diff --git "a/mce220_runner/dataset/audit/provenance.jsonl" "b/mce220_runner/dataset/audit/provenance.jsonl"
new file mode 100644--- /dev/null
+++ "b/mce220_runner/dataset/audit/provenance.jsonl"
@@ -0,0 +1,220 @@
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3H(X_{4,5,6,8,9}) + 9H(X_{3,5,7,8,9}) - 4H(X_{2,3,5,7,8,9}) - 4H(X_{1,2,3,6,7,8,9}) - H(X_{1,2,3,4,6,7,8,9}) + 7H(X_{3,4,5,6,7,8,9}) + H(X_{1,3,4,5,6,7,8,9}) + 9H(X_{10}) - 9H(X_{3,4,5,7,10}) + 4H(X_{6,7,10}) + 2H(X_{1,5,6,7,10}) - 2H(X_{1,4,5,6,7,10}) + 18H(X_{8,10}) - 9H(X_{2,8,10}) - 4H(X_{5,8,10}) - 9H(X_{1,4,7,8,10}) + 9H(X_{3,4,5,7,8,10}) + 6H(X_{1,3,4,5,6,7,8,10}) + 3H(X_{9,10}) - 3H(X_{1,3,5,7,9,10}) + 9H(X_{3,4,5,7,9,10}) - 4H(X_{6,7,9,10}) - 5H(X_{2,3,8,9,10}) - 5H(X_{5,8,9,10}) + 5H(X_{2,3,5,8,9,10}) + 9H(X_{3,6,8,9,10}) - 9H(X_{1,2,3,6,8,9,10}) - 3H(X_{4,6,8,9,10}) + 3H(X_{4,5,6,8,9,10}) - 8H(X_{7,8,9,10}) + 5H(X_{2,3,7,8,9,10}) - 5H(X_{2,3,5,7,8,9,10}) - 9H(X_{3,4,5,7,8,9,10}) + 8H(X_{6,7,8,9,10}) - 6H(X_{1,2,3,4,5,6,7,8,9,10})","witness_value":8849},"k":16,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{9}|X_{3})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{8};X_{9}|X_{3,4,5,7,10})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{4};X_{9}|X_{2,5,8})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{10}|X_{8})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{7}|X_{3,5,8,9})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{4};X_{8}|X_{3,5,9})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{5};X_{10}|X_{4,6,8,9})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{8}|X_{2,3,5,7})"}],"expression":[[4,1],[5,-1],[86,-2],[118,2],[128,3],[130,-3],[146,2],[154,-2],[214,2],[246,-2],[260,-1],[261,1],[276,-3],[284,3],[402,-2],[404,6],[406,-3],[410,2],[412,-3],[424,-1],[440,1],[468,-3],[470,3],[604,3],[640,-3],[642,3],[732,-3],[860,-3],[936,1],[952,-1],[988,3]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["H(X_{3}) - H(X_{1,3}) - 2H(X_{2,3,5,7}) + 2H(X_{2,3,5,6,7}) + 3H(X_{8}) - 3H(X_{2,8}) + 2H(X_{2,5,8}) - 2H(X_{2,4,5,8}) + 2H(X_{2,3,5,7,8}) - 2H(X_{2,3,5,6,7,8}) - H(X_{3,9}) + H(X_{1,3,9}) - 3H(X_{3,5,9}) + 3H(X_{3,4,5,9}) - 2H(X_{2,5,8,9}) + 6H(X_{3,5,8,9}) - 3H(X_{2,3,5,8,9}) + 2H(X_{2,4,5,8,9}) - 3H(X_{3,4,5,8,9}) - H(X_{4,6,8,9}) + H(X_{4,5,6,8,9}) - 3H(X_{3,5,7,8,9}) + 3H(X_{2,3,5,7,8,9}) + 3H(X_{3,4,5,7,10}) - 3H(X_{8,10}) + 3H(X_{2,8,10}) - 3H(X_{3,4,5,7,8,10}) - 3H(X_{3,4,5,7,9,10}) + H(X_{4,6,8,9,10}) - H(X_{4,5,6,8,9,10}) + 3H(X_{3,4,5,7,8,9,10})>=0","-H(X_{3}) + H(X_{1,3}) + 2H(X_{2,3,5,7}) - 2H(X_{2,3,5,6,7}) - 3H(X_{8}) + 3H(X_{2,8}) - 2H(X_{2,5,8}) + 2H(X_{2,4,5,8}) - 2H(X_{2,3,5,7,8}) + 2H(X_{2,3,5,6,7,8}) + H(X_{3,9}) - H(X_{1,3,9}) + 3H(X_{3,5,9}) - 3H(X_{3,4,5,9}) + 2H(X_{2,5,8,9}) - 6H(X_{3,5,8,9}) + 3H(X_{2,3,5,8,9}) - 2H(X_{2,4,5,8,9}) + 3H(X_{3,4,5,8,9}) + H(X_{4,6,8,9}) - H(X_{4,5,6,8,9}) + 3H(X_{3,5,7,8,9}) - 3H(X_{2,3,5,7,8,9}) - 3H(X_{3,4,5,7,10}) + 3H(X_{8,10}) - 3H(X_{2,8,10}) + 3H(X_{3,4,5,7,8,10}) + 3H(X_{3,4,5,7,9,10}) - H(X_{4,6,8,9,10}) + H(X_{4,5,6,8,9,10}) - 3H(X_{3,4,5,7,8,9,10})>=0"],"text":"H(X_{3}) - H(X_{1,3}) - 2H(X_{2,3,5,7}) + 2H(X_{2,3,5,6,7}) + 3H(X_{8}) - 3H(X_{2,8}) + 2H(X_{2,5,8}) - 2H(X_{2,4,5,8}) + 2H(X_{2,3,5,7,8}) - 2H(X_{2,3,5,6,7,8}) - H(X_{3,9}) + H(X_{1,3,9}) - 3H(X_{3,5,9}) + 3H(X_{3,4,5,9}) - 2H(X_{2,5,8,9}) + 6H(X_{3,5,8,9}) - 3H(X_{2,3,5,8,9}) + 2H(X_{2,4,5,8,9}) - 3H(X_{3,4,5,8,9}) - H(X_{4,6,8,9}) + H(X_{4,5,6,8,9}) - 3H(X_{3,5,7,8,9}) + 3H(X_{2,3,5,7,8,9}) + 3H(X_{3,4,5,7,10}) - 3H(X_{8,10}) + 3H(X_{2,8,10}) - 3H(X_{3,4,5,7,8,10}) - 3H(X_{3,4,5,7,9,10}) + H(X_{4,6,8,9,10}) - H(X_{4,5,6,8,9,10}) + 3H(X_{3,4,5,7,8,9,10})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{9}|X_{4,5,6,7})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{9}|X_{3})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{7};X_{9}|X_{2,3,5,8})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{4}|X_{5,9})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{9};X_{10}|X_{5,8})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{5};X_{7}|X_{2,3,8,9,10})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{7};X_{8}|X_{1,2,3,6,9})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{1};X_{2}|X_{9})"}],"expression":[[4,1],[5,-1],[120,-2],[124,2],[144,1],[150,-1],[214,1],[256,-2],[257,2],[258,2],[259,-2],[260,-1],[261,1],[272,1],[274,-1],[280,-1],[282,1],[295,1],[359,-1],[376,2],[380,-2],[400,-1],[406,1],[423,-1],[470,-1],[487,1],[656,-1],[902,1],[912,1],[918,-1],[966,-1],[982,1]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["H(X_{3}) - H(X_{1,3}) - 2H(X_{4,5,6,7}) + 2H(X_{3,4,5,6,7}) + H(X_{5,8}) - H(X_{2,3,5,8}) + H(X_{2,3,5,7,8}) - 2H(X_{9}) + 2H(X_{1,9}) + 2H(X_{2,9}) - 2H(X_{1,2,9}) - H(X_{3,9}) + H(X_{1,3,9}) + H(X_{5,9}) - H(X_{2,5,9}) - H(X_{4,5,9}) + H(X_{2,4,5,9}) + H(X_{1,2,3,6,9}) - H(X_{1,2,3,6,7,9}) + 2H(X_{4,5,6,7,9}) - 2H(X_{3,4,5,6,7,9}) - H(X_{5,8,9}) + H(X_{2,3,5,8,9}) - H(X_{1,2,3,6,8,9}) - H(X_{2,3,5,7,8,9}) + H(X_{1,2,3,6,7,8,9}) - H(X_{5,8,10}) + H(X_{2,3,8,9,10}) + H(X_{5,8,9,10}) - H(X_{2,3,5,8,9,10}) - H(X_{2,3,7,8,9,10}) + H(X_{2,3,5,7,8,9,10})>=0","-H(X_{3}) + H(X_{1,3}) + 2H(X_{4,5,6,7}) - 2H(X_{3,4,5,6,7}) - H(X_{5,8}) + H(X_{2,3,5,8}) - H(X_{2,3,5,7,8}) + 2H(X_{9}) - 2H(X_{1,9}) - 2H(X_{2,9}) + 2H(X_{1,2,9}) + H(X_{3,9}) - H(X_{1,3,9}) - H(X_{5,9}) + H(X_{2,5,9}) + H(X_{4,5,9}) - H(X_{2,4,5,9}) - H(X_{1,2,3,6,9}) + H(X_{1,2,3,6,7,9}) - 2H(X_{4,5,6,7,9}) + 2H(X_{3,4,5,6,7,9}) + H(X_{5,8,9}) - H(X_{2,3,5,8,9}) + H(X_{1,2,3,6,8,9}) + H(X_{2,3,5,7,8,9}) - H(X_{1,2,3,6,7,8,9}) + H(X_{5,8,10}) - H(X_{2,3,8,9,10}) - H(X_{5,8,9,10}) + H(X_{2,3,5,8,9,10}) + H(X_{2,3,7,8,9,10}) - H(X_{2,3,5,7,8,9,10})>=0"],"text":"H(X_{3}) - H(X_{1,3}) - 2H(X_{4,5,6,7}) + 2H(X_{3,4,5,6,7}) + H(X_{5,8}) - H(X_{2,3,5,8}) + H(X_{2,3,5,7,8}) - 2H(X_{9}) + 2H(X_{1,9}) + 2H(X_{2,9}) - 2H(X_{1,2,9}) - H(X_{3,9}) + H(X_{1,3,9}) + H(X_{5,9}) - H(X_{2,5,9}) - H(X_{4,5,9}) + H(X_{2,4,5,9}) + H(X_{1,2,3,6,9}) - H(X_{1,2,3,6,7,9}) + 2H(X_{4,5,6,7,9}) - 2H(X_{3,4,5,6,7,9}) - H(X_{5,8,9}) + H(X_{2,3,5,8,9}) - H(X_{1,2,3,6,8,9}) - H(X_{2,3,5,7,8,9}) + H(X_{1,2,3,6,7,8,9}) - H(X_{5,8,10}) + H(X_{2,3,8,9,10}) + H(X_{5,8,9,10}) - H(X_{2,3,5,8,9,10}) - H(X_{2,3,7,8,9,10}) + 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H(X_{6}) + H(X_{1,6}) + H(X_{5,6}) - H(X_{1,5,6}) + 2H(X_{2,5,6}) - 2H(X_{1,2,5,6}) - H(X_{7}) + H(X_{2,7}) + H(X_{6,7}) - H(X_{2,6,7}) - 2H(X_{6,7,9}) + 2H(X_{5,6,7,9}) - H(X_{10}) + H(X_{3,10}) - H(X_{6,10}) + H(X_{1,6,10}) - 2H(X_{2,5,6,10}) + 2H(X_{1,2,5,6,10}) + 2H(X_{6,7,9,10}) - 2H(X_{5,6,7,9,10}) + H(X_{7,8,11}) - H(X_{3,7,8,11}) - H(X_{7,8,9,11}) + H(X_{3,7,8,9,11}) + H(X_{6,10,11}) - 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2H(X_{2,3,4,5,6}) + 2H(X_{7}) - 2H(X_{1,7}) + 3H(X_{2,7}) + H(X_{1,3,5,7}) - H(X_{1,3,5,6,7}) + 2H(X_{4,6,9}) - 3H(X_{3,5,6,9}) - 2H(X_{7,9}) + 2H(X_{1,7,9}) - H(X_{1,4,5,8,9}) + 3H(X_{3,5,6,8,9}) + H(X_{1,4,5,6,8,9}) + H(X_{2,4,10}) - 2H(X_{2,4,5,6,10}) + 2H(X_{2,3,4,5,6,10}) + 3H(X_{2,11}) + H(X_{2,4,11}) + 2H(X_{4,6,11}) + 2H(X_{1,7,11}) - 3H(X_{2,7,11}) - 2H(X_{1,5,7,11}) - 2H(X_{1,6,7,11}) + 2H(X_{1,5,6,7,11}) + 2H(X_{8,11}) - 2H(X_{2,8,11}) - 2H(X_{7,8,11}) + 2H(X_{2,7,8,11}) + 2H(X_{5,9,11}) - 2H(X_{4,5,9,11}) - 2H(X_{4,6,9,11}) + 3H(X_{3,5,6,9,11}) + H(X_{1,4,5,8,9,11}) - 3H(X_{3,5,6,8,9,11}) - H(X_{1,4,5,6,8,9,11}) - H(X_{2,4,10,11}) - 2H(X_{5,9,10,11}) + 2H(X_{4,5,9,10,11}) - 2H(X_{3,12}) + 2H(X_{2,3,12})>=0"],"text":"3H(X_{2}) - 2H(X_{3}) + 2H(X_{2,3}) + H(X_{2,4}) + H(X_{1,3,5}) + 2H(X_{4,6}) - H(X_{1,3,5,6}) - 2H(X_{2,4,5,6}) + 2H(X_{2,3,4,5,6}) - 2H(X_{7}) + 2H(X_{1,7}) - 3H(X_{2,7}) - H(X_{1,3,5,7}) + H(X_{1,3,5,6,7}) - 2H(X_{4,6,9}) + 3H(X_{3,5,6,9}) + 2H(X_{7,9}) - 2H(X_{1,7,9}) + H(X_{1,4,5,8,9}) - 3H(X_{3,5,6,8,9}) - H(X_{1,4,5,6,8,9}) - H(X_{2,4,10}) + 2H(X_{2,4,5,6,10}) - 2H(X_{2,3,4,5,6,10}) - 3H(X_{2,11}) - H(X_{2,4,11}) - 2H(X_{4,6,11}) - 2H(X_{1,7,11}) + 3H(X_{2,7,11}) + 2H(X_{1,5,7,11}) + 2H(X_{1,6,7,11}) - 2H(X_{1,5,6,7,11}) - 2H(X_{8,11}) + 2H(X_{2,8,11}) + 2H(X_{7,8,11}) - 2H(X_{2,7,8,11}) - 2H(X_{5,9,11}) + 2H(X_{4,5,9,11}) + 2H(X_{4,6,9,11}) - 3H(X_{3,5,6,9,11}) - H(X_{1,4,5,8,9,11}) + 3H(X_{3,5,6,8,9,11}) + H(X_{1,4,5,6,8,9,11}) + H(X_{2,4,10,11}) + 2H(X_{5,9,10,11}) - 2H(X_{4,5,9,10,11}) + 2H(X_{3,12}) - 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2H(X_{6,9}) + 2H(X_{5,6,11}) - 2H(X_{5,6,8,11}) - H(X_{12}) - 2H(X_{1,12}) - 2H(X_{3,12}) + 2H(X_{1,4,12}) + 2H(X_{3,4,12}) - 3H(X_{6,12}) + 3H(X_{3,6,12}) + 2H(X_{3,7,12}) - 2H(X_{3,4,7,12}) + H(X_{8,12}) + 2H(X_{1,8,12}) - 3H(X_{2,3,8,12}) - 2H(X_{1,4,8,12}) + 3H(X_{2,3,5,8,12}) + H(X_{9,12}) + 2H(X_{2,7,9,12}) - 2H(X_{2,4,7,9,12}) - H(X_{8,9,12}) - 2H(X_{1,11,12}) + 2H(X_{1,5,11,12}) + 2H(X_{1,10,11,12}) - 2H(X_{1,5,10,11,12}) + 2H(X_{5,13}) - 2H(X_{5,9,13}) - 2H(X_{5,11,13}) + 2H(X_{5,9,11,13}) + 3H(X_{6,12,13}) - 3H(X_{3,6,12,13}) - 2H(X_{2,7,9,12,13}) + 2H(X_{2,4,7,9,12,13})>=0"],"text":"H(X_{2}) - 2H(X_{2,4}) + 2H(X_{1,2,4}) + 2H(X_{6}) + 2H(X_{5,6}) + 2H(X_{2,4,7}) - 2H(X_{1,2,4,7}) + H(X_{8}) - H(X_{2,8}) - 3H(X_{2,3,8}) + 3H(X_{2,3,5,8}) - 2H(X_{5,6,8}) + 2H(X_{9}) - 2H(X_{6,9}) - 2H(X_{5,6,11}) + 2H(X_{5,6,8,11}) + H(X_{12}) + 2H(X_{1,12}) + 2H(X_{3,12}) - 2H(X_{1,4,12}) - 2H(X_{3,4,12}) + 3H(X_{6,12}) - 3H(X_{3,6,12}) - 2H(X_{3,7,12}) + 2H(X_{3,4,7,12}) - H(X_{8,12}) - 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2H(X_{1,2,4}) + 2H(X_{2,3,4}) + 10H(X_{2,3,6}) - 10H(X_{1,2,3,6}) + H(X_{2,4,5,6}) + 9H(X_{7}) + 10H(X_{2,7}) + 6H(X_{2,3,5,6,7}) + 11H(X_{8}) - 27H(X_{1,8}) + 27H(X_{1,2,8}) + 6H(X_{1,3,8}) + H(X_{4,8}) - H(X_{2,4,5,6,8}) + 10H(X_{1,3,4,5,6,8}) - 10H(X_{1,2,3,4,5,6,7,8}) + 70H(X_{9}) - 30H(X_{4,5,9}) + 30H(X_{2,4,5,9}) + 2H(X_{1,6,9}) - 2H(X_{1,2,3,4,6,9}) - 6H(X_{1,3,8,9}) - 30H(X_{2,3,10}) + H(X_{1,2,6,7,10}) - 18H(X_{9,10}) - 9H(X_{11}) - 9H(X_{1,2,11}) - 30H(X_{2,3,11}) + 5H(X_{3,8,11}) + 9H(X_{1,2,9,11}) + 30H(X_{4,5,9,11}) - 30H(X_{2,4,5,9,11}) + 2H(X_{6,9,11}) - 5H(X_{3,8,9,11}) - 2H(X_{10,11}) + 30H(X_{2,3,10,11}) + 2H(X_{4,10,11}) + 7H(X_{2,3,5,10,11}) - 7H(X_{2,3,5,8,10,11}) + 9H(X_{3,6,8,10,11}) + 21H(X_{12}) - 9H(X_{4,12}) - 21H(X_{9,12}) - 9H(X_{10,12}) + H(X_{3,4,5,8,10,12}) + 9H(X_{1,5,6,8,9,10,12}) - 9H(X_{1,5,6,7,8,9,10,12}) - H(X_{11,12}) + 10H(X_{2,11,12}) + 6H(X_{3,11,12}) + 10H(X_{5,11,12}) - 10H(X_{2,5,11,12}) - 6H(X_{2,3,5,6,7,11,12}) - 5H(X_{1,3,5,7,8,9,11,12}) - 2H(X_{4,5,6,8,9,10,11,12}) + 2H(X_{1,4,5,6,8,9,10,11,12}) + 69H(X_{13}) - 18H(X_{2,13}) - 27H(X_{3,13}) + 4H(X_{2,3,13}) + 20H(X_{5,13}) - 30H(X_{1,6,13}) - 10H(X_{2,3,6,13}) + 10H(X_{1,2,3,6,13}) + 30H(X_{1,4,6,13}) + 30H(X_{1,6,7,13}) - 30H(X_{1,4,6,7,13}) + 7H(X_{8,13}) + 27H(X_{1,8,13}) - 27H(X_{1,2,8,13}) - 4H(X_{2,3,8,13}) - 20H(X_{5,8,13}) - 38H(X_{9,13}) + 18H(X_{9,10,13}) + 9H(X_{1,2,11,13}) - 9H(X_{1,2,9,11,13}) + 7H(X_{2,3,4,9,11,13}) - 2H(X_{6,9,11,13}) - 7H(X_{2,3,4,8,9,11,13}) + 2H(X_{1,2,3,10,11,13}) - 2H(X_{1,2,3,4,10,11,13}) - 9H(X_{2,3,6,8,10,11,13}) - 20H(X_{5,12,13}) + 3H(X_{1,2,3,4,5,12,13}) + 20H(X_{5,8,12,13}) + 9H(X_{1,2,4,7,8,10,12,13}) + H(X_{3,4,5,8,9,11,12,13}) + 5H(X_{1,3,5,7,8,9,11,12,13}) - H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13})","witness_value":1323},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{10};X_{11}|X_{2,3})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{9};X_{13})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{5}|X_{11,12})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{9};X_{12})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{4};X_{7}|X_{1,6,13})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{13}|X_{2,3,6})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{12}|X_{5,13})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{11}|X_{4,5,9})"}],"expression":[[6,3],[38,1],[39,-1],[256,5],[280,-3],[282,3],[518,-3],[1030,-3],[1304,3],[1306,-3],[1542,3],[2048,3],[2304,-3],[3072,-1],[3074,1],[3088,1],[3090,-1],[4096,2],[4112,2],[4129,-3],[4134,-1],[4135,1],[4137,3],[4193,3],[4201,-3],[4240,-2],[4352,-2],[6160,-2],[6288,2]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["3H(X_{2,3}) + H(X_{2,3,6}) - 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3H(X_{1,4,7,9,10,11,12}) - 16H(X_{13}) - 9H(X_{3,13}) - 20H(X_{4,13}) + 20H(X_{4,5,13}) + 6H(X_{2,6,13}) - 2H(X_{1,3,6,13}) + 6H(X_{7,13}) + 20H(X_{4,7,13}) - 20H(X_{4,5,7,13}) + 18H(X_{8,13}) - 9H(X_{6,9,13}) - H(X_{1,6,9,13}) + 9H(X_{5,6,9,13}) - 6H(X_{2,6,7,9,13}) + 9H(X_{3,10,13}) + H(X_{2,3,5,7,8,9,10,13}) - 9H(X_{11,13}) + 27H(X_{1,11,13}) + 27H(X_{2,11,13}) - 27H(X_{1,2,11,13}) + 2H(X_{3,6,11,13}) - 18H(X_{8,11,13}) - 27H(X_{2,9,11,13}) + 27H(X_{1,2,9,11,13}) + H(X_{3,5,7,10,11,13}) - 20H(X_{12,13}) + 9H(X_{1,3,4,12,13}) + 20H(X_{5,12,13}) - 9H(X_{1,3,4,8,12,13}) + 20H(X_{9,12,13}) - 20H(X_{5,9,12,13}) + H(X_{1,3,6,7,9,12,13}) - H(X_{2,3,5,7,8,9,10,12,13}) - 2H(X_{3,5,6,8,11,12,13}) - H(X_{3,5,7,10,11,12,13}) + 5H(X_{3,9,10,11,12,13}) - 5H(X_{3,4,9,10,11,12,13}) + 8H(X_{1,2,3,4,6,7,8,9,10,11,12,13}) - 8H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13})","witness_value":9793},"k":16,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{7}|X_{4,13})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{10};X_{12}|X_{9,11})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{12}|X_{1,6,9})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{9}|X_{12,13})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{2};X_{10}|X_{3})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{7}|X_{6})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{7};X_{11}|X_{4,12})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{4};X_{10}|X_{5,8,11})"}],"expression":[[4,-2],[6,2],[32,-3],[34,3],[96,3],[98,-3],[289,-3],[293,3],[516,2],[518,-2],[1168,-3],[1176,3],[1280,3],[1680,3],[1688,-3],[1792,-3],[2056,-1],[2120,1],[2337,3],[2341,-3],[3080,1],[3144,-1],[3328,-3],[3840,3],[4104,2],[4120,-2],[4168,-2],[4184,2],[6144,2],[6160,-2],[6400,-2],[6416,2]],"id":0,"kind":"composite_equality","label":"witness 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2H(X_{4,5,13}) + 2H(X_{4,7,13}) - 2H(X_{4,5,7,13}) - 2H(X_{12,13}) + 2H(X_{5,12,13}) + 2H(X_{9,12,13}) - 2H(X_{5,9,12,13})>=0"],"text":"-2H(X_{3}) + 2H(X_{2,3}) - 3H(X_{6}) + 3H(X_{2,6}) + 3H(X_{6,7}) - 3H(X_{2,6,7}) - 3H(X_{1,6,9}) + 3H(X_{1,3,6,9}) + 2H(X_{3,10}) - 2H(X_{2,3,10}) - 3H(X_{5,8,11}) + 3H(X_{4,5,8,11}) + 3H(X_{9,11}) + 3H(X_{5,8,10,11}) - 3H(X_{4,5,8,10,11}) - 3H(X_{9,10,11}) - H(X_{4,12}) + H(X_{4,7,12}) + 3H(X_{1,6,9,12}) - 3H(X_{1,3,6,9,12}) + H(X_{4,11,12}) - H(X_{4,7,11,12}) - 3H(X_{9,11,12}) + 3H(X_{9,10,11,12}) + 2H(X_{4,13}) - 2H(X_{4,5,13}) - 2H(X_{4,7,13}) + 2H(X_{4,5,7,13}) + 2H(X_{12,13}) - 2H(X_{5,12,13}) - 2H(X_{9,12,13}) + 2H(X_{5,9,12,13})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{5};X_{13}|X_{6,9})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{11}|X_{13})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{10};X_{13}|X_{3})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{1};X_{8}|X_{4,10})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{12}|X_{9})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{1};X_{13}|X_{11})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{12}|X_{11})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{1};X_{9}|X_{2,11,13})"}],"expression":[[4,1],[256,2],[272,-2],[288,1],[304,-1],[516,-1],[520,-2],[521,2],[648,2],[649,-2],[1024,4],[1025,-3],[1032,-1],[2304,-2],[2320,2],[3072,-1],[3080,1],[4096,-2],[4100,-1],[4224,2],[4384,-1],[4400,1],[4612,1],[5120,-1],[5121,3],[5122,3],[5123,-3],[5248,-2],[5378,-3],[5379,3]],"id":1,"kind":"composite_equality","label":"witness 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3H(X_{1,2,9,11,13})>=0"],"text":"H(X_{3}) + 2H(X_{9}) - 2H(X_{5,9}) + H(X_{6,9}) - H(X_{5,6,9}) - H(X_{3,10}) - 2H(X_{4,10}) + 2H(X_{1,4,10}) + 2H(X_{4,8,10}) - 2H(X_{1,4,8,10}) + 4H(X_{11}) - 3H(X_{1,11}) - H(X_{4,11}) - 2H(X_{9,12}) + 2H(X_{5,9,12}) - H(X_{11,12}) + H(X_{4,11,12}) - 2H(X_{13}) - H(X_{3,13}) + 2H(X_{8,13}) - H(X_{6,9,13}) + H(X_{5,6,9,13}) + H(X_{3,10,13}) - H(X_{11,13}) + 3H(X_{1,11,13}) + 3H(X_{2,11,13}) - 3H(X_{1,2,11,13}) - 2H(X_{8,11,13}) - 3H(X_{2,9,11,13}) + 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7H(X_{4}) + 21H(X_{1,3,4}) + 28H(X_{5}) + 7H(X_{2,5}) + 45H(X_{6}) - 10H(X_{5,6}) + 31H(X_{7}) - 21H(X_{1,3,4,7}) - 30H(X_{6,7}) + 10H(X_{8}) + 10H(X_{2,8}) - 10H(X_{6,8}) - 10H(X_{2,7,8}) - H(X_{9}) + 37H(X_{4,9}) + 10H(X_{8,9}) - 4H(X_{1,6,8,9}) + 5H(X_{10}) + 8H(X_{4,5,10}) - 21H(X_{7,10}) - 14H(X_{8,10}) + 14H(X_{1,8,10}) + 14H(X_{5,8,10}) - 14H(X_{1,5,8,10}) - 30H(X_{4,9,10}) - 21H(X_{1,3,4,11}) + 20H(X_{6,11}) - 20H(X_{5,6,11}) + 21H(X_{1,3,4,7,11}) - 10H(X_{9,11}) + 10H(X_{2,9,11}) - 20H(X_{4,9,11}) - 10H(X_{2,4,9,11}) + 30H(X_{4,9,10,11}) - 5H(X_{12}) + 7H(X_{4,12}) - 9H(X_{2,4,8,12}) - 9H(X_{3,6,10,12}) + 2H(X_{2,5,8,11,12}) + H(X_{3,4,5,6,7,8,11,12}) + 17H(X_{13}) - 21H(X_{3,13}) - 30H(X_{5,13}) + 9H(X_{7,13}) - 10H(X_{8,13}) + 10H(X_{6,8,13}) - 8H(X_{4,5,10,13}) + 21H(X_{7,10,13}) - 6H(X_{2,3,5,9,10,13}) + 2H(X_{1,2,4,5,7,11,13}) - 4H(X_{4,7,9,11,13}) + 4H(X_{1,4,6,8,9,11,13}) + 4H(X_{3,4,6,7,8,9,11,13}) - 23H(X_{12,13}) - 7H(X_{4,12,13}) + 30H(X_{5,12,13}) + 9H(X_{1,2,4,8,12,13}) + 9H(X_{1,3,4,5,6,8,10,12,13}) + 3H(X_{2,4,5,6,11,12,13}) - 3H(X_{1,2,3,4,5,6,11,12,13}) - 2H(X_{1,2,4,5,7,11,12,13}) - 2H(X_{2,5,8,11,12,13}) + 5H(X_{3,6,7,8,11,12,13}) - 5H(X_{3,5,6,7,8,11,12,13}) + 8H(X_{1,2,10,11,12,13}) - 8H(X_{1,2,3,10,11,12,13}) + 6H(X_{2,3,4,5,6,8,9,10,11,12,13}) - 4H(X_{14}) + 4H(X_{1,14}) - 21H(X_{3,14}) + 4H(X_{1,6,8,9,14}) - 21H(X_{10,14}) + 6H(X_{2,5,6,10,14}) + 21H(X_{8,10,14}) + 21H(X_{9,10,14}) - 6H(X_{2,5,6,9,10,14}) - 21H(X_{8,9,10,14}) - 20H(X_{2,12,14}) + 20H(X_{2,7,12,14}) + 4H(X_{4,8,9,12,14}) - 4H(X_{1,4,8,9,12,14}) + 21H(X_{3,13,14}) - 4H(X_{1,4,6,8,9,11,13,14}) + 20H(X_{2,12,13,14}) - 20H(X_{2,7,12,13,14}) + 5H(X_{2,4,5,11,12,13,14}) - 5H(X_{2,4,5,6,11,12,13,14}) + 5H(X_{1,2,3,4,5,6,7,8,9,11,12,13,14}) - 5H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14})","witness_value":2440},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{8};X_{9}|X_{10,14})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{13};X_{14}|X_{3})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{5})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{9})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{10};X_{13}|X_{7})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{1};X_{5}|X_{8,10})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{7};X_{11}|X_{1,3,4})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{13}|X_{12})"}],"expression":[[2,1],[4,-3],[8,1],[13,-3],[16,1],[18,-1],[64,-3],[77,3],[256,1],[264,-1],[576,3],[640,2],[641,-2],[656,-2],[657,2],[1037,3],[1101,-3],[2048,1],[2056,-1],[4100,3],[4160,3],[4672,-3],[6144,-1],[6152,1],[8196,3],[8704,3],[8832,-3],[8960,-3],[9088,3],[12292,-3]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["H(X_{2}) - 3H(X_{3}) + H(X_{4}) - 3H(X_{1,3,4}) + H(X_{5}) - H(X_{2,5}) - 3H(X_{7}) + 3H(X_{1,3,4,7}) + H(X_{9}) - H(X_{4,9}) + 3H(X_{7,10}) + 2H(X_{8,10}) - 2H(X_{1,8,10}) - 2H(X_{5,8,10}) + 2H(X_{1,5,8,10}) + 3H(X_{1,3,4,11}) - 3H(X_{1,3,4,7,11}) + H(X_{12}) - H(X_{4,12}) + 3H(X_{3,13}) + 3H(X_{7,13}) - 3H(X_{7,10,13}) - H(X_{12,13}) + H(X_{4,12,13}) + 3H(X_{3,14}) + 3H(X_{10,14}) - 3H(X_{8,10,14}) - 3H(X_{9,10,14}) + 3H(X_{8,9,10,14}) - 3H(X_{3,13,14})>=0","-H(X_{2}) + 3H(X_{3}) - H(X_{4}) + 3H(X_{1,3,4}) - H(X_{5}) + H(X_{2,5}) + 3H(X_{7}) - 3H(X_{1,3,4,7}) - H(X_{9}) + H(X_{4,9}) - 3H(X_{7,10}) - 2H(X_{8,10}) + 2H(X_{1,8,10}) + 2H(X_{5,8,10}) - 2H(X_{1,5,8,10}) - 3H(X_{1,3,4,11}) + 3H(X_{1,3,4,7,11}) - H(X_{12}) + H(X_{4,12}) - 3H(X_{3,13}) - 3H(X_{7,13}) + 3H(X_{7,10,13}) + H(X_{12,13}) - H(X_{4,12,13}) - 3H(X_{3,14}) - 3H(X_{10,14}) + 3H(X_{8,10,14}) + 3H(X_{9,10,14}) - 3H(X_{8,9,10,14}) + 3H(X_{3,13,14})>=0"],"text":"H(X_{2}) - 3H(X_{3}) + H(X_{4}) - 3H(X_{1,3,4}) + H(X_{5}) - H(X_{2,5}) - 3H(X_{7}) + 3H(X_{1,3,4,7}) + H(X_{9}) - H(X_{4,9}) + 3H(X_{7,10}) + 2H(X_{8,10}) - 2H(X_{1,8,10}) - 2H(X_{5,8,10}) + 2H(X_{1,5,8,10}) + 3H(X_{1,3,4,11}) - 3H(X_{1,3,4,7,11}) + H(X_{12}) - H(X_{4,12}) + 3H(X_{3,13}) + 3H(X_{7,13}) - 3H(X_{7,10,13}) - H(X_{12,13}) + H(X_{4,12,13}) + 3H(X_{3,14}) + 3H(X_{10,14}) - 3H(X_{8,10,14}) - 3H(X_{9,10,14}) + 3H(X_{8,9,10,14}) - 3H(X_{3,13,14})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{10};X_{11}|X_{4,9})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{11}|X_{6})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{5};X_{12}|X_{13})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{5};X_{6})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{6};X_{7})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{6};X_{13}|X_{8})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{7};X_{13})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{7};X_{13}|X_{2,12,14})"}],"expression":[[16,3],[32,4],[48,-1],[96,-3],[128,1],[160,-1],[264,3],[776,-3],[1056,2],[1072,-2],[1288,-3],[1800,3],[4112,-3],[4160,3],[4224,-1],[4256,1],[6144,-3],[6160,3],[10242,-2],[10306,2],[14338,2],[14402,-2]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["3H(X_{5}) + 4H(X_{6}) - H(X_{5,6}) - 3H(X_{6,7}) + H(X_{8}) - H(X_{6,8}) + 3H(X_{4,9}) - 3H(X_{4,9,10}) + 2H(X_{6,11}) - 2H(X_{5,6,11}) - 3H(X_{4,9,11}) + 3H(X_{4,9,10,11}) - 3H(X_{5,13}) + 3H(X_{7,13}) - H(X_{8,13}) + H(X_{6,8,13}) - 3H(X_{12,13}) + 3H(X_{5,12,13}) - 2H(X_{2,12,14}) + 2H(X_{2,7,12,14}) + 2H(X_{2,12,13,14}) - 2H(X_{2,7,12,13,14})>=0","-3H(X_{5}) - 4H(X_{6}) + H(X_{5,6}) + 3H(X_{6,7}) - H(X_{8}) + H(X_{6,8}) - 3H(X_{4,9}) + 3H(X_{4,9,10}) - 2H(X_{6,11}) + 2H(X_{5,6,11}) + 3H(X_{4,9,11}) - 3H(X_{4,9,10,11}) + 3H(X_{5,13}) - 3H(X_{7,13}) + H(X_{8,13}) - H(X_{6,8,13}) + 3H(X_{12,13}) - 3H(X_{5,12,13}) + 2H(X_{2,12,14}) - 2H(X_{2,7,12,14}) - 2H(X_{2,12,13,14}) + 2H(X_{2,7,12,13,14})>=0"],"text":"3H(X_{5}) + 4H(X_{6}) - H(X_{5,6}) - 3H(X_{6,7}) + H(X_{8}) - H(X_{6,8}) + 3H(X_{4,9}) - 3H(X_{4,9,10}) + 2H(X_{6,11}) - 2H(X_{5,6,11}) - 3H(X_{4,9,11}) + 3H(X_{4,9,10,11}) - 3H(X_{5,13}) + 3H(X_{7,13}) - H(X_{8,13}) + H(X_{6,8,13}) - 3H(X_{12,13}) + 3H(X_{5,12,13}) - 2H(X_{2,12,14}) + 2H(X_{2,7,12,14}) + 2H(X_{2,12,13,14}) - 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5H(X_{5}) + 3H(X_{1,5}) + 2H(X_{3,5}) + H(X_{5,6}) - H(X_{1,5,6}) + 4H(X_{5,7}) - 2H(X_{1,5,7}) - 2H(X_{4,5,7}) + 3H(X_{6,8}) - 3H(X_{3,6,8}) - 3H(X_{6,7,8}) + 3H(X_{3,6,7,8}) + 3H(X_{9}) - 3H(X_{3,9}) - 2H(X_{5,7,9}) + 2H(X_{4,5,7,9}) + 2H(X_{6,11,12}) - 2H(X_{6,9,11,12}) - 2H(X_{6,11,12,13}) + 2H(X_{6,9,11,12,13}) + 2H(X_{3,14,15}) - 2H(X_{3,9,14,15}) - 2H(X_{3,12,14,15}) + 2H(X_{3,9,12,14,15})>=0","-H(X_{3}) + 5H(X_{5}) - 3H(X_{1,5}) - 2H(X_{3,5}) - H(X_{5,6}) + H(X_{1,5,6}) - 4H(X_{5,7}) + 2H(X_{1,5,7}) + 2H(X_{4,5,7}) - 3H(X_{6,8}) + 3H(X_{3,6,8}) + 3H(X_{6,7,8}) - 3H(X_{3,6,7,8}) - 3H(X_{9}) + 3H(X_{3,9}) + 2H(X_{5,7,9}) - 2H(X_{4,5,7,9}) - 2H(X_{6,11,12}) + 2H(X_{6,9,11,12}) + 2H(X_{6,11,12,13}) - 2H(X_{6,9,11,12,13}) - 2H(X_{3,14,15}) + 2H(X_{3,9,14,15}) + 2H(X_{3,12,14,15}) - 2H(X_{3,9,12,14,15})>=0"],"text":"H(X_{3}) - 5H(X_{5}) + 3H(X_{1,5}) + 2H(X_{3,5}) + H(X_{5,6}) - H(X_{1,5,6}) + 4H(X_{5,7}) - 2H(X_{1,5,7}) - 2H(X_{4,5,7}) + 3H(X_{6,8}) - 3H(X_{3,6,8}) - 3H(X_{6,7,8}) + 3H(X_{3,6,7,8}) + 3H(X_{9}) - 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+ 7H(X_{2}) - 7H(X_{1,2}) + 14H(X_{4}) - 6H(X_{5}) - 12H(X_{6}) + 6H(X_{3,6}) - 14H(X_{7}) + 6H(X_{5,7}) + 14H(X_{4,9}) - 26H(X_{4,8,9}) + 12H(X_{4,5,8,9}) + 12H(X_{4,7,8,9}) - 12H(X_{4,5,7,8,9}) + 7H(X_{1,10}) + 6H(X_{5,10}) + 6H(X_{6,10}) - 6H(X_{3,6,10}) - 6H(X_{5,7,10}) - 7H(X_{1,8,10}) + 2H(X_{4,7,9,10}) + H(X_{2,6,8,9,10}) - 6H(X_{7,11}) - 9H(X_{2,5,7,11}) + 6H(X_{7,8,11}) - 14H(X_{2,6,9,11}) + 14H(X_{2,4,6,9,11}) + 14H(X_{2,5,6,9,11}) - 14H(X_{2,4,5,6,9,11}) + 9H(X_{2,4,5,7,9,11}) + 4H(X_{7,10,11}) + 2H(X_{3,6,7,10,11}) - 6H(X_{7,8,10,11}) + 3H(X_{12}) + 5H(X_{5,6,12}) - 5H(X_{4,5,6,12}) + 2H(X_{2,3,9,12}) - 9H(X_{3,6,8,10,12}) + 9H(X_{3,5,6,8,10,12}) + 2H(X_{7,9,10,11,12}) - 2H(X_{3,6,7,9,10,11,12}) - 14H(X_{13}) - 8H(X_{1,2,13}) + 14H(X_{7,13}) + 5H(X_{2,3,7,13}) - 5H(X_{2,3,4,7,13}) + 6H(X_{5,6,8,13}) - 4H(X_{1,2,3,6,10,13}) + 4H(X_{1,2,3,6,9,10,13}) + 9H(X_{2,5,7,11,13}) - 9H(X_{2,4,5,7,9,11,13}) + 2H(X_{1,6,7,10,11,13}) - 2H(X_{1,2,3,6,7,9,10,11,12,13}) + 12H(X_{5,14}) - 12H(X_{2,5,14}) - 12H(X_{4,5,14}) + 12H(X_{2,4,5,14}) - 14H(X_{4,9,14}) + 14H(X_{4,8,9,14}) - 2H(X_{1,3,5,12,14}) + 2H(X_{1,3,5,9,11,12,14}) + 6H(X_{2,4,6,10,11,12,14}) + 2H(X_{1,3,5,7,12,13,14}) - 2H(X_{1,3,5,7,9,11,12,13,14}) - 6H(X_{2,4,5,6,8,10,11,12,13,14}) + 17H(X_{15}) - 21H(X_{3,15}) - 14H(X_{4,15}) - 14H(X_{7,15}) + 12H(X_{3,7,15}) + 14H(X_{4,7,15}) - 12H(X_{3,5,7,15}) + 18H(X_{8,15}) - 12H(X_{3,7,9,15}) + 12H(X_{3,5,7,9,15}) - 7H(X_{1,10,15}) + 7H(X_{1,8,10,15}) + 7H(X_{4,8,10,15}) - 7H(X_{4,8,9,10,15}) - 12H(X_{3,11,15}) + 12H(X_{3,6,11,15}) + 12H(X_{3,10,11,15}) - 12H(X_{3,6,10,11,15}) - 7H(X_{4,8,10,11,15}) + 7H(X_{4,8,9,10,11,15}) + 18H(X_{12,15}) - 18H(X_{8,12,15}) + 9H(X_{1,2,3,4,6,8,10,11,12,15}) - 9H(X_{1,2,3,4,5,6,8,10,11,12,15}) + 8H(X_{1,2,3,4,5,6,8,12,13,15}) + H(X_{4,8,10,12,13,15}) - H(X_{1,4,8,10,12,13,15}) - 21H(X_{14,15}) + 21H(X_{3,14,15}) - 5H(X_{1,2,4,5,13,14,15}) + 4H(X_{1,2,3,6,7,10,13,14,15}) - 4H(X_{1,2,3,6,7,9,10,13,14,15}) + H(X_{1,3,4,5,7,11,12,13,14,15}) + 5H(X_{1,2,4,5,7,10,11,12,13,14,15}) - H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15})","witness_value":10237},"k":16,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{14}|X_{15})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{8};X_{15}|X_{1,10})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{9};X_{11}|X_{4,8,10,15})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{7};X_{13})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{2})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{14}|X_{4,9})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{4};X_{5}|X_{2,6,9,11})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{4};X_{7}|X_{15})"}],"expression":[[1,1],[2,1],[3,-1],[64,-2],[264,2],[392,-2],[513,1],[641,-1],[1314,-2],[1322,2],[1330,2],[1338,-2],[4096,-2],[4160,2],[8456,-2],[8584,2],[16384,5],[16388,-3],[16392,-2],[16448,-2],[16456,2],[16897,-1],[17025,1],[17032,1],[17288,-1],[18056,-1],[18312,1],[24576,-3],[24580,3]],"id":0,"kind":"composite_equality","label":"witness 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3H(X_{3,14,15})>=0"],"text":"H(X_{1}) + H(X_{2}) - H(X_{1,2}) - 2H(X_{7}) + 2H(X_{4,9}) - 2H(X_{4,8,9}) + H(X_{1,10}) - H(X_{1,8,10}) - 2H(X_{2,6,9,11}) + 2H(X_{2,4,6,9,11}) + 2H(X_{2,5,6,9,11}) - 2H(X_{2,4,5,6,9,11}) - 2H(X_{13}) + 2H(X_{7,13}) - 2H(X_{4,9,14}) + 2H(X_{4,8,9,14}) + 5H(X_{15}) - 3H(X_{3,15}) - 2H(X_{4,15}) - 2H(X_{7,15}) + 2H(X_{4,7,15}) - H(X_{1,10,15}) + H(X_{1,8,10,15}) + H(X_{4,8,10,15}) - H(X_{4,8,9,10,15}) - H(X_{4,8,10,11,15}) + H(X_{4,8,9,10,11,15}) - 3H(X_{14,15}) + 3H(X_{3,14,15})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{8};X_{10}|X_{7,11})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{9}|X_{3,7,15})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{3};X_{10}|X_{6})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{7}|X_{4,8,9})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{7};X_{10}|X_{5})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{2};X_{4}|X_{5,14})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{10}|X_{3,11,15})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{8};X_{12}|X_{15})"}],"expression":[[16,-1],[32,-1],[36,1],[80,1],[392,-2],[408,2],[456,2],[472,-2],[528,1],[544,1],[548,-1],[592,-1],[1088,-1],[1216,1],[1600,1],[1728,-1],[8208,2],[8210,-2],[8216,-2],[8218,2],[16384,-3],[16452,2],[16468,-2],[16512,3],[16708,-2],[16724,2],[17412,-2],[17444,2],[17924,2],[17956,-2],[18432,3],[18560,-3]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["-H(X_{5}) - H(X_{6}) + H(X_{3,6}) + H(X_{5,7}) - 2H(X_{4,8,9}) + 2H(X_{4,5,8,9}) + 2H(X_{4,7,8,9}) - 2H(X_{4,5,7,8,9}) + H(X_{5,10}) + H(X_{6,10}) - H(X_{3,6,10}) - H(X_{5,7,10}) - H(X_{7,11}) + H(X_{7,8,11}) + H(X_{7,10,11}) - H(X_{7,8,10,11}) + 2H(X_{5,14}) - 2H(X_{2,5,14}) - 2H(X_{4,5,14}) + 2H(X_{2,4,5,14}) - 3H(X_{15}) + 2H(X_{3,7,15}) - 2H(X_{3,5,7,15}) + 3H(X_{8,15}) - 2H(X_{3,7,9,15}) + 2H(X_{3,5,7,9,15}) - 2H(X_{3,11,15}) + 2H(X_{3,6,11,15}) + 2H(X_{3,10,11,15}) - 2H(X_{3,6,10,11,15}) + 3H(X_{12,15}) - 3H(X_{8,12,15})>=0","H(X_{5}) + H(X_{6}) - H(X_{3,6}) - H(X_{5,7}) + 2H(X_{4,8,9}) - 2H(X_{4,5,8,9}) - 2H(X_{4,7,8,9}) + 2H(X_{4,5,7,8,9}) - H(X_{5,10}) - H(X_{6,10}) + H(X_{3,6,10}) + H(X_{5,7,10}) + H(X_{7,11}) - H(X_{7,8,11}) - H(X_{7,10,11}) + H(X_{7,8,10,11}) - 2H(X_{5,14}) + 2H(X_{2,5,14}) + 2H(X_{4,5,14}) - 2H(X_{2,4,5,14}) + 3H(X_{15}) - 2H(X_{3,7,15}) + 2H(X_{3,5,7,15}) - 3H(X_{8,15}) + 2H(X_{3,7,9,15}) - 2H(X_{3,5,7,9,15}) + 2H(X_{3,11,15}) 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9H(X_{1,3}) - 6H(X_{4}) + 3H(X_{5}) - 9H(X_{2,5}) + 6H(X_{4,5}) + 4H(X_{6}) - 10H(X_{7}) + 10H(X_{1,7}) + 9H(X_{2,7}) + 2H(X_{3,7}) + 9H(X_{8}) - 3H(X_{2,4,6,7,8}) - 8H(X_{9}) + 3H(X_{6,7,9}) + 5H(X_{8,9}) - 6H(X_{6,8,9}) + 6H(X_{3,6,8,9}) - 9H(X_{10}) + 6H(X_{6,10}) - 6H(X_{2,6,10}) - 9H(X_{5,8,10}) + 2H(X_{2,9,10}) - 2H(X_{2,3,7,9,10}) + 10H(X_{3,8,9,10}) + 11H(X_{11}) - 4H(X_{2,11}) + 6H(X_{3,11}) + 4H(X_{2,6,11}) + 4H(X_{2,7,11}) - 9H(X_{8,11}) + 4H(X_{2,8,11}) - 4H(X_{2,7,8,11}) - H(X_{5,7,9,11}) - 6H(X_{6,10,11}) + 6H(X_{2,6,10,11}) + 3H(X_{2,4,6,7,8,10,11}) + H(X_{12}) + 9H(X_{1,9,12}) + 9H(X_{10,12}) + 9H(X_{5,8,10,12}) + 10H(X_{4,6,7,9,10,12}) - 10H(X_{1,4,6,7,9,10,12}) - 10H(X_{3,8,9,10,12}) + H(X_{13}) - 4H(X_{6,13}) + 4H(X_{1,3,6,13}) - 9H(X_{2,7,13}) + 3H(X_{2,4,6,7,8,13}) + 6H(X_{6,8,9,13}) - 6H(X_{3,6,8,9,13}) + 8H(X_{9,10,13}) - 2H(X_{11,13}) + H(X_{5,7,9,11,13}) - 3H(X_{2,4,6,7,8,10,11,13}) + 3H(X_{3,10,14}) - 3H(X_{3,6,7,9,10,14}) + 10H(X_{1,2,3,4,6,7,8,9,10,14}) + H(X_{5,6,7,9,10,11,14}) + 5H(X_{3,13,14}) - 2H(X_{2,7,9,13,14}) + 2H(X_{1,2,3,4,9,10,13,14}) - H(X_{5,6,7,9,10,11,13,14}) + 2H(X_{2,3,7,9,12,13,14}) + 16H(X_{15}) + 9H(X_{7,15}) - 9H(X_{2,7,15}) + 8H(X_{2,5,6,9,15}) + 9H(X_{5,8,10,15}) - 6H(X_{11,15}) - 9H(X_{1,7,9,12,15}) - 9H(X_{5,8,10,12,15}) + 9H(X_{2,7,13,15}) - 8H(X_{2,5,6,9,10,13,15}) + 5H(X_{2,3,6,7,8,9,10,11,13,15}) - 5H(X_{1,2,3,6,7,8,9,10,11,13,15}) + 5H(X_{2,3,8,9,14,15}) - 5H(X_{1,2,3,8,9,14,15}) - 10H(X_{1,2,3,4,6,7,8,9,10,14,15}) + 2H(X_{2,6,7,8,9,13,14,15}) + 7H(X_{4,5,7,9,10,11,13,14,15}) - 2H(X_{2,3,6,7,8,9,12,13,14,15}) + 11H(X_{16}) - 6H(X_{1,16}) - 5H(X_{8,9,16}) + 5H(X_{5,9,11,13,16}) - 5H(X_{3,5,9,11,13,14,16}) - 9H(X_{1,4,6,15,16}) - 3H(X_{1,2,3,7,9,11,15,16}) + 9H(X_{1,4,6,8,9,10,11,15,16}) + 4H(X_{12,15,16}) + 2H(X_{5,6,7,8,11,12,15,16}) + 4H(X_{1,12,13,15,16}) - 4H(X_{1,3,6,12,13,15,16}) - 4H(X_{1,2,6,11,12,13,15,16}) + 3H(X_{1,2,3,7,8,9,11,14,15,16}) - 2H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16})","witness_value":1687},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{5})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{8};X_{11})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{12};X_{15}|X_{5,8,10})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{1};X_{16})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{10};X_{12})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{13};X_{15}|X_{2,7})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{13}|X_{6,8,9})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{1};X_{3})"}],"expression":[[1,5],[2,3],[4,3],[5,-3],[16,3],[18,-3],[66,1],[128,3],[416,-2],[420,2],[512,-3],[656,-3],[1024,3],[1152,-3],[2048,-3],[2560,3],[2704,3],[4162,-1],[4512,2],[4516,-2],[16450,-1],[17040,3],[19088,-3],[20546,1],[32768,2],[32769,-2]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["5H(X_{1}) + 3H(X_{2}) + 3H(X_{3}) - 3H(X_{1,3}) + 3H(X_{5}) - 3H(X_{2,5}) + H(X_{2,7}) + 3H(X_{8}) - 2H(X_{6,8,9}) + 2H(X_{3,6,8,9}) - 3H(X_{10}) - 3H(X_{5,8,10}) + 3H(X_{11}) - 3H(X_{8,11}) - 3H(X_{12}) + 3H(X_{10,12}) + 3H(X_{5,8,10,12}) - H(X_{2,7,13}) + 2H(X_{6,8,9,13}) - 2H(X_{3,6,8,9,13}) - H(X_{2,7,15}) + 3H(X_{5,8,10,15}) - 3H(X_{5,8,10,12,15}) + H(X_{2,7,13,15}) + 2H(X_{16}) - 2H(X_{1,16})>=0","-5H(X_{1}) - 3H(X_{2}) - 3H(X_{3}) + 3H(X_{1,3}) - 3H(X_{5}) + 3H(X_{2,5}) - H(X_{2,7}) - 3H(X_{8}) + 2H(X_{6,8,9}) - 2H(X_{3,6,8,9}) + 3H(X_{10}) + 3H(X_{5,8,10}) - 3H(X_{11}) + 3H(X_{8,11}) + 3H(X_{12}) - 3H(X_{10,12}) - 3H(X_{5,8,10,12}) + H(X_{2,7,13}) - 2H(X_{6,8,9,13}) + 2H(X_{3,6,8,9,13}) + H(X_{2,7,15}) - 3H(X_{5,8,10,15}) + 3H(X_{5,8,10,12,15}) - H(X_{2,7,13,15}) - 2H(X_{16}) + 2H(X_{1,16})>=0"],"text":"5H(X_{1}) + 3H(X_{2}) + 3H(X_{3}) - 3H(X_{1,3}) + 3H(X_{5}) - 3H(X_{2,5}) + H(X_{2,7}) + 3H(X_{8}) - 2H(X_{6,8,9}) + 2H(X_{3,6,8,9}) - 3H(X_{10}) - 3H(X_{5,8,10}) + 3H(X_{11}) - 3H(X_{8,11}) - 3H(X_{12}) + 3H(X_{10,12}) + 3H(X_{5,8,10,12}) - H(X_{2,7,13}) + 2H(X_{6,8,9,13}) - 2H(X_{3,6,8,9,13}) - H(X_{2,7,15}) + 3H(X_{5,8,10,15}) - 3H(X_{5,8,10,12,15}) + H(X_{2,7,13,15}) + 2H(X_{16}) - 2H(X_{1,16})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{11};X_{15})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{7};X_{8}|X_{2,11})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{13})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{4};X_{5})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{11};X_{13})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{11})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{11}|X_{6,10})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{13};X_{15}|X_{2,7})"}],"expression":[[4,-3],[8,-3],[16,-3],[24,3],[32,2],[66,3],[544,3],[546,-3],[1024,1],[1026,-2],[1028,3],[1090,2],[1154,2],[1218,-2],[1568,-3],[1570,3],[4096,3],[4128,-2],[4162,-3],[5120,-1],[16384,3],[16450,-3],[17408,-3],[20546,3]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["-3H(X_{3}) - 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24H(X_{8,16}) + 4H(X_{3,8,16}) - 8H(X_{3,6,11,16}) + 16H(X_{12,16}) - 16H(X_{10,12,16}) - 8H(X_{3,6,13,16}) + 8H(X_{3,6,11,13,16}) + 2H(X_{2,9,11,13,16}) + 5H(X_{1,2,3,4,7,8,9,10,12,14,16}) - 2H(X_{2,9,10,11,13,14,16}) - 9H(X_{15,16}) - H(X_{4,5,6,8,9,12,15,16}) + 9H(X_{1,3,4,5,7,10,11,13,15,16}) - 9H(X_{1,3,4,5,7,9,10,11,13,15,16}) - 23H(X_{17}) - 3H(X_{1,17}) - 2H(X_{2,17}) + 3H(X_{1,2,17}) + 9H(X_{5,17}) + 2H(X_{2,3,7,17}) + 7H(X_{8,17}) + 6H(X_{10,17}) + 9H(X_{12,17}) - 9H(X_{5,12,17}) + 8H(X_{13,17}) - 10H(X_{2,8,13,17}) + 10H(X_{1,2,5,6,8,9,13,17}) + 5H(X_{3,5,6,7,10,13,17}) - 5H(X_{3,5,6,7,9,10,13,17}) + 10H(X_{2,8,10,12,13,17}) - 10H(X_{1,2,5,6,8,9,10,12,13,17}) + 4H(X_{1,5,7,8,9,10,12,14,17}) + 4H(X_{5,6,8,11,12,14,17}) - 4H(X_{2,5,6,8,11,12,14,17}) - 7H(X_{1,2,3,4,5,6,7,8,12,15,17}) + 3H(X_{2,3,5,6,8,11,12,13,14,15,17}) + 8H(X_{16,17}) + 2H(X_{5,9,10,16,17}) - 16H(X_{12,16,17}) + 16H(X_{10,12,16,17}) + 8H(X_{1,5,7,11,12,16,17}) - 8H(X_{1,3,5,7,11,12,16,17}) - 8H(X_{13,16,17}) + 7H(X_{1,4,6,7,9,10,13,16,17}) - 4H(X_{1,3,5,7,8,9,10,12,14,16,17}) + 2H(X_{2,4,6,8,11,12,15,16,17}) - 2H(X_{2,3,4,6,7,8,11,12,15,16,17}) - 7H(X_{1,4,5,6,7,9,10,13,15,16,17}) + 3H(X_{1,4,5,11,14,15,16,17}) - 3H(X_{1,2,4,5,6,11,13,14,15,16,17}) - 