An original quantum cosmology model is introduced which extends the author’s quantum intelligent cosmology research programme [1] [2] [3] [4]. The motivation is to further formalize the programme’s original quantum foundational concepts whilst exploring encouraging new paths.

Let us commence by reflecting on a couple of inspiring quotes from John Archibald Wheeler, “Existence, what we call reality, is built on the discrete” and “… the world has at bottom an information-theoretic character” [5]. I propose that the discrete dimensions of our quantum information world are the most fundamental foundations of everything. A dimension is a structure that categorizes discrete information. Minimising computational time, a shared dimension makes economic computational use of that degree of freedom, as a Natural resource for the broadband parallel processing of distinguishable information.

This paper focuses on the Natural selection, evolution and function of shared dimensions in a deep learning quantum cosmology model. These shared dimensions are conformed dimensions in computer science, which ensure consistency between Nature’s quantum information processing of its physics at macro, meso and microscopic scales; and ensure consistency spanning from deep evolutionary time, through observer lifetimes, to near-instantaneous causal physics.

The model presented proposes our deep learning quantum computational universe is founded on a common architecture of shared dimensions, defined by discrete and exceptional spherical codes, on a 2-sphere, S 2 . The shared dimensions are utilised by Nature to compute the entirety of physics, across the whole universe, at all scales. Nature’s quantum computations at the cosmological scale can share dimensions with its computations at the Planck scale because the extreme difference in scale protects against detrimental crosstalk. Nature’s two extreme scales come into perspective reality at the meso-scale of the sentient Wheelerian participatory observer. The information processed in these shared dimensions is parallelized, in coordinated superpositions, and decomposes into Nature’s innumerable emergent physical phenomena, through observations, and described by effective theories. Fundamental information processing is impelled by Epistemic Drive, the Natural appetite for information selected for advantageous knowledge. This quantum cosmology (QC) model is underpinned by spherical codes, notably including the 24-vertex snub cubical spherical code, which implicitly involves the tribonacci constant, hence the moniker Tribonacci-QC.

The discrete dimensions of the spherical codes resemble the fine bristles in the puffball seed heads of the common dandelion flower (Taraxacum officinale) (Figure 1).

“No creature is fully itself till it is, like the dandelion, opened in the bloom of pure relationship to the sun, the entire living cosmos”. Reflections on the death of a porcupine D. H. Lawrence (1925).

This paper is organized as follows:

Section 2 introduces the idea of a self-organizing sequence of optimal ⋆ , non-antipodal ↮ , spherical codes, C ↮ ⋆ ⊂ S 2 , in Tribonacci-QC. The codes configure degrees of freedom in quantum cosmological information processing, which are isomorphic to dimensions of 24-spacetime and to the 24 generators of the special unitary group of degree five, S U ( 5 ) . For background, the reader is referred to research on spherical codes [6] , quantum self-organization [7] [8] , geometrogenesis and quantum graphity [9] [10] [11] , emergent spacetime [12] , tropical geometry [13] [14] , minimizing measures of the causal variational principle on the sphere [15] [16] [17] and the algebra of grand unified theories [18].

Section 3 reflects on quantum mechanics’ profound relationship with the number 24. An inspiring lecture by mathematician John Baez [19] gives further context and deeper insight.

Section 4 considers the snub cubical characteristics of the 24-vertex optimal non-antipodal spherical code. This includes a discussion on the tribonacci constant, T, which derives from the third-order tribonacci recurrence sequence. The

metric properties of the snub cube profoundly involve the tribonacci constant. The characteristic twist angle and chirality of the snub cube is also discussed. The properties of the snub cube are given in [20] [21]. Pertinent contemporary studies of the tribonacci sequence are given in [22] [23] and include those relating the tribonacci sequence to octonions and quaternions [24] [25] , which in turn relate to fundamental interpretations of the standard model of particle physics [26] [27] [28] [29] [30].

Section 5 presents the model’s cosmological scale factor evolution, a ( t ) , which is concisely formalized as an Ostwald-coarsening function of time, t, scaled by the tribonacci constant, T, where a ( t ) = 3 ( 6 t ) T / 3 − 1 for t < 0 and a ( t ) = 3 t T / 3 − 1 for t > 0 . The 6-fold transition in the pace of time, at t = 0 , relates to geometrogenesis from 24-spacetime to 4(6)-spacetime (i.e. from a 24-vertex to a coarsened 4(6)-vertex optimal non-antipodal spherical code, as discussed in Section 2). The function takes as its origin, time t = 0 and scale factor a ( t ) = 0 , relative to the spherical horizon datum. The present cosmological scale is defined a ( t ) → 1 , which is imminent, as t → 1 for each observing agent in a Wheelerian participatory universe [5]. The Hubble parameter, deceleration-acceleration transition redshift and astronomical ladder-calibrated cosmological timescale are also formalized. Specifically, CMB H 0 / 100 ≡ h t → 1 ≡ a ˙ / a = T log ( 3 ) / 3 ≈ 0.67355 , Distance Ladder h = 4 T log ( 3 ) / 11 ≈ 0.73479 , redshift at the transition from a 24-vertex to a 4(6)-vertex code is defined z 24 → 4 ( 6 ) = 2 (coeval with peak star formation rate density [31] , quasar activity [32] [33] , CMB lensing [34] , ⁶Li genesis by cosmic rays [35] [36] [37] [38] and exoplanetary abiogenesis [39] of Wheelerian participatory universe observers), and time of last zero cosmological scale is t a = 0 ( − 1 ) = 13.8   Gyr , when calibrated by t z = 2 = 10.4   Gyr [40]. The redshift at the transition from deceleration to acceleration is z d a = 3 e 1 − 3 / T − 1 ≈ 0.59607 which compares well with supernovae observations [41] [42].

Section 6 presents this quantum cosmological model’s formalization of baryon density as precisely one sixth of the total matter density, due to 6-fold parallelization at the transition from a 24-vertex to a coarsened 4(6)-vertex optimal non-antipodal spherical code. Specifically, Ω b h 2 = arccos ( T / 2 ) / 4 π ≈ 0.02287 and Ω m = 6 Ω b = 27 arccos ( T / 2 ) / 2 π log 2 ( 3 ) T 2 ≈ 0.30248 . This formalization derives from a proposed equivalence between the physical baryon density parameter, Ω b h 2 , and the characteristic twist angle of the 24-vertex snub cube, as a fraction of a rotational identity, 4π.

Section 7 considers the cosmographic distance ratio γ t o / γ t c of the optical over cosmic microwave background tangential shear [43]. An equivalence in Tribonacci-QC is proposed, where γ t o / γ t c = ( T − 1 ) / 2 ≈ 0.41964 .

Section 8 considers astronomical observations [31] - [38] [44] - [56] which fit the Tribonacci-QC model and shows how the model circumvents cosmological singularity problems at the time of last zero cosmological scale t a = 0 ( − 1 ) . This section also discusses cosmological evolution leading up to that event and considers this model in terms of John Archibald Wheeler’s participatory universe [5]. Furthermore, DNA is proposed to have evolved from an 8 × 3-nucleotide (eight codon) primeval molecular code on the model’s 24 shared dimensions.

Section 9 provides a summary of the paper and Section 10 gives conclusions.

In this model, a spherical code is a vertex set configured on the 2-sphere, S 2 . A key objective of this current research is to identify a sequence of exceptional spherical codes which configure evolving degrees of freedom for deep learning, quantum cosmological, information processing by Nature. As such, they are isomorphic to evolving dimensions of spacetime and, in parallel, they are isomorphic to evolving shared dimensions of the Standard Model gauge group, G S M , particularly as it relates to the special unitary group S U ( 5 ) .

