1, 6, 7, 7], [1, 14, 15, 7], [9, 6, 11, 3], [9, 14, 3, 3]], [3, 7], No]

source roles = [1, 7, [1, 9], [7, 15]]

derived = [[[1, 7, 0, 0], [1, 15, 8, 0], [9, 7, 10, 2], [9, 15, 2, 2]], [0, 2], No]

source roles = [2, 0, [2, 10], [0, 8]]

derived = [[[2, 0, 2, 2], [2, 8, 10, 2], [10, 0, 10, 2], [10, 8, 2, 2]], [2], Yes]

source roles = [2, 1, [2, 10], [1, 9]]

derived = [[[2, 1, 3, 3], [2, 9, 11, 3], [10, 1, 9, 1], [10, 9, 1, 1]], [1, 3], No]

source roles = [2, 2, [2, 10], [2, 10]]

derived = [[[2, 2, 4, 4], [2, 10, 12, 4], [10, 2, 8, 0], [10, 10, 0, 0]], [0, 4], No]

source roles = [2, 3, [2, 10], [3, 11]]

derived = [[[2, 3, 5, 5], [2, 11, 13, 5], [10, 3, 15, 7], [10, 11, 7, 7]], [5, 7], No]

source roles = [2, 4, [2, 10], [4, 12]]

derived = [[[2, 4, 6, 6], [2, 12, 14, 6], [10, 4, 14, 6], [10, 12, 6, 6]], [6], Yes]

source roles = [2, 5, [2, 10], [5, 13]]

derived = [[[2, 5, 7, 7], [2, 13, 15, 7], [10, 5, 13, 5], [10, 13, 5, 5]], [5, 7], No]

source roles = [2, 6, [2, 10], [6, 14]]

derived = [[[2, 6, 0, 0], [2, 14, 8, 0], [10, 6, 12, 4], [10, 14, 4, 4]], [0, 4], No]

source roles = [2, 7, [2, 10], [7, 15]]

derived = [[[2, 7, 1, 1], [2, 15, 9, 1], [10, 7, 11, 3], [10, 15, 3, 3]], [1, 3], No]

source roles = [3, 0, [3, 11], [0, 8]]

derived = [[[3, 0, 3, 3], [3, 8, 11, 3], [11, 0, 11, 3], [11, 8, 3, 3]], [3], Yes]

source roles = [3, 1, [3, 11], [1, 9]]

derived = [[[3, 1, 4, 4], [3, 9, 12, 4], [11, 1, 10, 2], [11, 9, 2, 2]], [2, 4], No]

source roles = [3, 2, [3, 11], [2, 10]]

derived = [[[3, 2, 5, 5], [3, 10, 13, 5], [11, 2, 9, 1], [11, 10, 1, 1]], [1, 5], No]

source roles = [3, 3, [3, 11], [3, 11]]

derived = [[[3, 3, 6, 6], [3, 11, 14, 6], [11, 3, 8, 0], [11, 11, 0, 0]], [0, 6], No]

source roles = [3, 4, [3, 11], [4, 12]]

derived = [[[3, 4, 7, 7], [3, 12, 15, 7], [11, 4, 15, 7], [11, 12, 7, 7]], [7], Yes]

source roles = [3, 5, [3, 11], [5, 13]]

derived = [[[3, 5, 0, 0], [3, 13, 8, 0], [11, 5, 14, 6], [11, 13, 6, 6]], [0, 6], No]

source roles = [3, 6, [3, 11], [6, 14]]

derived = [[[3, 6, 1, 1], [3, 14, 9, 1], [11, 6, 13, 5], [11, 14, 5, 5]], [1, 5], No]

source roles = [3, 7, [3, 11], [7, 15]]

derived = [[[3, 7, 2, 2], [3, 15, 10, 2], [11, 7, 12, 4], [11, 15, 4, 4]], [2, 4], No]

source roles = [4, 0, [4, 12], [0, 8]]

derived = [[[4, 0, 4, 4], [4, 8, 12, 4], [12, 0, 12, 4], [12, 8, 4, 4]], [4], Yes]

source roles = [4, 1, [4, 12], [1, 9]]

derived = [[[4, 1, 5, 5], [4, 9, 13, 5], [12, 1, 11, 3], [12, 9, 3, 3]], [3, 5], No]

source roles = [4, 2, [4, 12], [2, 10]]

derived = [[[4, 2, 6, 6], [4, 10, 14, 6], [12, 2, 10, 2], [12, 10, 2, 2]], [2, 6], No]

source roles = [4, 3, [4, 12], [3, 11]]

derived = [[[4, 3, 7, 7], [4, 11, 15, 7], [12, 3, 9, 1], [12, 11, 1, 1]], [1, 7], No]

source roles = [4, 4, [4, 12], [4, 12]]

derived = [[[4, 4, 0, 0], [4, 12, 8, 0], [12, 4, 8, 0], [12, 12, 0, 0]], [0], Yes]

source roles = [4, 5, [4, 12], [5, 13]]

derived = [[[4, 5, 1, 1], [4, 13, 9, 1], [12, 5, 15, 7], [12, 13, 7, 7]], [1, 7], No]

source roles = [4, 6, [4, 12], [6, 14]]

derived = [[[4, 6, 2, 2], [4, 14, 10, 2], [12, 6, 14, 6], [12, 14, 6, 6]], [2, 6], No]

source roles = [4, 7, [4, 12], [7, 15]]

derived = [[[4, 7, 3, 3], [4, 15, 11, 3], [12, 7, 13, 5], [12, 15, 5, 5]], [3, 5], No]

source roles = [5, 0, [5, 13], [0, 8]]

