Translated cyclers: principle
The goal is to decide [Family] Translated cyclers.
Principle:
The main idea for recognising ``Translated Cyclers’’ is to find two configurations that break a record (i.e. visit a memory cell that was never visited before) in the same state (here state D, color green) such that the content of the memory tape at distance L from the record positions is the same in both record configurations. Distance L is defined as being the maximum distance to record position 1 (green cell in record 1) that was visited between the configuration of record 1 and record 2.
More details at https://github.com/bbchallenge/bbchallengeproofs/blob/main/deciders/correctnessdeciders.pdf.
Decider examples and counterexamples
 Examples: #59,645,887; #15,167,997; #59,090,563 see tests for more
 Counterexamples: see tests
Decider code
https://github.com/bbchallenge/bbchallengedeciders/tree/main/decidertranslatedcyclers
Decider tests
Results

01/04/22: an additional 1,318 after applying the decider after the results of [Decider] Backward reasoning where discarded because of a bug. Machines #14,494,506, #85,518,052 and #87,747,774 were for instance discovered. Indices of decided machines available http://docs.bbchallenge.org/bb5_decided_indexes/translatedcyclersrun85454189d410time10000space5000minIndex14322029maxIndex88664064.

05/03/22: an additional 1,664 were decided by increasing the parameters of the decider. For instance, machine #59,090,563 was discovered at that occasion. Indices of decided machines available http://docs.bbchallenge.org/bb5_decided_indexes/translatedcyclersrun725fd45b37ebtime10000space5000minIndex14322029maxIndex88664064.

29/01/22: 73,857,622 translated cyclers were decided. Indices of decided machines available http://docs.bbchallenge.org/bb5_decided_indexes/translatedcyclersrun8f5b2539279atime1000space500minIndex14322029maxIndex88664064.
Database subset of application

01/04/22 results were run on the 3,575,540 machines remaining after [Decider] Cyclers and the first passes of translated cyclers and after discarding the machines decided by [Decider] Backward reasoning because of a bug needing to be resolved in it.

05/03/22 results were run on all machines that remained undecided after [Decider] Cyclers, the first results of this decider and the backward reasoning decider. This corresponds to a total of 2,323,786 machines tested.

29/01/22 results were run on the machines starting at index 14,322,029 until final index 88,664,063 of the seed database which are the machines that exceeded the 12k memory limit. Indeed, it is unlikely that a translated cycler would exceed 47M time limit before reaching 12k visited memory cells. This corresponds to a total of 74,342,035 machines tested.
Decider correctness
Proof available at https://github.com/bbchallenge/bbchallengeproofs/blob/main/deciders/correctnessdeciders.pdf.