In Tribonacci-QC degrees of freedom in quantum cosmological information processing:

・ Create distinction (isolate crosstalk) [57] whilst computational dimensions are shared for parallel processing of information at cosmological and Planck scales;

・ Order moments in time;

・ Position locations in space;

・ Frame spacetime events (located moments) [58] ;

・ Constitute foundations for symmetrical quantum physics (Standard Model and dark sector) [59] [60] ;

・ Accommodate and explain asymmetrical physics (arrow of time and e.g. matter-antimatter);

・ Govern the quantum to classical transition [61] ;

・ Enable cosmological expansion (Hubble phenomenon) [62] ;

・ Relate to Einstein’s general relativity tensor equivalence [63] ;

Additionally, the model embraces:

・ Darwin’s universal and deep learning algorithm (variational selective replication) [1] [64] [65] [66] ;

・ Epistemic drive (Naturally selected information for advantageous knowledge) [1] ;

・ Wheeler’s participatory universe concept [5] ;

・ Whitehead’s perspective on occasions of experience [67] ;

・ Astronomical observations.

To qualify as exceptional spherical codes that configure evolving degrees of freedom in this Tribonacci-QC model of quantum cosmological information processing with deep learning, the codes satisfy conditions:

1) Symmetrical vertex sets;

2) Few vertices (≤24);

3) Geometric compatibility to Naturally self-organize in an evolutionary sequence;

4) Non-antipodal (no diametrically opposite vertices);

5) Maximise the minimum vertex separation (solving Tammes’ problem) [68] - [73] ;

6) Chirality in the maximum vertex set of the sequence of exceptional spherical codes.

Succinctly, they are optimal non-antipodal spherical codes. Let us further consider each of these six above qualifying conditions:

1) Symmetrical vertex sets configure isomorphic degrees of freedom for symmetrical quantum physics, the symmetries in general relativity and for those in emergent Natural phenomena.

2) Optimal non-antipodal spherical codes have few vertices (≤24) to create distinction between degrees of freedom and thus isolate crosstalk. Redundant or overcrowded vertices degrade information processing reliability and cause detrimental dimensional ambiguity; i.e. fewer channels are better for error free discretisation.

3) Geometric compatibility of optimal non-antipodal spherical codes to self-organize in sequence allows for Natural evolution towards spontaneous cosmological order. Consider analogous DNA which is geometrically comprised of nucleotide building blocks. Geometric compatibility facilitates self-organization. Kalinin et al. [74] seminally discuss self-organized criticality and pattern emergence through the lens of tropical geometry. I propose the geometry of shared dimensions in this Tribonacci-QC model will become more completely understood via application of the tools of tropical geometry [13] [14] which will lead to more profound formalizations of fundamental physics.

4) Non-antipodal vertex sets allow radial degrees of freedom in a spherical code, C ↮ ⋆ ⊂ S 2 , to have negative values without antipodal interference.

5) By maximising the minimum vertex separation optimal non-antipodal spherical codes effectively exhibit vertex-vertex “repulsion” which optimises their distinction and isolates degrees of freedom from detrimental crosstalk.

6) Chirality in the maximum vertex set of the sequence of exceptional spherical codes establishes a foundation for observed asymmetric physics (antimatter-matter) in our otherwise substantially symmetrical quantum physical cosmology. The maximum vertex set in Tribonacci-QC has 24 vertices in a spherical snub cubical configuration.

Applying the above conditions, rules and concepts, by elimination and Natural selection, the model invokes four optimal non-antipodal spherical codes: 1-vertex, 3-vertex (equilateral triangle on a great circle), 4-vertex (spherical tetrahedron) and 24-vertex (spherical chiral snub cube). The vertices are points of independent, coherent, einselected [61] [75] speciated information [1] [76] , maximising separation and surviving environmental decoherence on a quantum noisy spherical horizon. Radii through the codes’ points on the 2-sphere, S 2 , configure isomorphic degrees of freedom in quantum cosmological information processing. These degrees of freedom are shared dimensions, that is they are utilised by the universe for its parallel quantum information processing and discretised at Planck and cosmological scales. Envisage points coincident with the code vertices on the 2-sphere, S 2 , relate to Planckian computational discretisation, whilst vector heads of long radial vectors through the spherical code vertices relate to cosmological computational discretisation.

Spherical code evolution is a computationally intelligent, deep learning, self-organization process (Figure 2). Twenty-four 1-vertex codes self-organize into eight 3-vertex codes, which then self-organize into one 24-vertex code, isomorphic to the dimensions of 24-spacetime, in which antecedent quantum astrophysical structures emerge (albeit hard to completely visualise from and through subsequent 4(6)-spacetime). The 24-vertex code is simultaneously isomorphic to the dimensions of quantum mechanics (Section 3).

Quantum self-organization on the spherical horizon locks at 24-vertex snub cubical packing, however information complexity inexorably continues to compound under Nature’s epistemic drive [1]. The 24-vertex code coarsens into a broadband 4-vertex code, isomorphic to the dimensions of emergent 4(6)-spacetime. At this coarsening transition antecedent 24-spacetime quantum astrophysical structures metamorphose into their familiar 4(6)-spacetime quantum astrophysical structures. The transition from 24-spacetime to 4(6)-spacetime is a form of geometrogenesis, during which energy is conserved. Degrees of freedom are also conserved, though 6-fold parallelization occurs to accommodate the lower symmetry. This spontaneous phase transition may also be considered as a flash freezing via nucleation on the spherical horizon of the optimal non-antipodal codes, C ↮ ⋆ ⊂ S 2 . Analogously, when water flash freezes to ice, its degrees of freedom instantly reduce and its scale factor steps up. When the rotationally symmetric liquid water state has degrees of freedom removed at a phase transition it crystallises into ice with a regular lattice structure and its volume expands. In Tribonacci-QC, the transition from 24-spacetime to 4(6)-spacetime causes space to expand.

Each of the vertices of the 4-vertex code constitutes a quantum superposition of 6 parallelized corresponding isomorphic degrees of freedom (conserved from the prior 24-vertex code). Congruently, the total matter density parameter Ω m is 6-fold heavier than the baryon density parameter Ω b and an accompanying torrent of information-equivalent energy downloads from 6-fold faster 24-spacetime to emergent 4(6)-spacetime. Consequent stress on coarser 4(6)-spacetime causes it to resize its dynamic memory by eruptively expanding its cosmological scale (Section 5). For a related view on staged self-assembly of colloids see [73]. To deepen our understanding of the foregoing, we turn to graph theory and to Figures 3-5.

In Figure 3 we see illustrated a 24-vertex, 60-edge, snub cubical graph in its triangular perimeter, see also [77] (this graph is more commonly illustrated in its square perimeter [78] [79] ). In its triangular perimeter, note the nested 1-gon (central vertex), in 5-gon (purple), in 8-gon (green), in 7-gon (red), in 3-gon (light blue) which has the following Diophantine relations, via their integer representations:

1 + 5 + 8 + 7 + 3 = 24 (2.1)

1 − 5 + 8 − 7 + 3 = 0 (2.2)

1 2 − 5 2 + 8 2 − 7 2 + 3 2 = 0 (2.3)

The author presently offers no physical interpretation of this observation, though future research may link this pattern of integers to some fundamental properties in physics. I guess tropical geometric comparison of the snub cubical graph with the Standard Model gauge group, G S M , particularly as it relates to the special unitary group S U ( 5 ) , may provide such enlightenment.

Illustrated in Figure 4, eight 3-vertex sets self-organize into the 24-vertex set, isomorphic to the dimensions of 24-spacetime. In the equivalent spherical codes, before ultimately settling into one, compact, snub cubical, 24-vertex code, the eight 3-vertex codes are equilateral triangles on great circles, i.e. they comprise eight 3-vertex optimal non-antipodal spherical codes that key together, interlocking to form the 24-vertex code. Considering the quantum self-organization of the isomorphic 24-vertex spherical code, it is noted that a small settling compaction is required to ultimately form a tightly fitting snub cubical optimal non-antipodal spherical code from eight such component equilateral triangles. The compacted and settled equilateral triangles, forming the self-organized snub cube, are finally circumscribed by (near-great) small circles with radius approximately 0.99688 of a corresponding unit radius spherical great circle. The settling compaction, ∆, is given in (2.4) as a function of the tribonacci constant, T. Section 4 provides the geometric properties of the snub cube in terms of the tribonacci constant, from which this compaction factor is readily calculated.

