# Topology-Aware Exploration of Energy-Based Models Equilibrium: Toric QC-LDPC
Codes and Hyperbolic MET QC-LDPC Codes

Vasiliy Usatyuk  R&D Department, T8
Email<EMAIL_ADDRESS>Denis Sapozhnikov  R&D Department, T8
Email<EMAIL_ADDRESS>Sergey Egorov  Computer Science Department, South-West
State University
Email<EMAIL_ADDRESS>
###### Abstract

This paper presents a method for achieving equilibrium in the ISING
Hamiltonian when confronted with unevenly distributed charges on an irregular
grid. Employing (Multi-Edge) QC-LDPC codes and the Boltzmann machine, our
approach involves dimensionally expanding the system, substituting charges
with circulants, and representing distances through circulant shifts. This
results in a systematic mapping of the charge system onto a space,
transforming the irregular grid into a uniform configuration, applicable to
Torical and Circular Hyperboloid Topologies. The paper covers fundamental
definitions and notations related to QC-LDPC Codes, Multi-Edge QC-LDPC codes,
and the Boltzmann machine. It explores the marginalization problem in code on
the graph probabilistic models for evaluating the partition function,
encompassing exact and approximate estimation techniques. Rigorous proof is
provided for the attainability of equilibrium states for the Boltzmann machine
under Torical and Circular Hyperboloid, paving the way for the application of
our methodology. Practical applications of our approach are investigated in
Finite Geometry QC-LDPC Codes, specifically in Material Science. The paper
further explores its effectiveness in the realm of Natural Language Processing
Transformer Deep Neural Networks, examining Generalized Repeat Accumulate
Codes, Spatially-Coupled and Cage-Graph QC-LDPC Codes. The versatile and
impactful nature of our topology-aware hardware-efficient quasi-cycle codes
equilibrium method is showcased across diverse scientific domains without the
use of specific section delineations.

##  I Introduction

The integration of graph models has had a transformative impact across diverse
domains, playing a crucial role in fields such as forward error correction,
source coding, compressed sensing, classical and quantum machine
learning/computation, and control theory, [1, 2, 3, 4].

Despite notable successes in applying graph models to shallow networks and
demonstrating achievements in lazy regime machine learning [5], challenges
persist. These encompass issues like optimal neural network architectures,
adversarial robustness in deep neural networks (DNNs), optimal preconditions
for DNN training, efficient estimation of the Hessian matrix for quasi-Newton
DNN training, and achieving sparse structural low-complexity learning and
inference in DNNs. Additionally, there is a notable gap in establishing proofs
for both constructive and non-constructive forms of Generalized Shannon’s
Theorem for nonlinear channels.

These challenges, spanning information theory, control theory, topology,
machine learning, and cryptography, are effectively reduced to the framework
of the generalized Poincaré theorem, specifically the Novikov conjecture [6,
7, 8]. Through a topology surgery approach, involving a sufficient subspace
metric (grid) to represent the $n$-dimensional manifold using low-dimensional
decomposition (embedding), the paper tackles these challenges, [9, 10, 11, 12,
13].

The residual error from topological surgery becomes a guiding force, leading
toward the formulation and understanding of the generalized Shannon theorem.
This holistic approach addresses challenges spanning various scientific
domains, emphasizing the intricate connections between topology, information
theory, and machine learning.

The paper aims to provide partial answers to questions by narrowing down the
exploration of complex system dynamics to constructing topology-aware energy-
based codes on graphs. It delves into the application of topology in
information theory, particularly in Channel Coding, across diverse scientific
domains like Material Science, Deep Neural Networks, Control Theory, and
beyond. This dual perspective contributes to the advancement of information
theory and broader interdisciplinary scientific endeavors. The study reveals
that choosing Topology, whether Torical or Hyperbolic, for representing non-
linear fields inherently gives rise to quasi-cyclic (Low density parity-check)
codes on graphs. The codewords of these codes serve as energy minima, while
pseudo-codewords form local minima around these symmetry points. The size of
the circulant (Automorphism) of the quasi-cyclic code plays a pivotal role in
determining the number of equivalent energy minima, unveiling a novel
perspective on energy landscapes.

