Tribonacci Quantum Cosmology: Optimal Non-Antipodal Spherical Codes & Graphs

Degrees of freedom in deep learning, quantum cosmological, information processing are shared and evolve through a self-organizing sequence of optimal  , non-antipodal  , spherical codes, 2 C S ⊂   . This Tribonacci Quantum Cosmology model invokes four C   codes: 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, 1 24 C ×   , self-organize into eight 3-vertex codes, 3 8 C ×   , which self-organize into one 24-vertex code, 24 C   , isomorphic to dimensions of 24-spacetime and 12(2) generators of ( ) 5 SU . 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 Gμυ μυ κ =  . Cosmological scale factor and Hubble parameter evolution is formalized as an Ostwald-coarsening function of time, scaled by the tribonacci constant ( ) 1.839 T ≈ 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), ( ) 4 6 C   , so 4-spacetime is properly denoted 4(6)-spacetime. Cosmological parameters are formalized: CMB ( ) 0 100 log 3 3 0.674 h H T = = ≈ , Distance Ladder ( ) 4 log 3 11 0.735 h T = ≈ , 1 3 3e 1 0.596 T da z − = − ≈ , ( ) 2 arccos 2 4π 0.023 bh T Ω = ≈ and ( ) 1 2 0.420 o c t t T γ γ = − ≈ . Due to 6-fold parallelization, the total matter density parameter is 6-fold heavier than the baryon density parameter, How to cite 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 Received: February 11, 2019 Accepted: March 26, 2019 Published: March 29, 2019 Copyright © 2019 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/

. Due to 6-fold parallelization, the total matter density parameter is 6-fold heavier than the baryon density parameter,

Introduction
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.
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, . 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  for 0 t > . The 6-fold transition in the pace of time, at 0 t = , 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 0 t = and scale factor ( ) 0 a t = , relative to the spherical horizon datum. The present cosmological scale is defined ( ) 1 a t → , which is imminent, as 1 t → 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 ( ) 2 z → = (coeval with peak star formation rate density [31], quasar activity [32] [33], CMB lensing [34], 6 Li 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 , when calibrated by 2 10.4 Gyr z t = = [40]. The redshift at the transition from deceleration to acceleration is γ γ of the optical over cosmic microwave background tangential shear [43]. An equivalence in Tribonacci-QC is proposed, where ( ) 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

Optimal Non-Antipodal Spherical Codes
In this model, a spherical code is a vertex set configured on the 2-sphere, 2 S . 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, SM G , particularly as it relates to the special unitary group 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.
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); Journal of Quantum Information Science 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 selforganize 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, , 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: 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, . 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.
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 m Ω is 6-fold heavier than the baryon density parameter b Ω 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  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: Journal of Quantum Information Science Figure 2. Optimal non-antipodal spherical codes: self-organization process, illustrated in stereographic projections. Far left: Twenty-four 1-vertex codes. Middle left: eight 3-vertex codes (equilateral triangle on a great circle). Middle right: one 24-vertex code (chiral, spherical snub cube). Far right: one 4-vertex code (spherical tetrahedron) with 6 parallelized vertices superposed at each of the four tetrahedral vertices. The self-organization from twenty-four 1-vertex codes to eight 3-vertex codes and thence to one 24-vertex code achieves an optimal snub cubical packing. However, information complexity inexorably continues to compound, and the 24-vertex code coarsens into a broadband 4-vertex code. Figure 3. A 24-vertex, 60-edge, snub cubical graph in its triangular perimeter (this graph is more commonly illustrated in its square perimeter). In this 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). The colours simply accentuate characteristic sub-graph shapes inherent within the snub cubical graph.   (Table 1) and the superscripts are index numbers in the example Hamiltonian super-cycle (2.5). Yellow labelled vertices are designated time-like ( ) t and the red, green and blue labelled vertices are space-like, respectively ( ) , , x y z . The example Hamiltonian super-cycle (2.5) is included and is coloured consistently with the graph.

SU
, may provide such enlightenment. 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.