Δ = 2 T 2 + 1 T 2 + T − 1 T + 2 3 1 T 2 + 1 T − 3 T − 2 ≈ 0.99688 (2.4)

Shown in Figure 5, the 24-vertex, 60-edge, snub cubical graph is in its triangular perimeter (Figure 3) with vertices numbered and coloured. The numbers pertain to the following 5-valency, snub cubical graph vertex adjacency list (Table 1) [79] :

The snub cubical graph has Hamiltonian cycles. Let us consider the pertinent example Hamiltonian super-cycle, H ↻ , depicted in Figure 5:

H ↻ = { 1 1 → 3 2 → 19 3 → 24 4 → 7 5 → 9 6 → 16 7 → 14 8 → 11 9 → 10 10                     → 13 11 → 23 12 → 5 13 → 21 14 → 18 15 → 4 16 → 6 17 → 17 18                     → 12 19 → 15 20 → 8 21 → 20 22 → 22 23 → 2 24 → 1 1 } (2.5)

which starts and recycles through the central vertex, numbered 1 (labelled yellow) and with superscript 1 referring to its index number in this example Hamiltonian super-cycle. The second vertex in the cycle is numbered 3 (labelled red) and with superscript 2 denoting its index as the second vertex in this example Hamiltonian super-cycle. The third vertex in the cycle is numbered 19 (labelled green) and with superscript 3 denoting its index as the third vertex in this example Hamiltonian super-cycle. The fourth vertex in the cycle is numbered 24 (labelled blue) and with superscript 4 denoting its index as the fourth vertex in this example Hamiltonian super-cycle, and so on through this cycle (2.5).

Note there are 6 sub-cycles comprising 4 vertices which start and end on a yellow labelled vertex, V yellow :

V yellow = { 1 1 , 7 5 , 11 9 , 5 13 , 6 17 , 8 21 } (2.6)

A snub cube has 6 square faces and these 6 yellow labelled vertices each sit on a distinct square face (quadrilateral in the graph representation). The 6 corresponding vertex numbers from the above example Hamiltonian super-cycle (2.5) and (2.6) have the following Diophantine relations:

1 + 7 + 11 + 5 + 6 + 8 = 38 = F snubcube (2.7)

1 + 7 + 11 − 5 − 6 − 8 = 0 (2.8)

1 + 7 + 11 = 5 + 6 + 8 (2.9)

where F snubcube is the number of faces of the snub cube. The author presently offers no physical interpretation of this observation, though future research may also link this pattern of integers to some fundamental properties in physics. Again, tropical geometry may provide the key.

The 6 yellow labelled vertices (2.6) are defined as isomorphic to time-like degrees of freedom, due to their model-assigned role in clocking the 6 successive ordered sub-cycles in the 24-vertex example Hamilton super-cycle (2.5).

Whereas in this example Hamiltonian super-cycle (2.5) the yellow labelled vertices, 1, 7, 11, 5, 6, 8, are designated time-like, the red, green and blue labelled vertices are space-like. The vertices labelled red are designated to the x dimension of space and have vertex numbers 3, 14, 13, 21, 15, 22. The vertices labelled green are designated to the y dimension of space and have vertex numbers 19, 9, 23, 18, 17, 2. The vertices labelled blue are designated to the z dimension of space and have vertex numbers 24, 16, 10, 4, 12, 20.

Let us denote generic space-like vertices ( l ) , for length, whereas specific space-like vertices are ( x , y , z ) . Returning to Figure 4, we now recognise that the eight 3-vertex sets, in this example Hamiltonian super-cycle (5), which self-organize into the snub cubical graph depicted in Figure 5, are categorised in the following summation:

1 ( l 2 t ) + 4 ( 2 l t ) + 3 ( 3 l 0 ) ⇒ 6 ( 3 l t ) (2.10)

That is, for the example Hamiltonian super-cycle (2.5), one 3-vertex set with two time-like vertices, plus four 3-vertex sets with one time-like vertex and three 3-vertex sets which are entirely space-like, self-organize into a 24-vertex set comprising six time-like vertices and eighteen (6 × 3) space-like vertices.

Figure 6 shows a 4(6)-vertex tetrahedral graph derived from a 6-fold chromatic parallelization of the vertices depicted in the 24-vertex snub cubical graph in its triangular perimeter of Figure 5. The colours in the example Hamiltonian super-cycle (2.5) have self-organized into 4 monochromatic vertices (yellow is time-like ( t ) , whilst red ( x ) , green ( y ) and blue ( z ) are space-like). This transformation from the 24-vertex graph to the 4(6)-vertex tetrahedral multigraph represents a coarsening of the isomorphic dimensions of 24-spacetime to those of 4(6)-spacetime.

The 6 time-like yellow labelled vertices (Figure 6) are isomorphic to 6 corresponding parallelized degrees of freedom and are collocated centrally, with vertex number 1 (tetrahedron apex). The red, green and blue labelled vertices are space-like and collocated at respective vertices of the graph’s triangular perimeter (tetrahedron base). Referring to Figure 5 and Figure 6, the red labelled vertices are collocated with vertex number 14, the green labelled vertices are collocated with vertex number 9 and the blue labelled vertices are collocated with vertex number 16 (Figure 7).

A simplified version of Figure 6 is shown in Figure 7 which depicts the chromatic parallelization of vertices of a tetrahedral graph, isomorphic to dimensions of 4(6)-spacetime. This is the result of coarsening the snub cubical graph shown in Figure 5. The arrows are directed from the example Hamiltonian super-cycle (2.5). The yellow dot ( t ) represents 6 superposed time-like vertices. The red dot ( x ) represents 6 superposed space-like vertices. The green dot ( y ) represents 6 superposed space-like vertices. The blue dot ( z ) represents 6 superposed space-like vertices. I now deconstruct Figure 6 and Figure 7, to further illustrate the workings of the example Hamiltonian super-cycle (2.5).

Adopting the layout of Figure 7, I now exhibit, in Figure 8, a deconstruction of the example Hamiltonian super-cycle (2.5) into its 6 sub-cycles. A rotating pattern is evident.

In Figure 8 we can see the example Hamiltonian super-cycle (2.5), (Figure 5), coarsened and 6-fold parallelized, as in Figure 7. It is deconstructed to show its 6 tetrahedral subgraph sub-cycles, which register a 4π (720˚) identity circuit, in 6 ordered turns of 2π/3 (120˚), around the yellow-labelled time-like vertex, in a rotating pattern. Each of the 4 tetrahedral vertices (isomorphic to 1-time dimension and 3-space dimensions) comprises 6 parallelized corresponding degrees of freedom.

We may think of the above discussion and Hamiltonian super-cycle (2.5), which concludes with Figure 8, as formalizing a kind of cellular automaton, or vector space analysis, that frames spacetime events (located moments) [58] [80].

A complementary path to map the 24-vertex set snub cube to the 4-vertex set tetrahedron is via the 24-cell and the binary tetrahedral group [81] of order 24 (Section 3). The binary tetrahedral group is the group of units in the ring of Hurwitz integers, with its 24 units given by all sign combinations of

{ ± 1 , ± i , ± j , ± k , 1 2 ( ± 1 ± i ± j ± k ) } (2.11)

All 24 units have absolute value 1 and lie in the unit quaternion group Sp(1). The convex hull of the 24 elements in 4-dimensional space forms the 24-cell, which is a convex regular 4-polytope.

Figure 9 illustrates the same snub cubical graph as shown in Figure 5, with the same vertex number labels and same sequential indices, as per the example Hamiltonian super-cycle (2.5). Here, however, this configuration of the snub cubical graph accentuates a decomposition of the 24-vertex set into four 6-gons (solid black edges). Whilst the four 6-gons in Figure 9 might establish a definition of four vertices of an isomorphic tetrahedral graph, there is no obvious computational parallelization benefit, nor direction, within any of the 6-gons, hence the parallelized configuration depicted in Figures 6-8 is Naturally selected in the Tribonacci-QC model.