2H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17})","witness_value":992},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{11};X_{13}|X_{3,6,16})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{13};X_{14})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{8})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{13};X_{16}|X_{17})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{8};X_{16})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{10};X_{17}|X_{12,16})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{6};X_{9}|X_{10,15})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{11};X_{12}|X_{2,7,15})"}],"expression":[[16,-2],[128,1],[144,2],[4096,-3],[8192,-3],[12288,3],[16450,-3],[16896,1],[16928,-1],[17152,-1],[17184,1],[17474,3],[18498,3],[19522,-3],[32768,3],[32804,1],[32896,-3],[33828,-1],[34816,2],[35328,-2],[36900,-1],[37924,1],[65536,-1],[69632,1],[98304,1],[100352,-2],[100864,2],[102400,-1]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["-2H(X_{5}) + H(X_{8}) + 2H(X_{5,8}) - 3H(X_{13}) - 3H(X_{14}) + 3H(X_{13,14}) - 3H(X_{2,7,15}) + H(X_{10,15}) - H(X_{6,10,15}) - H(X_{9,10,15}) + H(X_{6,9,10,15}) + 3H(X_{2,7,11,15}) + 3H(X_{2,7,12,15}) - 3H(X_{2,7,11,12,15}) + 3H(X_{16}) + H(X_{3,6,16}) - 3H(X_{8,16}) - H(X_{3,6,11,16}) + 2H(X_{12,16}) - 2H(X_{10,12,16}) - H(X_{3,6,13,16}) + H(X_{3,6,11,13,16}) - H(X_{17}) + H(X_{13,17}) + H(X_{16,17}) - 2H(X_{12,16,17}) + 2H(X_{10,12,16,17}) - H(X_{13,16,17})>=0","2H(X_{5}) - H(X_{8}) - 2H(X_{5,8}) + 3H(X_{13}) + 3H(X_{14}) - 3H(X_{13,14}) + 3H(X_{2,7,15}) - H(X_{10,15}) + H(X_{6,10,15}) + H(X_{9,10,15}) - H(X_{6,9,10,15}) - 3H(X_{2,7,11,15}) - 3H(X_{2,7,12,15}) + 3H(X_{2,7,11,12,15}) - 3H(X_{16}) - H(X_{3,6,16}) + 3H(X_{8,16}) + H(X_{3,6,11,16}) - 2H(X_{12,16}) + 2H(X_{10,12,16}) + H(X_{3,6,13,16}) - H(X_{3,6,11,13,16}) + H(X_{17}) - H(X_{13,17}) - H(X_{16,17}) + 2H(X_{12,16,17}) - 2H(X_{10,12,16,17}) + H(X_{13,16,17})>=0"],"text":"-2H(X_{5}) + H(X_{8}) + 2H(X_{5,8}) - 3H(X_{13}) - 3H(X_{14}) + 3H(X_{13,14}) - 3H(X_{2,7,15}) + H(X_{10,15}) - H(X_{6,10,15}) - H(X_{9,10,15}) + H(X_{6,9,10,15}) + 3H(X_{2,7,11,15}) + 3H(X_{2,7,12,15}) - 3H(X_{2,7,11,12,15}) + 3H(X_{16}) + H(X_{3,6,16}) - 3H(X_{8,16}) - H(X_{3,6,11,16}) + 2H(X_{12,16}) - 2H(X_{10,12,16}) - H(X_{3,6,13,16}) + H(X_{3,6,11,13,16}) - H(X_{17}) + H(X_{13,17}) + H(X_{16,17}) - 2H(X_{12,16,17}) + 2H(X_{10,12,16,17}) - H(X_{13,16,17})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{12})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{11}|X_{14})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{15}|X_{1,4,12})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{15};X_{16})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{5};X_{12}|X_{17})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{17}|X_{1})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{10};X_{17})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{4};X_{9})"}],"expression":[[1,1],[3,-1],[8,3],[32,-2],[256,3],[264,-3],[512,-2],[2048,-2],[2057,-2],[2080,2],[2185,2],[8192,-1],[8193,1],[9216,1],[9217,-1],[16384,3],[18441,2],[18569,-2],[32768,3],[49152,-3],[65536,-5],[65537,-1],[65539,1],[65552,3],[66048,2],[67584,3],[67600,-3]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["H(X_{1}) - H(X_{1,2}) + 3H(X_{4}) - 2H(X_{6}) + 3H(X_{9}) - 3H(X_{4,9}) - 2H(X_{10}) - 2H(X_{12}) - 2H(X_{1,4,12}) + 2H(X_{6,12}) + 2H(X_{1,4,8,12}) - H(X_{14}) + H(X_{1,14}) + H(X_{11,14}) - H(X_{1,11,14}) + 3H(X_{15}) + 2H(X_{1,4,12,15}) - 2H(X_{1,4,8,12,15}) + 3H(X_{16}) - 3H(X_{15,16}) - 5H(X_{17}) - H(X_{1,17}) + H(X_{1,2,17}) + 3H(X_{5,17}) + 2H(X_{10,17}) + 3H(X_{12,17}) - 3H(X_{5,12,17})>=0","-H(X_{1}) + H(X_{1,2}) - 3H(X_{4}) + 2H(X_{6}) - 3H(X_{9}) + 3H(X_{4,9}) + 2H(X_{10}) + 2H(X_{12}) + 2H(X_{1,4,12}) - 2H(X_{6,12}) - 2H(X_{1,4,8,12}) + H(X_{14}) - H(X_{1,14}) - H(X_{11,14}) + H(X_{1,11,14}) - 3H(X_{15}) - 2H(X_{1,4,12,15}) + 2H(X_{1,4,8,12,15}) - 3H(X_{16}) + 3H(X_{15,16}) + 5H(X_{17}) + H(X_{1,17}) - H(X_{1,2,17}) - 3H(X_{5,17}) - 2H(X_{10,17}) - 3H(X_{12,17}) + 3H(X_{5,12,17})>=0"],"text":"H(X_{1}) - H(X_{1,2}) + 3H(X_{4}) - 2H(X_{6}) + 3H(X_{9}) - 3H(X_{4,9}) - 2H(X_{10}) - 2H(X_{12}) - 2H(X_{1,4,12}) + 2H(X_{6,12}) + 2H(X_{1,4,8,12}) - H(X_{14}) + H(X_{1,14}) + H(X_{11,14}) - H(X_{1,11,14}) + 3H(X_{15}) + 2H(X_{1,4,12,15}) - 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6H(X_{17,18}) + 6H(X_{7,17,18}) - 8H(X_{2,3,8,9,17,18}) - 9H(X_{2,3,5,6,9,11,17,18}) + 6H(X_{2,3,4,5,6,7,9,12,14,15,17,18}) - 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2H(X_{8,17}) + 2H(X_{6,8,17}) + 7H(X_{4,6,8,17}) - 3H(X_{9,17}) + 2H(X_{8,11,17}) - 2H(X_{6,8,11,17}) - 2H(X_{1,5,6,13,17}) + 3H(X_{2,3,5,6,9,10,11,14,17}) + 7H(X_{1,5,7,8,10,11,12,15,17}) + 2H(X_{2,3,5,6,8,10,13,15,17}) - 7H(X_{1,5,7,8,10,11,12,14,15,17}) - 6H(X_{16,17}) + 6H(X_{2,16,17}) + 6H(X_{13,16,17}) - 6H(X_{2,13,16,17}) + 8H(X_{2,4,6,7,11,15,16,17}) - 8H(X_{2,4,6,7,8,11,15,16,17}) + 6H(X_{1,2,7,8,10,14,15,16,17}) - 6H(X_{1,2,5,7,8,10,14,15,16,17}) + 2H(X_{18}) + 2H(X_{3,18}) - 3H(X_{6,18}) + 5H(X_{1,2,4,5,6,10,11,12,13,18}) + H(X_{2,4,5,7,9,10,12,13,14,18}) - H(X_{2,4,5,6,7,9,10,12,13,14,18}) + 2H(X_{15,18}) - 2H(X_{1,15,18}) + 2H(X_{2,3,15,18}) + 10H(X_{1,4,5,8,13,15,18}) - 10H(X_{1,4,5,8,11,12,13,14,15,18}) - 9H(X_{1,2,10,16,18}) + 8H(X_{1,6,11,16,18}) + 9H(X_{1,2,4,5,6,7,9,10,12,14,16,18}) - 8H(X_{1,2,4,6,11,12,14,16,18}) - H(X_{4,11,13,15,16,18}) + H(X_{1,2,4,6,8,11,13,15,16,18}) + 3H(X_{1,4,7,8,12,13,15,16,18}) + H(X_{4,7,11,12,13,14,15,16,18}) - H(X_{1,2,4,6,7,8,11,12,13,14,15,16,18}) - 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2H(X_{2,4,7,12,16}) + 2H(X_{2,3,4,7,12,16}) + 10H(X_{13,16}) - 10H(X_{4,13,16}) - 8H(X_{4,6,13,14,16}) - 9H(X_{12,15,16}) + 9H(X_{1,10,12,15,16}) + 9H(X_{12,13,15,16}) - 9H(X_{1,10,12,13,15,16}) - 9H(X_{6,7,9,10,14,15,16}) - 6H(X_{1,2,4,7,10,17}) - 10H(X_{4,8,9,11,12,17}) + 5H(X_{6,8,9,15,17}) - 5H(X_{3,6,8,9,15,17}) + 2H(X_{2,4,7,9,10,11,12,16,17}) - 2H(X_{2,3,4,7,9,10,11,12,16,17}) - 10H(X_{13,16,17}) + 10H(X_{4,13,16,17}) - 2H(X_{18}) - 20H(X_{12,18}) + 8H(X_{1,3,5,6,7,8,10,12,18}) - 8H(X_{1,2,3,5,6,7,8,10,12,18}) + 6H(X_{4,6,10,11,14,18}) - 6H(X_{3,4,6,10,11,14,18}) + 20H(X_{12,14,18}) + 8H(X_{1,4,7,11,15,18}) + 9H(X_{2,3,4,7,14,15,18}) - 8H(X_{9,14,15,18}) + 8H(X_{6,9,14,15,18}) - 9H(X_{2,3,4,7,11,14,15,18}) + 2H(X_{5,6,7,8,12,13,14,15,18}) + 9H(X_{2,3,4,5,6,8,9,10,12,13,16,18}) + 8H(X_{3,4,5,6,7,10,13,14,16,18}) + 8H(X_{9,14,15,16,18}) - 8H(X_{6,9,14,15,16,18}) + 2H(X_{17,18}) + 6H(X_{1,2,3,4,7,9,10,17,18}) - 2H(X_{5,6,7,8,12,13,14,15,17,18}) + 10H(X_{1,3,4,5,6,8,9,10,11,12,13,16,17,18}) + 8H(X_{1,2,9,13,14,16,17,18}) + 9H(X_{2,5,6,7,8,9,10,11,14,15,16,17,18}) + 10H(X_{2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}) - 10H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18})","witness_value":29511},"k":16,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{11};X_{12}|X_{2})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{8}|X_{15})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{13};X_{15}|X_{6,8})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{3};X_{13}|X_{10})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{6};X_{16}|X_{9,14,15,18})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{15}|X_{1,4,12})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{7};X_{14}|X_{15})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{12};X_{14}|X_{10,15})"}],"expression":[[2,-2],[160,2],[512,-1],[516,1],[1026,2],[2050,2],[2057,-2],[2061,2],[3074,-2],[4256,-2],[4608,1],[4612,-1],[16384,5],[16388,-2],[16448,-3],[16512,-2],[16516,2],[16544,-2],[16896,-1],[18441,2],[18445,-2],[18944,1],[20640,2],[24576,-3],[24640,3],[25088,1],[27136,-1],[155904,1],[155936,-1],[188672,-1],[188704,1]],"id":0,"kind":"composite_equality","label":"witness 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H(X_{10,12,14,15}) - H(X_{9,14,15,18}) + H(X_{6,9,14,15,18}) + H(X_{9,14,15,16,18}) - H(X_{6,9,14,15,16,18})>=0"],"text":"-2H(X_{2}) + 2H(X_{6,8}) - H(X_{10}) + H(X_{3,10}) + 2H(X_{2,11}) + 2H(X_{2,12}) - 2H(X_{1,4,12}) + 2H(X_{1,3,4,12}) - 2H(X_{2,11,12}) - 2H(X_{6,8,13}) + H(X_{10,13}) - H(X_{3,10,13}) + 5H(X_{15}) - 2H(X_{3,15}) - 3H(X_{7,15}) - 2H(X_{8,15}) + 2H(X_{3,8,15}) - 2H(X_{6,8,15}) - H(X_{10,15}) + 2H(X_{1,4,12,15}) - 2H(X_{1,3,4,12,15}) + H(X_{10,12,15}) + 2H(X_{6,8,13,15}) - 3H(X_{14,15}) + 3H(X_{7,14,15}) + H(X_{10,14,15}) - H(X_{10,12,14,15}) + H(X_{9,14,15,18}) - H(X_{6,9,14,15,18}) - H(X_{9,14,15,16,18}) + H(X_{6,9,14,15,16,18})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{17}|X_{13,16})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{14};X_{18}|X_{12})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{7};X_{8}|X_{9,12})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{3};X_{4}|X_{10})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{4};X_{5}|X_{2,8,13})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{10};X_{14}|X_{3})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{12}|X_{2})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{1};X_{5}|X_{12,13})"}],"expression":[[2,-2],[4,-1],[34,2],[512,1],[520,-1],[524,1],[2048,2],[2050,2],[2082,-2],[2304,-2],[2368,2],[2432,2],[2496,-2],[4226,3],[4234,-3],[4242,-3],[4250,3],[6144,-3],[6145,3],[6160,3],[6161,-3],[8196,1],[8708,-1],[10240,-2],[36864,1],[36872,-1],[102400,-1],[102408,1],[133120,-2],[141312,2]],"id":1,"kind":"composite_equality","label":"witness 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2H(X_{12,14,18})>=0"],"text":"-2H(X_{2}) - H(X_{3}) + 2H(X_{2,6}) + H(X_{10}) - H(X_{4,10}) + H(X_{3,4,10}) + 2H(X_{12}) + 2H(X_{2,12}) - 2H(X_{2,6,12}) - 2H(X_{9,12}) + 2H(X_{7,9,12}) + 2H(X_{8,9,12}) - 2H(X_{7,8,9,12}) + 3H(X_{2,8,13}) - 3H(X_{2,4,8,13}) - 3H(X_{2,5,8,13}) + 3H(X_{2,4,5,8,13}) - 3H(X_{12,13}) + 3H(X_{1,12,13}) + 3H(X_{5,12,13}) - 3H(X_{1,5,12,13}) + H(X_{3,14}) - H(X_{3,10,14}) - 2H(X_{12,14}) + H(X_{13,16}) - H(X_{4,13,16}) - H(X_{13,16,17}) + H(X_{4,13,16,17}) - 2H(X_{12,18}) + 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H(X_{2,3,4,11,14,16,17}) + 5H(X_{9,13,14,16,17}) - 2H(X_{3,6,8,9,14,15,16,17}) - H(X_{1,2,3,4,7,11,14,15,16,17}) - 16H(X_{4,6,13,18}) + 16H(X_{4,6,9,13,18}) + 7H(X_{1,2,12,13,18}) - 16H(X_{4,14,18}) + 16H(X_{3,4,14,18}) + H(X_{2,7,14,18}) + 16H(X_{4,9,14,18}) - 16H(X_{3,4,9,14,18}) - 7H(X_{1,2,9,12,13,14,18}) + 16H(X_{4,6,13,15,18}) - 16H(X_{4,6,9,13,15,18}) + H(X_{12,14,15,18}) + 9H(X_{7,8,9,10,14,16,18}) + 4H(X_{1,6,8,12,13,17,18}) - H(X_{2,5,7,14,15,17,18}) + 7H(X_{2,3,6,8,12,13,14,15,17,18}) - 7H(X_{1,2,3,6,8,12,13,14,15,17,18}) + 2H(X_{1,3,4,5,6,7,8,9,10,12,13,14,15,16,17,18}) + 10H(X_{19}) + 16H(X_{2,19}) - 10H(X_{7,19}) - 16H(X_{2,8,19}) + 4H(X_{7,11,19}) + 9H(X_{1,4,14,19}) - 16H(X_{16,19}) + 16H(X_{5,16,19}) + 16H(X_{9,16,19}) - 16H(X_{5,9,16,19}) + H(X_{7,17,19}) + 8H(X_{1,2,3,4,6,7,8,9,10,11,13,14,17,19}) - H(X_{1,5,6,7,8,14,15,17,19}) + 5H(X_{1,2,3,4,5,6,7,8,10,11,12,15,18,19}) - 9H(X_{1,4,7,8,9,10,14,16,18,19}) - 4H(X_{1,6,7,8,11,12,13,17,18,19}) - 5H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19})","witness_value":14226},"k":16,"logical_conditions":[{"component_width":12,"component_witness_values":[0,0,0,0,0,0,0,0,0,0,0,0],"components":[{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{13}|X_{1,10,11})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{7}|X_{1,9})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{15}|X_{10,13,14})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{6};X_{12}|X_{16})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{16}|X_{1})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{9};X_{15}|X_{4,6,13,18})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{10}|X_{8})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{19}|X_{2})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{8}|X_{2,11})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{6};X_{7}|X_{16})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{9}|X_{4,14,18})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{9}|X_{16,19})"}],"expression":[[1,2],[2,2],[128,-1],[129,-1],[130,-2],[257,-3],[261,3],[321,3],[325,-3],[640,1],[641,-1],[1026,-2],[1058,2],[1154,2],[1186,-2],[1537,-3],[1541,3],[5633,3],[5637,-3],[12800,-2],[12928,2],[29184,2],[29312,-2],[32768,-2],[32769,-2],[32800,2],[32832,3],[32864,-3],[32897,2],[34816,-1],[34848,1],[135208,2],[135464,-2],[139272,2],[139276,-2],[139528,-2],[139532,2],[151592,-2],[151848,2],[262146,-2],[262274,2],[294912,2],[294928,-2],[295168,-2],[295184,2]],"id":0,"kind":"composite_equality","label":"witness 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2H(X_{2,6,11}) - 2H(X_{2,8,11}) + 2H(X_{2,6,8,11}) + 3H(X_{1,10,11}) - 3H(X_{1,3,10,11}) - 3H(X_{1,10,11,13}) + 3H(X_{1,3,10,11,13}) + 2H(X_{10,13,14}) - 2H(X_{8,10,13,14}) - 2H(X_{10,13,14,15}) + 2H(X_{8,10,13,14,15}) + 2H(X_{16}) + 2H(X_{1,16}) - 2H(X_{6,16}) - 3H(X_{7,16}) + 3H(X_{6,7,16}) - 2H(X_{1,8,16}) + H(X_{12,16}) - H(X_{6,12,16}) - 2H(X_{4,6,13,18}) + 2H(X_{4,6,9,13,18}) - 2H(X_{4,14,18}) + 2H(X_{3,4,14,18}) + 2H(X_{4,9,14,18}) - 2H(X_{3,4,9,14,18}) + 2H(X_{4,6,13,15,18}) - 2H(X_{4,6,9,13,15,18}) + 2H(X_{2,19}) - 2H(X_{2,8,19}) - 2H(X_{16,19}) + 2H(X_{5,16,19}) + 2H(X_{9,16,19}) - 2H(X_{5,9,16,19})>=0"],"text":"2H(X_{1}) + 2H(X_{2}) - H(X_{8}) - H(X_{1,8}) - 2H(X_{2,8}) - 3H(X_{1,9}) + 3H(X_{1,3,9}) + 3H(X_{1,7,9}) - 3H(X_{1,3,7,9}) + H(X_{8,10}) - H(X_{1,8,10}) - 2H(X_{2,11}) + 2H(X_{2,6,11}) + 2H(X_{2,8,11}) - 2H(X_{2,6,8,11}) - 3H(X_{1,10,11}) + 3H(X_{1,3,10,11}) + 3H(X_{1,10,11,13}) - 3H(X_{1,3,10,11,13}) - 2H(X_{10,13,14}) + 2H(X_{8,10,13,14}) + 2H(X_{10,13,14,15}) - 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20H(X_{3,7}) + 20H(X_{8}) - 20H(X_{5,8}) + 20H(X_{9}) + 20H(X_{11}) - 20H(X_{9,11}) - 30H(X_{4,5,12}) + 30H(X_{1,4,5,12}) + 30H(X_{4,5,7,12}) - 30H(X_{1,4,5,7,12}) + 20H(X_{13}) - 30H(X_{3,15}) + 30H(X_{1,3,15}) + 10H(X_{4,15}) - 10H(X_{3,4,15}) - 10H(X_{4,13,15}) + 10H(X_{3,4,13,15}) - 10H(X_{16}) - 10H(X_{17}) - 20H(X_{7,17}) + 20H(X_{3,7,17}) + 10H(X_{16,17}) + 20H(X_{19}) - 20H(X_{13,19}) + 30H(X_{3,15,19}) - 30H(X_{1,3,15,19})","witness_value":0},{"coefficient":-9,"expression":[[1,1],[8,1],[9,-1],[32,-3],[256,-2],[1024,-4],[1056,3],[2072,-1],[2073,1],[2136,1],[2137,-1],[4096,-3],[4352,2],[4416,-2],[5120,1],[37184,2],[131074,-2],[131138,2],[131330,2],[131394,-2],[135488,2],[168256,-2],[262176,1],[262304,-1],[262432,-1],[262560,1]],"kind":"composite_equality","label":"witness equality 2=0","role":"condition","scaled_expression":[[1,-9],[8,-9],[9,9],[32,27],[256,18],[1024,36],[1056,-27],[2072,9],[2073,-9],[2136,-9],[2137,9],[4096,27],[4352,-18],[4416,18],[5120,-9],[37184,-18],[131074,18],[131138,-18],[131330,-18],[131394,18],[135488,-18],[168256,18],[262176,-9],[262304,9],[262432,9],[262560,-9]],"scaled_witness_value":0,"source_index":1,"text":"-9H(X_{1}) - 9H(X_{4}) + 9H(X_{1,4}) + 27H(X_{6}) + 18H(X_{9}) + 36H(X_{11}) - 27H(X_{6,11}) + 9H(X_{4,5,12}) - 9H(X_{1,4,5,12}) - 9H(X_{4,5,7,12}) + 9H(X_{1,4,5,7,12}) + 27H(X_{13}) - 18H(X_{9,13}) + 18H(X_{7,9,13}) - 9H(X_{11,13}) - 18H(X_{7,9,13,16}) + 18H(X_{2,18}) - 18H(X_{2,7,18}) - 18H(X_{2,9,18}) + 18H(X_{2,7,9,18}) - 18H(X_{7,9,13,18}) + 18H(X_{7,9,13,16,18}) - 9H(X_{6,19}) + 9H(X_{6,8,19}) + 9H(X_{6,9,19}) - 9H(X_{6,8,9,19})","witness_value":0}],"certificate_length":22,"certificate_order":"greedy_negative_mass_condition_late","condition_coordinate_supports":[28,26],"condition_rank":2,"condition_tokens":[311,310,299,299],"condition_width":8,"config_sha256":"1b1f298ed012071c85e9cfb286fe5dcf7c183383de92041754d8a7802f426421","equality_condition_count":2,"full_tokens":1947,"generator_dependency_sha256":{"data/finite_field.py":"c9738c10a16ea2cfb3cde346a23673d1536dbfce491f8871346f441b9b4c2a6e","data/witness_first.py":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","data/witness_v2_algebraic.py":"ffbd48eb673a3f6f842327d00431285bbae8963a7e20f6956e37e544fa37165f","data/witness_v2_graph.py":"c81adcfefcf6ffc0e816f1b338af13323bc80e06293e8cac6b28a3b0ae2efc8a"},"generator_schema_version":"witness-first-v2","generator_source_sha256":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","id":"e2n-witness-first-v2-n19-weighted_modular_v2-518fc3689321d4c1","initial_target":{"coordinate_support":102,"effective_n":19,"expression":[[1,-9],[4,-10],[8,-9],[9,9],[10,8],[16,20],[32,27],[64,20],[68,-20],[128,20],[144,-20],[256,38],[1024,59],[1028,4],[1056,-27],[1226,5],[1280,-20],[2072,-21],[2073,21],[2136,21],[2137,-21],[4096,60],[4352,-18],[4416,18],[5120,-9],[8192,-4],[8224,4],[16388,-30],[16389,30],[16392,10],[16396,-10],[20488,-10],[20492,10],[32768,3],[32778,-8],[32832,4],[33792,-5],[33860,-4],[33994,-5],[34562,5],[37184,-18],[65536,-10],[65600,-20],[65604,20],[84647,10],[98304,10],[102408,-2],[103512,-1],[111704,1],[131074,18],[131076,10],[131138,-18],[131330,-18],[131394,18],[131552,3],[132576,-3],[135488,-18],[159502,4],[159534,-4],[168256,18],[176148,-2],[176156,2],[177172,2],[177180,-2],[177381,-2],[179429,2],[215719,-10],[219716,5],[234776,2],[262144,28],[262176,-9],[262304,9],[262432,9],[262552,4],[262560,-9],[263168,5],[264200,-2],[264204,2],[266240,-20],[266648,-4],[274788,-2],[274796,2],[274804,2],[274812,-2],[278532,30],[278533,-30],[365948,1],[374140,-1],[381191,8],[386185,5],[386955,-5],[397233,8],[401329,-8],[468649,-1],[468907,1],[482884,-5],[485049,1],[485307,-1],[510504,2],[510508,-2],[520191,1],[524287,-1]],"text":"-9H(X_{1}) - 10H(X_{3}) - 9H(X_{4}) + 9H(X_{1,4}) + 8H(X_{2,4}) + 20H(X_{5}) + 27H(X_{6}) + 20H(X_{7}) - 20H(X_{3,7}) + 20H(X_{8}) - 20H(X_{5,8}) + 38H(X_{9}) + 59H(X_{11}) + 4H(X_{3,11}) - 27H(X_{6,11}) + 5H(X_{2,4,7,8,11}) - 20H(X_{9,11}) - 21H(X_{4,5,12}) + 21H(X_{1,4,5,12}) + 21H(X_{4,5,7,12}) - 21H(X_{1,4,5,7,12}) + 60H(X_{13}) - 18H(X_{9,13}) + 18H(X_{7,9,13}) - 9H(X_{11,13}) - 4H(X_{14}) + 4H(X_{6,14}) - 30H(X_{3,15}) + 30H(X_{1,3,15}) + 10H(X_{4,15}) - 10H(X_{3,4,15}) - 10H(X_{4,13,15}) + 10H(X_{3,4,13,15}) + 3H(X_{16}) - 8H(X_{2,4,16}) + 4H(X_{7,16}) - 5H(X_{11,16}) - 4H(X_{3,7,11,16}) - 5H(X_{2,4,7,8,11,16}) + 5H(X_{2,9,10,11,16}) - 18H(X_{7,9,13,16}) - 10H(X_{17}) - 20H(X_{7,17}) + 20H(X_{3,7,17}) + 10H(X_{1,2,3,6,8,10,12,15,17}) + 10H(X_{16,17}) - 2H(X_{4,13,16,17}) - H(X_{4,5,7,11,13,16,17}) + H(X_{4,5,7,11,13,14,16,17}) + 18H(X_{2,18}) + 10H(X_{3,18}) - 18H(X_{2,7,18}) - 18H(X_{2,9,18}) + 18H(X_{2,7,9,18}) + 3H(X_{6,7,8,9,18}) - 3H(X_{6,7,8,9,11,18}) - 18H(X_{7,9,13,18}) + 4H(X_{2,3,4,9,10,11,12,14,15,18}) - 4H(X_{2,3,4,6,9,10,11,12,14,15,18}) + 18H(X_{7,9,13,16,18}) - 2H(X_{3,5,13,14,16,18}) + 2H(X_{3,4,5,13,14,16,18}) + 2H(X_{3,5,11,13,14,16,18}) - 2H(X_{3,4,5,11,13,14,16,18}) - 2H(X_{1,3,6,7,8,11,13,14,16,18}) + 2H(X_{1,3,6,7,8,11,12,13,14,16,18}) - 10H(X_{1,2,3,6,8,10,12,15,17,18}) + 5H(X_{3,7,10,12,13,15,17,18}) + 2H(X_{4,5,9,11,13,16,17,18}) + 28H(X_{19}) - 9H(X_{6,19}) + 9H(X_{6,8,19}) + 9H(X_{6,9,19}) + 4H(X_{4,5,8,9,19}) - 9H(X_{6,8,9,19}) + 5H(X_{11,19}) - 2H(X_{4,12,19}) + 2H(X_{3,4,12,19}) - 20H(X_{13,19}) - 4H(X_{4,5,8,9,13,19}) - 2H(X_{3,6,7,9,13,14,19}) + 2H(X_{3,4,6,7,9,13,14,19}) + 2H(X_{3,5,6,7,9,13,14,19}) - 2H(X_{3,4,5,6,7,9,13,14,19}) + 30H(X_{3,15,19}) - 30H(X_{1,3,15,19}) + H(X_{3,4,5,6,7,9,11,13,16,17,19}) - H(X_{3,4,5,6,7,9,11,13,14,16,17,19}) + 8H(X_{1,2,3,9,13,15,16,17,19}) + 5H(X_{1,4,8,11,14,15,16,17,19}) - 5H(X_{1,2,4,8,9,10,11,14,15,16,17,19}) + 8H(X_{1,5,6,8,9,10,11,12,18,19}) - 8H(X_{1,5,6,8,9,10,11,12,13,18,19}) - H(X_{1,4,6,8,10,11,14,17,18,19}) + H(X_{1,2,4,6,8,9,10,11,14,17,18,19}) - 5H(X_{3,7,10,11,12,13,15,17,18,19}) + H(X_{1,4,5,6,8,10,11,14,15,17,18,19}) - H(X_{1,2,4,5,6,8,9,10,11,14,15,17,18,19}) + 2H(X_{4,6,10,12,15,16,17,18,19}) - 2H(X_{3,4,6,10,12,15,16,17,18,19}) + H(X_{1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,17,18,19}) - H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19})","witness_value":1230},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{3};X_{13}|X_{4,15})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{1};X_{7}|X_{4,5,12})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{17}|X_{7})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{9};X_{11})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{1};X_{19}|X_{3,15})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{16};X_{17})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{13};X_{19})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{8})"}],"expression":[[16,-2],[64,-2],[68,2],[128,-2],[144,2],[256,-2],[1024,-2],[1280,2],[2072,3],[2073,-3],[2136,-3],[2137,3],[4096,-2],[16388,3],[16389,-3],[16392,-1],[16396,1],[20488,1],[20492,-1],[32768,1],[65536,1],[65600,2],[65604,-2],[98304,-1],[262144,-2],[266240,2],[278532,-3],[278533,3]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["-2H(X_{5}) - 2H(X_{7}) + 2H(X_{3,7}) - 2H(X_{8}) + 2H(X_{5,8}) - 2H(X_{9}) - 2H(X_{11}) + 2H(X_{9,11}) + 3H(X_{4,5,12}) - 3H(X_{1,4,5,12}) - 3H(X_{4,5,7,12}) + 3H(X_{1,4,5,7,12}) - 2H(X_{13}) + 3H(X_{3,15}) - 3H(X_{1,3,15}) - H(X_{4,15}) + H(X_{3,4,15}) + H(X_{4,13,15}) - H(X_{3,4,13,15}) + H(X_{16}) + H(X_{17}) + 2H(X_{7,17}) - 2H(X_{3,7,17}) - H(X_{16,17}) - 2H(X_{19}) + 2H(X_{13,19}) - 3H(X_{3,15,19}) + 3H(X_{1,3,15,19})>=0","2H(X_{5}) + 2H(X_{7}) - 2H(X_{3,7}) + 2H(X_{8}) - 2H(X_{5,8}) + 2H(X_{9}) + 2H(X_{11}) - 2H(X_{9,11}) - 3H(X_{4,5,12}) + 3H(X_{1,4,5,12}) + 3H(X_{4,5,7,12}) - 3H(X_{1,4,5,7,12}) + 2H(X_{13}) - 3H(X_{3,15}) + 3H(X_{1,3,15}) + H(X_{4,15}) - H(X_{3,4,15}) - H(X_{4,13,15}) + H(X_{3,4,13,15}) - H(X_{16}) - H(X_{17}) - 2H(X_{7,17}) + 2H(X_{3,7,17}) + H(X_{16,17}) + 2H(X_{19}) - 2H(X_{13,19}) + 3H(X_{3,15,19}) - 3H(X_{1,3,15,19})>=0"],"text":"-2H(X_{5}) - 2H(X_{7}) + 2H(X_{3,7}) - 2H(X_{8}) + 2H(X_{5,8}) - 2H(X_{9}) - 2H(X_{11}) + 2H(X_{9,11}) + 3H(X_{4,5,12}) - 3H(X_{1,4,5,12}) - 3H(X_{4,5,7,12}) + 3H(X_{1,4,5,7,12}) - 2H(X_{13}) + 3H(X_{3,15}) - 3H(X_{1,3,15}) - H(X_{4,15}) + H(X_{3,4,15}) + H(X_{4,13,15}) - H(X_{3,4,13,15}) + H(X_{16}) + H(X_{17}) + 2H(X_{7,17}) - 2H(X_{3,7,17}) - H(X_{16,17}) - 2H(X_{19}) + 2H(X_{13,19}) - 3H(X_{3,15,19}) + 3H(X_{1,3,15,19})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{8};X_{9}|X_{6,19})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{11};X_{13})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{16};X_{18}|X_{7,9,13})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{7};X_{9}|X_{2,18})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{7}|X_{4,5,12})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{9};X_{13})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{4})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{6};X_{11})"}],"expression":[[1,1],[8,1],[9,-1],[32,-3],[256,-2],[1024,-4],[1056,3],[2072,-1],[2073,1],[2136,1],[2137,-1],[4096,-3],[4352,2],[4416,-2],[5120,1],[37184,2],[131074,-2],[131138,2],[131330,2],[131394,-2],[135488,2],[168256,-2],[262176,1],[262304,-1],[262432,-1],[262560,1]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["H(X_{1}) + H(X_{4}) - H(X_{1,4}) - 3H(X_{6}) - 2H(X_{9}) - 4H(X_{11}) + 3H(X_{6,11}) - H(X_{4,5,12}) + H(X_{1,4,5,12}) + H(X_{4,5,7,12}) - H(X_{1,4,5,7,12}) - 3H(X_{13}) + 2H(X_{9,13}) - 2H(X_{7,9,13}) + H(X_{11,13}) + 2H(X_{7,9,13,16}) - 2H(X_{2,18}) + 2H(X_{2,7,18}) + 2H(X_{2,9,18}) - 2H(X_{2,7,9,18}) + 2H(X_{7,9,13,18}) - 2H(X_{7,9,13,16,18}) + H(X_{6,19}) - H(X_{6,8,19}) - H(X_{6,9,19}) + H(X_{6,8,9,19})>=0","-H(X_{1}) - H(X_{4}) + H(X_{1,4}) + 3H(X_{6}) + 2H(X_{9}) + 4H(X_{11}) - 3H(X_{6,11}) + H(X_{4,5,12}) - H(X_{1,4,5,12}) - H(X_{4,5,7,12}) + H(X_{1,4,5,7,12}) + 3H(X_{13}) - 2H(X_{9,13}) + 2H(X_{7,9,13}) - H(X_{11,13}) - 2H(X_{7,9,13,16}) + 2H(X_{2,18}) - 2H(X_{2,7,18}) - 2H(X_{2,9,18}) + 2H(X_{2,7,9,18}) - 2H(X_{7,9,13,18}) + 2H(X_{7,9,13,16,18}) - H(X_{6,19}) + H(X_{6,8,19}) + H(X_{6,9,19}) - H(X_{6,8,9,19})>=0"],"text":"H(X_{1}) + H(X_{4}) - H(X_{1,4}) - 3H(X_{6}) - 2H(X_{9}) - 4H(X_{11}) + 3H(X_{6,11}) - H(X_{4,5,12}) + H(X_{1,4,5,12}) + H(X_{4,5,7,12}) - H(X_{1,4,5,7,12}) - 3H(X_{13}) + 2H(X_{9,13}) - 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- 6H(X_{1,2}) - 7H(X_{4}) - 6H(X_{3,4}) + 7H(X_{5}) - 7H(X_{1,5}) - 14H(X_{4,6,9}) + 14H(X_{4,5,6,9}) + 14H(X_{4,6,8,9}) - 14H(X_{4,5,6,8,9}) + 11H(X_{10}) - 21H(X_{1,3,10}) + 21H(X_{1,2,3,10}) + 4H(X_{8,10}) + 9H(X_{11}) - 8H(X_{1,6,11}) + 8H(X_{1,2,6,11}) + 8H(X_{1,3,6,11}) - 8H(X_{1,2,3,6,11}) - 8H(X_{10,11}) + 21H(X_{12}) - 21H(X_{4,12}) - 24H(X_{7,9,12}) - 21H(X_{10,12}) + 21H(X_{4,10,12}) + 24H(X_{7,9,10,12}) + 14H(X_{13}) - 24H(X_{1,13}) + 6H(X_{3,4,13}) + 7H(X_{6,13}) - 10H(X_{2,5,6,13}) + 10H(X_{1,2,5,6,8,9,11,13}) - 17H(X_{12,13}) + 3H(X_{4,12,13}) + 17H(X_{9,12,13}) - 3H(X_{7,11,12,13}) + 3H(X_{6,7,11,12,13}) + 4H(X_{14}) - 5H(X_{15}) + 5H(X_{8,15}) + 7H(X_{13,15}) - 7H(X_{6,13,15}) + 3H(X_{1,7,10,11,12,13,14,15}) - 3H(X_{1,6,7,10,11,12,13,14,15}) - 24H(X_{13,16}) + 24H(X_{1,13,16}) + 3H(X_{5,7,8,9,10,11,13,16}) - 3H(X_{4,5,7,8,9,10,11,12,13,16}) + H(X_{1,6,8,12,14,16}) - H(X_{1,6,8,11,12,14,16}) + 21H(X_{6,17}) - 21H(X_{2,6,17}) + 6H(X_{1,2,3,4,6,17}) + 7H(X_{2,3,4,7,8,17}) + 21H(X_{1,3,10,17}) - 21H(X_{1,2,3,10,17}) - 21H(X_{6,10,17}) + 21H(X_{2,6,10,17}) - 6H(X_{1,2,3,4,6,13,17}) + 5H(X_{9,10,11,12,15,17}) - 5H(X_{8,9,10,11,12,15,17}) - 16H(X_{18}) - 6H(X_{1,18}) + 6H(X_{1,2,18}) + 24H(X_{7,9,12,18}) - 24H(X_{7,9,10,12,18}) - H(X_{6,10,11,12,18}) + 17H(X_{12,13,18}) - 17H(X_{9,12,13,18}) - 8H(X_{10,15,18}) + 8H(X_{4,10,15,18}) + 8H(X_{7,10,15,18}) - 8H(X_{4,7,10,15,18}) + H(X_{6,10,11,12,15,18}) - 7H(X_{8,16,18}) + 4H(X_{4,6,7,11,12,13,15,16,18}) + H(X_{3,5,6,10,11,12,17,18}) - H(X_{3,5,6,10,11,12,15,17,18}) - 8H(X_{19}) + 7H(X_{4,19}) + 10H(X_{2,3,5,6,7,13,14,19}) - 10H(X_{1,2,3,5,6,7,8,9,11,13,14,19}) - 7H(X_{2,3,4,7,8,17,19}) + 3H(X_{1,2,3,6,7,11,13,14,16,17,19}) - 3H(X_{1,2,3,6,7,10,11,13,14,16,17,19}) + 16H(X_{18,19}) - 10H(X_{2,3,18,19}) + 10H(X_{2,3,6,12,14,16,18,19}) + 7H(X_{1,3,4,8,9,13,15,16,18,19}) - 6H(X_{2,5,8,9,12,14,17,18,19}) + 6H(X_{2,4,5,8,9,10,12,14,17,18,19}) + 4H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,15,16,17,18,19}) - 4H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19})","witness_value":16703},"k":16,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{9};X_{18}|X_{12,13})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{17}|X_{1,3,10})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{4};X_{10}|X_{12})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{5})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{10}|X_{6,17})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{2};X_{18}|X_{1})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{8}|X_{4,6,9})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{6};X_{15}|X_{13})"}],"expression":[[1,-3],[3,2],[16,-1],[17,1],[296,2],[312,-2],[424,-2],[440,2],[517,3],[519,-3],[2048,-3],[2056,3],[2560,3],[2568,-3],[4096,1],[4128,-1],[6144,-1],[6400,1],[20480,-1],[20512,1],[65568,-3],[65570,3],[66053,-3],[66055,3],[66080,3],[66082,-3],[131073,2],[131075,-2],[137216,1],[137472,-1]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["-3H(X_{1}) + 2H(X_{1,2}) - H(X_{5}) + H(X_{1,5}) + 2H(X_{4,6,9}) - 2H(X_{4,5,6,9}) - 2H(X_{4,6,8,9}) + 2H(X_{4,5,6,8,9}) + 3H(X_{1,3,10}) - 3H(X_{1,2,3,10}) - 3H(X_{12}) + 3H(X_{4,12}) + 3H(X_{10,12}) - 3H(X_{4,10,12}) + H(X_{13}) - H(X_{6,13}) - H(X_{12,13}) + H(X_{9,12,13}) - H(X_{13,15}) + H(X_{6,13,15}) - 3H(X_{6,17}) + 3H(X_{2,6,17}) - 3H(X_{1,3,10,17}) + 3H(X_{1,2,3,10,17}) + 3H(X_{6,10,17}) - 3H(X_{2,6,10,17}) + 2H(X_{1,18}) - 2H(X_{1,2,18}) + H(X_{12,13,18}) - H(X_{9,12,13,18})>=0","3H(X_{1}) - 2H(X_{1,2}) + H(X_{5}) - H(X_{1,5}) - 2H(X_{4,6,9}) + 2H(X_{4,5,6,9}) + 2H(X_{4,6,8,9}) - 2H(X_{4,5,6,8,9}) - 3H(X_{1,3,10}) + 3H(X_{1,2,3,10}) + 3H(X_{12}) - 3H(X_{4,12}) - 3H(X_{10,12}) + 3H(X_{4,10,12}) - H(X_{13}) + H(X_{6,13}) + H(X_{12,13}) - H(X_{9,12,13}) + H(X_{13,15}) - H(X_{6,13,15}) + 3H(X_{6,17}) - 3H(X_{2,6,17}) + 3H(X_{1,3,10,17}) - 3H(X_{1,2,3,10,17}) - 3H(X_{6,10,17}) + 3H(X_{2,6,10,17}) - 2H(X_{1,18}) + 2H(X_{1,2,18}) - H(X_{12,13,18}) + H(X_{9,12,13,18})>=0"],"text":"-3H(X_{1}) + 2H(X_{1,2}) - H(X_{5}) + H(X_{1,5}) + 2H(X_{4,6,9}) - 2H(X_{4,5,6,9}) - 2H(X_{4,6,8,9}) + 2H(X_{4,5,6,8,9}) + 3H(X_{1,3,10}) - 3H(X_{1,2,3,10}) - 3H(X_{12}) + 3H(X_{4,12}) + 3H(X_{10,12}) - 3H(X_{4,10,12}) + H(X_{13}) - H(X_{6,13}) - H(X_{12,13}) + H(X_{9,12,13}) - H(X_{13,15}) + H(X_{6,13,15}) - 3H(X_{6,17}) + 3H(X_{2,6,17}) - 3H(X_{1,3,10,17}) + 3H(X_{1,2,3,10,17}) + 3H(X_{6,10,17}) - 3H(X_{2,6,10,17}) + 2H(X_{1,18}) - 2H(X_{1,2,18}) + H(X_{12,13,18}) - H(X_{9,12,13,18})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{10};X_{11})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{9};X_{18}|X_{12,13})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{10};X_{18}|X_{7,9,12})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{18};X_{19})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{18}|X_{1})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{3}|X_{1,6,11})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{7}|X_{10,15,18})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{1};X_{16}|X_{13})"}],"expression":[[1,1],[3,-1],[512,-1],[1024,-1],[1057,1],[1059,-1],[1061,-1],[1063,1],[1536,1],[2368,3],[2880,-3],[4096,-3],[4097,3],[6144,3],[6400,-3],[36864,3],[36865,-3],[131072,2],[131073,-1],[131075,1],[133440,-3],[133952,3],[137216,-3],[137472,3],[147968,1],[147976,-1],[148032,-1],[148040,1],[262144,2],[393216,-2]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["H(X_{1}) - H(X_{1,2}) - H(X_{10}) - H(X_{11}) + H(X_{1,6,11}) - H(X_{1,2,6,11}) - H(X_{1,3,6,11}) + H(X_{1,2,3,6,11}) + H(X_{10,11}) + 3H(X_{7,9,12}) - 3H(X_{7,9,10,12}) - 3H(X_{13}) + 3H(X_{1,13}) + 3H(X_{12,13}) - 3H(X_{9,12,13}) + 3H(X_{13,16}) - 3H(X_{1,13,16}) + 2H(X_{18}) - H(X_{1,18}) + H(X_{1,2,18}) - 3H(X_{7,9,12,18}) + 3H(X_{7,9,10,12,18}) - 3H(X_{12,13,18}) + 3H(X_{9,12,13,18}) + H(X_{10,15,18}) - H(X_{4,10,15,18}) - H(X_{7,10,15,18}) + H(X_{4,7,10,15,18}) + 2H(X_{19}) - 2H(X_{18,19})>=0","-H(X_{1}) + H(X_{1,2}) + H(X_{10}) + H(X_{11}) - H(X_{1,6,11}) + H(X_{1,2,6,11}) + H(X_{1,3,6,11}) - H(X_{1,2,3,6,11}) - H(X_{10,11}) - 3H(X_{7,9,12}) + 3H(X_{7,9,10,12}) + 3H(X_{13}) - 3H(X_{1,13}) - 3H(X_{12,13}) + 3H(X_{9,12,13}) - 3H(X_{13,16}) + 3H(X_{1,13,16}) - 2H(X_{18}) + H(X_{1,18}) - H(X_{1,2,18}) + 3H(X_{7,9,12,18}) - 3H(X_{7,9,10,12,18}) + 3H(X_{12,13,18}) - 3H(X_{9,12,13,18}) - H(X_{10,15,18}) + H(X_{4,10,15,18}) + H(X_{7,10,15,18}) - H(X_{4,7,10,15,18}) - 2H(X_{19}) + 2H(X_{18,19})>=0"],"text":"H(X_{1}) - H(X_{1,2}) - H(X_{10}) - H(X_{11}) + H(X_{1,6,11}) - H(X_{1,2,6,11}) - H(X_{1,3,6,11}) + H(X_{1,2,3,6,11}) + H(X_{10,11}) + 3H(X_{7,9,12}) - 3H(X_{7,9,10,12}) - 3H(X_{13}) + 3H(X_{1,13}) + 3H(X_{12,13}) - 3H(X_{9,12,13}) + 3H(X_{13,16}) - 3H(X_{1,13,16}) + 2H(X_{18}) - H(X_{1,18}) + H(X_{1,2,18}) - 3H(X_{7,9,12,18}) + 3H(X_{7,9,10,12,18}) - 3H(X_{12,13,18}) + 3H(X_{9,12,13,18}) + H(X_{10,15,18}) - H(X_{4,10,15,18}) - H(X_{7,10,15,18}) + H(X_{4,7,10,15,18}) + 2H(X_{19}) - 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8H(X_{1,2,3,5,6,7,8,11,12,14,17,18}) + 10H(X_{2,3,6,9,10,16,17,18}) - 10H(X_{2,3,6,8,9,10,16,17,18}) + 8H(X_{1,2,3,5,7,8,11,12,14,16,17,18}) - 8H(X_{1,2,3,5,6,7,8,11,12,14,16,17,18}) + 13H(X_{19}) + 2H(X_{3,4,6,19}) - 4H(X_{9,19}) - 4H(X_{14,19}) + 4H(X_{9,14,19}) + 3H(X_{1,2,3,4,5,11,12,14,19}) - 3H(X_{1,2,3,4,5,8,11,12,14,17,19}) + 5H(X_{1,7,8,9,10,12,13,15,16,18,19}) + 8H(X_{1,2,3,5,6,7,8,11,12,13,14,15,16,17,18,19}) + 2H(X_{20}) - 4H(X_{15,20}) + 4H(X_{14,15,20}) - 4H(X_{1,8,14,15,20}) + 4H(X_{1,7,8,14,15,20}) + 4H(X_{1,8,14,15,16,20}) - 4H(X_{1,7,8,14,15,16,20}) + 5H(X_{2,3,4,5,6,11,14,17,20}) + 9H(X_{1,4,12,13,15,16,17,20}) + 2H(X_{2,8,12,18,20}) + 9H(X_{1,2,5,7,8,10,11,15,18,20}) - 9H(X_{1,2,3,5,7,8,10,11,15,18,20}) + 5H(X_{2,4,8,9,10,11,12,14,15,18,20}) - 5H(X_{2,4,6,8,9,10,11,12,14,15,18,20}) + H(X_{4,6,7,13,14,19,20}) - H(X_{4,6,7,9,13,14,17,19,20}) - 9H(X_{1,4,12,13,15,16,17,19,20}) - 2H(X_{2,3,4,6,8,12,18,19,20}) + 5H(X_{1,3,7,8,10,11,12,13,14,16,17,18,19,20}) - 5H(X_{1,3,7,8,10,11,12,13,14,15,16,17,18,19,20}) - 5H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20})","witness_value":18589},"k":16,"logical_conditions":[{"component_width":12,"component_witness_values":[0,0,0,0,0,0,0,0,0,0,0,0],"components":[{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{17}|X_{7,9})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{18}|X_{7})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{10}|X_{6})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{14};X_{20}|X_{15})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{14}|X_{8,12})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{5};X_{9}|X_{10})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{5};X_{17})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{8})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{6};X_{16}|X_{4,17})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{7};X_{16}|X_{1,8,14,15,20})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{9};X_{14}|X_{19})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{4};X_{13}|X_{15})"}],"expression":[[1,1],[16,3],[32,2],[64,3],[66,-3],[128,1],[129,-1],[160,-2],[320,3],[322,-3],[512,-3],[528,3],[544,-2],[672,2],[768,3],[784,-3],[2176,-1],[2184,1],[10368,1],[10376,-1],[16392,-2],[20480,-2],[20488,2],[24576,2],[65536,3],[65544,-3],[65552,-3],[65576,3],[65856,-3],[65858,3],[98312,3],[98344,-3],[131136,-3],[131138,3],[262144,-2],[262400,2],[270336,2],[270592,-2],[540672,2],[548864,-2],[548993,2],[549057,-2],[581761,-2],[581825,2]],"id":0,"kind":"composite_equality","label":"witness 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H(X_{8,12,14}) + H(X_{4,8,12,14}) + 2H(X_{4,15}) + 2H(X_{13,15}) - 2H(X_{4,13,15}) - 2H(X_{14,15}) - 3H(X_{17}) + 3H(X_{4,17}) + 3H(X_{5,17}) - 3H(X_{4,6,17}) + 3H(X_{7,9,17}) - 3H(X_{2,7,9,17}) - 3H(X_{4,16,17}) + 3H(X_{4,6,16,17}) + 3H(X_{7,18}) - 3H(X_{2,7,18}) + 2H(X_{19}) - 2H(X_{9,19}) - 2H(X_{14,19}) + 2H(X_{9,14,19}) - 2H(X_{15,20}) + 2H(X_{14,15,20}) - 2H(X_{1,8,14,15,20}) + 2H(X_{1,7,8,14,15,20}) + 2H(X_{1,8,14,15,16,20}) - 2H(X_{1,7,8,14,15,16,20})>=0"],"text":"H(X_{1}) + 3H(X_{5}) + 2H(X_{6}) + 3H(X_{7}) - 3H(X_{2,7}) + H(X_{8}) - H(X_{1,8}) - 2H(X_{6,8}) + 3H(X_{7,9}) - 3H(X_{2,7,9}) - 3H(X_{10}) + 3H(X_{5,10}) - 2H(X_{6,10}) + 2H(X_{6,8,10}) + 3H(X_{9,10}) - 3H(X_{5,9,10}) - H(X_{8,12}) + H(X_{4,8,12}) + H(X_{8,12,14}) - H(X_{4,8,12,14}) - 2H(X_{4,15}) - 2H(X_{13,15}) + 2H(X_{4,13,15}) + 2H(X_{14,15}) + 3H(X_{17}) - 3H(X_{4,17}) - 3H(X_{5,17}) + 3H(X_{4,6,17}) - 3H(X_{7,9,17}) + 3H(X_{2,7,9,17}) + 3H(X_{4,16,17}) - 3H(X_{4,6,16,17}) - 3H(X_{7,18}) + 3H(X_{2,7,18}) - 2H(X_{19}) + 2H(X_{9,19}) + 2H(X_{14,19}) - 2H(X_{9,14,19}) + 2H(X_{15,20}) - 2H(X_{14,15,20}) + 2H(X_{1,8,14,15,20}) - 2H(X_{1,7,8,14,15,20}) - 2H(X_{1,8,14,15,16,20}) + 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H(X_{2,4,5,6,7,8,9,10,11,15,18,20})","witness_value":0},{"a_mask":791952,"b_mask":131080,"c_mask":0,"coefficient":1,"expression":[[131080,1],[791952,1],[923032,-1]],"kind":"conditional_mutual_information","label":"I(X_{5,8,9,11,13,19,20};X_{4,18})","role":"shannon","scaled_expression":[[131080,1],[791952,1],[923032,-1]],"scaled_witness_value":0,"source_index":18,"text":"H(X_{4,18}) + H(X_{5,8,9,11,13,19,20}) - H(X_{4,5,8,9,11,13,18,19,20})","witness_value":0},{"a_mask":850858,"b_mask":0,"c_mask":0,"coefficient":9,"expression":[[850858,1]],"kind":"conditional_entropy","label":"H(X_{2,4,6,8,9,10,12,13,14,15,16,19,20})","role":"shannon","scaled_expression":[[850858,9]],"scaled_witness_value":1791,"source_index":9,"text":"9H(X_{2,4,6,8,9,10,12,13,14,15,16,19,20})","witness_value":199},{"coefficient":10,"expression":[[64,3],[2048,6],[2112,-3],[16384,-4],[16388,2],[16392,2],[16904,-1],[17408,2],[17416,-2],[18432,2],[18436,-2],[18952,1],[25096,1],[27144,-1],[33032,3],[33544,-3],[49416,-3],[49928,3],[66568,3],[67080,-3],[131332,1],[131333,-1],[135428,-1],[135429,1],[197640,-3],[198152,3],[524288,3],[526336,-3]],"kind":"composite_equality","label":"witness