Illustrated in
The compacted and settled equilateral triangles, forming the self-organized snub 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.
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 ( 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). 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: where snub cube F 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,  , , 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: 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. Journal of Quantum Information Science Figure 6. A 4(6)-vertex, 6(4)-edge, tetrahedral graph, derived from a 6-fold chromatic parallelization of the vertices depicted in the 24-vertex snub cubical graph in its triangular perimeter ( Figure 5).
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 Hamilto- 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.    Figure 7), is displayed here, coarsened and parallelized. It is deconstructed to show its 6 tetrahedral subgraph sub-cycles, which clock a 4π (720˚) identity circuit, in 6 ordered turns of 2π/3 (120˚), around the yellow-labelled time-like vertex. Each of the 4 tetrahedral vertices (isomorphic to 1-time dimension and 3-space dimensions) comprises 6 parallelized corresponding degrees of freedom.
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 ( )   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. 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

Quantum Mechanics with 12(2) = 24 Degrees of Freedom
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 1 2 3 4 12 It is now well known that the zeta function of −1 has the same value ( )  ω ζ ω This is one of Nature's many fundamental mathematical incidences of the number 24 and some other occurrences are summarised as follows.  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 4-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 ( ) 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 Tribonac- McCoss ci-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 ( ) 5 SU [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 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 ( )

SU
and Naturally selects the 12 generators of the Standard Model gauge group, SM G . 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 SM G each of these 3 vertices also accommodates 2 non-mixing generators of ( ) and a single indivisible non-mixing generator of ( ) 2 L SU ( Figure  10). The yellow apex of the tetrahedral graph ( Figure 6 and Figure 7) accommodates the time dimensions of spacetime and in SM G this apical vertex also accommodates 2 non-mixing generators of ( ) and a single indivisible non-mixing generator of ( ) Figure 10). The degree of parallelization through dimensional reduction of the 24 generators of ( )

SU
at each vertex has a limiting lowest common denominator of 2, set by the single indivisible generators at each vertex (an ( ) 2 L SU generator at each vertex on the triangular base and a ( ) 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 Journal of Quantum Information Science 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 gauge theory which is experimentally well established at presently probed scales. The Standard Model gauge groups group of special unitary transformations acting on 5 complex variables.

SU
theory. She considers the Standard Model represented by  is commonly modded out) and finds that the ( )

SU
ladder symmetries then reduce to it, where ladder operators arise from the division algebras  (real numbers),  (complex numbers),  (the quaternions), and  (the octonions).
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 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 Fig   ; 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 (coloured cheddar-orange, like stringy cheese between two halves of a grilled sandwich!). Each of the 12-vertex sub-graphs (blue and green) has 24 edges. Vertex numbering is consistent with the adjacency list in Table 1.  . The snub cubical graph as depicted in Figure 12, but in a triangular perimeter. The vertex numbering is consistent with Table 1, Figure 5 and Figure 9. The apical 12-vertex subgraph (blue vertices) is inscribed within a basal 12-vertex subgraph (green vertices). Consider the two 12-vertex sub-graphs are representations of the two sets of 12 generators, selected (non-mixing) and deselected (mixing), in ( ) 5

SU
. Let us consider the green vertices selected, and the blue vertices deselected. The number of coupling-edges connecting the two sub-graphs is minimized to twelve (coloured cheddar-orange). There are 24 blue edges connecting blue vertices and 24 green edges connecting green vertices.
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. Parenthetically, Borsten et al. [93] derive 8 E 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 8 E Lie group has applications in several definitions of sm G . 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) Borsten et al. [93] derive 8 E symmetry from eight 3-level qutrits: 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: Similarly, deeply related to the Standard Model of particle physics, the special

Snub Cube Characteristics
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 with an even number of plus signs, together with all the odd permutations with an odd number of plus signs, and where 1.8392867552 T ≈  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.

