1RJ1RH_1RC1RB_1LI0RD_1RC1LE_0LE1LF_1LG1RH_1RB0LF_0RA1LE_1RF1LJ_0LK1RZ_1LL1LK_1LM1LM_0LI0LL
(bbch) is a former BB(13) champion. This machine was found by Racheline using an 8-state
-level mechanism (in the Fast-Growing Hierarchy) shared by the previous BB(13) champion and BB(14) champion. Instead of using the 5 extra states to build a large input to which
would be applied once before halting, this machine applies
repeatedly with Collatz-like rules, which allows it to halt with over
ones on the tape.
Analysis
The
mechanism operates on lists of numbers, and simulates the rule

where
denotes
consecutive zeroes in the list, using configurations of the form
.
This computes a slightly modified Fast-Growing Hierarchy. In particular, letting

one can prove by induction on
that

for
. This motivates the use of
instead, so that

and by applying this repeatedly,

This leaves only the case

, which we write as

. The machine then simulates the Collatz-like rules

where
denotes
consecutive copies of
(
total entries),
, and
means that the machine halts with
ones on the tape.
Trajectory
This machine reaches the configuration
after 89 steps, and then follows this trajectory:

where
.
The trajectory can be computed in the following way.
First, note that
can be proven by induction on
. Next, note that
where
is obtained by applying
functions of the form
to the input 7, so
, and
where
is obtained by applying
functions of the form
to the input 4, so
. This means that after the first application of the
rule,
will always be even, and thus for
appearing after this point, we have
.
Now the only modular arithmetic left is finding the values of the first arguments of
modulo large powers of 3, so that we know the values of the second arguments of
modulo 3, and thus know which rules to apply. We have
and
. Since the computation of
for any
ends with something of the form
, all first arguments of
starting from
are of the form
for large
, and thus are
for
(note: This upper bound is much larger than what we need and significantly smaller than the exact maximum of
for which this statement holds).
With this, we can easily compute each second argument of
modulo 3 just from its expression as a linear combination (with relatively simple coefficients) of the previous first arguments of
, and thus we can compute the linear combination of first arguments that is equal to the second argument in the next step. Repeating this allows us to compute the whole trajectory.
Related machines
The analysis section applies to all machines obtained from this machine by varying the
transition and the starting state. If the search didn't miss any of these machines, this machine takes the longest to halt out of all machines in this family. This family is part of a larger family of machines obtained by additionally varying the
transition, but the Collatz-like rules depend on the
transition. Furthermore, we can change the order of the last 5 states to get 1RJ1RH_1RC1RB_1LI0RD_1RC1LE_0LE1LF_1LG1RH_1RB0LF_0RA1LE_1LJ1LJ_0LK0LI_1LL1LK_1RF1LM_0LI1RZ
(bbch), which also forms a family of similar size by varying the transitions
and the starting state. Perhaps even more families can be obtained by changing the
transition from 1RB to 0RB, but these also differ in the
part of their behavior, so they compute a different variant of the Fast-Growing Hierarchy (in particular, they simulate
).
The second place so far (not counting machines that also reach
) is taken by 1RL1RH_1RC1RB_1LI0RD_1RC1LE_0LE1LF_1LG1RH_1RB0LF_0RA1LE_1RF1LJ_0LK1RZ_1LL1LK_1LM1LM_0LI0LL
(bbch), which reaches
and halts with more than
ones on the tape, so it's a close competition. The search of these families for higher scores is ongoing.