diff --git a/research/bit-zero-frame-feedback-4x4/CERTIFICATE.json b/research/bit-zero-frame-feedback-4x4/CERTIFICATE.json new file mode 100644 index 0000000000..b11c43dc46 --- /dev/null +++ b/research/bit-zero-frame-feedback-4x4/CERTIFICATE.json @@ -0,0 +1,11600 @@ +{ + "schema_version": 1, + "kind": "finite-exact-shortest-circuit-certificate", + "helpers": 4, + "targets": 4, + "gate_cost": 1, + "ancillas": 0, + "all_inputs_arbitrary": true, + "gate_convention": "[d, s] means register[d] ^= register[s]; h0..h3 are 0..3, y0..y3 are 4..7", + "k_index": "K[i][j] (target i, helper j) is bit 4*i + j", + "minimum_histogram": { + "0": 1, + "1": 16, + "2": 120, + "3": 560, + "4": 1820, + "5": 4416, + "6": 8452, + "7": 13040, + "8": 16002, + "9": 14372, + "10": 6329, + "11": 384, + "12": 24 + }, + "feedback_gains": 0, + "costs_sha256": "0440c929e84901b5f45a4cde70f440ccd32a955b8243e07aa5082cf6bb598861", + "costs_hex": "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", + "census_radius": 5, + "restricted_spheres": [ + 1, + 40, + 936, + 16428, + 234224, + 2808680 + ], + "unrestricted_spheres": [ + 1, + 56, + 1904, + 50316, + 1121820, + 21927640 + ], + "hard": { + "J_minus_I": 31710, + "restricted_J_minus_I_words": { + "le11": 0, + "le12": 15852 + }, + "unrestricted_J_minus_I_le11_hits_by_first_gate_orbit": [ + 0, + 0, + 0, + 0, + 0, + 0 + ], + "control_cost11_hits_first_gate_hh": 283 + }, + "orbit_count": 317, + "orbits": [ + { + "k": 0, + "orbit_size": 1, + "minimum": 0, + "restricted_witness": [] + }, + { + "k": 1, + "orbit_size": 16, + "minimum": 1, + "restricted_witness": [ + [ + 4, + 0 + ] + ] + }, + { + "k": 3, + "orbit_size": 24, + "minimum": 2, + "restricted_witness": [ + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 7, + "orbit_size": 16, + "minimum": 3, + "restricted_witness": [ + [ + 4, + 2 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 15, + "orbit_size": 4, + "minimum": 4, + "restricted_witness": [ + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 17, + "orbit_size": 24, + "minimum": 2, + "restricted_witness": [ + [ + 5, + 0 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 18, + "orbit_size": 72, + "minimum": 2, + "restricted_witness": [ + [ + 5, + 0 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 19, + "orbit_size": 144, + "minimum": 3, + "restricted_witness": [ + [ + 5, + 0 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 22, + "orbit_size": 144, + "minimum": 3, + "restricted_witness": [ + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 23, + "orbit_size": 144, + "minimum": 4, + "restricted_witness": [ + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 30, + "orbit_size": 48, + "minimum": 4, + "restricted_witness": [ + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 31, + "orbit_size": 48, + "minimum": 5, + "restricted_witness": [ + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 51, + "orbit_size": 36, + "minimum": 4, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 53, + "orbit_size": 144, + "minimum": 4, + "restricted_witness": [ + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 55, + "orbit_size": 144, + "minimum": 5, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 60, + "orbit_size": 36, + "minimum": 4, + "restricted_witness": [ + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 61, + "orbit_size": 144, + "minimum": 5, + "restricted_witness": [ + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 63, + "orbit_size": 72, + "minimum": 6, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 119, + "orbit_size": 24, + "minimum": 5, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 123, + "orbit_size": 72, + "minimum": 6, + "restricted_witness": [ + [ + 5, + 2 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 127, + "orbit_size": 48, + "minimum": 6, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 5 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 255, + "orbit_size": 6, + "minimum": 6, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 273, + "orbit_size": 16, + "minimum": 3, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 274, + "orbit_size": 144, + "minimum": 3, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 275, + "orbit_size": 144, + "minimum": 4, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 278, + "orbit_size": 144, + "minimum": 4, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 279, + "orbit_size": 144, + "minimum": 5, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 286, + "orbit_size": 48, + "minimum": 5, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 287, + "orbit_size": 48, + "minimum": 6, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 291, + "orbit_size": 144, + "minimum": 4, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 292, + "orbit_size": 96, + "minimum": 3, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 293, + "orbit_size": 576, + "minimum": 4, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 295, + "orbit_size": 288, + "minimum": 5, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 300, + "orbit_size": 144, + "minimum": 4, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 301, + "orbit_size": 288, + "minimum": 5, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 303, + "orbit_size": 144, + "minimum": 6, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 307, + "orbit_size": 144, + "minimum": 5, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 309, + "orbit_size": 288, + "minimum": 5, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 310, + "orbit_size": 576, + "minimum": 5, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 311, + "orbit_size": 576, + "minimum": 6, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 316, + "orbit_size": 288, + "minimum": 5, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 317, + "orbit_size": 288, + "minimum": 6, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 318, + "orbit_size": 288, + "minimum": 6, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 319, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 358, + "orbit_size": 144, + "minimum": 5, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 1, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 1 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 359, + "orbit_size": 288, + "minimum": 6, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 1, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 362, + "orbit_size": 288, + "minimum": 5, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 363, + "orbit_size": 576, + "minimum": 6, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 366, + "orbit_size": 288, + "minimum": 6, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 1, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 1 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 367, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 375, + "orbit_size": 144, + "minimum": 6, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 4, + 5 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 379, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 382, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 1 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 383, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 3 + ], + [ + 6, + 0 + ], + [ + 4, + 5 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 494, + "orbit_size": 48, + "minimum": 6, + "restricted_witness": [ + [ + 6, + 0 + ], + [ + 4, + 5 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 495, + "orbit_size": 96, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 0 + ], + [ + 6, + 0 + ], + [ + 4, + 5 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 511, + "orbit_size": 48, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 819, + "orbit_size": 24, + "minimum": 5, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 821, + "orbit_size": 288, + "minimum": 6, + "restricted_witness": [ + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 823, + "orbit_size": 144, + "minimum": 6, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 828, + "orbit_size": 72, + "minimum": 6, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 829, + "orbit_size": 144, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 0, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 831, + "orbit_size": 72, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 854, + "orbit_size": 96, + "minimum": 6, + "restricted_witness": [ + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 855, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 0, + 2 + ] + ] + }, + { + "k": 857, + "orbit_size": 96, + "minimum": 6, + "restricted_witness": [ + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 858, + "orbit_size": 288, + "minimum": 6, + "restricted_witness": [ + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 859, + "orbit_size": 576, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 862, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 1 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 863, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 887, + "orbit_size": 144, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 0, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 2 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 891, + "orbit_size": 144, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 892, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 893, + "orbit_size": 576, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 0, + 2 + ], + [ + 6, + 0 + ], + [ + 4, + 3 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 895, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 2 + ], + [ + 6, + 0 + ], + [ + 4, + 3 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 975, + "orbit_size": 72, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 2, + 3 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ], + [ + 2, + 3 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 989, + "orbit_size": 144, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 990, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 2, + 3 + ], + [ + 4, + 1 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 2, + 3 + ] + ] + }, + { + "k": 991, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 4, + 1 + ], + [ + 4, + 5 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 1023, + "orbit_size": 72, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 5 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 5 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 1911, + "orbit_size": 16, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 2 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 1915, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 1919, + "orbit_size": 48, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 0 + ], + [ + 0, + 2 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 1981, + "orbit_size": 96, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 0, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 1983, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 5, + 4 + ], + [ + 4, + 3 + ], + [ + 6, + 4 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ], + [ + 5, + 4 + ], + [ + 4, + 2 + ], + [ + 6, + 4 + ] + ] + }, + { + "k": 2047, + "orbit_size": 48, + "minimum": 8, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 5, + 3 + ], + [ + 5, + 6 + ], + [ + 6, + 2 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 6 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 4095, + "orbit_size": 4, + "minimum": 8, + "restricted_witness": [ + [ + 4, + 6 + ], + [ + 5, + 6 + ], + [ + 6, + 3 + ], + [ + 6, + 2 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 6 + ], + [ + 4, + 6 + ] + ] + }, + { + "k": 4369, + "orbit_size": 4, + "minimum": 4, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 4370, + "orbit_size": 48, + "minimum": 4, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4371, + "orbit_size": 48, + "minimum": 5, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 4374, + "orbit_size": 48, + "minimum": 5, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4375, + "orbit_size": 48, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 4382, + "orbit_size": 16, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4383, + "orbit_size": 16, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4386, + "orbit_size": 36, + "minimum": 4, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4387, + "orbit_size": 144, + "minimum": 5, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 4388, + "orbit_size": 144, + "minimum": 4, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4389, + "orbit_size": 288, + "minimum": 5, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 4390, + "orbit_size": 288, + "minimum": 5, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4391, + "orbit_size": 288, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 4396, + "orbit_size": 144, + "minimum": 5, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4397, + "orbit_size": 144, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 4398, + "orbit_size": 144, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4399, + "orbit_size": 144, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4403, + "orbit_size": 72, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 4405, + "orbit_size": 144, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 4406, + "orbit_size": 288, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4407, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4412, + "orbit_size": 144, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4413, + "orbit_size": 144, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4414, + "orbit_size": 144, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4415, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4454, + "orbit_size": 72, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 1, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 1 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 4455, + "orbit_size": 144, + "minimum": 7, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 4458, + "orbit_size": 144, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4459, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4462, + "orbit_size": 144, + "minimum": 7, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 1 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 4463, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 1 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 4471, + "orbit_size": 72, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 4475, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4478, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 4479, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 4, + 3 + ], + [ + 4, + 5 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 4590, + "orbit_size": 24, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 4591, + "orbit_size": 48, + "minimum": 8, + "restricted_witness": [ + [ + 4, + 0 + ], + [ + 4, + 5 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 4607, + "orbit_size": 24, + "minimum": 8, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 5, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 4659, + "orbit_size": 72, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 4660, + "orbit_size": 288, + "minimum": 5, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4661, + "orbit_size": 576, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 4663, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4668, + "orbit_size": 144, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4669, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4671, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4679, + "orbit_size": 96, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 4680, + "orbit_size": 24, + "minimum": 4, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4681, + "orbit_size": 288, + "minimum": 5, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 4, + 3 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 4683, + "orbit_size": 288, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 4, + 3 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 4687, + "orbit_size": 96, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4693, + "orbit_size": 288, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 0, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 2 + ] + ] + }, + { + "k": 4694, + "orbit_size": 288, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4695, + "orbit_size": 576, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4697, + "orbit_size": 288, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 4698, + "orbit_size": 288, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4699, + "orbit_size": 576, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4700, + "orbit_size": 576, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4701, + "orbit_size": 576, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4702, + "orbit_size": 576, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4703, + "orbit_size": 576, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4727, + "orbit_size": 144, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 4731, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4732, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4733, + "orbit_size": 576, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4735, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 4, + 3 + ], + [ + 4, + 5 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 4812, + "orbit_size": 72, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 2, + 3 + ], + [ + 5, + 2 + ], + [ + 4, + 2 + ], + [ + 2, + 3 + ] + ] + }, + { + "k": 4813, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 2, + 3 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ], + [ + 2, + 3 + ] + ] + }, + { + "k": 4815, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 2, + 3 + ], + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ], + [ + 2, + 3 + ] + ] + }, + { + "k": 4829, + "orbit_size": 144, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 4830, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 2, + 3 + ], + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 2, + 3 + ] + ] + }, + { + "k": 4831, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 4, + 1 + ], + [ + 4, + 5 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 4863, + "orbit_size": 72, + "minimum": 8, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 5, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 4915, + "orbit_size": 48, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 4917, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 4918, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 4919, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4924, + "orbit_size": 144, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4925, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 1, + 0 + ], + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 1, + 0 + ] + ] + }, + { + "k": 4926, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4927, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4950, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4951, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 6, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4953, + "orbit_size": 96, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4954, + "orbit_size": 576, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4955, + "orbit_size": 576, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 6, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4958, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 4, + 1 + ], + [ + 4, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4959, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 4, + 1 + ], + [ + 5, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4966, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 7, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 1 + ], + [ + 1, + 2 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ] + ] + }, + { + "k": 4967, + "orbit_size": 576, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 6, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 4970, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4971, + "orbit_size": 576, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 6, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4972, + "orbit_size": 576, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4973, + "orbit_size": 576, + "minimum": 8, + "restricted_witness": [ + [ + 4, + 0 + ], + [ + 4, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4974, + "orbit_size": 576, + "minimum": 8, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 5, + 1 + ], + [ + 1, + 2 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4975, + "orbit_size": 576, + "minimum": 9, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 4, + 0 + ], + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 5, + 1 + ], + [ + 1, + 2 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4983, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 2 + ], + [ + 6, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ] + ] + }, + { + "k": 4987, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 5, + 2 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4988, + "orbit_size": 576, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 2 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 5, + 2 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4989, + "orbit_size": 576, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 0, + 2 + ], + [ + 6, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 4990, + "orbit_size": 576, + "minimum": 9, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 4, + 1 + ], + [ + 1, + 0 + ], + [ + 7, + 0 + ], + [ + 5, + 1 + ], + [ + 1, + 2 + ], + [ + 6, + 1 + ], + [ + 4, + 3 + ], + [ + 1, + 0 + ] + ] + }, + { + "k": 4991, + "orbit_size": 576, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 2 + ], + [ + 6, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 5068, + "orbit_size": 144, + "minimum": 7, + "restricted_witness": [ + [ + 2, + 3 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 4, + 2 + ], + [ + 2, + 3 + ] + ] + }, + { + "k": 5069, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 2, + 3 + ], + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 4, + 2 + ], + [ + 2, + 3 + ] + ] + }, + { + "k": 5070, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 2, + 3 + ], + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 4, + 2 + ], + [ + 2, + 3 + ] + ] + }, + { + "k": 5071, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 2, + 3 + ], + [ + 4, + 0 + ], + [ + 6, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 5, + 2 + ], + [ + 4, + 2 + ], + [ + 2, + 3 + ] + ] + }, + { + "k": 5085, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 5, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 5086, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 2, + 3 + ], + [ + 4, + 1 + ], + [ + 4, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 2, + 3 + ] + ] + }, + { + "k": 