View source for 1RB0LD 1RC0RC 1LA0RD 1RC1LE 0LF--- 1LA0RC

{{machine|1RB0LD_1RC0RC_1LA0RD_1RC1LE_0LF---_1LA0RC}}
{{TM|1RB0LD_1RC0RC_1LA0RD_1RC1LE_0LF---_1LA0RC}} is a non-halting [[BB(6)]] TM. It is equivalent to {{TM|1RB0RC_1RC1LE_1LD0RB_1RA0LB_0LF---_1LD0RF}}.

== Analysis by @nerdyjoe ==
These two machines are non-halting. After a brief stabilization period, it works as a ternary counter with some fun behavior. The head sweeps left and right. Each bounce off the left side extends the meta-tape by one, each pass to the right "adds one" and encodes the result, bounces off the right side, then re-encodes the result.

The state of the counter can be read off most easily when the meta-tape is mostly ...111010111010..., instead of when it's mostly ...001001001...

The tail end looks something like ...010111010101101, which i would read as 131121. Subtracting the baseline ...131313 from these gives -2 1 -2. Renormalizing signs gives 212. These are the first several numbers, the first after 115 steps:
2 100 101 102 200 201 202 212 222 10000 10001 10002 10100 10101 10102 10200 10201 10202 10212 10222 20000 20001

Formalizing the details is a little annoying, but it should be relatively straightforward to pick out about 6 modules of tape and state to characterize the full behavior.

I have reposted my original message above, with the specific machines in question.

In general, this thread can be for machines with similar behavior, where a left-right sweeping head increments some sort of counter on one side, and extends the tape on which the counter is written on the other side.

== Analysis by mxdys ==
On Jun 16 2025, mxdys gave a Rocq proof that this TM does not halt: https://github.com/ccz181078/busycoq/blame/c0724b3706d96fddb5c253ee2301e9efeb18f5aa/verify/BTCv1.v#L479-L497

The equivalent TM was proved non-halting in lines 455-473.

[[Category:BB(6)]]