Assembly Theory of Binary Messages (How to Assemble a Black Hole and Use it to Assemble New Binary Information?)

1. Introduction

Assembly Theory (AT) [1,2,3,4,5,6,7] provides a distinctive complexity measure, superior to established complexity measures used in information theory, such as Shannon entropy or Kolmogorov complexity [1,5]. AT does not alter the fundamental laws of physics [6]. Instead, it redefines objects on which these laws operate. In AT, objects are not considered as sets of point particles (as in most physics), but instead are defined by the histories of their formation (assembly pathways) as an intrinsic property, where, in general, there are multiple assembly pathways to create a given object.

AT explains and quantifies selection and evolution, capturing the amount of memory necessary to produce a given object [6]. This is because the more complex a given object is, the less likely an identical copy can be observed without the selection of some information-driven mechanism that generated that object. Formalizing assembly pathways as sequences of joining operations, AT begins with basic units (such as chemical bonds) and concludes with a final object. This conceptual shift captures evidence of selection in objects [1,2,6].

The Assembly Index, which represents the length of the shortest assembly pathway leading to an object, facilitates the quantification of the minimum memory required for its construction. In general, it increases with the object’s size, but decreases with symmetry, so large objects with repeating substructures may have lower complexity than smaller objects with greater heterogeneity [1]. The copy number specifies the number of copies of an object, essential for assessing its structural complexity.

AT has been experimentally confirmed in the case of molecules and probed directly experimentally with high accuracy with spectroscopy techniques, including mass spectroscopy, IR, and NMR spectroscopy [6]. It is a versatile concept with applications in various domains. Beyond its application in the field of biology and chemistry [7], its adaptability to different data structures, such as text, graphs, groups, music notations, image files, compression algorithms, etc., showcases its potential in diverse fields [2].

In this study, we investigate the assembly pathways of binary strings by joining individual bits present in the assembly pool [6] and strings that entered the pool as a result of previous joining operations.

 In particular, we investigate the assembly of black-body objects BBs (black holes (BHs), white dwarfs, and neutron stars) considered binary strings [8,9,10]. It is known [2,8,9,10,11,12,13,14,15,16,17,18,19] that information in the universe evolves toward increased structural complexity, decreasing information entropy.

 We use emphasis for object as this term, understood as a collection of matter, is a misnomer, as it neglects the (quantum) nonlocality [20]. Nonlocality is independent of the entanglement among particles [21], as well as the quantum contextuality [22], and increases as the number of particles [23] grows [24,25]. Furthermore, the ugly duckling theorem [26,27] asserts that every two objects we perceive are equally similar (or equally dissimilar).

 This study extends the findings of previous research [8,9,10,23] within the framework of AT [1,2,3,4,5,6,7] and emergent dimensionality [8,9,10,15,17,18,20,23]. However, our results generally apply to information theory. Therefore, we put the BB-related content in frames like this one. The reader not interested in BBs may skip the text in these frames and the additional results presented in Section 3.

2. Results

We consider binary strings $C_k^{\left(N\right)}$ containing symbols $\{0,1\}$, which are our basis AT objects[2], with $N_0$ zeros and $N_1$ ones, having a fixed length $N=N_0+N_1$. We consider strings to be messages transmitted through a communication channel between a source and a receiver, similarly to the Claude Shannon approach used in the derivation of information entropy [28], and consider the process of their formation within the AT framework.

Definition 1.

A string assembly index $a_N$ is the smallest number of steps s required to assemble a binary string $C_k^{\left(N\right)}$ of length N by joining two basic symbols contained in the initial assembly pool $P=\{0,1\}$ and strings joined in previous steps that are added to the assembly pool. Therefore, the assembly index $a_N\left(C_k\right)$ is a function of the string $C_k^{\left(N\right)}$.

For example, the 8-bit string

$$C_k^{\left(8\right)}=\left[00100101\right]$$

(1)

can be assembled in at most seven steps:

1.

join 0 with 0 to form $C_k^{\left(2\right)}=\left[00\right]$, adding $\left[00\right]$ to P,

2.

join $C_k^{\left(2\right)}=\left[00\right]$ with 1 to form $C_k^{\left(3\right)}=\left[001\right]$, adding $\left[001\right]$ to P,

3.

...

7.

join $C_k^{\left(7\right)}=\left[0010010\right]$ with 1 to form $C_k^{\left(8\right)}=\left[00100101\right]$,

six or five steps:

• join 0 with 0 to form $C_k^{\left(2\right)}=\left[00\right]$, adding $\left[00\right]$ to P,
• join $C_k^{\left(2\right)}=\left[00\right]$ with 1 to form $C_k^{\left(3\right)}=\left[001\right]$, adding $\left[001\right]$ to P,
• join $C_k^{\left(3\right)}=\left[001\right]$ with $\left[001\right]$ taken from P to form $C_k^{\left(6\right)}=\left[001001\right]$, adding $\left[001001\right]$ to P,
• join $C_k^{\left(6\right)}=\left[001001\right]$ with 0 to form $C_k^{\left(7\right)}=\left[0010010\right]$, adding $\left[0010010\right]$ to P,
• join $C_k^{\left(7\right)}=\left[0010010\right]$ with 1 to form $C_k^{\left(8\right)}=\left[00100101\right]$,

or at least four steps:

• join 0 with 1 to form $C_k^{\left(2\right)}=\left[01\right]$, adding $\left[01\right]$ to P,
• join $C_k^{\left(2\right)}=\left[01\right]$ with 0 to form $C_k^{\left(3\right)}=\left[001\right]$, adding $\left[001\right]$ to P,
• join $C_k^{\left(3\right)}=\left[001\right]$ with $\left[001\right]$ taken from P to form $C_k^{\left(6\right)}=\left[001001\right]$, adding $\left[001001\right]$ to P,
• join $C_k^{\left(6\right)}=\left[001001\right]$ with $\left[01\right]$ taken from P to form $C_k^{\left(8\right)}=\left[00100101\right]$,

which also ensures that the assembly pool P is a distinct set. Therefore, the string (1) has an assembly index $a_8\left(C_k\right)=4$ that represents the length of its shortest assembly pathway; $C_k^{\left(8\right)}$ cannot be assembled in fewer steps.

Definition 2.

A string $B_k^{\left(N\right)}$ is a balanced string if it has the same number of symbols, where $N_1=N_0-1$ or $N_0=N_1-1$ if N is odd.

Without loss of generality, we assume that if N is odd, $N_1<N_0$ (e.g., for $N=5$, $N_1=2$, and $N_0=3$). However, our results are equivalently applicable if we assume the opposite (i.e. a larger number of ones for an odd N).

