What limits mechanical watch precision? It seems like good mechanical watches gain or lose +/- a few seconds per day. 5 s/day corresponds to a fractional frequency stability of $5\times 10^{-5}$. What limits this stability?
Obvious candidates are temperature and mechanical jostling. Fine, take a standard mechanical watch and put it in an environmentally controlled chamber and don't mechanically move it. Would the frequency stability improve beyond $10^{-6}$ or $10^{-7}$? What limits it in that case?
I suppose an issue is the mechanical watch has to be wound. This presents two problems. (1) The act of winding the watch may be mechanically violent (violent being relative to the stability scales I'm curious about) and (2) The watch tick rate may be dependent on how wound the spring is, for example when the spring is less tightly wound the clock may tick more slowly. To get around these two issues include, in the temperature controlled environment, a motor that continuously winds the spring at a rate equal to the rate that it becomes unwound. This means the mechanical environment the watch is exposed to is constant (i.e. there aren't intermittent winding events, it is continuous) and the spring keeps a near constant tightness.
With thermal, mechanical, and winding controls, what would the stability of a mechanical watch be? What would be the limiting factor? Would the residual instability be explained by one of these effects and you could improve the instability so long as you can improve control of one of these environmental factors? Or would another effect become significant?
One more fundamental effect that I could see limiting watch stability would be thermal Brownian motion. That is, even if the temperature is held perfectly constant, there will be thermal Brownian motion. Perhaps the way to model this would be temperature dependent fluctuations in force felt by the pallet fork. This could probably be addressed with material choices and holding the watch in a cooler environment.
Specific questions:

*

*What limits the stability of a watch in standard "watch" environments (i.e. on someone's arm during a regular day or year)

*What continues to limit stability as various obvious and less obvious sources of noise are controlled for

*Are there any references exploring this question?

 A: mechanical clocks are accuracy-limited by accelerations that act differently on different parts of the clocks because the clock mechanism itself is asymmetric. This can be addressed by splitting the mechanism into two mirror-image halves to cancel those effects, but this makes the mechanism physically larger which magnifies the problem.
The key to making mechanical clocks accurate enough for solving longitude problems in navigation was by miniaturizing them enough to make the acceleration effects smaller than, for example, temperature effects.
See the book called Longitude by Sobel for the history of this process.
A: There is a channel on YouTube showing how to service a watch. From what I can see, friction due to oil viscosity and dust, and an unequilibrated balance are the main factors affecting the stability of a mechanical clock.
Intrinsically, the mechanical clock is based on a pendulum whose period for low amplitude is given by: $T=2 \pi  \sqrt{ \frac{L}{g} }$. Dilatation, due to temperature change, directly affects L. But this formula is mathematically an approximation. You can improve the precision of T by adding more terms to the formula but, to be coherent, you need to include effects like friction and acceleration due to vibrations in the Physics and the Math becomes very complicated.  Friction can be somewhat compensated by the spring, but a clock is full of gears, and you cannot completely suppress the internal vibrations. As a result, the formula for T changes and the whole system is intrinsically limited.
