Hello Michael and Friends,

In the thread above, I had mentioned smooth flows on S3 and S7 resembling a superfluid, as a smooth flow initiated anywhere will come to cover the entire surface of the object (which is elucidated in Joy's book). I had previously mentioned that the geometry of spacetime seems to change, as we approach the Planck scale, becoming first non-commutative and then non-associative, conjecturing the appearance of an octonionic phase. I'm thinking this will be relevant to your work, Michael.

I have provided a link in the long thread above, to a set of slides from a talk given at Brookhaven and Stonybrook, about a Chiral Superfluid on a Holographic Boundary in the Quark-Gluon Plasma. I think this work will be of interest. My guess is that the parallelized spheres in Joy's construction would induce a chiral flow, because they are twisted to encode a torsional component. But thinking further; I also imagine that eddys will never close to a point on the surface - which seems quizzical - so the relevance of these thoughts, and the properties of the QGP, to the theories of Michael and Joy deserve discussion.

Regards,

Jonathan

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    • Post #363

    Jonathan,

    What gets me is the extent to which temperature is a far more complex and varied dynamic than time, but time seems the more mysterious. Why? Are we missing something?

    I just want to point out that a (S0, S1, S3, S7) universe is not just a proposal. I claim that it is the conclusion when we follow the spirit of Rick's standard of letting the algebras do the talking. However, Rick's choice of listening to the octonions violates the meta-principle of "make no preference". Instead, we should listen to all the normed division algebras R, C, H, O and let the octonions tell us their role. Now as a physicist, I want some experimental facts, but I will only need the Standard Model table for left-handed particles and the mismatch with their right-handed counterparts. table

    I claim that you can read-off the structure of the reality in which these particles exist from the properties of the normed division algebras given in John Baez's paper - attached for easy reference. I really do mean read-off, as in number of calculations = 0. The first things to note are the left and right-handed spinor irreducible representations in 4 (quaternions) and 8 dimensions (octonions) in Table 4 (p161 or pg17 of PDF), and the normed trialities of irreducible representations that give the quaternions and octonions on p162 or pg18 of PDF. Specifically note the triality of the irreducible representations V_8, S_8, S-_8 and their relationship to the Dynkin diagram D4 (on p163 or pg19 of PDF) of Spin(8) described on p162 (pg18 of PDF).

    Now count up the number of particles in the SM table: in each column there are 2 quarks with 3 colours plus a lepton and lepton neutrino, giving 8. This matches the triality of the 8-dim irreducible representations V_8, S_8, S-_8 and so the octonions tells us that they are about the different particle charges: octonions=particle space, just as quatenions=real space. Returning to Table 4, we see that left and right handed spinors only occur for the octonions and quaternions. Now to get the 8-dim octonion spinors to have the spatial handedness of quaternion spinors - and so match the chirality of the SM particle table - the octonion space would need to be mapped to the quaternion space so as to acquire their spatial handedness. This requires first picking out a 3-vector from the octonion space - which is something that has to be done in order to define the cross-product in 7D space residing in the octonions - and then map it to spatial 3-space. This mapping of a 3-vector in the octonion space to 3-space gives the 3 8-dim irreducible representations a spatial chirality, and breaks the symmetry of the octonion space. Thus the normed division algebras have just told us what the Higgs field is really all about.

    Returning to the D4 Dynkin diagram on p163 (pg19 of PDF) to consider what the symmetry breaking for the SM particles must be, using the Dynkin diagrams: SU(2) is 1 node; SU(3) is 2 linked nodes; and SU(4) is 3 linked nodes. If we imagine breaking all the links of D4, we would have a central SU(2) and triality involving an outer SU(2) ~Spin(3), plus a U(1) symmetry between the central and outer SU(2) groups. Now symmetry breaking in a Dynkin diagram involves removing a node, which here would be the central SU(2), leaving intact the triality over the outer SU(2)~Spin(3) and the U(1) symmetry. A 3-dim colour representation - needed to get particles in the 8-dim of V_8, S_8, S-_8 - selects Spin(3) over SU(2), giving the symmetry breaking encoded in the SM particle table as:

    Spin(3)*SU(2)*U(1) -> Spin(3)*U(1)

    The normed division algebras are telling us that the colour group cannot be SU(3), but is actually Spin(3). The condition for getting particle-like objects in any symmetry breaking pattern demands closed spaces, which here means S7 -> S3*S3*S1 and the monopole homotopy group PI6(S2) = Z4*Z3 (S6 being left after the unbroken S1 is put to one side) confirms the triality of 3 families of 4 particles in the SM table. The homotopy group PI7(S3) =Z2 also confirms the chirality of the mapping from the closed octonion space (particle space S7) to the closed quaternion space (closed spatial universe S3). These homotopy groups obviously come from the structure of the corresponding algebras.

