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In looking at is from a theoretical aspect it reminds me a bit of Time Travel Chess, however, with no King revisiting its past self. With the revisiting King aspect removed, and indeed pieces moving into the future (beyond 1 move on a given turn) removed, then I see the theory as simply being little more than the chess tree concept with 'bad' and 'good' branches identified. But I can see no actual theory in this... at least not how it is currently presented.

If we take a pure mate-in-three chess position, which has only 1 correct [pure] solution, then any moves that deviate from that line are bad (or less good)... but not necessarily catastrophic for the initiator. However, the person on the receiving end of the mate obviously experienced a catastrophe in his or her game at an earlier point. With the mate-in-3 scenario, the solver may obtain a mate-in-4 or a mate-in-5, for example [thus, having made inferior moves still avoids catastrophe for him or herself].

The idea of chess as a fabric consisting of a material/time continuum in a constant state of flux which in most cases deviates from an initial state near of equilibrium to a state that can be viewed as catastrophic for the dark or light element is an interesting concept.

The game known to many as 'Take Back Chess' in which players get to take back their last move in hopes of avoiding catastrophe is related to this topic. Though that version often allows one to avoid certain immediate disasters (a knight fork, an overlooked checkmate, for example) ... it does not enable one to avoid disasters that occur due to the gradual culmination of small subtle errors.

I think in this is perhaps a way that could be used to demonstrate chess on television, and relative positions.  It likely needs to be reworked, and if there is math involved, the variables explained, else this falls into the Stanley Random Chess category, which is closer to the comic strip Calvinball than the theoretical Heraclitian Calvinball I had asked about.

Anyone have any ideas on how this could potentially be used to demonstrate positions in chess that are about to collapse and fail? Maybe I can come up with something, or other people have ideas.

Rich I think is more correct, this is an attempt to use catastrophe theory to chess. I'm not sure it succeed in anyway. Essentially the author is arguing  that if a move is bad if it crosses a fold in the 'evaluation surface', that  is the surface created by giving every square a value depending on its importance. The surface is then warped to show moves that would cause irreversible changes in evaluation. 

Missoum applies this to one move in one game which allows for the nice graphics he drew. However as a general theory I do not see how one would begin to create one. 

Personally some kind of quantum set theory or more classically combinatoric game theory is far more apt.

This page came up on the random link.

For a chess board with a cusp in it, would the squares within the cusp be
split vertically down the middle into two halves, so that the information
of from which general direction the piece arrived would be preserved
whilst the piece were within the cusp?

This could produce some interesting boards.