Summarising this Tribonacci-QC, optimal non-antipodal spherical code, model thus far, Figure 2 illustrates that twenty-four 1-vertex codes self-organize into eight 3-vertex codes (Figure 4, in graph representation), which then self-organize into one 24-vertex code (Figure 5, in graph representation), isomorphic to dimensions of 24-spacetime. This 24-vertex code then coarsens into a broadband 4(6)-vertex code that has a spherical tetrahedral arrangement with 6-fold parallelization of vertices. The 4(6)-vertex code is isomorphic to the dimensions of 4(6)-spacetime. The vertex sets in Figure 2 are optimal non-antipodal spherical codes which are einselected [61] centres of coherent quantum information that maximise their minimum separation and survive environmental decoherence on a noisy spherical horizon. As such they configure the isomorphic degrees of freedom of emergent spacetimes and the dimensions, or generators, of the S U ( 5 ) group (Section 3).

The 24-vertex code has 6 time-like vertices, one per each of the four square faces of the snub cubical configuration. Quantum cosmological information processing, with deep learning, executes 6-fold faster through its resource of 6 distinct time-like degrees of freedom which order moments in time. Clocks in 24-spacetime advance 6-fold faster than in the subsequent coarser 4(6)-spacetime.

6-fold parallelization of 1 time-like and 3 space-like degrees of freedom in coarser 4(6)-spacetime provides Nature with quantum cosmological information processing which is 6-fold deeper and emerges 6-fold heavier than classically apparent. It is proposed in the model, that this 6-fold parallelization is manifest in the classical realm via 4(6)-dimensional general relativity and in the parallel processed quantum microscopic realm via 24, or more strictly, 12(2)-dimensional quantum physics, with both sets of dimensions being referenced to the spherical horizon.

Although a hard problem, we can begin to imagine extending 4(6)-dimensional (3-space, 1-time) general relativity back into prior 24-spacetime, with its finer-grained 24-dimensional (18-space, 6-time) quantum general relativity.

4-dimensional spacetime and corresponding general relativity is very well established in modern physics. It has also been related to a 2-dimensional horizon, via the holographic principle [82] [83] [84] and AdS/CFT correspondence [85].

6-fold higher degrees of freedom in 12(2) = 24-dimensional quantum physics also feature in established research and models of particle physics. According to quantum mechanics, a harmonic oscillator that vibrates with frequency ω can have quantized energy

1 2 ω , ( 1 + 1 2 ) ω , ( 2 + 1 2 ) ω , ( 3 + 1 2 ) ω , ( 4 + 1 2 ) ω , ⋯ (3.1)

in units where Planck’s constant equals 1. The lowest energy is 1 2 ω , (not zero due to the uncertainty principle), which is the ground state of the oscillator.

Euler observed that the surprizing sum of the Natural numbers to infinity is

1 + 2 + 3 + 4 + ⋯ = − 1 12 (3.2)

It is now well known that the zeta function of −1 has the same value

ζ ( − 1 ) = − 1 12 (3.3)

A string oscillating in one direction is the same as an infinite collection of oscillators with frequencies

ω = 1 , 2 , 3 , 4 , ⋯ (3.4)

and their ground states add, so the ground state energy of the string is given by the zeta function regularization

1 2 ( 1 + 2 + 3 + 4 + ⋯ ) ω = 1 2 ζ ( − 1 ) = − 1 24 ω (3.5)

This is one of Nature’s many fundamental mathematical incidences of the number 24 and some other occurrences are summarised as follows.

The Leech lattice Λ 24 is one of the most profound. It can be defined using octonion triplets and the Lie group E 8 [86] [87] which directly relates it to the mathematical foundations of particle physics [26] [27] [28] [29] [30]. The Leech lattice Λ 24 is the densest hypersphere packing in 24 dimensional Euclidean space and is mathematically related to the binary Golay code, which is a renowned error-correcting code capable of efficiently detecting 4 errors and correcting up to 3 errors in each 24-bit word.

The Dedekind eta function, η ( τ ) , is also mathematically related to the Leech Lattice, where

η ( τ ) = q 1 24 ∏ n = 1 ∞ ( 1 − q n ) (3.6)

and q ≡ e 2 π i τ , from number theory, and τ is any complex number where Im ( τ ) > 0 . The exponent 1/24 is a consequence of the relationship with Λ 24 and furthermore the Dedekind eta function, η ( τ ) , is intrinsically related to Virasoro algebra, conformal field theory and string theory.

Chapline’s work on a matrix non-linear Schrödinger equation living on a Lorentzian extension of the 24-dimensional Leech lattice, as a potential framework for a quantum theory of gravity and elementary particles in -dimensions, is also insightful [88].

Also related to the Leech lattice, so called monstrous moonshine (moonshine theory) is the surprising relationship between modular functions (particularly the j-function) and the monster group [89] , again connecting the number 24 to string theory.

Parenthetically, the Higgs field in S U ( 5 ) is 24-dimensional. I return to S U ( 5 ) theory below.

Indeed, Nature’s above proclivity towards the number 24 was wonderfully explored by Baez in a Rankine Lecture in 2008, and a recording can now be viewed online [19]. The numerical connections between the 1, to 3, to 8-vertex set quantum self-organization in this quantum cosmology model, Tribonacci-QC, and the profound occurrences of unitarity, triality and the octonions in mathematical physics and quantum physics, is noteworthy.

Whilst the Tribonacci-QC model is not presented in the language of bosonic string theory, an association is conjectured through the fundamental relationship of both to 24 degrees of freedom. Physical degrees of freedom of the bosonic string are given as 24 transverse coordinates relative to a worldsheet, whereas the 24-vertex set, and its coarser form the 4(6)-vertex set, optimal non-antipodal spherical codes of the Tribonacci-QC have 24 degrees of freedom relative to their spherical horizon.

In the rest of this section we examine the shared dimensions of the Tribonacci-QC model by considering possible correspondences between shared dimensions of spacetime and the dimensions of particle physics. In doing so, we explore paths framed by the Tribonacci-QC model which potentially lead to the discovery of long-sought fundamental correspondences between general relativity and quantum mechanics. Again, tropical geometry may provide useful tools to further formalize the correspondences.

Referring to Figure 10 and recent work by Furey [90] and others, there are 24 generators of S U ( 5 ) [18] [91] ladder symmetries which split into two types: 12 mixing and 12 non-mixing. Also recall (3.2) and (3.3) above, where the significance of this integer, twelve, is emphasised.

Consider non-mixing generators analogous to the Naturally selected optimal non-antipodal spherical codes in this Tribonacci-QC model. Non-mixing and non-antipodal characteristics both bestow discrete identity. Tribonacci-QC considers 12(2)-dimensional quantum physics occurring in deep 24-spacetime, thence and presently parallelized in coarser 4(6)-spacetime.

The deselected 12 mixing generators (dark boxes placed in Figure 10), responsible for proton decay, are excluded by Furey because in her division algebraic construction the two types of ladder operators are clearly algebraically distinct. The Naturally selected non-mixing generators (white boxes placed in Figure 10) do not mix the two types of ladder operators. Transitions leading to proton decay are expected by Furey to be blocked, given they coincide with forbidden transformations which would incorrectly mix distinct algebraic actions.

The 12 Naturally selected non-mixing generators comprise 8 which generate S U ( 3 ) C , 3 which generate S U ( 2 ) L and 1 which coincides with hypercharge Y. In the Tribonacci-QC model, the tetrahedral 4(6)-spacetime state emerges from the coarsening of the snub cubical 24-spacetime state. In a quantum cosmological computational process, involving parallel processing in shared dimensions, the tetrahedral spherical code configuring isomorphic degrees of freedom also accommodates a parallelization of the 24 generators of S U ( 5 ) and Naturally selects the 12 generators of the Standard Model gauge group, G S M .

