In part 1, I pointed out that in quantum theory (QT):

1. all states are represented as vectors, i.e., 0-dimensional points, in a Hilbert space, and
2. states acquire spatial extent by virtue of the fact that the Hilbert space dimensions are function-valued, specifically, as functions of space.

Point 1 means that in QT, states are represented in localist fashion, i.e., the representation of each state is formally (symbolically, mathematically) disjoint from that of every other state. Each pure (basis) state is formally a column vector (ket) and updating the state requires multiplying a current state, a row vector (bra), by each ket. Consider an N-qubit system. The system has 2^N states each of which is formally a linear combination over all 2^N states. Thus, the state update operation involves multiplying by a 2^N x 2^N matrix. Quantum computing is proposed to alleviate this by simultaneously performing all 2^N dot products (and renormalizing as necessary) so that the resulting state vector has L2 norm 1 (so that it can be interpreted as a probability distribution). The physical quantum computer consists of N physical objects, each representing a qubit. For example, a qubit might be an atom of XX held at near absolute zero. That is, the 2^N possible states are not explicitly physically realized. That would require 2^N physically disjoint memory locations, minimally 2^N bits, but in general, the state represented by each memory location, will contain far more information than a single bit. In any case, as stated, what is physically present in a QC, i.e., the part of the QC that represents the state of the simulated system, is just N physical objects. How then does the rest of the circuitry of a QC physically execute an update of the N-qubit register? In general, there will be a physical circuit for each operator (though the circuits can share common parts). For example, Fig. 1 shows a Hadamard gate. A Hadamard gate takes the state of the n-qubit register and places it in the superposition where all 2^N states have equal probability. How does that physically occur?

Note, we are not talking about the distinction between pure states (eigenstates) and mixed states (separable or not).

Any mixed state, X, is formally expressed as a linear superposition of eigenstates. Even a pure state is linear superposition, just with coefficient zero on all dimensions but one. But the zero terms can’t be ignored by the formalism, i.e., by the unitary operator (Schrodinger wave equation) that updates the state. That is formally multiplication by a matrix. but in any physical realization that emulates QT, i.e., that implements the unitary update operation that is the Schrodinger wave equation, the

Consequently, in any physical realization of the formalism, each state would require a unique physical representor, i.e., one that is physically disjoint from the representors of all other physical states.

Relation to ETS Theory of Lev Goldfarb and colleagues

Over the past 20 years, Lev Goldfarb and colleagues have been developing what they call the Evolving Transformation System (ETS) theory as a fundamentally new physical theory. ETS appears to have the same fundamental motivation, specifically, the inadequacy of the vector-based, thus point-based, mathematical formalism underlying QT. Note however, that Goldfarb explains that this fundamental formalism underlies not just QT, but literally the mathematization of all physical theories, from the beginnings of western math, i.e., Egyptians of 3000 BC. He also describes the millennia-old mathematical formalism as continuum-based, and explains that in his view, we need a fundamentally new formalism, one which is not extended from the extant continuum-centric formalism, but wholly new.

In response to the continuous mathematical formalism, Goldfarb called for a fundamentally discrete formalism, but moreover, one in which structure is fundamentally quantified. That is, in ETS, the new primitive is a unit of structure, not of scalar value. However, in contrast to ETS, as well to the mainstream, continuous formalism, in my theory, the primitive unit is a set, specifically, a sparse subset. Thus, my theory also has irreducible structure, i.e., the set representing an entity (or state) has no simpler (reduced) representation. If entity X is fully present, e.g., as an extant physical state in the physical version of the my theory, or as the active code (in a coding field) in my information-processing theory, then all Q fundamental elements that comprise (represent) X must be active. So, whereas Goldfarb chose to move from points (vectors) to structs, i.e., irreducible spatiotemporal flows of

The physical ground from which space and time emerge

In this talk, Nima Arkady-Hamed describes a new formalism for explaining the physical world. In particular, he asserts that the overall spacetime-like nature of the world only appears at macroscopic scales and that the lowest-level entities that comprise the physical world are individually not spacetime-like. This is fully consistent with the set-based physical theory that I’m developing. In this theory,

In my theory, there is a sense in which the physical law is probabilistic at the lowest functional scale, the corpuscle, but deterministic at super-corpuscle scales. This is because the set of codes of matter states will be different in any two corpuscles. Therefore, the set of mappings in the recurrent matrix of any given corpuscle, which implements (represents) the dynamics (physical law) that operates within that corpuscle, will also be different. Similarly, the set of mappings in each of the six matrices from any given corpuscle to its six face-connected neighboring corpuscles, which, in concert with the corpuscle’s recurrent matrix, complete the full implementation of physical law, will also differ from one corpuscle to the next. Nevertheless, the characteristics of that physical law, i.e., inertia, gravity, electromagnetism, etc., will be the same (even, identical up to some coarseness) in all corpuscles. So, the sense in which physical law is probabilistic at the corpuscle scale is that we can consider the codes that represent the matter states in each corpuscle to have been probabilistically chosen (probably using Sparsey’s CSA algorithm) a priori. Once those are chosen, the matrices are determined, again, since the mappings in those matrices, must implement the macroscopic physical laws we observe, i.e., any one of the matter state codes in a corpuscle must transition to a particular next code representing a matter state that is correct from the standpoint of that physical law.

