Question 24935: Qualify Both Groups and Every Product Pair before Applying Kernel Theorems

Original question and givens

Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], target identity=0, and total mapping=[0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).

• G001 — Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], target identity=0, and total mapping=[0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, 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.

• mko-finite-group-homomorphism-kernel

Step-by-step derivation

source roles = [kernel-homomorphism-qualification]

derived = [[source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate]]

source roles = [Kernel ORIGINAL SOURCE group, 64, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], 0]

derived = [12, [[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]], Yes]

source roles = [Kernel ORIGINAL SOURCE group, 12, FOR EVERY x,y,z in0..n-1]

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]], 1728, Yes]

source roles = [Kernel ORIGINAL SOURCE group, 0]

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

source roles = [Kernel ORIGINAL TARGET group, 64, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], 0]

derived = [12, [[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]], Yes]

source roles = [Kernel ORIGINAL TARGET group, 12, FOR EVERY x,y,z in0..n-1]

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]], 1728, Yes]

source roles = [Kernel ORIGINAL TARGET group, 0]

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

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

derived = [[[0, 0, 0, 0, Yes], [0, 1, 11, 11, Yes], [0, 2, 10, 10, Yes], [0, 3, 9, 9, Yes], [0, 4, 8, 8, Yes], [0, 5, 7, 7, Yes], [0, 6, 6, 6, Yes], [0, 7, 5, 5, Yes], [0, 8, 4, 4, Yes], [0, 9, 3, 3, Yes], [0, 10, 2, 2, Yes], [0, 11, 1, 1, Yes], [1, 0, 11, 11, Yes], [1, 1, 10, 10, Yes], [1, 2, 9, 9, Yes], [1, 3, 8, 8, Yes], [1, 4, 7, 7, Yes], [1, 5, 6, 6, Yes], [1, 6, 5, 5, Yes], [1, 7, 4, 4, Yes], [1, 8, 3, 3, Yes], [1, 9, 2, 2, Yes], [1, 10, 1, 1, Yes], [1, 11, 0, 0, Yes], [2, 0, 10, 10, Yes], [2, 1, 9, 9, Yes], [2, 2, 8, 8, Yes], [2, 3, 7, 7, Yes], [2, 4, 6, 6, Yes], [2, 5, 5, 5, Yes], [2, 6, 4, 4, Yes], [2, 