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Speaking of which, the teleport symbol is actually the Aperture Science logo; more specifically, a version it claiming to be in the Public Domain. (https://en.wikipedia.org/wiki/Locations_of_Half-Life#/media/File:Aperture_Science.svg). The smaller one is scaled down by a factor of $\sqrt 5$, so its area is $\frac 1 5$ of the bigger one. Counting the triangles in the pinwheel tiling, this should be just about correct.

It's pretty easy to see that the puzzle has two separate sides, with the recursive part in between. To see this clearly, point your teleport aware paint bucket tool on the outside using one hideous colour, and on the inside using another. Like this:  

Speaking of which, the teleport symbol is actually the Aperture Science logo; more specifically, a version of it claiming to be in the Public Domain. (https://en.wikipedia.org/wiki/Locations_of_Half-Life#/media/File:Aperture_Science.svg). The smaller one is scaled down by a factor of $\sqrt 5$, so its area is $\frac 1 5$ of the bigger one. Counting the triangles in the pinwheel tiling, this should be just about correct.

It's pretty easy to see that the puzzle has two separate sides, with the recursive part in between. To see this clearly, point your teleport aware paint bucket tool on the outside using one hideous colour, and on the inside using another. Like this: 

| Action | Recursion level (N) | Side (S) | (N+S) mod 2 | |--------|---------------------|----------|-------------| | Walk around | no change | no change | no change | | Walk over green border | +1 if going inwards
-1 if going outwards | always changes | no change | | Teleport within level | no change | no change | no change | | Teleport to inner level | +1 | changes | no change | | Teleport to outer level | -1 | changes | no change | | | | | | | Deusovi | 0 | 1 | 1 | | Sweet freedom | 0 | 0 | 0 |

Action        | Recursion level (N)  | Side (S)  | (N+S) mod 2
---------------------------------------------------------------
Walk around   | no change            | no change | no change
---------------------------------------------------------------
walk over     | +1 if going inwards  | always    |
green border  | -1 if going outwards |   changes | no change
---------------------------------------------------------------
Teleport      |                      |           |
within level  | no change            | no change | no change
---------------------------------------------------------------
Teleport to   |                      |           |
inner level   | +1                   | changes   | no change 
---------------------------------------------------------------
Teleport to   |                      |           |
outer level   | -1                   | changes   | no change 
---------------------------------------------------------------
---------------------------------------------------------------
Deusovi       | 0                    | 1         | 1
---------------------------------------------------------------
Sweet freedom | 0                    | 0         | 0
---------------------------------------------------------------

Therefore, even though both sides allow for arbitrarily long or complex paths, never ever will the sides meet.

(My most sincere apologies to anyone trying to read that table on a mobile device.)

| Action | Recursion level (N) | Side (S) | (N+S) mod 2 | |--------|---------------------|----------|-------------| | Walk around | no change | no change | no change | | Walk over green border | +1 if going inwards
-1 if going outwards | always changes | no change | | Teleport within level | no change | no change | no change | | Teleport to inner level | +1 | changes | no change | | Teleport to outer level | -1 | changes | no change | | | | | | | Deusovi | 0 | 1 | 1 | | Sweet freedom | 0 | 0 | 0 |

Therefore, even though both sides allow for arbitrarily long or complex paths, never ever will the sides meet.