Each of the matter state codes mentioned above is, fundamentally, just a set of active binary Planckons. It formally has the semantics of a set, i.e., order does not matter, because of the all-to-all connectivity of all the matrices. Thus, consistent with Arkady-Hamed’s views, these fundamental representations of matter do not, individually, exist in space and time. They exist outside of space and time. They are a physical ground from which space and time emerge. Spatial dimensions emerge as patterns of intersection across sets of these sets, in the way I’ve described in multiple places.

Now, I have to resolve something. The complete connectivity of a corpuscle’s matrices with its coding field clearly explains how space is an emergent phenomena within any single corpuscle. However, my theory requires that the macroscopic space we observe, i.e., the universe, is a tiling of corpuscles. My current visualization is that the universe is a cubic tiling of corpuscles. Thus, the three macroscopic spatial dimensions we observe is given a priori by this cubic tiling. Can we avoid this? We could…if we assumed the entire universe was one big corpuscle. In this case, there would likely not be a fundamental speed limit for transfer of effects across any space (of any dimension) that emerges from (in) that corpuscle. That’s because in this case, all matter states of the entire universe would correspond to codes over the same coding field (the same universal set). Thus, two matter states, one in which particle A existed at one extreme end of one dimension (at one side of the universe), and the other in which A was at the other extreme, would just be two different sets (of size Q) with low intersection size. More precisely, assuming there were other emergent dimensions in this universe, the two codes in question would not have zero intersection, amongst other things, simply because of the commonality that the both codes involve the same particle, A. So, if two codes represent the same particle, but at opposite sides of the universe, the we could imagine there being a mapping in the recurrent matrix (and in this case, there is only the recurrent matrix since there is only one corpuscle), that maps one state to the other, i.e., in a single update of the coding field. This would mean there is effectively no speed limit. All places are reachable in one time step. Of course, we could also imagine that the set of codes defined for this one-corpuscle universe are such that there are no mappings corresponding to such huge single-time-step movements. That is, a speed limit could effectively be enforced by the set of mappings.

So, why can’t, or shouldn’t, we assume that the universe is just one big corpuscle? Well, my initial intuition comes from considerations of the density of connections compared to the density of coding field units. The number of wired needed, or more generally, the amount of space needed by the connections for complete connectivity, grows as the square of the number of units that need to be connected. It just seems the small-world assumption is required in order to achieve some overall, possibly multi-hop, connectedness. But is this intuition not already presuming the emergent 3D nature of the world? Perhaps, this intuition introduces a red herring.

So if we stick with the assumption that the universe is a cubic tiling of corpuscles, then we are assuming that at the super-corpuscle scale, there are three spatial dimensions. The speed of light then is immediately determined by fact that information from one corpuscle takes one fundamental computational cycle, “clock tick”, to reach any neighbor (or recurrently, itself). Since the Planck length is the distance light travels in one Planck time, the diameter of the corpuscle in Planck lengths must equal the duration of the clock tick in Planck times. In this case, the speed of light is enforced across macroscopic, i.e., super-corpuscle, scales.

The prior paragraph introduced the notion of “clock tick” to this theory. Essentially, time is discrete. The smallest unit of time is the duration that the algorithm that updates the state of the corpuscle takes. Since in other writings, I’ve hypothesized that the corpuscle might, for example, be 10¹⁵ Planck lengths on a side, that means the smallest unit of time, the fundamental clock tick at which physical reality updates would be 10¹⁵ Planck times, many order of magnitude shorter than than the shortest duration yet measured. I do not currently have a clear or definite idea as to how the algorithm explicitly physically operates. For example, the CSA, requires that every coding field unit, i.e., flanckon, must compute its total input via all its impinging matrices on every time step. In an actual chip implementation of the CSA, we could imagine that that summation, which is of binary inputs, could occur as a purely physical step, i.e., as the total voltage on the impinging bus, where the other end of the bus is connected to all of the source units. So this step, which is computationally the most intensive step of the algorithm, is literally done by the physics. But in the physical theory, we are below the level where voltage, or even any particle, is a defined concept. Again, those are emergent phenomena. In this case, could there be some analog of voltage operative at the smallest scale. It seems that any formal description of such an analog might literally look just like the purely logical description, i.e., where the flanckon’s total input is simply the sum of active impinging blanckons, i.e., a counting process. But I seem to be assuming something behind (underneath) it all that is doing the counting, i.e., a computer that exists outside the physical ground I’ve already posited for the entire universe. This might not actually be a problem. After all, we will always, presumably, have the problem of not being able to explain the “prime mover”. In this case, my theory is absolutely compatible with the idea that our world is literally a simulation running on a computer in some larger universe. Note however, that the numbers I use in my other essay, actually leaves quite a lot of Planckons left over, which could conceivably be the physical ground of the processor in each corpuscle.