equality 2=0","role":"condition","scaled_expression":[[64,30],[2048,60],[2112,-30],[16384,-40],[16388,20],[16392,20],[16904,-10],[17408,20],[17416,-20],[18432,20],[18436,-20],[18952,10],[25096,10],[27144,-10],[33032,30],[33544,-30],[49416,-30],[49928,30],[66568,30],[67080,-30],[131332,10],[131333,-10],[135428,-10],[135429,10],[197640,-30],[198152,30],[524288,30],[526336,-30]],"scaled_witness_value":0,"source_index":1,"text":"30H(X_{7}) + 60H(X_{12}) - 30H(X_{7,12}) - 40H(X_{15}) + 20H(X_{3,15}) + 20H(X_{4,15}) - 10H(X_{4,10,15}) + 20H(X_{11,15}) - 20H(X_{4,11,15}) + 20H(X_{12,15}) - 20H(X_{3,12,15}) + 10H(X_{4,10,12,15}) + 10H(X_{4,10,14,15}) - 10H(X_{4,10,12,14,15}) + 30H(X_{4,9,16}) - 30H(X_{4,9,10,16}) - 30H(X_{4,9,15,16}) + 30H(X_{4,9,10,15,16}) + 30H(X_{4,11,17}) - 30H(X_{4,10,11,17}) + 10H(X_{3,9,18}) - 10H(X_{1,3,9,18}) - 10H(X_{3,9,13,18}) + 10H(X_{1,3,9,13,18}) - 30H(X_{4,11,17,18}) + 30H(X_{4,10,11,17,18}) + 30H(X_{20}) - 30H(X_{12,20})","witness_value":0},{"coefficient":4,"expression":[[32,-1],[128,1],[144,-1],[1024,-6],[1056,3],[8704,2],[8960,-2],[73728,-3],[74240,3],[77824,3],[78336,-3],[131072,-2],[131584,-1],[132096,3],[163840,-1],[164352,1],[262144,1],[262160,1],[262176,-2],[270848,-2],[271104,2]],"kind":"composite_equality","label":"witness equality 1=0","role":"condition","scaled_expression":[[32,-4],[128,4],[144,-4],[1024,-24],[1056,12],[8704,8],[8960,-8],[73728,-12],[74240,12],[77824,12],[78336,-12],[131072,-8],[131584,-4],[132096,12],[163840,-4],[164352,4],[262144,4],[262160,4],[262176,-8],[270848,-8],[271104,8]],"scaled_witness_value":0,"source_index":0,"text":"-4H(X_{6}) + 4H(X_{8}) - 4H(X_{5,8}) - 24H(X_{11}) + 12H(X_{6,11}) + 8H(X_{10,14}) - 8H(X_{9,10,14}) - 12H(X_{14,17}) + 12H(X_{10,14,17}) + 12H(X_{13,14,17}) - 12H(X_{10,13,14,17}) - 8H(X_{18}) - 4H(X_{10,18}) + 12H(X_{11,18}) - 4H(X_{16,18}) + 4H(X_{10,16,18}) + 4H(X_{19}) + 4H(X_{5,19}) - 8H(X_{6,19}) - 8H(X_{10,14,19}) + 8H(X_{9,10,14,19})","witness_value":0}],"certificate_length":22,"certificate_order":"greedy_negative_mass_condition_late","condition_coordinate_supports":[21,28],"condition_rank":2,"condition_tokens":[240,240,378,379],"condition_width":8,"config_sha256":"1b1f298ed012071c85e9cfb286fe5dcf7c183383de92041754d8a7802f426421","equality_condition_count":2,"full_tokens":2018,"generator_dependency_sha256":{"data/finite_field.py":"c9738c10a16ea2cfb3cde346a23673d1536dbfce491f8871346f441b9b4c2a6e","data/witness_first.py":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","data/witness_v2_algebraic.py":"ffbd48eb673a3f6f842327d00431285bbae8963a7e20f6956e37e544fa37165f","data/witness_v2_graph.py":"c81adcfefcf6ffc0e816f1b338af13323bc80e06293e8cac6b28a3b0ae2efc8a"},"generator_schema_version":"witness-first-v2","generator_source_sha256":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","id":"e2n-witness-first-v2-n20-weighted_modular_v2-7283d8ee6c1e79bb","initial_target":{"coordinate_support":110,"effective_n":20,"expression":[[1,3],[4,-9],[32,-4],[57,8],[64,24],[128,4],[144,-4],[256,8],[264,-4],[313,-8],[1024,-24],[1056,12],[1330,6],[2048,60],[2112,-30],[4096,-9],[4368,8],[8200,5],[8704,8],[8960,-8],[8968,4],[16384,-40],[16388,20],[16392,20],[16904,-10],[17408,20],[17416,-20],[18432,20],[18436,-20],[18952,10],[25096,10],[27144,-10],[33032,30],[33544,-30],[36880,9],[49159,8],[49416,-30],[49928,30],[53527,-8],[66568,30],[67080,-30],[73728,-12],[74240,12],[77824,12],[78336,-12],[94465,9],[110419,6],[127249,-9],[131072,-8],[131080,1],[131332,10],[131333,-10],[131584,-4],[131972,9],[132096,12],[133891,-2],[135428,-10],[135429,10],[138003,2],[149178,1],[163840,-4],[164352,4],[197640,-30],[198152,30],[200802,8],[231428,-7],[240395,2],[244507,-2],[262144,6],[262160,4],[262176,-8],[262707,10],[264586,4],[270848,-8],[271104,8],[273290,-4],[344272,-9],[348924,9],[426496,6],[427826,-6],[475344,9],[479996,-9],[524288,30],[524384,6],[524608,1],[524631,5],[526336,-30],[532831,-5],[557056,10],[557568,-3],[557600,3],[561316,2],[593745,-9],[634739,-6],[641036,9],[673786,-1],[691147,3],[691179,-3],[773004,-9],[776566,7],[787604,8],[791952,1],[819763,-10],[823460,-2],[850858,9],[909141,9],[923032,-1],[988406,-8],[1048574,3],[1048575,-3]],"text":"3H(X_{1}) - 9H(X_{3}) - 4H(X_{6}) + 8H(X_{1,4,5,6}) + 24H(X_{7}) + 4H(X_{8}) - 4H(X_{5,8}) + 8H(X_{9}) - 4H(X_{4,9}) - 8H(X_{1,4,5,6,9}) - 24H(X_{11}) + 12H(X_{6,11}) + 6H(X_{2,5,6,9,11}) + 60H(X_{12}) - 30H(X_{7,12}) - 9H(X_{13}) + 8H(X_{5,9,13}) + 5H(X_{4,14}) + 8H(X_{10,14}) - 8H(X_{9,10,14}) + 4H(X_{4,9,10,14}) - 40H(X_{15}) + 20H(X_{3,15}) + 20H(X_{4,15}) - 10H(X_{4,10,15}) + 20H(X_{11,15}) - 20H(X_{4,11,15}) + 20H(X_{12,15}) - 20H(X_{3,12,15}) + 10H(X_{4,10,12,15}) + 10H(X_{4,10,14,15}) - 10H(X_{4,10,12,14,15}) + 30H(X_{4,9,16}) - 30H(X_{4,9,10,16}) + 9H(X_{5,13,16}) + 8H(X_{1,2,3,15,16}) - 30H(X_{4,9,15,16}) + 30H(X_{4,9,10,15,16}) - 8H(X_{1,2,3,5,9,13,15,16}) + 30H(X_{4,11,17}) - 30H(X_{4,10,11,17}) - 12H(X_{14,17}) + 12H(X_{10,14,17}) + 12H(X_{13,14,17}) - 12H(X_{10,13,14,17}) + 9H(X_{1,9,13,14,15,17}) + 6H(X_{1,2,5,7,9,10,11,12,14,16,17}) - 9H(X_{1,5,9,13,14,15,16,17}) - 8H(X_{18}) + H(X_{4,18}) + 10H(X_{3,9,18}) - 10H(X_{1,3,9,18}) - 4H(X_{10,18}) + 9H(X_{3,8,9,10,18}) + 12H(X_{11,18}) - 2H(X_{1,2,9,10,12,18}) - 10H(X_{3,9,13,18}) + 10H(X_{1,3,9,13,18}) + 2H(X_{1,2,5,9,10,12,13,18}) + H(X_{2,4,5,6,8,10,11,15,18}) - 4H(X_{16,18}) + 4H(X_{10,16,18}) - 30H(X_{4,11,17,18}) + 30H(X_{4,10,11,17,18}) + 8H(X_{2,6,7,13,17,18}) - 7H(X_{3,12,16,17,18}) + 2H(X_{1,2,4,9,10,12,14,16,17,18}) - 2H(X_{1,2,4,5,9,10,12,13,14,16,17,18}) + 6H(X_{19}) + 4H(X_{5,19}) - 8H(X_{6,19}) + 10H(X_{1,2,5,6,10,19}) + 4H(X_{2,4,8,9,12,19}) - 8H(X_{10,14,19}) + 8H(X_{9,10,14,19}) - 4H(X_{2,4,8,9,10,12,14,19}) - 9H(X_{5,7,8,15,17,19}) + 9H(X_{3,4,5,6,7,8,10,13,15,17,19}) + 6H(X_{10,16,18,19}) - 6H(X_{2,5,6,9,10,11,16,18,19}) + 9H(X_{5,7,8,15,17,18,19}) - 9H(X_{3,4,5,6,7,8,10,13,15,17,18,19}) + 30H(X_{20}) + 6H(X_{6,7,20}) + H(X_{7,9,20}) + 5H(X_{1,2,3,5,7,9,20}) - 30H(X_{12,20}) - 5H(X_{1,2,3,4,5,7,9,14,20}) + 10H(X_{16,20}) - 3H(X_{10,16,20}) + 3H(X_{6,10,16,20}) + 2H(X_{3,6,8,13,16,20}) - 9H(X_{1,5,7,9,10,11,12,17,20}) - 6H(X_{1,2,5,6,7,9,10,11,12,14,16,17,20}) + 9H(X_{3,4,12,15,16,17,20}) - H(X_{2,4,5,6,7,8,9,10,11,15,18,20}) + 3H(X_{1,2,4,7,8,9,10,12,16,18,20}) - 3H(X_{1,2,4,6,7,8,9,10,12,16,18,20}) - 9H(X_{3,4,8,9,10,12,15,16,17,18,20}) + 7H(X_{2,3,5,6,7,9,12,13,15,16,17,18,20}) + 8H(X_{3,5,8,11,19,20}) + H(X_{5,8,9,11,13,19,20}) - 10H(X_{1,2,5,6,10,16,19,20}) - 2H(X_{3,6,8,13,16,19,20}) + 9H(X_{2,4,6,8,9,10,12,13,14,15,16,19,20}) + 9H(X_{1,3,5,7,9,10,11,12,13,15,16,17,19,20}) - H(X_{4,5,8,9,11,13,18,19,20}) - 8H(X_{2,3,5,6,7,8,11,13,17,18,19,20}) + 3H(X_{2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}) - 3H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20})","witness_value":3404},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{11};X_{18})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{5};X_{19})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{5};X_{8})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{6};X_{11})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{19})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{9};X_{19}|X_{10,14})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{10};X_{16}|X_{18})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{10};X_{13}|X_{14,17})"}],"expression":[[32,-1],[128,1],[144,-1],[1024,-6],[1056,3],[8704,2],[8960,-2],[73728,-3],[74240,3],[77824,3],[78336,-3],[131072,-2],[131584,-1],[132096,3],[163840,-1],[164352,1],[262144,1],[262160,1],[262176,-2],[270848,-2],[271104,2]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["-H(X_{6}) + H(X_{8}) - H(X_{5,8}) - 6H(X_{11}) + 3H(X_{6,11}) + 2H(X_{10,14}) - 2H(X_{9,10,14}) - 3H(X_{14,17}) + 3H(X_{10,14,17}) + 3H(X_{13,14,17}) - 3H(X_{10,13,14,17}) - 2H(X_{18}) - H(X_{10,18}) + 3H(X_{11,18}) - H(X_{16,18}) + H(X_{10,16,18}) + H(X_{19}) + H(X_{5,19}) - 2H(X_{6,19}) - 2H(X_{10,14,19}) + 2H(X_{9,10,14,19})>=0","H(X_{6}) - H(X_{8}) + H(X_{5,8}) + 6H(X_{11}) - 3H(X_{6,11}) - 2H(X_{10,14}) + 2H(X_{9,10,14}) + 3H(X_{14,17}) - 3H(X_{10,14,17}) - 3H(X_{13,14,17}) + 3H(X_{10,13,14,17}) + 2H(X_{18}) + H(X_{10,18}) - 3H(X_{11,18}) + H(X_{16,18}) - H(X_{10,16,18}) - H(X_{19}) - H(X_{5,19}) + 2H(X_{6,19}) + 2H(X_{10,14,19}) - 2H(X_{9,10,14,19})>=0"],"text":"-H(X_{6}) + H(X_{8}) - H(X_{5,8}) - 6H(X_{11}) + 3H(X_{6,11}) + 2H(X_{10,14}) - 2H(X_{9,10,14}) - 3H(X_{14,17}) + 3H(X_{10,14,17}) + 3H(X_{13,14,17}) - 3H(X_{10,13,14,17}) - 2H(X_{18}) - H(X_{10,18}) + 3H(X_{11,18}) - H(X_{16,18}) + H(X_{10,16,18}) + H(X_{19}) + H(X_{5,19}) - 2H(X_{6,19}) - 2H(X_{10,14,19}) + 2H(X_{9,10,14,19})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{13}|X_{3,9,18})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{12};X_{20})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{4};X_{11}|X_{15})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{12}|X_{15})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{12};X_{14}|X_{4,10,15})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{10};X_{15}|X_{4,9,16})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{10};X_{18}|X_{4,11,17})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{7};X_{12})"}],"expression":[[64,3],[2048,6],[2112,-3],[16384,-4],[16388,2],[16392,2],[16904,-1],[17408,2],[17416,-2],[18432,2],[18436,-2],[18952,1],[25096,1],[27144,-1],[33032,3],[33544,-3],[49416,-3],[49928,3],[66568,3],[67080,-3],[131332,1],[131333,-1],[135428,-1],[135429,1],[197640,-3],[198152,3],[524288,3],[526336,-3]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["3H(X_{7}) + 6H(X_{12}) - 3H(X_{7,12}) - 4H(X_{15}) + 2H(X_{3,15}) + 2H(X_{4,15}) - H(X_{4,10,15}) + 2H(X_{11,15}) - 2H(X_{4,11,15}) + 2H(X_{12,15}) - 2H(X_{3,12,15}) + H(X_{4,10,12,15}) + H(X_{4,10,14,15}) - H(X_{4,10,12,14,15}) + 3H(X_{4,9,16}) - 3H(X_{4,9,10,16}) - 3H(X_{4,9,15,16}) + 3H(X_{4,9,10,15,16}) + 3H(X_{4,11,17}) - 3H(X_{4,10,11,17}) + H(X_{3,9,18}) - H(X_{1,3,9,18}) - H(X_{3,9,13,18}) + H(X_{1,3,9,13,18}) - 3H(X_{4,11,17,18}) + 3H(X_{4,10,11,17,18}) + 3H(X_{20}) - 3H(X_{12,20})>=0","-3H(X_{7}) - 6H(X_{12}) + 3H(X_{7,12}) + 4H(X_{15}) - 2H(X_{3,15}) - 2H(X_{4,15}) + H(X_{4,10,15}) - 2H(X_{11,15}) + 2H(X_{4,11,15}) - 2H(X_{12,15}) + 2H(X_{3,12,15}) - H(X_{4,10,12,15}) - H(X_{4,10,14,15}) + H(X_{4,10,12,14,15}) - 3H(X_{4,9,16}) + 3H(X_{4,9,10,16}) + 3H(X_{4,9,15,16}) - 3H(X_{4,9,10,15,16}) - 3H(X_{4,11,17}) + 3H(X_{4,10,11,17}) - H(X_{3,9,18}) + H(X_{1,3,9,18}) + H(X_{3,9,13,18}) - H(X_{1,3,9,13,18}) + 3H(X_{4,11,17,18}) - 3H(X_{4,10,11,17,18}) - 3H(X_{20}) + 3H(X_{12,20})>=0"],"text":"3H(X_{7}) + 6H(X_{12}) - 3H(X_{7,12}) - 4H(X_{15}) + 2H(X_{3,15}) + 2H(X_{4,15}) - H(X_{4,10,15}) + 2H(X_{11,15}) - 2H(X_{4,11,15}) + 2H(X_{12,15}) - 2H(X_{3,12,15}) + H(X_{4,10,12,15}) + H(X_{4,10,14,15}) - H(X_{4,10,12,14,15}) + 3H(X_{4,9,16}) - 3H(X_{4,9,10,16}) - 3H(X_{4,9,15,16}) + 3H(X_{4,9,10,15,16}) + 3H(X_{4,11,17}) - 3H(X_{4,10,11,17}) + H(X_{3,9,18}) - H(X_{1,3,9,18}) - H(X_{3,9,13,18}) + H(X_{1,3,9,13,18}) - 3H(X_{4,11,17,18}) + 3H(X_{4,10,11,17,18}) + 3H(X_{20}) - 3H(X_{12,20})","type":"equality","witness_value":0}],"m_eq":2,"master_seed":20260825,"method":"weighted_modular_v2","n":20,"next_action_index":18,"one_sided_condition_count":0,"prompt_tokens":2030,"recipe_index":2,"remaining_action_coordinate_l1":848.0,"remaining_certificate_length":4,"remaining_coefficient_l1":24.0,"remaining_coefficient_max":10.0,"remaining_condition_actions":2,"remaining_shannon_actions":2,"remaining_shannon_kind_counts":{"conditional_entropy":1,"conditional_mutual_information":1},"residual_mode":"condition","schema_version":"reviewer-benchmark-v1","selected_coordinate_l1":848.0,"selected_coordinate_support":53,"selected_effective_n":20,"selected_residual_sparse":[[32,-4],[64,30],[128,4],[144,-4],[1024,-24],[1056,12],[2048,60],[2112,-30],[8704,8],[8960,-8],[16384,-40],[16388,20],[16392,20],[16904,-10],[17408,20],[17416,-20],[18432,20],[18436,-20],[18952,10],[25096,10],[27144,-10],[33032,30],[33544,-30],[49416,-30],[49928,30],[66568,30],[67080,-30],[73728,-12],[74240,12],[77824,12],[78336,-12],[131072,-8],[131080,1],[131332,10],[131333,-10],[131584,-4],[132096,12],[135428,-10],[135429,10],[163840,-4],[164352,4],[197640,-30],[198152,30],[262144,4],[262160,4],[262176,-8],[270848,-8],[271104,8],[524288,30],[526336,-30],[791952,1],[850858,9],[923032,-1]],"selector_source_sha256":"83a328168eac42cedd3412a9c8b5624b8683a52a14042b5defbccf6d833efefe","sha256":"7283d8ee6c1e79bbe29d53f4cee84b81adb64dccb48f2a3b0b665b45265149d1","step_idx":18,"subset_size_profile":{"a":[7,13],"b":[2,0],"c":[0,0]},"target_tokens":781,"tokenizer":{"add_special_tokens":false,"is_fast":true,"kind":"huggingface","model":"Qwen/Qwen3-0.6B","requested_revision":"c1899de289a04d12100db370d81485cdf75e47ca","resolved_commit":"c1899de289a04d12100db370d81485cdf75e47ca","tokenizer_class":"Qwen2TokenizerFast","transformers_version":"4.55.2","trust_remote_code":true,"use_fast":true},"trajectory_length":23,"trajectory_sha256":"70af3f96b9e30581d12945c0058e5f25000faf611d5564f08d9848c65968306b","witness":{"construction":"independent uniform blocks","entropy_unit":"bit","family":"weighted_modular_v2","schema_version":"witness-v2-algebraic-1","weight_sampling":{"independent":true,"maximum":30,"minimum":1},"weights_bits":[21,3,9,17,17,3,29,5,2,21,15,21,22,26,21,7,14,6,23,28]},"witness_evaluation":{"checks":{"direct_equals_remaining_actions":true,"equality_contribution_zero":true,"shannon_contribution_nonnegative":true,"target_nonnegative":true},"direct_target_value":1791,"evaluation_source":"construction_entropy_oracle","pass":true,"remaining_action_value":1791,"remaining_equality_value":0,"remaining_shannon_value":1791},"witness_sha256":"3cc40fe3e01a15af37033b16a48ac38f7175462489e28f4fd236bf8f4322fdc5","witness_variant":"weighted"}
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4H(X_{2,5,6,8,9,10}) - 4H(X_{1,2,5,6,8,9,10}) - 8H(X_{2,3,5,6,8,9,10}) + 36H(X_{11}) + 8H(X_{1,11}) + 8H(X_{3,11}) - 8H(X_{1,3,11}) - 34H(X_{4,11}) + 4H(X_{3,4,11}) - 10H(X_{5,11}) + 10H(X_{4,5,11}) + 3H(X_{1,3,4,5,11}) + 4H(X_{1,2,3,6,11}) - 4H(X_{1,2,3,4,6,11}) - 3H(X_{2,5,6,11}) + 4H(X_{1,4,7,11}) + 3H(X_{2,3,4,5,6,7,11}) - 24H(X_{8,11}) + 24H(X_{4,8,11}) - 2H(X_{1,2,4,8,11}) - 3H(X_{1,3,4,5,8,11}) + 8H(X_{3,5,6,8,11}) - 8H(X_{2,3,5,6,8,11}) - 4H(X_{1,2,3,4,7,8,11}) - 3H(X_{2,3,4,5,6,7,8,11}) + 5H(X_{1,2,4,5,9,11}) + 2H(X_{1,2,4,7,8,9,11}) + 6H(X_{6,7,8,9,11}) - 6H(X_{5,6,7,8,9,11}) - 10H(X_{10,11}) + 3H(X_{1,2,3,4,5,6,8,10,11}) - 5H(X_{1,2,3,4,5,6,7,8,9,10,11})","witness_value":233},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{8})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{3};X_{9})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{6}|X_{7})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{4};X_{8}|X_{11})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{5};X_{7}|X_{2,6})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{7}|X_{1})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{8})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{3}|X_{11})"}],"expression":[[1,-4],[4,1],[5,3],[16,2],[34,1],[50,-1],[64,-2],[65,3],[69,-3],[80,2],[96,2],[98,-1],[112,-2],[114,1],[128,1],[129,1],[144,-2],[256,1],[260,-1],[1024,-2],[1025,-1],[1028,-1],[1029,1],[1032,3],[1152,3],[1160,-3]],"id":0,"kind":"composite_equality","label":"witness 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H(X_{9}) - H(X_{3,9}) - 2H(X_{11}) - H(X_{1,11}) - H(X_{3,11}) + H(X_{1,3,11}) + 3H(X_{4,11}) + 3H(X_{8,11}) - 3H(X_{4,8,11})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{10};X_{11})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{5}|X_{11})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{4};X_{10}|X_{3})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{4})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{2}|X_{5})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{1};X_{10})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{7};X_{10})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{5}|X_{7})"}],"expression":[[1,-2],[4,3],[8,1],[9,-1],[12,-3],[16,-1],[17,1],[18,1],[19,-1],[64,4],[66,-3],[80,-3],[82,3],[512,-3],[513,3],[516,-3],[524,3],[576,-1],[1024,-2],[1032,1],[1040,1],[1048,-1],[1536,1]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["-2H(X_{1}) + 3H(X_{3}) + H(X_{4}) - H(X_{1,4}) - 3H(X_{3,4}) - H(X_{5}) + H(X_{1,5}) + H(X_{2,5}) - H(X_{1,2,5}) + 4H(X_{7}) - 3H(X_{2,7}) - 3H(X_{5,7}) + 3H(X_{2,5,7}) - 3H(X_{10}) + 3H(X_{1,10}) - 3H(X_{3,10}) + 3H(X_{3,4,10}) - H(X_{7,10}) - 2H(X_{11}) + H(X_{4,11}) + H(X_{5,11}) - H(X_{4,5,11}) + H(X_{10,11})>=0","2H(X_{1}) - 3H(X_{3}) - H(X_{4}) + H(X_{1,4}) + 3H(X_{3,4}) + H(X_{5}) - H(X_{1,5}) - H(X_{2,5}) + H(X_{1,2,5}) - 4H(X_{7}) + 3H(X_{2,7}) + 3H(X_{5,7}) - 3H(X_{2,5,7}) + 3H(X_{10}) - 3H(X_{1,10}) + 3H(X_{3,10}) - 3H(X_{3,4,10}) + H(X_{7,10}) + 2H(X_{11}) - H(X_{4,11}) - H(X_{5,11}) + H(X_{4,5,11}) - H(X_{10,11})>=0"],"text":"-2H(X_{1}) + 3H(X_{3}) + H(X_{4}) - H(X_{1,4}) - 3H(X_{3,4}) - H(X_{5}) + H(X_{1,5}) + H(X_{2,5}) - H(X_{1,2,5}) + 4H(X_{7}) - 3H(X_{2,7}) - 3H(X_{5,7}) + 3H(X_{2,5,7}) - 3H(X_{10}) + 3H(X_{1,10}) - 3H(X_{3,10}) + 3H(X_{3,4,10}) - H(X_{7,10}) - 2H(X_{11}) + H(X_{4,11}) + H(X_{5,11}) - H(X_{4,5,11}) + 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9H(X_{2,3,4,5,7,10,11}) - 9H(X_{1,2,5,6,7,10,11}) + 3H(X_{2,3,5,8,10,11}) - 5H(X_{1,6,7,8,10,11}) + 5H(X_{1,3,6,7,8,10,11}) + 5H(X_{9,10,11}) - 5H(X_{5,6,9,10,11}) - 9H(X_{1,2,3,4,5,6,7,8,9,10,11})","witness_value":454},"k":16,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{9})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{4};X_{10}|X_{3})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{6};X_{7})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{4};X_{6})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{5})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{6};X_{11})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{3};X_{5}|X_{10})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{7};X_{9})"}],"expression":[[2,2],[4,-2],[8,-2],[12,2],[16,-1],[18,1],[32,-2],[40,2],[96,3],[256,6],[258,-3],[320,-3],[512,-1],[516,3],[524,-2],[528,1],[532,-1],[1024,3],[1056,-3]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["2H(X_{2}) - 2H(X_{3}) - 2H(X_{4}) + 2H(X_{3,4}) - H(X_{5}) + H(X_{2,5}) - 2H(X_{6}) + 2H(X_{4,6}) + 3H(X_{6,7}) + 6H(X_{9}) - 3H(X_{2,9}) - 3H(X_{7,9}) - H(X_{10}) + 3H(X_{3,10}) - 2H(X_{3,4,10}) + H(X_{5,10}) - H(X_{3,5,10}) + 3H(X_{11}) - 3H(X_{6,11})>=0","-2H(X_{2}) + 2H(X_{3}) + 2H(X_{4}) - 2H(X_{3,4}) + H(X_{5}) - H(X_{2,5}) + 2H(X_{6}) - 2H(X_{4,6}) - 3H(X_{6,7}) - 6H(X_{9}) + 3H(X_{2,9}) + 3H(X_{7,9}) + H(X_{10}) - 3H(X_{3,10}) + 2H(X_{3,4,10}) - H(X_{5,10}) + H(X_{3,5,10}) - 3H(X_{11}) + 3H(X_{6,11})>=0"],"text":"2H(X_{2}) - 2H(X_{3}) - 2H(X_{4}) + 2H(X_{3,4}) - H(X_{5}) + H(X_{2,5}) - 2H(X_{6}) + 2H(X_{4,6}) + 3H(X_{6,7}) + 6H(X_{9}) - 3H(X_{2,9}) - 3H(X_{7,9}) - H(X_{10}) + 3H(X_{3,10}) - 2H(X_{3,4,10}) + H(X_{5,10}) - H(X_{3,5,10}) + 3H(X_{11}) - 3H(X_{6,11})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{10}|X_{3})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{9})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{8})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{7}|X_{2})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{5}|X_{10})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{5})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{6}|X_{9})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{3})"}],"expression":[[1,1],[2,4],[4,-1],[5,-1],[6,-3],[16,1],[18,-1],[20,2],[32,-2],[66,-3],[70,3],[160,2],[260,2],[288,2],[292,-2],[384,-2],[512,3],[516,-1],[528,-3],[532,1]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["H(X_{1}) + 4H(X_{2}) - H(X_{3}) - H(X_{1,3}) - 3H(X_{2,3}) + H(X_{5}) - H(X_{2,5}) + 2H(X_{3,5}) - 2H(X_{6}) - 3H(X_{2,7}) + 3H(X_{2,3,7}) + 2H(X_{6,8}) + 2H(X_{3,9}) + 2H(X_{6,9}) - 2H(X_{3,6,9}) - 2H(X_{8,9}) + 3H(X_{10}) - H(X_{3,10}) - 3H(X_{5,10}) + H(X_{3,5,10})>=0","-H(X_{1}) - 4H(X_{2}) + H(X_{3}) + H(X_{1,3}) + 3H(X_{2,3}) - H(X_{5}) + H(X_{2,5}) - 2H(X_{3,5}) + 2H(X_{6}) + 3H(X_{2,7}) - 3H(X_{2,3,7}) - 2H(X_{6,8}) - 2H(X_{3,9}) - 2H(X_{6,9}) + 2H(X_{3,6,9}) + 2H(X_{8,9}) - 3H(X_{10}) + H(X_{3,10}) + 3H(X_{5,10}) - H(X_{3,5,10})>=0"],"text":"H(X_{1}) + 4H(X_{2}) - H(X_{3}) - H(X_{1,3}) - 3H(X_{2,3}) + H(X_{5}) - H(X_{2,5}) + 2H(X_{3,5}) - 2H(X_{6}) - 3H(X_{2,7}) + 3H(X_{2,3,7}) + 2H(X_{6,8}) + 2H(X_{3,9}) + 2H(X_{6,9}) - 2H(X_{3,6,9}) - 2H(X_{8,9}) + 3H(X_{10}) - H(X_{3,10}) - 3H(X_{5,10}) + H(X_{3,5,10})","type":"equality","witness_value":0}],"m_eq":2,"master_seed":20260825,"method":"linear_rank_gf61_v2","n":11,"next_action_index":2,"one_sided_condition_count":0,"prompt_tokens":1433,"recipe_index":3,"remaining_action_coordinate_l1":287.0,"remaining_certificate_length":16,"remaining_coefficient_l1":91.0,"remaining_coefficient_max":10.0,"remaining_condition_actions":0,"remaining_shannon_actions":16,"remaining_shannon_kind_counts":{"conditional_entropy":4,"conditional_mutual_information":12},"residual_mode":"shannon","schema_version":"reviewer-benchmark-v1","selected_coordinate_l1":287.0,"selected_coordinate_support":45,"selected_effective_n":11,"selected_residual_sparse":[[2,-9],[3,-9],[4,5],[8,11],[11,9],[16,1],[66,11],[70,-2],[131,9],[136,-5],[139,-9],[256,3],[260,-3],[439,9],[473,5],[576,-8],[582,8],[625,8],[631,-8],[648,5],[869,1],[885,-1],[964,-4],[985,-5],[990,4],[1024,-5],[1025,1],[1033,-1],[1072,5],[1282,8],[1478,10],[1486,-10],[1587,9],[1608,9],[1624,-9],[1626,9],[1628,9],[1630,-9],[1651,-9],[1686,3],[1761,-5],[1765,5],[1792,5],[1840,-5],[2047,-9]],"selector_source_sha256":"83a328168eac42cedd3412a9c8b5624b8683a52a14042b5defbccf6d833efefe","sha256":"57f0a358fa979821b02b99d6639bc837835feeeb6da72b42bd653c356d156584","step_idx":2,"subset_size_profile":{"a":[1,1,5,2,1,2,4,4,1,3,1,2,6,2,3,6],"b":[1,1,1,3,6,2,1,7,0,0,1,1,1,1,0,0],"c":[2,5,1,2,0,1,2,0,6,5,0,0,0,0,0,0]},"target_tokens":700,"tokenizer":{"add_special_tokens":false,"is_fast":true,"kind":"huggingface","model":"Qwen/Qwen3-0.6B","requested_revision":"c1899de289a04d12100db370d81485cdf75e47ca","resolved_commit":"c1899de289a04d12100db370d81485cdf75e47ca","tokenizer_class":"Qwen2TokenizerFast","transformers_version":"4.55.2","trust_remote_code":true,"use_fast":true},"trajectory_length":19,"trajectory_sha256":"fd1c90b4a36e059c84281dd044cf31af42173799eb7c6f9636dbb296b61010d4","witness":{"coefficient_representatives":[-30,30],"construction":"structured-random 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2H(X_{2,8,12}) + 4H(X_{1,8,9,12}) + 8H(X_{1,5,8,9,12}) - 8H(X_{1,3,5,6,7,8,9,12}) + 5H(X_{4,5,10,12}) + 6H(X_{1,7,10,12}) + 2H(X_{2,8,10,12}) - 5H(X_{4,5,8,10,12}) + 9H(X_{1,3,8,9,10,12}) - 4H(X_{1,2,3,4,5,8,9,10,12}) - 9H(X_{1,3,7,8,9,10,12}) + 9H(X_{3,6,7,8,9,10,12}) - 9H(X_{3,4,6,7,8,9,10,12}) + 6H(X_{11,12}) - 6H(X_{2,11,12}) - 6H(X_{5,11,12}) - H(X_{4,6,11,12}) + 6H(X_{5,6,11,12}) - 18H(X_{7,11,12}) + 18H(X_{6,7,11,12}) + 4H(X_{1,3,6,7,11,12}) - 4H(X_{1,2,3,6,7,11,12}) - 8H(X_{1,5,8,11,12}) + 2H(X_{1,2,5,6,8,11,12}) + H(X_{1,2,4,6,7,9,11,12}) - 6H(X_{1,7,10,11,12}) - 2H(X_{1,2,5,6,8,10,11,12}) + 5H(X_{2,3,4,7,9,10,11,12}) + 9H(X_{8,9,10,11,12}) - 9H(X_{7,8,9,10,11,12}) - 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2H(X_{5,10}) - 2H(X_{6,10}) - 3H(X_{11}) + 3H(X_{1,11}) - 3H(X_{1,7,11}) + 3H(X_{6,7,11}) + H(X_{12}) - H(X_{2,12}) - H(X_{11,12}) + H(X_{2,11,12}) + 3H(X_{7,11,12}) - 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- 4H(X_{3}) + 6H(X_{1,3}) - 2H(X_{5}) + 2H(X_{1,5}) + 2H(X_{6}) - 2H(X_{3,6}) + 2H(X_{5,7}) - 2H(X_{1,5,7}) + 2H(X_{8}) - 2H(X_{3,8}) - 7H(X_{1,7,8}) + 7H(X_{1,6,7,8}) - 4H(X_{9}) + 4H(X_{1,9}) - 7H(X_{3,9}) + 7H(X_{3,5,9}) + 7H(X_{2,3,4,6,9}) - 7H(X_{2,3,4,5,6,9}) - 10H(X_{10}) + 6H(X_{5,10}) + 7H(X_{1,2,4,5,6,7,8,10}) + H(X_{2,3,5,8,9,10}) - 6H(X_{11}) + 2H(X_{3,11}) + 9H(X_{5,11}) + 7H(X_{1,4,7,8,11}) - 7H(X_{1,4,6,7,8,11}) + 4H(X_{10,11}) + 9H(X_{4,13}) - 2H(X_{5,13}) + 2H(X_{2,3,4,5,6,7,13}) - 4H(X_{6,9,13}) + 6H(X_{10,13}) - 6H(X_{5,10,13}) - 6H(X_{2,4,9,10,13}) + 6H(X_{1,2,4,9,10,13}) + 6H(X_{2,4,5,9,10,13}) - 6H(X_{1,2,4,5,9,10,13}) + 2H(X_{5,11,13}) - 9H(X_{4,5,11,13}) - 2H(X_{2,3,4,5,6,7,11,13}) + 4H(X_{5,8,11,13}) - 9H(X_{6,7,8,11,13}) + 4H(X_{1,2,4,6,9,11,13}) + 4H(X_{2,7,9,10,11,13}) - 4H(X_{1,2,7,9,10,11,13}) + 9H(X_{1,5,6,7,8,9,10,11,13}) + 4H(X_{6,9,12,13}) + H(X_{1,4,6,7,11,12,13}) - 4H(X_{1,2,4,6,9,11,12,13}) + H(X_{2,3,4,9,10,11,12,13}) - 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4H(X_{1,3,4,6,8,9,10,11,12}) + 24H(X_{13}) + 10H(X_{6,7,9,13}) - 24H(X_{10,13}) - 2H(X_{1,2,4,8,9,10,13}) + H(X_{1,2,3,4,7,8,9,10,13}) - 8H(X_{3,11,13}) + 6H(X_{3,6,11,13}) + 8H(X_{1,3,4,7,11,13}) - 3H(X_{1,3,4,5,7,11,13}) - 6H(X_{3,4,6,8,11,13}) + 9H(X_{3,5,7,9,11,13}) - 9H(X_{2,3,5,7,9,11,13}) - H(X_{2,7,8,9,11,13}) + H(X_{1,2,4,6,7,8,9,11,13}) + 3H(X_{1,2,3,4,5,6,7,8,10,11,13}) + 3H(X_{2,4,10,12,13}) - 3H(X_{2,4,8,10,12,13}) + 2H(X_{1,2,4,7,8,9,10,12,13}) + 10H(X_{3,4,9,11,12,13}) - 10H(X_{1,3,4,9,11,12,13}) - 10H(X_{4,7,10,11,12,13}) + 10H(X_{1,4,7,10,11,12,13}) + 7H(X_{1,9,10,11,12,13}) - 7H(X_{1,4,9,10,11,12,13}) + 10H(X_{3,4,5,7,9,10,11,12,13}) - 10H(X_{1,3,4,5,7,9,10,11,12,13}) - 10H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13})","witness_value":389},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{12}|X_{10})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{5};X_{11}|X_{12})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{8};X_{10})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{11})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{6})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{8}|X_{4})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{7};X_{11})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{10};X_{11}|X_{12})"}],"expression":[[2,-1],[8,1],[24,-2],[32,-1],[40,1],[64,-2],[128,3],[136,-2],[152,2],[512,2],[520,1],[640,-3],[1024,-3],[1026,1],[1088,2],[2048,-1],[2064,3],[2560,-1],[2568,-1],[3072,1],[3088,-3],[3584,2]],"id":0,"kind":"composite_equality","label":"witness 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2H(X_{6,9,12,13})>=0"],"text":"7H(X_{1}) + 6H(X_{2}) - 3H(X_{1,2}) + 6H(X_{3}) - 3H(X_{1,3}) - 3H(X_{2,3}) + H(X_{5}) - H(X_{1,5}) - 3H(X_{3,5}) + H(X_{6}) - H(X_{1,6}) - H(X_{5,7}) + H(X_{1,5,7}) + H(X_{9}) - 2H(X_{6,9}) - H(X_{8,9}) + H(X_{10}) - H(X_{9,10}) + 3H(X_{11}) - 3H(X_{5,11}) + 3H(X_{3,5,11}) - H(X_{9,11}) + H(X_{8,9,11}) - H(X_{10,11}) + H(X_{9,10,11}) - H(X_{12}) + 3H(X_{5,12}) + H(X_{6,12}) + 2H(X_{6,9,12}) - 3H(X_{11,12}) + 2H(X_{6,9,13}) + H(X_{12,13}) - H(X_{6,12,13}) - 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6H(X_{1,2,5,7,8,9,13}) - 6H(X_{1,2,4,5,7,8,9,11,13}) + 9H(X_{4,8,10,12,13}) - 9H(X_{1,3,4,7,8,10,12,13}) + 3H(X_{2,3,7,11,12,13}) - 3H(X_{1,2,3,7,9,11,12,13}) + 51H(X_{14}) - 6H(X_{3,14}) - 6H(X_{4,14}) + 12H(X_{1,4,14}) + 6H(X_{3,4,14}) - 18H(X_{6,14}) + 4H(X_{6,7,14}) - 36H(X_{9,14}) + 9H(X_{3,9,14}) + 7H(X_{1,3,5,9,14}) - 4H(X_{1,7,8,10,14}) + 4H(X_{1,2,7,8,10,14}) + 3H(X_{2,4,9,10,14}) + 4H(X_{1,7,8,9,10,14}) - 4H(X_{1,2,7,8,9,10,14}) - 45H(X_{11,14}) - 12H(X_{1,4,11,14}) + 18H(X_{6,11,14}) + 27H(X_{9,11,14}) - 6H(X_{1,12,14}) - 4H(X_{6,7,8,12,14}) + 6H(X_{1,5,6,7,9,12,14}) - 8H(X_{10,12,14}) + 8H(X_{1,2,4,5,9,10,12,14}) - 7H(X_{1,3,4,5,6,9,10,12,14}) + 6H(X_{1,3,11,12,14}) - 6H(X_{1,3,5,6,7,9,11,12,14}) + 8H(X_{8,10,11,12,14}) - 8H(X_{1,2,4,5,8,9,10,11,12,14}) + 4H(X_{5,10,13,14}) - 4H(X_{2,5,10,13,14}) - 3H(X_{4,8,10,13,14}) + 3H(X_{4,7,8,10,13,14}) + H(X_{1,2,3,4,5,6,7,10,11,13,14}) + 3H(X_{2,3,4,8,9,10,11,12,13,14}) - 3H(X_{2,3,4,7,8,9,10,11,12,13,14}) - 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3H(X_{3,9,14}) + 2H(X_{1,4,11,14})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{11};X_{14}|X_{1,4})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{12})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{1};X_{5}|X_{13})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{5};X_{10})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{9};X_{11}|X_{14})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{6};X_{12})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{9};X_{12})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{11}|X_{14})"}],"expression":[[9,-2],[16,1],[32,-1],[128,-2],[256,3],[512,1],[528,-1],[1033,2],[2080,1],[2176,2],[2304,-3],[4096,-3],[4097,3],[4112,3],[4113,-3],[8192,5],[8201,2],[8224,-2],[8448,-3],[9216,-5],[9225,-2],[9248,2],[9472,3]],"id":1,"kind":"composite_equality","label":"witness 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- 12H(X_{1,3}) + 8H(X_{2,3}) - 12H(X_{4}) + 5H(X_{1,4}) + 12H(X_{5}) - 12H(X_{3,5}) + 12H(X_{1,3,5}) + 12H(X_{1,3,7}) - 12H(X_{5,7}) + 12H(X_{3,5,7}) - 12H(X_{1,3,5,7}) + 3H(X_{5,8}) - 16H(X_{9}) + 4H(X_{4,9}) + 8H(X_{6,8,9}) - 8H(X_{2,6,8,9}) - 4H(X_{10}) - 8H(X_{11}) + 19H(X_{2,11}) - 8H(X_{2,3,11}) - 3H(X_{2,5,8,11}) + 8H(X_{1,2,3,6,7,9,11}) - 10H(X_{3,4,6,8,9,11}) - 4H(X_{12}) + 8H(X_{4,12}) + 4H(X_{10,12}) - 8H(X_{1,2,3,6,7,9,11,12}) - 4H(X_{4,5,7,8,9,11,12}) - 2H(X_{13}) + 2H(X_{9,13}) + 8H(X_{1,4,5,6,8,11,12,13}) + 5H(X_{7,14}) + 8H(X_{11,14}) - 8H(X_{2,11,14}) - 5H(X_{1,8,9,11,14}) + 5H(X_{1,5,8,9,10,11,14}) + 4H(X_{4,5,7,8,9,11,12,14}) - 19H(X_{15}) - 5H(X_{1,4,15}) + 5H(X_{1,4,7,15}) + 5H(X_{1,2,6,7,15}) - 5H(X_{1,2,4,6,7,15}) + 12H(X_{9,15}) + 5H(X_{1,4,5,9,11,12,15}) - 5H(X_{1,4,5,7,9,11,12,15}) + 7H(X_{2,3,6,7,13,15}) + 2H(X_{2,4,8,11,13,15}) - 2H(X_{2,4,8,9,11,13,15}) + 5H(X_{2,5,6,12,13,15}) + 8H(X_{2,3,7,9,10,14,15}) + 10H(X_{1,3,4,5,6,8,9,11,14,15}) + 7H(X_{12,14,15}) - 7H(X_{2,3,6,7,12,13,14,15}) - 5H(X_{2,5,6,7,12,13,14,15}) - 8H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15})","witness_value":382},"k":12,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{5};X_{7}|X_{1,3})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{10};X_{12})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{9})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{4};X_{12})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{11}|X_{2})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{2};X_{14}|X_{11})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{9};X_{15})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{7}|X_{5})"}],"expression":[[2,-2],[5,-3],[6,2],[8,-3],[16,3],[20,-3],[21,3],[69,3],[80,-3],[84,3],[85,-3],[256,-4],[264,1],[512,-1],[1024,-2],[1026,4],[1030,-2],[2048,-3],[2056,2],[2560,1],[9216,2],[9218,-2],[16384,-3],[16640,3]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["-2H(X_{2}) - 3H(X_{1,3}) + 2H(X_{2,3}) - 3H(X_{4}) + 3H(X_{5}) - 3H(X_{3,5}) + 3H(X_{1,3,5}) + 3H(X_{1,3,7}) - 3H(X_{5,7}) + 3H(X_{3,5,7}) - 3H(X_{1,3,5,7}) - 4H(X_{9}) + H(X_{4,9}) - H(X_{10}) - 2H(X_{11}) + 4H(X_{2,11}) - 2H(X_{2,3,11}) - 3H(X_{12}) + 2H(X_{4,12}) + H(X_{10,12}) + 2H(X_{11,14}) - 2H(X_{2,11,14}) - 3H(X_{15}) + 3H(X_{9,15})>=0","2H(X_{2}) + 3H(X_{1,3}) - 2H(X_{2,3}) + 3H(X_{4}) - 3H(X_{5}) + 3H(X_{3,5}) - 3H(X_{1,3,5}) - 3H(X_{1,3,7}) + 3H(X_{5,7}) - 3H(X_{3,5,7}) + 3H(X_{1,3,5,7}) + 4H(X_{9}) - H(X_{4,9}) + H(X_{10}) + 2H(X_{11}) - 4H(X_{2,11}) + 2H(X_{2,3,11}) + 3H(X_{12}) - 2H(X_{4,12}) - H(X_{10,12}) - 2H(X_{11,14}) + 2H(X_{2,11,14}) + 3H(X_{15}) - 3H(X_{9,15})>=0"],"text":"-2H(X_{2}) - 3H(X_{1,3}) + 2H(X_{2,3}) - 3H(X_{4}) + 3H(X_{5}) - 3H(X_{3,5}) + 3H(X_{1,3,5}) + 3H(X_{1,3,7}) - 3H(X_{5,7}) + 3H(X_{3,5,7}) - 3H(X_{1,3,5,7}) - 4H(X_{9}) + H(X_{4,9}) - H(X_{10}) - 2H(X_{11}) + 4H(X_{2,11}) - 2H(X_{2,3,11}) - 3H(X_{12}) + 2H(X_{4,12}) + H(X_{10,12}) + 2H(X_{11,14}) - 2H(X_{2,11,14}) - 3H(X_{15}) + 3H(X_{9,15})","type":"equality","witness_value":0}],"m_eq":1,"master_seed":20260825,"method":"linear_rank_gf61_v2","n":15,"next_action_index":2,"one_sided_condition_count":0,"prompt_tokens":1285,"recipe_index":0,"remaining_action_coordinate_l1":406.0,"remaining_certificate_length":11,"remaining_coefficient_l1":57.0,"remaining_coefficient_max":8.0,"remaining_condition_actions":1,"remaining_shannon_actions":10,"remaining_shannon_kind_counts":{"conditional_entropy":2,"conditional_mutual_information":8},"residual_mode":"condition","schema_version":"reviewer-benchmark-v1","selected_coordinate_l1":374.0,"selected_coordinate_support":51,"selected_effective_n":15,"selected_residual_sparse":[[1,-5],[5,-12],[6,8],[8,-12],[9,5],[16,12],[20,-12],[21,12],[69,12],[80,-12],[84,12],[85,-12],[144,3],[256,-16],[264,4],[416,8],[418,-8],[512,-4],[1024,-8],[1026,19],[1030,-8],[1170,-3],[1383,8],[2048,-4],[2056,8],[2560,4],[3431,-8],[3544,-4],[4096,-2],[4352,2],[7353,8],[8256,5],[9216,8],[9218,-8],[9601,-5],[10129,5],[11736,4],[16384,-12],[16393,-5],[16457,5],[16483,5],[16491,-5],[16640,12],[19737,5],[19801,-5],[21642,2],[21898,-2],[22578,5],[25414,8],[30834,-5],[32767,-8]],"selector_source_sha256":"83a328168eac42cedd3412a9c8b5624b8683a52a14042b5defbccf6d833efefe","sha256":"c6abcea6b41f420b0a49cdf4c977fc1a57d8e62154685431679127a7d6c41c2a","step_idx":2,"subset_size_profile":{"a":[4,4,7,7,3,2,6,1,5,2],"b":[1,1,8,1,1,0,2,0,1,2],"c":[1,3,0,0,0,5,0,7,1,0]},"target_tokens":782,"tokenizer":{"add_special_tokens":false,"is_fast":true,"kind":"huggingface","model":"Qwen/Qwen3-0.6B","requested_revision":"c1899de289a04d12100db370d81485cdf75e47ca","resolved_commit":"c1899de289a04d12100db370d81485cdf75e47ca","tokenizer_class":"Qwen2TokenizerFast","transformers_version":"4.55.2","trust_remote_code":true,"use_fast":true},"trajectory_length":14,"trajectory_sha256":"c274b659a44ca09d1215bb0a3786adb7be120da68c10bed89e713921d41322a2","witness":{"coefficient_representatives":[-30,30],"construction":"structured-random 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7H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15})","witness_value":737},"k":16,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{10})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{8})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{8}|X_{15})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{13})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{6};X_{7}|X_{8})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{9};X_{15})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{15}|X_{1,2})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{8};X_{14})"}],"expression":[[3,2],[4,-2],[8,1],[16,2],[131,-2],[144,-2],[160,-1],[192,-1],[224,1],[256,-1],[512,-2],[516,2],[4096,1],[4104,-1],[8192,-3],[8320,3],[16384,-2],[16386,1],[16387,-2],[16512,1],[16514,-1],[16515,2],[16640,1]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["2H(X_{1,2}) - 2H(X_{3}) + H(X_{4}) + 2H(X_{5}) - 2H(X_{1,2,8}) - 2H(X_{5,8}) - H(X_{6,8}) - H(X_{7,8}) + H(X_{6,7,8}) - H(X_{9}) - 2H(X_{10}) + 2H(X_{3,10}) + H(X_{13}) - H(X_{4,13}) - 3H(X_{14}) + 3H(X_{8,14}) - 2H(X_{15}) + H(X_{2,15}) - 2H(X_{1,2,15}) + H(X_{8,15}) - H(X_{2,8,15}) + 2H(X_{1,2,8,15}) + H(X_{9,15})>=0","-2H(X_{1,2}) + 2H(X_{3}) - H(X_{4}) - 2H(X_{5}) + 