I am not sure which of these points are related to mith's post and which are related to mine.

denis_berthier wrote:having 11 (or any number of) candidates related by an OR relation doesn't make a CSP-Variable (2 or more of them can be True).

I believe that this reacts to my description of TO-Braids. I admit that I had forgotten B2B notation for a moment.

Initially I expected you to implement TO-Braids, where the two 11-cell patterns (or just the 12-cell contradiction pattern, at the very end) would also be used as rlcs.

denis_berthier wrote:a pattern is well defined if it can only be either present or absent in a resolution state. If you have to add conditions or processes to find it, it's not a pattern.

The conditions proposed by mith and me aren't necessary, they aim to filter out useless patterns.

There are a few things to be careful about:

If the TO leads to a contradiction with singles, it doesn't change the T&E depth and can be ignored, but only if only the contradictory pattern is used in the T&E (it could always be substituted by said contradiction with singles; if the 11-cell patterns are used as well, it may reduce the depth in some cases). It might not be a suitable filter for some applications.

mith wrote:When I say TO pattern here, I am not talking about the contradictory/impossible pattern of those 12-cells only containing 3 candidates; I am talking about the 4-chromatic pattern of 12-cells itself. Such patterns are obviously *always* present in sudoku grids, and in large numbers (5832 of them, if I've got my counting right). We could even just check all of these possible patterns (the majority of which could be skipped due to already containing givens - in this case, I believe something simpler is always available)...

This might be true, depending on how one defines "simpler", however I doubt that any of the TO-with-one-given patterns are checked for (unless we consider B2B simpler than TO).

(The number is correct, if we also take into account the 84 combinations of three digits, we are just short of half a milion.)

denis_berthier wrote:I'm not interested in patterns that do not support any resolution rule.

This might react to my suggestion of extending the pencilmarking.

In that case adding such a pencilmark would mean progress in the resolution process much like an elimination of a candidate does – there is only a finite amount of these pencilmarks you could have (as there is a finite amount of TOs they could come from), therefore it wouldn't produce any convergence issues (unlike for example relabeling back and forth) – provided other rules would be capable of using these pencilmarks.

Marek