From a mathematical perspective, quasi-cyclic codes can be conceptualized as
forming an irregular grid in space. Their inherent strong symmetry designates
them as linearizer manifolds, crucial in studying the Hamiltonian and the
conservation of physical quantities across spatial and temporal dimensions.
Linearizers have the capability to transform an irregular grid into a regular
one, as explored in works such as [14, 15, 16].

Introducing a pioneering Topology-Aware method for achieving equilibrium in
the Ising System Hamiltonian with unevenly distributed charges, the approach
leverages dimensional expansion through the substitution of charges with
circulants and distances with circulant shifts. This transformative process
results in a spatial mapping that converts the initially irregular charge grid
into a more uniform and structured configuration. The proposed methodology,
with the invariance of Quasi-Cyclic rotations and the symmetry of codewords,
coupled with the asymmetry of pseudocodewords, gives rise to structural
(quasi-cyclic) sparsity, offering computational efficiency and storage
optimization advantages. This makes the approach well-suited for applications
dealing with large-scale Dynamic Systems.

The paper is structured as follows. In Section II, we present fundamental
definitions and notations pertaining to QC-LDPC Codes, Multi-Edge QC-LDPC
codes, and the Boltzmann machine (Ising model). Section III provides an
overview of the marginalization problem in code on the graph probabilistic
models, addressing the evaluation of the partition function (normalization).
We explore exact estimation techniques using determinants and permanents, as
well as approximate estimation employing the Bethe-permanent. Section IV is
dedicated to proving that equilibrium states (ground states) of the Boltzmann
machine, characterized by unequal charges on an irregular grid, are attainable
under Torical and Circular Hyperboloid Topologies. This involves a systematic
expansion of the system’s charge dimension, with charges being replaced by
multiple circulants and distances represented through circulant shifts.
Section V delves into the practical application of this approach in Finite
Geometry QC-LDPC Codes and its relevance to Material Science. Finally, Section
VI investigates the application of this methodology to Generalized Repeat
Accumulate Codes ([17, 55]) and Cage-Graph QC-LDPC Codes ([56, 18, 19, 20]),
showcasing its effectiveness (LRA challenge, [21]) in the context of Natural
Language Processing Transformer Deep Neural Networks Architectures: "MEGA"
[22], "ChordMixer" [23], "CDIL" [24].

##  II Basic definition

LDPC (Low-Density Parity-Check) codes, with parameters $[N,K]$, are linear
error-correcting codes represented by sparse parity-check matrices. Here, $N$
is the length of the codeword, and $K$ is the number of information bits [25,
26]. The parity-check matrix $H$ specifies parity check equations and can be
visualized as a Tanner graph. In the case of the parity-check matrix given by:

$H=\begin{bmatrix}1&0&1&1&1\\\ 1&1&0&0&0\\\ 0&1&1&1&1\\\ \end{bmatrix},$ (1)

the corresponding Tanner graph is shown in Fig. 1, left.

Figure 1: (left) Tanner Graph of $H$ parity-check matrix (right) Multi-graph
of $H_{2}$ parity-check matrix

Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes, denoted as $[N,K]$,
belong to the class of linear block error-correcting codes with a special
quasi-cyclic parity-check matrix $H$. In an $(N,K)$ QC-LDPC code, $N$
represents the code length (number of codeword bits), and $K$ represents the
number of message bits (length of the original message). The remaining bits in
the codeword, i.e., $(N-K)$, are parity bits. The Tanner graph of a QC-LDPC
code is described by the parity-check matrix $H$, consisting of square blocks
that are either zero matrices or circulant permutation matrices.

The $L\times L$ circulant permutation matrix $P$ is defined as:

$P_{ij}=\begin{cases}1,&\text{if }i+1\equiv j\,\mathrm{mod}\,L\\\
0,&\text{otherwise}.\end{cases}$ (2)

Here, $P_{k}$ represents the circulant permutation matrix (CPM) that shifts
the identity matrix $I$ to the right by $k$ times for any $k$, $0\leq k\leq
L-1$. The zero matrix is denoted as $I^{-1}$, and the set
$\\{-1,0,1,\ldots,L-1\\}$ is denoted by $A_{L}$.