    Given the SM particle table, the normed division algebras are screaming out the structure of reality to anyone who is listening. Arguing against this conclusion is not arguing against me, but arguing against the fundamental fermions and the structure of the normed division algebras - that's an argument that's lost before it's even begun. Structure of reality: DONE.

    That's the easy way of how we arrive at what the structure of reality is, now how did the universe arrive at the answer?

    MichaelAttachment #1: Octonions.pdf

    Hello again,

    I note that the comment on a superfluid phase in the QGP, and the links above, considerably extend what I said on Dec. 6 at 18:14 GMT. I remarked that in his book, Joy explains "that a smooth flow initiated on the 3-sphere or 7-sphere will cover or come to involve the whole surface (as it is simply-connected)." Then I commented "This is analogous to the behavior of a superfluid. A recent paper by Bonder, Sudarsky, and Aguilar, suggests that a 'hydrodynamic' treatment of spacetime leads to a formulation of gravity that is Lorentz invariant. I think there should be a superfluid phase, however." The linked work of Kalaydzhyan certainly delves into that subject.

    Does the analogy of a smooth flow on 3-sheres and 7-spheres with a superfluid make sense? Would Joy's non-trivial twist in the topology of the parallelized 7-sphere induce chirality? Does the notion from my Mandelbrot Cosmology - that chirality comes from the symmetry breaking induced by choosing a direction in time - serve to explain things, or does it complicate the picture? It is to be noted that cosmologies based on M predict that there is a maximal extent and minimal temperature of the cosmos - making it a closed universe. It leaves open the question of whether it is terminal or cyclical, but tends to imply the latter.

    At this point; I am unclear whether my work with M has a definite connection with the ideas of Michael and Joy discussed here (so far as they are concerned). But I thought I'd put the idea out there, when I saw a clear connection with the discussion on symmetry breaking, as I am certain it will be a subject of active research for me.

    Kind Regards,

    Jonathan

    Hi Jonathan,

    In my full 11D GR theory, the geometry of spacetime would be expected to change as we approach the Planck scale as equivalence between the spatial and particle dimensions is restored. Unfortunately, in the full geometric theory this is a something of a nightmare, as the maths is horrific, with no obvious simplifying principle. Then there is my third-order Relativity where the definitions of physical quantities changes as we decrease in scale because the effective number of spatial dimensions changes.

    I follow the standard Kaluza-Klein reformulation of the full dimensional theory in a dimensionally reduced form, where the result looks like the Standard Model Lagrangian. This effectively gives 2 separate phases considered by the theory: the full 11D geometry away from the compactification; and the dimensionally reduced phase with compactification. Between the 2 is currently a sort of no-man's land I cannot see how to deal with. Even this clear division creates a problem for the dimensionally reduced theory - it is mathematically incomplete, in that there can exist features in the full dimensionally theory that cannot be derived in the dimensionally reduced theory. The wave property of the monopoles is such a property, which I added by hand in my paper, but I'm now fairly sure I will be able to derive it in the full theory using topological arguments. I currently seem to be the only one to acknowledge the existence of mathematical incompleteness in physics, and be prepared to just man-up and deal with it. Which is why I can derive QFT.

    Best,

    Michael

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    • Post #367

    Wow Michael!

    Your comment above at 16:44 GMT is awesome and much appreciated. I'll have a few things to say. First; I want to cross-reference it to the Baez paper - which I've probably already read - once I sign off on this comment. Thanks greatly for providing such an excellent road map.

    All the Best,

    Jonathan

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    • Post #368

    Hi Michael,

    I don't think it's just a matter of being man enough to deal with incompleteness, and tackle it you do, but you may be the first to notice that the doors have signs that says "Pull" and everyone before you has been trying to "Push" instead. I like the way use use Gödel's result as a door, rather than a window. To my view; you exploit incompleteness as a way to define the existence of essential degrees of freedom, where others view it as a limitation on possibilities only.

    This is decidedly a different way to proceed.

    All the Best,

    Jonathan

    Hi Michael,

    Thanks for the nice summary. If what you say is true, and if arguing against your conclusion is arguing against the fundamental fermions and the structure of the normed division algebras, then I am somewhat disappointed. Because I am convinced that despite your insistence and your overall philosophical justifications to the contrary your theory is manifestly non-local.