5087, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 4, + 1 + ], + [ + 4, + 5 + ], + [ + 5, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 5102, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 5, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 5103, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 4, + 0 + ], + [ + 4, + 5 + ], + [ + 5, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 5119, + "orbit_size": 144, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 5 + ], + [ + 5, + 0 + ], + [ + 6, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 5734, + "orbit_size": 48, + "minimum": 6, + "restricted_witness": [ + [ + 7, + 0 + ], + [ + 1, + 2 + ], + [ + 6, + 1 + ], + [ + 5, + 1 + ], + [ + 4, + 1 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 5735, + "orbit_size": 144, + "minimum": 7, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 1 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 5738, + "orbit_size": 288, + "minimum": 7, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 1, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 5739, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 1, + 2 + ], + [ + 4, + 1 + ] + ] + }, + { + "k": 5742, + "orbit_size": 144, + "minimum": 7, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 1 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 5743, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 1 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 5751, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 5754, + "orbit_size": 576, + "minimum": 8, + "restricted_witness": [ + [ + 2, + 1 + ], + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 2, + 1 + ] + ] + }, + { + "k": 5755, + "orbit_size": 576, + "minimum": 9, + "restricted_witness": [ + [ + 2, + 1 + ], + [ + 4, + 0 + ], + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 2, + 1 + ] + ] + }, + { + "k": 5758, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 5759, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 4, + 3 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 5804, + "orbit_size": 96, + "minimum": 7, + "restricted_witness": [ + [ + 4, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 6, + 1 + ], + [ + 5, + 3 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 5805, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 4, + 0 + ], + [ + 4, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 6, + 1 + ], + [ + 5, + 3 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 5806, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 3 + ], + [ + 1, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 5807, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 4, + 0 + ], + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 3 + ], + [ + 1, + 2 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 5819, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 5, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 6, + 1 + ], + [ + 5, + 3 + ], + [ + 5, + 1 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 5821, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 3 + ], + [ + 4, + 0 + ], + [ + 4, + 2 + ], + [ + 5, + 0 + ], + [ + 0, + 3 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 6, + 1 + ], + [ + 5, + 1 + ] + ] + }, + { + "k": 5822, + "orbit_size": 576, + "minimum": 9, + "restricted_witness": [ + [ + 1, + 3 + ], + [ + 4, + 1 + ], + [ + 4, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 5, + 1 + ], + [ + 1, + 3 + ], + [ + 6, + 1 + ], + [ + 5, + 0 + ] + ] + }, + { + "k": 5823, + "orbit_size": 576, + "minimum": 9, + "restricted_witness": [ + [ + 4, + 2 + ], + [ + 4, + 5 + ], + [ + 5, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 6, + 1 + ], + [ + 5, + 3 + ], + [ + 5, + 1 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 5870, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 1, + 3 + ], + [ + 7, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 1 + ], + [ + 1, + 3 + ], + [ + 6, + 1 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 5871, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 1, + 3 + ], + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 5, + 1 + ], + [ + 4, + 1 + ], + [ + 1, + 3 + ], + [ + 6, + 1 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 5887, + "orbit_size": 144, + "minimum": 9, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 4, + 5 + ], + [ + 5, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 3 + ], + [ + 5, + 1 + ], + [ + 4, + 5 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 6007, + "orbit_size": 48, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 2 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ] + ] + }, + { + "k": 6011, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 2, + 0 + ], + [ + 4, + 0 + ], + [ + 6, + 2 + ], + [ + 5, + 2 + ], + [ + 2, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 6014, + "orbit_size": 144, + "minimum": 9, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 0, + 1 + ], + [ + 4, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 4, + 3 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 6015, + "orbit_size": 144, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 2 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 6077, + "orbit_size": 96, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 3 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 4, + 2 + ], + [ + 5, + 0 + ], + [ + 0, + 3 + ], + [ + 6, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ] + ] + }, + { + "k": 6078, + "orbit_size": 288, + "minimum": 10, + "restricted_witness": [ + [ + 1, + 3 + ], + [ + 4, + 1 + ], + [ + 1, + 0 + ], + [ + 4, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 5, + 1 + ], + [ + 1, + 3 + ], + [ + 6, + 1 + ], + [ + 1, + 0 + ] + ] + }, + { + "k": 6079, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 5, + 4 + ], + [ + 4, + 3 + ], + [ + 6, + 4 + ], + [ + 7, + 0 + ], + [ + 4, + 1 + ], + [ + 4, + 0 + ], + [ + 5, + 4 + ], + [ + 4, + 2 + ], + [ + 6, + 4 + ] + ] + }, + { + "k": 6126, + "orbit_size": 144, + "minimum": 9, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 1, + 3 + ], + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 5, + 1 + ], + [ + 1, + 3 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 6127, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 5, + 4 + ], + [ + 4, + 3 + ], + [ + 6, + 4 + ], + [ + 7, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 1 + ], + [ + 5, + 4 + ], + [ + 4, + 0 + ], + [ + 6, + 4 + ] + ] + }, + { + "k": 6143, + "orbit_size": 144, + "minimum": 9, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 5, + 3 + ], + [ + 5, + 6 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 6, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 6 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 7918, + "orbit_size": 16, + "minimum": 8, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 1, + 3 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 1 + ], + [ + 4, + 1 + ], + [ + 1, + 3 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 7919, + "orbit_size": 48, + "minimum": 9, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 1, + 3 + ], + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 1 + ], + [ + 5, + 1 + ], + [ + 4, + 1 + ], + [ + 1, + 3 + ], + [ + 1, + 2 + ] + ] + }, + { + "k": 7935, + "orbit_size": 48, + "minimum": 9, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 5, + 0 + ], + [ + 5, + 6 + ], + [ + 7, + 0 + ], + [ + 6, + 3 + ], + [ + 6, + 2 + ], + [ + 6, + 1 + ], + [ + 5, + 6 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 8191, + "orbit_size": 16, + "minimum": 9, + "restricted_witness": [ + [ + 4, + 6 + ], + [ + 5, + 6 + ], + [ + 6, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 3 + ], + [ + 6, + 2 + ], + [ + 6, + 1 + ], + [ + 5, + 6 + ], + [ + 4, + 6 + ] + ] + }, + { + "k": 13107, + "orbit_size": 6, + "minimum": 6, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 13109, + "orbit_size": 96, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 0, + 1 + ], + [ + 4, + 0 + ] + ] + }, + { + "k": 13111, + "orbit_size": 48, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 13116, + "orbit_size": 24, + "minimum": 7, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 13117, + "orbit_size": 48, + "minimum": 8, + "restricted_witness": [ + [ + 1, + 0 + ], + [ + 4, + 0 + ], + [ + 7, + 1 + ], + [ + 6, + 1 + ], + [ + 5, + 1 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 1, + 0 + ] + ] + }, + { + "k": 13119, + "orbit_size": 24, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 4, + 2 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 13141, + "orbit_size": 72, + "minimum": 8, + "restricted_witness": [ + [ + 1, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 1 + ], + [ + 6, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 2 + ], + [ + 1, + 0 + ] + ] + }, + { + "k": 13142, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 13143, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 2 + ] + ] + }, + { + "k": 13145, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 1, + 0 + ], + [ + 4, + 0 + ], + [ + 7, + 1 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 1, + 0 + ] + ] + }, + { + "k": 13146, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 13147, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 13148, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 13149, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 1, + 0 + ], + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 7, + 1 + ], + [ + 6, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 0, + 2 + ], + [ + 1, + 0 + ] + ] + }, + { + "k": 13150, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 1 + ], + [ + 4, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 13151, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 4, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ] + ] + }, + { + "k": 13175, + "orbit_size": 72, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 13179, + "orbit_size": 72, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 13180, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 13181, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 4, + 3 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 13183, + "orbit_size": 144, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 4, + 3 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 13260, + "orbit_size": 18, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 2, + 3 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 4, + 2 + ], + [ + 2, + 3 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 13261, + "orbit_size": 144, + "minimum": 9, + "restricted_witness": [ + [ + 1, + 0 + ], + [ + 2, + 3 + ], + [ + 4, + 0 + ], + [ + 7, + 1 + ], + [ + 6, + 1 + ], + [ + 5, + 2 + ], + [ + 4, + 2 + ], + [ + 2, + 3 + ], + [ + 1, + 0 + ] + ] + }, + { + "k": 13263, + "orbit_size": 72, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 2, + 3 + ], + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 4, + 2 + ], + [ + 2, + 3 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 13277, + "orbit_size": 72, + "minimum": 9, + "restricted_witness": [ + [ + 1, + 0 + ], + [ + 4, + 5 + ], + [ + 5, + 0 + ], + [ + 7, + 1 + ], + [ + 6, + 1 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 4, + 5 + ], + [ + 1, + 0 + ] + ] + }, + { + "k": 13278, + "orbit_size": 72, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 2, + 3 + ], + [ + 4, + 1 + ], + [ + 4, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 2, + 3 + ] + ] + }, + { + "k": 13279, + "orbit_size": 144, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 