2H(X_{1,2,8}) + 2H(X_{5,8}) + H(X_{6,8}) + H(X_{7,8}) - H(X_{6,7,8}) + H(X_{9}) + 2H(X_{10}) - 2H(X_{3,10}) - H(X_{13}) + H(X_{4,13}) + 3H(X_{14}) - 3H(X_{8,14}) + 2H(X_{15}) - H(X_{2,15}) + 2H(X_{1,2,15}) - H(X_{8,15}) + H(X_{2,8,15}) - 2H(X_{1,2,8,15}) - H(X_{9,15})>=0"],"text":"2H(X_{1,2}) - 2H(X_{3}) + H(X_{4}) + 2H(X_{5}) - 2H(X_{1,2,8}) - 2H(X_{5,8}) - H(X_{6,8}) - H(X_{7,8}) + H(X_{6,7,8}) - H(X_{9}) - 2H(X_{10}) + 2H(X_{3,10}) + H(X_{13}) - H(X_{4,13}) - 3H(X_{14}) + 3H(X_{8,14}) - 2H(X_{15}) + H(X_{2,15}) - 2H(X_{1,2,15}) + H(X_{8,15}) - H(X_{2,8,15}) + 2H(X_{1,2,8,15}) + H(X_{9,15})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{3};X_{6})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{6};X_{15}|X_{8})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{7};X_{10})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{5};X_{9}|X_{14})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{7};X_{15})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{5};X_{13})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{11};X_{15}|X_{2,6})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{11};X_{14})"}],"expression":[[4,1],[16,1],[32,1],[34,3],[36,-1],[64,-2],[128,-1],[160,1],[512,1],[576,-1],[1024,-2],[1058,-3],[4096,1],[4112,-1],[8192,1],[8208,-3],[8448,-3],[8464,3],[9216,2],[16384,-3],[16418,-3],[16448,3],[16512,1],[16544,-1],[17442,3]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["H(X_{3}) + H(X_{5}) + H(X_{6}) + 3H(X_{2,6}) - H(X_{3,6}) - 2H(X_{7}) - H(X_{8}) + H(X_{6,8}) + H(X_{10}) - H(X_{7,10}) - 2H(X_{11}) - 3H(X_{2,6,11}) + H(X_{13}) - H(X_{5,13}) + H(X_{14}) - 3H(X_{5,14}) - 3H(X_{9,14}) + 3H(X_{5,9,14}) + 2H(X_{11,14}) - 3H(X_{15}) - 3H(X_{2,6,15}) + 3H(X_{7,15}) + H(X_{8,15}) - H(X_{6,8,15}) + 3H(X_{2,6,11,15})>=0","-H(X_{3}) - H(X_{5}) - H(X_{6}) - 3H(X_{2,6}) + H(X_{3,6}) + 2H(X_{7}) + H(X_{8}) - H(X_{6,8}) - H(X_{10}) + H(X_{7,10}) + 2H(X_{11}) + 3H(X_{2,6,11}) - H(X_{13}) + H(X_{5,13}) - H(X_{14}) + 3H(X_{5,14}) + 3H(X_{9,14}) - 3H(X_{5,9,14}) - 2H(X_{11,14}) + 3H(X_{15}) + 3H(X_{2,6,15}) - 3H(X_{7,15}) - H(X_{8,15}) + H(X_{6,8,15}) - 3H(X_{2,6,11,15})>=0"],"text":"H(X_{3}) + H(X_{5}) + H(X_{6}) + 3H(X_{2,6}) - H(X_{3,6}) - 2H(X_{7}) - H(X_{8}) + H(X_{6,8}) + H(X_{10}) - H(X_{7,10}) - 2H(X_{11}) - 3H(X_{2,6,11}) + H(X_{13}) - H(X_{5,13}) + H(X_{14}) - 3H(X_{5,14}) - 3H(X_{9,14}) + 3H(X_{5,9,14}) + 2H(X_{11,14}) - 3H(X_{15}) - 3H(X_{2,6,15}) + 3H(X_{7,15}) + H(X_{8,15}) - H(X_{6,8,15}) + 3H(X_{2,6,11,15})","type":"equality","witness_value":0}],"m_eq":2,"master_seed":20260825,"method":"linear_rank_gf61_v2","n":15,"next_action_index":4,"one_sided_condition_count":0,"prompt_tokens":1660,"recipe_index":3,"remaining_action_coordinate_l1":193.0,"remaining_certificate_length":14,"remaining_coefficient_l1":80.0,"remaining_coefficient_max":10.0,"remaining_condition_actions":0,"remaining_shannon_actions":14,"remaining_shannon_kind_counts":{"conditional_entropy":4,"conditional_mutual_information":10},"residual_mode":"shannon","schema_version":"reviewer-benchmark-v1","selected_coordinate_l1":193.0,"selected_coordinate_support":37,"selected_effective_n":15,"selected_residual_sparse":[[2,13],[772,8],[1568,9],[1677,7],[2048,1],[2185,4],[2187,-4],[4368,-3],[4595,3],[5504,7],[5646,4],[10792,-1],[10808,1],[11605,9],[11607,-9],[12568,3],[12795,-3],[16385,-5],[16400,4],[16484,10],[16568,8],[16641,5],[16898,7],[17340,-8],[20612,6],[22046,-4],[22402,-7],[25127,1],[26837,5],[27093,-5],[27175,-1],[27461,-6],[27477,6],[28200,1],[28216,-1],[31090,7],[32767,-7]],"selector_source_sha256":"83a328168eac42cedd3412a9c8b5624b8683a52a14042b5defbccf6d833efefe","sha256":"f87c66106e8fb47252e619e258d3e1cf77599c96e00d15f958c2dfceda3513e1","step_idx":4,"subset_size_profile":{"a":[6,8,5,9,4,1,5,6,4,1,7,4,3,4],"b":[1,1,3,6,3,0,2,2,1,2,1,0,0,0],"c":[2,0,0,0,0,8,3,0,0,5,0,0,0,0]},"target_tokens":741,"tokenizer":{"add_special_tokens":false,"is_fast":true,"kind":"huggingface","model":"Qwen/Qwen3-0.6B","requested_revision":"c1899de289a04d12100db370d81485cdf75e47ca","resolved_commit":"c1899de289a04d12100db370d81485cdf75e47ca","tokenizer_class":"Qwen2TokenizerFast","transformers_version":"4.55.2","trust_remote_code":true,"use_fast":true},"trajectory_length":19,"trajectory_sha256":"0f93bc0c304931aa08cc96ad0e22c2332e9b93c294e609ba85e0f0f732b2b784","witness":{"coefficient_representatives":[-30,30],"construction":"structured-random 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6H(X_{2,6,15})","witness_value":0}],"certificate_length":13,"certificate_order":"greedy_negative_mass_condition_late","condition_coordinate_supports":[21],"condition_rank":1,"condition_tokens":[193,192],"condition_width":8,"config_sha256":"1b1f298ed012071c85e9cfb286fe5dcf7c183383de92041754d8a7802f426421","equality_condition_count":1,"full_tokens":1157,"generator_dependency_sha256":{"data/finite_field.py":"c9738c10a16ea2cfb3cde346a23673d1536dbfce491f8871346f441b9b4c2a6e","data/witness_first.py":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","data/witness_v2_algebraic.py":"ffbd48eb673a3f6f842327d00431285bbae8963a7e20f6956e37e544fa37165f","data/witness_v2_graph.py":"c81adcfefcf6ffc0e816f1b338af13323bc80e06293e8cac6b28a3b0ae2efc8a"},"generator_schema_version":"witness-first-v2","generator_source_sha256":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","id":"e2n-witness-first-v2-n16-linear_rank_gf61_v2-e5e961867ee5d066","initial_target":{"coordinate_support":54,"effective_n":16,"expression":[[4,4],[16,-2],[32,-6],[34,6],[64,2],[80,-2],[128,-6],[132,-4],[272,-2],[512,-8],[528,4],[768,4],[1024,-2],[1040,2],[1152,-9],[2048,-11],[2176,6],[4096,-2],[4224,-3],[4352,2],[4742,9],[8196,-4],[8324,4],[9224,9],[13966,-9],[14521,8],[16416,6],[16418,-6],[17440,-8],[17441,-3],[18312,9],[18973,4],[19229,-4],[20545,-2],[23074,5],[28755,2],[32770,-2],[32774,2],[33349,-1],[33924,9],[34816,5],[41541,1],[45082,2],[45086,-2],[49733,1],[51014,8],[51084,-9],[55265,8],[55842,-5],[55905,1],[57925,-1],[62383,3],[65013,3],[65535,-8]],"text":"4H(X_{3}) - 2H(X_{5}) - 6H(X_{6}) + 6H(X_{2,6}) + 2H(X_{7}) - 2H(X_{5,7}) - 6H(X_{8}) - 4H(X_{3,8}) - 2H(X_{5,9}) - 8H(X_{10}) + 4H(X_{5,10}) + 4H(X_{9,10}) - 2H(X_{11}) + 2H(X_{5,11}) - 9H(X_{8,11}) - 11H(X_{12}) + 6H(X_{8,12}) - 2H(X_{13}) - 3H(X_{8,13}) + 2H(X_{9,13}) + 9H(X_{2,3,8,10,13}) - 4H(X_{3,14}) + 4H(X_{3,8,14}) + 9H(X_{4,11,14}) - 9H(X_{2,3,4,8,10,11,13,14}) + 8H(X_{1,4,5,6,8,12,13,14}) + 6H(X_{6,15}) - 6H(X_{2,6,15}) - 8H(X_{6,11,15}) - 3H(X_{1,6,11,15}) + 9H(X_{4,8,9,10,11,15}) + 4H(X_{1,3,4,5,10,12,15}) - 4H(X_{1,3,4,5,9,10,12,15}) - 2H(X_{1,7,13,15}) + 5H(X_{2,6,10,12,13,15}) + 2H(X_{1,2,5,7,13,14,15}) - 2H(X_{2,16}) + 2H(X_{2,3,16}) - H(X_{1,3,7,10,16}) + 9H(X_{3,8,11,16}) + 5H(X_{12,16}) + H(X_{1,3,7,10,14,16}) + 2H(X_{2,4,5,13,14,16}) - 2H(X_{2,3,4,5,13,14,16}) + H(X_{1,3,7,10,15,16}) + 8H(X_{2,3,7,9,10,11,15,16}) - 9H(X_{3,4,8,9,10,11,15,16}) + 8H(X_{1,6,7,8,9,10,11,13,15,16}) - 5H(X_{2,6,10,12,13,15,16}) + H(X_{1,6,7,10,12,13,15,16}) - H(X_{1,3,7,10,14,15,16}) + 3H(X_{1,2,3,4,6,8,9,10,13,14,15,16}) + 3H(X_{1,3,5,6,7,8,9,11,12,13,14,15,16}) - 8H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16})","witness_value":423},"k":12,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{8};X_{12})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{15}|X_{6})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{14}|X_{3})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{10})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{5};X_{7})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{9};X_{13})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{5};X_{11})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{5};X_{9})"}],"expression":[[4,-2],[16,1],[32,3],[34,-3],[64,-1],[80,1],[128,3],[132,2],[272,1],[512,2],[528,-2],[1024,1],[1040,-1],[2048,3],[2176,-3],[4096,1],[4352,-1],[8196,2],[8324,-2],[16416,-3],[16418,3]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["-2H(X_{3}) + H(X_{5}) + 3H(X_{6}) - 3H(X_{2,6}) - H(X_{7}) + H(X_{5,7}) + 3H(X_{8}) + 2H(X_{3,8}) + H(X_{5,9}) + 2H(X_{10}) - 2H(X_{5,10}) + H(X_{11}) - H(X_{5,11}) + 3H(X_{12}) - 3H(X_{8,12}) + H(X_{13}) - H(X_{9,13}) + 2H(X_{3,14}) - 2H(X_{3,8,14}) - 3H(X_{6,15}) + 3H(X_{2,6,15})>=0","2H(X_{3}) - H(X_{5}) - 3H(X_{6}) + 3H(X_{2,6}) + H(X_{7}) - H(X_{5,7}) - 3H(X_{8}) - 2H(X_{3,8}) - H(X_{5,9}) - 2H(X_{10}) + 2H(X_{5,10}) - H(X_{11}) + H(X_{5,11}) - 3H(X_{12}) + 3H(X_{8,12}) - H(X_{13}) + H(X_{9,13}) - 2H(X_{3,14}) + 2H(X_{3,8,14}) + 3H(X_{6,15}) - 3H(X_{2,6,15})>=0"],"text":"-2H(X_{3}) + H(X_{5}) + 3H(X_{6}) - 3H(X_{2,6}) - H(X_{7}) + H(X_{5,7}) + 3H(X_{8}) + 2H(X_{3,8}) + H(X_{5,9}) + 2H(X_{10}) - 2H(X_{5,10}) + H(X_{11}) - H(X_{5,11}) + 3H(X_{12}) - 3H(X_{8,12}) + H(X_{13}) - H(X_{9,13}) + 2H(X_{3,14}) - 2H(X_{3,8,14}) - 3H(X_{6,15}) + 3H(X_{2,6,15})","type":"equality","witness_value":0}],"m_eq":1,"master_seed":20260825,"method":"linear_rank_gf61_v2","n":16,"next_action_index":2,"one_sided_condition_count":0,"prompt_tokens":1167,"recipe_index":0,"remaining_action_coordinate_l1":194.0,"remaining_certificate_length":11,"remaining_coefficient_l1":43.0,"remaining_coefficient_max":9.0,"remaining_condition_actions":1,"remaining_shannon_actions":10,"remaining_shannon_kind_counts":{"conditional_entropy":5,"conditional_mutual_information":5},"residual_mode":"condition","schema_version":"reviewer-benchmark-v1","selected_coordinate_l1":194.0,"selected_coordinate_support":47,"selected_effective_n":16,"selected_residual_sparse":[[4,4],[16,-2],[32,-6],[34,6],[64,2],[80,-2],[128,-6],[132,-4],[272,-2],[512,-8],[528,4],[768,4],[1024,-2],[1040,2],[2048,-6],[2176,6],[4096,-2],[4224,-3],[4352,2],[4742,9],[8196,-4],[8324,4],[9224,9],[13966,-9],[14521,8],[16416,6],[16418,-6],[17440,-8],[17441,-3],[18973,4],[19229,-4],[20545,-2],[28755,2],[32770,-2],[32774,2],[33349,-1],[41541,1],[45082,2],[45086,-2],[49733,1],[51014,8],[55265,8],[55905,1],[57925,-1],[62383,3],[65013,3],[65535,-8]],"selector_source_sha256":"83a328168eac42cedd3412a9c8b5624b8683a52a14042b5defbccf6d833efefe","sha256":"e5e961867ee5d0662c3edebe609208249db0c4bff70ff607bd2b806857388354","step_idx":2,"subset_size_profile":{"a":[5,8,7,1,1,9,10,3,1,8],"b":[3,8,0,6,4,0,0,0,1,0],"c":[0,0,3,1,2,4,2,4,5,0]},"target_tokens":772,"tokenizer":{"add_special_tokens":false,"is_fast":true,"kind":"huggingface","model":"Qwen/Qwen3-0.6B","requested_revision":"c1899de289a04d12100db370d81485cdf75e47ca","resolved_commit":"c1899de289a04d12100db370d81485cdf75e47ca","tokenizer_class":"Qwen2TokenizerFast","transformers_version":"4.55.2","trust_remote_code":true,"use_fast":true},"trajectory_length":14,"trajectory_sha256":"0f8d93697042c45ea00e24d60d0f751e195adf95628471f01a7be4334ab0e769","witness":{"coefficient_representatives":[-30,30],"construction":"structured-random 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equality 2","prompt_rows":["3H(X_{2}) - 3H(X_{6}) + 3H(X_{4,6}) + 8H(X_{7}) - 3H(X_{1,7}) - 3H(X_{4,7}) - 3H(X_{2,9}) + H(X_{9,10}) - H(X_{2,9,10}) + 4H(X_{14}) - 2H(X_{7,14}) - 2H(X_{8,14}) - 3H(X_{2,15}) + 3H(X_{2,9,15}) - H(X_{9,10,15}) + H(X_{2,9,10,15}) - 3H(X_{16}) + 3H(X_{1,7,16}) + 3H(X_{13,16}) - 3H(X_{7,13,16}) - 2H(X_{14,16}) + 2H(X_{8,14,16})>=0","-3H(X_{2}) + 3H(X_{6}) - 3H(X_{4,6}) - 8H(X_{7}) + 3H(X_{1,7}) + 3H(X_{4,7}) + 3H(X_{2,9}) - H(X_{9,10}) + H(X_{2,9,10}) - 4H(X_{14}) + 2H(X_{7,14}) + 2H(X_{8,14}) + 3H(X_{2,15}) - 3H(X_{2,9,15}) + H(X_{9,10,15}) - H(X_{2,9,10,15}) + 3H(X_{16}) - 3H(X_{1,7,16}) - 3H(X_{13,16}) + 3H(X_{7,13,16}) + 2H(X_{14,16}) - 2H(X_{8,14,16})>=0"],"text":"3H(X_{2}) - 3H(X_{6}) + 3H(X_{4,6}) + 8H(X_{7}) - 3H(X_{1,7}) - 3H(X_{4,7}) - 3H(X_{2,9}) + H(X_{9,10}) - H(X_{2,9,10}) + 4H(X_{14}) - 2H(X_{7,14}) - 2H(X_{8,14}) - 3H(X_{2,15}) + 3H(X_{2,9,15}) - H(X_{9,10,15}) + H(X_{2,9,10,15}) - 3H(X_{16}) + 3H(X_{1,7,16}) + 3H(X_{13,16}) - 3H(X_{7,13,16}) 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6H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17})","witness_value":385},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{1};X_{14})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{16})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{6};X_{12})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{7};X_{17})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{2}|X_{13})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{8};X_{16}|X_{13})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{2};X_{15}|X_{16})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{10};X_{11})"}],"expression":[[1,3],[8,1],[32,-1],[64,2],[512,-2],[1024,-2],[1536,2],[2048,-1],[2080,1],[4097,1],[4098,1],[4099,-1],[4224,-1],[8192,3],[8193,-3],[32768,3],[32770,-2],[32776,-1],[36864,-1],[36992,1],[49152,-2],[49154,2],[65536,2],[65600,-2]],"id":0,"kind":"composite_equality","label":"witness 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5H(X_{2,3,4,5,7,8,9,10,11,12,14,15,16}) + 10H(X_{1,3,5,6,8,11,12,13,14,15,16}) - 10H(X_{1,2,3,5,6,8,11,12,13,14,15,16}) - 3H(X_{17}) + H(X_{7,17}) - H(X_{1,7,17}) - 2H(X_{1,8,17}) + 2H(X_{1,6,8,17}) + 2H(X_{5,9,11,17}) + 3H(X_{12,17}) + 8H(X_{1,4,5,8,11,12,13,17}) - 9H(X_{3,4,7,9,15,17}) + 9H(X_{2,3,4,7,9,15,17}) + 9H(X_{3,4,5,6,7,9,15,17}) - 9H(X_{2,3,4,5,6,7,9,15,17}) + H(X_{1,2,5,6,8,9,10,13,15,17}) + 2H(X_{1,8,11,12,13,15,17}) - 8H(X_{1,4,5,8,11,12,13,15,17}) - 2H(X_{1,6,8,11,12,13,15,17}) + 8H(X_{2,4,8,11,14,15,17}) - 8H(X_{2,4,8,10,11,14,15,17}) + 10H(X_{1,3,4,5,6,8,9,13,14,15,17}) - H(X_{1,2,3,5,6,8,9,10,11,13,14,15,17}) + 10H(X_{10,11,14,16,17}) - 8H(X_{1,2,4,5,7,9,10,12,13,14,16,17}) + 8H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,16,17}) - 10H(X_{2,4,5,6,7,8,9,10,11,14,15,16,17}) - 10H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17})","witness_value":390},"k":16,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{4};X_{14})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{3};X_{12}|X_{14})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{9})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{12};X_{17})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{17}|X_{7})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{10};X_{13})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{5};X_{12}|X_{15})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{9};X_{11})"}],"expression":[[8,2],[64,-1],[65,1],[128,-2],[256,-4],[384,2],[512,2],[1024,-2],[1280,2],[2048,-3],[4096,2],[4608,-2],[8192,1],[8196,1],[8200,-2],[10240,1],[10244,-1],[16384,-1],[16400,1],[18432,1],[18448,-1],[65536,-3],[65600,1],[65601,-1],[67584,3]],"id":0,"kind":"composite_equality","label":"witness 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8H(X_{2,5,6,10,13,15,16}) - 4H(X_{2,3,5,6,8,12,13,15,16}) + 5H(X_{1,2,3,5,13,17}) - 5H(X_{1,2,3,5,9,10,11,13,14,17}) - 9H(X_{2,4,6,7,8,16,17}) + 9H(X_{2,4,5,6,7,8,16,17}) + 4H(X_{10,16,17}) + 9H(X_{2,3,4,6,7,8,10,11,12,13,16,17}) - 9H(X_{2,3,4,5,6,7,8,10,11,12,13,16,17}) - 4H(X_{7,10,11,12,13,14,16,17}) + 10H(X_{4,8,10,15,16,17}) - H(X_{18}) - 21H(X_{6,18}) + 21H(X_{4,6,18}) + H(X_{4,5,7,18}) + 2H(X_{2,4,5,8,9,11,18}) + 10H(X_{1,2,14,18}) + 2H(X_{1,2,8,15,18}) - 2H(X_{1,2,7,8,14,15,18}) - H(X_{8,10,11,16,18}) + H(X_{2,3,8,11,12,16,18}) - H(X_{2,3,4,5,7,8,11,12,16,18}) + H(X_{2,3,8,10,11,12,16,18}) + 5H(X_{2,4,5,8,10,12,14,16,18}) - 5H(X_{2,4,5,8,9,10,12,14,16,18}) - 2H(X_{1,2,4,5,8,9,11,15,16,18}) + 8H(X_{2,5,6,8,9,12,13,15,16,17,18}) - 8H(X_{2,5,6,8,9,10,12,13,15,16,17,18}) - 10H(X_{1,2,4,8,10,14,15,16,17,18}) + 7H(X_{2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}) - 7H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18})","witness_value":260},"k":16,"logical_conditions":[{"component_width":12,"component_witness_values":[0,0,0,0,0,0,0,0,0,0,0,0],"components":[{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{3};X_{6}|X_{10})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{15})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{9};X_{16}|X_{4})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{1};X_{7}|X_{12})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{1};X_{4})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{4};X_{10})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{11};X_{12}|X_{7})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{7})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{4};X_{18}|X_{6})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{1};X_{6})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{13})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{4};X_{9}|X_{14,16})"}],"expression":[[2,3],[4,-1],[8,1],[9,-2],[32,-5],[33,2],[40,3],[64,2],[66,-3],[264,-1],[512,-1],[516,-1],[520,2],[544,-1],[548,1],[1088,1],[2048,2],[2049,-2],[2112,-1],[2113,2],[3136,-1],[4096,-3],[4100,3],[16384,2],[16388,-2],[32776,-1],[33032,1],[40960,2],[40968,-2],[41216,-2],[41224,2],[131104,3],[131112,-3]],"id":0,"kind":"composite_equality","label":"witness 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5H(X_{11,14,16}) + 16H(X_{15,16}) + 2H(X_{1,3,5,6,8,9,12,15,16}) - 2H(X_{1,3,4,5,6,7,8,9,12,15,16}) + 2H(X_{2,17}) - 6H(X_{5,17}) + 6H(X_{1,5,17}) + 6H(X_{5,12,17}) - 6H(X_{1,5,12,17}) - 2H(X_{2,6,8,11,14,17}) + 8H(X_{2,4,9,12,13,14,17}) + 9H(X_{1,3,4,7,8,9,10,12,13,15,17}) + 4H(X_{2,3,4,6,7,12,13,14,15,17}) - 4H(X_{1,2,3,4,6,7,10,12,13,14,15,17}) - 6H(X_{1,5,6,7,8,9,11,14,16,17}) + 6H(X_{1,5,6,7,8,9,11,12,14,16,17}) - H(X_{4,5,6,7,8,9,10,11,12,14,16,17}) - 6H(X_{2,18}) + 6H(X_{1,2,18}) + 6H(X_{2,9,18}) - 6H(X_{1,2,9,18}) + 5H(X_{1,2,4,5,6,7,11,12,18}) + 8H(X_{6,10,13,18}) - 6H(X_{8,14,15,18}) + 6H(X_{3,8,14,15,18}) - 9H(X_{1,2,3,5,8,9,11,14,15,18}) + 9H(X_{1,2,3,5,7,8,9,11,13,14,15,18}) + 5H(X_{5,9,14,16,18}) + 9H(X_{2,5,6,11,14,16,18}) - 5H(X_{1,2,4,5,6,7,11,12,14,16,18}) - 5H(X_{5,9,14,15,16,18}) - 8H(X_{2,4,6,9,10,12,13,14,17,18}) + 6H(X_{1,4,6,8,10,13,14,15,17,18}) - 6H(X_{1,3,4,6,8,10,13,14,15,17,18}) - 8H(X_{3,16,17,18}) + 8H(X_{3,9,16,17,18}) + 8H(X_{1,2,3,5,6,7,8,11,12,16,17,18}) - 8H(X_{1,2,3,5,6,7,8,9,11,12,16,17,18}) + H(X_{4,5,6,7,8,9,10,11,12,14,16,17,18}) - 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2H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18})","witness_value":357},"k":16,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{7})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{11};X_{14})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{8};X_{14}|X_{3})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{2};X_{14}|X_{7})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{8})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{3};X_{13})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{9};X_{10}|X_{8})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{1};X_{2})"}],"expression":[[1,-3],[2,-3],[3,3],[4,-2],[16,-2],[32,-2],[64,-4],[66,2],[80,2],[128,-5],[132,3],[160,2],[384,3],[640,3],[896,-3],[1024,-1],[4096,1],[4100,-1],[8192,-1],[8196,3],[8256,2],[8258,-2],[8324,-3],[9216,1]],"id":0,"kind":"composite_equality","label":"witness 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6H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19})","witness_value":659},"k":16,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{14})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{6};X_{13})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{6})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{12}|X_{18})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{9};X_{19}|X_{4})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{2};X_{15})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{11};X_{17})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{13})"}],"expression":[[1,-1],[2,2],[8,-1],[32,-6],[33,1],[128,-2],[264,1],[1024,2],[4096,-5],[4128,3],[4224,2],[8192,-2],[8224,2],[16384,2],[16386,-2],[65536,2],[66560,-2],[131072,-3],[131074,3],[133120,3],[133122,-3],[262152,1],[262408,-1]],"id":0,"kind":"composite_equality","label":"witness 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9H(X_{1,4,5,6,8,9,11,12,13,15,17,18})","witness_value":0},{"a_mask":524420,"b_mask":131072,"c_mask":40961,"coefficient":8,"expression":[[40961,-1],[172033,1],[565381,1],[696453,-1]],"kind":"conditional_mutual_information","label":"I(X_{3,8,20};X_{18}|X_{1,14,16})","role":"shannon","scaled_expression":[[40961,-8],[172033,8],[565381,8],[696453,-8]],"scaled_witness_value":24,"source_index":15,"text":"-8H(X_{1,14,16}) + 8H(X_{1,14,16,18}) + 8H(X_{1,3,8,14,16,20}) - 8H(X_{1,3,8,14,16,18,20})","witness_value":3},{"a_mask":740784,"b_mask":32770,"c_mask":64,"coefficient":6,"expression":[[64,-1],[32834,1],[740848,1],[773618,-1]],"kind":"conditional_mutual_information","label":"I(X_{5,6,8,9,11,12,15,17,18,20};X_{2,16}|X_{7})","role":"shannon","scaled_expression":[[64,-6],[32834,6],[740848,6],[773618,-6]],"scaled_witness_value":30,"source_index":10,"text":"-6H(X_{7}) + 6H(X_{2,7,16}) + 6H(X_{5,6,7,8,9,11,12,15,17,18,20}) - 6H(X_{2,5,6,7,8,9,11,12,15,16,17,18,20})","witness_value":5},{"a_mask":75884,"b_mask":0,"c_mask":283410,"coefficient":10,"expression":[[283410,-1],[359294,1]],"kind":"conditional_entropy","label":"H(X_{3,4,6,7,12,14,17}|X_{2,5,9,10,13,15,19})","role":"shannon","scaled_expression":[[283410,-10],[359294,10]],"scaled_witness_value":80,"source_index":13,"text":"-10H(X_{2,5,9,10,13,15,19}) + 10H(X_{2,3,4,5,6,7,9,10,12,13,14,15,17,19})","witness_value":8},{"a_mask":827390,"b_mask":221185,"c_mask":0,"coefficient":7,"expression":[[221185,1],[827390,1],[1048575,-1]],"kind":"conditional_mutual_information","label":"I(X_{2,3,4,5,6,7,8,9,10,11,12,13,16,19,20};X_{1,14,15,17,18})","role":"shannon","scaled_expression":[[221185,7],[827390,7],[1048575,-7]],"scaled_witness_value":56,"source_index":0,"text":"7H(X_{1,14,15,17,18}) + 7H(X_{2,3,4,5,6,7,8,9,10,11,12,13,16,19,20}) - 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- 6H(X_{3}) - 3H(X_{1,3}) + 5H(X_{4}) - 7H(X_{1,4}) + 5H(X_{5}) + 10H(X_{6}) - 17H(X_{7}) + 7H(X_{1,4,5,7}) - 9H(X_{8}) - 10H(X_{6,8}) + 9H(X_{7,8}) + 15H(X_{9}) - 5H(X_{4,9}) + 11H(X_{10}) - 5H(X_{5,10}) - 6H(X_{7,10}) - 15H(X_{11}) - 14H(X_{12}) + 9H(X_{1,12}) + 5H(X_{11,12}) - 4H(X_{13}) + 4H(X_{1,5,7,13}) + 10H(X_{8,13}) - 10H(X_{9,13}) - 20H(X_{14}) + 14H(X_{7,14}) + 3H(X_{3,7,9,14}) + 10H(X_{11,14}) - 3H(X_{9,13,14}) - 8H(X_{12,15}) - 3H(X_{1,3,10,12,15}) + 8H(X_{4,11,12,15}) + 3H(X_{1,3,6,9,10,11,12,14,15}) - 6H(X_{16}) + 6H(X_{3,16}) - 6H(X_{5,8,16}) + 3H(X_{1,7,9,16}) + 3H(X_{2,10,12,16}) - 3H(X_{2,3,7,9,10,12,14,16}) + 3H(X_{17}) + 3H(X_{8,9,12,13,14,17}) + 6H(X_{3,5,8,12,13,14,15,16,17}) - 6H(X_{18}) + 4H(X_{6,9,13,14,18}) - 4H(X_{1,5,6,7,9,13,14,18}) + 7H(X_{4,7,8,11,14,15,18}) + 6H(X_{5,8,16,18}) + 3H(X_{2,3,10,16,18}) + 6H(X_{17,18}) - 6H(X_{3,5,8,12,13,14,15,16,17,18}) - 6H(X_{19}) + 7H(X_{1,4,6,13,19}) - 7H(X_{1,4,5,6,7,13,19}) + 7H(X_{3,5,6,9,13,19}) + 6H(X_{14,19}) + 6H(X_{3,10,12,14,19}) + 3H(X_{2,6,7,9,13,14,19}) - 7H(X_{1,4,10,17,19}) + 7H(X_{1,3,4,7,10,17,19}) - 3H(X_{2,6,7,8,9,12,13,14,17,19}) + 7H(X_{1,4,8,9,10,13,15,17,19}) - 7H(X_{1,3,4,7,8,9,10,13,15,17,19}) - 7H(X_{3,4,5,6,7,8,9,11,13,14,15,18,19}) - 2H(X_{3,6,8,10,16,18,19}) + 6H(X_{20}) - 10H(X_{3,8,20}) - 4H(X_{10,20}) + 4H(X_{1,6,9,10,12,20}) - 6H(X_{14,20}) + 4H(X_{3,4,8,10,11,13,14,20}) - 4H(X_{1,3,4,6,8,9,10,11,12,13,14,20}) + 3H(X_{1,3,4,10,12,15,20}) - 7H(X_{1,2,3,5,6,9,14,15,20}) - 3H(X_{1,3,4,6,9,10,11,12,14,15,20}) + 7H(X_{1,2,3,5,6,9,13,14,15,20}) + 8H(X_{1,2,3,5,9,12,15,16,20}) - 8H(X_{1,2,3,4,5,9,11,12,15,16,20}) + 3H(X_{1,3,7,11,14,15,17,20}) + 5H(X_{3,5,6,8,10,12,13,15,18,20}) + 10H(X_{3,5,8,12,16,18,20}) + 6H(X_{1,2,4,5,6,7,8,9,11,13,15,16,17,18,20}) + H(X_{1,3,11,14,15,16,17,18,20}) - 3H(X_{1,2,3,7,10,11,14,15,16,17,18,20}) + 10H(X_{1,3,4,8,11,19,20}) + 9H(X_{1,2,5,6,8,11,12,14,15,16,19,20}) + 4H(X_{1,6,7,8,9,12,16,17,19,20}) - 4H(X_{1,6,7,8,9,12,14,16,17,19,20}) - 9H(X_{1,2,5,6,8,11,12,14,15,16,17,19,20}) - 10H(X_{1,3,4,5,8,11,12,16,18,19,20}) + 2H(X_{2,3,4,5,6,7,8,10,14,15,16,18,19,20}) - 6H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20})","witness_value":709},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{3})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{7};X_{10})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{14};X_{19})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{14};X_{20})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{1};X_{12})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{17};X_{18})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{16})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{7};X_{8})"}],"expression":[[1,2],[4,1],[5,1],[64,1],[128,3],[192,-3],[512,-2],[576,2],[2048,3],[2049,-3],[32768,2],[32772,-2],[65536,2],[131072,2],[196608,-2],[262144,2],[270336,-2],[524288,-2],[532480,2]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["2H(X_{1}) + H(X_{3}) + H(X_{1,3}) + H(X_{7}) + 3H(X_{8}) - 3H(X_{7,8}) - 2H(X_{10}) + 2H(X_{7,10}) + 3H(X_{12}) - 3H(X_{1,12}) + 2H(X_{16}) - 2H(X_{3,16}) + 2H(X_{17}) + 2H(X_{18}) - 2H(X_{17,18}) + 2H(X_{19}) - 2H(X_{14,19}) - 2H(X_{20}) + 2H(X_{14,20})>=0","-2H(X_{1}) - H(X_{3}) - H(X_{1,3}) - H(X_{7}) - 3H(X_{8}) + 3H(X_{7,8}) + 2H(X_{10}) - 2H(X_{7,10}) - 3H(X_{12}) + 3H(X_{1,12}) - 2H(X_{16}) + 2H(X_{3,16}) - 2H(X_{17}) - 2H(X_{18}) + 2H(X_{17,18}) - 2H(X_{19}) + 2H(X_{14,19}) + 2H(X_{20}) - 2H(X_{14,20})>=0"],"text":"2H(X_{1}) + H(X_{3}) + H(X_{1,3}) + H(X_{7}) + 3H(X_{8}) - 3H(X_{7,8}) - 2H(X_{10}) + 2H(X_{7,10}) + 3H(X_{12}) - 3H(X_{1,12}) + 2H(X_{16}) - 2H(X_{3,16}) + 2H(X_{17}) + 2H(X_{18}) - 2H(X_{17,18}) + 2H(X_{19}) - 2H(X_{14,19}) - 2H(X_{20}) + 2H(X_{14,20})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{5};X_{10})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{9};X_{13})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{13})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{11};X_{14})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{9})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{8})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{11};X_{12})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{7};X_{14})"}],"expression":[[8,1],[16,1],[32,2],[64,-2],[160,-2],[256,3],[264,-1],[512,1],[528,-1],[1024,-3],[2048,-1],[3072,1],[4224,2],[4352,-2],[8192,-4],[8256,2],[9216,2]],"id":1,"kind":"composite_equality","label":"witness 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7H(X_{3,4,5,7,8,10,12,15,16,17,18,19,20}) - 7H(X_{3,4,5,7,8,10,12,13,15,16,17,18,19,20}) - 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2H(X_{15,20})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{11};X_{17}|X_{19})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{14}|X_{12})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{10};X_{12}|X_{7})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{7};X_{11}|X_{4})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{15})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{18}|X_{10})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{5};X_{19}|X_{2})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{4};X_{7}|X_{6,12})"}],"expression":[[2,-1],[8,3],[18,1],[32,2],[64,-1],[72,-3],[512,-2],[576,1],[640,2],[1032,-3],[1096,3],[2048,-2],[2080,4],[2088,-2],[2112,1],[2144,-2],[2152,2],[2624,-1],[10240,2],[10272,-2],[16384,2],[16416,-2],[131584,2],[131712,-2],[262144,-3],[262146,1],[262162,-1],[263168,3],[327680,3],[328704,-3]],"id":1,"kind":"composite_equality","label":"witness 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2H(X_{1,3,5}) + 3H(X_{1,6}) - 3H(X_{1,5,6}) - 3H(X_{2,10}) + H(X_{3,5,10}) + 2H(X_{1,3,5,10}) + 3H(X_{2,6,10}) + 3H(X_{2,10,11}) - 3H(X_{2,6,10,11}) - 3H(X_{1,6,12}) + 3H(X_{1,5,6,12}) - 2H(X_{1,13}) - 2H(X_{3,13}) + 2H(X_{1,3,13}) + 2H(X_{5,13}) + 3H(X_{3,5,13}) + 2H(X_{7,13}) - 2H(X_{5,7,13}) - 3H(X_{8,13}) + 3H(X_{2,8,13}) + 3H(X_{5,8,13}) - 3H(X_{2,5,8,13}) - 3H(X_{9,13}) + 3H(X_{3,9,13}) - 3H(X_{3,5,10,13}) + 3H(X_{9,12,13}) - 3H(X_{3,9,12,13})>=0","H(X_{3,5}) + 2H(X_{1,3,5}) - 3H(X_{1,6}) + 3H(X_{1,5,6}) + 3H(X_{2,10}) - H(X_{3,5,10}) - 2H(X_{1,3,5,10}) - 3H(X_{2,6,10}) - 3H(X_{2,10,11}) + 3H(X_{2,6,10,11}) + 3H(X_{1,6,12}) - 3H(X_{1,5,6,12}) + 2H(X_{1,13}) + 2H(X_{3,13}) - 2H(X_{1,3,13}) - 2H(X_{5,13}) - 3H(X_{3,5,13}) - 2H(X_{7,13}) + 2H(X_{5,7,13}) + 3H(X_{8,13}) - 3H(X_{2,8,13}) - 3H(X_{5,8,13}) + 3H(X_{2,5,8,13}) + 3H(X_{9,13}) - 3H(X_{3,9,13}) + 3H(X_{3,5,10,13}) - 3H(X_{9,12,13}) + 3H(X_{3,9,12,13})>=0"],"text":"-H(X_{3,5}) - 2H(X_{1,3,5}) + 3H(X_{1,6}) - 3H(X_{1,5,6}) - 3H(X_{2,10}) + H(X_{3,5,10}) + 2H(X_{1,3,5,10}) + 3H(X_{2,6,10}) + 3H(X_{2,10,11}) - 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H(X_{2}) - H(X_{3}) - 5H(X_{1,3}) + H(X_{2,3}) - 14H(X_{5}) - 10H(X_{1,5}) + 8H(X_{2,5}) - H(X_{3,5}) + H(X_{3,6}) + 5H(X_{1,3,6}) - 12H(X_{5,6}) + 15H(X_{2,5,6}) + 2H(X_{7}) + 6H(X_{3,4,5,7}) - 5H(X_{8}) + 7H(X_{2,8}) + 10H(X_{5,8}) - 10H(X_{2,5,8}) - 17H(X_{9}) - 5H(X_{1,9}) + 18H(X_{4,9}) + 18H(X_{5,9}) + 11H(X_{1,5,9}) - 18H(X_{4,5,9}) - 5H(X_{1,10}) - 10H(X_{5,10}) - H(X_{3,5,10}) - H(X_{3,6,10}) + 10H(X_{5,7,10}) + 2H(X_{3,5,7,10}) + 4H(X_{9,10}) + 5H(X_{1,9,10}) + 2H(X_{5,9,10}) - 2H(X_{5,7,9,10}) + 2H(X_{11}) - 2H(X_{1,11}) + 15H(X_{5,6,11}) - 15H(X_{2,5,6,11}) + 6H(X_{1,5,8,9,11}) - 6H(X_{1,3,4,5,7,8,9,11}) + 11H(X_{12}) + 3H(X_{1,12}) + 3H(X_{8,12}) - 3H(X_{1,8,12}) - 4H(X_{9,10,12}) + H(X_{2,6,9,10,12}) + H(X_{13}) - 2H(X_{5,13}) + 2H(X_{2,5,13}) - 7H(X_{3,8,9,13}) + 2H(X_{2,3,4,5,6,8,10,13}) - 2H(X_{2,3,4,5,6,7,8,10,13}) + 6H(X_{4,11,13}) - 6H(X_{5,11,13}) + 6H(X_{5,6,11,13}) + 6H(X_{1,5,7,11,13}) - 6H(X_{1,5,6,7,11,13}) + 7H(X_{2,3,5,6,8,9,11,13}) - 5H(X_{4,5,12,13}) - H(X_{2,6,9,10,12,13}) + 5H(X_{2,3,4,5,6,7,11,12,13}) + 10H(X_{14}) + 5H(X_{1,3,14}) - 6H(X_{4,14}) + 6H(X_{1,3,4,14}) + 6H(X_{5,14}) + 10H(X_{1,5,14}) - 5H(X_{1,3,6,14}) + H(X_{2,3,7,14}) + 2H(X_{1,2,5,6,7,14}) - 2H(X_{1,2,5,6,7,8,14}) - 10H(X_{1,5,9,14}) - H(X_{2,3,7,9,14}) - 5H(X_{1,4,8,9,14}) + 5H(X_{1,2,4,5,6,7,8,9,14}) + 10H(X_{5,10,14}) - 10H(X_{5,7,10,14}) - 5H(X_{1,3,4,7,8,11,14}) + 5H(X_{1,3,4,6,7,8,9,11,14}) - 10H(X_{12,14}) + 7H(X_{3,4,6,7,8,9,12,14}) - 7H(X_{2,3,4,6,7,8,9,12,14}) - 9H(X_{10,12,14}) + 9H(X_{5,6,7,10,11,12,14}) + 6H(X_{4,11,13,14}) - 6H(X_{1,3,4,11,13,14}) - 6H(X_{4,5,11,13,14}) - 3H(X_{2,7,8,12,13,14}) + 9H(X_{10,12,13,14}) + 3H(X_{2,6,7,8,10,12,13,14}) - 9H(X_{5,6,7,10,11,12,13,14}) + 3H(X_{1,2,3,4,7,8,9,10,11,12,13,14}) - 3H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14})","witness_value":85.84838074595115},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[1.9984014443252818e-15,9.769962616701378e-15,1.1102230246251565e-15,1.5543122344752192e-15,-1.199040866595169e-14,6.661338147750939e-16,1.0769163338864018e-14,1.0658141036401503e-14],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{10};X_{5}|X_{3})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{13}|X_{5})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{10};X_{6}|X_{3})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{7};X_{9}|X_{5,10})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{3};X_{2})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{7};X_{3}|X_{5,10})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{5};X_{4}|X_{9})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{8};X_{1}|X_{12})"}],"expression":[[2,1],[4,1],[6,-1],[16,3],[18,-3],[20,1],[36,-1],[256,3],[264,-3],[272,-3],[280,3],[532,1],[548,1],[596,-2],[784,-2],[848,2],[2048,3],[2049,-3],[2176,-3],[2177,3],[4112,-3],[4114,3]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["H(X_{2}) + H(X_{3}) - H(X_{2,3}) + 3H(X_{5}) - 3H(X_{2,5}) + H(X_{3,5}) - H(X_{3,6}) + 3H(X_{9}) - 3H(X_{4,9}) - 3H(X_{5,9}) + 3H(X_{4,5,9}) + H(X_{3,5,10}) + H(X_{3,6,10}) - 2H(X_{3,5,7,10}) - 2H(X_{5,9,10}) + 2H(X_{5,7,9,10}) + 3H(X_{12}) - 3H(X_{1,12}) - 3H(X_{8,12}) + 3H(X_{1,8,12}) - 3H(X_{5,13}) + 3H(X_{2,5,13})>=0","-H(X_{2}) - H(X_{3}) + H(X_{2,3}) - 3H(X_{5}) + 3H(X_{2,5}) - H(X_{3,5}) + H(X_{3,6}) - 3H(X_{9}) + 3H(X_{4,9}) + 3H(X_{5,9}) - 3H(X_{4,5,9}) - H(X_{3,5,10}) - H(X_{3,6,10}) + 2H(X_{3,5,7,10}) + 2H(X_{5,9,10}) - 2H(X_{5,7,9,10}) - 3H(X_{12}) + 3H(X_{1,12}) + 3H(X_{8,12}) - 3H(X_{1,8,12}) + 3H(X_{5,13}) - 3H(X_{2,5,13})>=0"],"text":"H(X_{2}) + H(X_{3}) - H(X_{2,3}) + 3H(X_{5}) - 3H(X_{2,5}) + H(X_{3,5}) - H(X_{3,6}) + 3H(X_{9}) - 3H(X_{4,9}) - 3H(X_{5,9}) + 3H(X_{4,5,9}) + H(X_{3,5,10}) + H(X_{3,6,10}) - 2H(X_{3,5,7,10}) - 2H(X_{5,9,10}) + 2H(X_{5,7,9,10}) + 3H(X_{12}) - 3H(X_{1,12}) - 3H(X_{8,12}) + 3H(X_{1,8,12}) - 3H(X_{5,13}) + 3H(X_{2,5,13})","type":"equality","witness_value":-1.0613732115416497e-13},{"component_width":8,"component_witness_values":[8.881784197001252e-16,9.769962616701378e-15,7.549516567451064e-15,1.0769163338864018e-14,1.9984014443252818e-15,1.9984014443252818e-15,-1.8207657603852567e-14,1.2434497875801753e-14],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{14};X_{6}|X_{1,3})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{13}|X_{5})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{2};X_{8}|X_{5})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{5};X_{4}|X_{9})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{11};X_{2}|X_{5,6})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{7};X_{14}|X_{5,10})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{12};X_{14})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{9};X_{10}|X_{1})"}],"expression":[[1,1],[5,-1],[16,-1],[18,1],[37,1],[48,-3],[50,3],[144,2],[146,-2],[256,-3],[257,-1],[264,3],[272,3],[280,-3],[513,-1],[528,-2],[592,2],[769,1],[1072,3],[1074,-3],[2048,2],[4112,-1],[4114,1],[8192,2],[8197,1],[8229,-1],[8720,2],[8784,-2],[10240,-2]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["H(X_{1}) - H(X_{1,3}) - H(X_{5}) + H(X_{2,5}) + H(X_{1,3,6}) - 3H(X_{5,6}) + 3H(X_{2,5,6}) + 2H(X_{5,8}) - 2H(X_{2,5,8}) - 3H(X_{9}) - H(X_{1,9}) + 3H(X_{4,9}) + 3H(X_{5,9}) - 3H(X_{4,5,9}) - H(X_{1,10}) - 2H(X_{5,10}) + 2H(X_{5,7,10}) + H(X_{1,9,10}) + 3H(X_{5,6,11}) - 3H(X_{2,5,6,11}) + 2H(X_{12}) - H(X_{5,13}) + H(X_{2,5,13}) + 2H(X_{14}) + H(X_{1,3,14}) - H(X_{1,3,6,14}) + 2H(X_{5,10,14}) - 2H(X_{5,7,10,14}) - 2H(X_{12,14})>=0","-H(X_{1}) + H(X_{1,3}) + H(X_{5}) - H(X_{2,5}) - H(X_{1,3,6}) + 3H(X_{5,6}) - 3H(X_{2,5,6}) - 2H(X_{5,8}) + 2H(X_{2,5,8}) + 3H(X_{9}) + H(X_{1,9}) - 3H(X_{4,9}) - 3H(X_{5,9}) + 3H(X_{4,5,9}) + H(X_{1,10}) + 2H(X_{5,10}) - 2H(X_{5,7,10}) - H(X_{1,9,10}) - 3H(X_{5,6,11}) + 3H(X_{2,5,6,11}) - 2H(X_{12}) + H(X_{5,13}) - H(X_{2,5,13}) - 2H(X_{14}) - H(X_{1,3,14}) + H(X_{1,3,6,14}) - 2H(X_{5,10,14}) + 2H(X_{5,7,10,14}) + 2H(X_{12,14})>=0"],"text":"H(X_{1}) - H(X_{1,3}) - H(X_{5}) + H(X_{2,5}) + H(X_{1,3,6}) - 3H(X_{5,6}) + 3H(X_{2,5,6}) + 2H(X_{5,8}) - 2H(X_{2,5,8}) - 3H(X_{9}) - 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2H(X_{2,3,6}) - H(X_{1,4,7}) - 2H(X_{2,6,8}) + 2H(X_{2,3,6,8}) + 3H(X_{4,9}) - 3H(X_{1,4,9}) - 3H(X_{4,6,9}) + 3H(X_{1,4,6,9}) + 2H(X_{9,10}) - 2H(X_{7,9,10}) + 3H(X_{12}) - 3H(X_{9,12}) - 3H(X_{10,12}) + 3H(X_{9,10,12}) - 3H(X_{1,13}) + 2H(X_{1,2,13}) + H(X_{1,3,13}) - H(X_{1,4,13}) + 3H(X_{1,5,13}) - 2H(X_{1,2,5,13}) - H(X_{1,3,5,13}) + H(X_{1,4,7,13}) - 2H(X_{9,10,14}) + 2H(X_{7,9,10,14}) - 3H(X_{11,14}) + 3H(X_{4,11,14}) + 3H(X_{10,11,14}) - 3H(X_{4,10,11,14})>=0","-H(X_{1,4}) - 2H(X_{2,6}) + 2H(X_{2,3,6}) + H(X_{1,4,7}) + 2H(X_{2,6,8}) - 2H(X_{2,3,6,8}) - 3H(X_{4,9}) + 3H(X_{1,4,9}) + 3H(X_{4,6,9}) - 3H(X_{1,4,6,9}) - 2H(X_{9,10}) + 2H(X_{7,9,10}) - 3H(X_{12}) + 3H(X_{9,12}) + 3H(X_{10,12}) - 