The New Sudoku Players' Forum

[denis_berthier]

I get the impression that T&E(3) has a lot of 11.8 puzzles. In T&E(2), 11.8 is quite rare, with a lot more 11.7 and 11.6.
Now, when I find a 11.8, I am almost sure it is T&E(3) (they are all already in miths T&E(3) collection). And although I find only a few T&E(3) because I use T&E(2) seeds almost half of them is 11.8. Maybe another illustration of how the SE/PGX rating fails for T&E(3), also since I read the posts about how easy those puzzles sometimes are.

Such posts reflect a confusion between two different ways of rating puzzles. One way is the SER (that doesn't take tridagons into account). The other way uses tridagons. It's like comparing bananas and snails.

As for the SER 11.8:
- it is indeed rare in the current T&E(2) database (but remember it hasn't been searched with the same vicinity and expansion methods);
- the T&E(3) database of 847,778 minimals shows that 11.8 is very far from having any predominance in it (and it confirms an anomaly at SER 10.4 that I had already pointed out in [PBCS], plus one at SER 10.9 plus one in the 9.x range):

Code: Select all

-       7 have SER 11.9
-     316 have SER 11.8
-  95,482 have SER 11.7
-  63,706 have SER 11.6
-     989 have SER 11.5
-      13 have SER 11.4
-     825 have SER 11.3
-   5,525 have SER 11.2
-  27,422 have SER 11.1
-  63,306 have SER 11.0

- 109,033 have SER 10.9
-     369 have SER 10.8
-   2,855 have SER 10.7
-  51,816 have SER 10.6
-  52,857 have SER 10.5
- 214,743 have SER 10.4
- 101,502 have SER 10.3
-  35,108 have SER 10.2
-   1,653 have SER 10.1
-     900 have SER 10.0

- 16,429 have SER 9.x
-    842 have SER 8.x
-    528 have SER 7.x
-  1,005 have SER 6.x
-     24 have SER 5.x
-    523 have SER 4.x

The small values of the rating are due to rules of uniqueness in SER.

[Paquita]

Yes, in the T&E(2) database about 1 in 5000 is 11.8; in this T&E(3) collection it is about 1 in 2500. Not as much as I thought.

[denis_berthier]

Update as of 2022-11-06

The database "expanded_te3.db" currently holds 847778 depth 3 expanded forms*, with clue counts ranging from 24c-40c.
[...]
From this database, the min-expand** and max-expand*** puzzles have been determined.
[...]
max_expands_20221106 (48071 puzzles)
unix format
dos format
[...]
*** max-expand - these are expanded forms which cannot have clues added while remaining depth 3

There's a small error in the "max-expand" database: the following six puzzles can be further expanded in T&E(3)

Code: Select all

.23456......18923......74............35.62.41.1.....6.34....65.5.16...23.625..1.4;15829;673789
1.3.5.......18923......741...........35..2.41.1.....6.34....65.5.16...23.625..1.4;15829;710018
.234.6.......89.3......741...........35.62.41.1.....6.34....65.5.16...23.625..1.4;15829;403342
.2345........89.3......741...........35..2.41.1.....6.34....65.5.16...23.625..1.4;15829;419498
1...56.......8923......74............35.62.4..1.....6.34....65.5.16...23.625..1.4;15829;754675
....56.89...1.9..6.96.3....26...34.8.....469..49..8.23.32.....46.4......98..4..62;23048;452096

They have the following expansions by 1 clue (they may have further expansions; I didn't check:
(orig# is the place of the puzzle in the max-expand collection.)