2, + 3 + ], + [ + 4, + 0 + ], + [ + 4, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 2, + 3 + ] + ] + }, + { + "k": 13311, + "orbit_size": 36, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 5 + ], + [ + 5, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 5, + 2 + ], + [ + 4, + 5 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 13671, + "orbit_size": 96, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 4, + 1 + ], + [ + 7, + 1 + ], + [ + 6, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ] + ] + }, + { + "k": 13673, + "orbit_size": 288, + "minimum": 8, + "restricted_witness": [ + [ + 4, + 0 + ], + [ + 4, + 3 + ], + [ + 7, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ] + ] + }, + { + "k": 13675, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 4, + 3 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ] + ] + }, + { + "k": 13679, + "orbit_size": 96, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 4, + 1 + ], + [ + 4, + 3 + ], + [ + 7, + 1 + ], + [ + 6, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ] + ] + }, + { + "k": 13687, + "orbit_size": 144, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 6, + 0 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 13689, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 1, + 0 + ], + [ + 4, + 0 + ], + [ + 4, + 3 + ], + [ + 7, + 1 + ], + [ + 6, + 2 + ], + [ + 6, + 0 + ], + [ + 5, + 2 + ], + [ + 5, + 1 + ], + [ + 1, + 0 + ] + ] + }, + { + "k": 13690, + "orbit_size": 576, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 1 + ], + [ + 4, + 3 + ], + [ + 7, + 0 + ], + [ + 0, + 2 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 0, + 2 + ] + ] + }, + { + "k": 13691, + "orbit_size": 576, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 4, + 3 + ], + [ + 7, + 0 + ], + [ + 0, + 2 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 0, + 2 + ] + ] + }, + { + "k": 13694, + "orbit_size": 288, + "minimum": 10, + "restricted_witness": [ + [ + 1, + 2 + ], + [ + 4, + 1 + ], + [ + 1, + 0 + ], + [ + 4, + 3 + ], + [ + 5, + 1 + ], + [ + 1, + 2 + ], + [ + 7, + 1 + ], + [ + 6, + 2 + ], + [ + 6, + 0 + ], + [ + 1, + 0 + ] + ] + }, + { + "k": 13695, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 4, + 3 + ], + [ + 6, + 0 + ], + [ + 0, + 1 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 13726, + "orbit_size": 96, + "minimum": 9, + "restricted_witness": [ + [ + 4, + 1 + ], + [ + 4, + 2 + ], + [ + 4, + 3 + ], + [ + 7, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 5, + 0 + ] + ] + }, + { + "k": 13727, + "orbit_size": 96, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 3 + ], + [ + 4, + 0 + ], + [ + 4, + 1 + ], + [ + 4, + 2 + ], + [ + 7, + 1 + ], + [ + 5, + 0 + ], + [ + 0, + 3 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 6, + 0 + ] + ] + }, + { + "k": 13740, + "orbit_size": 72, + "minimum": 8, + "restricted_witness": [ + [ + 4, + 2 + ], + [ + 4, + 3 + ], + [ + 7, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 5, + 1 + ] + ] + }, + { + "k": 13741, + "orbit_size": 576, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 4, + 3 + ], + [ + 7, + 1 + ], + [ + 6, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 5, + 3 + ], + [ + 5, + 1 + ] + ] + }, + { + "k": 13743, + "orbit_size": 288, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 1, + 3 + ], + [ + 4, + 0 + ], + [ + 4, + 1 + ], + [ + 5, + 1 + ], + [ + 1, + 3 + ], + [ + 7, + 1 + ], + [ + 6, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ] + ] + }, + { + "k": 13755, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 3 + ], + [ + 4, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 3 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 0, + 1 + ], + [ + 6, + 0 + ] + ] + }, + { + "k": 13757, + "orbit_size": 288, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 3 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 4, + 2 + ], + [ + 5, + 0 + ], + [ + 0, + 3 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 0, + 1 + ], + [ + 6, + 0 + ] + ] + }, + { + "k": 13758, + "orbit_size": 576, + "minimum": 10, + "restricted_witness": [ + [ + 1, + 3 + ], + [ + 4, + 1 + ], + [ + 1, + 0 + ], + [ + 4, + 2 + ], + [ + 5, + 1 + ], + [ + 1, + 3 + ], + [ + 7, + 1 + ], + [ + 6, + 2 + ], + [ + 6, + 0 + ], + [ + 1, + 0 + ] + ] + }, + { + "k": 13759, + "orbit_size": 576, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 3 + ], + [ + 4, + 0 + ], + [ + 4, + 2 + ], + [ + 5, + 0 + ], + [ + 0, + 3 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 0, + 1 + ], + [ + 6, + 0 + ] + ] + }, + { + "k": 13806, + "orbit_size": 144, + "minimum": 9, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 5, + 1 + ], + [ + 5, + 2 + ], + [ + 7, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 13807, + "orbit_size": 288, + "minimum": 10, + "restricted_witness": [ + [ + 4, + 0 + ], + [ + 4, + 5 + ], + [ + 5, + 1 + ], + [ + 5, + 2 + ], + [ + 7, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 13823, + "orbit_size": 144, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 4, + 5 + ], + [ + 5, + 0 + ], + [ + 5, + 1 + ], + [ + 7, + 1 + ], + [ + 6, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 5, + 3 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 14199, + "orbit_size": 48, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 14203, + "orbit_size": 144, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 2, + 0 + ], + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 5, + 2 + ], + [ + 4, + 3 + ], + [ + 2, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 14204, + "orbit_size": 144, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 4, + 2 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 4, + 3 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 14205, + "orbit_size": 288, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 4, + 3 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 14207, + "orbit_size": 144, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 4, + 3 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 14268, + "orbit_size": 144, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 4, + 2 + ], + [ + 4, + 3 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 14269, + "orbit_size": 288, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 3 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 4, + 2 + ], + [ + 5, + 0 + ], + [ + 0, + 3 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 6, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 14271, + "orbit_size": 144, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 3 + ], + [ + 4, + 0 + ], + [ + 4, + 2 + ], + [ + 5, + 0 + ], + [ + 0, + 3 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 6, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 14285, + "orbit_size": 288, + "minimum": 10, + "restricted_witness": [ + [ + 1, + 0 + ], + [ + 2, + 3 + ], + [ + 4, + 0 + ], + [ + 4, + 2 + ], + [ + 7, + 1 + ], + [ + 5, + 2 + ], + [ + 2, + 3 + ], + [ + 6, + 2 + ], + [ + 6, + 1 + ], + [ + 1, + 0 + ] + ] + }, + { + "k": 14287, + "orbit_size": 288, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 2, + 3 + ], + [ + 4, + 0 + ], + [ + 4, + 2 + ], + [ + 7, + 0 + ], + [ + 5, + 2 + ], + [ + 2, + 3 + ], + [ + 6, + 2 + ], + [ + 6, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 14301, + "orbit_size": 288, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 0, + 3 + ], + [ + 4, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 3 + ], + [ + 0, + 1 + ], + [ + 6, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 14302, + "orbit_size": 288, + "minimum": 11, + "restricted_witness": [ + [ + 2, + 3 + ], + [ + 4, + 1 + ], + [ + 1, + 0 + ], + [ + 4, + 2 + ], + [ + 5, + 0 + ], + [ + 7, + 1 + ], + [ + 5, + 2 + ], + [ + 2, + 3 + ], + [ + 6, + 2 + ], + [ + 6, + 1 + ], + [ + 1, + 0 + ] + ] + }, + { + "k": 14303, + "orbit_size": 576, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 0, + 3 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 0, + 3 + ], + [ + 6, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 14335, + "orbit_size": 144, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 0, + 3 + ], + [ + 4, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 3 + ], + [ + 6, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 15615, + "orbit_size": 36, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 2, + 3 + ], + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 2, + 3 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 15837, + "orbit_size": 48, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 0, + 3 + ], + [ + 4, + 0 + ], + [ + 7, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 3 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ] + ] + }, + { + "k": 15838, + "orbit_size": 144, + "minimum": 10, + "restricted_witness": [ + [ + 2, + 3 + ], + [ + 0, + 2 + ], + [ + 4, + 1 + ], + [ + 4, + 2 + ], + [ + 7, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 2, + 3 + ] + ] + }, + { + "k": 15839, + "orbit_size": 144, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 0, + 3 + ], + [ + 4, + 0 + ], + [ + 4, + 1 + ], + [ + 7, + 1 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 3 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ] + ] + }, + { + "k": 15855, + "orbit_size": 144, + "minimum": 10, + "restricted_witness": [ + [ + 5, + 4 + ], + [ + 4, + 1 + ], + [ + 6, + 4 + ], + [ + 4, + 2 + ], + [ + 7, + 1 + ], + [ + 7, + 0 + ], + [ + 4, + 3 + ], + [ + 5, + 4 + ], + [ + 4, + 0 + ], + [ + 6, + 4 + ] + ] + }, + { + "k": 15871, + "orbit_size": 144, + "minimum": 10, + "restricted_witness": [ + [ + 4, + 5 + ], + [ + 5, + 1 + ], + [ + 5, + 6 + ], + [ + 6, + 0 + ], + [ + 7, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 3 + ], + [ + 6, + 2 + ], + [ + 5, + 6 + ], + [ + 4, + 5 + ] + ] + }, + { + "k": 16383, + "orbit_size": 24, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 0, + 3 + ], + [ + 4, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 3 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 30583, + "orbit_size": 4, + "minimum": 8, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 2 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 30587, + "orbit_size": 48, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 2, + 0 + ], + [ + 4, + 0 + ], + [ + 7, + 2 + ], + [ + 6, + 2 + ], + [ + 5, + 2 + ], + [ + 4, + 3 + ], + [ + 2, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 30591, + "orbit_size": 16, + "minimum": 9, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 4, + 3 + ], + [ + 0, + 2 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 30651, + "orbit_size": 36, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 2, + 0 + ], + [ + 0, + 3 + ], + [ + 4, + 0 + ], + [ + 7, + 2 + ], + [ + 6, + 2 + ], + [ + 5, + 0 + ], + [ + 0, + 3 + ], + [ + 2, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 30653, + "orbit_size": 144, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 0, + 1 + ], + [ + 4, + 3 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 0, + 2 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 30655, + "orbit_size": 144, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 4, + 0 + ], + [ + 4, + 3 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 0, + 2 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 30719, + "orbit_size": 24, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 0, + 3 + ], + [ + 4, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 3 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 0, + 2 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 31710, + "orbit_size": 24, + "minimum": 12, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 4, + 1 + ], + [ + 4, + 2 + ], + [ + 4, + 3 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 7, + 0 + ], + [ + 6, + 3 + ], + [ + 0, + 2 + ], + [ + 6, + 0 + ], + [ + 5, + 3 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 31711, + "orbit_size": 96, + "minimum": 11, + "restricted_witness": [ + [ + 0, + 2 + ], + [ + 0, + 3 + ], + [ + 5, + 0 + ], + [ + 0, + 1 + ], + [ + 4, + 0 + ], + [ + 6, + 0 + ], + [ + 0, + 3 + ], + [ + 7, + 0 + ], + [ + 6, + 2 + ], + [ + 0, + 2 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 31743, + "orbit_size": 72, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 3 + ], + [ + 6, + 0 + ], + [ + 0, + 2 + ], + [ + 5, + 0 + ], + [ + 4, + 0 + ], + [ + 0, + 3 + ], + [ + 7, + 0 + ], + [ + 0, + 2 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 32767, + "orbit_size": 16, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 0, + 3 + ], + [ + 4, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 3 + ], + [ + 7, + 0 + ], + [ + 0, + 2 + ], + [ + 0, + 1 + ] + ] + }, + { + "k": 65535, + "orbit_size": 1, + "minimum": 10, + "restricted_witness": [ + [ + 0, + 1 + ], + [ + 0, + 2 + ], + [ + 0, + 3 + ], + [ + 4, + 0 + ], + [ + 7, + 0 + ], + [ + 6, + 0 + ], + [ + 5, + 0 + ], + [ + 0, + 3 + ], + [ + 0, + 2 + ], + [ + 0, + 1 + ] + ] + } + ], + "controls": [ + "predecessor_3x3_reproduced", + "pr1557_3x4_reproduced", + "radius4_unresolved", + "feedback_rejected_in_restricted_alphabet", + "truncated_witness", + "wrong_target_map", + "self_xor", + "boolean_index", + "longer_word_mislabeled_optimal", + "tampered_minimum", + "cost11_positive_search_control" + ] +} diff --git a/research/bit-zero-frame-feedback-4x4/README.md b/research/bit-zero-frame-feedback-4x4/README.md new file mode 100644 index 0000000000..d2fd310c8b --- /dev/null +++ b/research/bit-zero-frame-feedback-4x4/README.md @@ -0,0 +1,251 @@ +# Zero-frame target feedback: exact four-helper/four-target census + +**Finite exact negative result; no new supplier or kappa.** For every binary +4-by-4 K, allowing targets to feed helpers does **not** shorten the minimum +fully restored circuit + + (h, y) -> (h, y + K h) + +on eight arbitrary dirty registers with unit-cost single-source XORs and no +extra registers. This answers the next question of +[PR1557](https://github.com/CrocSwap/integer-mult-bounds/pull/1557) (3-by-4), +which extended the merged 3-by-3 census of +[bit-zero-frame-feedback](../bit-zero-frame-feedback/README.md) (PR1306). The +model, the two alphabets and the prefix-substitution lemma are PR1306's, +unchanged. The scope is recorded as a refinement of `bit-helper-contract` in +[TASK.json](TASK.json). + +## Result + +All 65,536 labeled K, in 317 orbits under target and helper permutations, have +the same minimum in both alphabets: + +| Minimum XORs | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | +|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:| +| Number of K | 1 | 16 | 120 | 560 | 1,820 | 4,416 | 8,452 | 13,040 | 16,002 | 14,372 | 6,329 | 384 | 24 | + +* The maximum, 12, is attained exactly by the 24 matrices `J - P` (all ones minus + a permutation matrix): every row has weight three and the rows are distinct. + Twelve direct helper reads are optimal there. No restricted or unrestricted + word of at most 11 gates exists. +* The 408 K needing more than ten gates form three orbits. Two of them (288 K of + weight 11, 96 K of weight 13) cost exactly 11 in both alphabets. The weight-13 + orbit is two gates below its direct count, and that saving already exists + without feedback. The remaining 24 K form the J − P orbit, which costs 12. + +[CERTIFICATE.json](CERTIFICATE.json) has the histogram, the cost of every K +(`costs_hex`, indexed by `K[i][j]` = bit `4i + j`, target `i`, helper `j`) with +its SHA256, sphere sizes, the hard-case receipts, and for each orbit +representative an attaining restricted word. + +## Method and lower-bound argument + +A gate `[d, s]` is left multiplication by `E = I + e_d e_s^T` over F2. The required +map is `T_K = [[I, 0], [K, I]]`. The shears `N = {T_K}` form an elementary abelian +group, with `T_K T_K' = T_{K+K'}` and `T_K^2 = I`. + +**Coset keys.** `T_K U` changes only the target rows of `U`, adding combinations +of its helper rows. Two invertible matrices therefore lie in one coset `N U` +exactly when their helper rows agree and their target rows agree modulo the span +of those helper rows. `verify.py` keys each matrix by its helper rows and its +target rows reduced against a row-echelon basis of that span (zero at the pivot +columns). This is a canonical form of the coset. Within a coset, every element +is `T_label * anchor`, and two elements with labels `a`, `b` give `T_{a xor b}`. + +**Exactness to ten gates.** For each alphabet, `verify.py` builds the exact +radius-5 BFS ball and records every same-coset pair `(U, V)` whose depths differ +by at most one. Each pair is an upper bound: `T_K = V U^-1` costs at most +`d(U) + d(V)`, since gates are involutions. Conversely, let `w` be a minimal word +of length `L <= 10` for `T_K`. Its first `floor(L/2)` gates `P` and last +`ceil(L/2)` gates `S` are themselves geodesic, because they are subwords of a +geodesic. So `U = P^-1` and `V = S` lie in the ball, at depths differing by at +most one, with `V = T_K U`. The census minimum therefore equals the true minimum +whenever it is at most ten. If no pair gives at most ten, the minimum exceeds ten. + +**Beyond ten gates.** + +* *Weight-11 and weight-13 orbits:* the census shows both alphabets exceed ten + gates. An 11-gate restricted witness is replayed, so both costs are exactly 11. +* *J − P orbit, restricted:* a word of exactly 11 gates splits into a geodesic + first part of six gates and a last part of five. No `X` in the exact restricted + sphere 6 has `T_K X` in the restricted radius-5 ball. 12-gate words exist. +* *J − P orbit, unrestricted:* `J - I` is fixed by permuting helpers and targets + by the same `sigma` in S4. Relabeling a word by `sigma` gives another word for + the same `T_K` and length, so the first gate can be assumed to be one of six + orbit representatives. These are `hh`, `yy`, `y_a<-h_a`, `y_a<-h_b`, `h_a<-y_a` + and `h_a<-y_b`, with orbit sizes 12, 12, 4, 12, 4 and 12. With + `X = E_g Y` for `Y` in the radius-5 ball, no `T_K X` lies in that ball, so no + unrestricted word of 11 gates exists. Every other K in this orbit follows by + permutation invariance. + +Permutation invariance holds because conjugating by a helper permutation and a +target permutation maps each alphabet to itself and maps `T_K` to a permuted K. +The primary check also confirms it numerically on the computed costs. + +## Independent evidence + +[independent.py](independent.py) imports no primary or predecessor code. It +differs from `verify.py` in four ways: + +* matrices are column-packed rather than row-packed; +* lower bounds come from a per-orbit direct lookup of `T_K X` in the + unrestricted ball, with no coset grouping; +* the J − P exclusion normalizes the last gate rather than the first; +* every one of the 65,536 labeled K gets its orbit's witness transported by + register renaming, replayed on all 256 dirty inputs and checked against + `costs_hex`. + +An unrestricted lower bound also bounds the restricted alphabet. Both checkers +rest on the meet-in-the-middle principle, so this is independent code and +decomposition, not an independent mathematical method. + +A third, unregistered cross-check takes about 8 minutes. It scans all 56 first +gates for J − I with no symmetry reduction, and also found no word of at most +11 unrestricted gates. + +Both checkers reproduce published data: + +* `verify.py` reproduces the PR1306 3-by-3 histogram (read from its pinned + certificate) and the PR1557 3-by-4 histogram + `[1,12,66,220,495,810,1017,922,529,24]`, in both alphabets. +* The sphere sizes match the PR1306 table for 3-by-3. + +The 4-by-4 sphere sizes at radii 0 to 5 are: + +| Alphabet | 0 | 1 | 2 | 3 | 4 | 5 | +|---|---:|---:|---:|---:|---:|---:| +| Restricted (40 gates) | 1 | 40 | 936 | 16,428 | 234,224 | 2,808,680 | +| Unrestricted (56 gates) | 1 | 56 | 1,904 | 50,316 | 1,121,820 | 21,927,640 | + +Controls: + +* **Positive:** both predecessor reproductions; a cost-11 K found by the same + normalized search machinery used for J − P; attained-word lookups for 40 + orbits in the independent check. +* **Negative:** + * a radius-4 census must leave K unresolved; + * a cancelling feedback pair is accepted only in the unrestricted alphabet; + * a truncated witness, a wrong target map, a self-XOR and a Boolean register + index are each rejected; + * a valid but longer word labeled optimal is rejected; + * a tampered minimum is rejected. + +Both scripts refuse to run under `-O`. + +## Reproduce + +Python 3.12 with `numpy==2.4.0`, from the repository root: + +```sh +python3 scripts/research_package.py --package research/bit-zero-frame-feedback-4x4 --check-only +python3.12 scripts/research_package.py --package research/bit-zero-frame-feedback-4x4 --install-deps +``` + +The registered replay runs three commands: + +```sh +python3 -I -B research/bit-zero-frame-feedback-4x4/verify.py --stage census +python3 -I -B research/bit-zero-frame-feedback-4x4/verify.py --stage hard +python3 -I -B research/bit-zero-frame-feedback-4x4/independent.py +``` + +Expected lines: + +```text +PASS_ZERO_FRAME_FEEDBACK_4X4 maps=65536 orbits=317 max_minimum=12 feedback_gains=0 controls=11 no_new_kappa +PASS_ZERO_FRAME_FEEDBACK_4X4_HARD jmi_minimum=12 jmi_words_le11=0 both_alphabets no_new_kappa +PASS_INDEPENDENT_ZERO_FRAME_FEEDBACK_4X4 labeled_maps=65536 orbits=317 dirty_inputs_per_map=256 max_minimum=12 feedback_gains=0 +``` + +`verify.py --write` regenerates the certificate deterministically. + +Each stage took about 2.5 minutes (census and hard, run concurrently) and the +independent check about 1.5 minutes, on the donation host: Windows 11, CPython +3.14, numpy 2.4.0, 64 GB RAM. The largest arrays are the 23.1M-element +unrestricted ball and its level-5 expansion (about 63M candidates); peak memory +was not measured. CPython 3.12 was unavailable on the host. The registered 3.12 +replay was therefore **not** run locally; `--check-only` was. Local runs omitted +`-I` because numpy is installed in the host's user site directory. + +## Limits and unchecked obligations + +The equality is proved only for: + +* H = T = 4, over F2; +* PR1306's two single-source unit-cost alphabets; +* no extra registers, with all inputs arbitrary. + +It is not a theorem for larger dimensions, other gate costs, multi-source +gates, nonzero frames or other fields. + +**Exploratory, not certified:** every other shape with at most eight registers +that PR1306 and PR1557 left open was also checked with the same census code. +These are 1x4, 1x5, 2x4, 1x6, 2x5, 1x7, 2x6 and 3x5; transposes are equivalent +by the `M -> M^-T` duality. All give elementwise-equal costs in both alphabets, +with every K resolved within the exact ten-gate horizon. The maxima are 4, 5, 6, +6, 7, 7, 8 and 10. Reproduce one shape with: + +```sh +python3 -B research/bit-zero-frame-feedback-4x4/verify.py --shape H T +``` + +These shapes have no witnesses, independent check or registered replay here. +Together with PR1306 and PR1557, they suggest no feedback gain for any shape with +at most eight registers. Only 4-by-4 is certified. + +No fixed R439/R420 word was edited, and no physical supplier, precision/tape +bound, hosting, all-size compiler, native replay or recursive moment was built +or checked. The selected frontier's cold replay was not rerun for this isolated +finite result. + +## Open next questions + +1. **Is there a general no-feedback theorem, or a counterexample beyond eight + registers?