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[TCCE 2022_ Mathematical model of cognition.docm](https://mdr.nims.go.jp/filesets/8f7ebf6f-fbfc-4f4d-ae4d-e600ccae8a3b/download)

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Sudeshna Pramanik, Pushpendra Singh, Pathik Sahoo, Kanad Ray, [Anirban Bandyopadhyay](https://orcid.org/0000-0002-8823-4914)

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[1D to 20D Tensors Like Dodecanions and Icosanions to Model Human Cognition as Morphogenesis in the Density of Primes](https://mdr.nims.go.jp/datasets/0b620997-f956-4d7c-8ebf-7e3e325ebde0)

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18191D to 20D tensors like dodecanions and icosanions to model human cognition as morphogenesis in the density of primes Sudeshna Pramanik1, Pushpendra Singh2, Pathik Sahoo2, Kanad Ray1, and Anirban Bandyopadhyay2*1Amity School of Applied Science, Amity University Rajasthan, Kant Kalwar, NH-11C, Jaipur Delhi Highway, Jaipur, Rajasthan 303007, India.2National Institute for Materials Science, International Center for Materials Nanoarchitectronics, MANA; and Center for Advanced Measurement and Characterization, RCAMC; 1-2-1 Sengen, Tsukuba, Ibaraki-3050047, Japan.Abstract. From image processing to information retrieval from the brain structure and signal, brain researchers find common geometric shapes from the lower dimensional data to derive higher dimensional data and create the elements of higher dimensional data. We have challenged this culture and argued to replace it with a practice to find elements in the orthogonal space, which are conceptually invariants of lower dimensions. At the same time, we have argued to replace space-time with space-time-topology-prime-based invariants under the self-operating material universe, SOMU, since the density of primes is a bias-free infinite source to deliver unique symmetries perpetually. Here we have derived the topology or morphogenesis from the density of primes and estimated the framework of maniflats and manifolds derived from the 1D to 20D tensors holding the within-and-above network of invariants as conscious thoughts of a human brain.Keywords: Space-time, higher dimensional tensors, manifold, maniflats, human brain, the density of primes.IntroductionRecent findings that the brain operates in the 12 dimensions (12D, (Reimann et al., 2017)) have triggered a serious debate on developing higher dimensional information processing for human cognition and consciousness. However, we think that looking into more complex structures in the brain, if the decision is made that the brain is a 12D structure, then the approach is fundamentally wrong. There has been a fundamental mathematical protocol to derive a higher dimension that is not used in brain research. Rather, a wrong notion of the dimension is used, where deriving elements from a set with common dynamics is considered a new dimension. Right from the days of framing the string theory to create the theory of everything, constructing everything starting from a vibrating string, we have come to a point where the string is replaced by a helical spring, a fourth circuit element, Hinductor (Sahu et al., 2010 JP-511630; US 9019685B2, (2015). EU patent EP2562776B1)). While string theory bonded the strings to construct the universe bottom-up, we nested and twisted the spring to create everything, nesting the helical phases bottom-up. Once we fixed the fundamental element of the universe, the next step was to choose the metric. In 1916, Einstein introduced the concept of linking space and time together in one tensor, as if different constituents of space and time interact to create reality. In these tensors, always 3 dimensions of space and 1 dimension of time were taken into account, and the observer’s frame of reference was a key. We have changed the definition of a new dimension, suggesting that new higher dimension elements are orthogonal to the current dimension (or elements of the fifth dimension would be invariants of the fourth dimension) and given equal importance to space, time, topology and prime (3 dimensions x 4 = 12 dimensions), resulting in the metric, space-time-topology-prime, (STts) metric (Singh et al., 2021a, b). The reason for introducing two new variables, topology and prime, is that the instrument's measurement limits space and time.In contrast, topology, or morphogenesis, is a subject of study where the observer’s frame of reference is not a key. Therefore, the geometric elements of topology could be found in wide ranges of spatio-temporal events. For example, teardrop to the ellipsoid formation is found when a star is born, in the brain-spinal cord combination, protein folding, sperm-ovum duality etc. For prime, the density of prime has been the fundamental of the universe, which has no bias; deriving the geometric elements of morphogenesis provides us with a metric where instead of space-time, we take space-time-topology-prime