"""
"Use the pi digit sequence as a tape LUT": store data as the OFFSET where it
appears in pi, instead of the data itself. Test on REAL pi digits.

Claim under test: the offset is ~the same size as the data, so nothing is saved.
Reason: in a normal (random-looking) base-b sequence, an L-symbol string first
appears near position b^L on average, so the offset needs ~L*log2(b) bits =
exactly the data's size. Pointer-into-pi = base conversion = conservation.
"""
import mpmath, random, math

D = 1_000_000                      # decimal digits of pi to use as the tape
mpmath.mp.dps = D + 20
s = mpmath.nstr(mpmath.mp.pi, D + 10, strip_zeros=False).replace("3.", "", 1).replace(".", "")
s = s[:D]
print(f"pi tape: {len(s)} decimal digits\n")

random.seed(0)
print(f"{'k (data digits)':>16}{'trials found':>14}{'avg 1st-occ pos':>18}"
      f"{'log10(pos)':>12}{'offset digits':>15}")
print("-" * 76)
for k in range(1, 7):
    positions = []
    trials = 300
    found = 0
    for _ in range(trials):
        target = "".join(random.choice("0123456789") for _ in range(k))
        idx = s.find(target)
        if idx >= 0:
            positions.append(idx + 1)   # 1-based position
            found += 1
    if positions:
        avg = sum(positions) / len(positions)
        off_digits = math.log10(avg)
        print(f"{k:>16}{found:>10}/{trials}{avg:>18.0f}{math.log10(avg):>12.2f}"
              f"{off_digits:>15.2f}")
    else:
        print(f"{k:>16}{found:>10}/{trials}{'(none in tape)':>18}")

print("\nRead-off: 'offset digits' ~= 'k data digits' at every row -> the pointer into")
print("pi is the SAME size as the data it points to. Base conversion, zero gain.")
print("(k=6 needs ~10^6 positions ~ the whole 10^6-digit tape; beyond it you must")
print(" extend pi further, and the offset only grows to match the data exactly.)")