The number $|B^{\left(N\right)}|$ of balanced strings among all $2^N$ strings is1

$$\begin{array}{cc}\hfill |B^{\left(N\right)}|& =\left(\genfrac{}{}{0pt}{}{N}{⌊N/2⌋}\right)=\left(\genfrac{}{}{0pt}{}{N}{⌈N/2⌉}\right)\approx \sqrt{\frac{2}{\pi N}}{2}^N.\hfill \end{array}$$

(2)

This is OEIS A001405 sequence, the maximal number of subsets of an N-set such that no one contains another, as asserted by Sperner’s theorem, and approximated using Stirling’s approximation for large N.

 BBs emit Hawking black-body radiation having a continuous spectrum that depends only on one factor, the BB temperature $|T_{\mathrm{BB}}|=T_{\mathrm{P}}/\left(2\pi d_{\mathrm{BB}}\right)$ corresponding to the BB diameter $D_{\mathrm{BB}}{d}_{\mathrm{BB}}{\ell }_{\mathrm{P}},d_{\mathrm{BB}}\in \mathbb{R}$, where ${\ell }_{\mathrm{P}}$ and $T_{\mathrm{P}}$ is the Planck length and temperature [8].

 Triangulated BB surfaces contain a balanced number of Planck area triangles, each having binary potential $\delta {\phi }_k=-c^2·\{0,1\}$, where c denotes speed of light in vacuum, as has been shown [8,10], based on the Bekenstein-Hawking entropy [29,30,31] $S_{\mathrm{BB}}=k_{\mathrm{B}}{N}_{\mathrm{BB}}/4$. Here $k_{\mathrm{B}}$ is the Boltzmann constant and $N_{\mathrm{BB}}\pi D_{\mathrm{BB}}^2/{\ell }_{\mathrm{P}}^2=\pi d_{\mathrm{BB}}^2$ is the information capacity of the BB surface, i.e., the $⌊N_{\mathrm{BB}}⌋\in \mathbb{N}$ Planck triangles corresponding to bits of information [8,9,10,30,32,33], and the fractional part triangle(s) having the area $\left\{N_{\mathrm{BB}}\right\}{\ell }_{\mathrm{P}}^2=(N_{\mathrm{BB}}-⌊N_{\mathrm{BB}}⌋){\ell }_{\mathrm{P}}^2$ too small to carry a single bit of information [8,9].

 Therefore, a balanced string $B_k$ represents a BB surface comprising $N_1=⌊N_{\mathrm{BB}}⌋/2$ active Planck triangles (APTs) with binary potential equal to $-c^2$ [9].

Theorem 1.

A string having length $N=4$ is the shortest string having more than one string assembly index 1.

Proof.

The proof is trivial. For $N=1$ the assembly index $a_1\left(C\right)=0$, as all basis objects have a pathway assembly index of 0 [2] (they are not assembled). $N=2$ provides four available strings with $a_2\left(C\right)=1$. $N=3$ provides eight available strings with $a_3\left(C\right)=2$. Only $N=4$ provides 16 strings that include four stings with $a_4\left(C\right)=2$ and twelve strings with $a_4\left(C\right)=3$ including $|B^{\left(4\right)}|=6$ balanced strings, as shown in Table 1 and Table 2.

For example, to assemble the string $B_1=\left[0101\right]$ we need to assemble the string $\left[01\right]$ and reuse it. Therefore, $a_N\left(C_k\right)=N-1$ for $0<N<4,\forall k$ and $\underset{k}{min}\left(\left\{a_N\left(C_k\right)\right\}\right)<N-1$ for $N\ge 4$, where $\left\{a_N\left(C_k\right)\right\}$ denotes a set of different assembly indices. □

 Interestingly, Theorem 1 strengthens the meaning of $N_{\mathrm{BH}}=4$ as the minimum information capacity that provides a minimum thermodynamic (BH) entropy [29,30,31].

 There is no disorder or uncertainty in an object that can be assembled in the same number of steps $s\le 2$.

Table 1 and Table A3, Table A4, Table A5, Table A6, Table A7, Table A8, Table A9, Table A10 (Appendix C) show the distributions of the assembly indices among $2^N$ strings for $4\le N\le 12$ taking into account the number of ones $N_1$. The sums of each column form Pascal’s triangle read by rows (OEIS sequence A007318).

Conjecture 1

(Tight lower bound on the string assembly index). The smallest string assembly index $a_N\left(C_{\mathit{min}}\right)$ as a function of N corresponds to the shortest addition chain for N (OEIS sequence A003313).

Partial Proof.

Strings $C_{\min }$ for which $a_N\left(C_{\min }\right)=\underset{k}{min}\left(\left\{a_N\left(C_k\right)\right\}\right)$, $\forall k=\{1,2,\cdots ,2^N\}$ can be formed by joining two basic symbols, adding the pair to the pool and joining the longest strings taken from the pool until N is reached. This sets the bound for $N=2^s$, where s is the number of assembly steps and $\underset{k}{min}\left(\left\{a_{2^s}\left(C_k\right)\right\}\right)=s={log}_2\left(N\right)$. Only four strings $C_1^{2^s}=[00\cdots ]$, $C_2^{2^s}=[11\cdots ]$, $C_3^{2^s}=[0101\cdots ]$, and $C_4^{2^s}=[1010\cdots ]$ have such an assembly index in this case. □

Theorem 2

The length of an elegant binary program required to assemble a string that has the smallest assembly index is equal to its assembly index.

Proof.

An elegant program is the shortest program that produces a given output [34,35]. The assembly pool P is a distinct set, to which strings are added in subsequent assembly steps. Thus, we can define

 

Command 0: Take the last two elements from P, join them with each other, and output.

 

and

 

Command 1: Take the last element from P, join it with itself, and output.

as the only two commands of a binary program F of length $s_F$ applicable to the initial assembly pool $P=\{0,1\}$ containing only two basic symbols.

The $1^{\mathrm{st}}$ bit of the program F is irrelevant as $F=0$ assembles $C_3^{\left(2\right)}=\left[01\right]$ and $F=1$ assembles $C_1^{\left(2\right)}=\left[11\right]$ and we can denote it by $\ast =\{0,1\}$. Then the programs $F=\ast 1\cdots 1$ assemble the $2^{s_F}$-bit strings $C_{1,3}^{\left(2^s\right)}=[\ast 1\ast 1\cdots ]$ having the assembly index $a_{2^{s_F}}=s_F$, while strings $C_{2,4}^{2^{s_F}}$ with the smallest assembly index $a_{2^{s_F}}=s_F$ can be assembled with the same two programs starting with the pool to $P=\{1,0\}$.