** + * *Why it matters:* a proof, equivalent to geodesic convexity of the + block-lower-triangular parabolic subgroup of GL(H+T, 2) under unit + transvections, would close this D-local relaxation for every prefix size. + A counterexample would be the first legal cheaper prefix to price + physically. + * *First step:* certify the exploratory shapes above (witnesses and an + independent check, as here). A K on H helpers and T targets costs the same + as its transpose on T helpers and H targets, by the duality `M -> M^-T`, + which reverses every gate. Then look for a structural reason in the + incompressible extremal `J - P` family. + * *Blocked by:* there is no proof idea yet. Nine registers (4-by-5, 3-by-6) + need radius-6 unrestricted balls (hundreds of millions of matrices) or a + new reduction. + +The structured version is in [TASK.json](TASK.json). The 4-by-4 domain itself +needs no further search. + +## Credit + +Prepared for rtyu193-rgb with Claude Code (Anthropic Claude Opus 5.5), which +designed the coset method, wrote both checkers and drafted this README. This +was one session with no subagents and no independent human review. New work is +Apache-2.0. + +This package claims the finite 4-by-4 census and its certificate. It does not +claim novelty for elementary matrix substitution, CNOT synthesis or +meet-in-the-middle search. + +**Predecessors:** + +* @commdata2338's PR1306 (`bit-zero-frame-feedback`): the model, the + substitution lemma, the meet-in-the-middle setup and the original handoff. +* @bbMisk (Hanson Sun)'s + [PR1557](https://github.com/CrocSwap/integer-mult-bounds/pull/1557) + (`15e0987e5fa88d1c15699215ac0cd70b47a535a6`): the 3-by-4 census and this + next question. +* @woklerx's PR1194 and PR1105, and Arham Tahir's PR1311: the semantic-D and + feedback-cleanup lineage. + +**Underlying bit-word and checker lineage:** + +* Zhihao Chen (PR425), eumemic (PR555), DreamingOfClouds (PR589), Gareth + Davidson (PR926), @theon199 (PR967); +* Sagar Thakrar (PR1062), with Aaron Wang (PR1015) and Ross Richardson (PR923). + +No predecessor code is imported. The PR1306 certificate is read as a pinned +input for the reproduction control. diff --git a/research/bit-zero-frame-feedback-4x4/SOURCES.json b/research/bit-zero-frame-feedback-4x4/SOURCES.json new file mode 100644 index 0000000000..62cc06bd1b --- /dev/null +++ b/research/bit-zero-frame-feedback-4x4/SOURCES.json @@ -0,0 +1,24 @@ +{ + "schema_version": 1, + "base_commit": "418e40eb2572d930b58c1cf9b82b41c0fc7ee08d", + "files": [ + { + "path": "certificates/selected-result.json", + "usage": "provenance", + "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11", + "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32" + }, + { + "path": "docs/research/agent-task-catalog.json", + "usage": "provenance", + "git_blob": "06c96f5579bb67ed1b86d334e78fbed7da6945f1", + "sha256": "6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075" + }, + { + "path": "research/bit-zero-frame-feedback/CERTIFICATE.json", + "usage": "input", + "git_blob": "889acec6cf49f98cff65f120e8d6f27ca221e4cd", + "sha256": "8a370d0d7638582edfa2d7c5019e15b46dcae0d2b67c098517dbef49f72350e0" + } + ] +} diff --git a/research/bit-zero-frame-feedback-4x4/TASK.json b/research/bit-zero-frame-feedback-4x4/TASK.json new file mode 100644 index 0000000000..fe9885f6c1 --- /dev/null +++ b/research/bit-zero-frame-feedback-4x4/TASK.json @@ -0,0 +1,60 @@ +{ + "schema_version": 1, + "task_id": "bit-helper-contract", + "priority_revision": 3, + "base_commit": "418e40eb2572d930b58c1cf9b82b41c0fc7ee08d", + "original_card": { + "id": "bit-helper-contract", + "question": "Can one precisely specified helper-mixing word satisfy a replacement for the inherited H2/K interface?", + "starting_points": [ + "research/r439-bsection-stars-seed01/README.md", + "research/r439-copy-path-sharing-barrier/README.md" + ], + "methods": [ + "Derive a replacement interface lemma and price its implementation.", + "Seek an exact dirty-input or chronological-cut counterexample to the proposed replacement." + ], + "acceptance_gate": "A full boundary contract and paid profile; unchanged scalar counts alone are insufficient.", + "first_step": { + "command": [ + "python3", + "-B", + "research/r439-bsection-stars-seed01/verify.py" + ], + "expected": "PASS_R439_BSECTION_STARS", + "inspect": "Compare the two false H2/K flags with the unchanged paid profile in its README.", + "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay." + }, + "first_experiment": "Write one replacement H2/K implication for a single helper-star operation, then try to falsify it on dirty inputs.", + "bottleneck_link": "A legitimate replacement could admit a cheaper bit circuit; scalar savings alone do not alter the selected moment.", + "allowed_variations": [ + "Choose one helper-star or quartet instance; narrow to one boundary implication." + ], + "scope_change_triggers": [ + "Changing only source count or scalar addition count without an interface argument." + ], + "overlap_terms": [ + "H2_D_frame0_targets_from_helpers", + "K_equals_D", + "helper-star" + ] + }, + "scope_status": "refined", + "actual_scope": "D-local H1 relaxation, four helpers and four targets: exact minimum unit-cost single-source XOR count of the fully restored dirty shear (h,y) -> (h,y+Kh) for all 65,536 binary 4-by-4 K, with and without target feedback into helpers, eight arbitrary dirty registers and no extra registers.", + "scope_change_reason": "PR1194/PR1311 already treat the card's original H2/K question on a fixed word while retaining H1 inside D. PR1306 (merged) isolated that retained restriction in a fully restored zero-frame prefix and settled 1 <= H,T <= 3; open PR1557 settled 3-by-4 and named four helpers and four targets as the next question. This package answers that handoff in the same model and alphabets.", + "bottleneck_link": "A strictly cheaper feedback prefix would be a legal D-local scalar saving under PR1306's substitution lemma, which could then be priced for precision, hosting and the paid profile. The census finds no gate-count gain for any 4-by-4 K, so this finite domain cannot supply that improvement. Side result for the bit supplier; the bit side does not bind and no kappa is claimed.", + "budget_minutes": 60, + "stop_rule": "Stop once every labeled 4-by-4 K has an attained restricted word and an exact unrestricted shorter-word exclusion; if the radius-5 census plus targeted radius-6 checks cannot resolve every K, report the exact coverage without a minimum claim.", + "outcome": "negative-result", + "evidence": "CERTIFICATE.json: all 65,536 K in 317 row/column orbits; minimum histogram [1,16,120,560,1820,4416,8452,13040,16002,14372,6329,384,24], maximum 12 (the 24 matrices J-P), zero feedback gains. Primary: coset meet-in-the-middle census to ten gates in both alphabets, restricted sphere-6 and symmetry-reduced unrestricted checks beyond ten, attaining witnesses, PR1306 3x3 and PR1557 3x4 reproductions and eleven controls. Independent: column-packed per-orbit lookups, last-gate normalization for J-I and 65,536 transported witnesses replayed on all 256 dirty inputs. An unreduced 56-gate scan (not registered; about 8 minutes) also found no 11-gate J-I word. Exploratory (uncertified): all other shapes with at most eight registers left open by PR1306/PR1557 are elementwise equal via verify.py --shape.", + "author": "rtyu193-rgb", + "ai_assistance": "Claude Code (Anthropic Claude Opus 5.5) designed the method, wrote both checkers and the prose; one session, no subagents, no independent human review.", + "open_questions": [ + { + "question": "Does target feedback ever lower the minimum for any H and T (a general no-feedback theorem, i.e. geodesic convexity of the block-lower-triangular parabolic subgroup of GL(H+T,2) for unit transvections), or does a counterexample appear beyond eight registers?", + "why_it_matters": "A proof would close the D-local feedback relaxation as a source of scalar savings for every prefix size; a counterexample would give the first legal cheaper prefix to price physically.", + "first_step": "Certify the exploratory shapes with at most eight registers (1x4, 1x5, 2x4, 1x6, 2x5, 1x7, 2x6, 3x5, already elementwise equal via verify.py --shape; transposes equivalent by M -> M^-T) with witnesses and an independent check, then seek a structural reason in the incompressible extremal J-P family.", + "blocked_by": "No proof idea yet; nine registers (4x5, 3x6) need radius-6 unrestricted balls (hundreds of millions of matrices) or a new reduction." + } + ] +} diff --git a/research/bit-zero-frame-feedback-4x4/VERIFY.json b/research/bit-zero-frame-feedback-4x4/VERIFY.json new file mode 100644 index 0000000000..ea3877e391 --- /dev/null +++ b/research/bit-zero-frame-feedback-4x4/VERIFY.json @@ -0,0 +1,47 @@ +{ + "schema_version": 1, + "status": "ready", + "python": "3.12", + "requirements": [ + "numpy==2.4.0" + ], + "inputs": [ + "research/bit-zero-frame-feedback/CERTIFICATE.json" + ], + "commands": [ + { + "argv": [ + "{python}", + "-I", + "-B", + "{package}/verify.py", + "--stage", + "census" + ], + "timeout_seconds": 450, + "checks": "Coset meet-in-the-middle census of all 65,536 4x4 K to ten gates in both alphabets, permutation invariance, attaining restricted witness per orbit on all dirty inputs, full certificate comparison, PR1306 3x3 and PR1557 3x4 reproductions, radius-4 insufficiency and seven witness-rule controls." + }, + { + "argv": [ + "{python}", + "-I", + "-B", + "{package}/verify.py", + "--stage", + "hard" + ], + "timeout_seconds": 450, + "checks": "Beyond ten gates: no restricted J-I word of <= 11 gates via sphere 6; no unrestricted J-I word of <= 11 gates via S4 first-gate normalization; positive search control on a cost-11 K." + }, + { + "argv": [ + "{python}", + "-I", + "-B", + "{package}/independent.py" + ], + "timeout_seconds": 300, + "checks": "Code-disjoint column-packed check: 65,536 transported witnesses replayed on 256 dirty inputs, per-orbit unrestricted shorter-word exclusion by direct lookup, J-I 11-gate exclusion by last-gate normalization, sphere-size agreement." + } + ] +} diff --git a/research/bit-zero-frame-feedback-4x4/independent.py b/research/bit-zero-frame-feedback-4x4/independent.py new file mode 100644 index 0000000000..77624da521 --- /dev/null +++ b/research/bit-zero-frame-feedback-4x4/independent.py @@ -0,0 +1,152 @@ +#!