Tribonacci Constant and Sequence
Whereas the well-known Fibonacci numbers start with two given terms, the lesser The tribonacci constant, T, is the ratio toward which adjacent tribonacci numbers tend, which is also the real root of the polynomial  In Figure 14 we see how the permutations of the qutrit signature code (4.1.8) (4.1.9), corresponding to the zeroth, first and second index, affect the tribonacci sequences.
Let us consider this as an evolution from large negative indices (left) to increasingly large positive indices (right). We observe large and chaotic tribonacci numbers with large negative indices, which flip-flop between negative and positive values. This chaotic behaviour progressively dampens to the right, towards the signature code. The negative indices exhibit a double-sided decaying exponential signal. The signature code suppresses the chaotic oscillation, like a regulating gate.
Further to the right, through the gate (signature code), with increasing positive indices, the tribonacci numbers again become very large, but now the growth is smooth, exponentially rising and it is either positive or negative i.e. a single-sided (not double-sided) signal. The sign of the tribonacci sequence for positive indices depends on the numbers in the signature code. The signature code rectifies the sequence for positive indices.
I conjecture that a quantum computer with snub cubical graph architecture would exhibit behaviours related to the metric properties of the 24-vertex snub cube and thus to the tribonacci constant. Furthermore, I propose a balanced ternary, qutrit, deep learning, snub cubical quantum computer would suppress and harness primordial chaos and thence govern orderly cosmological expansion and the epistemic accumulation of advantageous knowledge, in the manner of the underlying tribonacci sequence evolution shown in Figure 14. Qutrit signature See [23] for a discussion on tribonacci matrices and a new coding theory. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. In Tribonacci-QC, cosmological and quantum mechanical information processing, based on fundamental tribonacci (rather than Fibonacci) sequences, are postulated to be Naturally selected.
The reader is also referred to the work of Cerda-Morales who formalizes tribonacci-type octonions and quaternions sequences and describes their properties [24] [25], which have profound occurrences in Nature and provides a link to octonion particle physics research [90].
Let us now consider this deep learning Tribonacci-QC as one in which Nature's fundamental computations are 3-level qutrit operations. The universe, or Triuniverse [1], is an evolving system of qutrit quantum computations, with innate ternary quantum deep learning. Fundamental information processing is driven by Epistemic Drive, the Natural appetite for information selected for advantageous knowledge. Evolution of the Triuniverse occurs through the operation of quantum Darwinian and classical Darwinian deep learning processes. Darwin's universal 3-step evolutionary algorithm of Variation, Selection and Replication is claimed to be fundamental to Natural deep learning and knowledge growth across all scales; from the macroscopic cosmos, through mesoscopic Life, to sub-microscopic quantum physics.
As in Dirac's belt trick, two loops of this evolutionary algorithm return the identity, which relates to the 4π identity rotation of a spinor. The two loops are here defined to be those of a directed trefoil torus knot ( Figure 15 and Figure 16). Figure 15 and Figure 16 illustrate the universal 3-step evolutionary algorithm of Variation, Selection and Replication. The captions of these figures guide the reader through the processing steps of the algorithm and explain its foundations on a 3-vertex directed graph which relates to a trefoil torus knot. The two-loop processing cycle, C, is an Eulerian cycle comprising subprocesses at its vertices: Variation, z w V ⇒ ⇒ with input z and output w, Selection, w x S ⇒ ⇒ with input w and output x and Replication, y R R ⇒ with input x and output z. The Replication subprocess (blue) is distinguished by its intrinsic replication sub-cycle, y.
The Darwinian Eulerian cycle of Variational-Selective-Replication, C, is defined as From graph theory, the adjacency matrix, L, corresponding to this Eulerian cycle, C, is   Podani et al. [103] for an account of Darwin's model of a growing population of elephants as a tribonacci sequence.
In closing this subsection, let us consider other characteristics of matrices M and L, specifically the eigenvalues: and may be compared with (4.1.6).
We note from (4.1.18)

Chirality and Twist Angle
Further important characteristics of the snub cube are its chirality and twist angle. Given the significance in this Tribonacci-QC model of the 24-vertex snub cubical optimal non-antipodal code and its isomorphism to 24-spacetime, we shall briefly discuss these characteristics. Their physical manifestations are discussed further in Sections 5, 6 and 8.
Unlike most uniform polyhedral, the snub cube has two enantiomorphs ( Figure 17), governed by their levo and dextro vertex configurations (4.1). In this Tribonacci-QC model, self-organization of optimal non-antipodal codes, as depicted in Figure 2, attains a 24-vertex configuration through combining eight 3-vertex codes on a noisy spherical horizon. That noise causes the dextro and levo enantiomorphs of the snub cube to be equiprobable outcomes of the quantum self-organization process. We shall return to this ratio, β , in Section 6, where a further relationship with matter is proposed (physical baryon density parameter).