7, 3, 3, Yes], [2, 8, 2, 2, Yes], [2, 9, 1, 1, Yes], [2, 10, 0, 0, Yes], [2, 11, 11, 11, Yes], [3, 0, 9, 9, Yes], [3, 1, 8, 8, Yes], [3, 2, 7, 7, Yes], [3, 3, 6, 6, Yes], [3, 4, 5, 5, Yes], [3, 5, 4, 4, Yes], [3, 6, 3, 3, Yes], [3, 7, 2, 2, Yes], [3, 8, 1, 1, Yes], [3, 9, 0, 0, Yes], [3, 10, 11, 11, Yes], [3, 11, 10, 10, Yes], [4, 0, 8, 8, Yes], [4, 1, 7, 7, Yes], [4, 2, 6, 6, Yes], [4, 3, 5, 5, Yes], [4, 4, 4, 4, Yes], [4, 5, 3, 3, Yes], [4, 6, 2, 2, Yes], [4, 7, 1, 1, Yes], [4, 8, 0, 0, Yes], [4, 9, 11, 11, Yes], [4, 10, 10, 10, Yes], [4, 11, 9, 9, Yes], [5, 0, 7, 7, Yes], [5, 1, 6, 6, Yes], [5, 2, 5, 5, Yes], [5, 3, 4, 4, Yes], [5, 4, 3, 3, Yes], [5, 5, 2, 2, Yes], [5, 6, 1, 1, Yes], [5, 7, 0, 0, Yes], [5, 8, 11, 11, Yes], [5, 9, 10, 10, Yes], [5, 10, 9, 9, Yes], [5, 11, 8, 8, Yes], [6, 0, 6, 6, Yes], [6, 1, 5, 5, Yes], [6, 2, 4, 4, Yes], [6, 3, 3, 3, Yes], [6, 4, 2, 2, Yes], [6, 5, 1, 1, Yes], [6, 6, 0, 0, Yes], [6, 7, 11, 11, Yes], [6, 8, 10, 10, Yes], [6, 9, 9, 9, Yes], [6, 10, 8, 8, Yes], [6, 11, 7, 7, Yes], [7, 0, 5, 5, Yes], [7, 1, 4, 4, Yes], [7, 2, 3, 3, Yes], [7, 3, 2, 2, Yes], [7, 4, 1, 1, Yes], [7, 5, 0, 0, Yes], [7, 6, 11, 11, Yes], [7, 7, 10, 10, Yes], [7, 8, 9, 9, Yes], [7, 9, 8, 8, Yes], [7, 10, 7, 7, Yes], [7, 11, 6, 6, Yes], [8, 0, 4, 4, Yes], [8, 1, 3, 3, Yes], [8, 2, 2, 2, Yes], [8, 3, 1, 1, Yes], [8, 4, 0, 0, Yes], [8, 5, 11, 11, Yes], [8, 6, 10, 10, Yes], [8, 7, 9, 9, Yes], [8, 8, 8, 8, Yes], [8, 9, 7, 7, Yes], [8, 10, 6, 6, Yes], [8, 11, 5, 5, Yes], [9, 0, 3, 3, Yes], [9, 1, 2, 2, Yes], [9, 2, 1, 1, Yes], [9, 3, 0, 0, Yes], [9, 4, 11, 11, Yes], [9, 5, 10, 10, Yes], [9, 6, 9, 9, Yes], [9, 7, 8, 8, Yes], [9, 8, 7, 7, Yes], [9, 9, 6, 6, Yes], [9, 10, 5, 5, Yes], [9, 11, 4, 4, Yes], [10, 0, 2, 2, Yes], [10, 1, 1, 1, Yes], [10, 2, 0, 0, Yes], [10, 3, 11, 11, Yes], [10, 4, 10, 10, Yes], [10, 5, 9, 9, Yes], [10, 6, 8, 8, Yes], [10, 7, 7, 7, Yes], [10, 8, 6, 6, Yes], [10, 9, 5, 5, Yes], [10, 10, 4, 4, Yes], [10, 11, 3, 3, Yes], [11, 0, 1, 1, Yes], [11, 1, 0, 0, Yes], [11, 2, 11, 11, Yes], [11, 3, 10, 10, Yes], [11, 4, 9, 9, Yes], [11, 5, 8, 8, Yes], [11, 6, 7, 7, Yes], [11, 7, 6, 6, Yes], [11, 8, 5, 5, Yes], [11, 9, 4, 4, Yes], [11, 10, 3, 3, Yes], [11, 11, 2, 2, Yes]], Yes, null]