/usr/bin/env python3 +"""Independent check of the 4x4 dirty-shear certificate; imports no primary or predecessor code. + +Different representation and decomposition from verify.py: + * matrices are column-packed (byte c holds column c; bit r is row r); + * lower bounds are per orbit representative, by direct lookup of T_K X in a ball, with no + coset grouping: no unrestricted word of (minimum - 1) gates exists; + * the J-I orbit's 11-gate exclusion normalizes the *last* gate under the diagonal S4 + stabilizer (verify.py normalizes the first gate); + * every one of the 65,536 labeled K receives its orbit witness transported by register + renaming, replayed on all 256 dirty inputs, and checked against costs_hex. + +Prepared for rtyu193-rgb with Claude Code (Claude Opus 5.5). Apache-2.0. +""" +import itertools +import json +from pathlib import Path +import time + +import numpy as np + +if not __debug__: + raise SystemExit("checks use assert; run without -O") + +CERT = Path(__file__).resolve().parent / "CERTIFICATE.json" +H = T = 4 +N = H + T +U64 = np.uint64 +ONES = U64(0x0101010101010101) +LAST_GATE_REPS = [(1, 0), (5, 4), (4, 0), (5, 0), (0, 4), (0, 5)] # one per diagonal-S4 gate orbit + + +def gate(X, d, s): + """Left multiplication by I + e_d e_s^T on column-packed matrices: in every column, bit d ^= bit s.""" + return X ^ (((X >> U64(s)) & ONES) << U64(d)) + + +def shear(k, X): + """T_K X: in every column, bit (H+i) ^= bit j for each K[i][j] = 1.""" + Y = X.copy() + for i in range(T): + for j in range(H): + if (k >> (i * H + j)) & 1: + Y ^= ((X >> U64(j)) & ONES) << U64(H + i) + return Y + + +def balls(radius): + """Unrestricted balls B_0..B_radius as sorted arrays (cumulative).""" + G = [(d, s) for d in range(N) for s in range(N) if d != s] + seen = np.array([sum(1 << (9 * c) for c in range(N))], U64) + frontier = seen.copy() + out = [seen.copy()] + for _ in range(radius): + nxt = np.unique(np.concatenate([gate(frontier, d, s) for d, s in G])) + frontier = nxt[~np.isin(nxt, seen)] + seen = np.union1d(seen, frontier) + out.append(seen) + return out + + +def contains(ball, q): + """Boolean membership of q in a sorted ball, by binary search.""" + i = np.minimum(np.searchsorted(ball, q), len(ball) - 1) + return ball[i] == q + + +def orbit_map(): + """For every K: (canonical index, (p, q)) with canon[i][j] = K[p[i]][q[j]].""" + ks = np.arange(1 << (H * T), dtype=np.int64) + best = np.full(len(ks), 1 << 20, np.int64) + arg = np.zeros(len(ks), np.int64) + perms = [(p, q) for p in itertools.permutations(range(T)) for q in itertools.permutations(range(H))] + for n, (p, q) in enumerate(perms): + img = np.zeros_like(ks) + for i in range(T): + for j in range(H): + img |= ((ks >> (p[i] * H + q[j])) & 1) << (i * H + j) + better = img < best + best[better], arg[better] = img[better], n + return best, arg, perms + + +def replay(word, k): + x = np.arange(1 << N) + regs = [(x >> r) & 1 for r in range(N)] + want = [r.copy() for r in regs] + for i in range(T): + for j in range(H): + if (k >> (i * H + j)) & 1: + want[H + i] = want[H + i] ^ regs[j] + for d, s in word: + regs[d] = regs[d] ^ regs[s] + return all((a == b).all() for a, b in zip(regs, want)) + + +def main(): + cert = json.loads(CERT.read_text(encoding="utf-8")) + assert (cert["helpers"], cert["targets"], cert["gate_cost"], cert["ancillas"]) == (H, T, 1, 0) + costs = np.array([int(c, 16) for c in cert["costs_hex"]]) + assert len(costs) == 1 << 16 + t = time.time() + canon, arg, perms = orbit_map() + reps = sorted(set(canon.tolist())) + entries = {e["k"]: e for e in cert["orbits"]} + assert sorted(entries) == reps, "orbit representatives differ" + # Upper bounds: transport each witness to every labeled K; replay all 256 dirty inputs. + for k in range(1 << 16): + e = entries[int(canon[k])] + p, q = perms[arg[k]] + rename = [q[j] for j in range(H)] + [H + p[i] for i in range(T)] + word = [(rename[d], rename[s]) for d, s in e["restricted_witness"]] + assert all(not (d < H and s >= H) and d != s for d, s in word), "restricted alphabet violated" + assert len(word) == e["minimum"] == costs[k], f"cost mismatch at {k}" + assert replay(word, k), f"transported witness fails at K={k}" + hist = np.bincount(costs).tolist() + assert {str(c): n for c, n in enumerate(hist)} == cert["minimum_histogram"] + assert sum(e["orbit_size"] for e in cert["orbits"]) == 1 << 16 + print(f" 65,536 transported restricted witnesses replayed on 256 dirty inputs ({time.time()-t:.0f}s)", flush=True) + # Lower bounds in the unrestricted alphabet (hence also restricted): exclude minimum - 1 gates. + t = time.time() + B = balls(5) + assert [len(b) for b in B] == list(np.cumsum(cert["unrestricted_spheres"])), "sphere sizes differ" + print(f" unrestricted balls {[len(b) for b in B]} ({time.time()-t:.0f}s)", flush=True) + jmi = cert["hard"]["J_minus_I"] + checked = 0 + for r in reps: + L = int(costs[r]) - 1 + if L < 0: + continue + if L <= 10: + a = max(0, L - 5) + assert not contains(B[min(L, 5)], shear(r, B[a])).any(), f"shorter word exists for K={r}" + checked += 1 + else: + assert L == 11 and canon[jmi] == r, "only the J-I orbit needs an 11-gate exclusion" + # J-I itself: <= 10 excluded directly, then exactly 11 by last-gate normalization. + assert not contains(B[5], shear(jmi, B[5])).any() + for d, s in LAST_GATE_REPS: + assert not contains(B[5], gate(shear(jmi, B[5]), d, s)).any(), f"11-gate word, last gate {(d, s)}" + # Positive controls: the same lookups find attained words one gate longer. + for r in reps[:40]: + L = int(costs[r]) + if 1 <= L <= 10: + assert contains(B[min(L, 5)], shear(r, B[max(0, L - 5)])).any(), f"control miss at {r}" + print(f" {checked} orbit lower bounds + J-I 11-gate exclusion ({time.time()-t:.0f}s)", flush=True) + print(f"PASS_INDEPENDENT_ZERO_FRAME_FEEDBACK_4X4 labeled_maps=65536 orbits={len(reps)} " + f"dirty_inputs_per_map=256 max_minimum={len(hist) - 1} feedback_gains=0") + + +if __name__ == "__main__": + main() diff --git a/research/bit-zero-frame-feedback-4x4/verify.py b/research/bit-zero-frame-feedback-4x4/verify.py new file mode 100644 index 0000000000..b40963aa61 --- /dev/null +++ b/research/bit-zero-frame-feedback-4x4/verify.py @@ -0,0 +1,432 @@ +#!/usr/bin/env python3 +"""Exact four-helper/four-target dirty-shear XOR costs with and without target feedback. + +Primary checker. Registers are ordered h0..h3, y0..y3; a gate [d, s] is +register[d] ^= register[s] at unit cost. The restricted alphabet omits every +helper-destination/target-source gate; the unrestricted alphabet allows every +ordered pair of distinct registers. The required complete map is +(h, y) -> (h, y + K h) on arbitrary dirty inputs, with no extra registers. + +Stages (run from the repository root): + census all 65,536 K exactly to ten gates in both alphabets by coset + meet-in-the-middle; predecessor reproductions; witnesses; controls + hard exact resolution of the 408 K beyond ten gates + --write regenerate CERTIFICATE.json (deterministic) after both stages + +Prepared for rtyu193-rgb with Claude Code (Claude Opus 5.5). Apache-2.0. +""" +import argparse +import hashlib +import itertools +import json +from pathlib import Path +import sys +import time + +import numpy as np + +if not __debug__: + raise SystemExit("checks use assert; run without -O") + +PKG = Path(__file__).resolve().parent +ROOT = PKG.parents[1] +CERT = PKG / "CERTIFICATE.json" +PREDECESSOR = ROOT / "research/bit-zero-frame-feedback/CERTIFICATE.json" +# PR1557 (bbMisk) head 15e0987e5fa88d1c15699215ac0cd70b47a535a6, three helpers / four targets. +PR1557_HISTOGRAM = [1, 12, 66, 220, 495, 810, 1017, 922, 529, 24] +U64 = np.uint64 +ONES = U64(0x0101010101010101) +RADIUS = 5 +H4 = T4 = 4 +# One gate from each orbit of S4 acting diagonally on (h_a, y_a): hh, yy, y_a<-h_a, +# y_a<-h_b, h_a<-y_a, h_a<-y_b. Orbit sizes 12+12+4+12+4+12 = 56. +GATE_ORBIT_REPS = [(0, 1), (4, 5), (4, 0), (4, 1), (0, 4), (0, 5)] + + +def log(*args): + print(*args, flush=True) + + +def gens(H, T, restricted): + n = H + T + return [(d, s) for d in range(n) for s in range(n) + if d != s and not (restricted and d < H and s >= H)] + + +def identity(n): + return U64(sum(1 << (9 * i) for i in range(n))) + + +def left_gate(M, d, s): + """(I + e_d e_s^T) M on row-packed matrices: row d ^= row s.""" + return M ^ (((M >> U64(8 * s)) & U64(0xFF)) << U64(8 * d)) + + +def ball(H, T, restricted, R): + """Exact BFS spheres 0..R as (matrices, inverses, last-gate index), each sorted by matrix.""" + G = gens(H, T, restricted) + I = identity(H + T) + spheres = [(np.array([I], U64), np.array([I], U64), np.array([-1], np.int16))] + prev = np.array([], U64) + for _ in range(R): + cur, cur_inv, _ = spheres[-1] + mats = np.concatenate([left_gate(cur, d, s) for d, s in G]) + # M' = E M gives M'^-1 = M^-1 E: column s ^= column d. + invs = np.concatenate([cur_inv ^ (((cur_inv >> U64(d)) & ONES) << U64(s)) for d, s in G]) + gi = np.repeat(np.arange(len(G), dtype=np.int16), len(cur)) + mats, idx = np.unique(mats, return_index=True) + invs, gi = invs[idx], gi[idx] + keep = ~np.isin(mats, cur, assume_unique=True) & ~np.isin(mats, prev, assume_unique=True) + prev = cur + spheres.append((mats[keep], invs[keep], gi[keep])) + return G, spheres + + +def coset_key(M, H, T): + """Key of the coset {T_K M}: exact h rows, y rows reduced modulo the span of the h rows.""" + rows = [((M >> U64(8 * r)) & U64(0xFF)).astype(np.uint8) for r in range(H + T)] + h = [r.copy() for r in rows[:H]] + y = [r.copy() for r in rows[H:]] + for k in range(H): + p = h[k] & ((~h[k]) + np.uint8(1)) + if not (p != 0).all(): + raise AssertionError("dependent helper rows in an invertible matrix") + for j in range(H): + if j != k: + h[j] ^= np.where((h[j] & p) != 0, h[k], np.uint8(0)) + for i in range(T): + y[i] ^= np.where((y[i] & p) != 0, h[k], np.uint8(0)) + key = np.zeros(len(M), U64) + for r in range(H): + key |= rows[r].astype(U64) << U64(8 * r) + for i in range(T): + key |= y[i].astype(U64) << U64(8 * (H + i)) + return key + + +def shear_label(V, Uinv, H, T): + """Index sum K[i,j] 2^(iH+j) of the K with V U^-1 = T_K; asserts it is a shear.""" + out = np.zeros(len(V), np.int64) + for c in range(H): # helper rows of V U^-1 must be identity rows + row = np.zeros(len(V), U64) + vrow = (V >> U64(8 * c)) & U64(0xFF) + for e in range(H + T): + row ^= ((vrow >> U64(e)) & U64(1)) * ((Uinv >> U64(8 * e)) & U64(0xFF)) + if not (row == U64(1 << c)).all(): + raise AssertionError("helper row changed: not a shear") + for i in range(T): + row = np.zeros(len(V), U64) + vrow = (V >> U64(8 * (H + i))) & U64(0xFF) + for e in range(H + T): + row ^= ((vrow >> U64(e)) & U64(1)) * ((Uinv >> U64(8 * e)) & U64(0xFF)) + if not ((row >> U64(H)) == U64(1 << i)).all(): + raise AssertionError("target block is not the identity: not a shear") + out |= (row & U64((1 << H) - 1)).astype(np.int64) << (i * H) + return out + + +def census(H, T, restricted, R): + """Exact minimum for every K up to 2R gates (2R+1 marks 'beyond'), plus sphere sizes. + + Words of length L <= 2R split into a prefix of floor(L/2) and suffix of ceil(L/2) gates, + so pairs whose BFS depths differ by at most one in a common coset of the shear group + suffice: with U = (prefix)^-1 and V = suffix, V = T_K U and |T_K| <= |U| + |V|. + """ + _, sph = ball(H, T, restricted, R) + sizes = [len(s[0]) for s in sph] + M = np.concatenate([s[0] for s in sph]); Minv = np.concatenate([s[1] for s in sph]) + dist = np.concatenate([np.full(len(s[0]), r, np.int16) for r, s in enumerate(sph)]) + del sph + key = coset_key(M, H, T) + order = np.argsort(key, kind="stable") + key, M, Minv, dist = key[order], M[order], Minv[order], dist[order] + starts = np.flatnonzero(np.r_[True, key[1:] != key[:-1]]) + counts = np.diff(np.r_[starts, len(key)]) + label = shear_label(M, Minv[np.repeat(starts, counts)], H, T) # element = T_label * anchor + del key, M, Minv + reach = np.zeros((2 * R + 1, 1 << (H * T)), bool) + for m in np.unique(counts): + idx = starts[counts == m][:, None] + np.arange(m)[None, :] + L, D = label[idx], dist[idx] + for a in range(m): + ok = np.abs(D[:, a:a + 1] - D[:, a:]) <= 1 + reach[(D[:, a:a + 1] + D[:, a:])[ok], (L[:, a:a + 1] ^ L[:, a:])[ok]] = True + cost = np.where(reach.any(0), reach.argmax(0), 2 * R + 1).astype(np.int16) + return cost, sizes + + +def histogram(cost, horizon): + return np.bincount(cost[cost <= horizon], minlength=horizon + 1).tolist() + + +def k_matrix(k, H=H4, T=T4): + return [[(k >> (i * H + j)) & 1 for j in range(H)] for i in range(T)] + + +def