Code: Select all

123456......18923......74............35.62.41.1.....6.34....65.5.16...23.625..1.4 orig# 17719
.23456......18923......741...........35.62.41.1.....6.34....65.5.16...23.625..1.4 orig# 17719

123.5.......18923......741...........35..2.41.1.....6.34....65.5.16...23.625..1.4 orig# 17720
1.345.......18923......741...........35..2.41.1.....6.34....65.5.16...23.625..1.4 orig# 17720
1.3.56......18923......741...........35..2.41.1.....6.34....65.5.16...23.625..1.4 orig# 17720
1.3.5.......18923......741...........35.62.41.1.....6.34....65.5.16...23.625..1.4 orig# 17720

1234.6.......89.3......741...........35.62.41.1.....6.34....65.5.16...23.625..1.4 orig# 17721
.23456.......89.3......741...........35.62.41.1.....6.34....65.5.16...23.625..1.4 orig# 17721
.234.6......189.3......741...........35.62.41.1.....6.34....65.5.16...23.625..1.4 orig# 17721
.234.6.......8923......741...........35.62.41.1.....6.34....65.5.16...23.625..1.4 orig# 17721

12345........89.3......741...........35..2.41.1.....6.34....65.5.16...23.625..1.4 orig# 17722
.23456.......89.3......741...........35..2.41.1.....6.34....65.5.16...23.625..1.4 orig# 17722
.2345.......189.3......741...........35..2.41.1.....6.34....65.5.16...23.625..1.4 orig# 17722
.2345........8923......741...........35..2.41.1.....6.34....65.5.16...23.625..1.4 orig# 17722
.2345........89.3......741...........35.62.41.1.....6.34....65.5.16...23.625..1.4 orig# 17722

12..56.......8923......74............35.62.4..1.....6.34....65.5.16...23.625..1.4 orig# 17723
1.3.56.......8923......74............35.62.4..1.....6.34....65.5.16...23.625..1.4 orig# 17723
1..456.......8923......74............35.62.4..1.....6.34....65.5.16...23.625..1.4 orig# 17723
1...56......18923......74............35.62.4..1.....6.34....65.5.16...23.625..1.4 orig# 17723
1...56.......8923......741...........35.62.4..1.....6.34....65.5.16...23.625..1.4 orig# 17723
1...56.......8923......74............35.62.41.1.....6.34....65.5.16...23.625..1.4 orig# 17723

...456.89...1.9..6.96.3....26...34.8.....469..49..8.23.32.....46.4......98..4..62 orig# 25533


[mith]

That's unexpected, but I was able to figure out what is going on here.

The actual max-expands for these puzzles are:

Code: Select all

12....78..571..2.66.82...152.....5..7.5.628.1............72165857..4....86.593...;15829;658138
....567...5718.2..8.62.71...65....1....52.697...6..5.35.286..7.67........817.....;23048;624629

Note that they look very different from the puzzles you've listed, and this is why they are listed despite being max. The current script only does a naive check of whether the solution minlex grid is a subgrid of a larger puzzle just by comparing the strings. However, these solution grids have a nontrivial automorphism, and it happens that the other "max-expand" puzzles listed are being canonicalized to the other morph of the solution grid, resulting in the naive check failing.

The second thing to note is that at least the first two expanded puzzles in your list are not in the database at all, because they are not singles expanded; both expand to the 36c puzzle above (placing 1 in either position makes the other placement of 1 a hidden single). The same sort of thing is happening for the rest (the puzzles listed either are actual max-expands or are intermediates that singles-expand to the actual max-expands).

So, there aren't any puzzles missing, but rather there are some extra puzzles in the max-expand list which should not be; the automorphism in the solution grids results in ambiguity when gsf canonicalizes in "solution_minlex" form. I'll look into addressing this in my next update, it should only require a change for solution grids with this property so it shouldn't add too much time to do a more thorough check for these grids.

[mith]

(I'm running a check now for any other solution grids with nontrivial automorphisms; it's over halfway through the solution grids included in the 2022-11-06 update, and these are the only two found. So this may be a really small task for correcting the old list - literally just removing the puzzles Denis listed from the max-expand file. Given how infrequent this issue is, it should be quick to just add special handling when the automorphism count is >1, rather than changing the process for all trees.

[edit]Confirmed, these are the only two out of the 44251 solution grids in the 2022-11 update. I'll check the remainder of the database when I next run the update and determine the max-expands.[/edit])

[denis_berthier]

.
Hi mith
Thanks for your quick answer.
Fortunately, this will not change what I was doing with the max-expands (see the tridagon thread).
.