The remaining $2^{s_F}-2$ programs will assemble some of the shorter strings with the assembly index $a_{\ast }=s_F$. In general, all programs F assemble strings having lengths expressible as a product of Fibonacci numbers (OEIS A065108) as shown in Table A2, wherein out of $2_F^s$ programs (cf. Table 6)

• $2^{s_F-1}$ programs $F=\ast 1\ast \cdots$ assemble $[\ast 1\ast 1\cdots ]$ strings including the strings $C_{1,3}^{\left(2^{s_F}\right)}$,
• $2^{s_F-2}$ programs $F=\ast 01\ast \cdots$ assemble $[1\ast 11\ast 1\cdots ]$ strings,
• $2^{s_F-3}$ programs $F=\ast 001\ast \cdots$ assemble $[\ast 11\ast 1\ast 11\ast 1\cdots ]$ strings,
• ⋯,
• the program $F=\ast 0\cdots 0$ assembles the shortest string having length belonging to the set of Fibonacci numbers, and
• the program $F=\ast 0\cdots 1$ assembles two such strings into a string of length being twice the Fibonacci sequence (OEIS A055389).

For $s_F\ge 4$, some of the programs are no longer elegant and some of the assembled strings do not have the minimal assembly index. For example, the program $F=\ast 000$ assembles the string $C^{\left(8\right)}=[1\ast 1\ast 11\ast 1]$, but if $\ast =1$ this string can be assembled by a shorter program $F=\ast 11$, and if $\ast =0$ this string does not have the minimal assembly index $a_8\left(C_{\min }\right)=3$ but $a_8=4$. For $s_F\ge 10$ and for the shortest string assembled by the program F the program F is not elegant for $\ast =1$ and the shortest string it assembles is not $C_{\min }$ for $\ast =0$. However, the length $s_F$ of any program F is not shorter than the assembly index of the string which this program assembles. □

The assembly programs F are listed in Table 3, Table 4 and Table 5 for one version of the assembly pool and for $3\le s_F\le 5$.

We note in passing that Theorem 2 would be violated if we defined the command "1" e.g. as "take the last element from the assembly pool, join it with itself, join with what you have already assembled (say at "the right"), and output". Then the 2-bit program "11" would produce $\left[111111\right]$ with the assembly index $a_6=3$. However, such a one-step command would violate the axioms of assembly theory since it would perform two assembly steps in one program step. An elegant program to output the gigabyte binary string of all zeros would take a few bits of code and would have a low Kolmogorov complexity [36]. However, such a string would be outputted, not assembled. Furthermore, the length of such a program that outputs the string $[1\cdots 1]$ would be shorter than the length of the program that outputs the string $[01\cdots ]$, while in AT, the lengths of these programs must be the same. Theorem 2 is about binputation2 of binary strings.

Table 6. 6-bit programs assembling strings with $a_{\ast }=\{5,6\}$.

F	$\mathit{C}({\mathit{s}}_{\mathit{F}}=6)$	N
$\ast 00000$	$1\ast 1\ast 11\ast 1\ast 11\ast 11\ast 1\ast 11\ast 1$	21
$\ast 00001$	$\ast 11\ast 11\ast 1\ast 11\ast 1\cdots$	26
$\ast 0001\ast$	$1\ast 1\ast 11\ast 1\cdots$	${24}^{†},{32}^{‡}$
$\ast 001\ast \ast$	$\ast 11\ast 1\cdots$	$25,30,40$
$\ast 01\ast \ast \ast$	$1\ast 1\cdots$	${24}^{†},\cdots$
$\ast 1\ast \ast \ast \ast$	$\ast 1\cdots$	$\cdots ,{32}^{†},\cdots$

^†. This program is not elegant. ^‡. This program is not elegant for $\ast =1$ and the assembled string is not $C_{\min }$ for $\ast =0$.

Table 6. 6-bit programs assembling strings with $a_{\ast }=\{5,6\}$.

F	$\mathit{C}({\mathit{s}}_{\mathit{F}}=6)$	N
$\ast 00000$	$1\ast 1\ast 11\ast 1\ast 11\ast 11\ast 1\ast 11\ast 1$	21
$\ast 00001$	$\ast 11\ast 11\ast 1\ast 11\ast 1\cdots$	26
$\ast 0001\ast$	$1\ast 1\ast 11\ast 1\cdots$	${24}^{†},{32}^{‡}$
$\ast 001\ast \ast$	$\ast 11\ast 1\cdots$	$25,30,40$
$\ast 01\ast \ast \ast$	$1\ast 1\cdots$	${24}^{†},\cdots$
$\ast 1\ast \ast \ast \ast$	$\ast 1\cdots$	$\cdots ,{32}^{†},\cdots$

^†. This program is not elegant. ^‡. This program is not elegant for $\ast =1$ and the assembled string is not $C_{\min }$ for $\ast =0$.

Theorem 2 is related to Gödel’s incompleteness theorems and the halting problem. N cases of the halting problem correspond only to $s={log}_2\left(N\right)$, not to N bits of information [37]. Therefore, N-bit elegant programs assemble all four strings $C^{\left(2^N\right)}$ with $a_{2^N}=N$ (with two versions of the assembly pool). Furthermore, we can consider all strings assembled by the N-bit assembly program as corresponding to provable theorems. Any formal axiomatic system only enables proving only true theorems [38].There is a more fundamental path to incompleteness that involves complexity, rather than self-reference [38].

In the following, we conjecture the form of the upper bound of the set of different string assembly indices 1.

In general, of all strings $C_k$ having a given assembly index, shown in Table 1 and Table A3, Table A4, Table A5, Table A6, Table A7, Table A8, Table A9, Table A10, most are those having $N_1=⌊N/2⌋$. The only exceptions are $N=8$ for $a_8=4$ ($4<8$) and for $a_8=6$ ($24<26$), $N=10$ for $a_{10}=4$ ($2<5$) and for $a_{10}=5$ ($32<33$), and $N=12$ for $a_{12}=4$ ($2<3$).

Introducing the definition 2 of a balanced string allows us to reduce the search space of possible strings with maximal assembly indices to balanced strings only. With the exception of $N=8$, of all strings $C_k^{\left(N\right)}$ having a maximum assembly index, most are balanced.

We can further restrict the search space to distinct strings.

Definition 3.

A string $D_k^{\left(N\right)}$ is a distinct string if a ring formed with this string by joining its beginning with its end is unique among the rings formed from the other distinct strings $D_l^{\left(N\right)},l\ne k$.

