After reading up on the "paradox" of Lavos' shell and his 1999 AD emergence / 12,000 BC demise, I have come to the conclusion that there is no paradox. I think we're simply applying theories of time travel (time bastard and TTI) to dimensional travel mistakenly.

Allow me to elaborate.

Here is an illustration of 2D space.

Note that in order to move between points B and C, you only move along one axis, while moving from either B or C to point A results in movement along 2 axes. This is in line with everything we know about spacial movement.

Now, let's look at 3D space.

Movement between points A, B, and C are preserved as-is from our 2D plane. Movement between points B - B' and A - A', however, represent 3D transformations. As I'm sure you all understand so far, this is like jumping in place and is not a protected act; all transformations here are basic 3D planar movements.

These three axes represent three dimensions of planar movement. Let's add a 4th axis. This axis, Omega, will represent a 4D space (length, width, depth, and time).

Each point along axis Omega represents another 3D space. This is what we call a timeline. All of the transformations from our 3D space are preserved; they all operate exactly the same. Any movement parallel to the Omega axis, however, represents a movement along the 4th dimension (time). These acts, as we know, are protected. Chrono Compendium has explored all of the mechanics of travel up until (and including) the 4th dimension. Here's where things get hairy.

We assume that Lavos creates a "pocket dimension" when he arrives on Earth in 65M BC. The very concept of a "pocket dimension", however, is flawed! You see, we can represent movement along a single axis as a line; I can move from a point at offset 1 to a point at offset 12 on a line easily. If I want to move any other way, however, I can't simply explain that this second point exists on a "pocket axis" that somehow inhabits the same axis but not; I have to create another axis to account for this difference in movement, and thus I create 2D space. This concept applies to dimensional travel. If we want more space to exist, we have to have room for it, and so to account for travel in 5 directions (length, width, depth, time, AND dimension), we would need to create a 5D space by adding another axis.

This other axis will represent all of the possible combinations of time and space parallel to each other. One more illustration.

Note that I have purposefully offset the two lines from each other. The reason for this is that two dimensions need not be equal. In 2D space, two points are referenced by their relation to each other and their surrounding space. This offset is the only relationship between spacial points. Much in this same way, two dimensions are only related by their offset from each other; nothing else needs to be equal. The two dimensions need not have the same mass, volume, energy... or time. Time in one dimension is free to run at a rate inequal to the rate of time in any surrounding dimensions. You see, all of time, in 5D space, is condensed into a single point (each point then becoming a dimension with its own personal timeline). If we think of dimensions this way, then Lavos' non-emergence in 1999 AD when killed in 12,000 BC is no longer a paradox because dimensional travel is not equivalent to time travel; the two kinds of travel are along separate and different axes!

When Lavos burrows into his "pocket dimension" (just a separate dimension from Crono's), he is breaking from 4D space into 5D space. He isn't traveling along the Omega axis, and as far as we know, only travel along an Omega axis is protected by TTI, since the Omega axis is the only axis that doesn't act as a spacial axis. Separate dimensions, however, are spacial in nature (they are offsets of space, not time) and so are not subject to TTI. When Lavos exits his dimension in 1999 AD, he is traveling along the 5th axis again. You see, all points in time are condensed when moving from 4D to 5D space, so when traveling 5ht dimensionally, you have access to ALL points along the Omega axis. You never move ALONG it. Only this movement along the Omega axis is protected (as far as we know), so Lavos' 1999 AD emergence is never protected by TTI!

Now, how about Lavos' shell? Again, when Crono and friends travel to Lavos' dimension to fight his shell, they have access to ALL points in time at once; the point of entrance for Crono's party is completely arbitrary. They have engaged Lavos in his own dimension, however, so when they defeat his shell, it is along his own timeline (an event logged on Lavos' Omega axis). This act isn't protected either, though. If Lavos had the ability to travel in time, he could theoretically go back ALONG HIS OWN OMEGA AXIS and prevent Crono's party from defeating his shell, thus allowing him to return to Crono's dimension with his shell intact; why he doesn't, we don't know. All we know is that when Crono and friends return to Lavos' dimension, they arrive AFTER the arbitrary time they did the first time around.

This allows for Lavos to lose his shell AND his 1999 AD emergence when Crono's party defeats him in 12,000 BC.