[mith]

Continued from hardest puzzle thread:

the 11.3 Denis posted earlier does *not* have a valid non-degenerate trivalue oddagon (at least by my definition); because r1c9 is limited to 123, the three marked cells in box 3 already cannot all contain 123, so the OR branching can be reduced to considering that r2c7 and r3c9 cannot both contain 123.

"My" 11.3 (indeed Paquita's) has a non-degenerate trivalue oddagon (it has indeed many different ones), unless one is willing to add to the definition conditions (such as those you're mentioning) that have never been defined and that are potentially in unlimited numbers. I think you're confusing the presence of the pattern (defined in the precise, simple way I gave in the tridagon thread) with its usefulness in specific circumstances.
In the present case, there are two independent circumstances that make it useless: the one you're mentioning and the very high number of guardians.
.

This condition was defined (insofar as it is part of the algorithm I use) in this thread: p328613 (fourth bullet point). This specific post does not exclude degenerate tridagons by your definition, though that is handled in a later algorithm (a trivalue oddagon can only contain all digits in all cells of the pattern if those digits are restricted as givens to a single box which is not aligned with the four boxes of the pattern; a given in any other box will necessarily remove that candidate from a cell of the pattern, since the pattern spans all rows and columns within each box it covers).

It is not merely a matter of usefulness; it is a matter of reduction to a smaller pattern. In this case, the set of cells r1c89+r2c7+r3c9 is a 4 cell pattern with a chromatic number of 4, and the guardians are a subset of the guardians in the trivalue oddagon. We can quickly identify such cases by noting that there is a cell in box 3 which is not part of the potential trivalue oddagon but which is limited to the three digits of the trivalue oddagon, and is quite limited in scope (I can think of one other similar limitation involving the rows/columns instead of the box, but it would be much more unlikely to occur; might add a check for that at some point anyway).

In fact, there is nothing precluding a case where the guardians of the smaller pattern are exactly the same as the guardians of the trivalue oddagon, and in such a case the trivalue oddagon would be exactly as useful as the smaller pattern. Even in cases where the guardians of the smaller pattern are a proper subset of the guardians of the trivalue oddagon, there is nothing precluding the smaller pattern from being useful.

You can choose to not define this case as degenerate, that's your prerogative. It's totally reasonable to draw the line at only considering cells and candidates which are part of the pattern. That said, one could perhaps make the argument that it is degenerate even by your definition in the sense that whatever (unknown) digit goes in r1c9 (which we know is from 123) is excluded from the other cells in the box (r1c8+r2c7+r3c9).

Anyway, I do consider this case degenerate, and specified in my post that my comment was according to my definition. And I have consistently defined it this way since starting to analyze the T&E(3) database for the pattern (it just happens to be the case that all puzzles in T&E(3) currently known satisfy this more restrictive condition as well).

[denis_berthier]

It is not merely a matter of usefulness; it is a matter of reduction to a smaller pattern. In this case, the set of cells r1c89+r2c7+r3c9 is a 4 cell pattern with a chromatic number of 4, and the guardians are a subset of the guardians in the trivalue oddagon. We can quickly identify such cases by noting that there is a cell in box 3 which is not part of the potential trivalue oddagon but which is limited to the three digits of the trivalue oddagon, and is quite limited in scope (I can think of one other similar limitation involving the rows/columns instead of the box, but it would be much more unlikely to occur; might add a check for that at some point anyway).
You can choose to not define such a case as degenerate, that's your prerogative. I do, and have done consistently since starting to analyze the T&E(3) database for the pattern (it just happens to be the case that all puzzles in T&E(3) currently known satisfy this more restrictive condition as well).

Adding conditions to my definition would:
1) weaken the result that the contradiction of the pattern (with no guardians) requires T&E(3)
2) lead to fewer puzzles having the pattern

What you're talking about here is the relation between two patterns, the tridagon and some (trivial) impossible pattern in the block b3.
As the second pattern is simpler than the first, it should have higher priority. Which in the present case would make the tridagon useless.
But I don't see this as a reason for restricting the definition of the tridagon itself.

