"""JAM 4: exact transition collisions for both L=6, M=54 states.

The earlier probe checked only whether an identified state occurred in
generations with different ignition outcomes.  A predictive quotient
also requires the identified histories to have the same next state.
These two witnesses live on Hofstadter Q's single realized trajectory.
"""

M = 54
L = 6

Q_PAIR = (188_410, 1_404_394)
E_PAIR = (95, 191)
N = max(*Q_PAIR, *E_PAIR) + 1


q = [0] * (N + 1)
q[1] = q[2] = 1
for n in range(3, N + 1):
    a = n - q[n - 1]
    b = n - q[n - 2]
    assert 1 <= a < n and 1 <= b < n
    q[n] = q[a] + q[b]


def generation_of(n: int) -> int:
    k = 2
    while 3 * (1 << k) <= n:
        k += 1
    return k


def phase(n: int) -> int:
    return (3 * (1 << generation_of(n)) - n) % M


def q_descriptor(n: int) -> tuple[int, ...]:
    tail = tuple(q[j] % M for j in range(n - L + 1, n + 1))
    return (phase(n), *tail)


def error_descriptor(n: int) -> tuple[int, ...]:
    tail = tuple(
        (2 * q[j] - j) % (2 * M)
        for j in range(n - L + 1, n + 1)
    )
    return (phase(n), *tail)


def show(name, pair, descriptor, next_residue):
    a, b = pair
    state_a, state_b = descriptor(a), descriptor(b)
    next_a, next_b = descriptor(a + 1), descriptor(b + 1)
    assert state_a == state_b
    assert next_a != next_b
    print(name)
    print(f"  positions: {a}, {b}")
    print(f"  common state: {state_a}")
    print(
        "  next incoming residues:",
        next_residue(a + 1),
        next_residue(b + 1),
    )
    print(f"  next states differ: {next_a != next_b}")


show(
    "D_Q = (phase, last six Q values mod 54)",
    Q_PAIR,
    q_descriptor,
    lambda n: q[n] % M,
)
show(
    "D_e = (phase, last six doubled errors mod 108)",
    E_PAIR,
    error_descriptor,
    lambda n: (2 * q[n] - n) % (2 * M),
)

print("VERDICT: both L=6, M=54 descriptors fail next-transition consistency.")