A. McCoss Journal of Quantum Information Science
I postulate that the snub cubical (and snub cubical coarsened into tetrahedral) optimal non-antipodal codes of Tribonacci-QC are isomorphic to corresponding chiral graphs which are stressed-graphs. The graph-stress partitions into shear graph-stress, ( ) The left-hand side, to coin Wheeler's phraseology, relates to spacetime, which tells matter how to move, and the right-hand side refers to matter, that tells spacetime how to curve. Journal of Quantum Information Science In Tribonacci-QC, foundational quantum information processing involves an error-correcting tribonacci coding-decoding procedure [23] that establishes dynamic equivalence in information-energy exchanges that are coded and decoded across a parallel computing partition (  ). I propose the left-hand side of that partition computes the normal graph-stress component, ( )  to yield rich future insights into the quantum gravity correspondence, QG, (4.2.5), and models of particle physics and cosmology.

Cosmological Scale Factor Evolution
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. 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 0 t = .
The ordinate intercept in Figure 18   Using an astronomical timescale [40], anchored on a cosmological age of 10.4 Gyr at redshift 2 z = , we thus calibrate 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  At the coarsening transition, antecedent, quantum astrophysical structures created in dimensions of 24-spacetime metamorphose into their familiar 4(6)-Journal of Quantum Information Science 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). Consider these two functions,  , 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,

Hubble Parameter
The dimensionless Hubble parameter, h, has a standard definition:   where 0 H is the Hubble "constant" and in Tribonacci-QC has value: which is close to the Planck 2018 estimate [104]. However, there is a significant tension in modern cosmology. The value of 0 H in (5.15) differs from that derived by the distance ladder [110].
Let us consider the fractional quantum Hall effect (FQHE) which is a property of a collective state with fractional quantum statistics. For example, Pan et al. [111] report a fractional quantum Hall effect at Landau level filling 4 11 ν = .
Let us take their experimental evidence to a more general level, where I postulate that fractional quantum effects also occur in quantum cosmology on a very large scale through deep spacetime.
Revisiting this Tribonacci-QC model's Ostwald coarsening (particularly the denominator of the Ostwald coarsening exponent in Equations (5.1) and (5.2)), within such a cosmological fractional quantum paradigm, then we may consider fractional quantum plateaus developing near e 3 =     , which has integer lowest radix economy [1] [107]. Specifically, let us search for economic fractional quantum plateaus between e 2.71828 which is close to the value of the Hubble "constant" measured using the distance ladder approach [110].

Deceleration-Acceleration Transition Redshift
The deceleration-acceleration transition occurs at model time da t when 0 a =  .
The second derivative of (5.1) when equal to zero and when 0 da t > and real is given by:

Matter Density Parameters
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, ( ) 24 4 6 σ , of the snub cubical graph.
Whereas, in this Section 6 we consider the shear graph-stress component, ( ) 24 Journal of Quantum Information Science The physical baryon density parameter, 2 b h Ω , 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, ( ) which is reconcilable with standard cosmic nucleosynthesis and the matter density parameter, m Ω , is:

Cosmographic Distance Ratio
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 back- , (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
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 Ω , (6.1), baryon density parameter, b Ω , (6.2), matter density parameter, m Ω , (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.

Fast Radio Bursts
The In Tribonacci-QC this event, 13.8 Gyr ago, marks a zero crossing between negative scale and positive scale, where scale is relative to the optimal non-antipodal codes' spherical horizon. Negative scale projects inside the horizon, whereas positive scale projects outside the horizon (Section 5) ( Figure   19).
Also recall that negative model time,  Figure 13.
FRBs, in Tribonacci-QC, are violent quantum tunnelling events, cascading into strong millisecond radio bursts, caused during the decay of 24-dimensional Planck stars, which are incompletely confined by their 4(6)-dimensional event horizons. This radioactive decay process will ultimately fully confine these 24-dimensional astrophysical objects within their 4(6)-dimensional cages.
For a discussion related to this Tribonacci-QC interpretation, the reader is referred to seminal research by Rovelli, Vidotto and Barrau [44] [45].