Actually, when i try to identify puzzles with tridagons, I do something similar: I apply Triplets before. This eliminates a few trivial cases.
.

[mith]

To be clear, I'm not talking about adding to the definition of a tridagon/trivalue oddagon. The pattern is clearly there, regardless of whether there is a smaller pattern with higher priority that can be used instead. I am only discussing here the "degenerate" classification.

We are already restricting the use of the pattern a great deal. At the most basic level of being an impossible pattern, we can say:

For any graph of cells which is isomorphic to the 12-cell 4-chromatic graph (which can be defined independently in purely graph terms)
And for any three digits
1. We cannot place digits such that all 12 cells contain only those three digits (by the chromatic number of the graph)
2. Therefore, at least one "guardian" digit (any candidate in one of the 12 cells which is not from the three digits in question) is true.

This is the form in which the pattern was discovered: not as a rigorously defined resolution rule applicable to CSP as you later provided, but as a 4-chromatic pattern leading to the conclusion (2) of at least one true guardian (exactly comparable to how guardians are used in bivalue oddagons, which are 3-chromatic).
 
In specific puzzles it may in fact be trivially true without considering the pattern at all. The definition above places no restriction at all on our choice of digits (the requirement that all three digits be candidates of all 12 cells is a degeneracy requirement, not a pattern requirement). As an extreme example, all of the chosen 12 cells could be filled by givens which are all from the other six digits; the conclusion is nevertheless true, even if it is also trivially true that none of the 12 cells can be from the three digits. I don't think you would consider such a case to be a tridagon at all (correct me if that's wrong) but either way you certainly wouldn't consider it to be a non-degenerate tridagon - nevertheless the pattern exists and the conclusion (2) holds, even if it is completely pointless.

In no way does considering certain puzzles degenerate weaken the result regarding the contradiction; it is true that the pattern with no guardians is contradicted, and that it requires T&E(3) to contradict, regardless of any specific puzzle, choice of cells, choice of digits. The result holds regardless of whether a specific deduction in a puzzle can be achieved through some other (not T&E(3)) elimination.

I would argue it would also not lead to fewer puzzles having the pattern (by the above less restrictive definition every puzzle has many examples of the pattern, albeit in some wildly degenerate form), it only leads to fewer puzzles have non-degenerate forms of the pattern.

As you say, we are doing similar things here; I am just removing more cases (which I consider trivial, and which you may not) from the non-degenerate pile.

------

I edited the above post before you responded but apparently before you quoted it, so I'll reiterate: I think you could consider this type of puzzle degenerate even by your definition, on the basis that whatever digit is in r1c9 - which we know is from 123 - cannot be in the three pattern cells in box 3.

I'll also mention that the slightly broader "degenerate" check I alluded to in the previous post can be summarized as follows:

For each row or column intersecting the pattern cells, there must not be multiple cells in that row or column which:
1. Occupy the same box
2. Are restricted to the three digits of the pattern

(If the box is one of the four pattern boxes, then at least one of the cells must not be part of the pattern, since pattern cells in a box never share a row or column, and therefore we have the box degeneracy above. On the other hand, if the box is one of the other four boxes sharing a band or stack with the pattern boxes, then we have the similar row/column degeneracy: the two pattern cells in that row or column cannot both contain pattern digits, because two pattern digits appear outside the pattern in that row or column.)

This should be pretty simple to code so I may check if any of the ph2010 puzzles have the row/column degeneracy. I would assume it's much rarer since the requirement involves two cells outside the pattern rather than one, if there are any examples at all.

[denis_berthier]

We are already restricting the use of the pattern a great deal. At the most basic level of being an impossible pattern, we can say:
For any graph of cells which is isomorphic to the 12-cell 4-chromatic graph (which can be defined independently in purely graph terms)
And for any three digits
1. We cannot place digits such that all 12 cells contain only those three digits (by the chromatic number of the graph)
2. Therefore, at least one "guardian" digit (any candidate in one of the 12 cells which is not from the three digits in question) is true.