    It can be made to be local, I hope, by tweaking it somewhere to accommodate my framework in such a manner that you can explicitly write down the four local functions, namely A(a, L), B(b, L), C(c, L), and D(d, L), for the four-particle GHZ state. The latter is of course just a simplest non-trivial example. What I mean is that if and only if you are able to accommodate the nested S7 of the kind both Rick and I are advocating (differently) can your theory be local. That is my claim in any case.

    I think it would be worthwhile if you reconsidered a theory based on S15 instead of S10. For S7 is uniquely linked to S15 octonions through a Hopf fibration of the latter by S8 (see, for example, the attached paper). Just a thought.

    Best,

    JoyAttachment #1: 1248.pdf

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    • Post #370

    Michael, Jonathan,

    Just a simple, basic thought here, but it seems math is inherently dynamic, while we look for static structures in it.

    Add, subtract, multiply, divide, etc. are verbs, not nouns.

    What seems to be, is this scale of complexity, that as you proceed upward, gathers ever more rules and structures. In the real world, this would make them more brittle, static and unstable. Could there be mathematical feedback loops occurring that our linear thought processes are not perceiving?

    That is an excellent comment John,

    And in my view, all of the static structures arise out of the dynamism of Maths. I tend to think that the dynamic number types octonions, and then quaternions, appeared first - and then this ultimately made the static Reals possible. People are used to numbers that just sit there, from normal Arithmetic, but this is atypical in higher Maths and as you point out - even the simple operations of add, subtract, multiply, and divide, bespeak the dynamism of Mathematics instead.

    And yes; that was my comment above, but I guess my login expired.

    All the Best,

    Jonathan

    Hi Michael,

    R, C and H are subalgebras of O, so if one "listens to the octonions", they very much are also listening to R, C and H. There is no preferential treatment for O over R, C and H at all really. There are however preferential choices made out of the clear vision both of us have on the supremacy of the division algebras, with O being the mother of all. I assume this position without apology.

    On H being "real space" and O being "particle space" I am OK with "particle space" if it is a statement of the fact it is the fundamental space (for me more *algebra* requiring the *space*) of reality. But I have shown gravitation to reside within an H subalgebra and electrodynamic observables reside within what John Baez calls a basic triple, which is not not an H triplet. O has 7 H subalgebras, and we can't assign all to be physical xyz triplets. Any of the 7 may be taken as one side of the physical space coin, so there is no *preferred* choice but we must make a choice none the less. Once again after this free choice is made we have 4 free choices of which 3 of the remaining 4 basis elements also map to the other side of the physical space coin. Traditionally, physical xyz is taken to be a "polar" representation, so it more closely (parallel projection) aligns with the basic triple than it does the chosen H triplet, which has multiplicative closure as do "axial" types. The left over basis element forms a C subalgebra and if we require everything to be analytic in this subalgebra, the 8D stuff cleanly partitions into the sum of what looks like 2 4D similar structures.

    I see your approach as not bothering with reading Nature's book from start to finish, but jumping ahead to the last few chapters. You fill in what you missed in the middle with fanciful notions like force = particle exchange, and reality *is* the symmetry groups rather than *descriptive of*, in order to make sense out of what you find. But I listen to your perspective with great interest, for it helps me put into context what I read as I progress one paragraph at a time to the same conclusion.

    Rick

    Hi Joy and Rick,

    As I have dimensional compactification, my theory has 2 distinct forms: the full dimensional GR which is manifestly local; and the dimensionally reduced phase for which the Lagrangian is still manifestly local. However, this doesn't include the topological monopoles that arise in the transition. These have to be added as classical physics objects subject to geodesic parallel transport, which can be done in a manifestly local way. Then comes the problem with the dimensionally reduced description, that it misses a bit. I am now fairly sure that the wave property (S1) of the topological monopoles (S0) can be arrived at by topological arguments in the full theory (to give the Hopf fibre-bundle S1), but this doesn't transfer to the dimensionally reduced theory and cannot be included in a simple classical physics description of objects because the wave property clashes with the object description.

    My reading of the switch in description from discrete particles to continuous particle fields - in order to describe this wave-particle duality - is that the local causation of the underlying metric field theory isn't messed up, and it is just the description which is non-local. So I totally agree that the description is manifestly non-local, but to me this looks like the QT paradox of non-local identity without non-local causation. Can you elucidate whether this is the case? If I am wrong about this, then I agree that some sort of nesting of S7 would be required.

