User:Sorceror Nobody/Decay Modus

The Decay Modus is a Fetch Modus used by Sorceror Nobody. Items captchalogued into this modus automatically occupy half of the available space, and the amount of space they use gradually decreases exponentially, with a well-defined "half-life". The modus undergoes multiple upgrades to circumvent certain critical flaws.

Decay Modus &alpha;
The initial version of the modus was created by &&-combining the Array Modus with a lump of Uranium. This is not to be confused with it costing Uranium Grist to create, although as it happens, it does. When an item is captchalogued into the modus, the number of unoccupied cards in the modus is halved (then rounded up), and the item is distributed over that number of cards. Over time, the number of cards the item occupies decays as follows:

$$N = \text{round} \left({N_0 \exp{\left(\frac{-t \ln{2}}{T_{1/2}}\right)}}\right)$$

...which is an exponential decay, rounded to the nearest integer. $$N$$ is the number of cards occupied by the item, $$N_0$$ is the number that it occupied at the time of captchaloguing, $$t$$ is the time that has passed since the item was captchalogued and $$T_{1/2}$$ is the "half-life" of the modus.

There are several acute problems with this version of the modus, most notable of which is that when the number of cards occupied by an item reaches zero, the item becomes "locked" into the modus. What this means is that, rather than vanishing from the modus entirely as might be expected – thankfully, items are not erased – locked items simply remain captchalogued, and occupy a single card, but they cannot be retrieved from the modus. At all. Not even by ejecting all of the modus's contents. Obviously, then, it is possible to completely clog up the modus with every card storing a locked item, rendering the sylladex completely inoperable. This is why the upgrades are crucial.

Incidentally, it may seem that an obvious solution would be to simply retrieve items before they decay to zero, then immediately re-captchalogue them. Unfortunately, the modus retains a "ghost" copy of the retrieved item, which continues to occupy the space the item was taking up, and continues to decay. If the item is re-captchalogued, it simply goes back into the space it was in, even if that space is smaller than when it was retrieved. When a ghost copy reaches zero, unlike actual items, it is erased.

Decay Modus &beta;
The second version of the modus was created by || -combining the &alpha; version with the artefacts used by the Prince of Time. This adds a sort of rewinding property to the decay, meaning that items no longer automatically lock in when they reach zero. Instead, they will resize back up to half of the available space (as usual, rounded up), not counting the card it occupies, which is added on. For example, if there are sixteen cards and only one item, that item will resize to occupy cards according to this formula:

$$\left\lceil \frac{16-1}{2} \right\rceil + 1 = 9$$

If the occupied card were counted as part of the "available space", this would present no problems, as the worst case scenario would be that the rest of the modus would be full, and so the item would simply jump back to occupying one card, then decay back to zero, then to one, et cetera until more space was available. However, the actual calculation means that an item may need more space than is available. If this happens, unfortunately, the item will again become locked in.

In case it wasn't obvious, the $$N_0$$ and $$t$$ in the decay equation are reset when item decays are "rewound".

Decay Modus &gamma;
The third version of the modus was created by &&-combining the &beta; version with a Gristwidget. This caused no changes to the fundamental principles of the modus, but it does allow locked-in items to be removed from the modus by converting them into Grist. Naturally, using this feature costs Boondollars.

Decay Modus X
The fourth and final version of the modus was created by || -combining the &gamma; version with one of Sorceror Nobody's Keyblades. The || combination incorporated the Keyblade's ability to unlock anything, meaning that locked-in items can finally be retrieved properly, although a complicating interaction with the Gristwidget component of the modus means that doing so comes at the cost of a small amount of both Grist and Boondollars. The ability to convert locked-in items into Grist is still available, and also still costs Boondollars.

Incidentally, Sorceror Nobody initially &&-combined the modus and Keyblade when attempting to make this upgrade; this produced the Fissile Spinstability Keyblade. Which is actually a pretty damn cool weapon, so he's not complaining. It did cost an insane amount of Grist to make it, mind you... but it was totally worth it.