There are at least two and at most N forms of a distinct string $D_k^{\left(N\right)}$ that differ in the position of the starting symbol. For example for $|B^{\left(4\right)}|=6$ balanced strings, shown in Table 2, two augmented strings with $a_4=2$ correspond to each other if we change the starting symbol

$$\begin{array}{cc}\hfill & [\cdots 1|0101\left|0101\right|01\cdots ]=\hfill \\ \hfill & [\cdots 10|1010\left|1010\right|1\cdots ].\hfill \end{array}$$

(3)

Similarly, four augmented strings with $a_4=3$ correspond to each other

$$\begin{array}{cc}\hfill & [\cdots |0110\left|0110\right|011\cdots ]=\hfill \\ \hfill & [\cdots 0|1100\left|1100\right|11\cdots ]=\hfill \\ \hfill & [\cdots 01|1001\left|1001\right|1\cdots ]=\hfill \\ \hfill & [\cdots 011|0011\left|0011\right|\cdots ],\hfill \end{array}$$

(4)

after a change in the position of the starting symbol. Thus, there are only two distinct strings for $N=4$

The number of distinct strings $|D^{\left(N\right)}|$ among all $2^N$ strings is given by the OEIS sequence A000031. In general (for $N\ge 3$), the number $|D^{\left(N\right)}|$ of distinct strings is much lower than the number $|B^{\left(N\right)}|$ of balanced strings.

 As asserted by the no-hair theorem [39], BH is characterized only by three parameters: mass, electric charge, and angular momentum.

 However, BHs are fundamentally uncharged and non-rotating, since the parameters of any conceivable BH, that is, charged (Reissner-Nordström), rotating (Kerr) and charged rotating (Kerr-Newman), can be arbitrarily altered, provided that the area of a BH surface does not decrease [40] using Penrose processes [41,42] to extract electrostatic and/or rotational energy of a BH [43].

 Thus, a BH is defined by a single real number, and no Planck triangle is distinct on a BH surface. We can define neither a beginning nor an end of a balanced distinct string $E_k^{\left(N_{\mathrm{BH}}\right)}$ that represents a given BH.

By neglecting the notion of the beginning and end of a string, we focus on its length and content. In Yoda’s language,

"complete, no matter where it begins. A message is".

The numbers of the balanced $|B_k|$, distinct $|D_k|$, and balanced distinct3 $|E_k|$ strings are shown in Table 7 and Figure 1.

We note that, in general, the starting symbol is relevant for the assembly index. Thus, different forms of a distinct string may have different assembly indices. For example, for $N=7$ balanced strings $B_{34}$ and $B_{35}$, shown in Table A13 have $a_7=6$. However, these strings are not distinct, since they correspond to each other and to the balanced strings $B_{13}$, $B_{18}$, $B_{20}$, $B_{28}$, and $B_{30}$ with $a_7=5$. They all have the same triplet of adjoining ones.

Definition 4.

The assembly index of a distinct string $D_k^{\left(N\right)}$ is the smallest assembly index among all forms of this string.

Thus, if different forms of a distinct string have different assembly indices, we assign the smallest assembly index to this string. In other words, we assume that the smallest number of steps

$$a_N\left(D_k\right)=\underset{l}{min}\left(\left\{a_N{\left(D_k\right)}_l\right\}\right),{\left(D_k\right)}_l\in D_k,$$

(5)

where ${\left(D_k\right)}_l$ denotes a particular form of a distinct string $D_k$, is the string assembly index of this distinct string.

 If an object that can be represented by a distinct string (a BB in particular) can be assembled in fewer steps, this procedure will be preferred by nature.

The distribution of the assembly indices of the balanced distinct strings $E_k$ is shown in Table 8.

If a string $C_{\min }$ for which $a_N\left(C_{\min }\right)=\underset{k}{min}\left(\left\{a_N\left(C_k\right)\right\}\right)$ is constructed from repeating patterns, then a string $C_{\max }$ for which $a_N\left(C_{\max }\right)=\underset{k}{max}\left(\left\{a_N\left(C_k\right)\right\}\right)$ must be the most patternless. The string assembly index must be bounded from above and $a_N\left(C_{\max }\right)$ must be a monotonically nondecreasing function of N that can increase at most by one between N and $N+1$.

Identifying the shortest pathway is known to be computationally challenging [3]. This problem has been proven to be at least as hard as NP-complete [44]. However, certain heuristic rules apply in our binary case. For example,

• for $N=7$ we cannot avoid two doublets (e.g. $2\times \left[00\right]$) within a distinct string $E_{28}^{\left(7\right)}=\left[0011100\right]$ and thus $a_7\left(C_{\max }\right)=5<6$,
• for $N=8$ we cannot avoid two pairs of doublets (e.g. $2\times \left[00\right]$ and $2\times \left[11\right]$) within a distinct string $E_7^{\left(8\right)}=\left[00001111\right]$ and thus $a_8\left(C_{\max }\right)=5<6$,
• for $N=12$ we cannot avoid three pairs of doublets (e.g. $2\times \left[00\right]$, $2\times \left[10\right]$, and $2\times \left[11\right]$) within a distinct string $E_k^{\left(12\right)}=\left[111000101100\right]$ and thus $a_{12}\left(C_{\max }\right)=8<9$,
• for $N=14$ we cannot avoid two pairs of doublets and one doublet three times (e.g. $2\times \left[00\right]$, $2\times \left[11\right]$, and $3\times \left[01\right]$, and thus $a_{14}\left(C_{\max }\right)=9<10$,
• etc.

Conjecture 2.

The problem of determining the assembly index of a given binary string $C_k^{\left(N\right)}$ is NP-complete [44], while the problem of creating the string so that it would have a predetermined maximum assembly index for this length of the string is NP-hard.

We found it much easier to determine an assembly index of a given binary string $C_k^{\left(N\right)}$ than to create a string so that it would have a maximum assembly index as a function of the length of the string. A proof of conjecture 2 would also be the proof of the following conjecture.

Conjecture 3.

$\mathrm{P}\ne \mathrm{NP}$

Every computable problem and every computable solution can be encoded as a finite binary string. Here, determining whether the assembly index of a given string has its known maximal value corresponds to checking the solution to a problem for correctness, whereas creating such a string corresponds to solving the problem. Thus, AT would solve the P versus NP problem in theoretical computer science. There is ample pragmatic justification for adding $\mathrm{P}\ne \mathrm{NP}$ as a new axiom [37].