The red-green-blue base of the tetrahedral graph (Figure 6 and Figure 7) accommodates the spatial dimensions of spacetime and, via shared dimensions, in G S M each of these 3 vertices also accommodates 2 non-mixing generators of S U ( 3 ) C and a single indivisible non-mixing generator of S U ( 2 ) L (Figure 10). The yellow apex of the tetrahedral graph (Figure 6 and Figure 7) accommodates the time dimensions of spacetime and in G S M this apical vertex also accommodates 2 non-mixing generators of S U ( 3 ) C and a single indivisible non-mixing generator of U ( 1 ) Y (Figure 10).

The degree of parallelization through dimensional reduction of the 24 generators of S U ( 5 ) at each vertex has a limiting lowest common denominator of 2, set by the single indivisible generators at each vertex (an S U ( 2 ) L generator at each vertex on the triangular base and a U ( 1 ) Y generator at the apical vertex). The deselected 12 mixing generators are located virtually at their corresponding vertices, however they are computationally overwritten in the Tribonacci-QC

model by the Naturally selected, principal, 12 non-mixing generators. Expanded mathematical definitions of the mixing and non-mixing generators are given by Furey [90].

Broadly, the Standard Model consensus theory of particle physics, is an S U ( 3 ) × S U ( 2 ) × U ( 1 ) gauge theory which is experimentally well established at presently probed scales. The Standard Model gauge groups S U ( 3 ) × S U ( 2 ) × U ( 1 ) fit in the S U ( 5 ) group of special unitary transformations acting on 5 complex variables.

Alleviating its proton decay problem, Furey [90] retains highly attractive features of S U ( 5 ) theory. She considers the Standard Model represented by G s m = S U ( 3 ) C × S U ( 2 ) L × U ( 1 ) Y / ℤ 6 (note the ℤ 6 is commonly modded out) and finds that the S U ( 5 ) ladder symmetries then reduce to it, where ladder operators arise from the division algebras ℝ (real numbers), ℂ (complex numbers), ℍ (the quaternions), and O (the octonions).

Koca et al. [92] formally describe chiral polyhedral, including the snub cube, as derived from coxeter diagrams and quaternions. Future studies may connect their work to that of Furey [90] and to others, including Cerda-Morales who formalizes (snub cubical) tribonacci-type octonions sequences and describes their properties [24] see Section 4.1 below.

In graph theory, the snub cubical graph is a quintic graph where all vertices have degree 5. See the vertex figure in Figure 11. I conjecture that a fundamental connection exists between the 5 complex variables in S U ( 5 ) and the 5 edges incident to each of the 24 vertices of the snub cubical graph of this Tribonacci-QC model (another lead for tropical geometricians). S U ( 5 ) has 24 corresponding generators, where 5 × 5 − 1 = 24 ; twelve of which are Naturally selected and twelve of which are virtual (Naturally deselected). Recall, Tribonacci-QC considers 12(2)-dimensional quantum physics occurring in deep 24-spacetime, thence and presently parallelized in coarser 4(6)-spacetime.

Alternatively, let us consider the representation of the snub cube, central to Tribonacci-QC, as two 12-vertex sets, one corresponding to the 12 Naturally selected generators and the other corresponding to the 12 virtual (Naturally deselected) generators. In Figure 12, we see the 24-vertex snub cubical graph configured in two sub-graphs, each comprising 12 vertices (one with vertices coloured blue and numbered 1, 2, 3, 4, 5, 6, 7, 19, 20, 22, 23, 24; and the other with vertices coloured green and numbered 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21). The number of coupling-edges, connecting the two sub-graphs, is minimized to twelve (in Figure 12 these 12 coupling-edges are coloured cheddar-orange, like stringy cheese between two halves of a grilled cheese sandwich!).

Each of the 12-vertex sub-graphs (blue subgraph and green subgraph) has 24 edges. The vertex numbering is consistent with the adjacency list in Table 1. Depicting an alternative configuration to Figure 10, these two 12-vertex sub-graphs in Figure 12 represent two sets of 12 generators, selected (non-mixing) and deselected (mixing), in S U ( 5 ) . Let us consider the green vertices selected, and the blue vertices deselected.

Consider Figure 13, which shows the snub cubical graph depicted in Figure 12, however here it is constructed within a triangular perimeter, like that in Figure 5. We see the blue vertices representing Naturally deselected mixing generators of S U ( 5 ) are located centrally, clustered around the apex of the corresponding (coarsened) tetrahedral graph configuration (Figure 7). Whereas the green vertices, representing Naturally selected non-mixing generators of S U ( 5 ) circumscribe the blue apex in a corresponding base-of-tetrahedral graph configuration. In Figure 7 the apex is defined to be time-like, whereas the basal triangle is space-like, with each vertex of the basal triangle corresponding to a dimension of 3-space. From this definition of blue and green vertices, we see that the green, Naturally selected 12 generators of S U ( 5 ) , are disposed to be more space-like in terms of vertex adjacency (Figure 13), although they are both spacelike and time-like in terms of the Hamiltonian super-cycle (2.5) depicted in Figure 5 and Figure 6.

See also Section 8.2, where Figure 13 is interpreted as a quantum mechanical configuration of a Planck star at the core of a black hole, where the enclosed, apical, time-like vertices 1, 2, 3, 4, 5, 6, 7, 19, 20, 22, 23, 24, correspond to quantum mechanics concealed within the black hole.

In this Tribonacci-QC model dimensions are shared. A challenge is to discover the correspondence between the 12(2)-generators that are isomorphic to particle physics and to the degrees of freedom written in spherical codes that are isomorphic to dimensions of 24-spacetime, and its coarser manifestation, 4(6)-spacetime.

Figure 10 depicts the placement of the 12 Naturally selected (and deselected) generators of S U ( 5 ) at the 4 vertices of the coarsened graph, i.e. the tetrahedral graph, representing the spherical tetrahedral spherical code C ↮ ⋆ 4 ( 6 ) . In Figure 10 three Naturally selected generators are placed at each of the 4 tetrahedral vertices. An alternative placement is depicted in Figure 13, in which the 12 Naturally selected generators of S U ( 5 ) are placed anti-clustered, enclosing the time-like apex, i.e. constituting the space-like base.

These alternative correspondences (Figure 10 versus Figure 13) are presented in this introduction of the Tribonacci-QC model to prompt further research, including by tropical geometricians and algebraists, into determining the best correspondences between the dimensions of spacetime and the dimensions of the Standard Model of particle physics. Furthermore, these two alternative correspondences are non-exclusive, further alternatives should be sought and contrasted.

Parenthetically, Borsten et al. [93] derive E 8 symmetry from 8 qutrits, which relates to the self-assembly of eight 3-vertex codes into the 24-vertex code of Tribonacci-QC (Figure 2 and Figure 4). The E 8 Lie group has applications in several definitions of G s m .

In Section 2, above, I point out that a complementary path to map the 24-vertex set snub cube to the 4(6)-vertex set tetrahedron is via the 24-cell and the binary tetrahedral group of order 24 (2.11). The group of unit quaternions is the double cover of the 3d rotation group. There are 12(2) = 24 unit quaternions that give rotational symmetries of the tetrahedron. These form the binary tetrahedral group. Unit quaternions thus link the tetrahedron to the number 24 and elliptic curves. The 24 unit quaternions in the binary tetrahedral group are also the Hurwitz integers of norm 1. The reader is directed to Braun’s recent work on the 24-Cell and Calabi-Yau threefolds with Hodge numbers (1, 1) [94] , Witten’s seminal contributions, for example [95] , those of Gross et al. [96] and to the works of Morrison and Vafa [97] [98] on compactifications on Calabi-Yau threefolds, which emphasise the fundamental relationship between those fields of research and the number 24.

I note the most promising approaches to formalize the foundations of physics deeply incorporate the number 24 and we should seek even deeper understanding of this exceptional number, particularly as it relates to computational degrees of freedom. I propose this future exploration be done with reference to the 24-vertex snub cube spherical code and its parallelized 4(6)-vertex tetrahedron spherical code of Tribonacci-QC.