    Rick, I am primarily concerned with the difference between the description of reality and reality itself. The problem I have with picking out one the triples from the octonions as being x,y,z - which would break the equivalence between the different octonion sub-algebras and give you a symmetry breaking - is that the triality of the octonions is just screaming out 3-fold flavour symmetry of the particle families. It appears to me that both your approach and Geoffrey Dixon's don't get the 3 families falling out naturally. Have I missed something? Trying to combine 3 flavours with colour group SU(3) just doesn't seem to fit the normed division algebras. Can you make it fit?

    I agree that S15 has a special look about it - being the remaining Hopf sphere - and I have given it some thought, but the difficulty is the same as for R, C, H, O directly: the 15 dimensions don't seem to match up with our reality in a natural way. The extra 4 dimensions don't seem to resolve the colour-space disagreement with the SM either, as that would still only give 7D for colour and not the 8D of the SU(3) group space. But if I'm wrong on my non-local description but local causation interpretation, then the natural end-point for an S7 embedding framework could well be S15.

    However, reading off the SM chiral particles existing in spacetime with spin ½ against the structure of the normed division algebras still naturally proceeds as I gave it, when you read all the factors: 3 flavours of 8 differently charged particles with spin ½ existing in spacetime with a chiral imbalance between left and right spins. I am reading the SM particle table from start to finish, and when I get to the end there's no room to stuff spacetime into the octonions.

    Best,

    Michael

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    • Post #374

    Jonathan,

    What's the old saying, "The opposite of small truths are false, while the opposite of large truths are also true."

    While I see math as inherently dynamic, I would also see the static as inherent as well. Above you made the following observation;

    "I think time has the fewest rules, needing to adhere only to the nature of process. Space needs to follow rules of geometrical evolution. Energy drives evolutionary dynamism, but must follow rules of propagation and circulation. And matter must follow a few more rules, or is the most constrained."

    I would ask; Why are the rules of geometrical evolution foundational to space? What if we were to consider the opposite, that space is foundational to the rules of geometrical evolution?

    To space I would assign only two properties, equilibrium and infinity. The absolute and the infinite. With gravity, we have the attraction of equilibrium and with radiation, we have the attraction of the void, the infinite. All other properties of matter and energy, circulation, interaction, attraction, repulsion, etc. could be considered as being defined by the interactions of these two polarities of space. Matter is constrained because constrainment defines form. To limit is to define, as to define is to limit.

    Consider the assumption in this observation;

    " I tend to think that the dynamic number types octonions, and then quaternions, appeared first - and then this ultimately made the static Reals possible."

    If anything is "first," then the sequence of real numbers would have to be. So even there, you have that foundational dichotomy of "being" and "doing."

    Or as Frank Sinatra so eloquently put it, "Dobedobedobedo."

    Hi John,

    I agree with the spirit of your comments on bottom-up vs. top-down causation. The way I would phrase it would be that bottom-up dynamics of the parts of a system generates emergent features at the level of the whole system that act as boundary conditions for the dynamics, i.e. top-down constraints. The top-down boundary conditions then alters what the bottom-up dynamics can be, which then alters the emergent properties acting as the top-down boundary conditions etc. This cycles around until there is consistency between the bottom-up dynamics of the system parts and the whole system emergent property that acts as a top-down boundary condition. When the system is disturbed, it is pushed out of its self-consistent bottom-up/top-down state and cycles around again. This gives a view of a dynamic system responding to external disturbances by switching between different self-consistent bottom-up dynamic states with emergent top-down boundary conditions. An entirely sensible physics view.

    I also agree with your comment on the comparison between the operations of maths and natural language, and the significance of your question "Could there be mathematical feedback loops occurring that our linear thought processes are not perceiving?" If our linear thought processes parallel the discrete logical steps of maths then for certain types of physical dynamic system, the answer is provably Yes.

    The comparison between maths and language underlies Wittgenstein's Tractus Logico-Philosophicus, but he didn't get it right. What he needed was Gödel's mathematical incompleteness proof, and then to find the sort of physical system where the classical physics theory of the system meets Gödel's conditions. I did this here in a way that parallels Wittgenstein's Tractus as much as possible. In proving that these systems could possess statements describing them which can't be derived within the maths of classical physics, it is proven that there could exist statements about real physical systems expressed in natural-language that cannot be explained by our thought processes in the way you ask.

