If the foundational assumptions are wrong or impossible to challenge, time t can extend indefinitely. Additionally, it is surely possible that whole sections of science is waylaid and diverges from truth on account of funding, legislation, big personalities, etc.
This is an important point, but fortunately nature has provided us with a solution. Namely that teenagers are defiant and look for ways to distance themselves from their elders.
This is Planck's principle that science advances one funeral at a time [0]
> A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it ...
> An important scientific innovation rarely makes its way by gradually winning over and converting its opponents: it rarely happens that Saul becomes Paul. What does happen is that its opponents gradually die out, and that the growing generation is familiarized with the ideas from the beginning: another instance of the fact that the future lies with the youth. — Max Planck, Scientific autobiography, 1950, p. 33, 97
The biggest impediments have historically been multi-generational organizations whose power requires limiting access to science. Famously the centralized church in the time of Galileo. More recently the cigarette and oil industries. Things like big personalities and legislative priorities tend to have much shorter time scale and usually allow for science to ratchet forward generationally. Big personalities die, legislators turn over every few years in the best case or in a few generations in the worst case.
Intra-generational power structures are a much larger impediment. Funerals take way too long to happen.
That is strange since to work in science today you have to be extremely compliant for close to a decade. Most people who aren't gets kicked out or removed in selection processes, what you have left is one of the most compliant parts of society.
Disclosure: Old person
There's actually quite a big assumption in here -- namely, that the truth is constant over time and throughout space. (This can possibly be weakened slightly, but that's the gist.)
I personally think the laws of physics probably are unchanging, and I certainly hope they are (because that means the scientific method converges). But whether they actually are is not only unknown, but a question that cannot be answered by empirical science.
But from a technical perspective, the math is perfectly capable of describing theoretical universes where the laws of physics change in time or space. For example, where physical constants are smoothly evolving or manifolds that don't look the same in all directions.
The math can tell us what observations we'd expect in those situations, and so far we haven't observed anything to indicate we should relax those assumptions.
If we did observe that the laws of physics were changing, that observation would be considered science in the traditional sense. So it's not that "truth" is shifting, it's that "truth" is a family of equations indexed by some parameter rather than a single equation. That's a similar flavor to the jump from Newtonian physics to relativity.
If you're interested in this topic, a relevant key word is "cosmological principle": https://en.wikipedia.org/wiki/Cosmological_principle
The current models include these assumptions for the same reason they include any other assumptions, because models where these assumptions aren't included don't do any better a job explaining any experimental results. If new experimental data shows these assumptions fail in some cases, then the models can be updated to handle them, same as any other failure has been handled in the past. (Granted each step becomes computationally more complex, so eventually we might hit a limit where it is too complex for a human to understand well enough to operate on.)
I think this is the best way to talk about the difference between science as a process vs science as what people in white lab coats do.
People have a tendency to conclude that we should abandon science if something isn't quite right. It's better to think in terms of what will get us un-stuck from a local minimum/maximum.
Isn't that generally what convergence means? At least in the math sense, we are talking about t at infinity, meaning that in any real world time limited application, there is always a gap.