So, we have this construct called Advancement, which refers to how the player moves from situation to situation through the game, gathering Knowledge and Experience, even if this requires replaying sections of it frequently. And we have this concept called Challenge, which is an intrinsic property of a particular situation.
Now, let's use some math, as an analogy. We have C(s), which represents the challenge C for a particular situation 's'. Technically, the challenge is represented in the 9-dimensional space defined in the article, but we will ignore that for the moment. And we have S(t) which represents advancement through time. The function of advancement returns a situation. So, we can have C(S(t)), which represents the challenge for a player at a particular point in time.
So, what is dC(S(t))/dt? For those who don't know Calculus, this means the rate of change of the function C(S(t)), or how C(S(t)) changes with time. If you were to graph C(S(t)), the rate of change at any time 't' would be the slope of the function at that time, not the point.
What does this rate of change represent? Well, because I'm getting sick of writing "rate of change", I'll name it: Pace.
What does it matter, and why does the title call it "the Foundation of Gaming?" Well, because it is.
Challenge is constantly in flux. The more you go through a particular level, the easier it gets. This is due to the accumulation of Knowledge and Experience, as depicted by Advancement. Advancement makes the game easier. Challenge, therefore, should increase as the player plays the game. The absolute challenge of a particular late-game situation should be far greater than that of an early game situation.
Pacing is about how you get there. It answers the very relevant question, "How much did the Challenge change between two Situations?"
Why does this question matter? Well, because challenge is a big part of what makes a game interesting. A game that lacks challenge does not give the player a sense of satisfaction upon playing it.
Here's the deal. Pacing, how challenge changes from situation to situation, is what defines the character of the game design. A game where challenge increases dramatically with player advancement is very difficult. However, it may not be considered enjoyably difficult. A game where challenge increases very little is easy, but not necessarily enjoyable.
A well-paced game is one in which the player, through advancement, learns something and integrates it cleverly in such a way as to meet the next challenge and surpass it.
Pacing can be controlled and varied over the course of the game. Boss encounters, for example, are really just substantial jumps in pacing. Suddenly changing the pacing can be refreshing for the player, particularly if it is a surprise. A nice change of... pace.
Pacing also strongly correlates to complexity. As the player progresses through a game, challenge increases need to use the complexity dimensions of challenge. The use of multiple facets of the game design, more clever use of previous knowledge, etc are all important in increasing challenge. So pacing increases often correlate with increases to complexity.
Videogaming is not the only artistic medium that has pacing. The pacing of a movie, for example, is about how the audience's tension changes with the progression of the narrative. Action sequences and dramatic moments are sudden changes of tension, so they represent times of higher pace than slower sections.
Most narrative formats have a similar construct. Narrative tends to build towards a designed climax, and how fast it builds to it is its pacing. Pacing is integral to these kinds of artistic media, and videogaming is no different, even when it doesn't have a narrative structure.
A key difference between videogame pacing and traditional narrative pacing. Note that pacing involves the function S(t), which we defined as Advancement. Further remember that Advancement is entirely determined by the player, not the game designer. Videogames, being interactive, have a component that other narrative media do not; the player has a very direct effect on pacing.
While a book reader can start a book over again, or put it down, or skip around, there's nothing an author can do about that, and therefore he/she does not write the book with that in mind. With games, there is something that a designer can do. While a designer cannot directly create pacing like those other media, a designer can take efforts to keep the pacing correct.
For example, several modern games actually make the game a bit easier if the player is having trouble in a certain spot. A more interesting (and, to my mind, better) way to change the pacing would be Wing Commander style: if you lose missions, the next missions you go on are different, and generally a bit easier. This more designed method allows the player to still Advance correctly through the game, but it may simply take longer.
Imagine a game where, if you demonstrate the repeated inability to pass some challenge, the game simply dumps you into a tutorial that walks you step-by-step through similar challenges. It doesn't ultimately let you pass the original one, but it gives slower-learning players the opportunity to advance through a section that the player is not ready for.
A brief note: outside of Challenge, Pacing is the one term whose definition I already had before even starting this design theory. What I found interesting is that I did not design this model to explain pacing; it simply fell out as I saw how challenge changes with advancement. Which strongly suggests that the Situation Model, if not the right answer, is at least pointing in the right direction.