I haven't kept track of your Charlie scenario, but I think it is equivalent to me saying that if the traveler (he) instantaneously changes his velocity at the turnpoint from +0.866 to ZERO, he will say that that doesn't instantaneously change the home twin's (her) age AT ALL. But her age WILL immediately begin to increase linearly. You can get the slope of the age correspondence diagram analytically, using the equation near the end of Section 10. Or you can easily get that slope graphically, as described in the paper.
You are not even using the most basic premises of SR, and as such, you are no longer doing SR at all.
It is one of the most basic premises of SR that two identical clocks at rest in the same reference frame tick at the same rate. This is easily derived from the premise that the speed of light is the same constant in all inertial frames. Simply consider a light-clock made from two mirrors a known distance apart, with a pulse of light bouncing between them. When two such identical clocks are at rest relative to each other, we conclude that they tick at the same rate. From this we can conclude that if the traveling twin changes his velocity at the turnpoint from +0.866 to zero, then after that moment, the home twin's clock and the traveling twin's clock will run at the same rate. There is nothing in SR about two identical clocks at rest in the same frame running at different rates, as it would violate the constancy of light speed if light-clocks are used.
Furthermore, it is one of the most basic premises of SR that two identical inertial clocks with uniform relative motion between them will each say that the other clock ticks at a rate of 1/gamma their own rate. This is easily derived from the premise that the speed of light is the same constant in all inertial frames. Simply consider a vertically oriented light-clock made from two mirrors a known distance apart, with a pulse of light bouncing up and down between them. When such a clock has a uniform relative motion, the light obviously follows a diagonal path instead of up and down, and we can use the Pythagorean theorem to derive the slowed rate that the "other" clock has. Note that this time dilation effect is 100% reciprocal, meaning that each clock will consider the "other" clock to be the one that is ticking slower, and we do not even need to know which one (or both) might have accelerated. Note that I am talking about the RATE of the clocks, not any particular time being displayed.
I will call this
the premise of reciprocal time dilation rates, as I will refer to it again later.
If you do nothing else, you should read the short proof I give that her age doesn't change at all during his instantaneous velocity change. It's very near the end of Section 7. It's a very simple proof.
Thank you for pointing me to this, as now I have some idea where you went wrong. If the traveling twin (he) wants to calculate her current age from the special image pulse that she sends to him, he will need two things. First he will need to know the distance between the emission point and the reception point. Note that this is not based on her current location relative to him, but rather where he reckons she was located in the past when she sent the image. Remember, he considers her to be moving and himself to be still.
Once he knows that distance, he cannot simply add the transit time to the time he sees on her clock in her image pulse. He must consider what
rate her clock has compared to his, and add in the amount of time that would have elapsed on her clock during the transit of the image. Luckily he knows her clock's rate from
the premise of reciprocal time dilation rates. In fact, since the premise of reciprocal time dilation rates is a basic premise of SR, he knows what her clock's rate will be
for the entire return leg of the journey. And since the journey ends with both twins located in the same place, he can take her age at the end, and work backwards to the moment just after the turnaround point, and figure out her age from the known rate he got from
the premise of reciprocal time dilation rates. Note that this makes calculating her age from the image pulse unnecessary or redundant. At the very least, calculating her age from the image pulse must give the same answer as working backwards from the end of the journey, and both methods use the known clock rate from
the premise of reciprocal time dilation rates.
You say, "The amount that she aged during the transit of that pulse (according to him) can’t have changed: it was (and forever is) an historical fact for him," but that is not entirely correct. Consider that the first thing he needs in order to calculate her current age from the special image pulse is the distance between the emission point and the reception point. Note that this is not based on her current location relative to him, but rather the location where he reckons she was located in the past when she sent the image. Remember, he considers her to be moving and himself to be still. Just before the turnaround point, he considers her to be moving away from him, so he would say that the location where she emitted the image (in the past) was closer to him than where she is currently located. And just after the turnaround point, he considers her to be moving toward him, so he would say that the location where she emitted the image (in the past) was when she was farther away than where she is currently located. This is how the same image pulse (just before/ just after) the turnaround point can result in two different calculations for her current age.
But again, calculating her age from the image pulse is not even necessary. At the very least, calculating her age from the image pulse must give the same answer as working backwards from the end of the journey using the known clock rate from
the premise of reciprocal time dilation rates.