Black Holes and Deep Learning Cosmological Cycles
Following the arrow of cosmological time, from the deep past towards the present, let us consider astronomical observations pertaining to the period 13.8 Gyr to 10 To observe these structures, post ( ) 1 3 a t = , we need to look through Einsteinian 4(6)-spacetime, into distant 24-spacetime.
Low mass primordial black holes [46] [55] [56] are hypothesised to originate from this period, beyond 2 z = , and may seed super massive black holes at the centres of galaxies in present 4(6)-spacetime. See also Kashlinsky's [50] analysis of what may have occurred if some primordial matter ("dark" matter in his analysis) comprised a population of primordial black holes. Such primordial black holes would distort the distribution of mass in the early universe, introducing a fluctuation with consequent effects when the first stars begin to form hundreds of millions of years later. Very old supermassive black holes have been observed and are in tension with standard Big Bang cosmology because their time of formation defies conventional explanation.
We may think of a black hole in terms of a further coarsening of the spherical code which is isomorphic to the dimensions of spacetime. Let us consider the formation of a black hole, wherein 4(6)-spacetime (tetrahedral spherical code in Figure 6) coarsens in two stages, as depicted in Figure 21. Firstly, the tetrahedral spherical code ( Figure 6) coarsens to a 3-vertex spherical code (great circle equilateral triangle, with edges coloured light blue, see left side of Figure 21). One way it could do this would be for the time-like (yellow) vertices to advance to their subsequent adjacent vertex in the example Hamiltonian super-cycle (2.5).
Thus, time-like (yellow) vertices 1 and 5 advance to the x space-like dimension (red), vertices 6 and 7 advance to the y space-like dimension (green) and vertices 8 and 11 advance to the z space-like dimension (blue). A similar chromatic coarsening arrangement can be derived when the time-like (yellow) vertices shift to their prior adjacent vertex in the example Hamiltonian super-cycle (2.5). The 3-vertex code is isomorphic to the dimensions of 3(8)-spacetime. I conjecture this 3-vertex code governs the physics of the event horizon of the black hole. A second stage of coarsening of the spherical code is shown in the right side of Figure 21. In this second stage, the 3-vertex spherical code (great circle equilateral triangle, with edges coloured light blue) coarsens to a 1-vertex spherical code (a point represented by a purple circle). The 1-vertex code is isomorphic to the dimensions of ultrabroadband 1(24)-spacetime. I conjecture this 1-vertex code governs the physics inside the event horizon of the black hole and that it has qualities which are those of eternal time (all vertices are yellow coloured, with their legacy space-like coloration depicted as red, green and blue coloured circles). The entire right side of Figure 21 is macroscopically manifest as eternally stretched time; however microscopically, it comprises 24 parallelized degrees of freedom.
In Section 8.3.7 below, on Wheelerian observers shaping our participatory universe, I propose that ultimate coarsening of apparent reality to which shows spherical code evolution as a computationally intelligent, deep learning, self-organization process. In this way, cosmological deep learning, fine-tuning and black hole to white hole rebirth [112] have parallel bases in the shared dimensions of Tribonacci-QC.
An alternative, or complementary, way to consider a black hole in Tribonacci-QC is depicted in Figure 13. That figure represents the corresponding quantum mechanical configuration of a black hole, wherein the central cluster of 12 apical vertices are isomorphic to 12(2) degrees of freedom in ( )

SU
inside the event horizon and the circumscribing 12 basal vertices ( Figure 13) are isomorphic to the degrees of freedom in ( )

SU
outside the event horizon. In concluding this sub-section, Tribonacci-QC provides novel graph-based models for astronomical observations associated with black holes, whether they formed in primordial cosmological times or more recently.

Transition Event (z = 2)
It is important to consider that our present observations are being made in 4(6)-spacetime. When we observe beyond 2 z = , to study objects formed before Indeed, it is via such a high resolution strategy that science is currently unlocking the marvels of quantum physics and the cosmology of deep spacetime.
Another important general characteristic of the 2 z = event is that it occurs at 0 t = in model time, (10.4 Gyr ago), that is when the first derivative of the cosmological scale factor ( ) a t is infinite for an infinitesimal moment of transition time. From (5.1) the first derivative of the scale factor is given by: In the discrete Tribonacci-QC model this quantum transformation (8.3.2) event is a high energy shock-spike, or bow shock, concurrent with the transition from 24-spacetime to 4(6)-spacetime. In the following sub-sections, we consider phenomena caused by this 2 z = high energy shock-spike, or bow shock.