    The issue is then finding those physical systems and those statements. I have shown that the physical dynamics of the chemical processes of a living cell meet the required conditions, and it is noticeable that there is no definition of what life is. This makes the state of Life a leading candidate to actually be a real example of this ... the Mind itself is obviously another leading candidate, as is Nature, and the economist's Market state ...

    Best,

    Michael

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    • Post #376

    Hi Michael,

    The answer to your question to me is today, no. sometime down the road perhaps. Things do not need to connect up at this point of my progression, for it may not be a superficial manifestation. There is a fair amount of work remaining to provide the full foundation for dynamics, and I think I have the pieces in place with my replacement for the algebra of tensor calculus; the ensemble derivative, as well as the purely algebraic considerations of algebraic invariance and how say electrodynamic form invariance restricts realistic transformation possibilities to as I showed in my essay forces a Lorentz-like hyperbolic transformations without a split signature metric.

    Things may not connect up until the solutions to the potential functions for stable particles are in hand. The structure you seek may be manifested by the potential solutions themselves rather than the algebra that demanded them. If I already had this, I would not be keeping it to myself.

    Rick

    Hi Michael,

    There are at least four types of non-localities that have to be distinguished for clarity to understand what either of us is doing:

    (1) Signalling (or relativistic) non-locality,

    (2) No-signalling (or quantum) non-locality,

    (3) Bertlmann's socks non-locality, and

    (4) Post-quantum non-locality.

    The fourth non-locality is just a science fiction that some non-locality fans are indulging in. It not only violates quantum principles, but also goes beyond the logic of division algebras, as I explain in the Chapter 7 of my book. In other words, we can safely ignore this type of fictional non-locality.

    A violation of signalling locality---number (1)---would of course undermine your theory rather seriously, and I am glad to be reassured that your theory does not violate such a relativistic locality.

    The third form of non-locality---the Bertlmann's socks non-locality---is a totally benign non-locality. Suppose you step out of your house to go somewhere on a cold winter night and halfway through you check your pockets to find only a right-hand glove. You would then instantly and non-locally know for sure that the glove you left at home is a left-hand glove. I don't think anyone is shocked by this kind of non-locality.

    This leaves number (2)---the no-signalling non-locality. This a more benign form of non-locality than the signalling non-locality (because you cannot send a signal exploiting it). But it is less benign than the Bertlmann's socks non-locality. It is this peculiar form of non-locality that is harboured by both quantum mechanics and your theory. It may be benign compared to the signalling non-locality, but it is by no means less disturbing. Let me explain this in concrete terms.

    Let A = +1 or -1 be a result observed by Alice in an EPR-Bohm experiment. And similarly let B = +1 or -1 be a result observed by Bob. Now in general A and B could be described by functions of the form

    A = A(a, b, B, L)

    and

    B = B(a, b, A, L),

    where a and b are the experimental parameters used by Alice and Bob respectively, and L is the common cause. As written, Alice's measurement result A(a, b, B, L) not only depends on her own experimental parameter a and the common cause L, but also on Bob's measurement result B and his experimental parameter b. And similarly for Bob's result. Thus the joint measurement results of Alice and Bob exhibit violations of both signalling and no-signalling localities.

    Now you claim that your theory does not violate relativistic causality. That means that in your theory Alice's measurement result does not depend on Bob's experimental parameter and Bob's measurement result does not depend on Alice's experimental parameter. In other words, in your theory the following expressions for A and B should be able to reproduce the quantum correlation:

    A = A(a, B, L)

    and

    B = B(b, A, L).

    Note that A still depends on B and B still depends on A. So this description is still non-local, and this non-locality is not as benign as Bertlmann's socks non-locality. To be benign, Alice's result should not depend on whether Bob performs his experiment or not. So it is in this sense that both your theory and quantum theory are non-local. For them to be local the functions A and B must be reducible to the forms

    A = A(a, L)

    and

    B = B(b, L)

    while still reproducing the strong correlation. I claim that this is not possible in your theory as it stands.

    Best,

    Joy

    Hi Joy, all,

    Sorry, but I still don't see why your framework doesn't work in Michael's theory after reading this thread.

    Now, in my concept there is "non-local" behavior at the individual particle level because spacetime is being created at that level. If spacetime is being created at that level, then I don't think we can even use the term "non-local" to describe what is going on. But it is impossible for there to be any "spooky action at a distance" beyond say the distance of the electron compton wavelength or perhaps even small distance than that. So your framework for me does explain EPR type scenarios classically.

    So why does Michael's theory allow "spooky action at a distance"?