3H(X_{9,10,12}) + 3H(X_{1,13}) - 2H(X_{1,2,13}) - H(X_{1,3,13}) + H(X_{1,4,13}) - 3H(X_{1,5,13}) + 2H(X_{1,2,5,13}) + H(X_{1,3,5,13}) - H(X_{1,4,7,13}) + 2H(X_{9,10,14}) - 2H(X_{7,9,10,14}) + 3H(X_{11,14}) - 3H(X_{4,11,14}) - 3H(X_{10,11,14}) + 3H(X_{4,10,11,14})>=0"],"text":"H(X_{1,4}) + 2H(X_{2,6}) - 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H(X_{1,3}) - 2H(X_{4}) + 3H(X_{2,4}) + H(X_{5}) - 2H(X_{1,2,6}) - 3H(X_{1,8}) - 3H(X_{2,10}) + 2H(X_{1,2,6,11}) - H(X_{1,12}) + 3H(X_{2,12}) + H(X_{1,3,12}) - 3H(X_{2,4,12}) - H(X_{13}) + 2H(X_{4,13}) - H(X_{5,13}) + 3H(X_{14}) - 3H(X_{2,14}) + 3H(X_{2,10,14}) - 3H(X_{11,14}) - 3H(X_{13,14}) + 3H(X_{11,13,14}) - 3H(X_{1,15}) + 2H(X_{1,2,6,15}) + 3H(X_{1,8,15}) - 2H(X_{1,2,6,11,15})","witness_value":-1.5654144647214707e-14}],"certificate_length":22,"certificate_order":"greedy_negative_mass_condition_late","condition_coordinate_supports":[28,26],"condition_rank":2,"condition_tokens":[302,302,289,290],"condition_width":8,"config_sha256":"1b1f298ed012071c85e9cfb286fe5dcf7c183383de92041754d8a7802f426421","equality_condition_count":2,"full_tokens":1942,"generator_dependency_sha256":{"data/finite_field.py":"c9738c10a16ea2cfb3cde346a23673d1536dbfce491f8871346f441b9b4c2a6e","data/witness_first.py":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","data/witness_v2_algebraic.py":"ffbd48eb673a3f6f842327d00431285bbae8963a7e20f6956e37e544fa37165f","data/witness_v2_graph.py":"c81adcfefcf6ffc0e816f1b338af13323bc80e06293e8cac6b28a3b0ae2efc8a"},"generator_schema_version":"witness-first-v2","generator_source_sha256":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","id":"e2n-witness-first-v2-n15-dag_v2-6eca5382dd600113","initial_target":{"coordinate_support":97,"effective_n":15,"expression":[[1,-3],[2,4],[4,-5],[5,-1],[8,-2],[10,7],[13,7],[16,4],[20,1],[32,12],[34,-8],[35,2],[129,-3],[193,2],[291,-4],[512,7],[514,-11],[520,-8],[536,8],[546,8],[584,8],[600,-8],[1033,8],[1059,2],[1245,1],[2049,-1],[2050,7],[2053,1],[2058,-7],[2304,-17],[2305,8],[2306,9],[2368,8],[2369,-8],[2445,-3],[2578,8],[3611,-8],[4096,3],[4100,4],[4104,2],[4112,-1],[4128,-8],[4675,-1],[4833,10],[5256,-3],[5457,4],[5489,-4],[6723,1],[7360,9],[7576,3],[8192,11],[8194,-3],[8244,7],[8320,-4],[8706,3],[8756,-7],[9216,-7],[9344,4],[9377,3],[9393,-3],[9934,1],[9950,-1],[12288,-3],[12320,4],[13312,3],[15680,-1],[16385,-3],[16389,9],[16419,-2],[16513,3],[16675,4],[17443,-2],[17923,4],[18064,-6],[18100,6],[18432,2],[19328,9],[19330,-9],[21203,1],[21705,2],[21798,-7],[22480,6],[22516,-6],[23251,-1],[23749,-9],[24721,7],[24733,-7],[26624,-2],[27648,2],[27841,-2],[28077,3],[29897,-2],[30243,-4],[30510,7],[31522,1],[32205,1],[32767,-1]],"text":"-3H(X_{1}) + 4H(X_{2}) - 5H(X_{3}) - H(X_{1,3}) - 2H(X_{4}) + 7H(X_{2,4}) + 7H(X_{1,3,4}) + 4H(X_{5}) + H(X_{3,5}) + 12H(X_{6}) - 8H(X_{2,6}) + 2H(X_{1,2,6}) - 3H(X_{1,8}) + 2H(X_{1,7,8}) - 4H(X_{1,2,6,9}) + 7H(X_{10}) - 11H(X_{2,10}) - 8H(X_{4,10}) + 8H(X_{4,5,10}) + 8H(X_{2,6,10}) + 8H(X_{4,7,10}) - 8H(X_{4,5,7,10}) + 8H(X_{1,4,11}) + 2H(X_{1,2,6,11}) + H(X_{1,3,4,5,7,8,11}) - H(X_{1,12}) + 7H(X_{2,12}) + H(X_{1,3,12}) - 7H(X_{2,4,12}) - 17H(X_{9,12}) + 8H(X_{1,9,12}) + 9H(X_{2,9,12}) + 8H(X_{7,9,12}) - 8H(X_{1,7,9,12}) - 3H(X_{1,3,4,8,9,12}) + 8H(X_{2,5,10,12}) - 8H(X_{1,2,4,5,10,11,12}) + 3H(X_{13}) + 4H(X_{3,13}) + 2H(X_{4,13}) - H(X_{5,13}) - 8H(X_{6,13}) - H(X_{1,2,7,10,13}) + 10H(X_{1,6,7,8,10,13}) - 3H(X_{4,8,11,13}) + 4H(X_{1,5,7,9,11,13}) - 4H(X_{1,5,6,7,9,11,13}) + H(X_{1,2,7,10,12,13}) + 9H(X_{7,8,11,12,13}) + 3H(X_{4,5,8,9,11,12,13}) + 11H(X_{14}) - 3H(X_{2,14}) + 7H(X_{3,5,6,14}) - 4H(X_{8,14}) + 3H(X_{2,10,14}) - 7H(X_{3,5,6,10,14}) - 7H(X_{11,14}) + 4H(X_{8,11,14}) + 3H(X_{1,6,8,11,14}) - 3H(X_{1,5,6,8,11,14}) + H(X_{2,3,4,7,8,10,11,14}) - H(X_{2,3,4,5,7,8,10,11,14}) - 3H(X_{13,14}) + 4H(X_{6,13,14}) + 3H(X_{11,13,14}) - H(X_{7,9,11,12,13,14}) - 3H(X_{1,15}) + 9H(X_{1,3,15}) - 2H(X_{1,2,6,15}) + 3H(X_{1,8,15}) + 4H(X_{1,2,6,9,15}) - 2H(X_{1,2,6,11,15}) + 4H(X_{1,2,10,11,15}) - 6H(X_{5,8,10,11,15}) + 6H(X_{3,5,6,8,10,11,15}) + 2H(X_{12,15}) + 9H(X_{8,9,10,12,15}) - 9H(X_{2,8,9,10,12,15}) + H(X_{1,2,5,7,8,10,13,15}) + 2H(X_{1,4,7,8,11,13,15}) - 7H(X_{2,3,6,9,11,13,15}) + 6H(X_{5,7,8,9,10,11,13,15}) - 6H(X_{3,5,6,7,8,9,10,11,13,15}) - H(X_{1,2,5,7,8,10,12,13,15}) - 9H(X_{1,3,7,8,11,12,13,15}) + 7H(X_{1,5,8,14,15}) - 7H(X_{1,3,4,5,8,14,15}) - 2H(X_{12,14,15}) + 2H(X_{11,12,14,15}) - 2H(X_{1,7,8,11,12,14,15}) + 3H(X_{1,3,4,6,8,9,11,12,14,15}) - 2H(X_{1,4,7,8,11,13,14,15}) - 4H(X_{1,2,6,10,11,13,14,15}) + 7H(X_{2,3,4,6,9,10,11,13,14,15}) + H(X_{2,6,9,10,12,13,14,15}) + H(X_{1,3,4,7,8,9,11,12,13,14,15}) - H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15})","witness_value":98.09823222147722},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[-1.6653345369377348e-15,-8.881784197001252e-16,-4.3298697960381105e-15,1.5543122344752192e-15,7.771561172376096e-16,1.3322676295501878e-15,3.3306690738754696e-16,-5.10702591327572e-15],"components":[{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{12}|X_{2})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{13})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{10};X_{6}|X_{2})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{7};X_{1}|X_{9,12})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{11};X_{8}|X_{14})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{15};X_{9}|X_{1,2,6})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{13};X_{3})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{7}|X_{4,10})"}],"expression":[[2,-1],[4,1],[10,-1],[32,-2],[34,2],[35,-1],[291,1],[514,2],[520,2],[536,-2],[546,-2],[584,-2],[600,2],[2050,-1],[2058,1],[2304,2],[2305,-2],[2368,-2],[2369,2],[4096,-1],[4100,-1],[4128,2],[8192,-1],[8320,1],[9216,1],[9344,-1],[16419,1],[16675,-1]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["-H(X_{2}) + H(X_{3}) - H(X_{2,4}) - 2H(X_{6}) + 2H(X_{2,6}) - H(X_{1,2,6}) + H(X_{1,2,6,9}) + 2H(X_{2,10}) + 2H(X_{4,10}) - 2H(X_{4,5,10}) - 2H(X_{2,6,10}) - 2H(X_{4,7,10}) + 2H(X_{4,5,7,10}) - H(X_{2,12}) + H(X_{2,4,12}) + 2H(X_{9,12}) - 2H(X_{1,9,12}) - 2H(X_{7,9,12}) + 2H(X_{1,7,9,12}) - H(X_{13}) - H(X_{3,13}) + 2H(X_{6,13}) - H(X_{14}) + H(X_{8,14}) + H(X_{11,14}) - H(X_{8,11,14}) + H(X_{1,2,6,15}) - H(X_{1,2,6,9,15})>=0","H(X_{2}) - H(X_{3}) + H(X_{2,4}) + 2H(X_{6}) - 2H(X_{2,6}) + H(X_{1,2,6}) - H(X_{1,2,6,9}) - 2H(X_{2,10}) - 2H(X_{4,10}) + 2H(X_{4,5,10}) + 2H(X_{2,6,10}) + 2H(X_{4,7,10}) - 2H(X_{4,5,7,10}) + H(X_{2,12}) - H(X_{2,4,12}) - 2H(X_{9,12}) + 2H(X_{1,9,12}) + 2H(X_{7,9,12}) - 2H(X_{1,7,9,12}) + H(X_{13}) + H(X_{3,13}) - 2H(X_{6,13}) + H(X_{14}) - H(X_{8,14}) - H(X_{11,14}) + H(X_{8,11,14}) - H(X_{1,2,6,15}) + H(X_{1,2,6,9,15})>=0"],"text":"-H(X_{2}) + H(X_{3}) - H(X_{2,4}) - 2H(X_{6}) + 2H(X_{2,6}) - H(X_{1,2,6}) + H(X_{1,2,6,9}) + 2H(X_{2,10}) + 2H(X_{4,10}) - 2H(X_{4,5,10}) - 2H(X_{2,6,10}) - 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H(X_{1,3}) - 2H(X_{4}) + 3H(X_{2,4}) + H(X_{5}) - 2H(X_{1,2,6}) - 3H(X_{1,8}) - 3H(X_{2,10}) + 2H(X_{1,2,6,11}) - H(X_{1,12}) + 3H(X_{2,12}) + H(X_{1,3,12}) - 3H(X_{2,4,12}) - H(X_{13}) + 2H(X_{4,13}) - H(X_{5,13}) + 3H(X_{14}) - 3H(X_{2,14}) + 3H(X_{2,10,14}) - 3H(X_{11,14}) - 3H(X_{13,14}) + 3H(X_{11,13,14}) - 3H(X_{1,15}) + 2H(X_{1,2,6,15}) + 3H(X_{1,8,15}) - 2H(X_{1,2,6,11,15})>=0","-4H(X_{1}) + H(X_{1,3}) + 2H(X_{4}) - 3H(X_{2,4}) - H(X_{5}) + 2H(X_{1,2,6}) + 3H(X_{1,8}) + 3H(X_{2,10}) - 2H(X_{1,2,6,11}) + H(X_{1,12}) - 3H(X_{2,12}) - H(X_{1,3,12}) + 3H(X_{2,4,12}) + H(X_{13}) - 2H(X_{4,13}) + H(X_{5,13}) - 3H(X_{14}) + 3H(X_{2,14}) - 3H(X_{2,10,14}) + 3H(X_{11,14}) + 3H(X_{13,14}) - 3H(X_{11,13,14}) + 3H(X_{1,15}) - 2H(X_{1,2,6,15}) - 3H(X_{1,8,15}) + 2H(X_{1,2,6,11,15})>=0"],"text":"4H(X_{1}) - H(X_{1,3}) - 2H(X_{4}) + 3H(X_{2,4}) + H(X_{5}) - 2H(X_{1,2,6}) - 3H(X_{1,8}) - 3H(X_{2,10}) + 2H(X_{1,2,6,11}) - H(X_{1,12}) + 3H(X_{2,12}) + H(X_{1,3,12}) - 3H(X_{2,4,12}) - H(X_{13}) + 2H(X_{4,13}) - 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9H(X_{3,6}) - 29H(X_{7}) - 9H(X_{3,7}) + 21H(X_{6,7}) + 9H(X_{3,6,7}) + H(X_{5,6,7}) + 4H(X_{8}) - 4H(X_{2,3,6,8}) + 2H(X_{7,8}) - 10H(X_{9}) + 27H(X_{7,10}) + 10H(X_{9,10}) + 4H(X_{2,3,6,8,11}) - 3H(X_{12}) + 5H(X_{1,12}) - 34H(X_{6,12}) + 27H(X_{3,6,12}) + 34H(X_{4,6,12}) - 27H(X_{3,4,6,12}) + 9H(X_{2,8,9,12}) + 14H(X_{6,10,12}) + 3H(X_{1,2,5,8,10,12}) - 14H(X_{6,8,10,12}) + H(X_{1,2,5,6,7,9,10,12}) - H(X_{1,2,3,5,6,7,9,10,12}) + 3H(X_{11,12}) - 21H(X_{4,10,11,12}) + 21H(X_{2,4,10,11,12}) - 3H(X_{1,2,5,8,10,11,12}) - 3H(X_{13}) + 4H(X_{2,13}) + 21H(X_{3,13}) + 18H(X_{1,3,13}) - 18H(X_{1,3,5,13}) - 21H(X_{7,13}) - 4H(X_{2,8,13}) - 5H(X_{1,3,7,9,13}) + 3H(X_{10,13}) + H(X_{3,10,13}) - H(X_{3,5,6,7,10,13}) - 6H(X_{5,7,8,10,13}) - 8H(X_{2,4,7,12,13}) - 14H(X_{6,10,12,13}) + 14H(X_{6,8,10,12,13}) + 6H(X_{2,4,5,7,8,10,11,12,13}) - 18H(X_{5,14}) + 18H(X_{5,6,14}) + 16H(X_{6,7,14}) + 4H(X_{1,2,3,6,8,14}) - 16H(X_{6,7,10,14}) + 18H(X_{5,11,14}) - 18H(X_{5,6,11,14}) - 4H(X_{1,2,3,6,8,11,14}) + 7H(X_{6,12,14}) - 7H(X_{4,6,12,14}) - 7H(X_{6,7,13,14}) + 3H(X_{3,4,6,7,9,13,14}) + 7H(X_{6,7,10,13,14}) - 3H(X_{3,4,6,7,9,10,13,14}) + 7H(X_{2,3,4,5,7,8,11,13,14}) - 18H(X_{5,15}) + 18H(X_{1,5,15}) + 48H(X_{7,15}) - 21H(X_{6,7,15}) + 18H(X_{5,9,15}) - 18H(X_{1,5,9,15}) + 10H(X_{3,6,9,15}) - 23H(X_{7,10,15}) - 4H(X_{2,5,7,10,15}) - 10H(X_{3,6,9,10,15}) - H(X_{2,3,6,7,12,15}) + H(X_{2,3,4,6,7,12,15}) + 7H(X_{1,6,9,10,12,15}) + H(X_{2,3,6,7,11,12,15}) - H(X_{2,3,4,6,7,11,12,15}) + 21H(X_{10,11,12,15}) - 21H(X_{2,10,11,12,15}) - 18H(X_{1,3,13,15}) + 18H(X_{1,3,5,13,15}) + 5H(X_{2,3,4,5,9,13,15}) + 8H(X_{2,4,6,7,12,13,15}) - 5H(X_{1,2,3,4,5,9,12,13,15}) - 18H(X_{5,14,15}) - 9H(X_{6,7,14,15}) + 18H(X_{5,10,14,15}) + 2H(X_{2,3,7,10,14,15}) + 9H(X_{6,7,10,14,15}) - 2H(X_{2,3,7,8,10,14,15}) + 18H(X_{5,11,14,15}) - 18H(X_{5,10,11,14,15}) + 5H(X_{1,2,3,5,6,7,9,10,12,13,14,15}) - 7H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15})","witness_value":107.43693008150909},"k":16,"logical_conditions":[{"component_width":8,"component_witness_values":[3.552713678800501e-15,-2.864375403532904e-14,3.552713678800501e-15,4.440892098500626e-16,1.7319479184152442e-14,1.7763568394002505e-15,2.220446049250313e-15,-2.1760371282653068e-14],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{10};X_{13}|X_{6,7,14})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{13};X_{7})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{13}|X_{6,10,12})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{14}|X_{6,12})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{15};X_{6}|X_{7})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{4}|X_{10,11,12})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{15}|X_{10,11,12})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{13})"}],"expression":[[4,3],[96,-3],[2080,1],[2088,-1],[2592,-2],[2720,2],[3592,3],[3594,-3],[4100,-3],[4160,3],[6688,2],[6816,-2],[8288,-1],[8800,1],[10272,-1],[10280,1],[12384,1],[12896,-1],[16448,-3],[16480,3],[19968,-3],[19970,3]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["3H(X_{3}) - 3H(X_{6,7}) + H(X_{6,12}) - H(X_{4,6,12}) - 2H(X_{6,10,12}) + 2H(X_{6,8,10,12}) + 3H(X_{4,10,11,12}) - 3H(X_{2,4,10,11,12}) - 3H(X_{3,13}) + 3H(X_{7,13}) + 2H(X_{6,10,12,13}) - 2H(X_{6,8,10,12,13}) - H(X_{6,7,14}) + H(X_{6,7,10,14}) - H(X_{6,12,14}) + H(X_{4,6,12,14}) + H(X_{6,7,13,14}) - H(X_{6,7,10,13,14}) - 3H(X_{7,15}) + 3H(X_{6,7,15}) - 3H(X_{10,11,12,15}) + 3H(X_{2,10,11,12,15})>=0","-3H(X_{3}) + 3H(X_{6,7}) - H(X_{6,12}) + H(X_{4,6,12}) + 2H(X_{6,10,12}) - 2H(X_{6,8,10,12}) - 3H(X_{4,10,11,12}) + 3H(X_{2,4,10,11,12}) + 3H(X_{3,13}) - 3H(X_{7,13}) - 2H(X_{6,10,12,13}) + 2H(X_{6,8,10,12,13}) + H(X_{6,7,14}) - H(X_{6,7,10,14}) + H(X_{6,12,14}) - H(X_{4,6,12,14}) - H(X_{6,7,13,14}) + H(X_{6,7,10,13,14}) + 3H(X_{7,15}) - 3H(X_{6,7,15}) + 3H(X_{10,11,12,15}) - 3H(X_{2,10,11,12,15})>=0"],"text":"3H(X_{3}) - 3H(X_{6,7}) + H(X_{6,12}) - H(X_{4,6,12}) - 2H(X_{6,10,12}) + 2H(X_{6,8,10,12}) + 3H(X_{4,10,11,12}) - 3H(X_{2,4,10,11,12}) - 3H(X_{3,13}) + 3H(X_{7,13}) + 2H(X_{6,10,12,13}) - 2H(X_{6,8,10,12,13}) - 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3H(X_{4,7,9}) - 18H(X_{10}) + 9H(X_{1,10}) + 3H(X_{2,10}) - 9H(X_{1,7,13}) + 9H(X_{1,3,7,13}) + 6H(X_{4,7,13}) - 6H(X_{2,4,7,13}) + 9H(X_{1,7,12,13}) - 9H(X_{1,3,7,12,13}) - 6H(X_{4,7,12,13}) + 6H(X_{2,4,7,12,13}) + 18H(X_{10,15}) - 9H(X_{1,10,15}) - 3H(X_{2,10,15}) - 6H(X_{2,16}) + 6H(X_{2,5,16}) + 6H(X_{2,6,16}) - 6H(X_{2,5,6,16}) + 6H(X_{10,16}) - 9H(X_{2,15,16}) + 9H(X_{2,9,15,16}) - 6H(X_{10,15,16}) + 9H(X_{2,13,15,16}) - 9H(X_{2,9,13,15,16})","witness_value":9.603429163007604e-14}],"certificate_length":22,"certificate_order":"greedy_negative_mass_condition_late","condition_coordinate_supports":[28,28],"condition_rank":2,"condition_tokens":[369,369,339,339],"condition_width":8,"config_sha256":"1b1f298ed012071c85e9cfb286fe5dcf7c183383de92041754d8a7802f426421","equality_condition_count":2,"full_tokens":2197,"generator_dependency_sha256":{"data/finite_field.py":"c9738c10a16ea2cfb3cde346a23673d1536dbfce491f8871346f441b9b4c2a6e","data/witness_first.py":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","data/witness_v2_algebraic.py":"ffbd48eb673a3f6f842327d00431285bbae8963a7e20f6956e37e544fa37165f","data/witness_v2_graph.py":"c81adcfefcf6ffc0e816f1b338af13323bc80e06293e8cac6b28a3b0ae2efc8a"},"generator_schema_version":"witness-first-v2","generator_source_sha256":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","id":"e2n-witness-first-v2-n16-dag_v2-f4c442edb00719db","initial_target":{"coordinate_support":108,"effective_n":16,"expression":[[1,-7],[8,-19],[9,17],[16,-1],[32,39],[34,-30],[64,-3],[128,6],[136,10],[137,-10],[160,-30],[162,30],[256,-3],[264,3],[272,7],[320,3],[328,-3],[512,-48],[513,9],[514,3],[520,-8],[1024,-10],[1025,30],[1026,-20],[1536,30],[1864,8],[2066,1],[4161,11],[4165,-11],[4168,26],[4170,-6],[4424,-10],[4546,-8],[4956,-1],[5120,10],[5121,-30],[5122,20],[5450,9],[5575,8],[6209,9],[6213,-9],[6216,-26],[6218,6],[6472,10],[8192,3],[8200,9],[8224,-3],[12353,-20],[12357,20],[12360,-10],[12657,3],[13642,-9],[14408,10],[16384,3],[16400,1],[16656,-1],[16783,-6],[16896,48],[16897,-9],[16898,-3],[17920,-30],[18450,-1],[18504,3],[20264,6],[20392,-6],[20912,1],[23535,6],[24859,1],[29115,-1],[29122,8],[29399,3],[30151,-8],[31097,-3],[32768,-6],[32770,-6],[32786,6],[32802,6],[32818,-6],[33280,6],[33713,7],[33721,-7],[34354,-6],[34817,6],[36136,3],[36635,3],[38024,-8],[42742,6],[44791,-6],[48008,6],[48040,-6],[49154,-9],[49410,9],[49664,-6],[50738,6],[51330,-9],[53019,-3],[53250,9],[53347,7],[53506,-9],[53619,-7],[55686,9],[55868,8],[57182,1],[57212,-8],[57932,-8],[58060,8],[63417,8],[65535,-3]],"text":"-7H(X_{1}) - 19H(X_{4}) + 17H(X_{1,4}) - H(X_{5}) + 39H(X_{6}) - 30H(X_{2,6}) - 3H(X_{7}) + 6H(X_{8}) + 10H(X_{4,8}) - 10H(X_{1,4,8}) - 30H(X_{6,8}) + 30H(X_{2,6,8}) - 3H(X_{9}) + 3H(X_{4,9}) + 7H(X_{5,9}) + 3H(X_{7,9}) - 3H(X_{4,7,9}) - 48H(X_{10}) + 9H(X_{1,10}) + 3H(X_{2,10}) - 8H(X_{4,10}) - 10H(X_{11}) + 30H(X_{1,11}) - 20H(X_{2,11}) + 30H(X_{10,11}) + 8H(X_{4,7,9,10,11}) + H(X_{2,5,12}) + 11H(X_{1,7,13}) - 11H(X_{1,3,7,13}) + 26H(X_{4,7,13}) - 6H(X_{2,4,7,13}) - 10H(X_{4,7,9,13}) - 8H(X_{2,7,8,9,13}) - H(X_{3,4,5,7,9,10,13}) + 10H(X_{11,13}) - 30H(X_{1,11,13}) + 20H(X_{2,11,13}) + 9H(X_{2,4,7,9,11,13}) + 8H(X_{1,2,3,7,8,9,11,13}) + 9H(X_{1,7,12,13}) - 9H(X_{1,3,7,12,13}) - 26H(X_{4,7,12,13}) + 6H(X_{2,4,7,12,13}) + 10H(X_{4,7,9,12,13}) + 3H(X_{14}) + 9H(X_{4,14}) - 3H(X_{6,14}) - 20H(X_{1,7,13,14}) + 20H(X_{1,3,7,13,14}) - 10H(X_{4,7,13,14}) + 3H(X_{1,5,6,7,9,13,14}) - 9H(X_{2,4,7,9,11,13,14}) + 10H(X_{4,7,12,13,14}) + 3H(X_{15}) + H(X_{5,15}) - H(X_{5,9,15}) - 6H(X_{1,2,3,4,8,9,15}) + 48H(X_{10,15}) - 9H(X_{1,10,15}) - 3H(X_{2,10,15}) - 30H(X_{10,11,15}) - H(X_{2,5,12,15}) + 3H(X_{4,7,12,15}) + 6H(X_{4,6,9,10,11,12,15}) - 6H(X_{4,6,8,9,10,11,12,15}) + H(X_{5,6,8,9,13,15}) + 6H(X_{1,2,3,4,6,7,8,9,10,12,13,15}) + H(X_{1,2,4,5,9,14,15}) - H(X_{1,2,4,5,6,8,9,13,14,15}) + 8H(X_{2,7,8,9,13,14,15}) + 3H(X_{1,2,3,5,7,8,10,13,14,15}) - 8H(X_{1,2,3,7,8,9,11,13,14,15}) - 3H(X_{1,4,5,6,7,9,12,13,14,15}) - 6H(X_{16}) - 6H(X_{2,16}) + 6H(X_{2,5,16}) + 6H(X_{2,6,16}) - 6H(X_{2,5,6,16}) + 6H(X_{10,16}) + 7H(X_{1,5,6,8,9,10,16}) - 7H(X_{1,4,5,6,8,9,10,16}) - 6H(X_{2,5,6,10,11,16}) + 6H(X_{1,12,16}) + 3H(X_{4,6,9,11,12,16}) + 3H(X_{1,2,4,5,9,10,11,12,16}) - 8H(X_{4,8,11,13,16}) + 6H(X_{2,3,5,6,7,8,10,11,14,16}) - 6H(X_{1,2,3,5,6,7,8,10,11,12,14,16}) + 6H(X_{4,8,9,10,12,13,14,16}) - 6H(X_{4,6,8,9,10,12,13,14,16}) - 9H(X_{2,15,16}) + 9H(X_{2,9,15,16}) - 6H(X_{10,15,16}) + 6H(X_{2,5,6,10,11,15,16}) - 9H(X_{2,8,12,15,16}) - 3H(X_{1,2,4,5,9,10,11,12,15,16}) + 9H(X_{2,13,15,16}) + 7H(X_{1,2,6,7,13,15,16}) - 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H(X_{4,9}) - H(X_{7,9}) + H(X_{4,7,9}) + 6H(X_{10}) - 3H(X_{1,10}) - H(X_{2,10}) + 3H(X_{1,7,13}) - 3H(X_{1,3,7,13}) - 2H(X_{4,7,13}) + 2H(X_{2,4,7,13}) - 3H(X_{1,7,12,13}) + 3H(X_{1,3,7,12,13}) + 2H(X_{4,7,12,13}) - 2H(X_{2,4,7,12,13}) - 6H(X_{10,15}) + 3H(X_{1,10,15}) + H(X_{2,10,15}) + 2H(X_{2,16}) - 2H(X_{2,5,16}) - 2H(X_{2,6,16}) + 2H(X_{2,5,6,16}) - 2H(X_{10,16}) + 3H(X_{2,15,16}) - 3H(X_{2,9,15,16}) + 2H(X_{10,15,16}) - 3H(X_{2,13,15,16}) + 3H(X_{2,9,13,15,16})>=0","-H(X_{9}) + H(X_{4,9}) + H(X_{7,9}) - H(X_{4,7,9}) - 6H(X_{10}) + 3H(X_{1,10}) + H(X_{2,10}) - 3H(X_{1,7,13}) + 3H(X_{1,3,7,13}) + 2H(X_{4,7,13}) - 2H(X_{2,4,7,13}) + 3H(X_{1,7,12,13}) - 3H(X_{1,3,7,12,13}) - 2H(X_{4,7,12,13}) + 2H(X_{2,4,7,12,13}) + 6H(X_{10,15}) - 3H(X_{1,10,15}) - H(X_{2,10,15}) - 2H(X_{2,16}) + 2H(X_{2,5,16}) + 2H(X_{2,6,16}) - 2H(X_{2,5,6,16}) + 2H(X_{10,16}) - 3H(X_{2,15,16}) + 3H(X_{2,9,15,16}) - 2H(X_{10,15,16}) + 3H(X_{2,13,15,16}) - 3H(X_{2,9,13,15,16})>=0"],"text":"H(X_{9}) - H(X_{4,9}) - H(X_{7,9}) + H(X_{4,7,9}) + 6H(X_{10}) - 3H(X_{1,10}) - H(X_{2,10}) + 3H(X_{1,7,13}) - 3H(X_{1,3,7,13}) - 2H(X_{4,7,13}) + 2H(X_{2,4,7,13}) - 3H(X_{1,7,12,13}) + 3H(X_{1,3,7,12,13}) + 2H(X_{4,7,12,13}) - 2H(X_{2,4,7,12,13}) - 6H(X_{10,15}) + 3H(X_{1,10,15}) + H(X_{2,10,15}) + 2H(X_{2,16}) - 2H(X_{2,5,16}) - 2H(X_{2,6,16}) + 2H(X_{2,5,6,16}) - 2H(X_{10,16}) + 3H(X_{2,15,16}) - 3H(X_{2,9,15,16}) + 2H(X_{10,15,16}) - 3H(X_{2,13,15,16}) + 3H(X_{2,9,13,15,16})","type":"equality","witness_value":9.603429163007604e-14},{"component_width":8,"component_witness_values":[2.6645352591003757e-15,-1.4432899320127035e-14,-1.8207657603852567e-14,-2.275957200481571e-14,-2.120525977034049e-14,-1.765254609153999e-14,4.884981308350689e-15,4.440892098500626e-15],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{12};X_{14}|X_{4,7,13})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{8}|X_{6})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{8}|X_{4})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{13};X_{2}|X_{11})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{13};X_{1}|X_{11})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{15};X_{11}|X_{10})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{14}|X_{1,7,13})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{12};X_{9}|X_{4,7,13})"}],"expression":[[8,1],[9,-1],[32,-3],[34,3],[136,-1],[137,1],[160,3],[162,-3],[512,3],[1024,1],[1025,-3],[1026,2],[1536,-3],[4161,-2],[4165,2],[4168,-2],[4424,1],[5120,-1],[5121,3],[5122,-2],[6216,2],[6472,-1],[12353,2],[12357,-2],[12360,1],[14408,-1],[16896,-3],[17920,3]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["H(X_{4}) - H(X_{1,4}) - 3H(X_{6}) + 3H(X_{2,6}) - H(X_{4,8}) + H(X_{1,4,8}) + 3H(X_{6,8}) - 3H(X_{2,6,8}) + 3H(X_{10}) + H(X_{11}) - 3H(X_{1,11}) + 2H(X_{2,11}) - 3H(X_{10,11}) - 2H(X_{1,7,13}) + 2H(X_{1,3,7,13}) - 2H(X_{4,7,13}) + H(X_{4,7,9,13}) - H(X_{11,13}) + 3H(X_{1,11,13}) - 2H(X_{2,11,13}) + 2H(X_{4,7,12,13}) - H(X_{4,7,9,12,13}) + 2H(X_{1,7,13,14}) - 2H(X_{1,3,7,13,14}) + H(X_{4,7,13,14}) - H(X_{4,7,12,13,14}) - 3H(X_{10,15}) + 3H(X_{10,11,15})>=0","-H(X_{4}) + H(X_{1,4}) + 3H(X_{6}) - 3H(X_{2,6}) + H(X_{4,8}) - H(X_{1,4,8}) - 3H(X_{6,8}) + 3H(X_{2,6,8}) - 3H(X_{10}) - H(X_{11}) + 3H(X_{1,11}) - 2H(X_{2,11}) + 3H(X_{10,11}) + 2H(X_{1,7,13}) - 2H(X_{1,3,7,13}) + 2H(X_{4,7,13}) - H(X_{4,7,9,13}) + H(X_{11,13}) - 3H(X_{1,11,13}) + 2H(X_{2,11,13}) - 2H(X_{4,7,12,13}) + H(X_{4,7,9,12,13}) - 2H(X_{1,7,13,14}) + 2H(X_{1,3,7,13,14}) - H(X_{4,7,13,14}) + H(X_{4,7,12,13,14}) + 3H(X_{10,15}) - 3H(X_{10,11,15})>=0"],"text":"H(X_{4}) - H(X_{1,4}) - 3H(X_{6}) + 3H(X_{2,6}) - H(X_{4,8}) + H(X_{1,4,8}) + 3H(X_{6,8}) - 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3H(X_{2,6,8}) + 2H(X_{9}) - 2H(X_{3,9}) - 2H(X_{9,12}) + 2H(X_{3,9,12}) - 2H(X_{11,12}) + 2H(X_{7,11,12}) + 2H(X_{11,12,13}) - 2H(X_{7,11,12,13}) + H(X_{1,14}) - H(X_{1,5,14}) - H(X_{1,13,14}) + H(X_{1,5,13,14}) - 2H(X_{15}) + 2H(X_{2,15}) - 4H(X_{10,15}) + 4H(X_{2,10,15}) + 3H(X_{6,10,15}) - 3H(X_{2,6,10,15}) + 2H(X_{11,15}) - 2H(X_{2,11,15}) + H(X_{10,14,15}) - H(X_{2,10,14,15}) + 3H(X_{1,7,16}) - 3H(X_{1,4,7,16}) - 3H(X_{2,8,16}) + 3H(X_{2,6,8,16})>=0","3H(X_{1,7}) - 3H(X_{1,4,7}) - 3H(X_{2,8}) + 3H(X_{2,6,8}) - 2H(X_{9}) + 2H(X_{3,9}) + 2H(X_{9,12}) - 2H(X_{3,9,12}) + 2H(X_{11,12}) - 2H(X_{7,11,12}) - 2H(X_{11,12,13}) + 2H(X_{7,11,12,13}) - H(X_{1,14}) + H(X_{1,5,14}) + H(X_{1,13,14}) - H(X_{1,5,13,14}) + 2H(X_{15}) - 2H(X_{2,15}) + 4H(X_{10,15}) - 4H(X_{2,10,15}) - 3H(X_{6,10,15}) + 3H(X_{2,6,10,15}) - 2H(X_{11,15}) + 2H(X_{2,11,15}) - H(X_{10,14,15}) + H(X_{2,10,14,15}) - 3H(X_{1,7,16}) + 3H(X_{1,4,7,16}) + 3H(X_{2,8,16}) - 3H(X_{2,6,8,16})>=0"],"text":"-3H(X_{1,7}) + 3H(X_{1,4,7}) + 3H(X_{2,8}) - 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4H(X_{4,6,9,15,16}) + 8H(X_{1,3,5,6,8,9,10,11,12,13,15,16}) + 8H(X_{12,14,15,16}) - 8H(X_{5,6,7,9,11,12,14,15,16}) - H(X_{1,3,8,9,17}) + 2H(X_{5,10,17}) - 2H(X_{5,9,10,17}) + H(X_{1,2,3,5,8,9,13,17}) - 2H(X_{5,10,13,17}) + 2H(X_{5,9,10,13,17}) + 8H(X_{2,4,7,14,17}) - 6H(X_{16,17}) + 7H(X_{4,6,16,17}) + 6H(X_{3,5,7,16,17}) + H(X_{1,3,8,9,16,17}) - 7H(X_{4,6,11,16,17}) - H(X_{1,2,3,5,8,9,13,16,17}) + 6H(X_{2,8,9,13,15,16,17}) - 6H(X_{2,3,5,7,8,9,13,15,16,17}) - 8H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17})","witness_value":9.261258278488958},"k":12,"logical_conditions":[{"component_width":8,"component_witness_values":[-3.1086244689504383e-15,7.327471962526033e-15,2.5091040356528538e-14,2.919886554764162e-14,2.1316282072803006e-14,-4.440892098500626e-15,7.771561172376096e-15,2.7422508708241367e-14],"components":[{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{4};X_{9}|X_{6,15,16})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{10}|X_{6,7})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{1};X_{2}|X_{7})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{12};X_{13}|X_{15})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{14}|X_{6})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{9};X_{13}|X_{5,10,17})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{14};X_{8}|X_{2,16})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{12};X_{3}|X_{15})"}],"expression":[[32,3],[36,-3],[64,-3],[65,3],[66,3],[67,-3],[96,-1],[98,1],[608,1],[610,-1],[8224,-3],[8228,3],[16384,1],[16388,-3],[18432,-1],[18436,3],[20480,2],[22528,-2],[32770,-1],[32898,1],[40962,1],[41090,-1],[49184,2],[49192,-2],[49440,-2],[49448,2],[66064,-1],[66320,1],[70160,1],[70416,-1]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["3H(X_{6}) - 3H(X_{3,6}) - 3H(X_{7}) + 3H(X_{1,7}) + 3H(X_{2,7}) - 3H(X_{1,2,7}) - H(X_{6,7}) + H(X_{2,6,7}) + H(X_{6,7,10}) - H(X_{2,6,7,10}) - 3H(X_{6,14}) + 3H(X_{3,6,14}) + H(X_{15}) - 3H(X_{3,15}) - H(X_{12,15}) + 3H(X_{3,12,15}) + 2H(X_{13,15}) - 2H(X_{12,13,15}) - H(X_{2,16}) + H(X_{2,8,16}) + H(X_{2,14,16}) - H(X_{2,8,14,16}) + 2H(X_{6,15,16}) - 2H(X_{4,6,15,16}) - 2H(X_{6,9,15,16}) + 2H(X_{4,6,9,15,16}) - H(X_{5,10,17}) + H(X_{5,9,10,17}) + H(X_{5,10,13,17}) - H(X_{5,9,10,13,17})>=0","-3H(X_{6}) + 3H(X_{3,6}) + 3H(X_{7}) - 3H(X_{1,7}) - 3H(X_{2,7}) + 3H(X_{1,2,7}) + H(X_{6,7}) - H(X_{2,6,7}) - H(X_{6,7,10}) + H(X_{2,6,7,10}) + 3H(X_{6,14}) - 3H(X_{3,6,14}) - H(X_{15}) + 3H(X_{3,15}) + H(X_{12,15}) - 3H(X_{3,12,15}) - 2H(X_{13,15}) + 2H(X_{12,13,15}) + H(X_{2,16}) - H(X_{2,8,16}) - H(X_{2,14,16}) + H(X_{2,8,14,16}) - 2H(X_{6,15,16}) + 2H(X_{4,6,15,16}) + 2H(X_{6,9,15,16}) - 2H(X_{4,6,9,15,16}) + H(X_{5,10,17}) - H(X_{5,9,10,17}) - H(X_{5,10,13,17}) + H(X_{5,9,10,13,17})>=0"],"text":"3H(X_{6}) - 3H(X_{3,6}) - 3H(X_{7}) + 3H(X_{1,7}) + 3H(X_{2,7}) - 3H(X_{1,2,7}) - H(X_{6,7}) + H(X_{2,6,7}) + H(X_{6,7,10}) - H(X_{2,6,7,10}) - 3H(X_{6,14}) + 3H(X_{3,6,14}) + H(X_{15}) - 3H(X_{3,15}) - H(X_{12,15}) + 3H(X_{3,12,15}) + 2H(X_{13,15}) - 2H(X_{12,13,15}) - H(X_{2,16}) + H(X_{2,8,16}) + H(X_{2,14,16}) - H(X_{2,8,14,16}) + 2H(X_{6,15,16}) - 2H(X_{4,6,15,16}) - 2H(X_{6,9,15,16}) + 2H(X_{4,6,9,15,16}) - H(X_{5,10,17}) + H(X_{5,9,10,17}) + H(X_{5,10,13,17}) - H(X_{5,9,10,13,17})","type":"equality","witness_value":4.551914400963142e-15}],"m_eq":1,"master_seed":20260825,"method":"dag_v2","n":17,"next_action_index":6,"one_sided_condition_count":0,"prompt_tokens":1502,"recipe_index":0,"remaining_action_coordinate_l1":186.0,"remaining_certificate_length":7,"remaining_coefficient_l1":26.0,"remaining_coefficient_max":8.0,"remaining_condition_actions":1,"remaining_shannon_actions":6,"remaining_shannon_kind_counts":{"conditional_mutual_information":6},"residual_mode":"condition","schema_version":"reviewer-benchmark-v1","selected_coordinate_l1":186.0,"selected_coordinate_support":50,"selected_effective_n":17,"selected_residual_sparse":[[1,4],[32,-6],[36,6],[64,6],[65,-6],[66,-6],[67,6],[96,2],[98,-2],[608,-2],[610,2],[1024,7],[2065,3],[8224,6],[8228,-6],[16384,-2],[16388,6],[16416,3],[16547,-1],[16551,1],[17408,4],[17409,-4],[18432,2],[18436,-6],[18481,-3],[19683,1],[19687,-1],[20480,-4],[22528,4],[32770,2],[32898,-2],[40962,-2],[41090,2],[49184,-4],[49192,4],[49440,4],[49448,-4],[57269,8],[65925,-1],[66064,2],[66320,-2],[70039,1],[70160,-2],[70416,2],[73802,8],[98344,7],[98693,1],[99368,-7],[102807,-1],[131071,-8]],"selector_source_sha256":"83a328168eac42cedd3412a9c8b5624b8683a52a14042b5defbccf6d833efefe","sha256":"8288cd37ffac7ba8bb23724470ddc1053bd9e81f2e03e8c229aaf3f9957c67c4","step_idx":6,"subset_size_profile":{"a":[5,4,2,3,3,1],"b":[12,1,1,2,1,3],"c":[0,0,0,0,5,5]},"target_tokens":769,"tokenizer":{"add_special_tokens":false,"is_fast":true,"kind":"huggingface","model":"Qwen/Qwen3-0.6B","requested_revision":"c1899de289a04d12100db370d81485cdf75e47ca","resolved_commit":"c1899de289a04d12100db370d81485cdf75e47ca","tokenizer_class":"Qwen2TokenizerFast","transformers_version":"4.55.2","trust_remote_code":true,"use_fast":true},"trajectory_length":14,"trajectory_sha256":"3cf318c3931e3578ee4dac766dd72cc63e27ccdf2bfe03def25e69a3167218b2","witness":{"alphabet":[0,1],"cpt_p_x1":[{"0":[1,5],"1":[1,4]},{"0":[1,2],"1":[2,5],"2":[1,3],"3":[3,5]},{"0":[2,5],"1":[1,2]},{"0":[3,4],"1":[2,3],"2":[2,3],"3":[1,3],"4":[1,5],"5":[1,3],"6":[4,5],"7":[1,4]},{"0":[1,5],"1":[3,4]},{"0":[1,4],"1":[3,4]},{"0":[1,4]},{"0":[1,3],"1":[1,3],"10":[3,5],"11":[2,3],"12":[1,2],"13":[1,2],"14":[3,5],"15":[4,5],"2":[3,4],"3":[2,5],"4":[4,5],"5":[1,5],"6":[2,3],"7":[1,5],"8":[3,4],"9":[1,3]},{"0":[2,3],"1":[3,5],"2":[1,2],"3":[4,5],"4":[3,5],"5":[4,5],"6":[1,4],"7":[2,3]},{"0":[1,3],"1":[1,4]},{"0":[1,3],"1":[2,3],"2":[2,5],"3":[4,5]},{"0":[1,2],"1":[3,5]},{"0":[3,5],"1":[3,5],"2":[2,5],"3":[3,4]},{"0":[2,3],"1":[4,5],"2":[1,4],"3":[1,5]},{"0":[1,4],"1":[4,5]},{"0":[3,5],"1":[2,5]},{"0":[3,5],"1":[1,2]}],"cpt_parent_assignment_order":"integer bits follow the listed parent order, least-significant first","edge_count":27,"family":"dag_v2","graph_family":"forest_plus","joint_state_count":131072,"max_indegree":4,"parents":[[6],[6,5],[5],[15,5,14],[12],[6],[],[9,14,3,4],[9,4,16],[15],[6,0],[14],[1,10],[15,1],[9],[6],[11]],"sampled_density":0.2,"state_cap":1048576,"topological_order":[6,15,5,0,9,14,1,3,10,12,11,4,2,16,7,13,8]},"witness_evaluation":{"checks":{"direct_equals_remaining_actions":true,"equality_contribution_zero":true,"shannon_contribution_nonnegative":true,"target_nonnegative":true},"direct_target_value":4.2403057621560984,"evaluation_source":"construction_entropy_oracle","pass":true,"remaining_action_value":4.2403057621561,"remaining_equality_value":-9.103828801926284e-15,"remaining_shannon_value":4.240305762156109},"witness_sha256":"ac4829ca7c4a70436c609028f1d2704e7c6c57047012c68048b92d6f7f9729d3","witness_variant":null}
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14H(X_{2,16}) + 7H(X_{3,16}) - 7H(X_{3,4,16}) + 10H(X_{2,3,4,5,6,9,10,16}) - 10H(X_{2,3,4,5,6,7,9,10,16}) + 21H(X_{2,3,4,12,16}) - 21H(X_{2,3,4,5,12,16}) + 6H(X_{4,9,11,12,13,16}) - 6H(X_{1,2,4,9,11,12,13,16}) + 5H(X_{7,15,16}) + 7H(X_{1,5,6,7,12,15,16}) + 10H(X_{1,8,12,15,16}) - 10H(X_{1,4,5,7,8,11,12,15,16}) - 5H(X_{1,3,5,6,7,11,14,15,16}) + 14H(X_{3,4,12,14,15,16}) - 7H(X_{1,5,6,7,12,14,15,16}) - 14H(X_{3,4,11,12,14,15,16}) + 4H(X_{3,4,5,6,7,8,9,11,12,14,15,16}) - 4H(X_{2,3,4,5,6,7,8,9,11,12,14,15,16}) + 8H(X_{17}) - H(X_{9,17}) + H(X_{7,9,17}) + 7H(X_{2,3,6,8,11,17}) - 7H(X_{2,3,5,6,7,8,9,11,17}) + 4H(X_{6,7,8,10,14,17}) - 4H(X_{6,7,8,9,10,14,17}) - 21H(X_{6,11,15,17}) + 7H(X_{2,6,11,15,17}) + H(X_{3,4,6,9,11,15,17}) - H(X_{3,4,6,7,9,11,15,17}) + 14H(X_{6,11,12,15,17}) - 8H(X_{3,5,7,8,14,15,17}) + 7H(X_{4,5,6,7,10,11,12,14,15,16,17}) - 7H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17})","witness_value":47.97214644004284},"k":16,"logical_conditions":[{"component_width":12,"component_witness_values":[-3.552713678800501e-15,1.7763568394002505e-15,-4.440892098500626e-15,3.885780586188048e-14,1.7763568394002505e-15,3.552713678800501e-15,4.618527782440651e-14,-2.220446049250313e-15,-6.183942247162122e-14,-7.16093850883226e-14,4.440892098500626e-16,-7.116529587847253e-14],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{17};X_{2}|X_{6,11,15})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{11};X_{16}|X_{3,4,12,14,15})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{10};X_{11}|X_{1,3,9,14})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{16}|X_{3})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{5};X_{16}|X_{2,3,4,12})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{11};X_{2}|X_{3,4,12,14,15})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{4};X_{14}|X_{3})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{17};X_{12}|X_{6,11,15})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{2};X_{16})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{14})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{6};X_{14}|X_{2,3,9})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{9};X_{8})"}],"expression":[[2,-1],[4,3],[12,-3],[128,-1],[256,-1],[262,-1],[294,1],[384,1],[2062,3],[2078,-3],[8192,1],[8194,-1],[8196,-2],[8204,2],[8453,-1],[8454,1],[8486,-1],[8965,1],[9477,1],[9989,-1],[17440,-3],[17442,1],[19488,2],[26636,5],[26638,-3],[27660,-5],[27662,3],[32768,-2],[32770,2],[32772,-1],[32780,1],[34830,-3],[34846,3],[59404,-2],[60428,2],[82976,3],[82978,-1],[85024,-2]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["-H(X_{2}) + 3H(X_{3}) - 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4H(X_{5}) + 4H(X_{4,5}) - 10H(X_{7}) + 6H(X_{4,7}) + 4H(X_{6,7}) + 4H(X_{8}) - 6H(X_{10}) + 6H(X_{6,8,12}) - 6H(X_{3,6,8,12}) + 2H(X_{13}) - 2H(X_{4,13}) + 10H(X_{7,13}) - 6H(X_{4,7,13}) - 4H(X_{6,7,13}) + 4H(X_{14}) - 4H(X_{8,14}) + 4H(X_{5,16}) - 4H(X_{4,5,16}) - 6H(X_{6,8,12,16}) + 6H(X_{3,6,8,12,16}) - 6H(X_{17}) + 6H(X_{10,17}) - 4H(X_{4,7,8,11,17}) + 4H(X_{4,5,7,8,11,17}) + 4H(X_{4,6,7,8,11,17}) - 4H(X_{4,5,6,7,8,11,17})","witness_value":6.716849298982197e-14}],"certificate_length":22,"certificate_order":"greedy_negative_mass_condition_late","condition_coordinate_supports":[27,25],"condition_rank":2,"condition_tokens":[330,330,319,320],"condition_width":8,"config_sha256":"1b1f298ed012071c85e9cfb286fe5dcf7c183383de92041754d8a7802f426421","equality_condition_count":2,"full_tokens":2082,"generator_dependency_sha256":{"data/finite_field.py":"c9738c10a16ea2cfb3cde346a23673d1536dbfce491f8871346f441b9b4c2a6e","data/witness_first.py":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","data/witness_v2_algebraic.py":"ffbd48eb673a3f6f842327d00431285bbae8963a7e20f6956e37e544fa37165f","data/witness_v2_graph.py":"c81adcfefcf6ffc0e816f1b338af13323bc80e06293e8cac6b28a3b0ae2efc8a"},"generator_schema_version":"witness-first-v2","generator_source_sha256":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","id":"e2n-witness-first-v2-n17-dag_v2-f5f3464f1c84b575","initial_target":{"coordinate_support":102,"effective_n":17,"expression":[[8,5],[16,60],[17,-24],[18,-24],[24,4],[48,-16],[64,-10],[72,6],[96,4],[128,4],[256,-4],[272,4],[512,-6],[768,-5],[1024,-8],[1032,24],[1152,8],[1536,6],[1544,-24],[2048,4],[2052,5],[2130,5],[2208,6],[2212,-6],[4096,2],[4104,-2],[4160,10],[4168,-6],[4192,-4],[7168,-4],[7172,4],[7496,4],[7500,-4],[8192,32],[8320,-4],[8832,5],[10240,-4],[10884,-5],[13525,-4],[13701,-4],[13797,4],[16401,5],[16416,-3],[16422,16],[16678,-16],[17152,5],[17496,5],[24614,-16],[24870,16],[26114,-10],[26122,10],[26146,10],[26154,-10],[26805,3],[26813,-3],[32768,16],[32784,-60],[32785,24],[32786,24],[32792,-4],[32816,16],[33553,5],[33800,-24],[34312,24],[34976,-6],[34980,6],[37441,-3],[40960,-16],[49937,-5],[52483,4],[52499,-4],[53281,8],[55037,3],[55417,9],[57376,3],[62941,4],[65536,2],[66048,6],[66568,8],[66696,-8],[66760,-4],[66776,4],[66792,4],[66808,-4],[67208,8],[70664,-8],[70792,8],[73728,-8],[73732,-9],[75654,9],[79239,4],[79335,-4],[80916,9],[82042,3],[89622,8],[89750,-8],[105506,5],[106500,9],[113684,-9],[121907,-5],[123002,-3],[131071,-9]],"text":"5H(X_{4}) + 60H(X_{5}) - 24H(X_{1,5}) - 24H(X_{2,5}) + 4H(X_{4,5}) - 16H(X_{5,6}) - 10H(X_{7}) + 6H(X_{4,7}) + 4H(X_{6,7}) + 4H(X_{8}) - 4H(X_{9}) + 4H(X_{5,9}) - 6H(X_{10}) - 5H(X_{9,10}) - 8H(X_{11}) + 24H(X_{4,11}) + 8H(X_{8,11}) + 6H(X_{10,11}) - 24H(X_{4,10,11}) + 4H(X_{12}) + 5H(X_{3,12}) + 5H(X_{2,5,7,12}) + 6H(X_{6,8,12}) - 6H(X_{3,6,8,12}) + 2H(X_{13}) - 2H(X_{4,13}) + 10H(X_{7,13}) - 6H(X_{4,7,13}) - 4H(X_{6,7,13}) - 4H(X_{11,12,13}) + 4H(X_{3,11,12,13}) + 4H(X_{4,7,9,11,12,13}) - 4H(X_{3,4,7,9,11,12,13}) + 32H(X_{14}) - 4H(X_{8,14}) + 5H(X_{8,10,14}) - 4H(X_{12,14}) - 5H(X_{3,8,10,12,14}) - 4H(X_{1,3,5,7,8,11,13,14}) - 4H(X_{1,3,8,9,11,13,14}) + 4H(X_{1,3,6,7,8,9,11,13,14}) + 5H(X_{1,5,15}) - 3H(X_{6,15}) + 16H(X_{2,3,6,15}) - 16H(X_{2,3,6,9,15}) + 5H(X_{9,10,15}) + 5H(X_{4,5,7,11,15}) - 16H(X_{2,3,6,14,15}) + 16H(X_{2,3,6,9,14,15}) - 10H(X_{2,10,11,14,15}) + 10H(X_{2,4,10,11,14,15}) + 10H(X_{2,6,10,11,14,15}) - 10H(X_{2,4,6,10,11,14,15}) + 3H(X_{1,3,5,6,8,12,14,15}) - 3H(X_{1,3,4,5,6,8,12,14,15}) + 16H(X_{16}) - 60H(X_{5,16}) + 24H(X_{1,5,16}) + 24H(X_{2,5,16}) - 4H(X_{4,5,16}) + 16H(X_{5,6,16}) + 5H(X_{1,5,9,10,16}) - 24H(X_{4,11,16}) + 24H(X_{4,10,11,16}) - 6H(X_{6,8,12,16}) + 6H(X_{3,6,8,12,16}) - 3H(X_{1,7,10,13,16}) - 16H(X_{14,16}) - 5H(X_{1,5,9,10,15,16}) + 4H(X_{1,2,9,11,12,15,16}) - 4H(X_{1,2,5,9,11,12,15,16}) + 8H(X_{1,6,13,15,16}) + 3H(X_{1,3,4,5,6,7,8,10,11,13,15,16}) + 9H(X_{1,4,5,6,7,12,13,15,16}) + 3H(X_{6,14,15,16}) + 4H(X_{1,3,4,5,7,8,9,11,13,14,15,16}) + 2H(X_{17}) + 6H(X_{10,17}) + 8H(X_{4,11,17}) - 8H(X_{4,8,11,17}) - 4H(X_{4,7,8,11,17}) + 4H(X_{4,5,7,8,11,17}) + 4H(X_{4,6,7,8,11,17}) - 4H(X_{4,5,6,7,8,11,17}) + 