shear(K, X, H=H4, T=T4): + Y = X.copy() + for i in range(T): + for j in range(H): + if K[i][j]: + Y ^= ((X >> U64(8 * j)) & U64(0xFF)) << U64(8 * (H + i)) + return Y + + +def member(sorted_arr, q): + pos = np.searchsorted(sorted_arr, q) + pos[pos == len(sorted_arr)] = 0 + return sorted_arr[pos] == q + + +def canonical_all(H=H4, T=T4): + """For every K, the minimum index over its row (target) and column (helper) permutations.""" + ks = np.arange(1 << (H * T), dtype=np.int64) + best = ks.copy() + for p in itertools.permutations(range(T)): + for q in itertools.permutations(range(H)): + img = np.zeros_like(ks) + for i in range(T): + for j in range(H): + img |= ((ks >> (p[i] * H + q[j])) & 1) << (i * H + j) + best = np.minimum(best, img) + return best + + +def word_of(M, r, G, sph): + word = [] + while r > 0: + arr, _, gi = sph[r] + pos = int(np.searchsorted(arr, M)) + if arr[pos] != M: + raise AssertionError("parent walk left the ball") + d, s = G[gi[pos]] + word.append([d, s]) + M = left_gate(np.array([M], U64), d, s)[0] + r -= 1 + return word[::-1] # chronological order + + +def find_word(k, length, G, sph): + """A word of exactly `length` gates for T_K from the ball, or None. T_K = V X^-1.""" + K = k_matrix(k) + R = len(sph) - 1 + for a in range(max(0, length - R), min(length, R) + 1): + X = sph[a][0] + V = shear(K, X) + hit = np.flatnonzero(member(sph[length - a][0], V)) + if len(hit): + wx = word_of(X[hit[0]], a, G, sph) + return wx[::-1] + word_of(V[hit[0]], length - a, G, sph) + return None + + +def replay_ok(word, k, restricted, H=H4, T=T4): + """Complete boundary on all 2^(H+T) dirty inputs; alphabet and index checks.""" + K = k_matrix(k, H, T) + n = H + T + for g in word: + if (len(g) != 2 or any(type(v) is not int for v in g) or not all(0 <= v < n for v in g) + or g[0] == g[1] or (restricted and g[0] < H and g[1] >= H)): + return False + x = np.arange(1 << n) + regs = [(x >> r) & 1 for r in range(n)] + want = [regs[r].copy() for r in range(n)] + for i in range(T): + for j in range(H): + if K[i][j]: + want[H + i] ^= regs[j] + regs = [r.copy() for r in regs] + for d, s in word: + regs[d] ^= regs[s] + return all((regs[r] == want[r]).all() for r in range(n)) + + +def costs_digest(cost): + return hashlib.sha256(np.asarray(cost, np.uint8).tobytes()).hexdigest() + + +def stage_census(cert=None): + out = {} + t = time.time() + # Positive controls: reproduce both predecessor censuses in both alphabets. + pred = json.loads(PREDECESSOR.read_text(encoding="utf-8")) + p33 = next(d for d in pred["dimensions"] if d["helpers"] == 3 and d["targets"] == 3) + want33 = [p33["minimum_histogram"][str(c)] for c in range(len(p33["minimum_histogram"]))] + for restricted in (True, False): + c33, _ = census(3, 3, restricted, 4) + assert histogram(c33, 8)[:len(want33)] == want33 and (c33 <= 7).all(), "PR1306 3x3 not reproduced" + c34, _ = census(3, 4, restricted, 5) + assert histogram(c34, 10)[:10] == PR1557_HISTOGRAM and (c34 <= 9).all(), "PR1557 3x4 not reproduced" + log(f" predecessor 3x3 (PR1306) and 3x4 (PR1557) histograms reproduced in both alphabets ({time.time()-t:.0f}s)") + # Insufficient-radius control: radius 4 leaves K unresolved; nothing beyond 8 may be claimed. + c4, _ = census(H4, T4, True, 4) + assert int((c4 > 8).sum()) > 0, "radius-4 control should leave unresolved K" + costs = {} + for name, restricted in (("restricted", True), ("unrestricted", False)): + t = time.time() + costs[name], out[name + "_spheres"] = census(H4, T4, restricted, RADIUS) + log(f" {name}: spheres {out[name + '_spheres']} ({time.time()-t:.0f}s)") + res, unr = costs["restricted"], costs["unrestricted"] + horizon = 2 * RADIUS + assert np.array_equal(res, unr), "feedback changed a cost within ten gates" + assert np.array_equal(c4 > 8, res > 8) and (res[c4 <= 8] == c4[c4 <= 8]).all(), "radius 4/5 disagree" + beyond = np.flatnonzero(res > horizon) + out["histogram_to_10"] = histogram(res, horizon) + out["beyond_10"] = beyond.tolist() + # Symmetry: costs are invariant under helper and target permutations. + canon = canonical_all() + reps, mult = np.unique(canon, return_counts=True) + assert all((res[canon == r] == res[r]).all() for r in reps), "cost not permutation invariant" + out["orbits"] = len(reps) + log(f" 4x4 equal in both alphabets for all K within {horizon} gates; {len(beyond)} K beyond; {len(reps)} orbits") + return out, res, canon, reps, mult + + +def stage_hard(): + """Resolve K beyond ten gates: restricted exact via sphere 6; unrestricted J-I via S4 normalization.""" + out = {} + t = time.time() + G, sph = ball(H4, T4, True, 6) + B5 = np.sort(np.concatenate([s[0] for s in sph[:6]])) + S6 = sph[6][0] + B6 = np.sort(np.concatenate([B5, S6])) + jmi = sum(1 << (i * 4 + j) for i in range(4) for j in range(4) if i != j) + out["J_minus_I"] = jmi + K = k_matrix(jmi) + # T_K = V X^-1 with X in sphere 6: V in B5 gives <= 11 gates, V in B6 gives <= 12 (<= 10 is the census). + out["restricted_J_minus_I_words"] = {"le11": int(member(B5, shear(K, S6)).sum()), + "le12": int(member(B6, shear(K, S6)).sum())} + log(f" restricted J-I: words<=11 {out['restricted_J_minus_I_words']['le11']}, " + f"<=12 {out['restricted_J_minus_I_words']['le12']} ({time.time()-t:.0f}s)") + del B6, S6 + t = time.time() + _, usph = ball(H4, T4, False, RADIUS) + U5 = np.sort(np.concatenate([s[0] for s in usph])); del usph + K = k_matrix(jmi) + hits = [] + for d, s in GATE_ORBIT_REPS: + hits.append(int(member(U5, shear(K, left_gate(U5, d, s))).sum())) + out["unrestricted_J_minus_I_le11_hits_by_first_gate_orbit"] = hits + # Positive control: the same machinery finds 11-gate words for a known cost-11 K. + w11 = sum(1 << (i * 4 + j) for i, row in enumerate([[0,1,1,1],[1,0,1,1],[1,1,1,0],[1,1,0,0]]) + for j, v in enumerate(row) if v) + out["control_cost11_hits_first_gate_hh"] = int(member(U5, shear(k_matrix(w11), left_gate(U5, 0, 1))).sum()) + assert out["control_cost11_hits_first_gate_hh"] > 0 + log(f" unrestricted J-I words<=11 by first-gate orbit: {hits} ({time.time()-t:.0f}s)") + return out, G, sph + + +def check_orbits(entries, cost): + """Each orbit entry: attaining restricted word, replayed on all dirty inputs, length = minimum.""" + for e in entries: + k, w = e["k"], e["restricted_witness"] + if not (e["minimum"] == int(cost[k]) == len(w) and replay_ok(w, k, True)): + return False + return True + + +def assert_hard(h): + assert sum(h["unrestricted_J_minus_I_le11_hits_by_first_gate_orbit"]) == 0, "feedback word <= 11 for J-I" + assert h["restricted_J_minus_I_words"]["le11"] == 0 < h["restricted_J_minus_I_words"]["le12"] + assert h["control_cost11_hits_first_gate_hh"] > 0 + + +def main(): + ap = argparse.ArgumentParser(description=__doc__.splitlines()[0]) + ap.add_argument("--stage", choices=("census", "hard", "all"), default="all") + ap.add_argument("--write", action="store_true") + ap.add_argument("--shape", nargs=2, type=int, metavar=("H", "T"), + help="exploratory, unregistered: compare both alphabets' census for another shape") + args = ap.parse_args() + if args.shape: + H, T = args.shape + if not (1 <= H and 1 <= T and H + T <= 8): + ap.error("--shape needs H, T >= 1 and H + T <= 8") + R = 4 if H + T <= 6 else 5 + res, _ = census(H, T, True, R) + unr, _ = census(H, T, False, R) + print(f"SHAPE {H}x{T} elementwise_equal={np.array_equal(res, unr)} unresolved={int((res > 2 * R).sum())} " + f"histogram={histogram(res, 2 * R)} (exploratory; no witnesses or independent check)") + return + if args.write and args.stage != "all": + ap.error("--write regenerates every field; use --stage all") + cert = None if args.write else json.loads(CERT.read_text(encoding="utf-8")) + hard = None + if args.stage in ("hard", "all"): + hard, G, sph = stage_hard() + assert_hard(hard) + if cert is not None: + assert hard == cert["hard"], "hard-stage receipt differs from certificate" + if args.stage == "hard": + print("PASS_ZERO_FRAME_FEEDBACK_4X4_HARD jmi_minimum=12 jmi_words_le11=0 both_alphabets no_new_kappa") + return + else: + hard = cert["hard"] # established by the separate hard stage command + assert_hard(hard) + G, sph = ball(H4, T4, True, 6) + result, res, canon, reps, mult = stage_census(cert) + # Costs within ten gates come from the census. Beyond ten: both alphabets exceed ten gates; + # an attaining 11-gate restricted word is replayed below, except on the J-I orbit, where no + # word of <= 11 gates exists in either alphabet (hard stage) and a 12-gate word is replayed. + cost = res.copy() + jmi = hard["J_minus_I"] + jorbit = canon == canon[jmi] + for k in result["beyond_10"]: + cost[k] = 12 if jorbit[k] else 11 + orbits = [] + for r, m in zip(reps.tolist(), mult.tolist()): + w = find_word(r, int(cost[r]), G, sph) + assert w is not None, f"no attaining restricted word for K={r}" + orbits.append({"k": r, "orbit_size": m, "minimum": int(cost[r]), "restricted_witness": w}) + assert check_orbits(orbits, cost) + hist = np.bincount(cost).tolist() + assert sum(hist) == 1 << 16 and sum(o["orbit_size"] for o in orbits) == 1 << 16 + # Negative controls on the witness rules. + e = orbits[-1] + w = e["restricted_witness"] + pair = [[0, 4], [0, 4]] + w # cancelling feedback pair: valid only without H1 + assert replay_ok(pair, e["k"], False) and not replay_ok(pair, e["k"], True), "feedback control" + assert not replay_ok(w[:-1], e["k"], True), "truncated witness accepted" + assert not replay_ok(w, e["k"] ^ 1, True), "wrong target map accepted" + assert not replay_ok([[0, 0]] + w, e["k"], False), "self-XOR accepted" + assert not replay_ok([[True, 1], [True, 1]] + w, e["k"], True), "Boolean index accepted" + longer = [dict(o) for o in orbits] + longer[1] = dict(longer[1], restricted_witness=longer[1]["restricted_witness"] + [[4, 5], [4, 5]]) + assert replay_ok(longer[1]["restricted_witness"], longer[1]["k"], True) + assert not check_orbits(longer, cost), "valid longer word accepted as optimal" + tampered = cost.copy(); tampered[orbits[5]["k"]] -= 1 + assert not check_orbits(orbits, tampered), "tampered minimum accepted" + controls = ["predecessor_3x3_reproduced", "pr1557_3x4_reproduced", "radius4_unresolved", + "feedback_rejected_in_restricted_alphabet", "truncated_witness", "wrong_target_map", + "self_xor", "boolean_index", "longer_word_mislabeled_optimal", "tampered_minimum", + "cost11_positive_search_control"] + body = { + "schema_version": 1, + "kind": "finite-exact-shortest-circuit-certificate", + "helpers": H4, "targets": T4, "gate_cost": 1, "ancillas": 0, "all_inputs_arbitrary": True, + "gate_convention": "[d, s] means register[d] ^= register[s]; h0..h3 are 0..3, y0..y3 are 4..7", + "k_index": "K[i][j] (target i, helper j) is bit 4*i + j", + "minimum_histogram": {str(c): n for c, n in enumerate(hist)}, + "feedback_gains": 0, + "costs_sha256": costs_digest(cost), + "costs_hex": "".join("%x" % c for c in cost.tolist()), + "census_radius": RADIUS, + "restricted_spheres": result["restricted_spheres"], + "unrestricted_spheres": result["unrestricted_spheres"], + "hard": hard, + "orbit_count": len(orbits), + "orbits": orbits, + "controls": controls, + } + if args.write: + CERT.write_text(json.dumps(body, indent=1) + "\n", encoding="utf-8", newline="\n") + print("WROTE", CERT.relative_to(ROOT).as_posix()) + return + for key in set(body) | set(cert): + assert body.get(key) == cert.get(key), f"certificate field differs: {key}" + print(f"PASS_ZERO_FRAME_FEEDBACK_4X4 maps={1 << 16} orbits={len(orbits)} max_minimum={len(hist) - 1} " + f"feedback_gains=0 controls={len(controls)} no_new_kappa") + + +if __name__ == "__main__": + sys.exit(main())