Finally, in this section, a recap of some profoundly simple mathematics related to the number 24. The number 24 has more divisors (exactly eight) than any smaller number, making it a highly composite number which provides computational utility with dimensional parsimony. 24 is the 4D kissing number [99]. 24 is also the factorial of 4 (24 = 4!) and the tesseract (the 4D analogue of the cube) has 24 two-dimensional faces, each of which is a 4-sided square. The product of any 4 consecutive numbers is divisible by 24 and 24 is the number of ways to order 4 distinct items.

The 12(2)-generators (i.e. 12 mixing and 12 non-mixing) in Furey’s work [90] on division algebras and S U ( 5 ) ladder symmetries yields:

2 × 12 = 24 (3.7)

Borsten et al. [93] derive E 8 symmetry from eight 3-level qutrits:

3 × 8 = 24 (3.8)

Baez makes a simple but profound observation, that 4-fold and 6-fold lattices in the plane have the most symmetries and the product of their fold is given by:

4 × 6 = 24 (3.9)

Similarly, deeply related to the Standard Model of particle physics, the special unitary group of degree 5, S U ( 5 ) , is the Lie group of 5 × 5 unitary matrices with determinant 1 and can be represented:

5 × 5 ¯ − 1 = 24 (3.10)

Finally, as we discuss in Section 4.1, the number of permutations of a signature code, of a certain third order recurrence sequence, the tribonacci sequence, drawn from the balanced ternary set { − 1 , 0 , 1 } and which are not single valued is

3 3 − 3 = 24 (3.11)

In this section I discuss the characteristics of the snub cube, firstly and briefly in an introductory way. Then two sub-sections cover details pertaining to the fundamentally related tribonacci constant and tribonacci sequence (Section 4.1) and to the chirality and twist angle of the snub cube (Section 4.2).

The snub cube is an Archimedean solid polyhedron. Kepler studied it in his Harmonices Mundi of 1619 and he called it the cubus simus. I humbly suggest Kepler’s intuitive search for a connection between the snub cube and cosmology is being continued in this present research, albeit building on four hundred years of scientific progress.

The snub cube has 24 vertices, 60 edges and 38 faces, 6 of which are squares and 32 of which are equilateral triangles. It can be represented as a spherical tiling and as a graph, as above. The symmetry group is octahedral, with order 24, and it is thus sometimes called the snub cuboctahedron. It is chiral, having two enantiomorphs, levo and dextro.

The tribonacci constant, T, is implicit in the snub cube, specifically, the Cartesian coordinates for the vertices of a snub cube are all the even permutations of

( ± 1 , ± 1 T , ± T ) (4.1)

with an even number of plus signs, together with all the odd permutations with an odd number of plus signs, and where T ≈ 1.8392867552 ⋯ is the tribonacci constant (Section 4.1). The even permutations with an odd number of plus signs and the odd permutations with an even number of plus signs, gives the enantiomorph.

Its circumradius, R, is given by [20]

R = 3 − T 8 − 4 T (4.2)

and (4.1) and (4.2), together with the geometrical properties of spherical equilateral triangles, lead to (2.4).

The preceding Sections 1 to 4 introduce the foundations of Tribonacci-QC and provide context to now define the cosmological scale factor evolution, a ( t ) . Referring to (4.2.3) (4.2.4) (4.2.5) we now contextualize the formalizations in this Section 5 within the normal graph-stress component, σ 4 ( 6 ) 24     , of the snub cubical graph.

From the following formalized cosmological scale factor evolution, equations are then derived for the Hubble parameter (Section 5.1) and deceleration-acceleration transition redshift (Section 5.2).

In Tribonacci-QC, the transition from dominantly 24-spacetime to dominantly 4(6)-spacetime (Section 2, Figure 2) necessitates, by definition, an abrupt change in a ( t ) at t = 0 , with an equation for a ( t ) when t < 0 , an equation for a ( t ) when t > 0 and the two equations Naturally splicing continuously at transition time t = 0 . Furthermore, a characteristic difference between the two equations is that during time t < 0 (in 24-spacetime) the pace of time is 6-fold faster than in parallelized 4(6)-spacetime.

Let us first focus on formalizing the cosmological scale factor evolution, a ( t ) , when t > 0 , i.e. in 4(6)-spacetime, because this is our spatio-temporal framework for present cosmological observations. We shall then modify its equation for time t < 0 (in 24-spacetime), when the pace of time was 6-fold faster.

The equation, in Tribonacci-QC, for cosmological scale factor evolution, a ( t ) , when t > 0 , is defined:

a ( t ) = 3 t T / 3 − 1 (5.1)

and when t < 0 , the cosmological scale factor evolution is defined in 6-fold faster time:

a ( t ) = 3 ( 6 t ) T / 3 − 1 (5.2)

In Figure 18, below, we see a plot of (5.1) and (5.2), for real (blue) and imaginary (orange) values and is Naturally smoothly spliced through the transition event at time t = 0 .

The ordinate intercept in Figure 18 (5.1) (5.2) for real values (blue) Re [ a ( t ) ] is:

a ( t ) = 1 3 ,       when   t = 0 (5.3)

A standard equation relates cosmological scale factor a ( t ) and cosmological redshift z:

a ( t ) = 1 1 + z (5.4)

Substituting (5.3) in (5.4) gives

z = 2 ,     when   t = 0 (5.5)

This result (5.5) is a novel and distinguishing characteristic of Tribonacci-QC and relates to the redshift z = 2 when dominantly 24-spacetime transitions into dominantly 4(6)-spacetime. In Section 8.3 the significance of z = 2 as a fundamental marker redshift in astronomical observations is discussed.

The real valued a ( t ) abscissa intercept in Figure 18 (5.2), nearest t = 0 , in negative model time, corresponds to the time of last zero cosmological scale, when a ( t ) = 0 :

t a = 0 ( − 1 ) ≈ − 0.33174 (5.6)

Let us now convert model time to look back time in billions of years (Gyr). Using an astronomical timescale [40] , anchored on a cosmological age of 10.4 Gyr at redshift z = 2 , we thus calibrate

t a = 0 ( − 1 ) ≈ a b s | − 0.33174 − 1 | × 10.4 ≈ 13.8   Gyr (5.7)

The time of last zero cosmological scale (5.7) matches the Planck 2018 estimate [104] for the age of the observable universe.

The abscissa intercept in Figure 18 (5.2), second nearest to t = 0 , in negative model time, corresponding to the 2^nd last time of zero cosmological scale:

t a = 0 ( − 2 ) ≈ − 1.99074 (5.8)

By the same calibration, anchored on a cosmological age of 10.4 Gyr at redshift z = 2 , we thus astronomically calibrate the 2^nd last time of zero cosmological scale as occurring at cosmological time:

t a = 0 ( − 2 ) ≈ a b s | − 1.99074 − 1 | × 10.4 ≈ 31   Gyr (5.9)

As a series, in negative model time, the real valued abscissa intercepts with zero cosmological scale factor (5.2) are:

t a = 0 ( − 1   to   − 10 ⋯ ) ≈ − 0.33174 , − 1.99074 , − 4.58000 , − 7.92884 , − 11.94626 ,                                   − 16.57218 , − 21.76279 , − 27.48413 , − 33.70878 , − 40.41394 , ⋯ (5.10)

which corresponds to the following cosmological ages anchored on 10.4 Gyr at redshift z = 2

t a = 0 ( − 1   to   − 10 ⋯ ) ≈ 13.8   Gyr , 31   Gyr , 58   Gyr , 93   Gyr , 135   Gyr ,                                   183   Gyr , 237   Gyr , 296   Gyr , 361   Gyr , 430   Gyr , ⋯ (5.11)

In Tribonacci-QC, the cosmological scale factor evolution a ( t ) , (5.1) (5.2), is an Ostwald coarsening with the cube root of time [105] [106] , scaled by the tribonacci constant ( T ≈ 1.83929 ) metric property of the snub cubical 24-vertex and parallelized 4(6)-vertex spherical codes. Tribonacci-QC is a quantum foundational model founded on the integer base of quantum ternary computing. The integer base and denominator of the Ostwald coarsening exponent in equations (5.1) and (5.2) is 3, which has integer lowest radix economy, i.e. ⌈ e ⌉ = 3 , [1] [107]. The −1 in the exponent (5.1) (5.2) simply sets a ( t ) = 1 at present time t → 1 , as a relevant frame of reference.