as a fundamental element, but the infinite series of the density of primes create a self-operating mathematical universe, SOMU (Singh et al., 2021a, b).Here in this work, we have concentrated only on the density of primes and how geometric elements could be derived from the density of primes. At the same time, we have revisited the 1D to 20D tensors to find the geometric maniflats that would estimate the linking of elements and interaction of elements in the space-time-topology-prime metric. First, we review historical efforts linking the tensors of different dimensions. Of course, the journey of string theory has been that a 4D tensor, quaternion and an 8D tensor, octonion, were coupled with a warp factor, and the sum of two tensors, 8 and 4, considered to estimate 12D tensor, which we consider a very wrong approach. Dixon algebra (Geoffrey M. Dixon (1994)) took a product of real numbers, complex numbers, octonions and quaternions to build a grand unification theory, GUT, from elementary particles. Why do we take the product of vectors of different dimensions to invent a new algebra? Because addition or subtraction does not take into account inter-dimensional interactions. Leonard Dickson showed in 1919 that octonions could be constructed as a combination of two-dimensional algebra where elements are quaternions (Dickson, L. E. (1919)). We have recently generalized the composition algebra (Jacobson, Nathan (1958)) for higher dimensional tensors. Instead of representing the composite tensor as a product, we replace elements of a tensor with other tensors. We have named it the decomposition of a tensor. We needed it to understand and analyze the maniflats, or the geometric structure where elements of different dimensions would link with each other and transmission between different elements could be made (Singh et al., 2021a, b).    Methods SummaryCalculation of invariants and space-time-topology-prime metric (STts) metric under SOMU.2.3.1. The mathematical operation that invents an invariant: 3D clock assembly representing spatio-temporal events is a polar decomposed structure, hence could be rewritten as a positive-definite tensor , ;  is the deformation of input  from memorized . Two 3D clock assemblies, one memorized () and the other unknown input () resonating with memory. Deformation  ( in the 3D clock assembly for train and test datasets are taken as differential  signal or  along three orthogonal axes , , , in general . The plot is , The invariant condition: partial derivatives of , with respect to ,  vanishes when  (…..(equation 3)). T denotes transpose,  trace delivered by . Seed molecule or supramolecule’s resonance frequency set  periodically oscillates, we get . When the output product of one self-assembly is used as a seed for the next phase of self-assembly, the orthogonal projection to determine the invariant  is satisfied. Both layers together generate the vortex that holds the geometric shape of an invariant (Ennis and Kindlmann, 2006). So, the definition of dimension used here could be found in solid-state physics literature in the 1960s.2.3.2. The space-time-topology-prime metric: We reported counting primes in a 3D clock arrangement to build an artificial brain (Reddy et al., 2018), the metric evolved over time (Bandyopadhyay, 2020). Finally, we reported the operational protocols for space-time-topology-prime (STts) metric (Singh et al., 2021 a, b). Here the metric measures distance in symmetry; for self-assembly of systems within and above, we could measure the differential space , differential time , differential topology  and finally, differential symmetry in the density of primes . Here the combinations of dual (either S or t), triple (either S, t or T) and quad features (S, T, t and s) are:Space-time , with 3+1 or 2=5 or 6 dimensions; Space-topology , with 3+5=8 dimensions; time-symmetry  with (1 or 2) + 7 = (8 or 9) dimensions; topology-symmetry , with 5+7=12 dimensions; , with 3+(1 or 2)+5+7=16 or 17 dimensions. Space-symmetry , with 3+7=10 dimensions; Space-topology , with (1 or 2) + 5 = (6 or 7) dimensions; space-symmetry-time  with 3+7+ 1 or 2=11 or 12 dimensions; space-time-topology  with 2+1 or 2+5 =8 or 9 dimensions; space-symmetry-topology  with 3+7+5 =15 dimensions; symmetry-time-topology  with (1 or 2)+7+5=13 or 14 dimensions; symmetry-space-time  3+7+1 or 2 = 11 or 12 dimensions. Since icosahedron has 12 corners, for the “within-and-above” universe, we would have a maximum of 12 dimensions. Since icosahedron has 20 planes, we would have 20 dimensions when dimension means adding new dynamics. Hence,  would be confined in one imaginary world, i.e., it would represent the projected and the feedback time crystal “to and from” the Phase Prime Metric PPM (Reddy et al., 2018; Bandyopadhyay, 2020; Singh et al., 2021a, b). The metric representing the polyatomic time crystal in  universe of 12 nested worlds is given by …. (equation 