Suppose the matrix $H$ of size $mL\times nL$ is defined as follows:

$H=\left[\begin{array}[]{cccc}{P_{a_{11}}}&{P_{a_{12}}}&{\cdots}&{P_{a_{1n}}}\\\
{P_{a_{21}}}&{P_{a_{22}}}&{\cdots}&{P_{a_{2n}}}\\\
{\vdots}&{\vdots}&{\ddots}&{\vdots}\\\
{P_{a_{m1}}}&{P_{a_{m2}}}&{\cdots}&{P_{a_{mn}}}\end{array}\right],$ (3)

where $a_{i,j}\in A_{L}$, and $L$ is the circulant size of $H$. A code $C$
with parity-check matrix $H$ is referred to as a QC-LDPC code. The exponent
matrix $E(H)=(E_{ij}(H))$ of $H$ is given by:

$E(H)=\left[\begin{array}[]{cccc}{a_{11}}&{a_{12}}&{\cdots}&{a_{1n}}\\\
{a_{21}}&{a_{22}}&{\cdots}&{a_{2n}}\\\ {\vdots}&{\vdots}&{\ddots}&{\vdots}\\\
{a_{m1}}&{a_{m2}}&{\cdots}&{a_{mn}}\end{array}\right],$ (4)

where $E_{ij}(H)=a_{ij}$. The protograph mother matrix or base graph $M(H)$ is
a binary matrix of size $m\times n$ obtained by replacing $-1$’s and other
integers with $0$ and $1$, respectively, in $E(H)$.

The Tanner graph of matrix $H$ forms a cycle if the equation
$\sum_{i=1}^{2l}(-1)^{i}a_{i}\equiv 0\,\mathrm{mod}\,L$ is satisfied. A sub-
graph of the Tanner graph, formed by cycles or cycle overlap in matrix $H$,
includes $a$ variable nodes and $b$ odd-degree checks, known as a trapping set
$TS(a,b)$, as defined in [27]. Fig. 2 illustrates examples such as TS(5,3)
generated by the overlap of three 8-cycles and TS(4,4) formed by cycle 8 in
the Tanner graph. The minimum codeword determining the code distance
($d_{min}$) of the LDPC code corresponds to $TS(a,0)$, where $a=d_{min}$.
Trapping sets form pseudocodewords [28, 29, 30, 31, 32].

Figure 2: Graphical representation of Trapping sets: a) TS(5,3), b) TS(4,4),
[27]

###  II-A Multi-edge Type (Protograph) QC-LDPC Codes

A QC-LDPC code and bipartite Tanner graph can be generalized to a multi-graph.
The multi-graph corresponding to the parity-check matrix:

$H_{2}=\left(\begin{array}[]{ccccc}{I^{1}+I^{2}+I^{7}}&{I^{9}}&{I^{23}}&{0}&{0}\\\
{I^{12}+I^{37}}&{I^{19}}&{0}&{I^{32}}&{I^{11}+I^{12}}\\\
{0}&{0}&{I^{33}}&{0}&{0}\end{array}\right)$ (5)

is shown in Figure 1 (right), where $I$ is the circulant permutation matrix
used in multi-edge (protograph) QC-LDPC codes. In the multi-graph
representation, the parity-check matrix is depicted as a collection of edges
connecting vertices (circulants of weight more than 1). The multi-graph
representation offers a more intuitive graphical perspective compared to
hypergraph representation, as it directly shows the connections between
variables and check nodes (constraints).

Multi-edge type low-density parity-check (LDPC) codes represent a class of
error-correcting codes that utilize multiple types of edges to connect
variable nodes and check nodes. Initially proposed by Richardson and Urbanke,
these codes were later extended by various researchers. The degree
distribution of multi-edge type LDPC codes is defined through a joint
distribution of variable and check node degrees, with variable node degrees
partitioned into multiple classes. This partitioning provides flexibility in
code design, enhancing performance. Protographs are employed to visualize the
structure of these codes, offering a graphical representation of the
connectivity pattern between variable and check nodes.

Protographs were introduced by David MacKay and Radford M. Neal in the context
of regular LDPC codes. Dariush Divsalar later proposed a protograph-based
approach for designing multi-edge type LDPC codes. Divsalar’s protograph
simplifies the representation of code structure, facilitating analysis and
design. The use of protographs has become a popular tool in the design and
analysis of LDPC codes. The two approaches, Richardson-Urbanke’s MET LDPC [33]
and MacKay-Neal-Divsalar Protograph LDPC [34], represent different ways of
describing the same type of LDPC code.

