> you arbitrarily chose the size of your LIF to make it too small.

No, I didn't. I didn't specify the size of the LIF at all, in terms of a standard unit like meters. I only specified it in terms of the mass of the black hole; its size has to be much smaller than M (or GM/c^2 in conventional units). That's not an arbitrary choice; it's required by the physics. See below.

> GR doesn't tell us the maximum size of a LIF, nor could it.

This is wrong as you state it. GR does set an upper limit on the size of the LIF, in terms of the spacetime curvature that is present, because if the LIF gets too large, the effects of spacetime curvature on the metric coefficients becomes equal in size to the metric coefficients themselves. At that point you can't have an LIF no matter what your measurement accuracy is, because the whole point of an LIF is that the corrections to the metric coefficients due to curvature are much smaller than the metric coefficients themselves.

In the particular case under discussion, for distances from the origin of the LIF that are of the same order as the mass of the black hole (GM/c^2 in conventional units), the curvature corrections to the metric coefficients are of order 1, i.e., the same size as the metric coefficients themselves. That's all my computation was showing, and it's enough to set an upper limit on the size of the LIF. In other words, the mass of the black hole determines the spacetime curvature at the horizon, and that determines the maximum possible size of an LIF falling through the horizon.

It's true that, by increasing the black hole mass M, you increase the maximum size of the LIF falling through the horizon; but I already took that into account in my analysis, as I said above.

> It's determined by the accuracy the experimenter wishes to achieve, e.g. 6 significant digits

This will determine an upper limit on the size of the LIF, as long as it's smaller than the limit I described above. But also, the more accurate the measurements, the smaller the upper limit on the size of the LIF. I'm not sure you realize this.

> When I compare the laws of physics between two LIFs I have even greater freedom to choose the LIF size.

No, you have exactly the freedom I specified, and no more: the LIF size over which you can do the comparison is the smaller of the two maximum sizes. So we have two possibilities in the case under discussion:

(1) The maximum size of the black hole LIF is larger than the maximum size of the skydiver LIF (which will be based on the spacetime curvature in the skydiver's vicinity). In that case, obviously the "catch-up distance" extrapolated from the LIF will be much larger than both LIF sizes.

(2) The maximum size of the skydiver LIF is larger than the maximum size of the black hole LIF. In that case, the "catch-up distance" will be much larger than the size of the black hole LIF; but if the skydiver LIF is large enough, that distance could be smaller than the size of the skydiver LIF. But the range of the comparison, for purposes of the equivalence principle, will be the size of the black hole LIF, since that's the smaller of the two, so there's no violation of the equivalence principle since there's no difference between the two cases within that range of comparison.

> I choose a black hole massive enough that the tidal force in the astronaut's frame is less than in the skydiver's frame.

And that just means the black hole LIF size will be larger than the skydiver's LIF size, which is the first of the two cases I discussed just above. In that case, since I've proved that the overtake can't happen within the black hole LIF size, it obviously can't happen within the skydiver LIF size either, since the latter is smaller.

> Now, when overtake happens in the skydiver's frame but not the astronaut's

Which it doesn't; see above. Once again, you are basing your reasoning on a false premise, so you're reaching a false conclusion.

If you wanted to make things more interesting, you could have chosen a black hole with tidal force in the astronaut's frame larger than in the skydiver's frame, i.e., the second of the two cases I discussed above. Then it would at least be possible that an overtake would happen within the skydiver's LIF; but, as I noted above, that doesn't violate the EP, because the comparison can only be done over the smaller of the two LIF sizes.

> let the probes be tiny rods. Fire them vertically-oriented, the first probe completely above the horizon, the second initially stradding the horizon and partially alongside the first probe. In the skydiver's frame the second probe will continue to overtake the first probe. In the astronaut's frame the second probe cannot continue to overtake the first probe, lest it be moving outward through the horizon, which GR disallows.

Incorrect. Someone brought up this case in the PhysicsForums thread I linked to earlier. My detailed answer is there. A quick summary: the second probe will overtake the first probe, within the range of comparison between LIFs (i.e., the smaller of the two LIF sizes). Only outside that range of comparison will the difference between the cases become apparent. And this will be true even though the second probe is indeed falling through the horizon, while the first probe is moving outward at escape velocity. (The upper end of the second probe will be moving faster than escape velocity at its initial altitude, since by hypothesis it is moving in the positive x direction in the LIF faster than the first probe, so the r coordinate of its upper end will be initially increasing, even though the r coordinate of its center of mass is decreasing, which is what is necessary for it to be falling through the horizon.)

> You say the probes can approach each other in the astronaut's frame. I say they can't. When the first probe's r-coordinate is increasing and the second probe's is decreasing, they are receding from each other as measured in any LIF that wholly contains them.

This is wrong and I've repeatedly explained why it's wrong. If you can't understand what I've already explained and why it's true, I'm sorry, that's the best you're going to get. I'm not going to teach you a course in relativity.

> We needn't refer to r-coordinates at all. Here is a simple law of physics (I just devised) for an LIF that you seemingly disagree with: Two free test particles each moving toward destinations beyond opposite sides of the LIF recede from each other.

Why do you think I would disagree with this? I don't. It's perfectly consistent with everything I've been saying, and with GR.

However, this claim is not equivalent to the claim I suspect you have in the back of your mind. I suspect the claim you have in the back of your mind is this: the first probe is moving upward from the horizon, while the second is moving downward from the horizon, therefore they are moving in opposite directions, therefore they must be receding from each other. The problem is that "upward from the horizon" and "downward from the horizon" are not the same as "moving toward destinations beyond opposite sides of the LIF".

First, "upward from the horizon" and "downward from the horizon" are global concepts, not local ones; they are defined in terms of the global r coordinate. You keep on making this mistake even though I've repeatedly explained why it's wrong, and even though you think you are not referring to r coordinates at all.

Second, the horizon is not a place in space. It's an outgoing light ray. And within the LIF, both probes are moving in the same spatial direction as that outgoing light ray! (The first probe is ahead of the light ray, and the second is behind it.) So within the LIF, the two probes are not moving in "opposite directions". So your argument, once again, is based on a false premise, so naturally you are reaching false conclusions.