4)Results and discussion3.1. A composition of five fractal patterns for the density of primes is key to cognition:We have argued earlier that the basic concept of the number system in the Vedic religious scriptures is a periodic power cycle of 10. Therefore, in Figure 1A, we have counted the number of primes at a gap of 10 integers, so in each loop, the possible number of primes available would be either 0, 1, 2, 3, or 4; only five values possible. We try to find how the prime number theorem predicts the number of primes and the actual density of primes in Figure 12B. We can see a huge discrepancy in the real density of primes with the prime number theorem, so we rejected the prime number theorem and investigated if there is any hidden periodicity or pattern in the density of primes. For that purpose, we calculated the frequency of finding a particular density of prime in the integer space. Say the density of primes is 3, and if we keep searching in the integer space, how frequently would we encounter a period of ten where the density of primes is 3? The gap between integers was normalized for all values 0, 1, 2, 3, 4, and 5 so that we could bring gap values for all five possible prime densities with similar values and differences in their frequency pattern to be compared. Figures 12C-G plotted half-the-integer-gap above and below a central axis, and C2 symmetry reveals a set of 6 waves or a triplet of wave pairs for each possible primes' possible densities. Thus, we get 5x6=30 waves that map the frequency pattern for the density of primes; the 30 clocks form an integrated architecture, as shown in Figure 1I. From the fitting plot of 30 waves, we discovered that the most dominating phase gap is 22.5o, Figure 1I. This is a very important angle, and we see it everywhere in nature. For each set of ten consecutive integers, there could be 4, 3, 2, 1, and 0 number of primes, and For these five density values, if isolated from integer series, we find three pairs of phase spheres for each of the five density values. This integrated network of 30 phase spheres, all with a specific phase gap and diameter, remains constant; it never evolves,  or counting takes place on the surface of this structure. It is the skeleton of SOMUThe data generation for Figures 1C-I. We search the prime numbers P1, P2, P3, …. Pn in every 10 numbers interval 0-10; 10-20; ... from 0 to 4000. Then we obtain the successive difference (Q1= P2-P1; Q2 = P3-P2.… Qn-1=Pn-Pn-1) between P1, P2, P3...Pn and frequently, we find the fixed numbers Q1, Q2, Q3, Q4, and Q5. After that, we select the numbers N1, N2, and N3…Nn where we get Q1, Q2, Q3, Q4, and Q5. For all Q, we obtain the number difference M1 = N2-N1; M2 = N3-N2;… Mn-1 = Nn-1-Nn. To get the smallest numbers S1, S2, S3 …. Sn, we divide the numbers M1, M2, ... Mn by a suitable number X. After that, we take the next difference level of smallest numbers T1 = S2-S1; T2 = S3-S2;….Tn-1 = Sn-Sn-1. We plot the curves between numbers 0-4000 and T1, T2, …Tn for each Q. Then, we fit the curves and get 3 sets of parodic equations for each Q.  Figure 1. The thirty infinite series of thirty invariant phases in the density of primes defines fifteen types of singularities and fifteen types of infinity. A. Density of primes calculated by counting the number of primes at a gap of 10, we get five possible values of density of primes, 0,1,2,3,4, and we find integer gap of the gap of gaps (three higher levels) to find positive and negative diversions in the integer series. B. All five densities for 0, 1, 2, 3 and 4 are plotted together. We consider five densities, 0,1,2,3,4 and build five plots in panels C, D, E, F, and G. Three pairs of waves fit the density of primes (6x5=30), and the fitting line is colored. Each pair has an opposite phase but with a phase gap. I. We plot the curves between numbers 0-4000 or 0-10000. 3.2. Implications of time crystal like the 3D architecture of clocks representing the density of primes3.2.1. Synthesis of system points and singularities:One of the important factors for clock-assembled architectures of the density of primes is that 30=2x3x5, 2x5x3, 3x5x2, 3x2x5, 5x3x2, 5x2x3; there are six topologies, it is a fusion of two triangles crossing over by facing each other, creating a singularity domain in the overlapping region. The density of primes has enormous complexity than the brain, which primarily uses 12 as the basis of metric variations (12 = 2x2x3, 2x3x2, 3x2x2). In the ordered factor metric, 12 cannot generate a system point (paired 2 is the reason), but 30 can (three separate integers, 2, 3 and 5). Most of the system points originated when we consider primes' density, which continues when we move downstream from 12D to 1D.  