###  II-B Spatially-Coupled Convolutional QC-LDPC Codes

The spatially-coupled MET QC-LDPC code is constructed by coupling together a
series of $L$ disjoint block MET QC-LDPC codes into a single coupled chain
through an edge-spreading operation [35], as depicted in Fig. 3. A $(W,C,N,L)$
tail-biting spatial-coupled QC-LDPC code $C(H)$ of length $W$ and multiple $L$
can be defined by a parity-check base matrix:

$H=\left[\begin{array}[]{cccc}{A_{0,0}}&{A_{1,0}}&{...}&{A_{WL-1,0}}\\\
{A_{0,1}}&{A_{1,1}}&{}\hfil&{A_{WL-1,1}}\\\
{\vdots}&{\vdots}&{\ddots}&{\vdots}\\\
{A_{0,CL-1}}&{A_{1,CL-1}}&{...}&{A_{WL-1,CL-1}}\end{array}\right]$ (6)

where $0\leq i\leq CL-1$, $1\leq j\leq WL-1$, and CPM $A_{i,j}$ represents
either an $N\times N$ zero matrix $Z$ or the $N\times N$ circulant permutation
matrix $I\left(p_{i,j}\right)$ obtained by cyclically right-shifting the
$N\times N$ identity matrix $I\left(0\right)$ by $p_{i,j}$ positions. Denote
by CPM-column the i-th column of $A_{0,i},A_{1,i},\dots,A_{CL-1,i}$.

For a specific Tail-biting Spatial-couple MET QC-LDPC code, we define the
corresponding CPM-shifts matrix as the matrix of circulant shifts that defines
the QC-LDPC code:

$B=\left[\begin{array}[]{cccc}{b_{0,0}}&{b_{1,0}}&{...}&{b_{W-1,0}}\\\
{b_{0,1}}&{b_{1,1}}&{}\hfil&{b_{W-1,1}}\\\
{\vdots}&{\vdots}&{\ddots}&{\vdots}\\\
{b_{0,C-1}}&{b_{1,C-1}}&{...}&{b_{W-1,C-1}}\end{array}\right].$ (7)

Define vector D of shifts, Fig. 4:
$D^{T}=\left[{d_{0}},{d_{1}},{\cdots},{d_{C-1}}\right],$ with conditions
$d_{0}=0$, $d_{i}<\ d_{j}\ {\rm iff\ }i<j,\ \forall i\ d_{i}<W$. If
$\frac{W}{C}\ $ is an integer, vector $D$ can be defined as $d_{i}=\frac{i\
W}{C}{\rm\ }\ $.

For $0\leq i\leq CL-1$ and $0\leq j\leq WL-1$, CPMs $A_{i,j}$ in parity-check
matrix $H$ can be calculated: if $i^{\prime}\ {\rm mod\ }\ WL\ \leq i\
<\left(i^{\prime}+W\right)\ {\rm mod\ }\ WL$, then $A_{i,j}=I(b_{i\ {\rm mod\
W},j\ {\rm mod\ C\ }})$, else $A_{i,j}=Z$. Summary $C(H)$ is represented in
Fig. 5. In papers [36, 37], it was shown that QC-LDPC codes are equivalent to
terminated Spatially-Coupled and Serial Turbo convolutional codes. Trapping
sets pseudocodewords and related Bethe-permanent reach their upper bound [38,
39, 28], energy-landscape local minimum, as we shall show later.