OK, but the question is, how can this "abstract" graph appear in Sudoku other than in the classical pattern of blocks, rows, columns? I don't think there's much restriction here.

I am only discussing here the "degenerate" classification.
[...]
(the requirement that all three digits be candidates of all 12 cells is a degeneracy requirement, not a pattern requirement).

This is our main disagreement, one of vocabulary.
For me, the main difference between the non-degenerate pattern and the degenerate ones is at which level of T&E the contradiction can be proven.
As I've proven in the tridagon thread, as soon as one of the three digits is missing in one of the twelve cells, the contradiction can be proven in T&E(2).
For me, (non-degenerate) tridagon and degenerate-cyclic tridagon are two different patterns (close to each other but different), based on the same pattern of cells but with different conditions on the 3 digits.
None of these two patterns involve any cell other than the 12 ones and, in particular, no condition about another cell in one of the 4 blocks having exactly the 3 digits can be part of their definition.

As an extreme example, all of the chosen 12 cells could be filled by givens which are all from the other six digits; the conclusion is nevertheless true, even if it is also trivially true that none of the 12 cells can be from the three digits. I don't think you would consider such a case to be a tridagon at all

First, don't confuse the abstract pattern (defined in terms of variables, not fixed rows, columns...) and its instantiations in a particular puzzle, in particular rows...
Your case is an instance of the non-degenerate tridagon pattern, with exactly 72 guardians, a totally useless instantiation and moreover an impossible situation.

In no way does considering certain puzzles degenerate

Puzzles can't be degenerate.

it is true that the pattern with no guardians is contradicted, and that it requires T&E(3) to contradict,

If one of the 3 digits is missing in one of the 12 cells, T&E(2) is enough to prove the contradiction of the abstract pattern, independently of any puzzle.

I'll also mention that the slightly broader "degenerate" check I alluded to in the previous post can be summarized as follows:

For each row or column intersecting the pattern cells, there must not be multiple cells in that row or column which:
1. Occupy the same box
2. Are restricted to the three digits of the pattern
(If the box is one of the four pattern boxes, then at least one of the cells must not be part of the pattern, since pattern cells in a box never share a row or column, and therefore we have the box degeneracy above. On the other hand, if the box is one of the other four boxes sharing a band or stack with the pattern boxes, then we have the similar row/column degeneracy: the two pattern cells in that row or column cannot both contain pattern digits, because two pattern digits appear outside the pattern in that row or column.)

But this would turn the 3-digit 12-cell pattern into something else, involving CSP-Variables that don't belong to the original pattern.This is a step I'm not ready to make.
There are smarter ways to keep my original formal definition (in the trdiagon thread) unchanged and to avoid the pattern to be used when other patterns make it useless. This is the general way resolution rules interact.
I've already given one example: apply Triplets before Tridagon rules. It doesn't take care of your cases, but similar rules can be written.

This should be pretty simple to code

No doubt, but see point above.
.

[mith]

We are already restricting the use of the pattern a great deal. At the most basic level of being an impossible pattern, we can say:
For any graph of cells which is isomorphic to the 12-cell 4-chromatic graph (which can be defined independently in purely graph terms)
And for any three digits
1. We cannot place digits such that all 12 cells contain only those three digits (by the chromatic number of the graph)
2. Therefore, at least one "guardian" digit (any candidate in one of the 12 cells which is not from the three digits in question) is true.

OK, but the question is, how can this "abstract" graph appear in Sudoku other than in the classical pattern of blocks, rows, columns? I don't think there's much restriction here.

In classic sudoku, it can't. It can absolutely appear in different forms in variant sudoku.
But in classic sudoku, our nodes are cells, our edges are houses (blocks/boxes, rows, columns), so naturally any subgraph of the sudoku graph is going to relate to the abstract 4-chromatic graph via the houses.

