"Spacetime is locally Lorentzian" is obviously an approximation of some sort, as also explained in the answers to your linked question. You're explicitly constructing a situation here where that approximation fails because there are no other effects left that the ignored "non-Lorentzian" parts could be small when compared to. I'm not sure what the question is.

From Gravitation, Page 21: "Statement of fact: The geometry of spacetime is locally Lorentzian everywhere" I've read this several times and I've yet to find room for ambiguity in that statement. Perhaps we have a different definition of everywhere?

Re The geometry of spacetime is locally Lorentzian see How can locally Euclidean space of zero curvature accumulate to non-zero global curvature?

@JohnRennie - The statement from Gravitation is that a 'local' patch of spacetime is Lorentzian. I follow your argument about the Christoffel symbol vanishing at a 'point', but we're not talking about 'points', unless the definition of 'local' is something that has a zero length. I don't know how you'd even test for Lorentzian geometry if you didn't have space between your test particles.

@GluonSoup local means a region small enough that the Christoffel symbols are experimentally indistinguishable from zero.

@JohnRennie - Thank you. That's the best definition I've seen so far. But I still have an issue with it. If the space in which my experiment takes place is curved (as in this example), but my instrument isn't sensitive enough to measure the curvature (that is, my experiment results agree with Newton's Second Law of Motion), what justification do I have to call that space Lorentzian? Unless this is some sort of Quantum Mechanics logic, my instruments don't change the reality of the geometry.

Locally Lorentzian is used exactly as in Riemannian geometry where one says that a spherical surface is locally Euclidean. It just means that the scalar product in the tangent space is the same as in the corresponding flat space. I know that quite often the terminology is used in a sloppy way. Physically speaking all that means that some properties tends to become the same as in flat spacetime when restricting around an event.

Some but not all, for instance curvatures remain even if restricting around a point. Relative acceleration of geodesics is a curvature property (geodesic deviation) so it cannot be completely cancelled at small scales.

@ValterMoretti - Yes, I understand how that works on a sphere, but please apply that logic to the example here: No matter how small I make my experiment, the two test particles will move away from each other at the same acceleration according to Einstein's equation. How is that flat on any scale?

Indeed it is a curvature property, it cannot be cancelled locally as it depends on derivatives Christoffel symbols and not from Christoffel symbols themselves.

It is not flat. It is false that the spacetime becomes flat at small scales. Only some properties become similar to the corresponding ones in flat space.

“Locally Lorentzian” does not mean “exactly Lorentzian in a sufficiently small region”. It means “Lorentzian to first order in normal coordinates”.

If you take all the matter and energy out of a significantly large volume of spacetime, what you'll be left with is a small chunk of spacetime that - in the assumptions of 𝜆𝐶𝐷𝑀 - will be Lorentzian. Who claims this?

@G.Smith - I'm trying to develop an intuitive understanding of 'locality'. I'm trying to understand what Gravitation means by "The local geometry of spacetime is Lorentzian everywhere". From reading the comments here, it appears to mean "remove all the other influences (e.g. a galaxy, rocks, elementary particles, anything that could cause space to curve locally) and take a section of space that is small compared to the global curvature.

@G.Smith - "Lorentzian to first order in normal coordinates" I think I understand what 'normal coordinates' are, but I still don't know what you mean by 'first order'. This implies a derivative, but a derivative of which function, exactly? The metric, the scale factor, the Christoffel symbol? An example would be useful.

@GluonSoup remove all the other influences... That understanding is incorrect. You do not have to remove anything. Except at singularities, spacetime is locally flat everywhere, no matter how much curvature-causing stuff is there.

@G.Smith - Yes, that's what I read in the books. How can that be if my two test particles accelerate away from each other no matter how small a scale I chose? Or are you saying that there's a limit beyond which the $\lambda$ is no longer a factor in the Einstein equation?

@GluonSoup Lorentzian to first order in normal coordinates When the metric is expanded in a Taylor series using normal coordinates around a point, it is Lorentzian to first order in those coordinates. There are no linear non-Lorentzian corrections. The first non-Lorentzian corrections are quadratic in these coordinates.

@GluonSoup How can that be... It seems impossible because you have an incorrect understanding of “locally flat”. Multiple times it has been explained that this only means “flat to first order”. Locally flat spacetime still has curvature. The curvature arises from the non-Lorentzian quadratic corrections to the metric.

@G.Smith - I have no doubt that my understand is incorrect, but to correct it, I need to find out exactly which of my concepts is flawed. Are you saying 1.) That in my example there's a scale beyond which the particles are no longer accelerating away from each other, 2.) The $\lambda$ in Einstein's equation disappears at some point or 3.) In the language of differential geometry, 'flat' spacetime can still result in acceleration between two test particles.

@GluonSoup The third is closest to being right, but it’s still wrong because “flat” and “locally flat” are not the same thing. “Flat” means “exactly flat” or “Lorentzian” and there is no acceleration. “Locally flat” means “flat to first order” or “Lorentzian to first order”; it is still curved and there is acceleration.

@GluonSoup So the correct statement is that locally flat spacetime can still result in acceleration between two test particles. This is not surprising because locally flat spacetime has curvature.

@G.Smith - Thank you. This is progress. Now, can you point me to a place to research "Lorentzian to first order". I've got Gravitation but haven't found a discussion of 'locality to the first order'. It looks like a derivative, but a derivative of what? The metric, the Christoffel, the scale factor?

This means that the Christoffel symbols in these coordinates are all zero there. But the second derivatives of the metric in these coordinates are not zero there, so the Riemann curvature tensor in these coordinates is not zero there.

See en.wikipedia.org/wiki/Normal_coordinates Other coordinates obscure the local flatness of spacetime. Normal coordinates make it obvious. But they are only useful in a neighborhood of whatever point you care about.