The density of primes intricately curves the probability of making a point undefined (ordered factor metric, Bandyopadhyay, A. 2020), the choices connecting lines for all integers from isolated discrete forms along the lines of integers, the isolation, superposition, overlaps and gaps are modulated by the density of primes. Connecting nearest neighbors of the density of primes plot derives C5xC3 symmetry of paired wave gives birth to the property of normalization, i.e., nearly a closed loop at 360o. 3.2.2. 12o solid angle for 30 wave packets on a phase sphere:The density of primes has fifteen paired periodic waves or thirty waves with a phase gap of 12o, totaling 360o, which introduced the concept of a circle or sphere when the density plot is projected to infinity. However, nine of fifteen density plots form a triplet of triplet group to demonstrate 108o phase cover. All these phase values are connected to integers 12, 36, and 108 forming the first triplet of triplet, and 12 points ordered factor metric forms the smallest triplet and smallest closed loop. Thus, the 360o value for a complete loop is not an accident, it originates from ~12o (11.75o or 23.5o/2) phase gap between paired waves, and there are 30 waves in the density of the primes plot. We have shown in Figure 4C that at N=12, the first clockwise rotation completes. Therefore, a solid angle of 12o creates a loop. 3.2.3. Fifteen vortices generated by 15 pair waves:The density of primes has fifteen paired periodic waves engaging in a unique journey of changing the shape of the 3D surface area covered. Similar to the process we generate the phase prime metric, 3D surface area, we build a change in the gap between primes for a particular density (0, 1, 2, 3, 4). Now, as the integer increases, the distance gets separated, and the waveforms generate a curvature following , in other words, a 3D solid angular twist by a pair of waves generating 15 conical cylinders are the evolution pathways for 12 classes of metrics generated for SOMU.3.2.4. Expanding clock architecture of density of primes:The density of primes creates 30 waveforms or 15 pairs of waveforms whose periodicities in the integer space have five quantized levels; since five types of prime densities are feasible. These polyatomic time crystals expand inhomogeneously. We need to correlate the density of primes derived invariant structure made of clocks, where instead of time, integers flow. One of the prime features of this invariant structure is the expansion of the spheres representing the clocks. The reason is continuously increasing separations between primes, yet, the geometric feature of the invariant 3D clock assembly remains constant. Therefore, we find that the nearest neighbors in the choices of prime arrangement plots (ordered factor metric) are unique. Interestingly, neighboring integers are not correlated, but the symmetry of prime positions brings choices of distantly located primes in one shape; neither magnitude of a prime is important nor the number of primes constituting an integer. The nearest numbers suggest that if symmetry breaks, where the system point would move. Therefore, neighboring points in the ordered factor metric loops or curved lines map a homogeneous gradient of several symmetries. Along these lines, if one moves, symmetry will not change dramatically. There will be the least changes in the system. Jumping between lines would be a phase transition while moving through the lines would be symmetry-breaking. 3.3. 108 fundamental constants made of 17 primes:The density of primes synthesizes and sets the condition for normalization. A triplet for each of the five density values of primes sets a triplet of three angular invariants. These angular invariants are conserved laws when thirty system points generated by paired waves of the density of primes jump from one isolated pattern of the metric space to another. The critical patterns of the density of primes generate curvatures as thirty waveforms converge at common regions. Note that 30 waveforms generated by connecting the density of primes do not start at a particular number, and by 109, all 30 waveforms get at least one point. For that reason, they do not converge to a singular point, but rather a domain. These domains generic  feature contributes to all the geometric patterns generated by invariants of all 12 dimensions. For this reason, these indices  generate fundamental constants . Thus, we get 95 fundamental constants for 47 bases ( to ), for the SOMU, we envision the ultimate universal engine would have a 17 prime base at 53, then we would have universal constants ( to ). Figure 2. Density of primes and its six fundamental features: The first metric accounts for the properties born directly from the 30 waves of the density of primes described in Figure 1. The density of primes gives birth to three types of normalizations applied on the ordered factor metric (Bandyopadhyay A., 2020, Chapter 3): Three types of normalization: e-pi-phi empty normalization (T), the density of primes: Prime contribution normalization, the density of primes induced (S): Polar plot makes clockwise and anti-clockwise rotation (R). Thus we get six features schematically summarised in six circles. Relative phases between 30 waves in the density of primes give rise to 23.5o, an angle responsible for multi-dimensional projections. It ensures the formation of a quadratic relation and, consequently, the fundamental constants (Bandyopadhyay, A., 2020).  