The three generations of particles

Now, what about the whole business about the analogy between a Sparsey hierarchy of levels of macs and the three generations of fundamental particles. I figured out how a 2D organization of macs, e.g., on a chip, could realize a hierarchy of tilings that allow shorter connection paths across 2D space. But I don’t know, i.e., can’t visualize, how this would work for the 3D case. Perhaps black holes correspond to “hubs” allowing shortcuts through space. Lot’s of work to do to visualize this. What’s needed to be able to visualize a 3D packing of corpuscles that would allow a higher-level corpuscle to be an immediate neighbor to many lower-level corpuscles. While for the 2D case, I can do this without requiring the higher-level corpuscles to be larger, i.e., larger tiles, than lower level corpuscles, I’m not sure this is possible for the 3D case. But maybe it doesn’t matter if we assume that higher-level corpuscles are larger than lower-level corpuscles. I mean…I can clearly visualize a 3D volume that has larger cubes embedded in a sea of smaller ones, and more generally, a range of different sizes of larger cubes embedded in the overall space.

Now, suppose we can imagine some type of packing of 3-space in which cubes (corpuscles) of two larger sizes, one for each of the two higher-energy particle generations, are embedded in a sea of the smallest size corpuscles. We must realize immediately that this means that we are saying that there are places in space where we cannot observe particles of any given generation. Now, it could be that corpuscles of the two larger sizes are intercalated at a fine enough scale that no experiment thus far will have detected such a constraint.

Note that assuming three different sizes of corpuscle does not allow, even if we can find an appropriate 3D intercalation, unlimited speed. Likely it would allow speeds up to perhaps two orders of magnitude higher than c, since, the two higher-energy particle generations are each about an order of magnitude larger than the prior generation. But, it could also be that even higher energy generations of particles exist, which would allow higher orders of magnitude speed increase over c, but we simply haven’t observed them yet. Note that in the brain model version of the hierarchy, I definitely think that the macs at the highest level span the entire input space (via the fan-in/out afforded by intervening levels). If the same is true for the physical theory version of the model, then there would in fact be a mechanism for, in one clock tick, traversing the entire universe. Oh what fun that would be!

What is Entanglement

The scientific use of word “entanglement” was introduced to the world by Schrodinger in 1935 to explain

More recently, we’ve seen entanglement described int the context of AdS/CFT, namely, stronger entanglement in the bulk corresponds to closer distance on the boundary. This corresponds nicely with my explanation of degree of entanglement corresponding to size of intersection. Two states are more entangled if simply if they have larger intersection.

And of course in the realm of information processing, entanglement is described as exactly the resource that is used by quantum computers to store information.

I have an extremely simple way to describe entanglement. It follows from an extremely simple way to represent information. Specifically, if items of information are represented by subsets, in particular, very sparse sets, all of the same cardinality, chosen from a common universal set, then the physical phenomenon described as entanglement is seen to be literally equivalent to the physical phenomena of set intersection. Two items of information, A and B, are entangled in proportion to the size of intersection of the sets, phi(A) and phi(B), which represent them. We refer to such a representation as a sparse distributed representation (SDR) or sparse distributed code (SDC). We’ll use SDC and refer to individual SDCs as codes, unless otherwise stated. Before focusing on explaining how the functionality of entanglement is achieved, w.l.o.g., in the context of an information processing model, and then bridging, by analogy, to a model of the physical world, I will first explain how the the concept of dimension naturally emerges from a set-based representation. Quite simply, a dimension emerges as a pattern of intersections over a set of sets, i.e., a set of codes, as shown in Fig. 1. Fig. 1 shows an SDC representational (coding) field and it shows five codes, i.e., for five items that we can assume to have been stored in the field. The intersections of the first item, A, with itself and the four other items, are shown at right. The intersection sizes are monotonically descending. We could call the dimension that is emergently represented by this pattern of intersections, “similarity to A”. Thus, A is most similar, in fact identical, to itself, and progressively less similar to the other four items. If you imagine for a moment that: a) some input field, i.e., a field of units receiving signals (energy) from an outside world, is connected to the coding field, say, via a complete matrix; and b) there is a method (algorithm, mechanism, process) by which inputs are assigned to codes so that some measure of similarity, S, in the input space is preserved–i.e., a method that, at least statistically, maps more similar inputs to more highly intersecting codes–then we would call the dimension represented by the emergent pattern of intersections S.