Table 9 shows the exemplary balanced strings $B_{\max }$ having maximal assembly indices that we created (cf. also Appendix B). To determine the assembly index $a_{18}=11$ of the string

$$E_k^{\left(18\right)}=\left[1\left(001\right)\left(11\right)\left(110\right)\left(110\right)\left(00\right)\left(001\right)0\right],$$

(6)

we look for the longest patterns that appear at least twice within the string, and we look for the largest number of these patterns. Here, we find that each of the two triplets $\left[001\right]$ and $\left[110\right]$ appear twice in $E_k^{\left(18\right)}$ and are based on the doublets $\left[00\right]$ and $\left[11\right]$ also appearing in $E_k^{\left(18\right)}$. Thus, we start with the assembly pool $\{0,1,[00],[001],[11],[110\left]\right\}$ made in four steps and join the elements of the pool in the following seven steps to arrive at $a_{18}\left(E_k\right)=11$. On the other hand, another form of this balanced distinct string

$$E_l^{\left(18\right)}=\left[\left(01\right)\left(11\right)\left(110\right)\left(110\right)00\left(001\right)\left(01\right)0\right],$$

(7)

has $a_{18}\left(E_l\right)=12$.

Conjecture 4

(Tight upper bound on a string assembly index). With exceptions for small N the largest string assembly index $a_N\left(C_{\max }\right)$ of a binary string as a function of N is given by a sequence formed by $\{+1,+1,k\times 0,+1,+1,k\times 0\}$ strings for $k\in {\mathbb{N}}_0$, where $+1$ denotes increasing $a_N\left(C_{\max }\right)$ by one, and 0 denotes maintaining it at the same level, and $a_0=-1$.

However, at this moment, we cannot state if this conjecture applies to distinct or non-distinct strings. The assembly indices for $N<3$ are unique, whereas the assembly indices for $4\le N\le 10$ were discussed above and are calculated in Appendix C for balanced and balanced distinct strings.

The conjectured sequence is shown in Figure 2 and Figure 3 starting with $a_0=-1$ (we note in passing that $n=-1$ is a dimension of the void, the empty set ∅, or (-1)-simplex). Subsequent terms are given by $\{0,1,2,3,4,5,5,6,7,7,8,9,9,9,10,\cdots \}$, which is periodic for $N=k(k+3)$ and flattens at $a_N\left(D_{\max }\right)=4k-3$, and $a_N\left(D_{\max }\right)=4k-1$, $k\in \mathbb{N}$, $k>1$.

This sequence can be generated using the following procedure

We note the similarity of this bound to the Aufbau rule4, the Janet sequence (OEIS A167268) and the monotonically non-decreasing Shannon entropy of chemical elements, including observable ones [23]. Perhaps the exceptions in the sequence 4 vanish as N increases.

The bounds 1 and 4 are shown in Table 7 and illustrated in Figure 2 and Figure 3. No binary string cannot be assembled in a smaller number of steps than given by a lower bound (cf. Conjecture 1). On the other hand, some strings cannot be assembled in a smaller number of steps than given by an upper bound (which for large N, as we suppose, has the form presented in Conjecture 4).

Conjecture 5.

There is no binary program that has a length shorter than the length of the string featuring the largest assembly index that could assemble this string.

Partial Proof.

In assembling the string featuring the assembly index (the largest complexity), we cannot rely solely on the last or two last strings in the assembly pool. Thus, we need to index the strings in the pool. However, we cannot predict in advance how many strings there will be in the assembly pool. Thus, we do not know how many bits will be needed to encode the indices. □

The Hamlet tragedy contains approximately 130,000 letters. Assigning five bits per letter (32 possibilities), the Hamlet tragedy can be encoded in a string having $N_{\mathrm{Hamlet}}=650000$ bits (81.25 kB) yielding the total number of possible strings $2^{N_{\mathrm{Hamlet}}}\approx 1\times {10}^{195312}$ (including $|B_{N_{\mathrm{Hamlet}}}|\approx 1\times {10}^{195309}$), and their assembly indices are bounded by

$$27\le a_{N_{\mathrm{Hamlet}}}\left(C_k\right)\ll 3217$$

(8)

The lower bound (8) can be estimated using the Conjecture 1. The upper bound (8) can be estimated by finding the smallest k that satisfies $k(k+3)\ge N_{\mathrm{Hamlet}}$ and using the relation $a_N\left(C_{\max }\right)=4⌈k⌉-1$ of Conjecture 4.

We assume that the assembly index of the string encoding the actual Hamlet tragedy is close to the upper bound. Even if the probability of random typing of the Hamlet tragedy is unfathomably small, when constrained to the bounds of the physical universe [5], as asserted by the infinite monkey theorem, this tragedy was once created by William Shakespeare.

SARS-CoV-2 genome sequence contains 29903 bases $\{\mathrm{A},\mathrm{C},\mathrm{G},\mathrm{T}\}$. Assigning two bits per base it can be encoded in a string of $N_{\mathrm{SARS}-\mathrm{CoV}-2}=59806$ bits having the assembly index bounded by

$$24\le a_{N_{\mathrm{SARS}-\mathrm{CoV}-2}}\left(C_k\right)\ll 971.$$

(9)

 The supermassive BH Sagittarius A ∗ has an estimated mass $M_{\mathrm{BH}}\approx 8.26\times {10}^{36}$ kg corresponding to the Schwarzschild diameter $D_{\mathrm{BH}}\approx 2.45\times {10}^{10}$ m and the information capacity $N_{\mathrm{Sagittarius}{\mathrm{A}}^{\ast }}\approx 7.24\times {10}^{90}$ [8]. In this case, its assembly index is bounded by

$$332\le a_{N_{\mathrm{Sagittarius}{\mathrm{A}}^{\ast }}}\ll 1.0763\times {10}^{46}.$$

(10)

 However, we conjecture that

Conjecture 6.

A BB surface is defined by a balanced distinct string that satisfies the upper bound of a distinct string assembly index.

 To be the most patternless [8], a balanced BB surface must minimize not only Shannon entropy and Kolmogorov complexity (the latter is uncomputable), but also maximize its assembly index. A BB cannot be assembled in a suboptimal way, since black-body radiation is informationless.

3. Additional results

The [perceivable] universe is not big enough to contain the future; it is deterministic going back in time and non-deterministic going forward in time [45]. But we know [2,8,9,10,11,12,13,14,15,16,17,18,19] that it has evolved to the present since the Big Bang.

For K subunits of an objectO the assembly index of AT is bounded [1] from below by

$$min\left(a_O\right)={log}_2\left(K\right),$$

(11)

and from above by

$$max\left(a_O\right)=K-1,$$

(12)

where in the latter case, the subunits must be distinct so that they could not be reused from the pool, decreasing the index. The lower bound (11) represents the fact that the simplest way to increase the size of an object in a pathway is to take the largest object so far and join it to itself [1]. However, ${log}_2\left(K\right)>K-1$ for $K\in \mathbb{R}$ and $1<K<2$.