3.3. Replotting density of primes and understanding the differentials in details:3.3.1. The importance of differentials:In Figure 1, we have observed that the periodicity we are applying forcefully to the system is an oversimplification of the true pattern that links the density of primes points. For that purpose, we have written an algorithm to find an integral gap between the two points in the integral system where a common density of primes (say, two instances of zero) is found and then what is the difference between counting (C). We call it dC/dN. When we get a singular differential plot, from that, we could plot dC/dN again. The derived plot would be . We have continued this process six or seven times. The reason is to find if there is a fractal pattern much more complex than simple periodic waves that we determined in Figure 1. In all the plots of Figure 3, where the differential dataset of the density of primes has been documented, we have got negative values.   3.3.2. The natural emergence of negative value.In Figure 1 and Figure 3, the reason we observe negative values is the fact that often the density of primes decreases. Therefore, it is not artificial but naturally embedded in the pattern of the density of primes.   Figure 3. The density of primes plot and differential features: There are five rows, and each row presents a particular density, L0 means zero number of primes in a gap of ten, and the first digit where zero density appears is taken. Similarly, all five possible density of primes (0, 1, 2, 3, 4) have been taken into account, and the first row is for zero, the second row is for density one, L1; the third row is for density 2, L2; the fourth row is for density 3, L3 and fifth row is for density four, L4. The first plot in every row is the density, C vs. integer, N. From second to sixth are differentials, DS2 means , DS3 means ; DS4 means . For the first two rows, Nmax is 4k~4000; for the third row, Nmax=10k; for the fourth and fifth rows, Nmax= 100k.  3.3.3. Fitting the density of primes differentialsFigure 4 shows only one example for each possible density of primes, and the nearest neighbors are connected. There are some remarkable features of the plots. Irrespective of the differentials of the second order or the third order, the typical branching pattern remains constant for a particular density of primes. Also, for all possible density of primes, we find that very particular branching out from a single point. Therefore, the density of primes plot Figure 1 delivers an oversimplified periodic feature made of 30 clocks. However, when we consider the differential plots in much more detail, we find that not only a typical branching out from a single point is embedded in the integer space, but the larger data we pack, the larger the size of the same fractal seed pattern. We draw two important conclusions. First, a within-and-above network of a “branching out from a point” structure should act as a seed dynamics. Second, there is a periodicity once again, like in Figure 1. The periodicity derived from the differential plot of Figure 3 and Fitted in Figure 4 is nearly similar to that obtained from Figure 1. The finding is a serious challenge to the prime number theorem that ignores all these intricate patterns and embedded invariants in the density of primes. Figure 4. Nearest neighbor connection of the density of primes plots: We have taken one of the differentials for the five densities of primes as noted to the right of each plot (density of primes C is along the vertical axis and N is along the horizontal axis) and connected the neighbors to create a connection. L0 is for Nmax~4k, L1 is for Nmax~10k, L2 is for Nmax=10k, L3 is for Nmax=100k and L4 is for Nmax=100k. A self-repeating branching pattern is seen, and we have noted the periodicity of the self-repeating or fractal seed pattern that is common for all densities of primes. For L0, the periodicity of N is 1k. For L1, the periodicity is 1k; for L2, the periodicity is 2k; for L3, the periodicity is 10k; for L4, it is an integral multiple of 10k, i.e., an expanding feature of the same seed pattern.    