based on Singularity Chess

The 2 blue regions that seem to be different in the image below are actually the same region! Square A is the same as the seemingly-other square A; indeed every square on the board is actually the same as the square 180 degrees around the hole from it. So there is only 1 red region as well. The 2 sides of this board to which Territories may be attached are the red and blue side; of course, a Territory attached to 1 blue side is attached to the every blue side as there is only 1 blue side. So there would appear on this diagram to be 2 of each Territory, but there is in actuality only 1. More rules are below the image.

Pieces may not move into or through the hole but may jump over it.

On the Battlefield, all Pieces consider the direction of Forwards to be directly away from the side their own controller's Territory is attached to or to be directly towards the side their enemy's Territory is attached to. Whichever edge is closer to the piece's starting square for that turn determines this; in the case both are the same distance away, the side the opponent's Territory is attached to is the determining factor. This means the Forwards direction is different depending on the location of a Piece; for passive effects that have a direction, the Forward direction as seen by the Piece causing the effect determines which squares are affected by the effect.

For example, the player whose Territory attaches to the blue side may use a Medieval Pawn on square A to Capture an enemy Piece on square B. Or they may Move a Medieval Pawn on square C to square D without Capturing. But the player whose Territory attaches to the red side would not be allowed to move a Medieval Pawn from square E to square F because that would be considered backwards for that Piece. Instead, they may use it to Capture to square C.

The Nifu rule and other rules that require the concept of a column to be globally defined are amended when this Battlefield is in play. Each of the columns referred to in the Nifu rule is the union of two Partial Columns, possibly a pink square, and possibly some portion of the Territories. A Partial Column is either entirely contained within the red region or entirely contained within the blue region and consists of the squares in that region that form an unbroken chain of being orthogonally vertically adjacent to each other with the vertical direction being that that would be seen by a Piece in that region. 2 Partial Columns are parts of the same column if Pieces in both would see the same pink square as orthogonally vertically adjacent to their Partial Column, and in this case, that pink square is also part of that column. For example, squares A, B, and E are in the same column. Also, the 2 Partial Columns orthogonally adjacent to the hole are part of the same column. The columns also extend into the Territories in the obvious way.

For notation, the letters that normally represent the File of a square represent which column it's in as defined for the Nifu rule, and the numbers that normally represent Ranks would represent distance from a square to the hole, measured in steps to an orthogonally or diagonally adjacent square. The column that contains the hole is always represented by an e, and columns that are further to the right where along the edge of the blue side are represented by letters further along in the alphabet. If letters before a are needed, the alphabet is considered to loop, with z before a.

Now, some ambiguity may arise due to each square appearing to have 2 different locations. For example, one may ask if an "alfil", a piece that jumps to the second square diagonally, may jump over the hole. For these ambiguities, it is useful to consider the apparently different locations of each square as actually different. In this hypothetical scenario, there would be 2 things occuring in each play, and it would have to be possible for 1 to occur and then the other. If that would be impossible, the actual play with only the single thing occuring is also illegal. So the "alfil" would not be able to jump across the hole.

The normal rules for Paradox Depths are amended when this Battlefield is in play. The Battlefield does not become a rectangle because Paradox Depths was chosen. If exactly 1 player picks Paradox Depths, their Territory's width will be 14 more than double the height of their opponent's Territory, and the vertical edges of only the Paradox Depths territory will be identified with each other, not those of any other part of the board. If 1 player picks Paradox Depths and their opponent picks Medieval Battalion, the Medieval Battalion Player may not use Byzantine Territory as their Territory choice. If both players pick Paradox Depths, the Paradox Titan's effect of identifying board edges no longer applies.