One might say that merely showing that a pattern of intersection sizes correlates with variation on some dimension is not the fully story. Namely, we must provide an explanation, a physical mechanism, by which the semantics of that dimension can be read out, or in other words, can affect any downstream computational step. This is simple. All we need to do is say there is a complete output matrix of connections from the coding field to some output, i.e., read out, field. Let’s call that read-out field, the “similarity-to-A-meter”. In this case, if we assume all the units in the coding field that comprise the subset of units that is the largest of the intersections are connected to the similarity-to-A-meter field, then when the code of A is active, the meter will read 100%, when B is active, it will read 80%, etc. So a pattern of physical intersections can represent an emergent dimension [in the parlance of information processing models, a discovered dimension, or a discovered latent (i.e., hidden) variable].

Now let’s look at Fig. 2. Fig. 2 shows a qualitatively identical situation. We have a set of codes over an SDC coding field. Suppose that the first code represents a particle, A, in some space (of some dimensionality >= 1) and that the second code represents another particle, B, at some distance in that space from A. In particular, note that both codes are approximately homogeneously distributed throughout the coding field. Thus, one could not tell anything about what the code represents, i.e.,. about what value it represents on any emergent dimension, by looking only at the one code. The values on any emergent dimensions can only be understood by seeing the patterns of intersections of all the codes and how that pattern changes from one time step to the next. Furthermore, we would be justified in saying that A is in a very real, physical sense, NOT localized. That is, the Q units that (when co-active) are A (and A at a particular location, and with particular values of any other emergent dimensions there might be) are dispersed approximately homogeneously throughout the entire coding field. Similarly for B. Now, note that the codes of A and B intersect. They have 6 units in common. Those 6 units comprising the intersection would collectively be the only physical resource available for representing ALL relationships of A and B, i.e., on all emergent dimensions. Note that not only are the whole codes, i.e., the full codes, of size Q, approximately homogeneously distributed throughout the coding field, but the subset corresponding to the intersection of A and B is also approximately homogeneously distributed throughout the coding field. This property is actually statistically ensured by Sparsey’s CSA. And, this property remains true for all progressively higher-order intersections.

Now we can explain entanglement. Suppose that, as an experimenter, I apply some signal/energy in the emergent world (that is, even the fact that there are particles A and B in this example must be considered to be emergent) that appears to impinge A. The application of the signal (energy, force) to A corresponds physically to changing some fraction of A’s Q units from 1 to 0 and the same number of units in the complement of A’s code from 0 to 1. Because, as explained above, the units comprising the intersection of A and B are approximately homogeneously distributed throughout the coding field, we can assume that with some non-negligible probability (depending on the specific parameters of the model, i.e., sizes of codes, etc.), one or more of the units that are changed by the perturbation are in the subset that is the intersection of A and B. In this case, then B is also immediately changed, even though, in the emergent world, B is spatially distant from A. Thus, intersection explains entanglement.

Although the above explanation was cast in terms of the information processing model, all we have to do to cast this to the physical model is to assume that the coding field now corresponds to physical units. But I guess, we’ve already essentially done that anyway in the information processing case, where we effectively assume the coding field units are just bits in a classical physical computer.

Entanglement refers to the phenomenon wherein spatially distant observations become correlated by the act of measurement. In QT, prior to observation, the state of the system that includes the two distant spatial locations is said to be indeterminate, i.e., in the singlet state. When a measurement is made at either one of the locations, thus making that value determinate, the value of the observable at the other distant location also simultaneously becomes determinate. Many people believe entanglement is nothing more than residual correlation (e.g., Gell-Mann). Specifically, the values of the observables at the two distant locations at time t become determined by a single physical event occurring at a single location at some point in the past. In this case, we don’t know the value that we will measure at the location at which we make the measurement until we make that measurement. Again, we immediately know the value of the observable at the other location as soon as the measurement is made at the measured location. But it is not the case that the value at the remote location was in any sense indeterminate at any time between the common causal event and t.

How does entanglement arise in Sparsey, specifically, in a Sparsey coding field. We’ll start with the simplest example. We assume that there is an input field (not a coding field) of binary units that is completely connected to the coding field as in Fig. ?. Note that the input field is 2D. We will further assume that all inputs will be of cardinality S, where S is approximately the diameter of the input field (i.e. of order of the square root of the number of discrete units comprising the input field). When input X is presented (activated) in the input field, signals are sent from those S units and simultaneously arrive at all units of the coding field. A particular code, consisting of Q units, is chosen (according the Sparsey’s Code Selection Algorithm). At the moment that all the Q units are selected to be the code of X, those Q units become entangled. The mechanism of that entanglement is simple. Specifically, each of those Q units will have the S active afferent weights (i.e., each one leading from one of the S active binary units in the input field) increased from 0 to 1. Thus, those Q coding field units that were chosen to be the code of X become completely correlated in the act of being chosen to be the code (and in the accompanying step of actually increasing the weights). Note that because each of those Q units is completely correlated with each of the others, they are all completely redundant: each of the Q units carries exactly the same information about what happened in the input space.