3.4 Generalization of the fractal seed pattern observed in the density of primes:3.4.1. A generic model that could establish the formation of a typical branching pattern in Figure 4:In Figure 5, we have attempted to deconstruct the “branching out from a point” feature observed in Figure 4. We start from a point in panel A of Figure 5, and we show that we could expand the branching out using three possibilities. Note that there are several different possibilities for branching out, as observed in various plots of Figure 4. For the time being, we have included all possible variations and created seed rules to generate 15 different morphogenesis that repeats with C2 symmetry or 30 different morphogenesis classes observed in the variations of the branching out the pattern of the density of primes in Figure 5 panel C.3.4.2. The necessity of structuring an undefined point: The growth of a pointHere we construct a philosophical basis for our SOMU model proposed earlier to incorporate our recent finding in Figure 4. Within a point or outside, we cannot have space, mass, potential, field, or even force at the beginning. Moreover, constructing SOMU requires choosing mathematical elements and constructs that use minimum assumptions.  If there are many points, we have to define how many, how they are correlated, and who links them. So, at the beginning of everything, we cannot take more than one point. We cannot go outside the point. The only possibility is going inside a point, exploring what we can do so that everything we do is normalized to 1 or unity and satisfy all criteria to be a point. If something starts moving from a point, it could go clockwise, anti-clockwise and knots or a composition of both clockwise and anti-clockwise rotations. A point's origin is not an entity, but a paradox, to exist or not exist. Instead of a point as a real physical entity, we would only have the probability of a point. Since that question must not have a finite answer, probabilities would deliver the basic elements of SOMU. An undefined point is not a null set. If it is, then it will be defined. It is the sum of all possibilities, an endless chain that cannot be defined.3.4.3. The birth of SRT follows a fusion with a point to create 15 morphogenesisNow at the second phase of the evolution of a point, when clockwise and anti-clockwise paths superpose, only the end path could exist, termed as T (of SRT); both the clockwise and anti-clockwise paths could cross and survive with a loop, R of SRT; finally, both clockwise and anticlockwise paths could coexist in a twisted path, S of SRT once we have SRT configuration, circle, ellipsoid and knots form. These are the origins of morphogenesis. SRT properties are defined at this stage very similarly, S=Fill or expand the pattern by repeating, R=Cross-over and inflate, and T=minimize corners to make it suitable for bonding. Bigyan-Vikshu argued in the 14th century that for several thousands of years, different schools and a wide range of scholars have argued for redefining S, R and T. However, the sense of SRT triplet S=projection, R=transformation and T=bonding were never changed. However, for morphogenesis, one requires to add a point and a bidirectional arrow to the circle, ellipsoid and knot triplet. Why do we need to add a point and bidirectional arrow? We have to insert a point inside a point; that is true. However, we need to deform the shape we have in hand so that the transformations continue. A bidirectional arrow refers to an expansion of the shape and thus incorporates a new pattern.We have outlined in Figure 5 at the bottom row how adding a point to the circle creates a teardrop, triangle and square. These three derived geometries combine to create wide ranges of geometries we observe as constituents for morphogenesis. When an ellipsoidal knot (R) undergoes a point and arrow transformation, it builds several asymptotes and divergent geometries, which are used as a basic element for morphogenesis. Finally, when knots interact with the point and the arrow, it generates several complex knots. The three-phase evolution of a point to basic morphogenesis constituents is the key for SOMU. Figure 5. A journey from point to elementary transitions observed in the density of primes in Figure 4:  Here, we describe the fundamental philosophical argument for developing the SOMU. A point generates paths and then continues to produce several dynamic paths by interacting with more points. A. There are only three possibilities for making a journey from a point; using an arrow, we describe three choices. B. The second phase of panel A is shown in panel B. Three events are noted by combining the output of first-phase products. C. The derivatives of the second phase described in panel B are combined to create third-generation progenies in panel C. In panel C, a new point is added as an attractor in all three generations. 