And I think you misunderstood what I meant by "we are already restricting" here; I wasn't saying this limited definition of a trivalue oddagon was restrictive, I was saying we are restricting (adding restrictions) on top of that very much unrestricted abstract graph approach.

I am only discussing here the "degenerate" classification.
[...]
(the requirement that all three digits be candidates of all 12 cells is a degeneracy requirement, not a pattern requirement).

This is our main disagreement, one of vocabulary.

I don't disagree that our disagreement is one of vocabulary. To be clear, I am not trying to convince you to change your definition.

For me, the main difference between the non-degenerate pattern and the degenerate ones is at which level of T&E the contradiction can be proven.
As I've proven in the tridagon thread, as soon as one of the three digits is missing in one of the twelve cells, the contradiction can be proven in T&E(2).
For me, (non-degenerate) tridagon and degenerate-cyclic tridagon are two different patterns (close to each other but different), based on the same pattern of cells but with different conditions on the 3 digits.

That's fair. I do not consider them to be different patterns, myself. For me, they are the same pattern, but examples of that pattern may or may not be degenerate in some way. (Calling them the same pattern or not isn't really a relevant distinction for me, though I appreciate that it may be for you and how you define pattern.)

None of these two patterns involve any cell other than the 12 ones and, in particular, no condition about another cell in one of the 4 blocks having exactly the 3 digits can be part of their definition.

I would argue that your degenerate tridagon pattern (under your definition), or at least its instantiation, does involve other cells; how else are you getting the restriction on a pattern cell that one of the digits is excluded as a candidate? We can certainly talk about the pattern in the abstract, and show that if we are missing a candidate the contradiction is in T&E(2), but such a pattern can't exist in the (classic) sudoku grid unless some given digit in another cell is imposing it.

As an extreme example, all of the chosen 12 cells could be filled by givens which are all from the other six digits; the conclusion is nevertheless true, even if it is also trivially true that none of the 12 cells can be from the three digits. I don't think you would consider such a case to be a tridagon at all

First, don't confuse the abstract pattern (defined in terms of variables, not fixed rows, columns...) and its instantiations in a particular puzzle, in particular rows...
Your case is an instance of the non-degenerate tridagon pattern, with exactly 72 guardians, a totally useless instantiation and moreover an impossible situation.

My example is an instance of a maximally degenerate (in that all 12 cells have none of the three pattern digits) trivalue oddagon with exactly 12 guardians (precisely the given digits in the 12 cells). It is (obviously) completely useless. The conclusion that at least one guardian is true is immediately proving by looking at the smaller pattern of any single cell. It is nevertheless a trivalue oddagon under my definition.

The non-degenerate tridagon with 72 guardians would appear in an empty sudoku grid (or empty in all boxes except for one). It's impossible in classic sudoku insofar as you can't have a unique solution with so much of the grid empty. However, it's also not an impossible situation in variant sudoku (I doubt it would be difficult to construct a sukaku with an example of this, say). And again, totally useless, sure. Maximally useless. But nevertheless a trivalue oddagon under my definition.

In no way does considering certain puzzles degenerate

Puzzles can't be degenerate.

Can we not snipe at little word choice things like this? I have no doubt you understood that my meaning was "considering certain puzzles to have degenerate trivalue oddagons"; I was responding to "lead to fewer puzzles having the pattern" and it's clear from the context of the rest of the sentence what I meant. Just as I understood you in the other thread to be saying "42,097 more tridagons in puzzles between 100,001 and 200,000, all [puzzles] from Paquita".

If I were publishing a paper and had sent it to you as an editor, this would have been a helpful and wanted comment. We're posting on a web forum. Shorthand and outright omissions are to be expected.

But this would turn the 3-digit 12-cell pattern into something else, involving CSP-Variables that don't belong to the original pattern.This is a step I'm not ready to make.

Wasn't asking you to. Keep in mind, I am responding to this:

unless one is willing to add to the definition conditions (such as those you're mentioning) that have never been defined and that are potentially in unlimited numbers

by pointing out that these conditions have been defined and have been part of my definition for degeneracy all along. Doesn't bother me if you don't want to include that in your definition of degeneracy, but at the same time your definition is not constraining how I choose to engage with this.