In Tribonacci-QC, Ostwald coarsening of optimal non-antipodal spherical codes broadens the bandwidth of quantum information processing on the spherical horizon. During coarsening, from the 24-vertex set to the parallelized 4(6)-vertex set, the average signal coarseness increases with the third root of time (as is characteristic of Ostwald coarsening), scaled by the tribonacci constant, t T / 3 . The concept of an evolving average coarseness is important.

It is an average signal coarseness related to two coexisting metastable states, 24-vertex and 4(6)-vertex, which spontaneously transition from dominantly 24-vertex, to dominantly 4(6)-vertex states. In the physical universe, mixed states dominated by the 24-vertex state transition (at t = 0 ) into mixed states dominated by the 4(6)-vertex state. In Tribonacci-QC, the isomorphic dimensionality of 24-spacetime persists into times when t > 0 , including the present time, t → 1 , where it is proposed to manifest in at least two remnant configurations.

One of these remnant configurations of the 24-vertex state persists as the pervasive 12(2)-dimensional quantum physics realm (Section 3), microscopically and quantum graphically circumscribed within the spacetime fabric of the parallelised 4(6)-vertex state. The other of these remnant configurations of the 24-vertex state persists as Planck stars [45] at the cores of black holes.

At the coarsening transition, antecedent, quantum astrophysical structures created in dimensions of 24-spacetime metamorphose into their familiar 4(6)-spacetime forms e.g. a 24-dimensional galaxy metamorphoses into a familiar 4(6)-dimensional galaxy. It is the same galaxy but is now manifest (conserving matter and energy) in fewer and parallelized degrees of freedom. Section 8, on astronomical observations, expands on these themes.

Also recall Figure 13 which may be interpreted as a quantum mechanical configuration of a Planck star at the core of a black hole, where the circumscribed, apical, time-like vertices correspond to quantum mechanics concealed within the black hole (Section 8.2, below).

Figure 19, shows the evolution (5.2) of the real (blue) and imaginary (orange) values of a ( t ) in dominantly 24-spacetime, during negative model time ( t < 0 ).

The real cosmological scale factor (blue), Re [ a ( t < 0 ) ] , and imaginary cosmological scale factor (orange), Im [ a ( t < 0 ) ] , evolve (5.2) in negative model time, from t → − ∞ to t = 0 , as positive and negative valued scale factor oscillations with a ( t ) amplitude increasing towards t = 0 . Positive valued scale factors project outside the optimal non-antipodal codes’ spherical horizon, whereas negative scale factors project inside that spherical horizon.

Consider these two functions, Re [ a ( t < 0 ) ] , Im [ a ( t < 0 ) ] , alternatively written as coupled differential equations, then their potential to represent an autocatalytic oscillating reaction [108] [109] could be studied. Alternatively, we may consider these two functions representing an entanglement of two energy-matter phases self-pumping in an epistemically-driven cavity bounded by the spherical horizon of the optimal non-antipodal codes, C ↮ ⋆ ⊂ S 2 .

Pending such reformalizations, the real function Re [ a ( t < 0 ) ] (blue) is postulated to relate to matter, whereas the imaginary function Im [ a ( t < 0 ) ] (orange) relates to antimatter, and while the proposed autocatalytic oscillating reaction consumes all imaginary antimatter during negative time, from t → − ∞ to t = 0 (Figure 19), only real matter persists into positive time t > 0 (Figure 18).

It is also informative to consider Figure 20 which shows a parametric plot of equation (5.2), when time t < 0 , during the evolution of dominantly 24-spacetime. The imaginary values of a ( t ) are represented on the ordinate. The real values of a ( t ) are represented on the abscissa. The origin ( 0 , 0 ) of this plot represents the origin of time, when t → − ∞ .

Tribonacci-QC postulates that cosmological scale a ( t ) → 0 as t → − ∞ , for both real and imaginary values. The arrow of time advances clockwise from the origin and the right-hand end of the spiral curve intersects the abscissa at a ( t ) = 1 / 3 , z = 2 , when t = 0 , (5.3) (5.5). The parametric plot shows that infinite negative time is an attracting fixed point of the reverse time evolution.

Referring to (4.2.5) and the above discussion in Section 4 on a postulated quantum gravity correspondence in Tribonacci-QC, we contextualize the spacetime scale factor evolution formalizations in Section 5 (preceding) within the normal graph-stress component, σ 4 ( 6 ) 24     , of the snub cubical graph.

Whereas, in this Section 6 we consider the shear graph-stress component, τ 4 ( 6 ) 24     , of the snub cubical graph, which I claim relates to the matter side of Einstein’s spacetime-matter tensor equivalence (4.2.5).

The physical baryon density parameter, Ω b h 2 , is deeply fundamental to cosmology, and I commence this section with its definition in Tribonacci-QC. Returning to the 24-vertex snub cube, a fundamental shear graph-stress parameter is the snubification twist angle, θ , [20] which is formalized in (4.2.1).

Referring to Dirac’s belt trick (Figure 15) and a 4π identity rotation, in Tribonacci-QC the twist angle, θ , divided by 4π defines a ratio, β , (4.2.2). I postulate this ratio, β , relates fundamentally to the shear graph-stress component, τ 4 ( 6 ) 24     , (4.2.5) and is equivalent to the physical baryon density parameter, Ω b h 2 , as follows:

Ω b h 2 = β = θ / 4 π = arccos ( T / 2 ) / 4 π ≈ 0.02287 (6.1)

Due to the 6-fold parallelization, from 24-spacetime to 4(6)-spacetime (Figure 2), the matter density parameter, Ω m , in Tribonacci-QC is 6-fold heavier than the baryon density parameter, Ω b . Inserting the formalization of the Hubble parameter h t → 1 (5.14) in (6.1) gives:

Ω b = arccos ( T / 2 ) / 4 π ( 1 3 T log ( 3 ) ) 2 ≈ 0.50414 (6.2)

which is reconcilable with standard cosmic nucleosynthesis and the matter density parameter, Ω m , is:

Ω m = 6 Ω b = 27 arccos ( T / 2 ) / 2 π log 2 ( 3 ) T 2 ≈ 0.30248 (6.3)

which is close to the Planck 2018 result [104].

It is interesting to note the recent work of Miyatake et al. [43] on a measurement of the cosmographic distance ratio, r, with galaxy and cosmic microwave background lensing, where

r = γ t o / γ t c (7.1)

and where γ t o and γ t c are the optical and cosmic microwave background tangential shear. Their measurement of r leads me to propose another equivalence in Tribonacci-QC:

| p 1 | = | p 2 | = r = γ t o γ t c = T − 1 2 ≈ 0.419643 (7.2)

where | p 1 | = | p 2 | is defined in (4.1.18) (4.1.19) (4.1.20) which relates to the real valued part of the second eigenvalue of the tribonacci polynomial companion matrix, M, (4.1.5).

Decompositions of the snub cubical graph adjacency matrix (4.2.6) and decompositions of the corresponding 24-vertex spherical code, C ↮ ⋆ 24 , (4.1) are conjectured above to yield rich insights into the quantum gravity correspondence (4.2.5), and models of particle physics and cosmology. The cosmographic distance ratio (7.2) is proposed to be one of these and future research aims to discover more.

