Barrel

 The Barrel  is not a box, but it is a concept that links the subsections of nothingness in the The Box. Each barrel in the Barrel series is hardly accessible from the last. Notice that all of the box contains recursive functions of itself, so to go outside of it it needs to be inaccessible to the inside of the Omniverse. That's how the size is determined.

Every barrels have their own "Barrel Index". Barrels almost certainly form chain above themselves, which are kind of hard to distinguish without special classifier. and that classifier is Barrel Index explained below. The index is based on "nothingness level", and works for pretty much every barrel of every box.

Barrel Index
The Barrel Index denotes "freedom of nothingness ", where Barrel-0 is The Outside, and Barrel-x has a noticeable increase in freedom compared to a structure of Barrel-(x-1) that it contains. The biggest Barrel-x value possible is chosen, and only structures that don't exist and are directly/indirectly contained by The Box count.

If a barrel doesn't directly or indirectly contain the main The Outside, then it is instead compared relative to similar freedom-wise structures in the main The Barrel hierarchy. If a structure is somehow less free than The Outside or more free than The Box, then it doesn't have a Barrel-x value.

The Barrel is Barrel-1, the Barrelplex is Barrel-2, etc.

Layers of The Barrel
Sometimes, these fractals destroy nothingness that interacts with it due to fractional dimensionality converting normal dimensionality into undefined structures, causing huge explosions that branch out throughout dimensions and can even make the dimensionality of some objects negative, causing things like this to happen.
 * Layer ∅: Layer ∅ is pre-nothing structures, like the Omniverse and other structures. It is defined as "layer null", due to the fact that it isn't nothing, so can't be really assigned a number value. Oh and this means finite verses are almost always at Layer ∅.
 * Layer 0: Layer 0 is nothingness, but not the Barrel yet. This area is known as The Outside. This layer is the base of all barrel chains, since without it, barrel chain is not a barrel chain anymore lol
 * Layer 1: Layer 1 is where the Barrel starts, and is very loosely connected with the other layers. There appear to be tons of "invisible walls" here, but they turn out to be gaps in nothingness, the container of things, that Layer 1 hasn't covered. The invisible walls get worse until the end of Layer 1, where entities that can't change existence state will get trapped. Changing existence state is very hard due to the tiny size of xoꓭ ǝʜT, but exposure to the "invisible walls" helps a tiny bit due to the fact that skipping over holes in space can break some objects. Entities that escape Layer 1 are pretty lucky.
 * Layer 2: Nothingness returns, but as little orbs scattered throughout this layer. These orbs each have a different density of nothingness, but it is evenly distributed throughout an orb. The total capacity of these orbs range from only containing a point to containing whole miniature Barrels, giving a fractal-like structure to The Barrel.
 * Layer 3: Space is extremely broken here. Most finite things cannot withstand more than 10 minutes of Layer 3 before they get stuck in a Heaa. They only get stuck in a Heaa as a failsafe; they would otherwise get vaporized due to broken space.
 * Layer 4+: These layers are just nested versions of Heaa and sub-Heaa structures, which ocassionally have mixes of other structures within them. This even goes all the way to Blueprint Particles at layer omega, causing a mess of a bunch of identical structures that are even identical in position due to duplicated information.  This makes the whole layer omega just seem like a portal to a singularity that you can't escape, similar to a black hole. Your information is also slowly deleted until you dissolve into a blueprint particle as well, adding on to the "singularity".

Barrel Chains' locations
Beyond the Barrel, we might see bigger Barrels that lie in The Box. Since the nesting can be purely infinite, one can expect to find every possible forms of barrels. Normally however, after infinite chaining of barrels, all chains of barrels terminate inside Barrelverse. Of course, not literally  EVERY  chains of barrel ends here, but they are usually chained only 2 to 21000 layers outside barrelverse.

Here is a table organizing them: