Markus Hanke

Yes, absolutely. That is why you need to remain invisible, undetectable, and silent yourself, so all anybody ever sees is at most shadows moving in the dark. If done right, your opponents won’t know where to direct that second shot at. It’s called ‘Dark Forest’ for precisely this reason - it’s a game of wiping out all potential threats, ideally before they even become aware of your own existence, in order to ensure your own survival. Like being lost in a deep dark forest, surrounded by silent and unseen predators. In this scenario, the resolution to the question of whether or not aliens exist might one day appear in the form of a small but massive projectile coming at us at high-relativistic speeds, and we’ll never even know who took the shot or where it came from. It’s pretty terrifying, and I sincerely hope this is not how things actually are out there. But unfortunately it is a rational scenario backed by game theory and fully compatible with the Fermi paradox, so one cannot simply dismiss it.

You probably mean gravitational waves - gravity waves are a phenomenon in fluid dynamics, and have no relation to black holes. The answer to your question is threefold: 1. Gravitational radiation during BH mergers does not originate only at the event horizon, but results from the quadrupole moment of the binary system as a whole. Any kind of binary system - irrespective of what kind of objects it is comprised of - will be a source of such radiation. It is, in that sense, a global phenomenon of such spacetimes, and its source cannot be localised to any one particular point or region, including the event horizon. That being said, the geometry of the horizons reflect the geometry of all the rest of this spacetime (in very complex ways), so observing the wave forms of the radiation field allows you to extrapolate what happens at the horizon during the merger. This whole process is really a global one, and doesn’t just happen at the horizon. 2. The diverging in-fall time you are referring to applies to Schwarzschild black holes, but the spacetime in a binary system of in-spiralling BHs is not of the Schwarzschild type, not even approximately. Figuring out the precise in-fall time of a test particle from far away into one of these BHs is a highly non-trivial task, that can only be done numerically, but my guess is that it wouldn’t be infinite at all (one of the necessarily prerequisites for an infinite in-fall time is asymptotic flatness, which does not hold in this type of spacetime); it would also explicitly depend on where and when the test particle begins its free fall. 3. Even for Schwarzschild BHs, the infinite in-fall time applies only to test particles moving on time-like or null geodesics of the undisturbed background, ie it applies only to test particles whose own gravity can be neglected. This, however, is not the case for gravitational waves, which will couple to the background curvature in non-linear ways. To put it differently, a spacetime that contains gravitational radiation cannot have Schwarzschild geometry, and thus the infinite in-fall time does not necessarily follow. Even in cases where the gravitational waves are weak enough so that the background could still approximately be treated as Schwarzschild (which is not the case for a binary BH system!), the wave fronts wouldn’t propagate along the same trajectories as free-falling test particles, due to non-linear interactions with the background curvature. No it doesn’t. The length of the world line of a test particle free-falling from far away and crossing the horizon is finite and well defined; spacetime at and around the event horizon is smooth and regular, so time proceeds as normal there. The thing with this is that in curved spacetimes, there is an important difference between coordinate in-fall time and proper in-fall time. The coordinate in-fall time is what a distant observer will calculate and observe, based on his own instruments, which are not themselves located at the horizon; the numerical value of this will depend on which observer you choose. The proper in-fall time, on the other hand, is what is directly measured by a clock that is attached to the freely falling test particle itself; by definition, it equals the length of that particle’s world line through spacetime. For the case of a test particle freely falling into a Schwarzschild black hole, a far-away stationary observer will determine a coordinate in-fall time that diverges (goes to infinity). However, the proper in-fall time of that same test particle is finite and well defined, so the particle reaches the horizon in a finite amount of time as measured by its own clock, and continues falling through the horizon and into the singularity (also in a finite, well defined amount of time). Because of the way that proper time is defined mathematically, all observers agree on it. On the other hand though, coordinate time is always specific to a chosen observer, and not valid anywhere else. In curved spacetimes, time is a purely local phenomenon; a far-away observer does not share any concept of simultaneity with processes that happen at the (for him) distant horizon.

Yes, that’s a good and valid point. However, one must remember that we ourselves don’t hesitate to annihilate large groups of sentient individuals belonging to other species, if our own interests are threatened. Consider - just as a random example - spraying large swathes of agricultural land with insecticides, which is still common practice in many places. Such acts lead to the death of millions of insects, and we don’t bat an eyelid. Why? Because very many people do not consider insects to be worthy objects of moral concern, since we regard them as primitive, expendable, undeveloped, unintelligent, and a direct threat to our own interests. If they annoy or threaten us, we simply annihilate them. The crucial factor in the DF scenario is incompleteness of information. If two civilisations are, say, 10000LY apart (not unreasonable if there are only a handful per galaxy), and assuming the laws of relativity as we know them cannot be circumvented in some way, then there simply isn’t any workable method for these civilisations to talk to each other in any meaningful sense. As a consequence, neither one of them can truly know what the intentions of the other one are, and (more importantly) what their current level of development and technology capability is, since the latest available information about them will be 10000 years old. You cannot extrapolate how a civilisation might develop over such long periods of time, especially not based on limited data. So you are pretty much left completely in the dark - you have no way of knowing what they are up to, what their intentions are, and how they think about you. Of course you can assume that they are moral beings, but that’s a huge gamble to take, and if you’re wrong then that’s the last mistake you’ll ever make. So that’s the basic conundrum - the speed of light is very slow when taken to even just galactic scales.

True. And personally I’m partial to the third option - that we simply aren’t going to see any alien civilisations any time soon. But we may well see more primitive forms of extraterrestrial life in the near future, perhaps even within our own solar system. I’d also like to point out that, in a Dark Forest scenario, a civilisation acting in the most rational way within the confines of that mathematical game (ie eliminating other civilisations) does not imply malevolence on their part. They simply do what they need to in order to maximise their own evolutionary potential in what is a scenario with few other options, given the constraints imposed by the laws of physics. Even an otherwise benevolent and ethical civilisation may find it necessary to take such drastic steps. Also, just because we place a high ethical value on life (do we??) does not necessarily mean that others share this concept.

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No. Dark Forest would suggest that in the first instance, civilisations remain silent. When you have a substantial number of intelligent races all confined into a galaxy where the total amount of available resources remains constant while civilisations continue to evolve and expand, 2-way communications are subject to the laws of relativity, and all these races are motivated by a basic desire to ensure survival of their own species, then you have what is a called a sequential and incomplete information game. You can now apply the tools of game theory to this scenario, and the result is that the most advantageous and rational course of action (relative to the basic motivation of survival) for each civilisation is to eliminate all known competitors as soon as they are discovered. Conversely, in order to avoid getting eliminated yourself by others, you remain silent and undetectable. So this isn’t about conquering others, but about annihilating them from the game - which can be done easily and effectively from a distance, once you know where they are. In fact, you would want to do this so long as they are still largely planet-bound. The resulting situation would be a galaxy that is potentially full of advanced races, but appears empty and devoid of intelligent life, since these races use their technologies to ensure that their presence remains undetectable by all others. Within such a state of affairs, a newcomer to the scene such as ourselves, who is unaware of the situation and is loudly advertising our presence through various acts of carelessness, will eventually find themselves getting annihilated. This is a pretty bleak scenario, and the question then remains whether such civilisations are subject to ethical considerations that are strong enough to override following the most rational course of action as described above. An interesting conundrum.

Hm. To be honest, I think to a neutral and external observer, the current state of humanity mightn’t look so appealing. There’s a lot of greed, hatred, delusion and ignorance. A very large percentage of our current population still lives in abject poverty. At any given time, there’ll be something like half a dozen armed conflicts in progress in different parts of the world. There are at present on the order of 13000 nuclear warheads available for launch (~90% of which in just two countries). Nationalism, militarism and various forms of religious and cultural extremism are rife everywhere. Even in developed nations with good education systems, the average person’s knowledge of scientific, philosophical, ethical, sociological and teleological concepts is at best rudimentary. Of course, our species has also achieved many positive things in the sciences, humanities, arts, etc etc - but it would seem to me that these sadly are not representative of our overall state of affairs, at least not at this point in time. Would I, as being an advanced civilisation, and based on what I am observing here on Earth right now, be inclined to share my knowledge and technology? I think probably not. A ultra-sharp kitchen knife is a really useful tool that enables you to cook a delicious meal - but you wouldn’t hand it to a toddler, would you?

That’s true, though it is not unreasonable to assume that at a minimum we share the basic motivation to ensure survival of our species. All known forms of life on Earth - irrespective of where and how they evolved - share this. Without such “motivation” (or more generally: biological mechanism), it is doubtful that any form of life would ever reach the stage of becoming a civilisation. I think such discussions serve no useful purpose, because we simply have no data available to us to base it on. It’s pretty much all conjecture. The only reason why I personally tend towards Dark Forest is that it is based on a minimal set of basic assumptions, along with the mathematical tools of game theory; it’s the most mathematical conclusion stemming from the smallest possible set of assumptions. I’m not saying that Dark Forest must be right; only that, among a vast array of guesswork and conjecture, it is presently the most scientific approach we have to this subject - though admittedly the outcome isn’t exactly pretty, and one might even say that it is somewhat scary. It would, however, resolve Fermi’s paradox quite neatly.

Yes. By “opposite” I simply meant the sign; the squared interval is the difference between the squared time part and the squared space part. The crucial bit is that whatever change occurs, it can only be such that the overall interval remains invariant.

https://www.quantamagazine.org/asymmetry-detected-in-the-distribution-of-galaxies-20221205/ https://arxiv.org/abs/2206.03625 Now this is very interesting! Despite the 7-sigma significance mentioned in the paper, this will of course require further corroboration. But if this turns out to be a real thing, then it would have far-reaching consequences. Definitely one to watch!

They do count as indications of some unusual event having taken place, but on their own they don’t allow us to conclude that said event was of extraterrestrial origin. As I said, the fact that there are unidentifiable objects and events out there isn’t in contention. No. As I have tried to explain, my assumption is that those photos which can be taken to be genuine (not photoshopped, not an intentional hoax) show unidentified objects. My problem is only that too many people jump to the conclusion “unidentified”=“extraterrestrial”, because this does not automatically follow. I agree with you, and at no point am I completely dismissing the possibility of us being visited by another civilisation. The probability for such a thing is not zero. I do maintain, however, that that possibility is extremely remote, even if intelligent civilisations are commonplace - which is why substantiating that claim requires some very hard evidence. Ultimately, to gain complete certainty that this is what is taking place, you’d need both one their crafts/probes to examine, as well as a specimen sample of alien biology. What we are seeing from all these reports really isn’t very consistent with how I’d expect an advanced technological race on an observational and scientific mission to behave - at a minimum they’d remain out of sight and detection, so as not to contaminate their observational data. Unless of course they don’t care for some reason, as in the scenario put forward in the fictional work Roadside Picnic by the Strugatsky brothers, but that’s even less probable. In fact, I see no reason why they would need to physically come here at all, since surely someone who has mastered interstellar travel would also have mastered observational techniques that enable them to collect the data that is of interest to them from a distance, as we ourselves are slowly starting to do with instruments such as the JWST. That’s a lot less resource intensive, which is likely an important consideration no matter how advanced you are. There are also important reasons to think that it might be a wise policy for you not to advertise your presence to anyone, as in Dark Forest Theory for example. I’m personally a bit partial to this.

Yes, I agree, that’s really all I am trying to say. It’s the uncritical assumption that “unidentified”=“extraterrestrial” that I have a problem with. To clarify my personal stance on this - I think it is unlikely that Earth is the only place in the universe (which is big) where some form of life has evolved, especially in light of recent discoveries about exoplanets and how prevalent they really are. I would nearly go so far as to say that some form of life being out there is extremely likely (just my personal opinion though), though that would of course overwhelmingly be primitive forms of life. Even primitive life elsewhere in our own solar system is not out of the question. How likely it is that some of these might have evolved into intelligent civilisations is a different - and much harder - question to answer. I don’t know what to think about this, tbh, because there really isn’t enough information available to us to come to an informed opinion, because we have only one single data point available, so we don’t know what causes and conditions must be in place for this to happen. My personal feeling is that yes, there might be other civilisations out there, but they will be rare and correspondingly far apart from one another. I wouldn’t be surprised at all if it turned out that we are (at present) the only ones within our galaxy; more likely I’d guesstimate there might perhaps at most be 5-6, including ourselves, at different stages of techno-cultural development. It is wise to remember that timing is of crucial importance here - in our own case, the difference between humanity in the year 1700 and in the year 2000 is huge so far as technological development is concerned, and 300 years is a tiny fraction of a blink of an eye on cosmic scales. So chances are that, if there’s anyone else out there in our general vicinity, they are either far ahead or far behind us, statistically speaking. The next question then is how many of these 5 or so civilisations are spacefaring - that’s basically impossible to answer, mostly because we cannot guess as to the psychological motivators such an alien civilisation is subject to. In our case, we are driven by curiosity and an urge to know things, as well as economic, military and religious considerations, so engaging in space travel would be pretty much natural to us. But again, that’s a single data point - we have no way of knowing if other civilisations are driven by similar concerns. Perhaps we are an outlier, and the normal state of affairs is that civilisations just aren’t interested in what anyone else is up to (unless they are directly threatened etc, which is a whole different scenario), and instead pre-occupied with their own concerns (which might be centred around local cultural, philosophical, religious etc pursuits, rather than exploration) - even if they do in principle have the technological ability for interstellar travel. Even among those who have both the ability and the willingness to engage in interstellar travel - how likely is it that someone will actually hop onto a spacecraft (or send unmanned probes) and spend 100s or 1000s of years travelling, never even mind the amount of time it would take to communicate their findings back home? And given the large variety of very differently-looking objects we are seeing on all those many UFO pictures, how likely is it that many different civilisations are visiting us simultaneously, each using their own unique technology? The entire concept just doesn’t seem plausible to me, unless I’m wrong and the galaxy really is teeming with intelligent civilisations who have somehow found ways to work around the numerous challenges of interstellar travel. I can’t rule this out of course - perhaps the evolution of intelligence, technology and culture really is commonplace. But then I would have expected them to make sure their presence here remains undetected by us (which wouldn’t be hard for someone who has mastered interstellar travel), instead of blatantly zipping about Earth in their disk-shaped crafts, plainly visible to anyone of us who cares to look. This makes no sense, no matter how you look at it, because it risks contaminating any data they have painstakingly come to here to collect. Thus, I am all for the idea of extraterrestrial life in general - but I’m very sceptical about alien civilisations physically visiting us (and why not just observe from a distance, using their far superior technology?). That’s quite an extraordinary claim that will require equally extraordinary evidence to substantiate it.

None of these points are “done” even remotely - that’s the problem. I think I was quite explicit in that the only admissible evidence here is of the physical kind; due to the magnitude of the overall claim (these objects being extraterrestrial), eyewitness accounts, photos, videos etc just won’t cut it here. We need physical evidence that can be examined independently by public research institutions. Something being unidentified doesn’t imply it must be ‘extraordinary’, in the sense of the claim being presented here. It just means exactly what it says on the tin - that the object couldn’t be readily identified. A photo being ‘genuine’ (ie not photoshopped) does not imply the object shown being of extraterrestrial origin. A hubcap flying sideways off a speeding car on the interstate and being photographed by someone in the adjacent field who doesn’t realise that it is in fact a hubcap in the process of flying off, would look much like ~95% of UFO photos out there; not photoshopped, not an intentional hoax, but crucially also not an extraterrestrial craft. You are still missing the point. No one has ever denied that there are sightings of objects that, based on circumstances and available technology at the time of the sighting, couldn’t be readily identified. That is not in contention. The problem is the assertion that, because we couldn’t identify them unambiguously at the time, they must be of extraterrestrial origin, because that does not follow at all.

But when you see a plane the make/model and livery of which you can’t immediately identify, you wouldn’t automatically assume it is of extraterrestrial origin just based on your inability to readily identify it, would you? That would be silly. The existence of unidentified objects in the air (and also under water) isn’t controversial - what is controversial are all the assumptions that people jump to when encountering such a thing: 1. They cannot be natural phenomena, based on some perceived inability to explain their characteristics or behaviour 2. They must thus be engineered and constructed machines 3. The creators of these machines must be intelligent beings 4. The intelligent beings who built these machines cannot be human 5. The non-human builders cannot be of terrestrial origin 6. The extraterrestrial builders use these machines for interstellar travel So far as I am concerned, the fact that there are numerous sightings (visual, radar,…) of objects in air and water that are difficult to identify, along with media such as photos and videos depicting such sightings, is not in contention. What is also not in contention is the possibility of life having arisen elsewhere (personally I would be very surprised if it hadn’t), and thus the - much smaller - possibility of some such life being intelligent. There may then also be a - very tiny - possibility of at least some such intelligent races engaging in interstellar travel. However, connecting sightings of hitherto unidentified aerial and marine objects with extraterrestrial spacefarers (robotic or in-person) is very much in contention, because this conclusion simply does not follow. The probability of unidentified objects (unidentified based on circumstance, current technology, and knowledge of the observer, mind you) actually being of extraterrestrial origin is in fact much, much smaller than the probability of extraterrestrial civilisations existing per se. Unidentified objects are simply just that - hitherto unidentified objects. “Unidentified” does not necessarily imply extraterrestrial - that’s a very extraordinary claim. Thus, so far as I personally am concerned, the only admissible evidence that would allow me to actually make this connection is evidence that: 1. Is in the public domain, in the sense that independent groups of researches have ready access to it, which enables application of the scientific method 2. Explicitly excludes all more probable explanations, ie misidentification, natural phenomena, experimental (terrestrial) technologies etc etc 3. Provides an explicit example of technology or biology that is manifestly not terrestrial This kind of evidence would thus have to be of the physical kind (photos, eyewitness accounts, leaked papers, radar sightings etc etc simply aren’t good enough here, and never will be, no matter how good the quality), and would at a minimum include all or part of one of those alleged machines. Ideally it would consist of a specimen of some form of extraterrestrial organism, but that’s not necessarily required - though it would make the evidence pretty much incontrovertible. So yes, that’s a pretty high standard - but then, the original claim is extraordinary, so the evidence to support it will have to be likewise, especially if one considers the scientific, philosophical, cultural and religious implications of such a find. You simply cannot be lax and laisser-faire with such a pivotal issue.

Of course there is - it requires both a quantum channel and a classical channel. The speed is of course relative to a given observer in vacuum. The point here is that it doesn’t matter which observer you choose, the numerical value of said speed will always come out the same, ie it is invariant. Yes, I said the statistical correlation is fixed from the start (which is what ‘entanglement’ means) - I did not say that the direction of the spin is fixed. Those are not the same things at all, and they are in fact mutually exclusive, as I have already explained in some detail; you couldn’t have entanglement with a fully determined spin at the same time. I have been very explicit on several occasions about this lack of local realism. This thread is really about entanglement, not teleportation, these two being related but distinct concepts. If you wish to discuss quantum teleportation, it would be best if you opened a new thread. There is no ‘signal’ of any kind - what gives the spin direction a definite value (as opposed to a statistical probability) is the act of measuring it, meaning the process of it becoming coupled to suitable degrees of freedom of its immediate environment. Which is, of course, always a purely local process, irrespective of whether or not the particle is entangled at all.

Just to add to joigus’ excellent post (+1) - spin also characterises how the mathematical object that represents the particle in question behaves under Lorentz transformations; it reflects what class of mathematical object you need to describe all relevant degrees of freedom of such a particle. Spin-0 particles are described by scalars, spin-½ particles by spinors, spin-1 particles by vectors, and spin-2 particles by rank-2 tensors. Other spins (such as 3/2 or 5/2) would correspond to multi-component spinors. These classes of objects are representations of the Lorentz group, and are valid solutions to the Bargmann-Wigner equations. Thus, spin is intrinsically a relativistic phenomenon - in a spacetime with Euclidean geometry, particles wouldn’t have spin. The elementary particles we know of have spins 0, ½ or 1; it is also possible to create composite particles with other spins such as 3/2 or 5/2. In flat spacetime, the Weinberg no-go theorem and the Coleman-Mandula theorem imply that no elementary particles with spin-3 or greater can exist.

As a general reminder, I think it is wise to remember that the concepts of ‘particle’ and ‘field’ are first and foremost mathematical models - to be precise, they are irreducible representations of symmetry groups. In physics, when we say that something ‘is’ a particle or field, then the verb ‘is’ should not be understood in a strictly ontological manner, but rather in the sense that these concepts are useful models for some observable aspect of reality. They are ways to make useful and functional models of the world, not irreducible ontologies. Consider the following effects: 1. The Unruh effect: for a given volume of space, the number of particles contained therein depends on the observer. An accelerated observer will figure that the volume contains more particles than does an inertial observer, even though they are looking at the same 3-volume. 2. The Aharonov-Bohm effect: there are measurable physical effects on charges due to an electromagnetic potential, even though the charges are located in a region where the electromagnetic field is absent (F=0). So clearly, our naive intuitions of what ‘particle’ and ‘field’ might mean can be very misleading. Ultimately, the question as to what exists in an irreducible way on the most fundamental level is very much open, in physics as well as philosophy. To me, it isn’t even clear what ‘exists’ and ‘most fundamental’ actually means, since these terms would need to be defined in a completely observer-independent way, which is difficult to say the least. Modern physics takes quantum fields and spacetime to be the most fundamental categories of (physical) existence that we know of, but there is certainly no claim that there couldn’t be something else that is even more fundamental - in fact, it is quite safe to claim that there most likely will be. There are also other ways to make models of what we already know - Rovelli’s relational QM, where the central category is a network of interactions rather than a collection of entities with given properties, is a good example. With all that being said, particles and fields are far to useful as computational models in order to simply abandon them.

Yes, precisely. Entanglement requires the absence of counterfactual definiteness - once the property in question can be given a definite state (no matter if we know what this state is or not), entanglement cannot exist. All these principles remain valid irrespective of the geometry of the background spacetime; even if we allow one of the entangled particles to (e.g.) fall past an event horizon, the entanglement relationship will persist. Another important observation (that may or may not have been mentioned already on this thread) is that placing a barrier between the entangled particles does not in any way affect the statistical correlation between them. This is irrespective of the precise nature of the barrier.

I’m afraid I don’t have much to offer on this either. Educated guess: first of all, the mathematical form of an entangled state does not in any way depend on either the metric of the background spacetime, nor on the spatial trajectories of the particle pair. Hence I don’t see any a priori reason why simply following a geodesic path (which is what “scattering” on a BH really is) should in itself break the entanglement relationship. In order to break the entanglement, you need to interact with any part of the entangled system in a way that constitutes a measurement of whatever property has been entangled. For example, if you have a pair of particles whose spins have been entangled, then the way to break that correlation is to do something to either of these particles that allows you to figure out its direction of spin, which (if you follow the Copenhagen interpretation) “collapses the wave function”. Just scattering these particles in a curved spacetime does not itself allow you to figure out their spin directions, so in this case the entanglement should persist. By “you” I mean the environment, so by “measurement” I mean that the entangled system’s respective degree of freedom in question becomes coupled to a suitable degree of freedom in its environment. Thus, even if the two entangled particles meet again, the entanglement should persist so long as the system remains isolated from its environment. That’s my thoughts on this - someone correct me please if I’m overlooking something obvious here.