8H(X_{4,8,10,11,17}) - 8H(X_{4,11,13,17}) + 8H(X_{4,8,11,13,17}) - 8H(X_{14,17}) - 9H(X_{3,14,17}) + 9H(X_{2,3,8,9,10,11,14,17}) + 4H(X_{1,2,3,8,9,11,13,14,17}) - 4H(X_{1,2,3,6,7,8,9,11,13,14,17}) + 9H(X_{3,5,11,12,13,14,17}) + 3H(X_{2,4,5,6,7,15,17}) + 8H(X_{2,3,5,10,11,12,13,15,17}) - 8H(X_{2,3,5,8,10,11,12,13,15,17}) + 5H(X_{2,6,11,12,13,16,17}) + 9H(X_{3,14,16,17}) - 9H(X_{3,5,11,12,13,14,16,17}) - 5H(X_{1,2,5,6,11,12,13,15,16,17}) - 3H(X_{2,4,5,6,7,14,15,16,17}) - 9H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17})","witness_value":157.77995960085238},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[4.440892098500626e-15,4.107825191113079e-15,2.6645352591003757e-15,1.6986412276764895e-14,-3.3306690738754696e-16,2.9976021664879227e-15,-8.881784197001252e-15,7.216449660063518e-15],"components":[{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{16}|X_{6,8,12})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{13};X_{6}|X_{7})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{5}|X_{4,7,8,11,17})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{13})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{16};X_{4}|X_{5})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{13};X_{4}|X_{7})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{17};X_{10})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{14};X_{8})"}],"expression":[[8,1],[16,-2],[24,2],[64,-5],[72,3],[96,2],[128,2],[512,-3],[2208,3],[2212,-3],[4096,1],[4104,-1],[4160,5],[4168,-3],[4192,-2],[8192,2],[8320,-2],[32784,2],[32792,-2],[34976,-3],[34980,3],[65536,-3],[66048,3],[66760,-2],[66776,2],[66792,2],[66808,-2]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["H(X_{4}) - 2H(X_{5}) + 2H(X_{4,5}) - 5H(X_{7}) + 3H(X_{4,7}) + 2H(X_{6,7}) + 2H(X_{8}) - 3H(X_{10}) + 3H(X_{6,8,12}) - 3H(X_{3,6,8,12}) + H(X_{13}) - H(X_{4,13}) + 5H(X_{7,13}) - 3H(X_{4,7,13}) - 2H(X_{6,7,13}) + 2H(X_{14}) - 2H(X_{8,14}) + 2H(X_{5,16}) - 2H(X_{4,5,16}) - 3H(X_{6,8,12,16}) + 3H(X_{3,6,8,12,16}) - 3H(X_{17}) + 3H(X_{10,17}) - 2H(X_{4,7,8,11,17}) + 2H(X_{4,5,7,8,11,17}) + 2H(X_{4,6,7,8,11,17}) - 2H(X_{4,5,6,7,8,11,17})>=0","-H(X_{4}) + 2H(X_{5}) - 2H(X_{4,5}) + 5H(X_{7}) - 3H(X_{4,7}) - 2H(X_{6,7}) - 2H(X_{8}) + 3H(X_{10}) - 3H(X_{6,8,12}) + 3H(X_{3,6,8,12}) - H(X_{13}) + H(X_{4,13}) - 5H(X_{7,13}) + 3H(X_{4,7,13}) + 2H(X_{6,7,13}) - 2H(X_{14}) + 2H(X_{8,14}) - 2H(X_{5,16}) + 2H(X_{4,5,16}) + 3H(X_{6,8,12,16}) - 3H(X_{3,6,8,12,16}) + 3H(X_{17}) - 3H(X_{10,17}) + 2H(X_{4,7,8,11,17}) - 2H(X_{4,5,7,8,11,17}) - 2H(X_{4,6,7,8,11,17}) + 2H(X_{4,5,6,7,8,11,17})>=0"],"text":"H(X_{4}) - 2H(X_{5}) + 2H(X_{4,5}) - 5H(X_{7}) + 3H(X_{4,7}) + 2H(X_{6,7}) + 2H(X_{8}) - 3H(X_{10}) + 3H(X_{6,8,12}) - 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3H(X_{4}) + 3H(X_{6}) - 3H(X_{1,6}) - 3H(X_{3,4,6}) + H(X_{7}) + 5H(X_{8}) - 5H(X_{9}) + 2H(X_{2,9}) - 15H(X_{10}) + 10H(X_{5,10}) - 5H(X_{7,10}) + 5H(X_{1,7,10}) + 5H(X_{8,10}) + 10H(X_{9,10}) - 10H(X_{5,9,10}) - 6H(X_{8,9,10}) + 17H(X_{11}) - 17H(X_{3,11}) - 10H(X_{4,11}) + 10H(X_{3,4,11}) - 8H(X_{2,4,9,10,11}) - 2H(X_{12}) + 5H(X_{9,12}) - 6H(X_{1,5,8,9,12}) + 8H(X_{2,4,5,7,9,10,11,12}) + 2H(X_{13}) + 10H(X_{3,4,5,8,13}) - 2H(X_{2,9,13}) - 2H(X_{12,13}) - 23H(X_{14}) + 3H(X_{4,14}) + 5H(X_{3,5,14}) + 20H(X_{6,14}) + 3H(X_{8,14}) + 15H(X_{10,14}) - 15H(X_{6,10,14}) + 3H(X_{11,14}) - 3H(X_{3,11,14}) - 3H(X_{4,8,12,13,14}) + 3H(X_{4,5,8,10,12,13,14}) + 5H(X_{5,7,8,9,10,12,13,14}) + 3H(X_{3,4,6,8,11,12,13,14}) - 3H(X_{3,4,5,6,8,10,11,12,13,14}) - H(X_{15}) + 5H(X_{1,15}) - 8H(X_{5,8,15}) + H(X_{12,15}) - 5H(X_{1,5,7,8,9,10,12,13,14,15}) + 7H(X_{16}) + 8H(X_{7,11,16}) - 8H(X_{7,8,11,16}) + 3H(X_{2,3,4,6,7,9,12,16}) + 6H(X_{1,2,6,8,9,10,14,16}) + 8H(X_{3,4,5,6,8,9,10,12,13,15,16}) + 2H(X_{17}) - 2H(X_{2,9,17}) + 5H(X_{10,17}) + 5H(X_{7,10,17}) - 5H(X_{1,7,10,17}) - 5H(X_{8,10,17}) - 10H(X_{11,17}) + 10H(X_{3,11,17}) + H(X_{1,5,9,11,17}) - H(X_{1,2,5,9,11,17}) - H(X_{1,5,9,10,11,17}) + H(X_{1,2,5,9,10,11,17}) - 2H(X_{12,17}) + 5H(X_{1,4,6,9,12,17}) + 2H(X_{2,9,13,17}) + 5H(X_{14,17}) - 5H(X_{6,14,17}) - 5H(X_{1,3,4,5,6,9,12,14,17}) + 8H(X_{5,7,8,15,17}) + 10H(X_{6,9,15,17}) + 8H(X_{2,4,9,10,11,15,17}) - 8H(X_{2,4,5,7,9,10,11,12,15,17}) - 10H(X_{3,4,5,6,8,9,13,15,17}) + 3H(X_{1,2,3,4,6,7,8,12,13,15,17}) - 4H(X_{3,4,10,11,12,13,15,17}) - 3H(X_{1,2,3,4,6,7,8,12,13,14,15,17}) + 7H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17}) - 3H(X_{3,15,16,17}) + 2H(X_{1,2,7,8,15,16,17}) + 3H(X_{3,8,9,10,12,15,16,17}) - 8H(X_{3,4,5,6,7,8,9,10,12,13,15,16,17}) + 4H(X_{1,2,3,4,5,6,7,10,11,12,13,15,16,17}) + 6H(X_{1,3,4,5,6,7,8,9,10,12,13,14,15,16,17}) - 7H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17})","witness_value":140.57758130770128},"k":16,"logical_conditions":[{"component_width":8,"component_witness_values":[2.6645352591003757e-15,-1.965094753586527e-14,-1.1879386363489175e-14,-1.965094753586527e-14,4.6629367034256575e-15,6.8833827526759706e-15,-1.2212453270876722e-15,3.552713678800501e-15],"components":[{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{10}|X_{1,5,9,11,17})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{14}|X_{11})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{12};X_{15})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{1};X_{6})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{13};X_{17}|X_{2,9})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{12};X_{17})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{12};X_{13})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{4};X_{14})"}],"expression":[[1,3],[8,-3],[32,3],[33,-3],[258,2],[1024,-3],[1028,3],[2048,3],[4096,2],[4354,-2],[6144,-2],[8192,-3],[8200,3],[9216,3],[9220,-3],[16384,-1],[18432,1],[65536,2],[65794,-2],[66833,1],[66835,-1],[67345,-1],[67347,1],[67584,-2],[69890,2]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["3H(X_{1}) - 3H(X_{4}) + 3H(X_{6}) - 3H(X_{1,6}) + 2H(X_{2,9}) - 3H(X_{11}) + 3H(X_{3,11}) + 3H(X_{12}) + 2H(X_{13}) - 2H(X_{2,9,13}) - 2H(X_{12,13}) - 3H(X_{14}) + 3H(X_{4,14}) + 3H(X_{11,14}) - 3H(X_{3,11,14}) - H(X_{15}) + H(X_{12,15}) + 2H(X_{17}) - 2H(X_{2,9,17}) + H(X_{1,5,9,11,17}) - H(X_{1,2,5,9,11,17}) - H(X_{1,5,9,10,11,17}) + H(X_{1,2,5,9,10,11,17}) - 2H(X_{12,17}) + 2H(X_{2,9,13,17})>=0","-3H(X_{1}) + 3H(X_{4}) - 3H(X_{6}) + 3H(X_{1,6}) - 2H(X_{2,9}) + 3H(X_{11}) - 3H(X_{3,11}) - 3H(X_{12}) - 2H(X_{13}) + 2H(X_{2,9,13}) + 2H(X_{12,13}) + 3H(X_{14}) - 3H(X_{4,14}) - 3H(X_{11,14}) + 3H(X_{3,11,14}) + H(X_{15}) - H(X_{12,15}) - 2H(X_{17}) + 2H(X_{2,9,17}) - H(X_{1,5,9,11,17}) + H(X_{1,2,5,9,11,17}) + H(X_{1,5,9,10,11,17}) - H(X_{1,2,5,9,10,11,17}) + 2H(X_{12,17}) - 2H(X_{2,9,13,17})>=0"],"text":"3H(X_{1}) - 3H(X_{4}) + 3H(X_{6}) - 3H(X_{1,6}) + 2H(X_{2,9}) - 3H(X_{11}) + 3H(X_{3,11}) + 3H(X_{12}) + 2H(X_{13}) - 2H(X_{2,9,13}) - 2H(X_{12,13}) - 3H(X_{14}) + 3H(X_{4,14}) + 3H(X_{11,14}) - 3H(X_{3,11,14}) - 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2H(X_{5,10}) + H(X_{7,10}) - H(X_{1,7,10}) - H(X_{8,10}) - 2H(X_{9,10}) + 2H(X_{5,9,10}) - 4H(X_{11}) + 4H(X_{3,11}) + 2H(X_{4,11}) - 2H(X_{3,4,11}) + H(X_{12}) - H(X_{9,12}) + 4H(X_{14}) - 4H(X_{6,14}) - 3H(X_{10,14}) + 3H(X_{6,10,14}) - H(X_{10,17}) - H(X_{7,10,17}) + H(X_{1,7,10,17}) + H(X_{8,10,17}) + 2H(X_{11,17}) - 2H(X_{3,11,17}) - H(X_{14,17}) + H(X_{6,14,17})>=0","-H(X_{9}) - 3H(X_{10}) + 2H(X_{5,10}) - H(X_{7,10}) + H(X_{1,7,10}) + H(X_{8,10}) + 2H(X_{9,10}) - 2H(X_{5,9,10}) + 4H(X_{11}) - 4H(X_{3,11}) - 2H(X_{4,11}) + 2H(X_{3,4,11}) - H(X_{12}) + H(X_{9,12}) - 4H(X_{14}) + 4H(X_{6,14}) + 3H(X_{10,14}) - 3H(X_{6,10,14}) + H(X_{10,17}) + H(X_{7,10,17}) - H(X_{1,7,10,17}) - H(X_{8,10,17}) - 2H(X_{11,17}) + 2H(X_{3,11,17}) + H(X_{14,17}) - H(X_{6,14,17})>=0"],"text":"H(X_{9}) + 3H(X_{10}) - 2H(X_{5,10}) + H(X_{7,10}) - H(X_{1,7,10}) - H(X_{8,10}) - 2H(X_{9,10}) + 2H(X_{5,9,10}) - 4H(X_{11}) + 4H(X_{3,11}) + 2H(X_{4,11}) - 2H(X_{3,4,11}) + H(X_{12}) - H(X_{9,12}) + 4H(X_{14}) - 4H(X_{6,14}) - 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H(X_{9}) + 10H(X_{1,9}) + 5H(X_{4,9}) - 8H(X_{6,10}) - 16H(X_{2,9,10}) + 16H(X_{2,3,9,10}) + 16H(X_{2,6,9,10}) - 16H(X_{2,3,6,9,10}) + 10H(X_{11}) + 7H(X_{9,11}) - 10H(X_{1,9,11}) + 2H(X_{10,11}) + 8H(X_{12}) - 8H(X_{7,12}) + H(X_{1,4,6,8,12,13}) - H(X_{1,4,6,7,8,12,13}) + 7H(X_{5,14}) + 32H(X_{2,5,14}) - 8H(X_{2,4,5,14}) - 32H(X_{2,5,7,14}) + 8H(X_{2,4,5,7,14}) - 24H(X_{2,5,8,14}) + 24H(X_{2,5,7,8,14}) + 9H(X_{9,14}) + 8H(X_{6,10,14}) + 16H(X_{8,11,14}) - 16H(X_{1,8,11,14}) + 8H(X_{4,8,11,14}) - 8H(X_{1,4,8,11,14}) + 16H(X_{15}) - 16H(X_{5,15}) - 16H(X_{9,15}) + 16H(X_{5,9,15}) + 8H(X_{6,10,15}) - 16H(X_{2,5,6,12,15}) + H(X_{4,11,13,15}) - 8H(X_{6,10,14,15}) + 16H(X_{2,5,6,12,14,15}) + H(X_{7,10,16}) + 2H(X_{1,8,9,15,16}) + 9H(X_{3,4,6,8,10,15,16}) - 2H(X_{1,8,9,10,11,15,16}) - H(X_{4,7,10,11,13,15,16}) - 9H(X_{3,4,6,8,9,10,14,15,16}) + 5H(X_{11,12,17}) - 5H(X_{4,9,11,12,17}) - 16H(X_{14,17}) + 16H(X_{8,14,17}) - 24H(X_{8,11,14,17}) + 24H(X_{1,8,11,14,17}) + 2H(X_{12,14,17}) + 16H(X_{13,14,17}) - 16H(X_{8,13,14,17}) + 6H(X_{1,3,5,7,8,15,17}) - 6H(X_{1,3,5,7,8,9,15,17}) + 16H(X_{2,5,6,12,15,17}) - 16H(X_{2,5,6,12,14,15,17}) - 2H(X_{1,5,7,12,14,15,16,17}) + 2H(X_{18}) - 5H(X_{4,9,11,18}) - 8H(X_{12,18}) + 8H(X_{7,12,18}) - 40H(X_{13,14,18}) + 16H(X_{4,13,14,18}) + 7H(X_{1,3,4,9,10,12,14,15,18}) - 7H(X_{1,3,4,9,10,11,12,14,15,18}) + 5H(X_{2,4,5,9,10,11,13,16,18}) + 40H(X_{13,14,16,18}) - 16H(X_{4,13,14,16,18}) + 7H(X_{3,4,5,6,8,9,12,15,16,18}) - 7H(X_{3,4,5,6,8,9,12,14,15,16,18}) + 5H(X_{4,9,11,17,18}) - 2H(X_{12,14,17,18}) + 24H(X_{13,14,17,18}) + 5H(X_{1,5,16,17,18}) - 5H(X_{2,4,5,9,10,11,13,16,17,18}) - 24H(X_{13,14,16,17,18}) + 2H(X_{1,2,3,5,7,8,12,14,15,16,17,18}) + 6H(X_{1,2,5,11,12,13,14,15,16,17,18}) - 6H(X_{1,2,5,8,11,12,13,14,15,16,17,18}) + 10H(X_{2,3,4,5,6,7,9,10,11,12,13,14,15,16,17,18}) - 10H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18})","witness_value":38.89597452830022},"k":16,"logical_conditions":[{"component_width":12,"component_witness_values":[7.105427357601002e-15,5.329070518200751e-15,-4.9182879990894435e-14,-1.4765966227514582e-14,8.881784197001252e-16,1.2434497875801753e-14,4.440892098500626e-15,-2.220446049250313e-15,-1.6431300764452317e-14,7.105427357601002e-15,-1.021405182655144e-14,-1.0658141036401503e-14],"components":[{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{6}|X_{2,9,10})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{17};X_{14}|X_{2,5,6,12,15})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{9}|X_{15})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{18};X_{7}|X_{12})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{7};X_{4}|X_{2,5,14})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{16};X_{4}|X_{13,14,18})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{4}|X_{8,11,14})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{7};X_{8}|X_{2,5,14})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{13};X_{8}|X_{14,17})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{16};X_{17}|X_{13,14,18})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{14};X_{15}|X_{6,10})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{1};X_{17}|X_{8,11,14})"}],"expression":[[544,1],[770,2],[774,-2],[802,-2],[806,2],[2048,-1],[2112,1],[8210,-4],[8218,1],[8274,4],[8282,-1],[8338,3],[8402,-3],[8736,-1],[9344,-2],[9345,2],[9352,-1],[9353,1],[16384,-2],[16400,2],[16640,2],[16656,-2],[16928,-1],[18482,2],[25120,1],[26674,-2],[73728,2],[73856,-2],[74880,3],[74881,-3],[77824,-2],[77952,2],[84018,-2],[92210,2],[133120,1],[133184,-1],[143360,5],[143368,-2],[176128,-5],[176136,2],[208896,-3],[241664,3]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["H(X_{6,10}) + 2H(X_{2,9,10}) - 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2H(X_{1,8,11,14}) + H(X_{4,8,11,14}) - H(X_{1,4,8,11,14}) + 2H(X_{15}) - 2H(X_{5,15}) - 2H(X_{9,15}) + 2H(X_{5,9,15}) + H(X_{6,10,15}) - 2H(X_{2,5,6,12,15}) - H(X_{6,10,14,15}) + 2H(X_{2,5,6,12,14,15}) - 2H(X_{14,17}) + 2H(X_{8,14,17}) - 3H(X_{8,11,14,17}) + 3H(X_{1,8,11,14,17}) + 2H(X_{13,14,17}) - 2H(X_{8,13,14,17}) + 2H(X_{2,5,6,12,15,17}) - 2H(X_{2,5,6,12,14,15,17}) - H(X_{12,18}) + H(X_{7,12,18}) - 5H(X_{13,14,18}) + 2H(X_{4,13,14,18}) + 5H(X_{13,14,16,18}) - 2H(X_{4,13,14,16,18}) + 3H(X_{13,14,17,18}) - 3H(X_{13,14,16,17,18})>=0"],"text":"H(X_{6,10}) + 2H(X_{2,9,10}) - 2H(X_{2,3,9,10}) - 2H(X_{2,6,9,10}) + 2H(X_{2,3,6,9,10}) - H(X_{12}) + H(X_{7,12}) - 4H(X_{2,5,14}) + H(X_{2,4,5,14}) + 4H(X_{2,5,7,14}) - H(X_{2,4,5,7,14}) + 3H(X_{2,5,8,14}) - 3H(X_{2,5,7,8,14}) - H(X_{6,10,14}) - 2H(X_{8,11,14}) + 2H(X_{1,8,11,14}) - H(X_{4,8,11,14}) + H(X_{1,4,8,11,14}) - 2H(X_{15}) + 2H(X_{5,15}) + 2H(X_{9,15}) - 2H(X_{5,9,15}) - H(X_{6,10,15}) + 2H(X_{2,5,6,12,15}) + H(X_{6,10,14,15}) - 2H(X_{2,5,6,12,14,15}) + 2H(X_{14,17}) - 2H(X_{8,14,17}) + 3H(X_{8,11,14,17}) - 3H(X_{1,8,11,14,17}) - 2H(X_{13,14,17}) + 2H(X_{8,13,14,17}) - 2H(X_{2,5,6,12,15,17}) + 2H(X_{2,5,6,12,14,15,17}) + H(X_{12,18}) - H(X_{7,12,18}) + 5H(X_{13,14,18}) - 2H(X_{4,13,14,18}) - 5H(X_{13,14,16,18}) + 2H(X_{4,13,14,16,18}) - 3H(X_{13,14,17,18}) + 3H(X_{13,14,16,17,18})","type":"equality","witness_value":-1.8196555373606316e-13}],"m_eq":1,"master_seed":20260825,"method":"dag_v2","n":18,"next_action_index":3,"one_sided_condition_count":0,"prompt_tokens":2024,"recipe_index":1,"remaining_action_coordinate_l1":187.0,"remaining_certificate_length":14,"remaining_coefficient_l1":66.0,"remaining_coefficient_max":10.0,"remaining_condition_actions":0,"remaining_shannon_actions":14,"remaining_shannon_kind_counts":{"conditional_entropy":3,"conditional_mutual_information":11},"residual_mode":"shannon","schema_version":"reviewer-benchmark-v1","selected_coordinate_l1":187.0,"selected_coordinate_support":38,"selected_effective_n":18,"selected_residual_sparse":[[32,2],[64,1],[128,6],[129,10],[256,6],[257,10],[264,5],[1024,10],[1281,-10],[1536,2],[6313,1],[6377,-1],[8448,9],[21512,1],[33344,1],[49537,2],[49836,9],[51073,-2],[54856,-1],[58284,-9],[68608,5],[68872,-5],[75776,2],[82133,6],[82389,-6],[125009,-2],[131072,2],[132360,-5],[169754,5],[197896,5],[206848,-2],[229393,5],[235290,-5],[256215,2],[261139,6],[261267,-6],[262014,10],[262143,-10]],"selector_source_sha256":"83a328168eac42cedd3412a9c8b5624b8683a52a14042b5defbccf6d833efefe","sha256":"a46b0a45e094577418c60ecfdc24eb8e2d8078eae6b7ff8f86971bdb59066af1","step_idx":3,"subset_size_profile":{"a":[2,2,5,7,11,7,3,3,5,4,4,6,5,1],"b":[16,1,1,2,1,1,2,1,2,0,3,1,0,0],"c":[0,0,4,0,0,0,0,0,0,8,0,0,0,0]},"target_tokens":762,"tokenizer":{"add_special_tokens":false,"is_fast":true,"kind":"huggingface","model":"Qwen/Qwen3-0.6B","requested_revision":"c1899de289a04d12100db370d81485cdf75e47ca","resolved_commit":"c1899de289a04d12100db370d81485cdf75e47ca","tokenizer_class":"Qwen2TokenizerFast","transformers_version":"4.55.2","trust_remote_code":true,"use_fast":true},"trajectory_length":18,"trajectory_sha256":"34fadeba58d0d8a8fa5d6ca33cda7eccaad643a38836b5f58c2fec8887247a53","witness":{"alphabet":[0,1],"cpt_p_x1":[{"0":[1,2],"1":[4,5],"2":[1,3],"3":[3,5],"4":[3,5],"5":[1,2],"6":[3,5],"7":[1,5]},{"0":[3,4],"1":[3,5],"2":[2,3],"3":[3,5],"4":[2,3],"5":[1,4],"6":[4,5],"7":[2,3]},{"0":[1,2],"1":[2,3],"2":[3,5],"3":[3,4],"4":[3,5],"5":[2,5],"6":[3,5],"7":[1,2]},{"0":[4,5]},{"0":[4,5],"1":[2,5]},{"0":[3,5]},{"0":[1,3],"1":[3,5],"2":[3,5],"3":[3,4],"4":[1,3],"5":[2,3],"6":[1,3],"7":[3,4]},{"0":[2,3],"1":[1,2],"10":[1,5],"11":[1,4],"12":[2,3],"13":[3,4],"14":[1,3],"15":[1,3],"2":[2,3],"3":[3,4],"4":[1,3],"5":[2,3],"6":[1,2],"7":[1,2],"8":[4,5],"9":[2,3]},{"0":[1,2],"1":[1,2],"2":[1,5],"3":[2,3],"4":[1,4],"5":[3,5],"6":[2,3],"7":[1,4]},{"0":[1,5]},{"0":[2,5],"1":[3,4]},{"0":[4,5],"1":[1,2],"2":[3,5],"3":[1,3]},{"0":[1,2],"1":[3,5],"2":[1,2],"3":[1,4]},{"0":[1,5],"1":[3,5],"2":[1,3],"3":[2,5]},{"0":[1,3]},{"0":[1,2],"1":[1,5],"2":[1,2],"3":[1,4],"4":[1,5],"5":[1,4],"6":[3,5],"7":[1,4]},{"0":[3,4],"1":[2,3],"10":[2,3],"11":[3,4],"12":[2,3],"13":[1,3],"14":[2,3],"15":[1,3],"16":[2,3],"17":[1,2],"18":[1,3],"19":[2,5],"2":[1,3],"20":[2,5],"21":[2,5],"22":[1,2],"23":[1,3],"24":[3,5],"25":[1,2],"26":[1,4],"27":[3,4],"28":[1,2],"29":[1,3],"3":[3,4],"30":[3,5],"31":[2,3],"4":[3,5],"5":[3,4],"6":[1,3],"7":[3,5],"8":[2,3],"9":[1,3]},{"0":[3,5],"1":[1,4]}],"cpt_parent_assignment_order":"integer bits follow the listed parent order, least-significant first","edge_count":36,"family":"dag_v2","graph_family":"layered","joint_state_count":262144,"max_indegree":5,"parents":[[13,10,7],[3,9,14],[9,1,8],[],[14],[],[4,1,13],[13,10,2,16],[3,5,14],[],[3],[9,14],[13,16],[9,5],[],[13,17,12],[5,14,4,11,1],[11]],"sampled_density":0.12,"state_cap":1048576,"topological_order":[3,9,5,14,4,11,1,8,13,10,2,16,17,7,12,6,0,15]},"witness_evaluation":{"checks":{"direct_equals_remaining_actions":true,"equality_contribution_zero":true,"shannon_contribution_nonnegative":true,"target_nonnegative":true},"direct_target_value":38.12444244477631,"evaluation_source":"construction_entropy_oracle","pass":true,"remaining_action_value":38.12444244477631,"remaining_equality_value":0,"remaining_shannon_value":38.12444244477631},"witness_sha256":"f49b8897bb2352ea354d8d203fcec29cfd8fc3d3ad2a4fe6f2f637d95c4e2f66","witness_variant":null}
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8H(X_{1,3,5,8,10,13,17,18})","witness_value":0.0042846633202772155},{"a_mask":131072,"b_mask":66569,"c_mask":10322,"coefficient":8,"expression":[[10322,-1],[76891,1],[141394,1],[207963,-1]],"kind":"conditional_mutual_information","label":"I(X_{18};X_{1,4,11,17}|X_{2,5,7,12,14})","role":"shannon","scaled_expression":[[10322,-8],[76891,8],[141394,8],[207963,-8]],"scaled_witness_value":-3.552713678800501e-14,"source_index":7,"text":"-8H(X_{2,5,7,12,14}) + 8H(X_{1,2,4,5,7,11,12,14,17}) + 8H(X_{2,5,7,12,14,18}) - 8H(X_{1,2,4,5,7,11,12,14,17,18})","witness_value":-4.440892098500626e-15},{"a_mask":5120,"b_mask":131072,"c_mask":26996,"coefficient":8,"expression":[[26996,-1],[32116,1],[158068,1],[163188,-1]],"kind":"conditional_mutual_information","label":"I(X_{11,13};X_{18}|X_{3,5,6,7,9,12,14,15})","role":"shannon","scaled_expression":[[26996,-8],[32116,8],[158068,8],[163188,-8]],"scaled_witness_value":0.0001109520712176959,"source_index":12,"text":"-8H(X_{3,5,6,7,9,12,14,15}) + 8H(X_{3,5,6,7,9,11,12,13,14,15}) + 8H(X_{3,5,6,7,9,12,14,15,18}) - 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10H(X_{1,2,7,8,11,13,16,18})","witness_value":0.0008168498189062845},{"a_mask":3866,"b_mask":0,"c_mask":213120,"coefficient":10,"expression":[[213120,-1],[216986,1]],"kind":"conditional_entropy","label":"H(X_{2,4,5,9,10,11,12}|X_{8,15,17,18})","role":"shannon","scaled_expression":[[213120,-10],[216986,10]],"scaled_witness_value":62.27238621491936,"source_index":11,"text":"-10H(X_{8,15,17,18}) + 10H(X_{2,4,5,8,9,10,11,12,15,17,18})","witness_value":6.227238621491936},{"a_mask":38159,"b_mask":0,"c_mask":149728,"coefficient":8,"expression":[[149728,-1],[187887,1]],"kind":"conditional_entropy","label":"H(X_{1,2,3,4,9,11,13,16}|X_{6,7,8,12,15,18})","role":"shannon","scaled_expression":[[149728,-8],[187887,8]],"scaled_witness_value":55.31046297695498,"source_index":14,"text":"-8H(X_{6,7,8,12,15,18}) + 8H(X_{1,2,3,4,6,7,8,9,11,12,13,15,16,18})","witness_value":6.9138078721193725},{"a_mask":103399,"b_mask":16,"c_mask":8192,"coefficient":4,"expression":[[8192,-1],[8208,1],[111591,1],[111607,-1]],"kind":"conditional_mutual_information","label":"I(X_{1,2,3,6,7,8,9,10,13,16,17};X_{5}|X_{14})","role":"shannon","scaled_expression":[[8192,-4],[8208,4],[111591,4],[111607,-4]],"scaled_witness_value":0.13520571170269768,"source_index":1,"text":"-4H(X_{14}) + 4H(X_{5,14}) + 4H(X_{1,2,3,6,7,8,9,10,13,14,16,17}) - 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8H(X_{5}) + 8H(X_{1,5}) + 8H(X_{3,5}) + 16H(X_{1,6}) - 8H(X_{1,5,6}) - 4H(X_{11}) + 4H(X_{2,11}) + 4H(X_{3,11}) - 12H(X_{2,3,11}) - 8H(X_{2,3,14}) + 4H(X_{9,14}) - 4H(X_{5,9,14}) + 8H(X_{2,3,11,14}) - 4H(X_{9,13,14}) + 4H(X_{5,9,13,14}) + 12H(X_{1,15}) - 12H(X_{1,6,15}) - 4H(X_{1,16}) + 4H(X_{1,6,16}) + 8H(X_{7,9,16}) - 8H(X_{4,7,9,16}) - 8H(X_{7,8,9,16}) + 8H(X_{4,7,8,9,16}) + 8H(X_{5,18}) - 8H(X_{3,5,18})","witness_value":-1.8540724511240114e-14}],"certificate_length":22,"certificate_order":"greedy_negative_mass_condition_late","condition_coordinate_supports":[29,27],"condition_rank":2,"condition_tokens":[317,318,309,310],"condition_width":8,"config_sha256":"1b1f298ed012071c85e9cfb286fe5dcf7c183383de92041754d8a7802f426421","equality_condition_count":2,"full_tokens":2012,"generator_dependency_sha256":{"data/finite_field.py":"c9738c10a16ea2cfb3cde346a23673d1536dbfce491f8871346f441b9b4c2a6e","data/witness_first.py":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","data/witness_v2_algebraic.py":"ffbd48eb673a3f6f842327d00431285bbae8963a7e20f6956e37e544fa37165f","data/witness_v2_graph.py":"c81adcfefcf6ffc0e816f1b338af13323bc80e06293e8cac6b28a3b0ae2efc8a"},"generator_schema_version":"witness-first-v2","generator_source_sha256":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","id":"e2n-witness-first-v2-n18-dag_v2-094f7989d42b7b0c","initial_target":{"coordinate_support":103,"effective_n":18,"expression":[[1,6],[3,-15],[4,11],[5,10],[6,3],[8,-2],[16,-18],[17,8],[18,1],[20,18],[33,1],[35,15],[49,-8],[70,5],[132,7],[261,-10],[517,-10],[528,4],[773,10],[1024,-4],[1026,4],[1028,-6],[1030,-12],[2048,30],[2052,-15],[2560,-15],[4096,10],[7460,3],[8192,-4],[8198,-3],[8208,4],[8262,-5],[8448,4],[8464,-4],[9222,8],[10322,-8],[12544,-4],[12560,4],[12941,7],[13312,4],[16385,12],[16417,-12],[26996,-8],[32116,8],[32768,5],[32769,-4],[32776,-5],[32784,10],[32788,-10],[32801,4],[33024,1],[33042,-1],[33088,3],[33096,-8],[33216,-3],[33224,8],[33600,5],[33728,-5],[34816,-15],[35328,15],[65560,7],[66069,8],[66466,4],[76891,8],[77060,10],[78493,-7],[79778,-4],[90368,2],[90386,7],[90387,-7],[90634,-5],[90654,5],[111591,4],[111607,-4],[116725,2],[131088,8],[131092,-8],[135300,8],[136559,4],[136772,10],[137087,-4],[141394,8],[145418,2],[147456,3],[149728,-8],[154916,-3],[158068,8],[163188,-8],[165059,10],[169155,-10],[172642,4],[172646,-4],[177152,-7],[177168,7],[177664,7],[177680,-7],[187887,8],[201365,-8],[207963,-8],[212804,-10],[213120,-10],[216986,10],[262143,-2]],"text":"6H(X_{1}) - 15H(X_{1,2}) + 11H(X_{3}) + 10H(X_{1,3}) + 3H(X_{2,3}) - 2H(X_{4}) - 18H(X_{5}) + 8H(X_{1,5}) + H(X_{2,5}) + 18H(X_{3,5}) + H(X_{1,6}) + 15H(X_{1,2,6}) - 8H(X_{1,5,6}) + 5H(X_{2,3,7}) + 7H(X_{3,8}) - 10H(X_{1,3,9}) - 10H(X_{1,3,10}) + 4H(X_{5,10}) + 10H(X_{1,3,9,10}) - 4H(X_{11}) + 4H(X_{2,11}) - 6H(X_{3,11}) - 12H(X_{2,3,11}) + 30H(X_{12}) - 15H(X_{3,12}) - 15H(X_{10,12}) + 10H(X_{13}) + 3H(X_{3,6,9,11,12,13}) - 4H(X_{14}) - 3H(X_{2,3,14}) + 4H(X_{5,14}) - 5H(X_{2,3,7,14}) + 4H(X_{9,14}) - 4H(X_{5,9,14}) + 8H(X_{2,3,11,14}) - 8H(X_{2,5,7,12,14}) - 4H(X_{9,13,14}) + 4H(X_{5,9,13,14}) + 7H(X_{1,3,4,8,10,13,14}) + 4H(X_{11,13,14}) + 12H(X_{1,15}) - 12H(X_{1,6,15}) - 8H(X_{3,5,6,7,9,12,14,15}) + 8H(X_{3,5,6,7,9,11,12,13,14,15}) + 5H(X_{16}) - 4H(X_{1,16}) - 5H(X_{4,16}) + 10H(X_{5,16}) - 10H(X_{3,5,16}) + 4H(X_{1,6,16}) + H(X_{9,16}) - H(X_{2,5,9,16}) + 3H(X_{7,9,16}) - 8H(X_{4,7,9,16}) - 3H(X_{7,8,9,16}) + 8H(X_{4,7,8,9,16}) + 5H(X_{7,9,10,16}) - 5H(X_{7,8,9,10,16}) - 15H(X_{12,16}) + 15H(X_{10,12,16}) + 7H(X_{4,5,17}) + 8H(X_{1,3,5,10,17}) + 4H(X_{2,6,8,9,10,17}) + 8H(X_{1,2,4,5,7,11,12,14,17}) + 10H(X_{3,9,11,12,14,17}) - 7H(X_{1,3,4,5,8,10,13,14,17}) - 4H(X_{2,6,8,9,10,11,13,14,17}) + 2H(X_{9,14,15,17}) + 7H(X_{2,5,9,14,15,17}) - 7H(X_{1,2,5,9,14,15,17}) - 5H(X_{2,4,10,14,15,17}) + 5H(X_{2,3,4,5,10,14,15,17}) + 4H(X_{1,2,3,6,7,8,9,10,13,14,16,17}) - 4H(X_{1,2,3,5,6,7,8,9,10,13,14,16,17}) + 2H(X_{1,3,5,6,7,8,9,10,11,15,16,17}) + 8H(X_{5,18}) - 8H(X_{3,5,18}) + 8H(X_{3,8,13,18}) + 4H(X_{1,2,3,4,6,7,9,11,13,18}) + 10H(X_{3,7,10,11,13,18}) - 4H(X_{1,2,3,4,5,6,7,9,10,11,13,18}) + 8H(X_{2,5,7,12,14,18}) + 2H(X_{2,4,12,13,14,18}) + 3H(X_{15,18}) - 8H(X_{6,7,8,12,15,18}) - 3H(X_{3,6,9,11,12,13,15,18}) + 8H(X_{3,5,6,7,9,12,14,15,18}) - 8H(X_{3,5,6,7,9,11,12,13,14,15,18}) + 10H(X_{1,2,7,8,11,16,18}) - 10H(X_{1,2,7,8,11,13,16,18}) + 4H(X_{2,6,7,10,14,16,18}) - 4H(X_{2,3,6,7,10,14,16,18}) - 7H(X_{11,13,14,16,18}) + 7H(X_{5,11,13,14,16,18}) + 7H(X_{10,11,13,14,16,18}) - 7H(X_{5,10,11,13,14,16,18}) + 8H(X_{1,2,3,4,6,7,8,9,11,12,13,15,16,18}) - 8H(X_{1,3,5,8,10,13,17,18}) - 8H(X_{1,2,4,5,7,11,12,14,17,18}) - 10H(X_{3,7,9,10,11,12,13,14,17,18}) - 10H(X_{8,15,17,18}) + 10H(X_{2,4,5,8,9,10,11,12,15,17,18}) - 2H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18})","witness_value":152.03383598074677},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[4.440892098500626e-15,-4.9960036108132044e-15,4.6629367034256575e-15,-2.220446049250313e-15,2.5757174171303632e-14,7.549516567451064e-15,-1.2656542480726785e-14,-1.0103029524088925e-14],"components":[{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{16}|X_{5})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{6};X_{2}|X_{1})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{14};X_{7}|X_{2,3})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{10};X_{9}|X_{1,3})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{12};X_{3})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{8};X_{10}|X_{7,9,16})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{16})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{16};X_{10}|X_{12})"}],"expression":[[1,3],[3,-3],[4,3],[5,2],[6,-1],[8,1],[16,-2],[20,2],[33,-3],[35,3],[70,1],[261,-2],[517,-2],[773,2],[2048,6],[2052,-3],[2560,-3],[8198,1],[8262,-1],[32768,1],[32776,-1],[32784,2],[32788,-2],[33088,-1],[33216,1],[33600,1],[33728,-1],[34816,-3],[35328,3]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["3H(X_{1}) - 3H(X_{1,2}) + 3H(X_{3}) + 2H(X_{1,3}) - H(X_{2,3}) + H(X_{4}) - 2H(X_{5}) + 2H(X_{3,5}) - 3H(X_{1,6}) + 3H(X_{1,2,6}) + H(X_{2,3,7}) - 2H(X_{1,3,9}) - 2H(X_{1,3,10}) + 2H(X_{1,3,9,10}) + 6H(X_{12}) - 3H(X_{3,12}) - 3H(X_{10,12}) + H(X_{2,3,14}) - H(X_{2,3,7,14}) + H(X_{16}) - H(X_{4,16}) + 2H(X_{5,16}) - 2H(X_{3,5,16}) - H(X_{7,9,16}) + H(X_{7,8,9,16}) + H(X_{7,9,10,16}) - H(X_{7,8,9,10,16}) - 3H(X_{12,16}) + 3H(X_{10,12,16})>=0","-3H(X_{1}) + 3H(X_{1,2}) - 3H(X_{3}) - 2H(X_{1,3}) + H(X_{2,3}) - H(X_{4}) + 2H(X_{5}) - 2H(X_{3,5}) + 3H(X_{1,6}) - 3H(X_{1,2,6}) - H(X_{2,3,7}) + 2H(X_{1,3,9}) + 2H(X_{1,3,10}) - 2H(X_{1,3,9,10}) - 6H(X_{12}) + 3H(X_{3,12}) + 3H(X_{10,12}) - H(X_{2,3,14}) + H(X_{2,3,7,14}) - H(X_{16}) + H(X_{4,16}) - 2H(X_{5,16}) + 2H(X_{3,5,16}) + H(X_{7,9,16}) - H(X_{7,8,9,16}) - H(X_{7,9,10,16}) + H(X_{7,8,9,10,16}) + 3H(X_{12,16}) - 3H(X_{10,12,16})>=0"],"text":"3H(X_{1}) - 3H(X_{1,2}) + 3H(X_{3}) + 2H(X_{1,3}) - H(X_{2,3}) + H(X_{4}) - 2H(X_{5}) + 2H(X_{3,5}) - 3H(X_{1,6}) + 3H(X_{1,2,6}) + H(X_{2,3,7}) - 2H(X_{1,3,9}) - 2H(X_{1,3,10}) + 2H(X_{1,3,9,10}) + 6H(X_{12}) - 3H(X_{3,12}) - 3H(X_{10,12}) + H(X_{2,3,14}) - H(X_{2,3,7,14}) + H(X_{16}) - H(X_{4,16}) + 2H(X_{5,16}) - 2H(X_{3,5,16}) - H(X_{7,9,16}) + H(X_{7,8,9,16}) + H(X_{7,9,10,16}) - H(X_{7,8,9,10,16}) - 3H(X_{12,16}) + 3H(X_{10,12,16})","type":"equality","witness_value":1.3633538742396922e-13},{"component_width":8,"component_witness_values":[-4.107825191113079e-15,1.3322676295501878e-15,6.217248937900877e-15,3.3306690738754696e-15,2.7644553313166398e-14,-5.551115123125783e-16,2.3314683517128287e-15,-6.661338147750939e-16],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{6};X_{16}|X_{1})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{18}|X_{5})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{4}|X_{7,9,16})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{14};X_{11}|X_{2,3})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{3}|X_{11})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{6};X_{15}|X_{1})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{5}|X_{1})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{13};X_{5}|X_{9,14})"}],"expression":[[1,4],[6,-2],[16,2],[17,-2],[20,-2],[33,-4],[49,2],[1024,1],[1026,-1],[1028,-1],[1030,3],[8198,2],[8448,-1],[8464,1],[9222,-2],[12544,1],[12560,-1],[16385,-3],[16417,3],[32769,1],[32801,-1],[33088,-2],[33096,2],[33216,2],[33224,-2],[131088,-2],[131092,2]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["4H(X_{1}) - 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9H(X_{1,4}) + 7H(X_{5}) - 4H(X_{4,6}) + 4H(X_{7}) + 2H(X_{4,5,7}) + 52H(X_{10}) + 2H(X_{1,10}) + 2H(X_{3,5,8,10}) - 9H(X_{4,11}) + 9H(X_{1,4,11}) + 6H(X_{8,9,12}) - 6H(X_{4,8,9,12}) - 6H(X_{5,8,9,12}) + 6H(X_{4,5,8,9,12}) - 9H(X_{10,12}) + 9H(X_{7,10,12}) - 7H(X_{3,4,8,11,12}) + 7H(X_{3,4,7,8,11,12}) + 4H(X_{5,13}) - 4H(X_{5,7,13}) - 2H(X_{4,5,7,13}) + 9H(X_{1,2,3,4,5,8,9,10,13}) - 4H(X_{5,11,13}) + 5H(X_{6,11,13}) + 4H(X_{5,7,11,13}) + 9H(X_{10,12,13}) - 9H(X_{7,10,12,13}) + H(X_{14}) + 5H(X_{1,3,14}) - 4H(X_{4,14}) + 4H(X_{4,6,14}) + 6H(X_{2,9,10,14}) - 8H(X_{3,7,11,14}) + 8H(X_{3,7,8,11,14}) - 9H(X_{8,13,14}) - 7H(X_{9,13,14}) - 5H(X_{1,3,6,11,13,14}) + 7H(X_{3,4,6,9,10,12,13,14}) - 52H(X_{10,15}) - 2H(X_{1,10,15}) + 6H(X_{1,6,11,15}) + 9H(X_{2,3,6,7,9,10,11,15}) - 9H(X_{2,3,6,7,8,9,10,11,15}) - 6H(X_{1,2,6,9,10,11,14,15}) - 23H(X_{16}) + 27H(X_{4,16}) - 4H(X_{7,16}) + 9H(X_{8,16}) - 27H(X_{10,16}) + H(X_{1,8,9,13,16}) + 2H(X_{1,6,9,12,13,16}) - 2H(X_{1,3,5,6,8,9,10,12,13,16}) + 7H(X_{9,13,14,16}) - H(X_{1,8,9,13,14,16}) - 7H(X_{3,4,6,9,10,12,13,14,16}) + 27H(X_{10,15,16}) - 6H(X_{4,5,7,12,13,14,15,16}) + 6H(X_{4,5,6,7,10,12,13,14,15,16}) + 9H(X_{8,17}) - 27H(X_{10,17}) + 3H(X_{3,4,14,17}) + 27H(X_{10,15,17}) + 4H(X_{12,15,17}) - 4H(X_{2,12,15,17}) - 4H(X_{12,13,15,17}) + 4H(X_{2,12,13,15,17}) + 2H(X_{8,16,17}) - 18H(X_{1,8,16,17}) - 11H(X_{2,8,16,17}) + 18H(X_{1,2,8,16,17}) + 9H(X_{8,9,16,17}) - 9H(X_{2,8,9,16,17}) - 5H(X_{18}) - 2H(X_{4,5,7,18}) + 5H(X_{11,18}) + 3H(X_{5,6,8,11,12,18}) + 2H(X_{4,5,7,13,18}) + 8H(X_{3,4,5,6,7,11,14,18}) - 8H(X_{3,4,5,6,7,8,11,14,18}) + 7H(X_{1,3,4,5,8,10,11,12,14,18}) - 7H(X_{1,3,4,5,7,8,10,11,12,14,18}) - 3H(X_{2,7,8,10,12,13,15,18}) + 3H(X_{2,4,7,8,10,12,13,15,18}) + 3H(X_{2,5,7,8,10,12,13,15,18}) - 3H(X_{2,4,5,7,8,10,12,13,15,18}) + 7H(X_{1,3,4,6,7,8,14,16,18}) - 7H(X_{1,3,4,5,6,7,8,14,16,18}) + 9H(X_{1,2,4,5,7,8,11,12,13,14,15,16,18}) - 9H(X_{1,3,10,12,13,17,18}) - 3H(X_{3,4,5,6,8,11,12,14,17,18}) + 5H(X_{1,3,5,6,8,9,10,13,14,17,18}) - 5H(X_{1,3,5,6,8,9,10,11,13,14,17,18}) - 2H(X_{8,16,17,18}) + 2H(X_{2,8,16,17,18}) + 9H(X_{1,2,3,6,7,8,9,10,12,13,16,17,18}) + 9H(X_{6,7,11,12,14,15,16,17,18}) - 9H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18})","witness_value":145.18691662091356},"k":16,"logical_conditions":[{"component_width":8,"component_witness_values":[8.881784197001252e-16,-1.3433698597964394e-14,3.1086244689504383e-15,2.6645352591003757e-15,1.4876988529977098e-14,7.882583474838611e-15,1.1435297153639112e-14,-7.549516567451064e-15],"components":[{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{18}|X_{8,16,17})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{16};X_{7})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{4};X_{5}|X_{8,9,12})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{13};X_{2}|X_{12,15,17})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{11};X_{7}|X_{5,13})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{14};X_{6}|X_{4})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{15};X_{1}|X_{10})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{18};X_{13}|X_{4,5,7})"}],"expression":[[8,2],[40,-2],[64,2],[88,1],[512,-1],[513,1],[2432,3],[2440,-3],[2448,-3],[2456,3],[4112,2],[4176,-2],[4184,-1],[5136,-2],[5200,2],[8200,-2],[8232,2],[16896,1],[16897,-1],[32768,2],[32832,-2],[83968,2],[83970,-2],[88064,-2],[88066,2],[98432,1],[98434,-1],[131160,-1],[135256,1],[229504,-1],[229506,1]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["2H(X_{4}) - 2H(X_{4,6}) + 2H(X_{7}) + H(X_{4,5,7}) - H(X_{10}) + H(X_{1,10}) + 3H(X_{8,9,12}) - 3H(X_{4,8,9,12}) - 3H(X_{5,8,9,12}) + 3H(X_{4,5,8,9,12}) + 2H(X_{5,13}) - 2H(X_{5,7,13}) - H(X_{4,5,7,13}) - 2H(X_{5,11,13}) + 2H(X_{5,7,11,13}) - 2H(X_{4,14}) + 2H(X_{4,6,14}) + H(X_{10,15}) - H(X_{1,10,15}) + 2H(X_{16}) - 2H(X_{7,16}) + 2H(X_{12,15,17}) - 2H(X_{2,12,15,17}) - 2H(X_{12,13,15,17}) + 2H(X_{2,12,13,15,17}) + H(X_{8,16,17}) - H(X_{2,8,16,17}) - H(X_{4,5,7,18}) + H(X_{4,5,7,13,18}) - H(X_{8,16,17,18}) + H(X_{2,8,16,17,18})>=0","-2H(X_{4}) + 2H(X_{4,6}) - 2H(X_{7}) - H(X_{4,5,7}) + H(X_{10}) - H(X_{1,10}) - 3H(X_{8,9,12}) + 3H(X_{4,8,9,12}) + 3H(X_{5,8,9,12}) - 3H(X_{4,5,8,9,12}) - 2H(X_{5,13}) + 2H(X_{5,7,13}) + H(X_{4,5,7,13}) + 2H(X_{5,11,13}) - 2H(X_{5,7,11,13}) + 2H(X_{4,14}) - 2H(X_{4,6,14}) - H(X_{10,15}) + H(X_{1,10,15}) - 2H(X_{16}) + 2H(X_{7,16}) - 2H(X_{12,15,17}) + 2H(X_{2,12,15,17}) + 2H(X_{12,13,15,17}) - 2H(X_{2,12,13,15,17}) - H(X_{8,16,17}) + H(X_{2,8,16,17}) + H(X_{4,5,7,18}) - H(X_{4,5,7,13,18}) + H(X_{8,16,17,18}) - H(X_{2,8,16,17,18})>=0"],"text":"2H(X_{4}) - 2H(X_{4,6}) + 2H(X_{7}) + H(X_{4,5,7}) - H(X_{10}) + H(X_{1,10}) + 3H(X_{8,9,12}) - 3H(X_{4,8,9,12}) - 3H(X_{5,8,9,12}) + 3H(X_{4,5,8,9,12}) + 2H(X_{5,13}) - 2H(X_{5,7,13}) - H(X_{4,5,7,13}) - 2H(X_{5,11,13}) + 2H(X_{5,7,11,13}) - 2H(X_{4,14}) + 2H(X_{4,6,14}) + H(X_{10,15}) - H(X_{1,10,15}) + 2H(X_{16}) - 2H(X_{7,16}) + 2H(X_{12,15,17}) - 2H(X_{2,12,15,17}) - 2H(X_{12,13,15,17}) + 2H(X_{2,12,13,15,17}) + H(X_{8,16,17}) - H(X_{2,8,16,17}) - H(X_{4,5,7,18}) + H(X_{4,5,7,13,18}) - H(X_{8,16,17,18}) + H(X_{2,8,16,17,18})","type":"equality","witness_value":-7.205347429817266e-14},{"component_width":8,"component_witness_values":[3.1086244689504383e-15,-1.587618925213974e-14,1.6764367671839864e-14,1.1213252548714081e-14,3.9968028886505635e-15,4.440892098500626e-15,4.884981308350689e-15,1.587618925213974e-14],"components":[{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{17};X_{16}|X_{8})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{16};X_{4})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{11}|X_{4})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{15};X_{17}|X_{10})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{7};X_{13}|X_{10,12})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{9}|X_{8,16,17})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{2};X_{1}|X_{8,16,17})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{15};X_{16}|X_{10})"}],"expression":[[8,2],[9,1],[128,1],[512,-6],[1032,1],[1033,-1],[2560,1],[2624,-1],[6656,-1],[6720,1],[16896,6],[32768,3],[32776,-3],[32896,-1],[33280,3],[49664,-3],[65664,-1],[66048,3],[82432,-3],[98433,2],[98434,1],[98435,-2],[98688,-1],[98690,1]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["2H(X_{4}) + H(X_{1,4}) + H(X_{8}) - 6H(X_{10}) + H(X_{4,11}) - H(X_{1,4,11}) + H(X_{10,12}) - H(X_{7,10,12}) - H(X_{10,12,13}) + H(X_{7,10,12,13}) + 6H(X_{10,15}) + 3H(X_{16}) - 3H(X_{4,16}) - H(X_{8,16}) + 3H(X_{10,16}) - 3H(X_{10,15,16}) - H(X_{8,17}) + 3H(X_{10,17}) - 3H(X_{10,15,17}) + 2H(X_{1,8,16,17}) + H(X_{2,8,16,17}) - 2H(X_{1,2,8,16,17}) - H(X_{8,9,16,17}) + H(X_{2,8,9,16,17})>=0","-2H(X_{4}) - H(X_{1,4}) - H(X_{8}) + 6H(X_{10}) - H(X_{4,11}) + H(X_{1,4,11}) - H(X_{10,12}) + H(X_{7,10,12}) + H(X_{10,12,13}) - H(X_{7,10,12,13}) - 6H(X_{10,15}) - 3H(X_{16}) + 3H(X_{4,16}) + H(X_{8,16}) - 3H(X_{10,16}) + 3H(X_{10,15,16}) + H(X_{8,17}) - 3H(X_{10,17}) + 3H(X_{10,15,17}) - 2H(X_{1,8,16,17}) - H(X_{2,8,16,17}) + 