.
1) T&E-depth is a universal classification (not rating) of all the instances of all the finite binary CSPs.
Each level of T&E requires specific but universal resolution rules (chain rules) and allows precise sub-classifications and/or ratings;
- T&E(1) sub-classified and rated by braids[n];
- T&E(2) sub-classified by Bn-B (computable) and rated by B-braids[n] (impractical);
- ...
In the most general context, it is much more useful to use pairs (T&E-depth, sub-classification within this T&E level) than a single rating for all.
2) 9x9 Sudoku puzzles in T&E(2) are a very small fraction of all the 9x9 Sudoku puzzles (1 in a few millions);
9x9 Sudoku puzzles in T&E(3) are probably also a very small fraction of all the 9x9 Sudoku puzzles in T&E(≥2).
Why would one care to define a unique general rating based on such exceptional cases?
3) Exotic patterns (such as sk-loops, J-Exocets, anti-tridagons ...) are very brittle: if some of the defining candidates disappears, what remains is a degenerated form of the pattern. [The situation is similar to some uniqueness rules.]
The degenerated form may be totally impossible to find (for a manual solver and for a computer as well). This is true of the above 3 examples.
And for the anti-tridagon case, this is true in spite of the non-degenerated form being in T&E(3) and the degenerated one in T&E(2).
4) CSP-Rules deals with this degeneracy problem by making the anti-tridagon ORk-relations ultra-persistent, thus allowing to "keep in mind" the fact that an anti-tridagon pattern has been identified and that it may loose some of its defining candidates and/or some of its guardians during the resolution process.
This works well for puzzles in mith's collection because all the known 9x9 T&E(3) puzzles have a full anti-tridagon pattern at the start.
5) At this point, nobody knows if this is true of all the T&E(3) 9x9 puzzles and nobody knows if there are puzzles in T&E(2) with a degenerated form of the anti-tridagon pattern that would not derive from any non-degenerated form.
The fact is, T&E(2) is a very ill-known land. The "hardest collection" is strongly biased towards puzzles with specific patterns (sk-loops...), but we have no idea of any set of patterns that would allow to simplify all the puzzles in T&E(2) in the same way as anti-tridagons may simplify puzzles in T&E(3).
I'm not aware of any systematic search for patterns that would make a puzzle potentially be in T&E(2). Such search would be much more interesting to me than talks about defining a unique rating.
6) For puzzles in T&E(3) (with the still limited knowledge we have of this level), for any k, Wn+ORkWn is a good rating system. You can change Wn to gWn or to Bn or to gBn; you'll get slightly different ratings. Or you can change Wn to FWn. Or you can use Wn+ORkWn+ORkFWn. See the tables I've given in the "tridagon rule" thread (http://forum.enjoysudoku.com/the-tridagon-rule-t39859.html). They clearly show what can be hoped for each value of k and n.
7) About defining the complexity of the tridagon rule.
My first approach was to say: it is based on 12 CSP-Variables, so that it must be 12. However, mith's additional criterion (3 digits + a block such that the 3 digits are not decided in any other block) drastically simplifies the search. Once such a block has been found, the rest is mere checking of the other conditions.
[The absence of such an additional criterion is also what makes the degenerated forms impossible to find.]
As a result, I've now granted the anti-tridagon pattern complexity 3: it will be found immediately after the Subsets[3]. If you want to revert to the first view (complexity 12), SudoRules has a control variable to allow this (but the default is now 3) - see the forthcoming release.
There's another reason for detecting the anti-tridagon pattern as soon as possible: prevent it from degenerating before being found.
8) Complexity verus priority
Because the default strategy in CSP-Rules is simplest-first, the priority of a rule is normally defined inversely to its complexity. But this can easily be changed and I could have granted the anti-tridagon detection rule a high priority while keeping its complexity 12.
The problem with this would be, most puzzles with this pattern would be granted the same complexity 12. But in reality, the hard part of solving is in the Wn+ORkWn.