In the typical QT explanation of entanglement, the observable is spin. For example, a single event, e.g., the decay of an electron from one energy level to another, gives rise to two photons, with opposite spin, that head out in the exact opposite directions from each other, from the single point of origin, at the speed of light. Thus, the two photons become entangled in the origin event. We don’t know whether the up photon is heading west or east, but we know that if the photon heading west has spin up, then the photon heading east has spin down.

In my model, the only observable that a coding field unit has, is its binary state, on or off. So, how do we set up the analogy to making a measurement of a physical state? In my physical theory, a state of a corpuscle is a set of Q active fermion-planckons. The state

What are the eigenstates in Sparsey?

Sparsey’s analog of QT’s eigenstates are the codes that have been stored (learned). Each eigenstate is a set of Q active binary units. Only a vanishingly small fraction of the possible unique eigenstates (codes) can actually be stored. Yet, the number that can be stored rises superlinearly in the number of units comprising the coding field and linearly in the number of weights in the matrices afferent to the coding field. In comparison in QT, the number of eigenstates is exponential in the number of particles comprising the system. However, it is generally explained that the probabilities of almost all eigenstates is so close to zero that they can be ignored (cf. Feynman diagrams). The situation is quite similar in Sparsey. Specifically, although the space of possible codes in exponential in Sparsey, the number actually needed to very accurately reflect (emulate) the input domain, is a vanishingly small fraction of the number of possible codes. This is essentially a consequence of the strong recursive part-whole structure of physical kinds, both in space and time. In stark contrast to Sparsey, QT is NOT a learning theory. There is no concept of the underlying physical law (dynamics) changing over time. Rather, the rules of dynamics are given a priori and permanently fixed. Those dynamics, i.e., the Schrodinger wave equation, which in matrix form is the multiplication of the 2^N state vector by 2^Nx2^N matrix (unitary operator), are fixed. So, formally, there are 2^N eigenstates, all of which is formally possible, though most of which have such small probabilities that they can be ignored in computations of the wave equation. It is really exactly the same physical reason that underlies the need for storing (learning) only a small number of eigenstates in a Sparsey coding field and the allowance to disregard the vast majority of technically-possible eigenstates of the physical system in QT. That reason is just the extreme low likelihoods of such states.

In the case of Sparsey, which is an unsupervised learning model, several of the model’s rules (properties) combine to cause noisy inputs or spurious inputs, i.e., those due to serendipitous alignments (e.g., compositions) of features (where the features themselves are not noisy, to have low probability of being assigned to permanent codes, while allowing non-noisy, non-spurious features to have high probability of being assigned to permanent codes. That is, a noisy, spurious input will cause a code to be activated and the input features comprising that input will have their weights increased onto the units comprising that code. However, since the input is spurious, the likelihood of it occurring again any time soon is tiny, which means that the weights increased when the spurious input occurred, will tend to decay back to zero.

The question we can then ask is: for a given input space (a given “window” onto a real physical world), how many prototypes, i.e., prototypical input patterns, are needed to adequately represent that space, where by adequately, I mean, ‘with sufficiently highly fidelity’? Stated differently, how many prototypes are needed so that the expected distance between the actual input and the closest-matching stored input, is less then some threshold? Suppose that that number is relatively small, in particular, smaller than the number of codes that can safely be stored in a coding field. Well, that’s good. That means that we can expect to allow the coding field to experience a steady input stream, assigning codes to each experienced input, and simply shut off the ability to learn (stored) new codes just as the safe-storage capacity is reached. By the principle stated in the prior paragraph, we can expect the set of codes present at any given time to consist predominantly of codes of non-noisy, non-spurious, thus generally meaningful (ecologically relevant) inputs. There may be some codes of spurious inputs that have not faded yet, but all of the permanent codes will be of non-spurious inputs. And by the principle in the first half of this paragraph, whatever that set of non-spurious, permanent codes is, the inputs that are represented by those codes are sufficiently diverse and have sufficiently high coverage, so that we expect every future input to be adequately close, i.e., have high enough fidelity (similarity), to at least one of those stored inputs, so that the coding field functions well as a model of the input space, in perpetuity.

