So, keeping with your example, v=0.866c (gamma = 2.0)...

She has an array of helper-clocks synched to her own. So everyone at rest in that reference frame agrees that all of these clocks display a time that is the same as her age.

When he passes one of those clocks, his age is half the age on the helper-clock. So he might be 20 years old, and he might be passing one of her helper clocks which displays 40 years, even though he calculates that she is only 10 years old at that time.

OK, up to this instant, he and she have never accelerated, so they are effectively both perpetually inertial. So we don't have to specify a simultaneity method in that case, because we know that in the perpetually-inertial case, the only correct simultaneity is that given by the Lorentz equations and/or the Time Dilation Equation (TDE) You have specified that he is 20 at the instant immediately before he changes his speed to zero. And we are assuming they were each zero years old when they were momentarily co-located. Therefore, by the TDE, he says she is 10 when he is 20. She says she is 40 when he is 20.

So let's say he stops there.

OK. Now you have specified that he has accelerated (with a Dirac delta acceleration that instantaneously reduces his speed relative to her to zero). So now you DO have to specify which simultaneity method you want to use. You can't say anything now about simultaneity until you've chosen a simultaneity method. From your conclusions about what happens at that instant (when he changes their relative speed to zero), you have clearly chosen the CMIF (Co-Moving Inertial Frames) simultaneity method. The defining assumption of the CMIF method is that the observer ALWAYS agrees with the perpetually-inertial observer (the PIO) with whom he is currently co-stationary and co-located at that instant. The PIO says that she is 40 at that instant (when he is 20).

He is now standing still relative to the helper clock, and he is also standing still relative to her. He knows he himself is currently 20 years old, and he knows that the helper clock indicates that she is currently 40 years old. According to your own argument, there is no reason for him to wait for any light signals to travel from her to him. And so he can immediately abandon his prior belief that she was 10 years old, and accept that as he decelerated quickly [instantaneously] and stopped, she must have changed from 10 to 40 years old.

He doesn't abandon his prior belief. I.e., he doesn't say "I must have been wrong before". What he says is that she instantaneously aged by 30 years, from 10 years old to 40 years old. (And if, instead of changing their relative speed to zero, he had instantaneously increased his speed, the CMIF method would say that her age had instantaneously DECREASED by some amount ... i.e., she had instantaneously gotten YOUNGER).

But if, instead of choosing the CMIF simultaneity method to get your answer, you had chosen my simultaneity method, then he would NOT conclude that her age changed instantaneously when he changed their relative speed to zero. He would NOT agree with the PIO for some (determinable) time after his speed change. That amount of time in his life when he disagrees with the PIO is called the Disagreement Interval (DI). The magnitude of the disagreement is largest immediately after his speed change, and decreases after that until it reaches zero at the end of the DI.

And in the alternative scenario above where he instantaneously increases his speed (rather that instantly making it zero), my simultaneity method gives NO instantaneous decrease in her age. In fact, it never gives ANY decrease in her age at all, not even a gradual decrease.

If you plot an Age Correspondence Diagram (ACD) for your scenario, using the CMIF simultaneity method, it will look like this:

(I recommend either drawing all these diagrams I'm going to describe below very accurately, or at least sketching them well enough to see what's going on ... otherwise, the words alone are hard to follow.)

The ACD always just plots her age "tau" according to him, on the vertical axis, versus his age "t" on the horizontal axis. So during the initial segment, the line starts at the origin, and rises linearly with slope 1/2, until it reaches the point (t = 20, tau = 10).

Then, when he instantaneously changes his speed to zero, the plot rises vertically from tau = 10 to tau = 40. That vertical piece of the plot is the second segment of the plot.

Then, from there, the third segment rises linearly with the slope 1.0, because they are now ageing at the same rate, according to him (and also according to her in this case).

In my simultaneity method, the first segment of the plot is the same as for the first segment of the CMIF plot: a straight line rising from the origin with a slope of 0.5, until the point (t = 20, tau = 10) is reached. And we know that, for my method, immediately after his speed change, there is NO change in her age ... i.e., there is no discontinuity. This is the beginning of the disagreement interval (the DI), where he disagrees with the PIO. But before we can do anything further, we have to determine the end of the disagreement interval (the DI), where he once again agrees with the PIO. To do that, we need to draw a Minkowski diagram, which plots her age, tau, horizontally, and their separation X (according to her), vertically.

The first segment of the diagram is a straight line rising from the origin, with a slope of 0.866. When she is 40, he is (40)(0.866) = 34.64 ly from her, according to her. So the coordinates of the end of that first segment on the Minkowski diagram are (tau = 40, X = 34.64).

The second (and final) segment of the Minkowski diagram is just a horizontal line, because their relative speed is zero from then on, so their separation doesn't change any more.

To determine the end of the disagreement interval, we draw a straight line on the Minkowki diagram, starting from tau = 40 on the horizontal axis, rising to the right with slope 1.0 (an angle of 45 degrees wrt the horizontal axis). That line represents a light pulse that she emits toward him when she is 40. Extend that line upward to the right until it intersects the horizontal segment of his worldline. Consider the right triangle formed by that light pulse line, together with the segment of the horizontal axis to the right of the point tau = 40, together with the vertical line going upward from that point. The two equal sides of that right triangle each have a "length" of 34.64. Therefore her age increases by 34.64 years, according to her, while the pulse is in transit. So she is 74.64 years old when the pulse reaches him (according to her, and in this particular case, also according to him). And since their relative speed is zero during the transit of the pulse, he likewise ages by 34.64 years during the pulse transit, so he is 54.64 years old when he receives the pulse. (They both agree about that, because the arrival of the pulse is an event.)

The end of the disagreement interval (the DI) occurs when he receives the pulse. (And the DI started when he changed his speed). So we now have enough information to finish the age correspondence diagram. The DI ends when he is 54.64 years old, and she is 74.64 years old. WARNING: Note that, if he had changed their relative speed to anything other than zero, we would need to do a little work at this point to determine his line of simultaneity (LOS) that passes through his worldline where he receives the pulse, and then determine where that LOS intersects the horizontal axis. That point of intersection is her age when he is 54.64, according to him. That's what is needed for the ACD. But in this simple case, because their relative speed is zero, his LOS is just a vertical line, and so she and he both agree that when he is 54.64 years old, she is 74.64 years old.

SO, to complete the ACD for my method, we just find the point on the ACD for the CMIF method where he is 54.64 years old and she is 74.64 years old (which denotes the the end of the DI), and then draw a straight line between that point and the point where he is 20 and she is 10 (the beginning of the DI), because that beginning point gives their ages immediately after he changed his speed, and her age didn't change there, according to my method).

The reason it's fairly easy to draw that middle segment in my method is because it's always a straight line, so we only need to determine its two end points. If it were a curved line, it would still be possible to determine it (as described in my monograph in the definition of my method), but it would be MUCH more time-consuming. Sometimes you get lucky.

So, we now have the solution for your scenario for both the CMIF simultaneity method and for my simultaneity method. Which one do we choose? Obviously, we want to choose the correct one. And I'm convinced (for philosophical reasons) that there IS a single correct answer. But there doesn't seem to be any way to experimentally determine the correct answer. So there seems to be no way to tell what the correct simultaneity method is. It may be the CMIF simultaneity method, or it may be my method, or it may be neither. I don't like that situation, but I think we're stuck with it. I would prefer that the CMIF method be the correct one, because it is simpler, and because I don't have a problem with negative ageing. But many people can't accept the negative ageing implied by the CMIF method, so for them, my method may be more comforting.