3.5. Derivation of an engine that generates all possible branching out patterns of Figure 4:We observe in Figure 4 and Figure 5 that all journeys must begin from a point and branch out. Therefore, in Figures 6, 7, 8 and 9, we have created a point and two concentric circles. The outer ring decides when the system returns to the initial point, or the periodicity of the fractal seed pattern observed in the density of primes plot in Figure 4 is derived in the second ring of Figures 6-9. Now, we have chosen a triangular path and remained strict to the rules that our maximum journey would be limited to dimension 20 or D20 because we would not go beyond 12 dimensions. Now, 12 dimensions could be written in the planes of the dodecahedron (12D) or along the corner diagonals of the icosahedron (20D for 20 planes). Therefore, under the current model of SOMU, where we limit ourselves to the minimum dimension we need to create a self-operation, we do not need to go beyond 20D. However, we kept the possibilities open for future SOMU (last figure of the book Bandyopadhyay, A. 2020).Figure 6. Generic manifolds for NxN tensors where the density of primes paths form as one moves from the center of the maniflat architecture: We have drawn two concentric circles representing the product determination and signature determination routes. For taking the product of two vectors of a particular dimension, one should start from the center, which could be the point of Figure 5 and the row value of the tensor, and move along the line to get the column value. While completing the triangle, the third value should be placed as the product at the coordinate (row, column). If the arrow direction is followed, its positive and opposite directions would be negative. The dimension of the tensor is written as the NxN value. Figure 7. Generic manifolds for NxN tensors where the density of primes paths form as one moves from the center of the maniflat architecture: We have drawn two concentric circles representing the product determination and signature determination routes. For taking the product of two vectors of a particular dimension, one should start from the center, which could be the point of Figure 5 and the row value of the tensor, and move along the line to get the column value. While completing the triangle, the third value should be placed as the product at the coordinate of (row, column). If the arrow direction is followed, its positive and opposite directions would be negative. The dimension of the tensor is written as the NxN value.Figure 8. Generic manifolds for NxN tensors where the density of primes paths form as one moves from the center of the maniflat architecture: We have drawn two concentric circles representing the product determination and signature determination routes. For taking the product of two vectors of a particular dimension, one should start from the center, which could be the point of Figure 5 and the row value of the tensor, and move along the line to get the column value. While completing the triangle, the third value should be placed as the product at the coordinate of (row, column). If the arrow direction is followed, its positive and opposite directions would be negative. The dimension of the tensor is written as the NxN value.Figure 9. Generic manifolds for NxN tensors where the density of primes paths form as one moves from the center of the maniflat architecture: We have drawn two concentric circles representing the product determination and signature determination routes. For taking the product of two vectors of a particular dimension, one should start from the center, which could be the point of Figure 5 and the row value of the tensor, and move along the line to get the column value. While completing the triangle, the third value should be placed as the product at the coordinate of (row, column). If the arrow direction is followed, its positive and opposite directions would be negative. The dimension of the tensor is written as the NxN value.4. Conclusion: The prime number theorem suggested a homogeneous law to explain the density of primes, which ignores the plethora of geometric invariants and laws of nature. We have created generic engines of 20D as generic alternatives of the Fano plane for octonions to predict products of all 1D to 20D vectors and, at the same time, predict dynamic behaviors of the branching out fractal seed patterns in the density of primes. Our finding paves the way to analyze morphogenesis observed in nature using the density of primes, wherein morphogenesis could eventually generate all possible spatio-temporal events happening in nature. Therefore, we provide an essential tool for bias-free self-operating modules from the density of primes to SOMU.         Acknowledgements: Authors acknowledge the Asian office of Aerospace R&D (AOARD) a part of United States Air Force (USAF) for the Grant no. FA2386-16-1-0003 (2016–2019) on the electromagnetic resonance based communication and intelligence of biomaterials.Competing interests statement: The authors declare that they have no competing financial interest.Resources: All algorithms used here to build the maniflats and tensors and density of primes differentials are available through the joint NIMS-Amity university free resources in the GitHub.  ReferencesBandyopadhyay, A. (2020). Nanobrain: the making of an artificial brain from a time crystal. 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