One might say, what about non-stationarity, i.e., the possibility that after learning has been shut down, the physical world underlying the input space will change sufficiently much so that the set of inputs that were permanently stored no longer does a good job modeling that world? But, we’ve already assumed that is not the case. So what allows us to assume that non-stationarity will never be an issue? The answer is the strongly recursive hierarchical part-whole (compositional) structure of the natural world, specifically of the entities in that world. By ‘entities’, I basically mean things with structured bodies, or at least ‘hard’ things, e.g., rocks, geologic forms, which I intend to specifically not include things like (large expanses of) water, air, sand, large homogeneous (thus, featureless) expanses of anything. It is against the backdrop (negative space) of these homogeneous expanses, that entities appear, interact, move, etc. In the first instance, the prototypes we perceive and learn occur at the borders of entities, where the negative space is often a homogeneous field, of water, of air (sky), of sand, etc. So, returning to the question, why isn’t non-stationarity a problem? Well, just as I’ve described the important things in the natural world having strongly recursive part-whole structure, the brain, specifically, the cortex, and even more specifically, the sensory cortex, is arranged in a strongly hierarchical way. For example, the visual cortex is organized as a hierarchy of several levels, where each level is a tiling of autonomous modules. The modules at the lowest level, V1, are the hypercolumns (macrocolumns). Each hypercolumn sees only a small portion (“aperture”) of the total visual field. The job of each hypercolumn is to represent, with sufficiently high fidelity, the events transpiring in its aperture. But again, the V1 aperture is small, e.g., 1-2 degrees of visual angle. Thus, the typical entity present in a person’s overall visual field might span many such apertures. For example, the body of the person that someone is talking to might span a 20 degree diameter. Assuming that the V1 apertures have some overlap, that might correspond to order 100-200 V1 apertures. One can then ask, how detailed of a set of features do we need for any of those apertures in order for the collective effect of all those apertures to represent that there is a human body present in the subsumed region, or a human body with this or that characteristic, e.g., fat, skinny, tall, short, or a human body in a particular configuration, e.g., standing, sitting, legs crossed, etc, or even a particular (i.e., familiar) human being.

I assume that physical law is the same throughout the universe. E.g., at all locales in the universe, it is possible for an electron to be moving in any of some large number of directions (in 3-space), and it is possible for a neutrino moving in (any of those directions) to interact with a proton, etc. In other words, the same set of possible fundamental physical phenomena exists at all places in the universe. This is tantamount to saying that the blanckon partitions of all corpuscles throughout the universe are functionality equivalent. However, as suggested already, the number of possible flanckon patterns for representing (physically grounding) the possible matter states of a corpuscle is vast (10^15,000,000). But, the corpuscle is hypothesized to be only 10¹⁵ Planck lengths (10^-20 m) on a side. Do we really need 10^15,000,000 unique matter states to explain all known physical phenomena, i.e., of the scale of the Standard Model (SM) or higher? I doubt it. In fact, the number of unique matter states required to explain all SM-scale (and higher) phenomena occurring in a volume of space that is 10^-20m on a side could plausibly be relatively tiny compared to the combinatorially large space of flanckon patterns, perhaps 10⁵. In that case, the amount of Hamming distance between the members of a set of 10⁵ such Blanckon activation patterns is huge. That means that the number of such sets of 10⁵ patterns that meets a given set of criteria over the intersections amongst the members of the set could also be large (or vast). If we take the set of criteria over the intersections to be what essentially determines physical law, then any one of the sets of blanckon patterns that meets the intersection criteria could be a corpuscle’s actual set of blanckon patterns. Thus one particular set of 10⁵ flanckon patterns might be used in one corpuscle A and a different set in any of its neighbors, B. The binary pattern over the blanckon matrix connecting A to B implements all of the state transitions that characterize physical law. Of course, the particular pattern over the blanckon matrix will be determined by those sets of flanckon patterns in A and in B. But if, for any particular corpuscle, A, there exists a set of seven mappings, i.e., from A to itself, and from A to each of its six neighbors, which implements observed physical law (dynamics), then there exists a set of seven mappings implementing that same physical law for every other corpuscle. Thus, we conclude that the ultimate physical grounding, at the Planck scale, of physical reality (its matter states and transitions), could differ from one locale to another, even though physical law, at the SM-scale and higher, is uniform throughout the universe.

However: a) the vector-based theories of quantum physics have never provided a physically intuitive, i.e., consistent with macroscopic common sense, explanation of physical phenomena; and b) not coincidentally, quantum physics views the individual particles of the SM (and all higher-level composites), as well as the forces themselves, as being being localized (subject to uncertainty) and moving through space (i.e., photons move, gluons move, etc.). The set-based theory developed here solves the first problem and fundamentally changes to a view where nothing moves: rather all apparent localizations (of matter) and apparent movements are simply correlations, i.e., patterns of intersection and discrete transitions of such, over binary activation patterns of flanckons and blanckons. Nothing actually moves in the set-based theory; there is just change of activation patterns from one moment to the next, just is as the case for the pixels of the TV when you watch television.