Perceivable information about any object can be encoded by a binary string [26,27]. This does not imply that a binary string defines an object. Information that defines a chemical compound, a virus, a computer program, etc. can be encoded by a binary string. However, a dissipative structure [12] such as a living biological cell (or its conglomerate such as a human, for example) cannot be represented by a binary string (even if its genome can). This information can only be perceived (so this is not an object defining information). Each of us is given to ourselves as a mystery [46]. Therefore, since one bit is the smallest amount and the quantum of information, the lower bound and the upper bound of the string assembly index define the allowed region of the assembly indices for binary strings.

The bounds 1, 4, (11), and (12) on the assembly index are shown also in Figure 4 (adopted from [1] and modified). According to the authors of [1], the "green portion of the figure is illustrative of the location in the complexity space where life might reasonably be found. Regions below can be thought of as being potentially naturally occurring, and regions above being so complex that even living systems might have been unlikely to create them. This is because they represent structures with limited internal structure and symmetries, which would require vast amounts of effort to faithfully reproduce." [1].

We disagree with this statement. It is obvious that a binary string itself is neither dissipative nor creative. It is its assembly process that can be dissipative or creative. Evolution is about assembling new information and optimizing it until it reaches its assembly index.

That is why, we found determining the assembly index of a given binary string $C_k^{\left(N\right)}$ is easier than creating a string with a maximum assembly index for this length of the string (Conjecture 2). Once the new information is assembled (by a dissipative structure operating far from thermodynamic equilibrium, or created by humans) increasing the information entropy according to the 2^nd law of infodynamics [16], it enters the realm of the 2^nd law of thermodynamics, and nature seeks how to optimize its assembly pathway decreasing information entropy. And only humans are gifted with creativity. Any creation is required to be shaped by the unique personality of the creator to such an extent that it is statistically one-time in nature [47]; it is an imprint of the author’s personality.

The total entropy of the universe S is constant and is the sum of the information entropy $S_{info}$ and the physical entropy $S_{phys}$. Therefore, over time [19]

$$\frac{d{S}_{info}}{dt}+\frac{d{S}_{phys}}{dt}=0.$$

(13)

The time corresponds to an increasing information capacity. Bit by bit: $dt=(N+1)-N=1$

At first, the newly assembled information corresponds to the discovery by groping [11]. However, its assembly pathway does not attain its most economical or efficient form all at once. For a certain period of time, its evolution gropes about within itself. The try-out follows the try-out, not being finally adopted. Then finally perfection comes within sight, and from that moment the rhythm of change slows down [11]. The new information, having reached the limit of its potentialities, enters the phase of conquest. Stronger now than its less perfected neighbours, the new information multiplies and consolidates. When the assembly index is reached, new information attains its equilibrium (not necessarily a BH equilibrium) and its evolution terminates. It becomes stable.

There is a certain minimum amount of information $N_C$ required to establish a creation, as shown in Figure 4. Sixteen possibilities provided by the minimum of thermodynamic entropy [29,30,31] bifurcate the assembly pathways (cf. Theorem 1) but none of these possibilities can be considered a creation. However, the boundary between the green region of dissipative structures [12] and the red region of human creativity remains to be discovered.

"Thanks to its characteristic additive power, living matter (unlike the matter of the physicists) finds itself ’ballasted’ with complications and instability. It falls, or rather rises, towards forms that are more and more improbable. Without orthogenesis life would only have spread; with it there is an ascent of life that is invincible." [11]

BB having the energy given by mass-energy equivalence

$$\begin{array}{cc}\hfill & E_{\mathrm{BB}}=\frac{k}{2}{M}_{\mathrm{BB}}{c}^2=\frac{k}{2}{m}_{\mathrm{BB}}{E}_{\mathrm{P}}=\frac{d_{\mathrm{BB}}}{4}{E}_{\mathrm{P}}=\frac{1}{4}\sqrt{\frac{N_{\mathrm{BB}}}{\pi }}{E}_{\mathrm{P}},\hfill \\ \hfill & 2\le k\le k_{\max }=\frac{2{\alpha }^2}{\sqrt{{\alpha }^4-{\alpha }_2^4}}\approx 6.7933\hfill \end{array}$$

(14)

where $M_{\mathrm{BB}}:=m_{\mathrm{BB}}{m}_{\mathrm{P}}$, $m_{\mathrm{BB}}\in \mathbb{R}$ denote the BB mass, and $E_{\mathrm{P}}$, $m_{\mathrm{P}}$ denote the Planck energy and mass, $\alpha \approx 1/137.036$ is the fine-structure constant and ${\alpha }_2\approx -1/140.178$ is the $2^{\mathrm{nd}}$ fine-structure constant related to $\alpha$ by $(\alpha +{\alpha }_2)/\left(\alpha {\alpha }_2\right)=-\pi$, and k is the BB size-to-mass ratio (STM) [10] ($k=2$ if BB is BH).

It was shown [9] based on the Mandelstam-Tamm [48], Margolus–Levitin [49], and Levitin-Toffoli [50] theorems on the quantum orthogonalization interval that BBs generate (or rather assemble) a pattern forming nonequilibrium shell (VS) through the solid-angle correspondence, as shown in Figure 5. The BB entropic work

$$\begin{array}{cc}\hfill W_{\mathrm{BB}}& =T_{\mathrm{BB}}{S}_{\mathrm{BB}}=T_{\mathrm{BB}}\frac{1}{4}{k}_{\mathrm{B}}{N}_{\mathrm{BB}}=T_{\mathrm{BB}}\frac{1}{4}{k}_{\mathrm{B}}\pi d_{\mathrm{BB}}^2\hfill \\ \hfill & =\frac{E_{\mathrm{P}}{d}_{\mathrm{BB}}}{4k}\left(1\pm i\sqrt{\frac{k^2}{4}-1}\right),\hfill \end{array}$$

(15)

is the work done by all APTs of a BB. It is the product of the BB entropy [29,30,31] and the general, complex BB temperature

$$T_{\mathrm{BB}}=\frac{T_{\mathrm{P}}}{k\pi d_{\mathrm{BB}}}\left(1\pm i\sqrt{\frac{k^2}{4}-1}\right),$$

(16)

which in modulus and for a BH ($k=2$) reduces [10] to Hawking temperature

$$T_{\mathrm{BH}}=\frac{\hslash c^3}{8\pi G{M}_{\mathrm{BH}}{k}_{\mathrm{B}}}=\frac{T_{\mathrm{P}}}{2\pi d_{\mathrm{BH}}},$$