Astronomical observations are used to further discuss and test Tribonacci-QC. We have already discussed the good fits between the model and observations as they pertain to the present value of the Hubble parameter, h t → 1 , (5.14) (5.16), the deceleration-acceleration transition redshift, z d a , (5.21), the physical baryon density parameter, Ω b h 2 , (6.1), baryon density parameter, Ω b , (6.2), matter density parameter, Ω m , (6.3), and cosmographic distance ratio, r, (7.2). These parameter-based fits are covered in above Sections 5, 6 and 7.

In this Section 8 we discuss astronomical observations framed by the above parameters, which also relate to and support the Tribonacci-QC model. Cosmological ages are specified in billions of years (Gyr) ago, as lookback times.

In this paper it is proposed that the discrete dimensions of our quantum information world are the most fundamental foundations of everything. A dimension is a structure that categorizes discrete information. Minimising computational time, a shared dimension makes economic computational use of that degree of freedom, as a natural resource for the broadband parallel processing of distinguishable information. I focus on the Natural selection, evolution and function of shared dimensions in a deep learning quantum cosmology model.

These shared dimensions are conformed dimensions in computer science, which ensure consistency between Nature’s quantum information processing of its physics at macro, meso and microscopic scales, and ensure consistency spanning from deep evolutionary time, through observer lifetimes, to near-instantaneous causal particle physics. Nature’s quantum computations at the cosmological scale can share dimensions with its computations at the Planck scale because the extreme difference in scale (a preposterous hierarchy) protects against detrimental crosstalk. Nature’s two extreme scales come into Reality at the meso-scale nexus of the sentient Wheelerian participatory observer.

The Tribonacci-QC model presented proposes our deep learning quantum computational universe is founded on a common architecture of shared dimensions, defined by discrete and exceptional spherical codes, on a 2-sphere, S 2 . The shared dimensions are utilised by Nature to compute the entirety of physics, across the whole universe, at all scales. The information processed in these shared dimensions is parallelized, in coordinated superpositions, and decomposes into Nature’s innumerable emergent physical phenomena, through observations described by fundamental and effective theories.

This Tribonacci-QC model invokes four optimal , non-antipodal , spherical codes, C ↮ ⋆ ⊂ S 2 : 1-vertex, 3-vertex (great circle equilateral triangle), 4-vertex (spherical tetrahedron) and 24-vertex (spherical snub cube). The vertices are einselected centres of coherent quantum information that maximise their minimum separation and survive environmental decoherence on a noisy horizon. Twenty-four 1-vertex codes, 24 × C ↮ ⋆ 1 , self-organize into eight 3-vertex codes, 8 × C ↮ ⋆ 3 , which self-organize into one 24-vertex code, C ↮ ⋆ 24 , isomorphic to dimensions of 24-spacetime and the 12(2), or 24, generators of S U ( 5 ) .

Snub cubical 24-vertex code chirality causes matter asymmetries and the corresponding graph-stress has normal and shear components relating to respective sides of Einstein’s tensor equivalence G μ υ = κ T μ υ . Cosmological scale factor and Hubble parameter evolution is formalized as an Ostwald-coarsening function of time, scaled by the tribonacci constant ( T ≈ 1.839 ) property of the snub cube. The 24-vertex code coarsens to a broadband 4-vertex code, isomorphic to emergent 4-spacetime and antecedent structures in 24-spacetime metamorphose to familiar 4-spacetime forms. Each of the coarse code’s 4-vertices has 6-fold parallelized degrees of freedom (conserved from the 24-vertex code), C ↮ ⋆ 4 ( 6 ) , so 4-spacetime is properly denoted 4(6)-spacetime.

Cosmological parameters are formalized: CMB h = H 0 / 100 = T log ( 3 ) / 3 ≈ 0.674 , Distance Ladder h = 4 T log ( 3 ) / 11 ≈ 0.735 , z d a = 3 e 1 − 3 / T − 1 ≈ 0.596 , Ω b h 2 = arccos ( T / 2 ) / 4 π ≈ 0.023 and γ t o / γ t c = ( T − 1 ) / 2 ≈ 0.420 . Due to 6-fold parallelization, the total matter density parameter is 6-fold heavier than the baryon density parameter, Ω m = 6 Ω b = 27 arccos ( T / 2 ) / 2 π log 2 ( 3 ) T 2 ≈ 0.302 . A torrent of information-equivalent energy downloads from 6-fold faster 24-spacetime to 4(6)-spacetime. Consequent stress on 4(6)-spacetime causes it to resize its dynamic memory, expanding its cosmological scale. Ultimate coarsening of apparent reality to C ↮ ⋆ 1 , isomorphic to eternal time, is imminent for each observing agent in a Wheelerian participatory universe. Life, even microbial, is outcome-selecting. The onset of outcome-selection marks the dawn of observer-participancy and Agency, and the beginning of its cascading influence over the quantum information structure of our entangled Universe. DNA is proposed to have evolved from an 8 × 3-nucleotide (eight codon) primeval molecular code on the model’s 24 shared dimensions.

The significance of the model’s cosmological transition event, from 24-spacetime to 4(6)-spacetime is emphasized. It occurs at redshift #Math_476#, that is at model time t = 0 , when the scale factor a ( t ) = 1 / 3 . This aspect of the Tribonacci-QC model clearly distinguishes it from other models. I present an introductory case for the extraordinariness of redshift z = 2 in astronomy. This redshift marks an event which is related to several phenomena; including peak star formation rate density, quasar activity, CMB lensing, ⁶Li genesis by cosmic rays and exoplanetary abiogenesis of Wheelerian participatory universe observers.

This model also makes certain testable assertions about the dimensionality of particle physics. Tribonacci-QC describes a quantum mechanical realm for particle physics which has 12(2) degrees of freedom and claims these dimensions, which are shared with those of 24-spacetime, are circumscribed within the coarsened 4(6)-spacetime. Furthermore 12 of these are Naturally selected whilst 12 are de-selected yet virtual. Exceptional, self-organizing spherical codes, that are simultaneously optimal and non-antipodal, are isomorphic to the shared dimensions which provide degrees of freedom in Nature’s entangled quantum information processing of General Relativity, Darwinian Life and Quantum Mechanics. This feature of Tribonacci-QC presents a novel path towards their long-sought unification.

Having set out a summary in the above Section 9, I now succinctly conclude with some overarching high-level reflections about this Tribonacci-QC model. These can be viewed from three perspectives.

Firstly, the most significant structure of our quantum computational Universe is its fundamental dimensionality, which in this model is a dimensionality shared across all scales. The preposterous hierarchy between quantum mechanics and relativistic cosmology is bridged by the meso-scaled Agency of Life, the participatory observers who collapse wave functions and bring about Reality, all of which can be related to the same shared dimensions. Indeed, it is the preposterous hierarchy of scale which allows for sharing of dimensions without detrimental crosstalk between its end members.

Secondly, the factor pairs of 24 have innate utility in the description of Nature. This is particularly where they characterise optimal, non-antipodal, spherical codes (self-isolated against detrimental crosstalk) and corresponding graphs. The factor pairs of 24 characterise quantum particle physics, a prebiotic precursor of DNA, and the geometrogenesis of spacetime (all on 24 shared dimensions, with some factors parallelized and Naturally selected in certain situations).

Thirdly, the 24-vertex spherical code which is optimal and non-antipodal, is snub cubical, and this configuration is intrinsically related to the tribonacci constant and its recurrence sequence. The tribonacci sequence innately regulates and rectifies chaos into orderly expansion which underpins such fundamental characteristics of our emergent Universe. Remarkably, many cosmological parameters can be simply expressed in terms of the tribonacci constant.

Shared dimensions, factor pairs of 24 and the tribonacci constant are Naturally mutual to this original model of our World.

This original research is self-funded and I thank my reviewers and editors for their valuable support.

The author declares no conflicts of interest regarding the publication of this paper.

McCoss, A. (2019) Tribonacci Quantum Cosmology: Optimal Non-Antipodal Spherical Codes & Graphs. Journal of Quantum Information Science, 9, 41-97. https://doi.org/10.4236/jqis.2019.91004