Figure 3: Example of protograph and spatial coupled set of protograph Figure
4: One $W\times C$ CPM-block of circulants $A(b_{(i,j)})$ shifted by values of
vector D Figure 5: Tail-biting Spatial-Couple QC-LDPC code parity-check matrix
$C(H)$

###  II-C Boltzmann machine

A Boltzmann machine is a network comprising interconnected units (refer to
Figure 6). Let $N$ denote the number of units, each assuming a binary value (0
or 1). The random variable $X_{i}$ represents the value of the $i$-th unit
($i\in[1,N]$). We use the column vector $\bm{X}$ to denote the random values
of all $N$ units. The Boltzmann machine involves two types of parameters: bias
($b_{i}$ for the $i$-th unit) and weight ($w_{i,j}$ between units $i$ and $j$
for $(i,j)\in[1,N-1]\times[i+1,N]$). The bias for all units is represented by
the column vector $\mathbf{b}$, and the weight for all unit pairs is
represented by the matrix $\mathbf{W}$, where the $(i,j)$-th element of
$\mathbf{W}$ is $w_{i,j}$, with $w_{i,j}=0$ for $i\geq j$ and disconnected
unit pairs. The parameters are collectively denoted as:

$\theta\equiv(b_{1},\ldots,b_{N},w_{1,2},\ldots,w_{N-1,N}),$ (8)

which is also expressed as $\theta=(\mathbf{b},\mathbf{W})$.

Figure 6: A Boltzmann machine, [40].

The Boltzmann machine’s energy, denoted as $E_{\theta}(\mathbf{x})$, is
computed using the following expression:

$E_{\theta}(\mathbf{x})=-\sum_{i=1}^{N}b_{i}\,x_{i}-\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}w_{i,j}\,x_{i}\,x_{j}=\\\
-\mathbf{b}^{\top}\mathbf{x}-\mathbf{x}^{\top}\mathbf{W}\mathbf{x}.$ (9)

Based on this energy formulation, the Boltzmann machine establishes the
probability distribution over binary patterns through the expression:

$P_{\theta}(\mathbf{x})=\frac{\exp\left(-E_{\theta}(\mathbf{x})\right)}{\sum_{\mathbf{\tilde{x}}}\exp\left(-E_{\theta}(\mathbf{\tilde{x}})\right)},$
(10)

where the summation with respect to $\mathbf{\tilde{x}}$ spans all possible
$N$-bit binary values. The denominator, referred to as the partition function,
is denoted as:

$Z\equiv\sum_{\mathbf{\tilde{x}}}\exp\left(-E_{\theta}(\mathbf{x})\right).$
(11)

A Boltzmann machine can be utilized to model the probability distribution
$P_{\rm target}(\cdot)$ for target patterns. By optimally configuring the
values of $\theta$, we can approximate $P_{\rm target}(\cdot)$ with
$P_{\theta}(\cdot)$. In this context, hidden units, not directly associated
with target patterns, can be introduced (refer to Figure 7, b). Units that
directly correspond to target patterns are designated as visible, and these
visible units can be further classified into input and output categories (see
Figure 7, c). The Boltzmann machine is then capable of modeling the
conditional distribution of output patterns given an input pattern.

(a) visible only

(b) visible and hidden

(c) input and output

Figure 7: Boltzmann machines with hidden units, input, and output, [40].

##  III Marginalization Problem

Probabilistic models often face challenges in computing the partition
function, especially in normalizing to achieve a total probability of 1. To
overcome this, probabilistic models can be transformed into energy-based
models, eliminating the need for normalization and providing computational
advantages. However, designing suitable objective functions for energy-based
models remains a challenge, as discussed in [43, 44].

In various Boltzmann machine training approaches, there exists an equation
linking the probability ($p$) of the desired random variable distribution to
the bond energy function ($E$) of input parameters, normalized by the known
partition function ($Z$):

$p(x)=\frac{\exp(-E(x))}{Z}.$ (12)

Different methods for determining the normalization factor are proposed. In
Approach 1, the normalization factor relies on the determinant of the weight
matrix $W$:

$Z=\frac{1}{|\det(W)|}.$ (13)

Another approach, as in [46], also uses binding energy and determines the
normalizing factor with a specific relation:

$Z(x;\theta)=\sum_{v}\sum_{h}\frac{1}{\exp(E(v,h,x;\theta))}.$ (14)

For binding energy in a bipartite graph matrix, [46] proposes an analytical
relation, called the marginal distribution:

$E(v,x;\theta)=\sum_{j=1}^{|h|}\log[1+\exp(v^{T}W_{\bullet
j}^{vh}+W_{j\bullet}^{vh}x+b_{j}^{h})]+\\\ v^{T}W^{vx}x+v^{T}b^{v}.$ (15)