(17)

where $\hslash =h/\left(2\pi \right)$ is the reduced Planck constant, G is the gravitational constant, and $T_{\mathrm{P}}$ is the Planck temperature. In particular [10]

$$T_{\mathrm{BB}}\left(k_{\max }\right)=\frac{T_{\mathrm{P}}}{2\pi d_{\mathrm{BB}}{\alpha }^2}\left(\sqrt{{\alpha }^4-{\alpha }_2^4}\pm i{\alpha }_2^2\right),$$

(18)

$$\begin{array}{cc}\hfill & T_{\mathrm{BB}}\left(k_{\mathrm{eq}}\right)=\frac{T_{\mathrm{P}}}{2\pi d_{\mathrm{BB}}}\frac{{\alpha }^2\pm i{\alpha }_2^2}{\sqrt{{\alpha }^4+{\alpha }_2^4}},\hfill \end{array}$$

(19)

where

$$k_{\mathrm{eq}}=\frac{2}{{\alpha }^2}\sqrt{{\alpha }^4+{\alpha }_2^4}\approx 2.7665.$$

(20)

is the energy equilibrium STM.

A VS has the information capacity bounded by

$$\begin{array}{cc}\hfill N_{\mathrm{BB}}& \le N_{\mathrm{VS}}\le 4N_{\mathrm{BB}},\hfill \\ \hfill N_{\mathrm{VS}}:=& l{N}_{\mathrm{BB}},1\le l\le 4,\hfill \end{array}$$

(21)

where l is a VS defining factor. The number of APTs is bounded by

$$⌊\frac{1}{4}{N}_{\mathrm{BB}}⌋\le N_1\le ⌊\frac{1}{2}{N}_{\mathrm{BB}}⌋,$$

(22)

as shown in Figure 6, and thus its binary potential $\delta {\phi }_{\mathrm{VS}}=-N_1{c}^2/N_{VS}$ [8,9] is bounded by

$$-\frac{1}{2}{c}^2\le \delta {\phi }_{\mathrm{VS}}<\left\{\begin{array}{cc}\left(\frac{1}{N_{\mathrm{BB}}}-\frac{1}{4}\right)c^2>0,\hfill & N_{\mathrm{BB}}<4\hfill \\ \left(\frac{1}{4N_{\mathrm{BB}}}-\frac{1}{16}\right)c^2<0,\hfill & N_{\mathrm{BB}}>4\hfill \end{array}\right..$$

(23)

and the theoretical probability $p_1:=N_1/N_{\mathrm{VS}}$ for a triangle on a VS to be an active Planck triangle is also bounded [9] by

$$\frac{1}{16}-\frac{1}{4N_{\mathrm{BB}}}<p_1\le \frac{1}{2}.$$

(24)

On the other hand, the entropy variation [8,51] $\delta S/k_{\mathrm{B}}=-c\delta \phi$ so that for $N_{\mathrm{BB}}<4$ the lower bound (24) is negative and the upper bound (23) is positive ($N_1\le 1$ in this range). The Planck triangle of VS is located somewhere on the VS surface defined by a solid angle

$$\mathrm{\Omega }=\frac{{\ell }_{\mathrm{P}}^2}{R_{\mathrm{BB}}^2}=\frac{4\pi {\ell }_{\mathrm{P}}^2}{4\pi R_{\mathrm{BB}}^2}=\frac{4\pi }{N_{\mathrm{BB}}},$$

(25)

that corresponds to the BB Planck triangle.

The BB information capacity is dictated by its diameter and the BB energy (14) as a function of its diameter is the same for all BBs (it is independent on k). However, the BB mass and density

$${\rho }_{\mathrm{BB}}=\frac{M_{\mathrm{BB}}}{V_{\mathrm{BB}}}=\frac{3}{k{N}_{\mathrm{BB}}}{\rho }_{\mathrm{P}},$$

(26)

are not.

Based on the orbiting condition $V_O^2\le V_R^2\le V_E^2$, where $V_O=\sqrt{G{M}_{\mathrm{C}}/R_{\mathrm{avg}}}$ is the orbital, and $V_E=\sqrt{2G{M}_{\mathrm{C}}/R_{\mathrm{avg}}}$ the escape speed of an orbiting object, $R_{\mathrm{avg}}$ is the average distance from the center of the central object to the center of the orbiting object, and $M_{\mathrm{C}}$ is the mass of the central object, the bounds

$$N_{\mathrm{BB}}\le 4v_R^4{N}_{\mathrm{VS}}\le 4N_{\mathrm{BB}},$$

(27)

containing the velocity term $V_R=v_{R}c$, $v_R\in \{\mathbb{R},\mathbb{I}\}$ were also derived [9]. Plugging $N_{\mathrm{VS}}$ from the bounds (21) into the bounds (27) we arrive at

$$\frac{1}{4l}\le v_R^4\le \frac{1}{l},$$

(28)

which is satisfied by real and imaginary (but not complex) velocities (for example, for $l=1$ by $-1\le v_R\le -1/\sqrt{2}$, $1/\sqrt{2}\le v_R\le 1$, $-i\le v_R\le -i/\sqrt{2}$, and $i/\sqrt{2}\le v_R\le i$). Taking the square root of the bounds (28), using $v_{LL}^2+v_{RR}^2=1$, $v_R\in \{\mathbb{R},\mathbb{I}\}$ [9], and squaring again, we arrive at

$$\frac{l-2\sqrt{l}+1}{l}\le v_L^4\le \frac{4l-4\sqrt{l}+1}{4l}.$$

(29)

The bounds (28) and (29), shown in Figure 7, meet at $v=1/\sqrt{2}$, where de Broglie and Compton wavelengths of mass M are the same

$${\lambda }_{\mathrm{dB}}=\frac{h}{p}=h\frac{\sqrt{1-\frac{V^2}{c^2}}}{MV}={\lambda }_{\mathrm{C}}=\frac{h}{Mc}\iff \frac{V}{c}=\frac{1}{\sqrt{2}},$$

(30)

where p is the relativistic momentum. The same is the ratio of orbital to escape speed: $\frac{V_O}{V_E}=\frac{1}{\sqrt{2}}$.

Furthermore, the bounds (28) and (29) do not overlap only for $l=\{1,4\}$. Therefore, $1<l<4$ defines the dissipativity or the assembly range. Furthermore, the intersection of the bounds (28) and (29) is the common region for both velocities. If $v_L$ is within this region, then $v_R$ is as well. We note that the average orbital velocity of each orbiting object only slightly exceeds its orbital speed $V_O$. This implies that the average VS defining factor $l_{\mathrm{avg}}\gtrsim 1$ in (21) for a VS orbiting object (cf. Appendix A).

