Dear Rafael,
thanks for having a look at my essay. I will try to find some time to read yours.
As for finiteness and extensibility, perhaps it helps to consider Spekkens' toy model, which demonstrates many of the features of quantum mechanics. The basic idea there is the 'knowledge balance principle': "the number of questions about the physical state of a system that are answered must always be equal to the number that are unanswered in a state of maximal knowledge."
In other words, if your have one bit of information about the state of the system, you must also lack one bit of information that could be gained by additional measurement. But if you perform that measurement, then you'd have two bits of knowledge, and no further knowledge could be obtained; hence, to make things come out right with the knowledge balance principle, the state of the system must change so that the previous knowledge no longer applies.
It's similar with my model. You can think of this as trying to localize a system further within its state space: some amount of knowledge will allow you to perform this localization to a certain degree of precision. If only 'finiteness' were true, then well, that might just be it: you've localized the system as well as it's possible to localize it.
But 'extensibility' implies that you can obtain additional information. For instance, if the system (in state space) is localized to some degree along one axis, you can try to increase its localization along that axis by making a more highly fine-grained measurement; but to cope with the finiteness-requirement, its localization must consequently decrease along another axis. You can visualize this like squeezing a bubble, or a squishy ball: the volume (the total localization/total information you have) stays the same (equal to a power of Planck's constant), but the shape will deform, yielding information gain in one property, compensated by loss along another.
Does this make matters more clear?
Cheers,
Jochen