Reverse checks

• 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

(12, 12, [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 11, 11, Yes], [0, 2, 10, 10, Yes], [0, 3, 9, 9, Yes], [0, 4, 8, 8, Yes], [0, 5, 7, 7, Yes], [0, 6, 6, 6, Yes], [0, 7, 5, 5, Yes], [0, 8, 4, 4, Yes], [0, 9, 3, 3, Yes], [0, 10, 2, 2, Yes], [0, 11, 1, 1, Yes], [1, 0, 11, 11, Yes], [1, 1, 10, 10, Yes], [1, 2, 9, 9, Yes], [1, 3, 8, 8, Yes], [1, 4, 7, 7, Yes], [1, 5, 6, 6, Yes], [1, 6, 5, 5, Yes], [1, 7, 4, 4, Yes], [1, 8, 3, 3, Yes], [1, 9, 2, 2, Yes], [1, 10, 1, 1, Yes], [1, 11, 0, 0, Yes], [2, 0, 10, 10, Yes], [2, 1, 9, 9, Yes], [2, 2, 8, 8, Yes], [2, 3, 7, 7, Yes], [2, 4, 6, 6, Yes], [2, 5, 5, 5, Yes], [2, 6, 4, 4, Yes], [2, 7, 3, 3, Yes], [2, 8, 2, 2, Yes], [2, 9, 1, 1, Yes], [2, 10, 0, 0, Yes], [2, 11, 11, 11, Yes], [3, 0, 9, 9, Yes], [3, 1, 8, 8, Yes], [3, 2, 7, 7, Yes], [3, 3, 6, 6, Yes], [3, 4, 5, 5, Yes], [3, 5, 4, 4, Yes], [3, 6, 3, 3, Yes], [3, 7, 2, 2, Yes], [3, 8, 1, 1, Yes], [3, 9, 0, 0, Yes], [3, 10, 11, 11, Yes], [3, 11, 10, 10, Yes], [4, 0, 8, 8, Yes], [4, 1, 7, 7, Yes], [4, 2, 6, 6, Yes], [4, 3, 5, 5, Yes], [4, 4, 4, 4, Yes], [4, 5, 3, 3, Yes], [4, 6, 2, 2, Yes], [4, 7, 1, 1, Yes], [4, 8, 0, 0, Yes], [4, 9, 11, 11, Yes], [4, 10, 10, 10, Yes], [4, 11, 9, 9, Yes], [5, 0, 7, 7, Yes], [5, 1, 6, 6, Yes], [5, 2, 5, 5, Yes], [5, 3, 4, 4, Yes], [5, 4, 3, 3, Yes], [5, 5, 2, 2, Yes], [5, 6, 1, 1, Yes], [5, 7, 0, 0, Yes], [5, 8, 11, 11, Yes], [5, 9, 10, 10, Yes], [5, 10, 9, 9, Yes], [5, 11, 8, 8, Yes], [6, 0, 6, 6, Yes], [6, 1, 5, 5, Yes], [6, 2, 4, 4, Yes], [6, 3, 3, 3, Yes], [6, 4, 2, 2, Yes], [6, 5, 1, 1, Yes], [6, 6, 0, 0, Yes], [6, 7, 11, 11, Yes], [6, 8, 10, 10, Yes], [6, 9, 9, 9, Yes], [6, 10, 8, 8, Yes], [6, 11, 7, 7, Yes], [7, 0, 5, 5, Yes], [7, 1, 4, 4, Yes], [7, 2, 3, 3, Yes], [7, 3, 2, 2, Yes], [7, 4, 1, 1, Yes], [7, 5, 0, 0, Yes], [7, 6, 11, 11, Yes], [7, 7, 10, 10, Yes], [7, 8, 9, 9, Yes], [7, 9, 8, 8, Yes], [7, 10, 7, 7, Yes], [7, 11, 6, 6, Yes], [8, 0, 4, 4, Yes], [8, 1, 3, 3, Yes], [8, 2, 2, 2, Yes], [8, 3, 1, 1, Yes], [8, 4, 0, 0, Yes], [8, 5, 11, 11, Yes], [8, 6, 10, 10, Yes], [8, 7, 9, 9, Yes], [8, 8, 8, 8, Yes], [8, 9, 7, 7, Yes], [8, 10, 6, 6, Yes], [8, 11, 5, 5, Yes], [9, 0, 3, 3, Yes], [9, 1, 2, 2, Yes], [9, 2, 1, 1, Yes], [9, 3, 0, 0, Yes], [9, 4, 11, 11, Yes], [9, 5, 10, 10, Yes], [9, 6, 9, 9, Yes], [9, 7, 8, 8, Yes], [9, 8, 7, 7, Yes], [9, 9, 6, 6, Yes], [9, 10, 5, 5, Yes], [9, 11, 4, 4, Yes], [10, 0, 2, 2, Yes], [10, 1, 1, 1, Yes], [10, 2, 0, 0, Yes], [10, 3, 11, 11, Yes], [10, 4, 10, 10, Yes], [10, 5, 9, 9, Yes], [10, 6, 8, 8, Yes], [10, 7, 7, 7, Yes], [10, 8, 6, 6, Yes], [10, 9, 5, 5, Yes], [10, 10, 4, 4, Yes], [10, 11, 3, 3, Yes], [11, 0, 1, 1, Yes], [11, 1, 0, 0, Yes], [11, 2, 11, 11, Yes], [11, 3, 10, 10, Yes], [11, 4, 9, 9, Yes], [11, 5, 8, 8, Yes], [11, 6, 7, 7, Yes], [11, 7, 6, 6, Yes], [11, 8, 5, 5, Yes], [11, 9, 4, 4, Yes], [11, 10, 3, 3, Yes], [11, 11, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)

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.