BBs define a perfect thermodynamic equilibrium, and the bounds (21) and (22) show that nature uses optimally assembled information (cf. Conjecture 6) to assemble new information. Figure 8 shows the bounds on the string assembly indices, and Figure 9 shows the BB temperature (17), energy (14), and entropic work (15) for $0\le N_{\mathrm{BB}}\le 5$. $k_{\mathrm{B}}\left|T_{\mathrm{BB}}\right|/E_{\mathrm{BB}}=2/N_{\mathrm{BB}}$ is a rational number for natural $N_{\mathrm{BB}}$.

Let us examine this process starting from the Big Bang during the Planck epoch and shortly thereafter, and for continuous $N_{\mathrm{BB}}\in \mathbb{R}$ (i.e., including fractional Planck triangle(s)).

$N_{\mathrm{BB}}=0$

There is nothing to talk about. It is a mystery.

$0<N_{\mathrm{BB}}<1$

The Big Bang has occurred, forming the 1^st BB. At $N_{\mathrm{BB}}\left(k_{\max }\right)=({\alpha }^4-{\alpha }_2^4)/\left(4\pi {\alpha }^4\right)\approx 0.0069$ the BB temperature (17) and subsequently at $N_{\mathrm{BH}}=1/\left(4\pi \right)\approx 0.0796$ the BH temperature (17) become equal to the Planck temperature, but any BB in this range is still too small to carry a single bit of information and cannot be triangulated. However, independent BBs merge [9,10] summing their entropies and increasing the information capacity.

$N_{\mathrm{BB}}=1$

The first bit (a degree of freedom [9]) becomes available, and the BH energy reaches the limit of the equipartition theorem for one bit ($E_{\mathrm{BH}}=\frac{1}{2}{k}_{\mathrm{B}}{T}_{\mathrm{BH}}$). APTs on BBs begin to fluctuate. However, the bounds (22) make them unable to generate any APTs on a VS ($N_1=0$).

$1<N_{\mathrm{BB}}<2$

This is the only range in which the lower AT bound (11) is greater than the upper AT bound (12).

The BH temperature (17) exceeds its energy (14) ($\frac{1}{2}{k}_{\mathrm{B}}{T}_{\mathrm{BH}}<E_{\mathrm{BH}}<k_{\mathrm{B}}{T}_{\mathrm{BH}}$) [9]. At $N_{\mathrm{BH}}=2ln\left(2\right)$ the BH energy (14) is equal to the Landauer limit $E_{\mathrm{BH}}=k_{\mathrm{B}}{T}_{\mathrm{BH}}ln\left(2\right)\approx 1.3863$ [52]. Shortly thereafter, at $N_{\mathrm{BH}}=1.5$, the BH density reaches the level of the Planck density For a BB [10] Still $N_1=0$. Merging BBs expand fractional Planck triangle(s) to form the 2^nd bit.

$N_{\mathrm{BB}}=2$

The first nonvanishing $N_1=1$ becomes available on a VS generated by a BB. The BH temperature (17) is equal to its energy (14) ($k_{\mathrm{B}}{T}_{\mathrm{BH}}=E_{\mathrm{BH}}=E_{\mathrm{P}}/\left(2\sqrt{2\pi }\right)$).

$2<N_{\mathrm{BB}}<3$

At $N_{\mathrm{BB}}=4ln\left(2\right)$ the BH entropic work (15) is equal to the Landauer limit ($k_{\mathrm{B}}{T}_{\mathrm{BH}}=W_{\mathrm{BH}}=E_{\mathrm{P}}/\left(4\sqrt{\pi }\right)$). At $N_{\mathrm{BB}}>2.4507$ the density of the least dense BB ($k_{\max }\approx 6.7933$) drops below the modulus of its temperature. $N_1=\{0,1\}$.

$N_{\mathrm{BB}}=3$

$3<N_{\mathrm{BB}}<4$

With $N_{\mathrm{BB}}>3$ BBs can finally be triangulated. Yet, containing only one APT ($N_1=\{0,1\}$), they are not ergodic [9].

At $N_{\mathrm{BH}}>\pi$ the BH surface gravity $g_{\mathrm{BH}}=1/d_{\mathrm{BH}}$ drops below the Planck acceleration and the tangential acceleration [8,9] becomes real ($a_L\in \mathbb{R}$).

$N_{\mathrm{BB}}=4$

The BB assembly index bifurcates, minimal thermodynamic entropy [30] is reached, and the relation (22) provides the second bit on a VS ($N_1=2$). At this moment BB can be assembled in a different number of steps and nature seeks to minimize this number following the dynamics induced by the relation (13). The BH temperature (17) is equal to its entropic work (15) ($k_{\mathrm{B}}{T}_{\mathrm{BH}}=W_{\mathrm{BH}}$).

$4<N_{\mathrm{BB}}<6$

The BH temperature (17) finally drops below the entropic work (15) limit and $N_1\ge 2$.

$N_{\mathrm{BB}}=6$

A BB reaches the upper bound on distinct assembly index.

$6<N_{\mathrm{BB}}<7$

The imaginary Planck time appears at the BH surface [8] heralding the end of the Planck epoch. After crossing this threshold, the VSs begin to operate with $1\le N_1\le 3$ on $2\pi <N_{\mathrm{VS}}\le 8\pi$, and the first dissipative structures can be assembled.

Nature enters a directed exploration phase ($\alpha <1$) and selectivity emerges, limiting the discovery of new objects [6].

$N_{\mathrm{BB}}=7$

A BB reaches the upper bound on nondistinct assembly index.

⋯

$N_{\mathrm{BB}}>12$

At $N_{\mathrm{BB}}=4\pi$ a first precise diameter relation can be established between the vertices of the BB surface. Furthermore, for $N_{\mathrm{BB}}=4\pi$, the solid angle (25) equals one steradian.

⋯

$N_{\mathrm{BB}}=N_C$

The onset of human creativity.

4. Conclusions

The results reported here can be applied in the fields of cryptography, data compression methods, stream ciphers, approximation algorithms [53], reinforcement learning algorithms [54], information-theoretically secure algorithms, etc. Another possible application of the results of this study could be molecular physics and crystallography.

 Overall, the results reported here support the assembly theory [1,2,3,4,5,6,7], the Bekenstein’s minimum of thermodynamic entropy [29,30,31], the holographic principle [32], entropic gravity [33], emergent dimensionality [8,9,10,15,17,18,20,23], the second law of infodynamics [16,19], and invite further research.