2H(X_{1,2,8,16,17}) + H(X_{8,9,16,17}) - H(X_{2,8,9,16,17})>=0"],"text":"2H(X_{4}) + H(X_{1,4}) + H(X_{8}) - 6H(X_{10}) + H(X_{4,11}) - H(X_{1,4,11}) + H(X_{10,12}) - H(X_{7,10,12}) - H(X_{10,12,13}) + H(X_{7,10,12,13}) + 6H(X_{10,15}) + 3H(X_{16}) - 3H(X_{4,16}) - H(X_{8,16}) + 3H(X_{10,16}) - 3H(X_{10,15,16}) - 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7H(X_{4}) + H(X_{2,4}) + 10H(X_{6}) + 20H(X_{3,6}) + 13H(X_{4,9}) - 11H(X_{2,4,9}) + 10H(X_{2,4,8,9}) + 33H(X_{10}) - 20H(X_{6,10}) + 8H(X_{4,7,10}) - H(X_{8,10}) + 10H(X_{11}) - 2H(X_{2,9,10,11}) + 2H(X_{1,2,9,10,11}) + 2H(X_{2,8,9,10,11}) - 2H(X_{1,2,8,9,10,11}) - 41H(X_{10,12}) + 20H(X_{6,10,12}) - 2H(X_{2,5,7,10,12}) + H(X_{8,10,12}) + 2H(X_{2,5,7,8,10,12}) + 6H(X_{1,2,3,4,6,8,11,12}) + 10H(X_{4,13}) - 10H(X_{4,9,13}) - 10H(X_{4,9,12,13}) + 10H(X_{6,8,14}) - 20H(X_{10,14}) + 20H(X_{10,12,14}) + 9H(X_{2,6,7,11,13,14}) - 7H(X_{4,8,15}) - H(X_{4,5,6,13,14,15}) + H(X_{4,5,6,10,13,14,15}) + 7H(X_{1,2,3,4,6,8,12,13,14,15}) - 4H(X_{16}) + 17H(X_{3,16}) - H(X_{2,3,16}) + H(X_{1,2,3,16}) - 20H(X_{3,6,16}) + 5H(X_{2,10,16}) + 4H(X_{5,11,16}) + H(X_{4,5,6,8,13,14,15,16}) - H(X_{4,5,6,8,10,13,14,15,16}) + 10H(X_{2,3,4,5,7,9,12,17}) - 10H(X_{2,3,4,5,7,8,9,12,17}) + 7H(X_{4,5,8,15,17}) - 6H(X_{2,3,8,9,12,15,17}) - 9H(X_{2,11,12,13,15,17}) + 9H(X_{1,2,3,11,12,13,15,17}) + 6H(X_{2,3,8,9,12,14,15,17}) - 7H(X_{1,2,3,4,5,6,8,12,13,14,15,17}) + 4H(X_{2,12,16,17}) - 4H(X_{2,5,11,12,16,17}) + 6H(X_{3,7,9,10,12,15,16,17}) - 6H(X_{3,4,7,9,10,12,15,16,17}) + 9H(X_{9,18}) + 2H(X_{2,5,6,7,10,12,18}) - 2H(X_{2,5,6,7,8,10,12,18}) + 8H(X_{2,3,4,6,7,8,10,11,13,18}) + 10H(X_{4,9,11,12,13,18}) - 20H(X_{14,18}) + 20H(X_{10,14,18}) - 9H(X_{9,15,18}) + H(X_{9,16,18}) - H(X_{9,15,16,18}) - 10H(X_{9,17,18}) + 10H(X_{9,15,17,18}) + 6H(X_{1,2,3,5,8,9,12,15,17,18}) - 6H(X_{1,2,3,5,8,9,12,14,15,17,18}) + 10H(X_{2,3,4,6,7,8,10,16,17,18}) - 10H(X_{2,3,4,6,7,8,10,11,16,17,18}) + 2H(X_{19}) - 3H(X_{3,19}) + 2H(X_{4,19}) - 2H(X_{4,9,19}) - 2H(X_{11,19}) - 21H(X_{2,3,11,19}) + 30H(X_{2,3,4,11,19}) + H(X_{2,3,5,11,19}) - 10H(X_{13,19}) + 10H(X_{4,9,12,13,19}) + 8H(X_{9,10,15,19}) + 3H(X_{3,16,19}) + 8H(X_{1,2,3,5,6,11,16,19}) - 8H(X_{1,2,3,4,5,6,7,10,11,16,19}) + 21H(X_{2,3,11,17,19}) - 30H(X_{2,3,4,11,17,19}) - H(X_{2,3,5,11,17,19}) + 5H(X_{1,3,4,6,7,9,11,13,15,17,19}) + H(X_{2,3,4,8,10,12,15,16,17,19}) - H(X_{1,2,3,4,8,10,12,15,16,17,19}) - 5H(X_{1,2,3,4,6,7,9,10,11,13,15,16,17,19}) + 10H(X_{1,2,3,5,6,7,8,9,11,12,13,15,16,17,19}) - 2H(X_{18,19}) + 2H(X_{11,18,19}) - 10H(X_{2,3,11,18,19}) + 10H(X_{5,9,10,11,18,19}) - 10H(X_{5,6,9,10,11,18,19}) - 10H(X_{4,9,11,12,13,18,19}) + 20H(X_{14,18,19}) - 20H(X_{10,14,18,19}) - 8H(X_{2,3,4,6,7,8,9,10,11,13,15,18,19}) - 6H(X_{2,4,17,18,19}) + 6H(X_{2,4,10,17,18,19}) + 10H(X_{2,3,11,17,18,19}) + 6H(X_{2,4,5,7,8,12,17,18,19}) - 6H(X_{2,4,5,7,8,10,12,17,18,19}) + 10H(X_{1,2,3,5,13,17,18,19}) - 10H(X_{1,2,3,5,6,8,13,14,17,18,19}) + 6H(X_{5,7,9,10,13,14,15,16,17,18,19}) - 6H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19})","witness_value":195.25643667435637},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[-3.6415315207705135e-14,2.298161660974074e-14,9.769962616701378e-15,2.1094237467877974e-15,1.1102230246251565e-15,1.9984014443252818e-15,1.1546319456101628e-14,-3.4083846855992306e-14],"components":[{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{16};X_{6}|X_{3})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{12};X_{6}|X_{10})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{17};X_{4}|X_{2,3,11,19})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{12};X_{14}|X_{10})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{15};X_{17}|X_{9,18})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{10};X_{19}|X_{14,18})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{17};X_{18}|X_{2,3,11,19})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{9};X_{13}|X_{4})"}],"expression":[[4,-2],[8,-1],[36,2],[264,1],[512,4],[544,-2],[2560,-4],[2592,2],[4104,1],[4360,-1],[8704,-2],[10752,2],[32772,2],[32804,-2],[131328,1],[139264,-2],[139776,2],[147712,-1],[196864,-1],[213248,1],[263174,-2],[263182,3],[328710,2],[328718,-3],[394246,-1],[401408,2],[401920,-2],[459782,1]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["-2H(X_{3}) - H(X_{4}) + 2H(X_{3,6}) + H(X_{4,9}) + 4H(X_{10}) - 2H(X_{6,10}) - 4H(X_{10,12}) + 2H(X_{6,10,12}) + H(X_{4,13}) - H(X_{4,9,13}) - 2H(X_{10,14}) + 2H(X_{10,12,14}) + 2H(X_{3,16}) - 2H(X_{3,6,16}) + H(X_{9,18}) - 2H(X_{14,18}) + 2H(X_{10,14,18}) - H(X_{9,15,18}) - H(X_{9,17,18}) + H(X_{9,15,17,18}) - 2H(X_{2,3,11,19}) + 3H(X_{2,3,4,11,19}) + 2H(X_{2,3,11,17,19}) - 3H(X_{2,3,4,11,17,19}) - H(X_{2,3,11,18,19}) + 2H(X_{14,18,19}) - 2H(X_{10,14,18,19}) + H(X_{2,3,11,17,18,19})>=0","2H(X_{3}) + H(X_{4}) - 2H(X_{3,6}) - H(X_{4,9}) - 4H(X_{10}) + 2H(X_{6,10}) + 4H(X_{10,12}) - 2H(X_{6,10,12}) - H(X_{4,13}) + H(X_{4,9,13}) + 2H(X_{10,14}) - 2H(X_{10,12,14}) - 2H(X_{3,16}) + 2H(X_{3,6,16}) - H(X_{9,18}) + 2H(X_{14,18}) - 2H(X_{10,14,18}) + H(X_{9,15,18}) + H(X_{9,17,18}) - H(X_{9,15,17,18}) + 2H(X_{2,3,11,19}) - 3H(X_{2,3,4,11,19}) - 2H(X_{2,3,11,17,19}) + 3H(X_{2,3,4,11,17,19}) + H(X_{2,3,11,18,19}) - 2H(X_{14,18,19}) + 2H(X_{10,14,18,19}) - H(X_{2,3,11,17,18,19})>=0"],"text":"-2H(X_{3}) - H(X_{4}) + 2H(X_{3,6}) + H(X_{4,9}) + 4H(X_{10}) - 2H(X_{6,10}) - 4H(X_{10,12}) + 2H(X_{6,10,12}) + H(X_{4,13}) - H(X_{4,9,13}) - 2H(X_{10,14}) + 2H(X_{10,12,14}) + 2H(X_{3,16}) - 2H(X_{3,6,16}) + H(X_{9,18}) - 2H(X_{14,18}) + 2H(X_{10,14,18}) - H(X_{9,15,18}) - H(X_{9,17,18}) + H(X_{9,15,17,18}) - 2H(X_{2,3,11,19}) + 3H(X_{2,3,4,11,19}) + 2H(X_{2,3,11,17,19}) - 3H(X_{2,3,4,11,17,19}) - H(X_{2,3,11,18,19}) + 2H(X_{14,18,19}) - 2H(X_{10,14,18,19}) + H(X_{2,3,11,17,18,19})","type":"equality","witness_value":-1.3644640972643174e-13},{"component_width":8,"component_witness_values":[2.609024107869118e-14,-6.472600233564663e-14,-1.8207657603852567e-14,3.552713678800501e-15,4.6629367034256575e-15,-3.219646771412954e-14,7.993605777301127e-15,-3.5860203695392556e-14],"components":[{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{12};X_{8}|X_{10})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{9};X_{2}|X_{4})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{11};X_{18}|X_{19})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{1};X_{8}|X_{2,9,10,11})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{15};X_{16}|X_{9,18})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{16};X_{19}|X_{3})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{17};X_{5}|X_{2,3,11,19})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{9};X_{19}|X_{4})"}],"expression":[[4,3],[8,-3],[10,1],[264,3],[266,-1],[512,1],[640,-1],[1794,-2],[1795,2],[1922,2],[1923,-2],[2560,-1],[2688,1],[32772,-3],[131328,-1],[147712,1],[164096,1],[180480,-1],[262144,2],[262148,-3],[262152,2],[262408,-2],[263168,-2],[263174,-1],[263190,1],[294916,3],[328710,1],[328726,-1],[393216,-2],[394240,2]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["3H(X_{3}) - 3H(X_{4}) + H(X_{2,4}) + 3H(X_{4,9}) - H(X_{2,4,9}) + H(X_{10}) - H(X_{8,10}) - 2H(X_{2,9,10,11}) + 2H(X_{1,2,9,10,11}) + 2H(X_{2,8,9,10,11}) - 2H(X_{1,2,8,9,10,11}) - H(X_{10,12}) + H(X_{8,10,12}) - 3H(X_{3,16}) - H(X_{9,18}) + H(X_{9,15,18}) + H(X_{9,16,18}) - H(X_{9,15,16,18}) + 2H(X_{19}) - 3H(X_{3,19}) + 2H(X_{4,19}) - 2H(X_{4,9,19}) - 2H(X_{11,19}) - H(X_{2,3,11,19}) + H(X_{2,3,5,11,19}) + 3H(X_{3,16,19}) + H(X_{2,3,11,17,19}) - H(X_{2,3,5,11,17,19}) - 2H(X_{18,19}) + 2H(X_{11,18,19})>=0","-3H(X_{3}) + 3H(X_{4}) - H(X_{2,4}) - 3H(X_{4,9}) + H(X_{2,4,9}) - H(X_{10}) + H(X_{8,10}) + 2H(X_{2,9,10,11}) - 2H(X_{1,2,9,10,11}) - 2H(X_{2,8,9,10,11}) + 2H(X_{1,2,8,9,10,11}) + H(X_{10,12}) - H(X_{8,10,12}) + 3H(X_{3,16}) + H(X_{9,18}) - H(X_{9,15,18}) - H(X_{9,16,18}) + H(X_{9,15,16,18}) - 2H(X_{19}) + 3H(X_{3,19}) - 2H(X_{4,19}) + 2H(X_{4,9,19}) + 2H(X_{11,19}) + H(X_{2,3,11,19}) - H(X_{2,3,5,11,19}) - 3H(X_{3,16,19}) - H(X_{2,3,11,17,19}) + H(X_{2,3,5,11,17,19}) + 2H(X_{18,19}) - 2H(X_{11,18,19})>=0"],"text":"3H(X_{3}) - 3H(X_{4}) + H(X_{2,4}) + 3H(X_{4,9}) - H(X_{2,4,9}) + H(X_{10}) - H(X_{8,10}) - 2H(X_{2,9,10,11}) + 2H(X_{1,2,9,10,11}) + 2H(X_{2,8,9,10,11}) - 2H(X_{1,2,8,9,10,11}) - H(X_{10,12}) + H(X_{8,10,12}) - 3H(X_{3,16}) - H(X_{9,18}) + H(X_{9,15,18}) + H(X_{9,16,18}) - H(X_{9,15,16,18}) + 2H(X_{19}) - 3H(X_{3,19}) + 2H(X_{4,19}) - 2H(X_{4,9,19}) - 2H(X_{11,19}) - H(X_{2,3,11,19}) + H(X_{2,3,5,11,19}) + 3H(X_{3,16,19}) + H(X_{2,3,11,17,19}) - H(X_{2,3,5,11,17,19}) - 2H(X_{18,19}) + 2H(X_{11,18,19})","type":"equality","witness_value":-9.880984919163893e-15}],"m_eq":2,"master_seed":20260825,"method":"dag_v2","n":19,"next_action_index":19,"one_sided_condition_count":0,"prompt_tokens":2206,"recipe_index":2,"remaining_action_coordinate_l1":581.0,"remaining_certificate_length":3,"remaining_coefficient_l1":20.0,"remaining_coefficient_max":10.0,"remaining_condition_actions":2,"remaining_shannon_actions":1,"remaining_shannon_kind_counts":{"conditional_entropy":1},"residual_mode":"condition","schema_version":"reviewer-benchmark-v1","selected_coordinate_l1":565.0,"selected_coordinate_support":49,"selected_effective_n":19,"selected_residual_sparse":[[4,-17],[8,-13],[10,1],[36,20],[264,13],[266,-1],[512,41],[544,-20],[640,-1],[1794,-2],[1795,2],[1922,2],[1923,-2],[2560,-41],[2592,20],[2688,1],[4104,10],[4360,-10],[8704,-20],[10752,20],[13410,9],[32772,17],[32804,-20],[131328,9],[139264,-20],[139776,20],[147712,-9],[164096,1],[180480,-1],[196864,-10],[213248,10],[262144,2],[262148,-3],[262152,2],[262408,-2],[263168,-2],[263174,-21],[263182,30],[263190,1],[294916,3],[328710,21],[328718,-30],[328726,-1],[393216,-2],[394240,2],[394246,-10],[401408,20],[401920,-20],[459782,10]],"selector_source_sha256":"83a328168eac42cedd3412a9c8b5624b8683a52a14042b5defbccf6d833efefe","sha256":"a2f3664415151c3e9c8716e37f3692170bc15cabb77835501d42641542bdbb84","step_idx":19,"subset_size_profile":{"a":[6],"b":[0],"c":[0]},"target_tokens":704,"tokenizer":{"add_special_tokens":false,"is_fast":true,"kind":"huggingface","model":"Qwen/Qwen3-0.6B","requested_revision":"c1899de289a04d12100db370d81485cdf75e47ca","resolved_commit":"c1899de289a04d12100db370d81485cdf75e47ca","tokenizer_class":"Qwen2TokenizerFast","transformers_version":"4.55.2","trust_remote_code":true,"use_fast":true},"trajectory_length":23,"trajectory_sha256":"fe917862509f6d992d8220c16578a6e1f06ef24f237ab43e1392f6eec3aac443","witness":{"alphabet":[0,1],"cpt_p_x1":[{"0":[2,5],"1":[2,5],"10":[1,2],"11":[2,3],"12":[2,5],"13":[1,5],"14":[1,3],"15":[1,3],"2":[3,5],"3":[1,3],"4":[1,5],"5":[1,3],"6":[1,4],"7":[1,4],"8":[2,5],"9":[1,5]},{"0":[3,4],"1":[4,5]},{"0":[1,5],"1":[3,4]},{"0":[2,3],"1":[3,4],"2":[3,5],"3":[4,5],"4":[3,5],"5":[3,4],"6":[4,5],"7":[1,5]},{"0":[2,5],"1":[4,5],"2":[3,5],"3":[1,5],"4":[4,5],"5":[1,2],"6":[1,3],"7":[1,3]},{"0":[1,4],"1":[1,3],"10":[3,4],"11":[4,5],"12":[1,2],"13":[3,4],"14":[3,4],"15":[2,5],"2":[1,4],"3":[3,4],"4":[1,3],"5":[2,5],"6":[3,4],"7":[1,3],"8":[1,3],"9":[1,2]},{"0":[2,3],"1":[1,2]},{"0":[1,5],"1":[4,5],"2":[2,3],"3":[3,5]},{"0":[1,2],"1":[3,5]},{"0":[3,5],"1":[3,4],"2":[1,4],"3":[4,5]},{"0":[1,2],"1":[2,3]},{"0":[1,5],"1":[1,4]},{"0":[1,4],"1":[1,3]},{"0":[4,5]},{"0":[1,5],"1":[2,5],"2":[1,2],"3":[2,5]},{"0":[4,5],"1":[1,5]},{"0":[3,5],"1":[1,4],"10":[3,5],"11":[1,4],"12":[1,4],"13":[2,3],"14":[1,5],"15":[1,5],"2":[1,3],"3":[1,5],"4":[1,4],"5":[2,5],"6":[1,4],"7":[2,3],"8":[1,2],"9":[1,2]},{"0":[1,2],"1":[3,4]},{"0":[1,4],"1":[2,5],"2":[3,5],"3":[4,5]}],"cpt_parent_assignment_order":"integer bits follow the listed parent order, least-significant first","edge_count":35,"family":"dag_v2","graph_family":"forest_plus","joint_state_count":524288,"max_indegree":4,"parents":[[9,8,10,1],[10],[13],[13,2,11],[3,18,10],[13,2,9,1],[15],[13,17],[3],[13,17],[18],[9],[17],[],[17,8],[2],[2,18,10,1],[2],[12,7]],"sampled_density":0.3,"state_cap":1048576,"topological_order":[13,2,17,9,11,12,7,3,8,15,18,6,10,14,1,16,4,5,0]},"witness_evaluation":{"checks":{"direct_equals_remaining_actions":true,"equality_contribution_zero":true,"shannon_contribution_nonnegative":true,"target_nonnegative":true},"direct_target_value":47.74385677138605,"evaluation_source":"construction_entropy_oracle","pass":true,"remaining_action_value":47.74385677138606,"remaining_equality_value":-1.3743450821834813e-12,"remaining_shannon_value":47.743856771387435},"witness_sha256":"9b12278590fd7062f6a7be796ec1e30b8a21fccf864ad4db518fe46bd1d9a3b8","witness_variant":null}
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2H(X_{1,2,4,8,9,14,15,17,19,20}) - 21H(X_{18,19,20}) + 7H(X_{2,18,19,20}) + 14H(X_{5,18,19,20}) + 4H(X_{2,3,4,5,6,7,8,9,10,11,15,16,18,19,20}) + 7H(X_{1,2,7,9,10,12,14,17,18,19,20}) - 7H(X_{1,2,7,9,10,12,13,14,16,17,18,19,20}) - 5H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20})","witness_value":133.1482030372623},"k":16,"logical_conditions":[{"component_width":12,"component_witness_values":[-4.551914400963142e-14,2.708944180085382e-14,4.440892098500626e-16,5.773159728050814e-15,-8.43769498715119e-15,-6.350475700855895e-14,1.5987211554602254e-14,4.929390229335695e-14,1.283417816466681e-13,-5.6621374255882984e-14,1.8207657603852567e-14,-3.4861002973229915e-14],"components":[{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{2};X_{7}|X_{16,20})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{14};X_{19}|X_{1,8,17})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{18};X_{5}|X_{19,20})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{18};X_{2}|X_{19,20})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{12};X_{11}|X_{3,10})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{7};X_{11}|X_{1})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{9};X_{7}|X_{3,4,11,12})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{16};X_{3}|X_{12,17,19})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{11})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{10}|X_{16,20})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{11};X_{19}|X_{4,10,13})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{19};X_{4}|X_{8,10})"}],"expression":[[1,2],[4,-3],[65,-2],[516,-1],[640,-2],[648,2],[1024,-3],[1025,-2],[1028,3],[1089,2],[1540,1],[2564,1],[3084,-3],[3148,3],[3340,3],[3404,-3],[3588,-1],[4616,-2],[5640,2],[65665,-1],[73857,1],[262784,2],[262792,-2],[266760,2],[267784,-2],[327809,1],[329728,-1],[329732,1],[336001,-1],[362496,1],[362500,-1],[557056,-3],[557058,3],[557120,2],[557122,-2],[557568,1],[557570,-1],[786432,3],[786434,-1],[786448,-2],[917504,-3],[917506,1],[917520,2]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["2H(X_{1}) - 3H(X_{3}) - 2H(X_{1,7}) - H(X_{3,10}) - 2H(X_{8,10}) + 2H(X_{4,8,10}) - 3H(X_{11}) - 2H(X_{1,11}) + 3H(X_{3,11}) + 2H(X_{1,7,11}) + H(X_{3,10,11}) + H(X_{3,10,12}) - 3H(X_{3,4,11,12}) + 3H(X_{3,4,7,11,12}) + 3H(X_{3,4,9,11,12}) - 3H(X_{3,4,7,9,11,12}) - H(X_{3,10,11,12}) - 2H(X_{4,10,13}) + 2H(X_{4,10,11,13}) - H(X_{1,8,17}) + H(X_{1,8,14,17}) + 2H(X_{8,10,19}) - 2H(X_{4,8,10,19}) + 2H(X_{4,10,13,19}) - 2H(X_{4,10,11,13,19}) + H(X_{1,8,17,19}) - H(X_{12,17,19}) + H(X_{3,12,17,19}) - H(X_{1,8,14,17,19}) + H(X_{12,16,17,19}) - H(X_{3,12,16,17,19}) - 3H(X_{16,20}) + 3H(X_{2,16,20}) + 2H(X_{7,16,20}) - 2H(X_{2,7,16,20}) + H(X_{10,16,20}) - H(X_{2,10,16,20}) + 3H(X_{19,20}) - H(X_{2,19,20}) - 2H(X_{5,19,20}) - 3H(X_{18,19,20}) + H(X_{2,18,19,20}) + 2H(X_{5,18,19,20})>=0","-2H(X_{1}) + 3H(X_{3}) + 2H(X_{1,7}) + H(X_{3,10}) + 2H(X_{8,10}) - 2H(X_{4,8,10}) + 3H(X_{11}) + 2H(X_{1,11}) - 3H(X_{3,11}) - 2H(X_{1,7,11}) - H(X_{3,10,11}) - H(X_{3,10,12}) + 3H(X_{3,4,11,12}) - 3H(X_{3,4,7,11,12}) - 3H(X_{3,4,9,11,12}) + 3H(X_{3,4,7,9,11,12}) + H(X_{3,10,11,12}) + 2H(X_{4,10,13}) - 2H(X_{4,10,11,13}) + H(X_{1,8,17}) - H(X_{1,8,14,17}) - 2H(X_{8,10,19}) + 2H(X_{4,8,10,19}) - 2H(X_{4,10,13,19}) + 2H(X_{4,10,11,13,19}) - H(X_{1,8,17,19}) + H(X_{12,17,19}) - H(X_{3,12,17,19}) + H(X_{1,8,14,17,19}) - H(X_{12,16,17,19}) + H(X_{3,12,16,17,19}) + 3H(X_{16,20}) - 3H(X_{2,16,20}) - 2H(X_{7,16,20}) + 2H(X_{2,7,16,20}) - H(X_{10,16,20}) + H(X_{2,10,16,20}) - 3H(X_{19,20}) + H(X_{2,19,20}) + 2H(X_{5,19,20}) + 3H(X_{18,19,20}) - H(X_{2,18,19,20}) - 2H(X_{5,18,19,20})>=0"],"text":"2H(X_{1}) - 3H(X_{3}) - 2H(X_{1,7}) - H(X_{3,10}) - 2H(X_{8,10}) + 2H(X_{4,8,10}) - 3H(X_{11}) - 2H(X_{1,11}) + 3H(X_{3,11}) + 2H(X_{1,7,11}) + H(X_{3,10,11}) + H(X_{3,10,12}) - 3H(X_{3,4,11,12}) + 3H(X_{3,4,7,11,12}) + 3H(X_{3,4,9,11,12}) - 3H(X_{3,4,7,9,11,12}) - H(X_{3,10,11,12}) - 2H(X_{4,10,13}) + 2H(X_{4,10,11,13}) - H(X_{1,8,17}) + H(X_{1,8,14,17}) + 2H(X_{8,10,19}) - 2H(X_{4,8,10,19}) + 2H(X_{4,10,13,19}) - 2H(X_{4,10,11,13,19}) + H(X_{1,8,17,19}) - H(X_{12,17,19}) + H(X_{3,12,17,19}) - H(X_{1,8,14,17,19}) + H(X_{12,16,17,19}) - 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14H(X_{10,13}) - 14H(X_{7,15}) - 7H(X_{7,8,15}) - 14H(X_{13,16}) + 14H(X_{10,13,16}) + 21H(X_{4,17}) - 21H(X_{4,12,17}) + 7H(X_{3,18}) - 7H(X_{1,3,18}) + 7H(X_{13,18}) + 7H(X_{6,13,18}) - 7H(X_{8,13,18}) - 7H(X_{6,8,13,18}) + 21H(X_{7,15,18}) - 14H(X_{13,17,18}) + 14H(X_{8,13,17,18}) + 14H(X_{7,15,19}) + 7H(X_{7,8,15,19}) - 21H(X_{7,15,18,19}) - 21H(X_{17,20}) + 21H(X_{12,17,20})","witness_value":7.793765632868599e-14}],"certificate_length":22,"certificate_order":"greedy_negative_mass_condition_late","condition_coordinate_supports":[25,24],"condition_rank":2,"condition_tokens":[296,295,296,296],"condition_width":8,"config_sha256":"1b1f298ed012071c85e9cfb286fe5dcf7c183383de92041754d8a7802f426421","equality_condition_count":2,"full_tokens":1933,"generator_dependency_sha256":{"data/finite_field.py":"c9738c10a16ea2cfb3cde346a23673d1536dbfce491f8871346f441b9b4c2a6e","data/witness_first.py":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","data/witness_v2_algebraic.py":"ffbd48eb673a3f6f842327d00431285bbae8963a7e20f6956e37e544fa37165f","data/witness_v2_graph.py":"c81adcfefcf6ffc0e816f1b338af13323bc80e06293e8cac6b28a3b0ae2efc8a"},"generator_schema_version":"witness-first-v2","generator_source_sha256":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","id":"e2n-witness-first-v2-n20-dag_v2-351e1ad88626aefc","initial_target":{"coordinate_support":108,"effective_n":20,"expression":[[4,-7],[5,7],[32,-4],[65,20],[128,-9],[176,4],[256,-5],[352,-8],[544,4],[576,-20],[1024,-72],[1028,10],[1040,20],[1056,30],[4096,14],[4608,-14],[9216,50],[9220,-10],[9232,-20],[16448,-4],[16502,3],[16576,-7],[28672,8],[32832,-10],[33088,10],[33792,20],[36864,-14],[37376,14],[41984,-20],[49216,-10],[65544,21],[65600,20],[65601,-20],[65664,9],[67592,-21],[73952,1],[82210,1],[98306,-2],[98564,-5],[98816,-5],[99302,5],[131076,7],[131077,-7],[131136,-20],[131648,20],[131944,8],[132096,30],[132128,-30],[135168,7],[135200,7],[135296,-7],[135328,-7],[139586,8],[143552,-10],[143556,10],[147520,21],[149826,4],[150002,-4],[163904,10],[164160,-10],[174372,10],[200704,-14],[200832,14],[233992,5],[262496,5],[268416,10],[278592,4],[278720,7],[295171,10],[303104,3],[311360,10],[319606,-3],[332480,10],[409664,-21],[471232,10],[471236,-10],[506852,-10],[515179,2],[526028,8],[533696,10],[540704,-3],[540832,3],[547119,-1],[554700,-8],[555311,1],[589824,-21],[591872,21],[622868,5],[623106,2],[623232,5],[623606,-5],[634226,8],[684304,5],[690394,9],[755930,-9],[758408,-5],[765818,-8],[828867,-10],[946544,-5],[966365,1],[993318,8],[994342,-8],[998700,4],[999212,-4],[1008175,3],[1008303,-3],[1039979,-2],[1048575,-1]],"text":"-7H(X_{3}) + 7H(X_{1,3}) - 4H(X_{6}) + 20H(X_{1,7}) - 9H(X_{8}) + 4H(X_{5,6,8}) - 5H(X_{9}) - 8H(X_{6,7,9}) + 4H(X_{6,10}) - 20H(X_{7,10}) - 72H(X_{11}) + 10H(X_{3,11}) + 20H(X_{5,11}) + 30H(X_{6,11}) + 14H(X_{13}) - 14H(X_{10,13}) + 50H(X_{11,14}) - 10H(X_{3,11,14}) - 20H(X_{5,11,14}) - 4H(X_{7,15}) + 3H(X_{2,3,5,6,7,15}) - 7H(X_{7,8,15}) + 8H(X_{13,14,15}) - 10H(X_{7,16}) + 10H(X_{7,9,16}) + 20H(X_{11,16}) - 14H(X_{13,16}) + 14H(X_{10,13,16}) - 20H(X_{11,14,16}) - 10H(X_{7,15,16}) + 21H(X_{4,17}) + 20H(X_{7,17}) - 20H(X_{1,7,17}) + 9H(X_{8,17}) - 21H(X_{4,12,17}) + H(X_{6,7,8,14,17}) + H(X_{2,6,9,15,17}) - 2H(X_{2,16,17}) - 5H(X_{3,9,16,17}) - 5H(X_{10,16,17}) + 5H(X_{2,3,6,7,8,9,10,16,17}) + 7H(X_{3,18}) - 7H(X_{1,3,18}) - 20H(X_{7,18}) + 20H(X_{7,10,18}) + 8H(X_{4,6,7,9,10,18}) + 30H(X_{11,18}) - 30H(X_{6,11,18}) + 7H(X_{13,18}) + 7H(X_{6,13,18}) - 7H(X_{8,13,18}) - 7H(X_{6,8,13,18}) + 8H(X_{2,7,9,14,18}) - 10H(X_{7,8,13,14,18}) + 10H(X_{3,7,8,13,14,18}) + 21H(X_{7,15,18}) + 4H(X_{2,7,9,12,15,18}) - 4H(X_{2,5,6,7,8,9,12,15,18}) + 10H(X_{7,16,18}) - 10H(X_{7,9,16,18}) + 10H(X_{3,6,9,12,14,16,18}) - 14H(X_{13,17,18}) + 14H(X_{8,13,17,18}) + 5H(X_{4,10,13,16,17,18}) + 5H(X_{6,7,9,19}) + 10H(X_{8,12,13,19}) + 4H(X_{7,15,19}) + 7H(X_{7,8,15,19}) + 10H(X_{1,2,9,16,19}) + 3H(X_{14,16,19}) + 10H(X_{7,15,16,19}) - 3H(X_{2,3,5,6,7,14,15,16,19}) + 10H(X_{7,8,10,13,17,19}) - 21H(X_{7,15,18,19}) + 10H(X_{7,8,13,14,17,18,19}) - 10H(X_{3,7,8,13,14,17,18,19}) - 10H(X_{3,6,7,8,9,10,12,13,14,16,17,18,19}) + 2H(X_{1,2,4,6,7,11,12,13,15,16,17,18,19}) + 8H(X_{3,4,7,8,10,11,20}) + 10H(X_{7,8,11,14,20}) - 3H(X_{6,15,20}) + 3H(X_{6,8,15,20}) - H(X_{1,2,3,4,6,9,12,13,15,20}) - 8H(X_{3,4,7,8,10,11,13,14,15,20}) + H(X_{1,2,3,4,6,9,12,13,14,15,20}) - 21H(X_{17,20}) + 21H(X_{12,17,20}) + 5H(X_{3,5,9,16,17,20}) + 2H(X_{2,10,16,17,20}) + 5H(X_{8,10,16,17,20}) - 5H(X_{2,3,5,6,7,8,9,10,16,17,20}) + 8H(X_{2,5,6,7,9,11,12,14,16,17,20}) + 5H(X_{5,9,13,14,15,18,20}) + 9H(X_{2,4,5,7,8,12,16,18,20}) - 9H(X_{2,4,5,7,8,12,16,17,18,20}) - 5H(X_{4,8,10,13,16,17,18,20}) - 8H(X_{2,4,5,6,7,9,10,11,12,14,16,17,18,20}) - 10H(X_{1,2,7,8,9,11,14,16,19,20}) - 5H(X_{5,6,7,9,13,14,15,18,19,20}) + H(X_{1,3,4,5,7,8,10,11,12,13,14,16,18,19,20}) + 8H(X_{2,3,6,12,14,17,18,19,20}) - 8H(X_{2,3,6,11,12,14,17,18,19,20}) + 4H(X_{3,4,6,9,11,12,13,14,17,18,19,20}) - 4H(X_{3,4,6,9,10,11,12,13,14,17,18,19,20}) + 3H(X_{1,2,3,4,6,10,14,15,17,18,19,20}) - 3H(X_{1,2,3,4,6,8,10,14,15,17,18,19,20}) - 2H(X_{1,2,4,6,7,10,11,12,13,15,16,17,18,19,20}) - H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20})","witness_value":94.10469677315476},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[-9.43689570931383e-14,2.0872192862952943e-14,2.1316282072803006e-14,-1.141309269314661e-13,2.7533531010703882e-14,-1.141309269314661e-13,-1.3322676295501878e-14,-1.212363542890671e-13],"components":[{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{6};X_{18}|X_{11})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{19};X_{16}|X_{7,15})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{10};X_{18}|X_{7})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{14};X_{16}|X_{11})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{17};X_{1}|X_{7})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{14};X_{5}|X_{11})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{9};X_{18}|X_{7,16})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{14};X_{3}|X_{11})"}],"expression":[[65,-2],[576,2],[1024,8],[1028,-1],[1040,-2],[1056,-3],[9216,-5],[9220,1],[9232,2],[16448,-1],[32832,1],[33088,-1],[33792,-2],[41984,2],[49216,1],[65600,-2],[65601,2],[131136,2],[131648,-2],[132096,-3],[132128,3],[163904,-1],[164160,1],[278592,1],[311360,-1]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["-2H(X_{1,7}) + 2H(X_{7,10}) + 8H(X_{11}) - H(X_{3,11}) - 2H(X_{5,11}) - 3H(X_{6,11}) - 5H(X_{11,14}) + H(X_{3,11,14}) + 2H(X_{5,11,14}) - H(X_{7,15}) + H(X_{7,16}) - H(X_{7,9,16}) - 2H(X_{11,16}) + 2H(X_{11,14,16}) + H(X_{7,15,16}) - 2H(X_{7,17}) + 2H(X_{1,7,17}) + 2H(X_{7,18}) - 2H(X_{7,10,18}) - 3H(X_{11,18}) + 3H(X_{6,11,18}) - H(X_{7,16,18}) + H(X_{7,9,16,18}) + H(X_{7,15,19}) - H(X_{7,15,16,19})>=0","2H(X_{1,7}) - 2H(X_{7,10}) - 8H(X_{11}) + H(X_{3,11}) + 2H(X_{5,11}) + 3H(X_{6,11}) + 5H(X_{11,14}) - H(X_{3,11,14}) - 2H(X_{5,11,14}) + H(X_{7,15}) - H(X_{7,16}) + H(X_{7,9,16}) + 2H(X_{11,16}) - 2H(X_{11,14,16}) - H(X_{7,15,16}) + 2H(X_{7,17}) - 2H(X_{1,7,17}) - 2H(X_{7,18}) + 2H(X_{7,10,18}) + 3H(X_{11,18}) - 3H(X_{6,11,18}) + H(X_{7,16,18}) - H(X_{7,9,16,18}) - H(X_{7,15,19}) + H(X_{7,15,16,19})>=0"],"text":"-2H(X_{1,7}) + 2H(X_{7,10}) + 8H(X_{11}) - H(X_{3,11}) - 2H(X_{5,11}) - 3H(X_{6,11}) - 5H(X_{11,14}) + H(X_{3,11,14}) + 2H(X_{5,11,14}) - H(X_{7,15}) + H(X_{7,16}) - H(X_{7,9,16}) - 2H(X_{11,16}) + 2H(X_{11,14,16}) + H(X_{7,15,16}) - 2H(X_{7,17}) + 2H(X_{1,7,17}) + 2H(X_{7,18}) - 2H(X_{7,10,18}) - 3H(X_{11,18}) + 3H(X_{6,11,18}) - H(X_{7,16,18}) + H(X_{7,9,16,18}) + H(X_{7,15,19}) - H(X_{7,15,16,19})","type":"equality","witness_value":8.826273045769994e-13},{"component_width":8,"component_witness_values":[-5.417888360170764e-14,-3.9968028886505635e-15,-5.340172748447003e-14,-1.865174681370263e-14,-3.552713678800501e-14,-3.019806626980426e-14,9.769962616701378e-15,-2.842170943040401e-14],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{18}|X_{3})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{19};X_{8}|X_{7,15})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{16};X_{10}|X_{13})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{8};X_{17}|X_{13,18})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{12};X_{4}|X_{17})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{12};X_{20}|X_{17})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{19};X_{18}|X_{7,15})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{8};X_{6}|X_{13,18})"}],"expression":[[4,-1],[5,1],[4096,2],[4608,-2],[16448,-2],[16576,-1],[36864,-2],[37376,2],[65544,3],[67592,-3],[131076,1],[131077,-1],[135168,1],[135200,1],[135296,-1],[135328,-1],[147520,3],[200704,-2],[200832,2],[278592,2],[278720,1],[409664,-3],[589824,-3],[591872,3]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["-H(X_{3}) + H(X_{1,3}) + 2H(X_{13}) - 2H(X_{10,13}) - 2H(X_{7,15}) - H(X_{7,8,15}) - 2H(X_{13,16}) + 2H(X_{10,13,16}) + 3H(X_{4,17}) - 3H(X_{4,12,17}) + H(X_{3,18}) - H(X_{1,3,18}) + H(X_{13,18}) + H(X_{6,13,18}) - H(X_{8,13,18}) - H(X_{6,8,13,18}) + 3H(X_{7,15,18}) - 2H(X_{13,17,18}) + 2H(X_{8,13,17,18}) + 2H(X_{7,15,19}) + H(X_{7,8,15,19}) - 3H(X_{7,15,18,19}) - 3H(X_{17,20}) + 3H(X_{12,17,20})>=0","H(X_{3}) - H(X_{1,3}) - 2H(X_{13}) + 2H(X_{10,13}) + 2H(X_{7,15}) + H(X_{7,8,15}) + 2H(X_{13,16}) - 2H(X_{10,13,16}) - 3H(X_{4,17}) + 3H(X_{4,12,17}) - H(X_{3,18}) + H(X_{1,3,18}) - H(X_{13,18}) - H(X_{6,13,18}) + H(X_{8,13,18}) + H(X_{6,8,13,18}) - 3H(X_{7,15,18}) + 2H(X_{13,17,18}) - 2H(X_{8,13,17,18}) - 2H(X_{7,15,19}) - H(X_{7,8,15,19}) + 3H(X_{7,15,18,19}) + 3H(X_{17,20}) - 3H(X_{12,17,20})>=0"],"text":"-H(X_{3}) + H(X_{1,3}) + 2H(X_{13}) - 2H(X_{10,13}) - 2H(X_{7,15}) - H(X_{7,8,15}) - 2H(X_{13,16}) + 2H(X_{10,13,16}) + 3H(X_{4,17}) - 3H(X_{4,12,17}) + H(X_{3,18}) - H(X_{1,3,18}) + H(X_{13,18}) + H(X_{6,13,18}) - H(X_{8,13,18}) - 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5H(X_{3,6}) + 10H(X_{7}) - 6H(X_{8}) - 12H(X_{2,8}) + 6H(X_{5,8}) - 6H(X_{2,9}) - 30H(X_{5,9}) + 12H(X_{5,8,9}) - 2H(X_{4,6,10}) + 2H(X_{4,6,8,10}) + 5H(X_{1,5,7,8,10}) + 6H(X_{2,9,10}) + 18H(X_{5,9,11}) - 18H(X_{12}) + 18H(X_{8,12}) - 10H(X_{8,13}) + 10H(X_{6,8,13}) - 2H(X_{2,5,6,9,13}) + H(X_{14}) + 30H(X_{5,9,14}) - 12H(X_{5,8,9,14}) + 3H(X_{4,7,10,14}) - 18H(X_{5,9,11,14}) - 12H(X_{2,16}) + 5H(X_{3,6,7,16}) - 4H(X_{4,6,7,16}) - 10H(X_{8,16}) + 12H(X_{2,8,16}) + 10H(X_{7,8,16}) + 6H(X_{2,9,16}) - 6H(X_{2,9,10,16}) + 2H(X_{1,2,3,4,6,7,10,11,16}) - 2H(X_{1,2,3,4,6,7,8,10,11,16}) + 4H(X_{4,6,7,15,16}) + 4H(X_{2,17}) + 7H(X_{5,17}) + 18H(X_{6,17}) + 6H(X_{8,17}) - 6H(X_{5,8,17}) + 3H(X_{12,17}) - 18H(X_{6,12,17}) - 3H(X_{4,7,10,12,14,17}) + 4H(X_{1,4,5,6,7,10,16,17}) + 10H(X_{2,7,8,9,10,13,16,17}) - 10H(X_{2,6,7,8,9,10,13,16,17}) - 4H(X_{1,4,5,6,7,10,15,16,17}) - 18H(X_{6,18}) + 18H(X_{6,12,18}) + 9H(X_{3,11,12,15,16,17,18}) + 18H(X_{19}) - 54H(X_{2,19}) + 18H(X_{2,3,19}) - 42H(X_{5,19}) + 6H(X_{2,5,19}) + 18H(X_{4,5,19}) + 18H(X_{7,19}) + 18H(X_{2,7,19}) - 18H(X_{2,3,7,19}) + 12H(X_{8,19}) - 12H(X_{7,8,19}) + 36H(X_{5,9,19}) - 18H(X_{4,5,9,19}) + 18H(X_{5,11,19}) - 18H(X_{5,9,11,19}) + 2H(X_{1,2,4,5,6,7,8,9,13,19}) - 18H(X_{14,19}) + 18H(X_{2,14,19}) - H(X_{1,3,4,7,14,19}) + H(X_{1,3,4,5,6,7,8,14,19}) + H(X_{1,3,4,7,13,14,19}) - H(X_{1,3,4,5,6,7,8,13,14,19}) + 6H(X_{15,19}) - 6H(X_{7,15,19}) - 8H(X_{1,4,5,9,12,14,15,16,19}) + 8H(X_{1,2,4,5,9,12,14,15,16,19}) + 8H(X_{1,4,5,9,12,14,15,16,17,19}) - 8H(X_{1,2,4,5,9,12,14,15,16,17,19}) - 12H(X_{18,19}) + 12H(X_{2,18,19}) + 4H(X_{20}) + 18H(X_{12,20}) - 18H(X_{8,12,20}) + 9H(X_{6,14,20}) + 10H(X_{4,5,8,9,10,11,16,20}) - 10H(X_{4,5,7,8,9,10,11,16,20}) - 4H(X_{2,17,20}) + H(X_{4,5,8,10,11,13,17,20}) - H(X_{4,5,8,10,11,13,14,17,20}) + 10H(X_{6,8,9,11,14,15,16,17,20}) - 10H(X_{6,7,8,9,11,14,15,16,17,20}) - 8H(X_{1,5,8,9,10,11,16,17,18,20}) - 9H(X_{3,6,11,12,14,15,16,17,18,20}) + 5H(X_{3,5,6,12,19,20}) + 7H(X_{9,11,13,15,19,20}) - 5H(X_{3,5,6,7,12,16,19,20}) - 7H(X_{5,9,11,13,15,17,19,20}) + 8H(X_{1,5,8,9,10,11,14,15,16,17,18,19,20}) + 5H(X_{2,3,4,6,9,11,12,13,14,15,16,17,18,19,20}) - 5H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20})","witness_value":31.353644499248485},"k":16,"logical_conditions":[{"component_width":8,"component_witness_values":[1.3034018309099338e-13,-1.2512213487525514e-13,6.372680161348399e-14,4.063416270128073e-14,2.283728761653947e-13,1.7030821197749901e-13,-1.1957101975212936e-13,1.6409096303959814e-13],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{5};X_{17}|X_{8})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{5}|X_{19})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{9};X_{4}|X_{5,19})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{9};X_{11}|X_{5,19})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{20};X_{8}|X_{12})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{10};X_{16}|X_{2,9})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{14}|X_{19})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{12};X_{18}|X_{6})"}],"expression":[[32,3],[128,-1],[144,1],[258,-1],[770,1],[2048,-3],[2080,-3],[2176,3],[33026,1],[33538,-1],[65664,1],[65680,-1],[131104,-3],[133152,3],[262144,4],[262146,-4],[262160,-7],[262162,1],[262168,3],[262416,6],[262424,-3],[263184,3],[263440,-3],[270336,-3],[270338,3],[526336,3],[526464,-3]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["3H(X_{6}) - H(X_{8}) + H(X_{5,8}) - H(X_{2,9}) + H(X_{2,9,10}) - 3H(X_{12}) - 3H(X_{6,12}) + 3H(X_{8,12}) + H(X_{2,9,16}) - H(X_{2,9,10,16}) + H(X_{8,17}) - H(X_{5,8,17}) - 3H(X_{6,18}) + 3H(X_{6,12,18}) + 4H(X_{19}) - 4H(X_{2,19}) - 7H(X_{5,19}) + H(X_{2,5,19}) + 3H(X_{4,5,19}) + 6H(X_{5,9,19}) - 3H(X_{4,5,9,19}) + 3H(X_{5,11,19}) - 3H(X_{5,9,11,19}) - 3H(X_{14,19}) + 3H(X_{2,14,19}) + 3H(X_{12,20}) - 3H(X_{8,12,20})>=0","-3H(X_{6}) + H(X_{8}) - H(X_{5,8}) + H(X_{2,9}) - H(X_{2,9,10}) + 3H(X_{12}) + 3H(X_{6,12}) - 3H(X_{8,12}) - H(X_{2,9,16}) + H(X_{2,9,10,16}) - H(X_{8,17}) + H(X_{5,8,17}) + 3H(X_{6,18}) - 3H(X_{6,12,18}) - 4H(X_{19}) + 4H(X_{2,19}) + 7H(X_{5,19}) - H(X_{2,5,19}) - 3H(X_{4,5,19}) - 6H(X_{5,9,19}) + 3H(X_{4,5,9,19}) - 3H(X_{5,11,19}) + 3H(X_{5,9,11,19}) + 3H(X_{14,19}) - 3H(X_{2,14,19}) - 3H(X_{12,20}) + 3H(X_{8,12,20})>=0"],"text":"3H(X_{6}) - H(X_{8}) + H(X_{5,8}) - H(X_{2,9}) + H(X_{2,9,10}) - 3H(X_{12}) - 3H(X_{6,12}) + 3H(X_{8,12}) + H(X_{2,9,16}) - H(X_{2,9,10,16}) + H(X_{8,17}) - 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10H(X_{1,2,4,5,7,10,12,13,14,15,16}) - 9H(X_{1,4,8,10,12,13,14,15,16}) - 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4H(X_{2,3,8,10,11,13,14,16,17,18,19}) + 8H(X_{1,2,7,9,10,11,12,13,15,16,17,18,19}) - 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30H(X_{3,5,7}) + 9H(X_{2,3,5,7}) - 3H(X_{3,6,7}) + 10H(X_{8}) - 10H(X_{7,8}) - 12H(X_{10}) + H(X_{1,2,5,10}) + 8H(X_{5,6,10}) + 6H(X_{2,3,4,7,10}) - 6H(X_{2,3,4,5,7,10}) + 8H(X_{4,8,10}) + 4H(X_{3,6,8,10}) - 8H(X_{4,5,6,8,10}) + 19H(X_{11}) - 12H(X_{1,11}) - 10H(X_{2,11}) + 4H(X_{1,2,11}) - 22H(X_{4,11}) + 10H(X_{2,4,11}) - 30H(X_{3,5,11}) + 12H(X_{6,11}) + 8H(X_{1,6,11}) - 4H(X_{1,2,6,11}) + 14H(X_{7,11}) - 70H(X_{1,7,11}) - 10H(X_{2,7,11}) + 6H(X_{1,2,7,11}) + 40H(X_{1,3,7,11}) - 4H(X_{5,7,11}) + 4H(X_{1,5,7,11}) + 30H(X_{3,5,7,11}) + 4H(X_{1,2,6,7,11}) + 20H(X_{8,11}) + 3H(X_{3,4,5,8,11}) - 3H(X_{1,3,4,5,8,11}) - 20H(X_{6,8,11}) + 6H(X_{7,8,11}) + 38H(X_{1,7,8,11}) - 8H(X_{9,11}) + 4H(X_{1,9,11}) + 6H(X_{1,2,4,5,7,9,11}) - 6H(X_{1,2,4,5,6,7,9,11}) - 6H(X_{10,11}) + 12H(X_{4,10,11}) - H(X_{1,2,5,10,11}) - 10H(X_{1,7,10,11}) - 20H(X_{1,3,7,10,11}) + 4H(X_{9,10,11}) + 8H(X_{2,8,9,10,11}) - 8H(X_{2,5,8,9,10,11}) + 6H(X_{6,12}) - 4H(X_{2,5,7,12}) + 4H(X_{2,4,5,6,7,12}) + 6H(X_{1,2,3,7,8,12}) - 6H(X_{1,2,3,5,7,8,12}) - 9H(X_{4,5,8,9,12}) + 3H(X_{2,6,8,9,12}) - 3H(X_{1,2,6,8,9,12}) + 9H(X_{4,5,6,8,9,12}) + 10H(X_{3,4,7,8,9,12}) + 4H(X_{1,2,6,10,12}) - 4H(X_{1,2,3,6,10,12}) + 3H(X_{5,11,12}) - 3H(X_{2,5,11,12}) + 28H(X_{1,7,11,12}) - 20H(X_{1,3,7,11,12}) - 30H(X_{1,7,8,11,12}) - 6H(X_{6,7,8,11,12}) + 30H(X_{1,7,10,11,12}) + 4H(X_{3,6,8,9,10,11,12}) - 10H(X_{3,4,7,8,9,10,11,12}) + 4H(X_{1,3,4,7,8,9,10,11,12}) - 4H(X_{1,2,3,4,5,6,7,8,9,10,11,12})","witness_value":1863},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{10};X_{12}|X_{1,7,11})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{7};X_{11}|X_{3,5})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{10}|X_{1,7,11})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{6};X_{8}|X_{11})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{8};X_{12}|X_{1,7,11})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{12}|X_{1,7,11})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{4}|X_{11})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{2}|X_{7,11})"}],"expression":[[20,3],[84,-3],[1024,-1],[1026,-1],[1032,-1],[1034,1],[1044,-3],[1056,2],[1088,1],[1089,-5],[1090,-1],[1091,1],[1093,4],[1108,3],[1152,2],[1184,-2],[1217,3],[1601,-1],[1605,-2],[3137,2],[3141,-2],[3265,-3],[3649,3]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["3H(X_{3,5}) - 3H(X_{3,5,7}) - H(X_{11}) - H(X_{2,11}) - H(X_{4,11}) + H(X_{2,4,11}) - 3H(X_{3,5,11}) + 2H(X_{6,11}) + H(X_{7,11}) - 5H(X_{1,7,11}) - H(X_{2,7,11}) + H(X_{1,2,7,11}) + 4H(X_{1,3,7,11}) + 3H(X_{3,5,7,11}) + 2H(X_{8,11}) - 2H(X_{6,8,11}) + 3H(X_{1,7,8,11}) - H(X_{1,7,10,11}) - 2H(X_{1,3,7,10,11}) + 2H(X_{1,7,11,12}) - 2H(X_{1,3,7,11,12}) - 3H(X_{1,7,8,11,12}) + 3H(X_{1,7,10,11,12})>=0","-3H(X_{3,5}) + 3H(X_{3,5,7}) + H(X_{11}) + H(X_{2,11}) + H(X_{4,11}) - H(X_{2,4,11}) + 3H(X_{3,5,11}) - 2H(X_{6,11}) - H(X_{7,11}) + 5H(X_{1,7,11}) + H(X_{2,7,11}) - H(X_{1,2,7,11}) - 4H(X_{1,3,7,11}) - 3H(X_{3,5,7,11}) - 2H(X_{8,11}) + 2H(X_{6,8,11}) - 3H(X_{1,7,8,11}) + H(X_{1,7,10,11}) + 2H(X_{1,3,7,10,11}) - 2H(X_{1,7,11,12}) + 2H(X_{1,3,7,11,12}) + 3H(X_{1,7,8,11,12}) - 3H(X_{1,7,10,11,12})>=0"],"text":"3H(X_{3,5}) - 3H(X_{3,5,7}) - H(X_{11}) - H(X_{2,11}) - H(X_{4,11}) + H(X_{2,4,11}) - 3H(X_{3,5,11}) + 2H(X_{6,11}) + H(X_{7,11}) - 5H(X_{1,7,11}) - H(X_{2,7,11}) + H(X_{1,2,7,11}) + 4H(X_{1,3,7,11}) + 3H(X_{3,5,7,11}) + 2H(X_{8,11}) - 2H(X_{6,8,11}) + 3H(X_{1,7,8,11}) - H(X_{1,7,10,11}) - 2H(X_{1,3,7,10,11}) + 2H(X_{1,7,11,12}) - 2H(X_{1,3,7,11,12}) - 3H(X_{1,7,8,11,12}) + 