It’s a very interesting approach, thanks for sharing +1 I’m definitely looking forward to seeing how this develops further, and personally I’m quite open to an approach such as this - on the other hand, extraordinary claims require extraordinary evidence, and just doing away with unitarity as a fundamental symmetry is most certainly extraordinary. So I’m open to this, but not in any way convinced.

Precisely +1

In the quantum realm, when you talk to a wall, there is a small but non-zero probability that at least some of what you say gets through to the other side: \[T^{-1} =1+\frac{V_{0}^{2}}{4E( E+V_{0})} sin^{2}\left(\frac{2a}{\hbar }\sqrt{2m( E+V_{0}}\right)\] This thread here, alas, appears to be a purely classical situation.

Quantum entanglement is also a series of local events, starting from the instant when the entanglement was first created - joigus was quite right in bringing up this (classical) analogy. The difference between them is simply counterfactual definiteness.

No it doesn’t - the probability of measuring either state on any of the two particles is exactly ½, and that’s a fixed probability and not a function of some distant variable. It’s both constant and invariant. There’s no meaningful sense in which you can call this an example of non-locality, because the probability isn’t a function of anything, it’s simply constant. To state this differently, there is no local experiment you can perform that will tell you whether or not a particle is entangled, because no local measurement you perform depends in any way on distant coordinates. You will always observe a probability of ½ for either local outcome of a binary state, regardless of whether the particle you perform the measurement on is entangled or not - and, more importantly, regardless of what happens anywhere else in the universe. It is only when you compare the outcomes - which you can only physically do in accord with the laws of SR - that you see the correlation. And of course, that correlation exists as the result of a purely local process in the past, which is how the entanglement came to be in the first place. If the situation were non-local, then it wouldn’t be possible to get a statistical probability of exactly ½, because the outcome would explicitly depend on some distantly variable. This is evidently not so. No you can’t, unless you already know that the particle is entangled - which you can only do if you were either present in the past when the entanglement was created, or if you bring together and compare the outcomes of measurements in the present. Both cases ensure that your prediction/correlation remains a purely local process - if neither of these local cases apply, then the probability of the distant measurement coming out in any particular way remains ½. IOW, if you don’t know whether or not a local particle is entangled, then you can’t predict the outcome of the distant measurement with certainty; and conversely, if you can predict the distant outcome with certainty, then that means the information about the pre-existing correlation is already available to you, thus ensuring that the prediction is a local process. Not only is entanglement not an example of non-locality, but non-locality and entanglement are mutually exclusive concepts! Yes, exactly - and that’s not the same as non-locality. See above, because none of the probabilities involved in the measurement outcomes is a function of distant coordinates - the correlation between outcomes exists because these particles underwent a local process in the past whereby they became entangled, and have remained that way ever since. This whole thing is a good demonstration of counterfactual definiteness, the absence of local realism, and the non-separability of specific classes of wave functions - but not of non-locality.

The original Banach-Tarski “paradox” explicitly concerns a solid ball, not a sphere, in Euclidean 3-space (see first sentence): https://en.wikipedia.org/wiki/Banach–Tarski_paradox Not that this really makes any difference, because any 2D surface element of a sphere can likewise be considered a non-measurable ensemble of points, in which case the surface area of that subset wouldn’t be self-consistently defined either. So you could - using a similar process as in the original “paradox” - deconstruct a sphere and re-construct it in some other way, without the original surface area being conserved. There still isn’t an inconsistency, because this has to do with measures.