In the proposed solution, the energy determination is similar but numerical,
derived from the Bethe approximation of Gibbs energy, [29, 73, 79]:

$\exp(-E(\chi))\sim\text{perm}_{\text{bethe}}(\chi).$ (16)

Following the normalization condition, it is known that:

$p(x)=\frac{\exp(-E(x))}{Z},\\\ \quad 0\leq p(x)\leq 1\Rightarrow
Z=\sum\exp(-E(x)).$ (17)

These energy calculations are computationally feasible for sparse structural
bipartite graphs, such as quasi-cyclic codes, and have been successfully
employed in quasi-Newtonian methods for training deep neural networks via
belief propagation, as demonstrated in [45, 46]. In particular, Bethe-energy
learning could be accomplished using Bethe-permanent message-passing
estimation, [30, 31, 32].

##  IV Topology-Aware Equilibrium

######  Definition 1.

A particle is considered to be in equilibrium (ground state) if no force acts
on it, or the sum of the forces acting on the particle is zero:

$\sum_{i=1}^{n}F_{i}^{q_{j}}=0.$ (18)

As geodesic lines on a torus are circles, we establish preliminary statements
regarding circles, representing them as geodesic lines.

######  Definition 2.

Let $d_{l}$ denote the left distance to the particle when walking
counterclockwise from the left, and $d_{r}$ denote the right distance when
walking counterclockwise from the right.

######  Lemma 1.

Two particles with the same charge on a circle are in equilibrium if the left
distance between the particles equals the right distance.

###### Proof.

According to the equilibrium criteria in Equation (18), a particle is in
equilibrium when the net force acting on it is zero. Therefore,

$F^{q_{1}}=F^{q_{2}}=\frac{kq_{1}q_{2}}{r_{r}^{2}}-\frac{kq_{1}q_{2}}{r_{l}^{2}}=0\Leftrightarrow
r_{r}=r_{l}.$ (19)

∎

######  Lemma 2.

$n$ identical particles on a circle are in a state of equilibrium if all
distances between neighboring particles are equal.

###### Proof.

By analogy with Lemma 1, we express the conditions for particle equilibrium:

$\displaystyle F^{q_{1}}$
$\displaystyle=\frac{kq_{1}q_{2}}{d_{r1}^{2}}-\frac{kq_{1}q_{n}}{d_{l1}^{2}}=0\Leftrightarrow
d_{r1}=d_{l1},$ $\displaystyle F^{q_{2}}$
$\displaystyle=\frac{kq_{2}q_{3}}{d_{r2}^{2}}-\frac{kq_{2}q_{1}}{d_{l2}^{2}}=0\Leftrightarrow
d_{r2}=d_{l2},\,d_{r1}\equiv r_{l2},$ $\displaystyle\dots$ $\displaystyle
F^{q_{n-1}}$
$\displaystyle=\frac{kq_{n-1}q_{n}}{d_{rn-1}^{2}}-\frac{kq_{n-1}q_{n-2}}{d_{l\,n-1}^{2}}=0\Leftrightarrow
d_{rn-1}=d_{l\,n-1},$ $\displaystyle d_{rn-2}\equiv d_{ln-1},$ $\displaystyle
F^{q_{n}}$
$\displaystyle=\frac{kq_{n}q_{1}}{d_{rn}^{2}}-\frac{kq_{n}q_{n-1}}{d_{l\,n}^{2}}=0\Leftrightarrow
d_{rn}=d_{l\,n},\,d_{rn-1}\equiv d_{ln}.$ (20)

These conditions are satisfied only when all distances are equal. ∎

The arrangement of particles on the circle is clearly a uniform grid. With
these auxiliary statements, we are now prepared to prove the main theorem.

Figure 8: Quadrangular multicell of four cells on the surface of a torus.

######  Theorem 1.

A collection of identical particles situated on the surface of a torus
achieves equilibrium if and only if it configures into a uniform grid.

###### Proof.

Necessity. Initially, we establish that a set of particles in equilibrium on
the torus surface adopts a uniform grid configuration. As per Lemma 2, the
orthogonal nature of directions $x_{1}$ and $x_{2}$ on the torus ensures that
a sufficiently large group of particles forms a uniform grid in each
direction. Equilibrium conditions (IV) are met for both orthogonal directions.
Thus, the assertion stands proven.