As stated earlier, the actual sparsity of the sets is likely orders of magnitude greater than suggested by the figures. Suppose we assume a corpuscle side is 10¹⁵ Planck lengths (10^-20 m), smaller than any physical size measurement ever made. Suppose we assume that the CM side is 10⁵ Planck lengths, and therefore that the number of flanckons in a CM is 10¹⁵. And as suggested above, let’s assume that there are 10⁶ CMs in a corpuscle. Thus, there are N = 10⁶ x 10¹⁵ = 10²¹ flanckons in a corpuscle. Then we need N² = (10²¹)² = 10⁴² blanckons for each of the seven full connectivity matrices mentioned above . Since we’ve assumed the corpuscle side length is 10¹⁵ Planck lengths, the corpuscle contains (10¹⁵)³ = 10⁴⁵ total planckons. We’d only need a total of 7 x 10⁴² boson-planckons, which is less than 1% of the 10⁴⁵ planckons in the corpuscle, to implement full self and face-adjacent connectivity. So, with these estimates, the state, specifically, the fermionic (matter) state, of any corpuscle, is always a set of exactly 10⁶ active flanckons. That’s out of a total of 10²¹ flanckons in the corpuscle, and moreover, out of a total of 10⁴⁵ planckons, so incredibly sparse, i.e., a density of 10^-15 or 10^-39, respectively. And, as stated earlier, the total number of unique fermionic states of the corpuscle is 10^15,000,000 (i.e., a state is a choice of one out of 10¹⁵ flanckons to be active in each of those CMs and there are 10⁶ CMs). That is, the theory has (10¹⁵)⁶= 10^15,000,000 possible underlying physical states, i.e., flanckon activation patterns, each one a set of 10⁶ active flanckons (out of 10¹⁵ x 10⁶ =10²¹ total flanckons) to explain the higher-scale, i.e., standard-model (SM) scale, fermionic state of each corpuscle-sized (i.e., 10^-20 m on a side) region of space in the universe. The question then becomes: with that many possible flanckon states for any given corpuscle, can we choose/construct a subset of them whose intersection structure can explain all the possible SM-scale) measurements / observations that might conceivably be made of that corpuscle? Further, can we choose/construct such subsets in any given corpuscle and in each of its six immediate neighbors, such that the patterns of intersection (within each subset) explain–i.e., correlate in some commonsense fashion with–all observables, e.g., movements of fermions, changes of spin, charge, etc., that have been made and might plausibly be made across such expanses?

At the ends of papers and talks on the “learned indexes” concept described by Kraska et al in 2017, an exciting future research direction referred to as “multidimensional indexes” is suggested.

I described a multidimensional index concept about 10 years ago (unpublished) but referred to it as learning representations that were “simultaneously physically ordered on multiple uncorrelated dimensions”. The crucial property that allows informational items, e.g., records of a database consisting of multiple fields (dimensions, features), to be simultaneously ordered on multiple dimensions is that they have the semantics of extended bodies, as opposed to point masses.  Formally, this means that items must be represented as sets, not vectors.

Why is this? Fig. 1 gives the immediate intuition. Let there be a coding field consisting of 12 binary units and let the representation, or code, which we will denote with Greek letter φ, of an item be a subset consisting of 5 of the 12 units.  First consider Fig. 1a.  It shows a case where three inputs, X, Y, and Z, have been stored.  To the right of the codes, we show the pairwise intersections of the three codes.  In this possible world, PW1, the code of X is more similar to the code of Y than to that of Z.  We have not shown you the specific inputs and we have not described the learning process that mapped these inputs to these codes.  But, we do assume that that learning process preserves similarity, i.e., it maps more similar input to more highly intersecting codes.  Given this assumption, we know that

sim(X,Y) > sim(X,Z) and also that sim(Y,Z) > sim(X,Z).

Fig. 1

That is, this particular pattern of code intersections imposes constraints on the statistical (and thus, physical) structure of PW1.  Thus, we have some partial ordering information over the items of PW1. We don’t know the nature of the physical dimensions that have led to this pattern of code intersections (since we haven’t shown you the inputs or the input space).  We only know that there are physical dimensions on which items in PW1 can vary and that that X, Y, and Z have the relative similarities, relative orders, given above.  But note that given only what has been said so far, we could attach names to these underlying physical dimensions (regardless of what they actually are).  That is, there is some dimension of the input space on which Y is more similar to X than is Z.  Thus, we could call this dimension, “X-ness”.  Y has more X-ness than Z does.  Similarly, there is another physical dimension present that we can call “Y-ness”, and Z has more Y-ness than X does.  Or, we could label that dimension “Z-ness”, in which case, we’d say that Y has more Z-ness than X does.