[denis_berthier]

We are already restricting the use of the pattern a great deal. At the most basic level of being an impossible pattern, we can say:
For any graph of cells which is isomorphic to the 12-cell 4-chromatic graph (which can be defined independently in purely graph terms)
And for any three digits
1. We cannot place digits such that all 12 cells contain only those three digits (by the chromatic number of the graph)
2. Therefore, at least one "guardian" digit (any candidate in one of the 12 cells which is not from the three digits in question) is true.

OK, but the question is, how can this "abstract" graph appear in Sudoku other than in the classical pattern of blocks, rows, columns? I don't think there's much restriction here.

In classic sudoku, it can't. It can absolutely appear in different forms in variant sudoku.
But in classic sudoku, our nodes are cells, our edges are houses (blocks/boxes, rows, columns), so naturally any subgraph of the sudoku graph is going to relate to the abstract 4-chromatic graph via the houses.

And I think you misunderstood what I meant by "we are already restricting" here; I wasn't saying this limited definition of a trivalue oddagon was restrictive, I was saying we are restricting (adding restrictions) on top of that very much unrestricted abstract graph approach.

I wouldn't call this a restriction. It's just an injection of the abstract graph into the Sudoku graph. It can easily be checked that it defines a bijection of the abstract graph to the sudoku subgraph defined by the 12 cells.
It doesn't add more conditions to the pattern.

None of these two patterns involve any cell other than the 12 ones and, in particular, no condition about another cell in one of the 4 blocks having exactly the 3 digits can be part of their definition.

I would argue that your degenerate tridagon pattern (under your definition), or at least its instantiation, does involve other cells; how else are you getting the restriction on a pattern cell that one of the digits is excluded as a candidate? We can certainly talk about the pattern in the abstract, and show that if we are missing a candidate the contradiction is in T&E(2), but such a pattern can't exist in the (classic) sudoku grid unless some given digit in another cell is imposing it.

The 'why" of candidates being present or absent in a grid has nothing to do with the presence or not of a pattern.
Do you ask why in a Triplets there are only 3 candidates?

As an extreme example, all of the chosen 12 cells could be filled by givens which are all from the other six digits; the conclusion is nevertheless true, even if it is also trivially true that none of the 12 cells can be from the three digits. I don't think you would consider such a case to be a tridagon at all

First, don't confuse the abstract pattern (defined in terms of variables, not fixed rows, columns...) and its instantiations in a particular puzzle, in particular rows...
Your case is an instance of the non-degenerate tridagon pattern, with exactly 72 guardians, a totally useless instantiation and moreover an impossible situation.

My example is an instance of a maximally degenerate (in that all 12 cells have none of the three pattern digits) trivalue oddagon with exactly 12 guardians (precisely the given digits in the 12 cells). It is (obviously) completely useless. The conclusion that at least one guardian is true is immediately proving by looking at the smaller pattern of any single cell. It is nevertheless a trivalue oddagon under my definition.

I had misunderstood your example. If all the 3x12 tridagon candidates are absent, for me it's not a tridagon at all... for the same reason that two Pairs in the same row are not a Quad.
This extreme case is a perfect example of why I've defined the degenerate cyclic tridagon pattern: in order to put some limit on allowed degeneracy, before we fall into absurd situations.
I'm surprised you add conditions involving other cells (which I would indeed call "restrictions"), but you don't add obvious ones that involve only the 12 cells and that can be defined at the level of the abstract graph.

unless one is willing to add to the definition conditions (such as those you're mentioning) that have never been defined and that are potentially in unlimited numbers

by pointing out that these conditions have been defined and have been part of my definition for degeneracy all along. Doesn't bother me if you don't want to include that in your definition of degeneracy, but at the same time your definition is not constraining how I choose to engage with this.

My point was not about your specific conditions, but about the potentially unlimited number of such conditions (plus the already mentioned fact that they involve other cells).
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