3H(X_{1,7,10,11,12})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{9}|X_{11})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{1};X_{5}|X_{7,11})"},{"coefficient":2,"kind":"conditional_entropy","label":"H(X_{12}|X_{1,7,11})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{1};X_{6}|X_{11})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{6};X_{7}|X_{1,2,11})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{9};X_{10}|X_{11})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{4};X_{10}|X_{11})"},{"coefficient":2,"kind":"conditional_entropy","label":"H(X_{8}|X_{1,7,11})"}],"expression":[[1024,7],[1025,-3],[1027,1],[1032,-3],[1056,-2],[1057,2],[1059,-1],[1088,1],[1089,-5],[1091,-1],[1104,-1],[1105,1],[1123,1],[1217,2],[1280,-2],[1281,1],[1536,-4],[1544,3],[1792,1],[3137,2]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["7H(X_{11}) - 3H(X_{1,11}) + H(X_{1,2,11}) - 3H(X_{4,11}) - 2H(X_{6,11}) + 2H(X_{1,6,11}) - H(X_{1,2,6,11}) + H(X_{7,11}) - 5H(X_{1,7,11}) - H(X_{1,2,7,11}) - H(X_{5,7,11}) + H(X_{1,5,7,11}) + H(X_{1,2,6,7,11}) + 2H(X_{1,7,8,11}) - 2H(X_{9,11}) + H(X_{1,9,11}) - 4H(X_{10,11}) + 3H(X_{4,10,11}) + H(X_{9,10,11}) + 2H(X_{1,7,11,12})>=0","-7H(X_{11}) + 3H(X_{1,11}) - H(X_{1,2,11}) + 3H(X_{4,11}) + 2H(X_{6,11}) - 2H(X_{1,6,11}) + H(X_{1,2,6,11}) - H(X_{7,11}) + 5H(X_{1,7,11}) + H(X_{1,2,7,11}) + H(X_{5,7,11}) - H(X_{1,5,7,11}) - H(X_{1,2,6,7,11}) - 2H(X_{1,7,8,11}) + 2H(X_{9,11}) - H(X_{1,9,11}) + 4H(X_{10,11}) - 3H(X_{4,10,11}) - H(X_{9,10,11}) - 2H(X_{1,7,11,12})>=0"],"text":"7H(X_{11}) - 3H(X_{1,11}) + H(X_{1,2,11}) - 3H(X_{4,11}) - 2H(X_{6,11}) + 2H(X_{1,6,11}) - H(X_{1,2,6,11}) + H(X_{7,11}) - 5H(X_{1,7,11}) - H(X_{1,2,7,11}) - H(X_{5,7,11}) + H(X_{1,5,7,11}) + H(X_{1,2,6,7,11}) + 2H(X_{1,7,8,11}) - 2H(X_{9,11}) + H(X_{1,9,11}) - 4H(X_{10,11}) + 3H(X_{4,10,11}) + H(X_{9,10,11}) + 2H(X_{1,7,11,12})","type":"equality","witness_value":0}],"m_eq":2,"master_seed":20260825,"method":"abelian_product_v2","n":12,"next_action_index":11,"one_sided_condition_count":0,"prompt_tokens":1886,"recipe_index":2,"remaining_action_coordinate_l1":748.0,"remaining_certificate_length":11,"remaining_coefficient_l1":48.0,"remaining_coefficient_max":10.0,"remaining_condition_actions":2,"remaining_shannon_actions":9,"remaining_shannon_kind_counts":{"conditional_entropy":3,"conditional_mutual_information":6},"residual_mode":"condition","schema_version":"reviewer-benchmark-v1","selected_coordinate_l1":704.0,"selected_coordinate_support":55,"selected_effective_n":12,"selected_residual_sparse":[[1,6],[2,3],[4,4],[20,30],[32,3],[68,3],[84,-30],[86,9],[100,-3],[512,-4],[531,1],[676,4],[1024,19],[1025,-12],[1026,-10],[1027,4],[1032,-22],[1034,10],[1044,-30],[1056,12],[1057,8],[1059,-4],[1088,14],[1089,-70],[1090,-10],[1091,6],[1093,40],[1104,-4],[1105,4],[1108,30],[1123,4],[1152,20],[1180,3],[1181,-3],[1184,-20],[1217,38],[1280,-8],[1281,4],[1536,-16],[1544,12],[1555,-1],[1601,-10],[1605,-20],[1792,4],[2466,3],[2467,-3],[2595,4],[2599,-4],[3088,3],[3090,-3],[3137,28],[3141,-20],[3265,-30],[3649,30],[4004,4]],"selector_source_sha256":"83a328168eac42cedd3412a9c8b5624b8683a52a14042b5defbccf6d833efefe","sha256":"b6e1b8157bd45b4718a4287411f3640776516f702d12fab4eaa32471e442411f","step_idx":11,"subset_size_profile":{"a":[5,6,3,1,5,1,4,4,4],"b":[1,0,1,5,1,2,1,0,0],"c":[0,1,0,0,0,0,0,0,0]},"target_tokens":784,"tokenizer":{"add_special_tokens":false,"is_fast":true,"kind":"huggingface","model":"Qwen/Qwen3-0.6B","requested_revision":"c1899de289a04d12100db370d81485cdf75e47ca","resolved_commit":"c1899de289a04d12100db370d81485cdf75e47ca","tokenizer_class":"Qwen2TokenizerFast","transformers_version":"4.55.2","trust_remote_code":true,"use_fast":true},"trajectory_length":23,"trajectory_sha256":"29dfaaf58b6c9afcdb4ec87ab3e637549789238c36c7f738cead5d176f79ee7e","witness":{"construction":"quotients of a uniform element of a direct product of cyclic 2-groups","cover_variables":[[10,6,2],[10,6],[10,6,0],[10],[10,6],[10,6],[10,0,1],[10,6,0],[10,6],[10,6,0],[2,4],[10,6,0]],"entropy_formula":"H(X_S)=sum_j max_{i in S} a_ij bits","entropy_unit":"bit","family":"abelian_product_v2","group_factors":[{"exponent_bits":5,"order":32,"type":"cyclic"},{"exponent_bits":7,"order":128,"type":"cyclic"},{"exponent_bits":2,"order":4,"type":"cyclic"},{"exponent_bits":2,"order":4,"type":"cyclic"},{"exponent_bits":3,"order":8,"type":"cyclic"},{"exponent_bits":6,"order":64,"type":"cyclic"},{"exponent_bits":5,"order":32,"type":"cyclic"}],"incomparable_profile_pairs":[[0,1],[0,2],[0,3],[0,4],[0,5],[0,6],[0,7],[0,8],[0,9],[0,10],[0,11],[1,2],[1,3],[1,4],[1,5],[1,6],[1,7],[1,8],[1,9],[1,10],[1,11],[2,3],[2,4],[2,5],[2,6],[2,7],[2,8],[2,9],[2,10],[2,11],[3,4],[3,5],[3,6],[3,7],[3,8],[3,9],[3,11],[4,5],[4,6],[4,7],[4,8],[4,9],[4,10],[4,11],[5,6],[5,7],[5,8],[5,9],[5,10],[5,11],[6,7],[6,8],[6,9],[6,10],[6,11],[7,8],[7,9],[7,10],[7,11],[8,9],[8,10],[8,11],[9,10],[9,11],[10,11]],"quotient_exponent_profiles":[[5,0,2,1,1,6,2],[0,7,2,2,0,0,2],[0,7,1,2,3,6,3],[2,5,1,2,3,4,0],[5,1,2,2,2,3,5],[0,7,0,2,0,2,5],[5,7,2,1,3,0,4],[0,4,0,1,2,6,5],[2,6,0,1,0,2,5],[0,3,2,0,3,6,1],[2,5,1,2,3,5,5],[1,0,0,1,3,6,5]],"schema_version":"witness-v2-algebraic-1"},"witness_evaluation":{"checks":{"direct_equals_remaining_actions":true,"equality_contribution_zero":true,"shannon_contribution_nonnegative":true,"target_nonnegative":true},"direct_target_value":700,"evaluation_source":"construction_entropy_oracle","pass":true,"remaining_action_value":700,"remaining_equality_value":0,"remaining_shannon_value":700},"witness_sha256":"568d4602e10b1bfca7e52bb08bb62c25e05091835222df96bafe16e113c98509","witness_variant":null}
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10H(X_{2,5,8,11,12,13,14}) + 10H(X_{2,4,5,8,11,12,13,14}) - 9H(X_{15}) + 8H(X_{4,15}) + 10H(X_{9,15}) - 15H(X_{4,9,15}) - H(X_{11,15}) - 8H(X_{4,11,15}) + H(X_{6,7,11,15}) + 15H(X_{4,9,11,15}) + 4H(X_{1,2,3,5,9,11,15}) + H(X_{4,10,11,15}) - H(X_{4,6,7,10,11,15}) - 4H(X_{1,2,3,5,9,10,11,15}) - 10H(X_{3,4,5,7,8,9,10,11,15}) + 9H(X_{5,10,12,15}) + 8H(X_{3,11,12,15}) - 7H(X_{2,6,9,14,15}) + 7H(X_{4,9,10,14,15}) - 7H(X_{4,9,10,11,14,15}) + 2H(X_{2,3,5,6,8,9,11,13,14,15}) + 7H(X_{6,7,9,11,12,13,14,15}) + H(X_{16}) + 2H(X_{1,16}) + 32H(X_{9,16}) + 7H(X_{1,2,3,4,5,8,10,16}) - 24H(X_{9,13,16}) + 8H(X_{1,4,5,6,14,16}) - 8H(X_{9,14,16}) + 9H(X_{4,7,10,11,14,16}) - 9H(X_{4,7,8,10,11,12,14,16}) - 7H(X_{6,13,14,16}) - H(X_{10,12,13,14,16}) + 9H(X_{4,7,9,15,16}) - 2H(X_{8,10,15,16}) - 9H(X_{4,5,7,9,10,12,15,16}) + 7H(X_{2,4,6,7,9,10,14,15,16}) - 8H(X_{1,3,4,5,6,11,12,14,15,16}) + 7H(X_{1,3,5,6,7,9,10,13,14,15,16}) + 2H(X_{4,5,6,7,8,9,10,13,14,15,16}) - 2H(X_{1,2,3,5,6,8,9,11,13,14,15,16}) - 7H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16})","witness_value":687},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{1};X_{4}|X_{9})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{2};X_{7}|X_{3,9})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{3}|X_{4,9})"},{"coefficient":-1,"kind":"conditional_entropy","label":"H(X_{11}|X_{2,9})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{14}|X_{2,9})"},{"coefficient":3,"kind":"conditional_entropy","label":"H(X_{6}|X_{2,9})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{10}|X_{9})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{2};X_{9}|X_{11})"}],"expression":[[256,-5],[257,2],[258,2],[260,2],[262,-2],[264,3],[265,-2],[266,-2],[268,-1],[270,1],[290,3],[324,-2],[326,2],[768,3],[770,-3],[1024,-2],[1026,2],[1280,2],[1282,-3],[8450,-1],[8458,1]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["-5H(X_{9}) + 2H(X_{1,9}) + 2H(X_{2,9}) + 2H(X_{3,9}) - 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15H(X_{1,2}) - 8H(X_{1,2,3}) - 3H(X_{4}) + 39H(X_{1,2,4}) + 3H(X_{1,2,5}) + 9H(X_{1,2,3,5}) - 21H(X_{1,2,4,5}) + 8H(X_{7}) - 7H(X_{1,2,7}) - 14H(X_{1,2,3,7}) + 4H(X_{4,7}) + 6H(X_{5,6,7}) + 27H(X_{1,8}) - 27H(X_{1,2,8}) - 5H(X_{1,7,8}) + 5H(X_{1,5,6,7,8}) + 14H(X_{1,2,9}) - 6H(X_{1,2,10}) + 27H(X_{1,2,3,10}) - 21H(X_{1,2,8,10}) - 18H(X_{1,8,11}) + 18H(X_{1,2,8,11}) - 2H(X_{12}) - 30H(X_{1,2,12}) + 2H(X_{3,12}) + 9H(X_{1,2,5,12}) + 21H(X_{1,2,7,12}) - 9H(X_{1,8,12}) + 9H(X_{1,2,8,12}) + 6H(X_{2,8,9,11,12}) - 6H(X_{2,5,6,7,8,9,11,12}) - H(X_{13}) + H(X_{1,13}) + 20H(X_{1,2,13}) - 14H(X_{1,2,3,13}) - 18H(X_{1,2,4,13}) + 12H(X_{1,2,8,13}) + H(X_{5,6,9,13}) - H(X_{1,5,6,9,13}) + 2H(X_{1,4,7,8,9,10,11,12,13}) - 2H(X_{1,3,4,7,8,9,10,11,12,13}) - 9H(X_{1,2,14}) + 9H(X_{1,2,8,14}) + 9H(X_{10,14}) + 5H(X_{9,12,14}) + 7H(X_{15}) + 14H(X_{1,2,15}) - 14H(X_{1,2,9,15}) + H(X_{1,2,11,15}) + 8H(X_{12,15}) - 4H(X_{3,7,11,12,15}) + 9H(X_{1,3,4,5,6,9,11,12,13,15}) + 8H(X_{2,3,10,12,13,14,15}) - 8H(X_{2,3,7,10,12,13,14,15}) - 9H(X_{1,3,4,5,6,9,10,11,12,13,14,15}) - 7H(X_{1,16}) - 10H(X_{1,3,6,7,9,10,11,13,16}) + 10H(X_{1,3,5,6,7,9,10,11,13,14,16}) + 4H(X_{1,2,3,5,6,7,8,9,10,11,12,13,15,16}) + H(X_{1,2,3,4,6,7,8,9,10,13,14,15,16}) - 7H(X_{1,17}) - 10H(X_{1,3,4,6,9,12,17}) + 10H(X_{1,3,4,5,6,9,12,17}) + H(X_{5,11,12,17}) + 10H(X_{1,3,4,6,9,11,12,17}) - 10H(X_{1,3,4,5,6,9,11,12,17}) + 8H(X_{2,4,5,7,8,10,11,13,17}) + 7H(X_{2,7,8,13,14,17}) - 8H(X_{2,4,5,7,8,10,11,12,13,15,17}) - 7H(X_{2,7,8,13,14,15,17}) + 7H(X_{1,16,17}) + H(X_{1,4,5,16,17}) + 5H(X_{5,11,16,17}) + 3H(X_{3,4,5,6,8,9,11,13,16,17}) + 5H(X_{1,7,8,10,11,13,16,17}) - 5H(X_{1,5,6,7,8,10,11,13,16,17}) - 5H(X_{5,9,11,12,14,16,17}) - H(X_{1,2,4,5,11,15,16,17}) - H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17})","witness_value":1379},"k":16,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{4};X_{13}|X_{1,2})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{5};X_{12}|X_{1,2})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{8};X_{14}|X_{1,2})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{10}|X_{1,2})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{8};X_{13}|X_{1,2})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{3};X_{5}|X_{1,2})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{2};X_{11}|X_{1,8})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{12}|X_{1,8})"}],"expression":[[3,3],[7,-4],[11,2],[19,-2],[23,1],[129,3],[131,-3],[515,-3],[519,3],[1153,-2],[1155,2],[2051,-1],[2067,1],[2177,-1],[2179,1],[4099,3],[4107,-2],[4227,-1],[8195,-1],[8323,1]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["3H(X_{1,2}) - 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+ 10H(X_{7}) + 7H(X_{3,9}) - 10H(X_{5,7,10}) + 5H(X_{11}) - 10H(X_{3,13}) - 2H(X_{14}) + 6H(X_{4,5,7,12,14}) - 2H(X_{1,2,13,14}) + 10H(X_{2,3,10,13,14}) + 25H(X_{7,10,15}) - 10H(X_{7,10,14,15}) - 5H(X_{2,4,7,11,14,15}) - 9H(X_{16}) + 9H(X_{1,3,4,16}) - 10H(X_{1,13,16}) + 10H(X_{3,13,16}) + 9H(X_{14,16}) - 9H(X_{1,3,4,14,16}) + 7H(X_{2,7,14,16}) + 7H(X_{1,4,5,6,8,12,14,16}) - 7H(X_{1,3,4,5,6,8,9,12,14,16}) - 10H(X_{2,3,10,13,14,16}) - 8H(X_{7,8,11,13,14,16}) + 8H(X_{5,7,8,9,11,13,14,16}) + 8H(X_{7,8,10,11,13,14,16}) - 8H(X_{5,7,8,9,10,11,13,14,16}) - H(X_{1,2,3,5,6,14,15,16}) + 5H(X_{1,3,4,5,6,8,9,14,15,16}) - 5H(X_{1,3,4,5,6,8,9,11,14,15,16}) + 15H(X_{17}) - 15H(X_{4,17}) - 5H(X_{8,17}) + 5H(X_{3,8,17}) + 5H(X_{5,8,17}) + 15H(X_{6,8,17}) - 20H(X_{2,6,8,17}) - 5H(X_{8,10,17}) - 5H(X_{8,11,17}) + 10H(X_{1,2,3,5,7,9,10,11,17}) + 5H(X_{8,12,17}) - 5H(X_{3,8,12,17}) - 15H(X_{6,8,12,17}) + 15H(X_{2,6,8,12,17}) - 20H(X_{8,13,17}) + 15H(X_{8,11,13,17}) - 20H(X_{14,17}) + 5H(X_{3,14,17}) + 15H(X_{6,14,17}) + 35H(X_{8,14,17}) - 5H(X_{3,8,14,17}) - 20H(X_{6,8,14,17}) + 5H(X_{2,6,8,14,17}) - H(X_{1,4,10,13,14,17}) + H(X_{1,4,8,10,13,14,17}) + 10H(X_{8,15,17}) - 25H(X_{7,10,15,17}) - 10H(X_{8,11,15,17}) + 10H(X_{7,10,14,15,17}) + 7H(X_{4,6,9,13,14,15,17}) + H(X_{1,4,6,10,13,14,15,17}) - H(X_{1,4,6,8,10,13,14,15,17}) - 15H(X_{16,17}) + 15H(X_{4,16,17}) + H(X_{3,4,6,9,11,12,14,16,17}) - H(X_{1,3,4,6,9,11,12,14,16,17}) + 2H(X_{1,2,3,4,5,7,8,10,11,12,13,14,16,17}) + 4H(X_{2,4,5,6,8,11,13,15,16,17}) + H(X_{1,2,3,5,6,14,15,16,17}) + 5H(X_{1,2,4,6,7,8,9,10,11,12,14,15,16,17}) - 7H(X_{2,4,6,7,9,13,14,15,16,17}) - H(X_{1,6,11,18}) + H(X_{1,2,6,11,18}) + H(X_{1,6,7,8,9,10,11,18}) - H(X_{1,2,6,7,8,9,10,11,18}) + 4H(X_{1,3,7,9,10,12,14,18}) + 10H(X_{2,3,13,14,18}) - 10H(X_{2,3,7,13,14,18}) + 6H(X_{1,3,11,15,18}) - 6H(X_{1,3,4,5,7,11,12,14,15,18}) + 5H(X_{1,5,7,8,11,15,16,18}) - 5H(X_{1,5,7,8,11,14,15,16,18}) + 10H(X_{1,3,4,6,7,8,9,10,11,12,13,14,15,16,18}) - 5H(X_{8,17,18}) + 5H(X_{6,8,17,18}) + 3H(X_{3,4,8,9,17,18}) + 5H(X_{8,10,17,18}) + 10H(X_{8,13,17,18}) - 15H(X_{8,14,17,18}) - 6H(X_{5,8,16,17,18}) + 6H(X_{1,3,4,5,7,8,9,10,14,15,16,17,18}) - 4H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18})","witness_value":1563},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{3};X_{8}|X_{14,17})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{2};X_{12}|X_{6,8,17})"},{"coefficient":1,"kind":"conditional_entropy","label":"H(X_{5}|X_{8,17})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{4};X_{16}|X_{17})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{14};X_{17}|X_{7,10,15})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{3};X_{12}|X_{8,17})"},{"coefficient":-3,"kind":"conditional_entropy","label":"H(X_{17}|X_{7,10,15})"},{"coefficient":1,"kind":"conditional_entropy","label":"H(X_{13}|X_{8,17})"}],"expression":[[16960,5],[25152,-2],[65536,3],[65544,-3],[65664,-3],[65668,1],[65680,1],[65696,3],[65698,-3],[67712,1],[67716,-1],[67744,-3],[67746,3],[69760,1],[73728,-1],[73732,1],[73856,1],[73860,-1],[82496,-5],[90688,2],[98304,-3],[98312,3]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["5H(X_{7,10,15}) - 2H(X_{7,10,14,15}) + 3H(X_{17}) - 3H(X_{4,17}) - 3H(X_{8,17}) + H(X_{3,8,17}) + H(X_{5,8,17}) + 3H(X_{6,8,17}) - 3H(X_{2,6,8,17}) + H(X_{8,12,17}) - H(X_{3,8,12,17}) - 3H(X_{6,8,12,17}) + 3H(X_{2,6,8,12,17}) + H(X_{8,13,17}) - H(X_{14,17}) + H(X_{3,14,17}) + H(X_{8,14,17}) - H(X_{3,8,14,17}) - 5H(X_{7,10,15,17}) + 2H(X_{7,10,14,15,17}) - 3H(X_{16,17}) + 3H(X_{4,16,17})>=0","-5H(X_{7,10,15}) + 2H(X_{7,10,14,15}) - 3H(X_{17}) + 3H(X_{4,17}) + 3H(X_{8,17}) - H(X_{3,8,17}) - H(X_{5,8,17}) - 3H(X_{6,8,17}) + 3H(X_{2,6,8,17}) - H(X_{8,12,17}) + H(X_{3,8,12,17}) + 3H(X_{6,8,12,17}) - 3H(X_{2,6,8,12,17}) - H(X_{8,13,17}) + H(X_{14,17}) - H(X_{3,14,17}) - H(X_{8,14,17}) + H(X_{3,8,14,17}) + 5H(X_{7,10,15,17}) - 2H(X_{7,10,14,15,17}) + 3H(X_{16,17}) - 3H(X_{4,16,17})>=0"],"text":"5H(X_{7,10,15}) - 2H(X_{7,10,14,15}) + 3H(X_{17}) - 3H(X_{4,17}) - 3H(X_{8,17}) + H(X_{3,8,17}) + H(X_{5,8,17}) + 3H(X_{6,8,17}) - 3H(X_{2,6,8,17}) + H(X_{8,12,17}) - H(X_{3,8,12,17}) - 3H(X_{6,8,12,17}) + 3H(X_{2,6,8,12,17}) + H(X_{8,13,17}) - H(X_{14,17}) + H(X_{3,14,17}) + H(X_{8,14,17}) - H(X_{3,8,14,17}) - 5H(X_{7,10,15,17}) + 2H(X_{7,10,14,15,17}) - 3H(X_{16,17}) + 3H(X_{4,16,17})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{14};X_{18}|X_{8,17})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{11};X_{13}|X_{8,17})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{13};X_{18}|X_{8,17})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{14}|X_{6,8,17})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{10};X_{18}|X_{8,17})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{6};X_{8}|X_{14,17})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{11};X_{15}|X_{8,17})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{6};X_{18}|X_{8,17})"}],"expression":[[65664,2],[65698,-1],[66176,-1],[66688,-1],[69760,-5],[70784,3],[73728,-3],[73760,3],[73856,6],[73888,-4],[73890,1],[82048,2],[83072,-2],[196736,-1],[196768,1],[197248,1],[200832,2],[204928,-3]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["2H(X_{8,17}) - H(X_{2,6,8,17}) - H(X_{8,10,17}) - H(X_{8,11,17}) - 5H(X_{8,13,17}) + 3H(X_{8,11,13,17}) - 3H(X_{14,17}) + 3H(X_{6,14,17}) + 6H(X_{8,14,17}) - 4H(X_{6,8,14,17}) + H(X_{2,6,8,14,17}) + 2H(X_{8,15,17}) - 2H(X_{8,11,15,17}) - H(X_{8,17,18}) + H(X_{6,8,17,18}) + H(X_{8,10,17,18}) + 2H(X_{8,13,17,18}) - 3H(X_{8,14,17,18})>=0","-2H(X_{8,17}) + H(X_{2,6,8,17}) + H(X_{8,10,17}) + H(X_{8,11,17}) + 5H(X_{8,13,17}) - 3H(X_{8,11,13,17}) + 3H(X_{14,17}) - 3H(X_{6,14,17}) - 6H(X_{8,14,17}) + 4H(X_{6,8,14,17}) - H(X_{2,6,8,14,17}) - 2H(X_{8,15,17}) + 2H(X_{8,11,15,17}) + H(X_{8,17,18}) - H(X_{6,8,17,18}) - H(X_{8,10,17,18}) - 2H(X_{8,13,17,18}) + 3H(X_{8,14,17,18})>=0"],"text":"2H(X_{8,17}) - H(X_{2,6,8,17}) - H(X_{8,10,17}) - H(X_{8,11,17}) - 5H(X_{8,13,17}) + 3H(X_{8,11,13,17}) - 3H(X_{14,17}) + 3H(X_{6,14,17}) + 6H(X_{8,14,17}) - 4H(X_{6,8,14,17}) + H(X_{2,6,8,14,17}) + 2H(X_{8,15,17}) - 2H(X_{8,11,15,17}) - H(X_{8,17,18}) + H(X_{6,8,17,18}) + 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5H(X_{12,15,17}) + 5H(X_{10,12,15,17}) - 4H(X_{16,17}) + 9H(X_{9,16,17}) - 5H(X_{15,16,17}) + 8H(X_{1,3,4,6,7,13,14,15,16,17}) - 8H(X_{1,2,3,4,6,7,13,14,15,16,17}) + 6H(X_{18}) + 3H(X_{2,4,18}) + 4H(X_{8,11,15,18}) + 8H(X_{8,14,15,18}) + 6H(X_{1,5,7,9,10,13,14,15,18}) + 9H(X_{4,10,16,18}) - 9H(X_{1,4,5,10,15,16,18}) - 8H(X_{1,5,8,12,14,15,16,18}) + 10H(X_{17,18}) - 3H(X_{2,4,17,18}) - 5H(X_{6,17,18}) + 8H(X_{12,17,18}) - 3H(X_{3,12,17,18}) - 5H(X_{11,12,17,18}) - 5H(X_{13,17,18}) + 10H(X_{10,14,17,18}) - 10H(X_{2,6,7,10,12,14,17,18}) - 6H(X_{16,17,18}) - 10H(X_{3,4,5,6,8,9,11,13,16,17,18}) + 5H(X_{2,3,4,5,6,7,9,10,19}) + 8H(X_{2,4,6,7,8,9,10,16,19}) - 8H(X_{2,4,5,6,7,8,9,10,16,19}) - 3H(X_{17,19}) + 3H(X_{4,17,19}) - 5H(X_{2,3,4,5,6,7,9,10,11,17,19}) + 4H(X_{8,11,13,15,17,19}) - 10H(X_{5,6,8,9,10,11,14,16,17,19}) + 10H(X_{1,2,5,6,8,9,10,11,14,16,17,19}) - 4H(X_{8,11,13,15,17,18,19}) + 2H(X_{2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,19}) + 6H(X_{1,2,3,4,7,8,10,11,12,16,17,18,19}) + 10H(X_{3,4,5,6,8,9,11,13,16,17,18,19}) - 2H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19})","witness_value":1642},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{3};X_{11}|X_{12,17})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{6};X_{18}|X_{17})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{14}|X_{12,17})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{10};X_{15}|X_{12,17})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{7}|X_{17})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{15};X_{16}|X_{17})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{11};X_{18}|X_{12,17})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{13};X_{18}|X_{17})"}],"expression":[[65536,4],[65544,-1],[65568,-1],[65600,-1],[65608,1],[67584,-4],[67588,4],[68096,1],[68612,-1],[69632,-1],[75776,3],[75780,-3],[81920,-1],[83968,1],[84480,-1],[98304,-1],[114688,1],[196608,-2],[196640,1],[198656,-1],[199680,1],[200704,1]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["4H(X_{17}) - H(X_{4,17}) - H(X_{6,17}) - H(X_{7,17}) + H(X_{4,7,17}) - 4H(X_{12,17}) + 4H(X_{3,12,17}) + H(X_{10,12,17}) - H(X_{3,11,12,17}) - H(X_{13,17}) + 3H(X_{12,14,17}) - 3H(X_{3,12,14,17}) - H(X_{15,17}) + H(X_{12,15,17}) - H(X_{10,12,15,17}) - H(X_{16,17}) + H(X_{15,16,17}) - 2H(X_{17,18}) + H(X_{6,17,18}) - H(X_{12,17,18}) + H(X_{11,12,17,18}) + H(X_{13,17,18})>=0","-4H(X_{17}) + H(X_{4,17}) + H(X_{6,17}) + H(X_{7,17}) - H(X_{4,7,17}) + 4H(X_{12,17}) - 4H(X_{3,12,17}) - H(X_{10,12,17}) + H(X_{3,11,12,17}) + H(X_{13,17}) - 3H(X_{12,14,17}) + 3H(X_{3,12,14,17}) + H(X_{15,17}) - H(X_{12,15,17}) + H(X_{10,12,15,17}) + H(X_{16,17}) - H(X_{15,16,17}) + 2H(X_{17,18}) - H(X_{6,17,18}) + H(X_{12,17,18}) - H(X_{11,12,17,18}) - H(X_{13,17,18})>=0"],"text":"4H(X_{17}) - H(X_{4,17}) - H(X_{6,17}) - H(X_{7,17}) + H(X_{4,7,17}) - 4H(X_{12,17}) + 4H(X_{3,12,17}) + H(X_{10,12,17}) - H(X_{3,11,12,17}) - H(X_{13,17}) + 3H(X_{12,14,17}) - 3H(X_{3,12,14,17}) - H(X_{15,17}) + H(X_{12,15,17}) - H(X_{10,12,15,17}) - H(X_{16,17}) + H(X_{15,16,17}) - 2H(X_{17,18}) + H(X_{6,17,18}) - H(X_{12,17,18}) + H(X_{11,12,17,18}) + H(X_{13,17,18})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{17}|X_{2,4})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{19}|X_{17})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{4};X_{14}|X_{17})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{1};X_{3}|X_{12,17})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{7};X_{14}|X_{12,17})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{3};X_{18}|X_{12,17})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{9};X_{16}|X_{17})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{17};X_{18}|X_{2,4})"}],"expression":[[10,3],[14,-2],[65536,-5],[65544,2],[65546,-3],[65550,2],[65792,3],[67584,-2],[67585,2],[67588,1],[67589,-2],[67648,1],[73728,1],[73736,-1],[75776,1],[75840,-1],[98304,3],[98560,-3],[131082,-1],[196618,1],[198656,-1],[198660,1],[327680,1],[327688,-1]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["3H(X_{2,4}) - 2H(X_{2,3,4}) - 5H(X_{17}) + 2H(X_{4,17}) - 3H(X_{2,4,17}) + 2H(X_{2,3,4,17}) + 3H(X_{9,17}) - 2H(X_{12,17}) + 2H(X_{1,12,17}) + H(X_{3,12,17}) - 2H(X_{1,3,12,17}) + H(X_{7,12,17}) + H(X_{14,17}) - H(X_{4,14,17}) + H(X_{12,14,17}) - H(X_{7,12,14,17}) + 3H(X_{16,17}) - 3H(X_{9,16,17}) - H(X_{2,4,18}) + H(X_{2,4,17,18}) - H(X_{12,17,18}) + H(X_{3,12,17,18}) + H(X_{17,19}) - H(X_{4,17,19})>=0","-3H(X_{2,4}) + 2H(X_{2,3,4}) + 5H(X_{17}) - 2H(X_{4,17}) + 3H(X_{2,4,17}) - 2H(X_{2,3,4,17}) - 3H(X_{9,17}) + 2H(X_{12,17}) - 2H(X_{1,12,17}) - H(X_{3,12,17}) + 2H(X_{1,3,12,17}) - H(X_{7,12,17}) - H(X_{14,17}) + H(X_{4,14,17}) - H(X_{12,14,17}) + H(X_{7,12,14,17}) - 3H(X_{16,17}) + 3H(X_{9,16,17}) + H(X_{2,4,18}) - H(X_{2,4,17,18}) + H(X_{12,17,18}) - H(X_{3,12,17,18}) - H(X_{17,19}) + H(X_{4,17,19})>=0"],"text":"3H(X_{2,4}) - 2H(X_{2,3,4}) - 5H(X_{17}) + 2H(X_{4,17}) - 3H(X_{2,4,17}) + 2H(X_{2,3,4,17}) + 3H(X_{9,17}) - 2H(X_{12,17}) + 2H(X_{1,12,17}) + H(X_{3,12,17}) - 2H(X_{1,3,12,17}) + H(X_{7,12,17}) + H(X_{14,17}) - 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21H(X_{5,16}) + 21H(X_{5,8,16}) + 14H(X_{5,14,16}) - 21H(X_{5,17}) + 21H(X_{5,9,17}) - 14H(X_{5,14,16,17}) + 14H(X_{5,18}) - 14H(X_{1,5,18}) + 7H(X_{10,14,18}) - 7H(X_{5,10,14,18}) + 14H(X_{10,14,15,19}) - 14H(X_{5,10,14,15,19})","witness_value":0},{"coefficient":-8,"expression":[[16,-5],[18,1],[20,1],[22,-1],[272,2],[512,2],[528,-2],[768,-2],[784,2],[2064,2],[32784,-3],[32788,2],[32848,1],[33296,3],[33300,-2],[33360,-1],[40976,-1],[106512,1],[262160,-1],[264208,-2],[270352,3],[294928,3],[303120,-3]],"kind":"composite_equality","label":"witness equality 1=0","role":"condition","scaled_expression":[[16,40],[18,-8],[20,-8],[22,8],[272,-16],[512,-16],[528,16],[768,16],[784,-16],[2064,-16],[32784,24],[32788,-16],[32848,-8],[33296,-24],[33300,16],[33360,8],[40976,8],[106512,-8],[262160,8],[264208,16],[270352,-24],[294928,-24],[303120,24]],"scaled_witness_value":0,"source_index":0,"text":"40H(X_{5}) - 8H(X_{2,5}) - 8H(X_{3,5}) + 8H(X_{2,3,5}) - 16H(X_{5,9}) - 16H(X_{10}) + 16H(X_{5,10}) + 16H(X_{9,10}) - 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2H(X_{2,4,6,7,9,12,13,18}) + 7H(X_{10,14,18}) - 7H(X_{5,10,14,18}) + 10H(X_{1,6,7,9,11,12,15,18}) + 8H(X_{5,19}) + 6H(X_{4,5,6,7,11,19}) - 2H(X_{1,3,6,9,11,19}) + 2H(X_{1,3,6,9,10,11,19}) + 16H(X_{5,12,19}) + 2H(X_{2,3,4,6,7,8,9,12,19}) - 2H(X_{2,3,4,6,7,8,9,11,12,19}) + 7H(X_{6,10,13,19}) - 24H(X_{5,14,19}) + 14H(X_{10,14,15,19}) - 14H(X_{5,10,14,15,19}) - 24H(X_{5,16,19}) + 24H(X_{5,14,16,19}) - 6H(X_{1,4,5,6,7,11,13,14,16,19}) + 6H(X_{1,2,5,6,9,11,12,13,14,15,16,19}) - 6H(X_{1,2,5,6,8,9,11,12,13,14,15,16,19}) + 2H(X_{1,2,3,4,6,8,9,11,12,13,17,19}) - 2H(X_{1,2,3,4,6,8,9,10,11,12,13,17,19}) - 3H(X_{2,3,7,8,9,13,16,17,19}) + 10H(X_{2,3,4,5,8,10,13,14,16,17,19}) - 7H(X_{1,6,10,12,13,14,16,17,19}) + 10H(X_{2,4,10,12,13,15,18,19}) - 10H(X_{2,4,5,8,10,12,13,14,15,18,19}) - 8H(X_{4,6,7,8,10,13,17,18,19}) - 10H(X_{3,4,5,6,9,11,13,17,18,19}) + 10H(X_{2,3,4,5,6,7,9,11,13,17,18,19}) + 8H(X_{4,5,6,7,8,10,13,14,17,18,19}) + 5H(X_{1,4,10,11,14,15,17,18,19}) - 5H(X_{1,2,4,10,11,14,15,17,18,19}) + 3H(X_{2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19}) - 10H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19})","witness_value":1695},"k":16,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{2};X_{3}|X_{5})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{12};X_{19}|X_{5})"},{"coefficient":2,"kind":"conditional_entropy","label":"H(X_{9}|X_{5})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{14};X_{16}|X_{5,19})"},{"coefficient":1,"kind":"conditional_entropy","label":"H(X_{17}|X_{5,14,16})"},{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{3};X_{10}|X_{5,16})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{9}|X_{10})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{7};X_{10}|X_{5,16})"}],"expression":[[16,-5],[18,1],[20,1],[22,-1],[272,2],[512,2],[528,-2],[768,-2],[784,2],[2064,2],[32784,-3],[32788,2],[32848,1],[33296,3],[33300,-2],[33360,-1],[40976,-1],[106512,1],[262160,-1],[264208,-2],[270352,3],[294928,3],[303120,-3]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["-5H(X_{5}) + H(X_{2,5}) + H(X_{3,5}) - H(X_{2,3,5}) + 2H(X_{5,9}) + 2H(X_{10}) - 2H(X_{5,10}) - 2H(X_{9,10}) + 2H(X_{5,9,10}) + 2H(X_{5,12}) - 3H(X_{5,16}) + 2H(X_{3,5,16}) + H(X_{5,7,16}) + 3H(X_{5,10,16}) - 2H(X_{3,5,10,16}) - H(X_{5,7,10,16}) - H(X_{5,14,16}) + H(X_{5,14,16,17}) - H(X_{5,19}) - 2H(X_{5,12,19}) + 3H(X_{5,14,19}) + 3H(X_{5,16,19}) - 3H(X_{5,14,16,19})>=0","5H(X_{5}) - H(X_{2,5}) - H(X_{3,5}) + H(X_{2,3,5}) - 2H(X_{5,9}) - 2H(X_{10}) + 2H(X_{5,10}) + 2H(X_{9,10}) - 2H(X_{5,9,10}) - 2H(X_{5,12}) + 3H(X_{5,16}) - 2H(X_{3,5,16}) - H(X_{5,7,16}) - 3H(X_{5,10,16}) + 2H(X_{3,5,10,16}) + H(X_{5,7,10,16}) + H(X_{5,14,16}) - H(X_{5,14,16,17}) + H(X_{5,19}) + 2H(X_{5,12,19}) - 3H(X_{5,14,19}) - 3H(X_{5,16,19}) + 3H(X_{5,14,16,19})>=0"],"text":"-5H(X_{5}) + H(X_{2,5}) + H(X_{3,5}) - H(X_{2,3,5}) + 2H(X_{5,9}) + 2H(X_{10}) - 2H(X_{5,10}) - 2H(X_{9,10}) + 2H(X_{5,9,10}) + 2H(X_{5,12}) - 3H(X_{5,16}) + 2H(X_{3,5,16}) + H(X_{5,7,16}) + 3H(X_{5,10,16}) - 2H(X_{3,5,10,16}) - H(X_{5,7,10,16}) - H(X_{5,14,16}) + H(X_{5,14,16,17}) - 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2H(X_{1,5}) - 3H(X_{4,5}) + 2H(X_{1,4,5}) - H(X_{5,13}) + H(X_{5,15}) - H(X_{5,16}) + 3H(X_{1,5,16}) - 3H(X_{1,5,8,16}) - H(X_{5,15,17}) + 6H(X_{16,17}) - 5H(X_{5,16,17}) - 3H(X_{11,16,17}) + 3H(X_{5,11,16,17}) - 4H(X_{5,18}) + 3H(X_{4,5,18}) + H(X_{5,13,18}) + 2H(X_{1,5,19}) - 2H(X_{1,4,5,19}) - 3H(X_{16,17,20}) + 3H(X_{5,16,17,20})","witness_value":0}],"certificate_length":22,"certificate_order":"greedy_negative_mass_condition_late","condition_coordinate_supports":[26,21],"condition_rank":2,"condition_tokens":[328,327,262,261],"condition_width":8,"config_sha256":"1b1f298ed012071c85e9cfb286fe5dcf7c183383de92041754d8a7802f426421","equality_condition_count":2,"full_tokens":1951,"generator_dependency_sha256":{"data/finite_field.py":"c9738c10a16ea2cfb3cde346a23673d1536dbfce491f8871346f441b9b4c2a6e","data/witness_first.py":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","data/witness_v2_algebraic.py":"ffbd48eb673a3f6f842327d00431285bbae8963a7e20f6956e37e544fa37165f","data/witness_v2_graph.py":"c81adcfefcf6ffc0e816f1b338af13323bc80e06293e8cac6b28a3b0ae2efc8a"},"generator_schema_version":"witness-first-v2","generator_source_sha256":"7af2a1d6096862a776e9c8530d0c6a3ae2b83aa7ecd8bb1026256d6f1cdc3560","id":"e2n-witness-first-v2-n20-abelian_product_v2-2eebffaf09cdf17d","initial_target":{"coordinate_support":105,"effective_n":20,"expression":[[1,-1],[3,5],[4,-5],[16,-14],[17,-11],[20,9],[24,-3],[25,2],[81,9],[272,9],[512,-8],[516,8],[1024,8],[2064,9],[2068,-9],[4096,-2],[4112,-1],[8193,-10],[8224,4],[8811,9],[16400,10],[16656,-9],[32784,-1],[32785,21],[32789,-27],[32849,9],[32896,4],[32913,-3],[33297,9],[33361,-9],[36864,2],[49169,-27],[49173,27],[65553,9],[65572,-9],[65588,9],[65617,-9],[67652,5],[68109,1],[81936,-1],[98304,6],[98320,-5],[98340,9],[98356,-9],[98368,18],[98384,-18],[99328,-3],[99344,3],[102400,-18],[102416,18],[131088,-4],[131096,3],[135184,1],[152459,-4],[197993,-10],[204816,-1],[205855,1],[221067,4],[221186,-3],[254480,1],[255519,-1],[262144,1],[262161,2],[262169,-2],[271617,10],[315597,4],[323821,-4],[328392,5],[328395,-5],[360536,8],[361560,-8],[459656,8],[459660,-8],[460137,10],[547532,2],[557393,1],[578566,-6],[578694,6],[580300,-2],[589851,-4],[595970,-8],[596034,8],[606491,4],[612407,8],[612471,-8],[622592,-3],[622608,3],[625501,-1],[655360,9],[664171,-9],[666131,10],[673293,5],[740941,-5],[906286,6],[906414,-6],[919616,-4],[921028,4],[929555,-10],[930137,-6],[936000,4],[937412,-4],[1014342,3],[1015679,4],[1015807,6],[1048575,-4]],"text":"-H(X_{1}) + 5H(X_{1,2}) - 5H(X_{3}) - 14H(X_{5}) - 11H(X_{1,5}) + 9H(X_{3,5}) - 3H(X_{4,5}) + 2H(X_{1,4,5}) + 9H(X_{1,5,7}) + 9H(X_{5,9}) - 8H(X_{10}) + 8H(X_{3,10}) + 8H(X_{11}) + 9H(X_{5,12}) - 9H(X_{3,5,12}) - 2H(X_{13}) - H(X_{5,13}) - 10H(X_{1,14}) + 4H(X_{6,14}) + 9H(X_{1,2,4,6,7,10,14}) + 10H(X_{5,15}) - 9H(X_{5,9,15}) - H(X_{5,16}) + 21H(X_{1,5,16}) - 27H(X_{1,3,5,16}) + 9H(X_{1,5,7,16}) + 4H(X_{8,16}) - 3H(X_{1,5,8,16}) + 9H(X_{1,5,10,16}) - 9H(X_{1,5,7,10,16}) + 2H(X_{13,16}) - 27H(X_{1,5,15,16}) + 27H(X_{1,3,5,15,16}) + 9H(X_{1,5,17}) - 9H(X_{3,6,17}) + 9H(X_{3,5,6,17}) - 9H(X_{1,5,7,17}) + 5H(X_{3,7,12,17}) + H(X_{1,3,4,10,12,17}) - H(X_{5,15,17}) + 6H(X_{16,17}) - 5H(X_{5,16,17}) + 9H(X_{3,6,16,17}) - 9H(X_{3,5,6,16,17}) + 18H(X_{7,16,17}) - 18H(X_{5,7,16,17}) - 3H(X_{11,16,17}) + 3H(X_{5,11,16,17}) - 18H(X_{13,16,17}) + 18H(X_{5,13,16,17}) - 4H(X_{5,18}) + 3H(X_{4,5,18}) + H(X_{5,13,18}) - 4H(X_{1,2,4,8,9,10,13,15,18}) - 10H(X_{1,4,6,7,9,11,17,18}) - H(X_{5,14,17,18}) + H(X_{1,2,3,4,5,11,14,17,18}) + 4H(X_{1,2,4,8,9,10,11,12,13,15,17,18}) - 3H(X_{2,14,15,17,18}) + H(X_{5,10,14,15,16,17,18}) - H(X_{1,2,3,4,5,10,11,14,15,16,17,18}) + H(X_{19}) + 2H(X_{1,5,19}) - 2H(X_{1,4,5,19}) + 10H(X_{1,9,11,14,19}) + 4H(X_{1,3,4,7,8,13,15,16,19}) - 4H(X_{1,3,4,6,7,8,13,14,15,16,19}) + 5H(X_{4,7,8,10,17,19}) - 5H(X_{1,2,4,7,8,10,17,19}) + 8H(X_{4,5,7,16,17,19}) - 8H(X_{4,5,7,11,16,17,19}) + 8H(X_{4,8,9,10,17,18,19}) - 8H(X_{3,4,8,9,10,17,18,19}) + 10H(X_{1,4,6,7,9,11,17,18,19}) + 2H(X_{3,4,7,8,10,12,13,15,20}) + H(X_{1,5,7,9,16,20}) - 6H(X_{2,3,11,13,15,16,20}) + 6H(X_{2,3,8,11,13,15,16,20}) - 2H(X_{3,4,7,8,10,12,13,15,16,20}) - 4H(X_{1,2,4,5,17,20}) - 8H(X_{2,12,13,17,20}) + 8H(X_{2,7,12,13,17,20}) + 4H(X_{1,2,4,5,9,15,17,20}) + 8H(X_{1,2,3,5,6,12,13,15,17,20}) - 8H(X_{1,2,3,5,6,7,12,13,15,17,20}) - 3H(X_{16,17,20}) + 3H(X_{5,16,17,20}) - H(X_{1,3,4,5,7,9,10,12,16,17,20}) + 9H(X_{18,20}) - 9H(X_{1,2,4,6,7,10,14,18,20}) + 10H(X_{1,2,5,10,12,14,18,20}) + 5H(X_{1,3,4,10,11,15,18,20}) - 5H(X_{1,3,4,7,10,11,12,15,17,18,20}) + 6H(X_{2,3,4,6,11,13,15,16,17,19,20}) - 6H(X_{2,3,4,6,8,11,13,15,16,17,19,20}) - 4H(X_{7,12,18,19,20}) + 4H(X_{3,7,8,9,11,12,18,19,20}) - 10H(X_{1,2,5,9,10,11,12,14,18,19,20}) - 6H(X_{1,4,5,7,9,13,14,18,19,20}) + 4H(X_{7,12,15,18,19,20}) - 4H(X_{3,7,8,9,11,12,15,18,19,20}) + 3H(X_{2,3,7,10,12,13,14,15,17,18,19,20}) + 4H(X_{1,2,3,4,5,6,7,9,10,11,12,13,14,15,17,18,19,20}) + 6H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,19,20}) - 4H(X_{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20})","witness_value":994},"k":20,"logical_conditions":[{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{7}|X_{16,17})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{7};X_{10}|X_{1,5,16})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{7};X_{17}|X_{1,5})"},{"coefficient":-3,"kind":"conditional_mutual_information","label":"I(X_{3};X_{15}|X_{1,5,16})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{9};X_{15}|X_{5})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{5};X_{16}|X_{3,6,17})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{5};X_{13}|X_{16,17})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{3};X_{12}|X_{5})"}],"expression":[[16,-2],[17,-1],[20,1],[81,1],[272,1],[2064,1],[2068,-1],[16400,1],[16656,-1],[32785,2],[32789,-3],[32849,1],[33297,1],[33361,-1],[49169,-3],[49173,3],[65553,1],[65572,-1],[65588,1],[65617,-1],[98340,1],[98356,-1],[98368,2],[98384,-2],[102400,-2],[102416,2]],"id":0,"kind":"composite_equality","label":"witness equality 1","prompt_rows":["-2H(X_{5}) - H(X_{1,5}) + H(X_{3,5}) + H(X_{1,5,7}) + H(X_{5,9}) + H(X_{5,12}) - H(X_{3,5,12}) + H(X_{5,15}) - H(X_{5,9,15}) + 2H(X_{1,5,16}) - 3H(X_{1,3,5,16}) + H(X_{1,5,7,16}) + H(X_{1,5,10,16}) - H(X_{1,5,7,10,16}) - 3H(X_{1,5,15,16}) + 3H(X_{1,3,5,15,16}) + H(X_{1,5,17}) - H(X_{3,6,17}) + H(X_{3,5,6,17}) - H(X_{1,5,7,17}) + H(X_{3,6,16,17}) - H(X_{3,5,6,16,17}) + 2H(X_{7,16,17}) - 2H(X_{5,7,16,17}) - 2H(X_{13,16,17}) + 2H(X_{5,13,16,17})>=0","2H(X_{5}) + H(X_{1,5}) - H(X_{3,5}) - H(X_{1,5,7}) - H(X_{5,9}) - H(X_{5,12}) + H(X_{3,5,12}) - H(X_{5,15}) + H(X_{5,9,15}) - 2H(X_{1,5,16}) + 3H(X_{1,3,5,16}) - H(X_{1,5,7,16}) - H(X_{1,5,10,16}) + H(X_{1,5,7,10,16}) + 3H(X_{1,5,15,16}) - 3H(X_{1,3,5,15,16}) - H(X_{1,5,17}) + H(X_{3,6,17}) - H(X_{3,5,6,17}) + H(X_{1,5,7,17}) - H(X_{3,6,16,17}) + H(X_{3,5,6,16,17}) - 2H(X_{7,16,17}) + 2H(X_{5,7,16,17}) + 2H(X_{13,16,17}) - 2H(X_{5,13,16,17})>=0"],"text":"-2H(X_{5}) - H(X_{1,5}) + H(X_{3,5}) + H(X_{1,5,7}) + H(X_{5,9}) + H(X_{5,12}) - H(X_{3,5,12}) + H(X_{5,15}) - H(X_{5,9,15}) + 2H(X_{1,5,16}) - 3H(X_{1,3,5,16}) + H(X_{1,5,7,16}) + H(X_{1,5,10,16}) - H(X_{1,5,7,10,16}) - 3H(X_{1,5,15,16}) + 3H(X_{1,3,5,15,16}) + H(X_{1,5,17}) - H(X_{3,6,17}) + H(X_{3,5,6,17}) - H(X_{1,5,7,17}) + H(X_{3,6,16,17}) - H(X_{3,5,6,16,17}) + 2H(X_{7,16,17}) - 2H(X_{5,7,16,17}) - 2H(X_{13,16,17}) + 2H(X_{5,13,16,17})","type":"equality","witness_value":0},{"component_width":8,"component_witness_values":[0,0,0,0,0,0,0,0],"components":[{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{16};X_{17}|X_{5})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{5};X_{11}|X_{16,17})"},{"coefficient":3,"kind":"conditional_entropy","label":"H(X_{8}|X_{1,5,16})"},{"coefficient":-2,"kind":"conditional_mutual_information","label":"I(X_{4};X_{19}|X_{1,5})"},{"coefficient":-1,"kind":"conditional_mutual_information","label":"I(X_{15};X_{17}|X_{5})"},{"coefficient":1,"kind":"conditional_mutual_information","label":"I(X_{13};X_{18}|X_{5})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{5};X_{20}|X_{16,17})"},{"coefficient":3,"kind":"conditional_mutual_information","label":"I(X_{4};X_{18}|X_{5})"}],"expression":[[16,-4],[17,2],[24,3],[25,-2],[4112,1],[16400,-1],[32784,1],[32785,-3],[32913,3],[81936,1],[98304,-6],[98320,5],[99328,3],[99344,-3],[131088,4],[131096,-3],[135184,-1],[262161,-2],[262169,2],[622592,3],[622608,-3]],"id":1,"kind":"composite_equality","label":"witness equality 2","prompt_rows":["-4H(X_{5}) + 2H(X_{1,5}) + 3H(X_{4,5}) - 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2H(X_{2,3,4,6,7,8,10,12,13,16,18,19}) - 9H(X_{1,2,4,6,7,11,15,17,18,19}) - 5H(X_{1,3,7,8,9,10,13,16,17,18,19}) - 5H(X_{7,12,20}) + 16H(X_{1,13,20}) - 18H(X_{1,8,13,20}) + 5H(X_{4,6,7,12,13,14,20}) + 18H(X_{2,15,20}) + 5H(X_{7,12,15,20}) + 2H(X_{1,13,15,20}) - 18H(X_{2,13,15,20}) - 5H(X_{4,6,7,12,13,14,15,20}) + 9H(X_{1,2,6,7,15,16,20}) - 4H(X_{6,10,17,20}) + 9H(X_{3,5,7,9,10,13,17,20}) - 9H(X_{1,3,5,7,9,10,13,17,20}) + 4H(X_{6,10,14,17,20}) + 2H(X_{1,5,9,11,14,15,17,20}) + 4H(X_{5,6,10,16,17,20}) - 4H(X_{5,6,10,14,16,17,20}) + 7H(X_{2,7,8,10,18,20}) + 2H(X_{5,15,18,20}) - 2H(X_{5,14,15,18,20}) - 7H(X_{1,2,7,8,10,14,17,18,20}) + 8H(X_{1,8,11,14,15,19,20}) + H(X_{1,3,5,8,12,13,14,16,17,19,20}) - H(X_{1,3,5,8,12,13,14,15,16,17,19,20}) - 8H(X_{1,8,11,14,15,18,19,20}) - 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3H(X_{2,11,13,15}) - 6H(X_{1,13,16}) + 3H(X_{1,3,13,16}) - 3H(X_{1,13,17}) + 3H(X_{1,13,14,17}) - H(X_{13,18}) + H(X_{4,13,18}) - H(X_{1,13,19}) - 2H(X_{1,13,14,19}) + 3H(X_{1,13,16,19}) + 2H(X_{1,13,20}) - 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