Peak Star formation Rate Density (z = 2)
The Fermi Large Area Telescope Collaboration 2018 [31] reports that the light emitted by all galaxies over the history of the Universe produces the extragalactic background light, at infrared, optical and ultraviolet wavelengths. This extragalactic background light is a source of opacity for γ-rays via photon-photon interactions, leaving a record in the spectra of distant γ-ray sources. They measured this attenuation using over seven hundred active galaxies and one γ-ray burst.
This analysis allows them to explain the evolution of the extragalactic background light and determine the star-formation history of the Universe, over 90% of cosmic time. Their star formation history is consistent with independent measurements from galaxy surveys, which clearly peaks at redshift 2 z = . See also [52].
In the context of Tribonacci-QC peak star formation rate density relates to the coarsening transition from 24-spacetime to 4(6)-spacetime at 2 z = . Consider triggered star formation during which events compress a molecular cloud initiating gravitational collapse [53]. Molecular clouds may collide or a supernova explosion may trigger cloud collapse, by sending shocked matter into it at high velocity. The new stars thus born may then produce supernovae, leading to self-propagating star formation.
A source of gravitational shock throughout the entire cosmos is proposed to be the abrupt (8.3.1) (8.3.2) coarsening transition from 24-spacetime to 4(6)-spacetime at 2 z = . Referring to Figure 18, we see the cosmological scale factor at 0 t = , . This is proposed to be a direct signature in Tribonacci-QC of a 2 z = transition gravitational shock having an abrupt transition effect on the scale of the entire cosmos. The model informs us that gravitational binding of the cosmos spontaneously weakens upon coarsening from 24-spacetime to 4(6)-spacetime which is manifest in an instantaneous inflation of cosmologi-Journal of Quantum Information Science event is epistemically driven in a runaway self-realisation, with positive feedback through cosmological cycles, propelled by physics but directionally nudged by its own emergent agents. In other words, the Agency of Life triggers Schrödinger's cat to pounce out of its box and fine-tunes the fundamental parameters of Nature through evolutionary cosmological cycles.

Wheelerian Observers Shape Our Participatory Universe (z = 2)
Wheeler explains his seminal idea of observer-participancy by describing the world as a self-synthesizing, quantum networking, system of existences [5]  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.
We know from fossil evidence that Life initiated on Earth about 3.5 Gyr ago, only some 0.5 Gyr after the Earth became stable and cool enough for life to develop and evolve. There has therefore been plenty of time since the 2 z = transition event for three back-to-back episodes of similar evolution towards sentience elsewhere in the cosmos. Now consider the potential for parallel biogenesis and biological evolution across innumerable life-bearing planets and moons and we reach the conclusion that observer-participancy, including observer-observer communication (e.g. observer-consensus about constellations and the workings of physics), has the capacity to physically shape the cosmos. Add to that my proposal that the cosmos is deep-learning and is in a present, parallel-opitmized, cycle which was preceded by innumerable prior cycles. Furthermore, that optimization is considered to occur across the shared dimensions of Tribonacci-QC.
Indeed, in the language of Schrödinger, and in Tribonacci-QC, I envisage First Life collapsing the wave function of 24-spacetime, from its superposed eigenstates, through widespread participatory observation into a 4(6)-spacetime eigenstate. I consider the pervasiveness of participating, communicating and consensus-building observers as being an omnipresent Agency. This Agency is in-Journal of Quantum Information Science trinsic to the environment for decoherence and thus einselection occurs via its ever-present outcome-selection and broadband, consensus-affirming, classical outcomes.
Finally, consider that Life occupies and fulfils a pivotal participatory position amidst the scale of everything, which spans from the miniscule quantum mechanical dominion, to the vastness of our relativistic cosmos. I claim Life provides existential outcome-selecting Agency, spanning all scales, and operates at the measurement-nexus of a preposterous scale hierarchy. See Paul Davies' recent book, "The Demon in the Machine: How Hidden Webs of Information Are Finally Solving the Mystery of Life" [113], for an inciteful overview of the connections between physics and biology in terms of information science, and for a useful bibliography. I propose the eight 3-vertex sets are isomorphic to eight codons (sequences of 3 nucleotides in a unit of genetic code). They self-organise to constitute a snub cubical 24-vertex set, with a Hamiltonian super-cycle which I claim is isomorphic to the most primeval molecular algorithmic-loop, which is Nature's prebiotic precursor to DNA (and RNA).
Thus Tribonacci-QC proposes that quantum mechanics, abiogenesis and cosmology are all governed by the 24 shared dimensions of snub cubical, optimal, non-antipodal spherical codes, operating at three inter-related entangled scales across a preposterous hierarchy, with Life being the pivotal measurement-nexus, i.e. the participatory observer (Schrödinger's box opener), in the quantum information processing of our entire Universe.

Summary
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.

A. McCoss Journal of Quantum Information Science
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, 2 S .
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 obser- 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.

Conclusions
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.

A. McCoss Journal of Quantum Information Science
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.