    Best,

    Fred

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    • Post #379

    Michael,

    The best example of what you describe is the relationship between organisms and ecosystems. Markets and the various public, corporate, and private entities making them up are a reflection of this organic relationship. In order to sustain existence, these entities need to keep adding that bottom up input, without it being overly disruptive. Life adapts to the internal and external disruptions by regenerating and resetting the genetic code.

    Any bounded system, no matter how large or small, is subject to this entropy and dissolution of energy. In an infinite, or open system though, conservation of energy trumps entropy. Energy lost to one system is being gained by others.

    Currently, for various reasons of observation and assumption, cosmology has devised an essentially closed cosmic model. With the initial singularity, inflation, dark energy and whatever dissipative process is proposed, it isn't hermetically sealed though. I think many aspects of this theory could be explained by other means, as the conceptual issue of the universe as an entity, rather than an environment is problematic from my view. For one thing, the background radiation would simply be the solution to Olber's paradox, if light is actually redshifted as an effect of expanding to fill the compounding volume of space. What does hold a quanta of light to a point particle anyway? Wouldn't it expand as the nature of light?

    Galaxies and their various groupings, would be the cosmic structure in the environment of space.

    Having gotten that particular obsessive rant of mine out of the way, it does seem our formal logic is like those particular entities in a larger system. While the various models might seem superficially whole and complete in themselves, they tie into a larger and effectively unbounded logical system or environment. In my own entry in the recent contest, I conclude by arguing that perception and logic are inherently subjective. That combining different models may expand applicability, but it destroys specific information in the process. The generalized, vs. specialized dichotomy.

    • [deleted]

    • Post #380

    " The generalized, vs. specialized dichotomy."

    Which ties back into top down, vs. bottom up.....

    Hi Joy,

    I like your 4 classes of non-locality. When using the term non-local, I would add a fifth class for the distinction between the local structure of a manifold and its non-local, i.e. global, structure. This is because QT correlations in your hidden variable framework (HVF) are due to global (non-local) topological structure.

    The topological monopoles in my theory are necessarily around a hole in 3D space, which means that in the GR part of the dimensionally reduced theory the monopoles must be of the form of rotating (spin defects) black holes (event horizon marks the S2 boundary of the hole in space) and so have the Kerr metric. This necessarily means that they have an ergo-region, and this is why my theory - prior to the representational shift to QFT - is manifestly local, but may not appear so. This takes us back to the local vs. non-local issue we discussed in August and my PDF (attached) illustrating how manifestly local physics crossing from Minkowski space M4 to Euclidean space E4 (inside the ergo-region) and back again, appears to be non-local when interpreted solely - and incorrectly - from M4. In order to make the representational shift from these discrete objects to a continuous field description in flat M4 - i.e. QFT - this extended spacetime structure including an E4 region has to be approximated as being a point in M4. This necessarily results in a non-local description (non-local class 2) that is not really non-local (class 1).

    This is the reason for my unwavering insistence - much to your annoyance I know - that my theory is manifestly local. However, the manifestly local nature of the full theory isn't currently of any practical value, because I cannot relate the full dimensional geometric formulation to practical experimental results - which is your point. A shift in representation to continuous fields represents one path to a practical theory - namely QFT - but it isn't manifestly local because of the approximation involved. So I am correct in saying that my theory is manifestly local - before I get QFT - but you are correct in saying that I cannot currently describe the experiments in question in a manifestly local way.

    The second path to a practical theory - and I think ONLY other alternative to QFT - is to enclose my topological monopoles within an S2 surface, and treat it as a physical hidden domain described by the full geometric theory in a manifestly local way. For this to be a relativistic HVF that is manifestly local, the first condition is for spacetime to be M4 at the S2 surface, but for the S2 domain to contain an E4 region. This would give the required appearance of non-locality from the exterior M4 perspective outside of the S2 domain, but be manifestly local inside the S2 domain. For the conditions that spacetime be homogeneous, isotropic and continuous this M4-E4 condition looks like the only way to match QT within a HVF.

    I would argue that your HVF will have to meet this condition when it is formulated in relativistic form. In terms of practical formulation for your HVF, I think this brings us back to the issue you mentioned with the awkwardness of the spacetime split in the formulation of spacetime algebra for M4. The M4-E4 condition may require being able to denote both M4 and E4 within the same algebra framework. So instead of the S7 nesting you see in octonion terms, I see the main issue as being quaternion in character, i.e. spacetime.

    Best,

    MichaelAttachment #1: 3_Local_to_nonlocal.pdf