bbchallenge - #bb3x3

It turns out [`1RB2LC1RC_2LC---2RB_2LA0LB0RA`](https://wiki.bbchallenge.org/wiki/1RB2LC1RC_2LC---2RB_2LA0LB0RA) (ID 758) and its sibling [`1RB0LB0RC_2LC2LA1RA_1RA1LC---`](https://bbchallenge.org/1RB0LB0RC_2LC2LA1RA_1RA1LC---) (ID 153) are exactly equivalent. Take the config of ID 153 after `1790901` steps, and exchange `A → C → B → A, 1 ↔ 2, L ↔ R`. Then, you obtain the config of ID 758 after `1841608` steps, which is `0^∞ 12 (01)^84 C> 22 (20)^255 0^∞`.

I was feeling bored, so I went and augmented [the rules](https://discord.com/channels/960643023006490684/1259770474897080380/1259968221218607145) for the 3×3 halting cryptid [`1RB2LC1RC_2LC---2RB_2LA0LB0RA`](https://wiki.bbchallenge.org/wiki/1RB2LC1RC_2LC---2RB_2LA0LB0RA) with their step counts, and double-checked their validity while I was at it. They came out to: ``` A(a,c) = A1(a,0,c) = (0^∞ 1 2^a <C (20)^c 0^∞) start: A(0,1) [2 steps] A(a,c) ->[3c^2+8c+5] A(a-c-1,3c/2+2) [c ≤ a-1, c ≡ 0 (mod 2)] A(a,c) ->[(6c^2+31c+37)/2] A(a-c-2,3(c-1)/2+5) [c ≤ a-2, c ≡ 1 (mod 4)] A(a,c) ->[3c^2+29c+65] A(a-c-4,3(c-1)/2+8) [c ≤ a-4, c ≡ 3 (mod 4)] A(a,c) ->[(6c^2+43c+75)/2] Halt(3(c-1)/2+7) [c = a-3, c ≡ 3 (mod 4)] A(a,c) ->[(6c^2+37c+63)/2] A(3c+8,1) [c = a-2, c ≡ 3 (mod 4)] A(a,c) ->[3c^2+8c+6] Halt(3(c-1)/2+4) [c = a-1, c ≡ 1 (mod 2)] A(a,c) ->[3a^2+5a+9c+24] Halt(a/2+c+4) [c ≥ a, a ≡ 0 (mod 2), c ≡ 0 (mod 2)] A(a,c) ->[3a^2+3a+5c+11] A(1,a/2+c+2) [c ≥ a+1, a ≡ 0 (mod 2), c ≡ 1 (mod 2)] A(a,c) ->[3a^2-9a+14c+30] A(3a,c-a+4) [c ≥ a+1, a ≡ 1 (mod 2), c ≡ 0 (mod 2)] A(a,c) ->[3a^2+5c+6] A(3a+2,c-a+1) [c ≥ a, a ≡ 1 (mod 2), c ≡ 1 (mod 2)] ``` Using the numbers from where my [direct simulation](https://discord.com/channels/960643023006490684/960643023530762341/1275459473112170569) left off, I get s(M) ≥ 1052884886930621967473049580 as an (extremely naive) lower bound, about 8.8 billion times the current S(3,3) champion. So it's nice to know that it's definitely both the S and Σ champion, if it halts at all.

I worked out RTL cascading rules for that machine some months ago. It's just as chaotic as the LTR 'Wily Coyote' machine, and the halting condition is the same, except the rules and patterns are slightly different.

This file has the part of the analysis where I transformed it. After I did that, I fed it into my simulator and confirmed that it's no better than the other one.

If I had to put down a timeline for [`1RB2LC1RC_2LC---2RB_2LA0LB0RA`](https://wiki.bbchallenge.org/wiki/1RB2LC1RC_2LC---2RB_2LA0LB0RA), it would be something like: 1. The machine (and its [sibling](https://wiki.bbchallenge.org/wiki/1RB2LC1RC_2LC---2RB_2LA0LB0RA#Equivalence_to_1RB0LB0RC_2LC2LA1RA_1RA1LC---)) pass straight through various rounds of cleverly-written deciders 2. dyuan posts his set of rules and looks into it a bit, Justin notes that he'd briefly looked at it some time back 3. Shawn and savask write up some acceleration schemes, find empirical evidence that it repeatedly comes close to halting, and speculate that it may actually halt after some period of time 4. I take a crack at it myself, this time in terms of chains starting from `A(1,c)` configs, find some `c` values leading to a halt (showing that there's no obvious structural barrier), and conjecture some heuristics for how long it should take to reach such a halting `c` value 5. dyuan and I nail down the ideas of 'halter families' (collections of infinitely many halting `c` values so we don't have to brute-force them) and 'reachability analysis' (obtaining a strong probvious guess of whether a `c` value will or won't occur in the machine's actual forward behavior, assuming it hasn't halted beforehand) 6. dyuan and I race to find a halting `c` value that 'explodes' (passes the reachability analysis), first within the 'trivial' halter family to no avail, then within all halter families (writing a program to enumerate halter families is extremely fiddly, at this point I treat some of my own code as a black box not to be touched) 7. By dint of a somewhat more performant program, I find (and dyuan confirms) the `c = (3^307-8282708212)/13959275` we all know and love, probviously sealing the machine's fate

Well, I did simulate up to A(1, 1059491543272883) today - file with some details here: https://discord.com/channels/960643023006490684/1363377386066018445/1390772553806774292. Took ~28 hours to get there. I didn't work out how many steps that was though.