derived = [[[5, 0, 5, 5], [5, 8, 13, 5], [13, 0, 13, 5], [13, 8, 5, 5]], [5], Yes]

source roles = [5, 1, [5, 13], [1, 9]]

derived = [[[5, 1, 6, 6], [5, 9, 14, 6], [13, 1, 12, 4], [13, 9, 4, 4]], [4, 6], No]

source roles = [5, 2, [5, 13], [2, 10]]

derived = [[[5, 2, 7, 7], [5, 10, 15, 7], [13, 2, 11, 3], [13, 10, 3, 3]], [3, 7], No]

source roles = [5, 3, [5, 13], [3, 11]]

derived = [[[5, 3, 0, 0], [5, 11, 8, 0], [13, 3, 10, 2], [13, 11, 2, 2]], [0, 2], No]

source roles = [5, 4, [5, 13], [4, 12]]

derived = [[[5, 4, 1, 1], [5, 12, 9, 1], [13, 4, 9, 1], [13, 12, 1, 1]], [1], Yes]

source roles = [5, 5, [5, 13], [5, 13]]

derived = [[[5, 5, 2, 2], [5, 13, 10, 2], [13, 5, 8, 0], [13, 13, 0, 0]], [0, 2], No]

source roles = [5, 6, [5, 13], [6, 14]]

derived = [[[5, 6, 3, 3], [5, 14, 11, 3], [13, 6, 15, 7], [13, 14, 7, 7]], [3, 7], No]

source roles = [5, 7, [5, 13], [7, 15]]

derived = [[[5, 7, 4, 4], [5, 15, 12, 4], [13, 7, 14, 6], [13, 15, 6, 6]], [4, 6], No]

source roles = [6, 0, [6, 14], [0, 8]]

derived = [[[6, 0, 6, 6], [6, 8, 14, 6], [14, 0, 14, 6], [14, 8, 6, 6]], [6], Yes]

source roles = [6, 1, [6, 14], [1, 9]]

derived = [[[6, 1, 7, 7], [6, 9, 15, 7], [14, 1, 13, 5], [14, 9, 5, 5]], [5, 7], No]

source roles = [6, 2, [6, 14], [2, 10]]

derived = [[[6, 2, 0, 0], [6, 10, 8, 0], [14, 2, 12, 4], [14, 10, 4, 4]], [0, 4], No]

source roles = [6, 3, [6, 14], [3, 11]]

derived = [[[6, 3, 1, 1], [6, 11, 9, 1], [14, 3, 11, 3], [14, 11, 3, 3]], [1, 3], No]

source roles = [6, 4, [6, 14], [4, 12]]

derived = [[[6, 4, 2, 2], [6, 12, 10, 2], [14, 4, 10, 2], [14, 12, 2, 2]], [2], Yes]

source roles = [6, 5, [6, 14], [5, 13]]

derived = [[[6, 5, 3, 3], [6, 13, 11, 3], [14, 5, 9, 1], [14, 13, 1, 1]], [1, 3], No]

source roles = [6, 6, [6, 14], [6, 14]]

derived = [[[6, 6, 4, 4], [6, 14, 12, 4], [14, 6, 8, 0], [14, 14, 0, 0]], [0, 4], No]

source roles = [6, 7, [6, 14], [7, 15]]

derived = [[[6, 7, 5, 5], [6, 15, 13, 5], [14, 7, 15, 7], [14, 15, 7, 7]], [5, 7], No]

source roles = [7, 0, [7, 15], [0, 8]]

derived = [[[7, 0, 7, 7], [7, 8, 15, 7], [15, 0, 15, 7], [15, 8, 7, 7]], [7], Yes]

source roles = [7, 1, [7, 15], [1, 9]]

derived = [[[7, 1, 0, 0], [7, 9, 8, 0], [15, 1, 14, 6], [15, 9, 6, 6]], [0, 6], No]

source roles = [7, 2, [7, 15], [2, 10]]

derived = [[[7, 2, 1, 1], [7, 10, 9, 1], [15, 2, 13, 5], [15, 10, 5, 5]], [1, 5], No]

source roles = [7, 3, [7, 15], [3, 11]]

derived = [[[7, 3, 2, 2], [7, 11, 10, 2], [15, 3, 12, 4], [15, 11, 4, 4]], [2, 4], No]

source roles = [7, 4, [7, 15], [4, 12]]

derived = [[[7, 4, 3, 3], [7, 12, 11, 3], [15, 4, 11, 3], [15, 12, 3, 3]], [3], Yes]

source roles = [7, 5, [7, 15], [5, 13]]

derived = [[[7, 5, 4, 4], [7, 13, 12, 4], [15, 5, 10, 2], [15, 13, 2, 2]], [2, 4], No]

source roles = [7, 6, [7, 15], [6, 14]]

derived = [[[7, 6, 5, 5], [7, 14, 13, 5], [15, 6, 9, 1], [15, 14, 1, 1]], [1, 5], No]

source roles = [7, 7, [7, 15], [7, 15]]

derived = [[[7, 7, 6, 6], [7, 15, 14, 6], [15, 7, 8, 0], [15, 15, 0, 0]], [0, 6], No]

source roles = [[0, 1, [1, 7], No], [0, 8], [1, 9]]

derived = [[0, 1, 0, 1, 1, 8, 1, 7], No, No]

Reverse checks

• S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

([0, 8], No, [[0, 8], [1, 9], [2, 10], [3, 11], [4, 12], [5, 13], [6, 14], [7, 15]], [0, 1, 0, 1, 1, 8, 1, 7], No, No, explicit_representative_dependence_blocks_quotient_group_multiplication)

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.