Congratulations for such a brilliant essay, which be more remembered for the food for thought it presents than its actual content. I thoroughly enjoyed it.
A prediction is a statement about the future based upon past experience and knowledge derived from such experience. Hence something is predictable, if 1) we have similar experience in the past and 2) there are no additional factors that can influence the possible outcome. While 1) is simple, the 2) is not. Uncertainty is inherent in Nature. Our knowledge of the universe is never complete. This makes the field wide open. We may not and cannot know all factors that may affect the outcome of any event. This makes everything unpredictable. You are absolutely right here.
Having said that, there may be limits on such unpredictability. We cannot pin point the precise position of an electron at any specific moment. Yet, we can predict that it will be found somewhere within a fixed band. Experiment proves us right. Thus, these are predictable unpredictable. We use probability and average to predict the outcome based on past experience based on measurement and knowledge derived from such experience. While the laws of physics are same everywhere, the same law is not applicable everywhere. This leads to model-dependent definition of predictability. You are right again: "Something that is predictable in one model can be unpredictable in another".
But is it same in mathematics - "what is decidable in one model of mathematics can be undecidable in another"? I beg to differ. Your statement is true for mathematical physics, but not for pure mathematics.
Language is the unambiguous transposition of one's/a system's thoughts/command on another person's/system's mind/CPU. Mathematics does that with numbers. Hence mathematics is a language of Nature. But it only depicts quantitative aspect of Nature. It does not cover all aspects. Hence what is true for mathematics may not be true for physics or biology. For example, if by paying $3000 we can get a bike, by paying $1000 we can get 1/3 of a bike. This statement is mathematically correct. But physically it does not make any sense. There is no equation for the observer, but it has an important role in physics. Thus, physics beyond mathematics cannot be denied. Extending the limited scope of mathematics (scalar numbers) to (vector) physics or biology, makes us fall in the trap of reductionism.
In mathematics, we have irrational numbers like square-root of 2 or such numbers likeпЃ°. We cannot use it as a fraction of whole numbers. But in physics, we cannot use inderminables. We must chose a limit up to which we can go to determine the nearest fraction. This is the subject matter of my essay in this forum.
Your poser: "whether a non-standard model can predict (correctly!) some things that are unpredictable in the standard model" is very interesting. In fact, it has the potential to change the direction of modern research. Let me give one example.
From the times of Newton, measuring the exact value of G, the gravitational constant, is one of the priority areas, as it is used widely in many equations as a constant and any change to its value will make many theories crumble. Yet there was wide disagreement on the precise value of G. Now it has been noticed that it G exhibit a 5.9-year periodic oscillation value. Interestingly, this closely matches the 5.9-year oscillation in length of day (LOD - defined as a frequency measurement such that a positive increase in LOD values means slower Earth rotation rates and therefore longer days) measurements. The aforementioned period is also about half of a solar activity cycle of about 11 years, but the correlation is far less convincing because of the big difference in values. The 5.9 year periodic signal in LOD has previously been interpreted as due to fluid core motions and inner-core coupling. Again, this is not convincing. The G/LOD correlation, whose statistical significance is 0.99764 assuming no difference in phase, points to a totally new direction.
It is known that unlike other interactions, gravity is a polygamous force. This is because the other interactions are intra-body and follow a symmetry, whereas gravity is interbody and asymmetrical (that is why planetary orbits wobble). That it is related to mass and distance shows that not only the bodies concerned, but all others that influence the two bodies, play an important role in it. While it may not be evident always, the bodies between the Sun and Earth (like Mercury, Venus) are affected by Earth, but the Earth is not affected by these bodies to that extent - the forces being directed at Sun. But what about Jupiter, which is massive enough and the Earth falls in the path of its interaction with the Sun? There is gravitational interaction between Jupiter and Earth. When these are in a straight line with Sun, Jupiter drags Earth away from Sun, which reduces and reaches a peak when Sun is between Earth and Jupiter. This starts to increase again to reach a peak after 11.86 years - the period of Jupiter. Thus, the change is periodic with a 5.93 year periodic cycle.
This forces us to look at the theory of gravitation afresh. It is also another example of predictable unpredictable bordering the unpredictable unpredictables.
Now, I will like to give another thought to your unpredictable unpredictables. Decay and Evolution are established facts of Nature. Decay is related to time evolution. Evolution is limited transformation of something without losing its basic characteristics. Can it lead to the rabbit having horns? No one can tell with certainty. It is unpredictable unpredictables. Yet, when look at it closely, does it fulfil the criteria of prediction? We have seen rabbits. We have seen horns on many animals. We make a statement about the future based upon past experience and knowledge derived from such experience. So based on our past experience of a horn being a natural tool to provide protection from danger, and looking at the increasing threat to the rabbit population, they may evolve to grow horns. What we do here is extend a relation (natural mechanism for protection from danger) to two unrelated fields (rabbit and horn), without properly analyzing and without taking into account all possible aspects. Nature has provided a safety mechanism to the rabbit - its speed to run away from danger. Having a pair of horns will hamper that speed. With its limited physical volume and soft body, it will not be able to defend itself from predators. Nature functions creatively. It will not jeopardize safety to life while providing food for others. Hence, horns of a rabbit will never materialize. This way, we can explain one case of unpredictable unpredictables.