I will note that the original proof I offered for the "trivalue oddagon" pattern's impossibility explicitly uses bivalue oddagons.

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Note that in each box, we can pick two cells which are part of the 8-loop (not part of the rectangle r25c25) of row/column links. WLOG, let's choose the cells in b5 (r4c4, r6c6) and say these are 2 and 3 in that order, with 1 in the other cell (r5c5). Note that in all cases, the two cells chosen form a 5-cell oddagon by picking one cell from each of the other three boxes (this is an inherent property of the 8-loop; we are just choosing three cells between the two sharing a box, in either direction of the loop) - for example, r1c3, r1c4, r6c3.

Since we have 23 in two cells, at least one of the other cells must contain a 1 (or we have a broken bivalue oddagon).

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In the first case (the cell in the opposite box from the 23 pair, in this case r1c3), we immediately get a 23 remote pair in the other cells of the oddagon, in turn giving a 23 remote pair in the b24 cells of the rectangle, and now r2c2 sees all three digits.

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`. . * | 1 . .`

. * . | . 3 .

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The other cases are messier, but we can focus on the rectangle to get through: one of the rectangle cells (r25c2) is 2. However, whichever it is results in an oddagon through the other. For example, if r2c2 is 2, we have the following loop of cells all forced to be from 13: r5c2 - r5c5 - r2c5 - r1c4 - r1c3 - r3c1 - r4c1 - r5c2.

The other link to oddagons is in the parity flow proof - an odd number of parity changes in the loop is broken.

All that said, I would argue there are as clear if not clearer analogues to the bivalue oddagon in some of the other patterns. Patto Patto's pattern is essentially a 5-cell bivalue oddagon expanded in each row and column to cover a third cell. The other 10-cell pattern is perhaps best explained as a "Broken Wing" (single-digit oddagon) on each digit, with an odd number of links (b124, r16, c16).

We can also think of bivalue oddagons themselves as simple examples of non-2-colorable graphs. In some sense, all of these patterns are analogues to the bivalue or bilocal oddagons, generalized to trivalue cells. (I would say this is similar to how an X-Wing/2-fish is a continuous nice loop, but its extension the Swordfish/3-fish is not any type of AIC, because of the reliance on binary propositions. Perhaps some sort of trinary links are necessary to relate these things to existing patterns.)