Now, consider Fig 1b.  It shows an alternative set of codes for X, Y and Z, that would result if the world had a slightly different physical structure.  Actually, the only change is that Y has a slightly different code.  Thus, the only difference between PW2 and PW1 is that in PW2, whatever physical dimension X-ness corresponds to, item Y has more of it than it does in PW1. That’s because |{φ(X) ∩ φ(Y)| = 3 in PW2, but equals 2 in PW1. ALL other pairwise relations are the same in PW2 as they are in PW1.  Thus, what this example shows, is that the representation has the degrees of freedom to allow ordering relations on one dimension to vary without impacting orderings on other dimensions. While it is true that this is a small example, I hope it is clear that this principle will scale to higher dimensions and much larger numbers of items.  Essentially, the principle elaborated here leverages the combinatorial space of set intersections (and intersections of intersections, etc.) to counteract the curse of dimensionality.

The example of Fig. 1 shows that when items are represented as sets, they have the internal degrees of freedom to allow their degrees of physical similarity on one dimension to be varied while maintaining their degrees of similarity on another dimension¹In contrast, a vector, which has the semantics of a 0-dimensional point, has not internal degrees of freedom..  We actually made a somewhat stronger claim at the outset, i.e., that items represented as sets can simultaneously exist in physical order on multiple uncorrelated dimensions.  Fig. 2 shows this directly, for the case where the items are in fact simultaneously ordered on two completely anti-correlated dimensions.  

In Fig. 2, the coding field consists of 32 binary units and the convention is that all codes stored in the field will consist of exactly 8 active units. We show the codes of four items (entities), A to D, which we have handpicked to have a particular intersection structure.  The dashed line shows that the units can be divided into two disjoint subsets, each representing a different “feature” (latent variable) of the input space, e.g., Height (H) and IQ (Q).    Thus, as the rest of the figure shows, the pattern of code intersections simultaneously represents both the order, A > B > C > D, for Height and the anti-correlated order, D > C > B > A, for IQ.

Fig. 1

The units comprising this coding field may generally be connected, via weight matrices, to any number of other, downstream coding fields, which could “read out” different functions of this source field, e.g., access the ordering information on either of the two sub-fields, H or Q.

The point of these examples is simply to show that a set of extended objects, i.e., sets, can simultaneously be ordered on multiple uncorrelated dimensions.  But there are other key points including the following.

1. Although we hand-picked the codes for these examples, the model, Sparsey, which is founded on using a particular format of fixed-size sparse distributed representation (SDR), and which gave rise to the realization described in this essay, is a single-trial, unsupervised learning model that allows the ordering (similarity) relations on multiple latent variables to emerge automatically.  Sparsey is described in detail in several publications: 1996 thesis, 2010, 2014, 2017 arxiv.
2. While conventional, localist DBs use external indexes (typically trees, e.g., B-trees, KD-trees) to realize log time best-match retrieval, the set-based representational framework described here actually allows fixed-time (no serial search) approximate best-match retrieval on the multiple, uncorrelated dimensions (as well as allowing fixed-time insertion).  And, crucially, there are no external indexes: all the “indexing” information is internal to the representations of the items themselves.  In other words, there is no need for these set objects to exist in an external coordinate system in order for the similarity/ordering relations to be represented and used.

Finally, I underscore two major corollary realizations that bear heavily on understanding the most expedient way forward in developing “learned indexes”.

1. A localist representation cannot be simultaneously ordered on more than one dimension.  That’s because localist representations have point mass semantics.  All commercial DBs are localist: the records of a DB are stored physically disjointly.  True, records may generally have fields pointing to other records, which can therefore be physically shared by multiple records.  But any record must have at least some portion that is physically disjoint from all other records.  The existence of that portion implies point mass semantics and (ignoring the trivial case where two or more fields of the records of a DB are completely correlated) a set of points can be simultaneously ordered (arranged) on at most one dimension at a time.  This is why a conventional DB generally needs a unique external index (typically some kind of tree structure) for each dimension or tuple on which the records need to be ordered so as to allow fast, i.e., log time, best-match retrieval.
2. In fact, dense distributed representations (DDR), e.g., vectors of reals, as for example present in the internal fields of most mainstream machine learning / deep learning models, also formally have point mass semantics.  Intersection is formally undefined for vectors over reals.  Thus, any similarity measure between vectors (points) must also formally have point mass semantics, e.g., Euclidean distance.  Consequently, DDR also  precludes simultaneous ordering on multiple uncorrelated dimensions.

Fig. 3 gives final example showing the relation of viewing items in terms of a point representation to set representation.  Here the three stored items are purple, blue, and green.  Fig. 3a begins showing the three items as points with no internal structure sitting in a vector space and having some particular similarity (distance) relationships, namely that purple and blue are close and they are both  far away from green.  In Fig. 3b, we now have set representations of the three items.  There is one coding field here, consisting of 6×7=42 binary units and red units show intersection with the purple item’s representation. Fig 3c shows that the external coordinate system is no longer needed to represent the similarity (distance) relationships, and Fig. 3d just reinforces the fact that there is really only one coding field here and that the three codes are just different activation patterns over that single field. The change from representing information formally as points in an external space to representing them as sets (extended bodies) that require no external space will revolutionize AI / ML.

 Fig. 3