Sufficiency. Next, we demonstrate that particles positioned at the nodes of a
uniform grid are, indeed, in a state of equilibrium. We examine a quadrangular
multicell (Fig. 8) on the torus surface, articulate the sum of forces acting
on the nodal element, and confirm that it satisfies the equilibrium
conditions.

$\displaystyle
X1:\left|F_{X1}^{q0}=kq_{0}\left(\frac{q_{1}}{r_{x10}^{2}}+\frac{q_{8}}{r_{x80}^{2}}+\frac{q_{7}}{r_{x70}^{2}}-\frac{q_{3}}{r_{x30}^{2}}-\frac{q_{4}}{r_{x40}^{2}}-\frac{q_{5}}{r_{x50}^{2}}\right)\right.=0,$
(21) $\displaystyle
X2:\left|F_{X2}^{q0}=kq_{0}\left(\frac{q_{1}}{r_{y10}^{2}}+\frac{q_{2}}{r_{y20}^{2}}+\frac{q_{3}}{r_{y30}^{2}}-\frac{q_{5}}{r_{y50}^{2}}-\frac{q_{6}}{r_{y60}^{2}}-\frac{q_{7}}{r_{y70}^{2}}\right)\right.=0.$

.

Examine a quadrangular mesh. The force equations in axis projections are
defined by equations 21. It is clear that equilibrium conditions are fulfilled
when either all projections of distances along one axis are identical or there
is pairwise equality of projections with one having a positive sign and the
other a negative sign. This results in solutions corresponding to either an
orthogonal quadrilateral uniform mesh or a skewed quadrilateral uniform mesh,
respectively. ∎

Similarly, the theorem can be demonstrated for a triangular uniform mesh. A
more robust assertion states that these relationships are valid for a regular
grid, as evident from Theorem 1 and non-constructive proof from [41].

Nevertheless, under the considered hypothesis, an alternative method for
achieving equilibrium with unequal charges on an irregular grid is suggested.
This approach entails a linear expansion of the charge system’s dimension,
substituting charges with multiple circulants and distances with circulant
shifts. This results in a transformation of the charge system into a space
where the charges constitute a uniform grid.

######  Definition 3.

According to the Ising model, each cell of the lattice corresponds to a spin
value equal to +1 or -1. Each arrangement of particles and spins is associated
with energy:

$E\left(\sigma\right)=\sum_{i,j}C_{ij}J_{ij}\sigma_{i}\sigma_{j},$ (22)

where $C_{ij}$ is the connectivity matrix of the spin element, and $J_{ij}$ is
the interaction energy between spins $\sigma_{i},\sigma_{j}$.

######  Definition 4.

Energy potential is defined by considering the interaction of particles, all
having charges of the same sign (for convenience of presentation), in the one-
dimensional case (Fig. 9), using the Ising model.

Figure 9: Three particles on one line.

Charges 1 and 3, having the same sign, create an electrostatic field. The
field energy created by charge $i$ at point $j$ is directly proportional to
the magnitude of the charge and inversely proportional to the square of the
distance between the points:

$J_{ij}=k\frac{q_{i}}{r_{ij}^{2}}$ (23)

Quantizing charge and space, we define the values $q_{1}$, $q_{3}$,
$r_{12}^{2}=R_{1}$, and $r_{32}^{2}=R_{3}$, where the electric field created
by charge 1 at point 2 balances the field created at point 2 by charge 3. To
determine possible states of the system where the field magnitude created by
neighboring charges at the desired point reaches a minimum, we introduce the
concept of a circulant $I_{n}$—a unit matrix of size $e\times e$, subjected to
a cyclic shift of magnitude $n$.

