"""THE TILL WHEEL - instrument (2026-07-22).
Claim (PROVED in the landmark, two lines): along any fertile
compulsory-doubling ray, consecutive branch values satisfy
b_{j+1} = 4 * b_j exactly, hence their classes mod 18 rotate through
the three tills in the fixed order 4 -> 16 -> 10 -> 4 forever.
Corollary checked here too: the ray's ENTRY till is a function of the
seed's residue u mod 9 alone.

This script verifies both on a large census of fertile odd seeds.
"""

def branch_values(u, n_branches):
    """Branch values (b = 2^k * u with b = 4 mod 6) along u's doubling ray."""
    out, v = [], u
    while len(out) < n_branches:
        v *= 2
        if v % 6 == 4:
            out.append(v)
    return out

def main():
    N_SEEDS = 200_000
    N_BR = 12
    wheel = {4: 16, 16: 10, 10: 4}
    entry_by_u9 = {}
    checked = rays = 0
    for u in range(1, 2 * N_SEEDS, 2):
        if u % 3 == 0:
            continue  # sterile ray (THE FERTILE RAY): never branches
        rays += 1
        bs = branch_values(u, N_BR)
        cls = [b % 18 for b in bs]
        assert all(c in (4, 10, 16) for c in cls), (u, cls)
        for a, b in zip(cls, cls[1:]):
            assert wheel[a] == b, ("wheel broken", u, cls)
            checked += 1
        # integer relation, not just modular
        for x, y in zip(bs, bs[1:]):
            assert y == 4 * x, ("4x broken", u)
        entry_by_u9.setdefault(u % 9, set()).add(cls[0])
    print("rays checked: %d | wheel transitions verified: %d | zero exceptions"
          % (rays, checked))
    print("entry till by u mod 9 (each must be a single class):")
    for r in sorted(entry_by_u9):
        assert len(entry_by_u9[r]) == 1, (r, entry_by_u9[r])
        print("  u = %d (mod 9)  ->  enters at till %d" % (r, entry_by_u9[r].pop()))

if __name__ == "__main__":
    main()