quadratic deviation...does it make sense?

Between the turns.

Pardon my cryptic response, but a mainspring powered watch will be most accurate in that state.

Al has posted here before on this subject, try a search for isochronism (I think).

Isochronism (Greek for same, chronos for time) refers to the ability of a highly adjusted chronometer to maintain a constant rate, regardless of the amplitude of the balance wheel. As the mainspring runs down, amplitude of the balance wheel tends to drop off, and on a watch that is not adjusted for isochronism, the rate may be expected to vary more than would be the case with a chronometer. Even highly adjusted chronometers may be expected to vary a bit, particularly when close to being run down. Over all condition of a highly adjusted watch has an effect on isochronism. Rules on isochronism go out the window if condition is not tops.

This refers to the period (rate) of an oscillator staying the same, regardless of the distance (amplitude) that oscillator travels.

I don't know that there is accepted "typical" pattern to how the timing will change, other than to generalize that as the distance the balance has to travel becomes shorter, the period is likely to become shorter, so the watch will tend to speed up as it runs down as you have observed.

However this is a very broad generalization, because how a specific watch that is adjusted a specific way will react as the torque from the mainspring lowers and the balance amplitude starts to drop, will depend on a lot of different factors. It will depend on various "disturbances" that are affecting the balance, and I don't mean external to the watch, but things in the design/execution of the specific watch that will affect how it behaves.

One is if the balance is free sprung (variable inertia) or uses a regulator. Another is the shape/type of balance spring (flat v overcoil). Location of pinning point can be a factor. And then a watch may react very differently if left in a horizontal position compared to a vertical position. So if the watch is in a vertical position, then the C of G of the balance now comes into play, so if the C of G at rest is above the axis of the balance (the staff) then as the amplitude drops it will tend to slow down up. But if the C of G is below the axis at rest, the watch will tend to run faster as the amplitude drops - this is related to the balance either having positive or negative acceleration as the impulse pin travels through the lift angle.

These are just a small number of factors that are involved in how the oscillator behaves - it gets quite complex from a physics point of view very quickly, and quite frankly in a typical watchmaking school students won't get into these issues any further than is practically necessary from a bench level, where you are expected to minimize these errors. I even dug out my old textbook (The Theory of Horology) and they really don't discuss any specific pattern, because there are just too many variables at play.

In reality you could have two identical movements made by the same manufacturer, and due to the way they are adjusted one might gain as it runs down and one might lose time.

Anyway, hope this helps.

Cheers, Al

This isn't quite on topic but I am fascinated by the engineering that has gone into watch making over the years. Isochronism has been known about for a long time and is why the fusee was originally invented. In a fusee movement the mainspring barrel does not directly drive the movement, it is connected via a chain to a fusee (essentially a cone shaped pulley) and the fusee drives the movement. As the mainspring unwinds and the torque becomes less the chain unwinds from around the fusee going from the narrow end (which decreases the amount of initial torque applied to the movement) to the wider end (gradually increasing the amount of torque derived from the mainspring). This counteracts the the loss of torque in the mainspring as it unwinds.

The fuse was only one method used to give more constant torque - another solution used quite a lot was the Geneva stopworks. That limited the section of the mainspring that was used to run the watch to the center section where the torque curve was more consistent.

These days more constant torque is achieved by the shape and materials used for mainsprings. Springs are now shaped like an "S" in the unwound state, and the materials they are made of are much more stable. Very few watches use a fuse or stopworks, and if they do it's more for show than for go...