######  Statement 1.

Since the distances between particles are rational numbers, represented as
proportional integers, we introduce the size of the circulant $e$, considered
as the maximum possible analogue of the distance between charges 1 and 3:

$e=\sum(R_{1},R_{3})$ (24)

######  Statement 2.

The magnitudes of the charges are multiples of the elementary charge,
represented as an integer number of elementary charges. The integer ratios of
possible charges to the square of the distance, not exceeding the size of the
circulant, will be equal to the cyclic shift of the circulant:

$\frac{q_{i}}{R_{i}}=n,\,0\leq n<e.$ (25)

Thus, the minimum field energy at the desired point forms a set of pairs of
circulants of the form:

$\begin{array}[]{l}{\frac{q_{1}}{R_{1}}=n_{1};\frac{q_{3}}{R_{3}}=n_{3}},\\\
{n_{1}+n_{3}=e-1}.\end{array}$

The matrix describing the interaction of electric fields 1 and 3 at point 2,
where the fields at point 2 are balanced, looks like this:

$H=\left[I_{n_{1}}|0|I_{n_{3}}\right].$ (26)

In the case of three particles, the zero circulant can be reduced.

Numerical Example 1. The idea illustrating Statement is demonstrated for
$e=\sum(R_{1},R_{3})=5$ in Figure 10:

Figure 10: Coulomb’s Law rules Quasi-Cyclic [42]

In this example, we have a set of possible circulants that minimizes the field
energy at point 2. The circulants are of the following form:

$H=\left(\begin{array}[]{l}{\left[I_{0}|I_{5}\right]}\\\
{\left[I_{1}|I_{4}\right]}\\\ {\left[I_{2}|I_{3}\right]}\\\
{\left[I_{3}|I_{2}\right]}\\\ {\left[I_{4}|I_{1}\right]}\\\
{\left[I_{5}|I_{0}\right]}\end{array}\right).$ (27)

However, a circulant with zero shift (unit matrix) represents an equilibrium
state of the system, corresponding to an infinite number of charge options.
Let’s choose from this set the states that can accurately describe the final
system:

$H=\begin{array}[]{l}{\left(\begin{array}[]{l}{\left[I_{0}|I_{5}\right]}\\\
{\left[I_{1}|I_{4}\right]}\\\ {\left[I_{2}|I_{3}\right]}\\\
{\left[I_{3}|I_{2}\right]}\\\ {\left[I_{4}|I_{1}\right]}\\\
{\left[I_{5}|I_{0}\right]}\end{array}\right)\Rightarrow\left(\begin{array}[]{l}{\left[I_{1}|I_{4}\right]}\\\
{\left[I_{2}|I_{3}\right]}\\\ {\left[I_{3}|I_{2}\right]}\\\
{\left[I_{4}|I_{1}\right]}\end{array}\right)}\end{array}.$ (28)

Figure 11: Projection of a two-dimensional problem onto a one-dimensional case

###  IV-A Two-dimensional Ising Equilibrium Problem

Let’s examine the scenario of four particles positioned on a straight line,
representing the projection of the two-dimensional Ising problem. This is
depicted in Figure 11. Assume the squares of distances between particles are
given by:

$\begin{array}[]{l}{r_{12}^{2}=a},\\\ {r_{23}^{2}=b},\\\
{r_{34}^{2}=a+b}.\end{array}$ (29)

Here, $a$ and $b$ denote the squares of distances between the particles.

Due to the interaction screening (localization) of particles, particle 1 does
not influence particle 3, and particle 4 does not affect particle 2.

Now, let’s explore solutions for two scenarios: the equilibrium/ground state
of particles 1-3 and the equilibrium/ground state of particles 2-4.

For the 1-3 system, the matrix describing the interaction between charges can
be expressed as:

$H=\left(\begin{array}[]{cccc}{I_{a}^{a+b}}&{0}&{I_{b}^{a+b}}&{0}\\\
{0}&{I_{b}^{a+2b}}&{0}&{I_{a+b}^{a+2b}}\end{array}\right).$ (30)

This matrix comprises four circulants. The initial circulant signifies the
interaction between particles 1 and 2, the second circulant characterizes the
interaction between particles 2 and 3, the third circulant delineates the
interaction between particles 3 and 4, and the fourth circulant illustrates
the interaction between particles 1 and 4. In these expressions, $I_{p}^{o}$
represents a circulant of size $o$ with a cyclic shift of $p$. To standardize
the circulants to the same size, we can utilize the following relations:

$\begin{array}[]{l}{c=\text{LCM}\left(a+b,a+2b\right)},\\\
{\overline{a}=\frac{a\cdot c}{a+b};\overline{b_{1}}=\frac{b\cdot c}{a+b}},\\\
{\overline{b_{2}}=\frac{b\cdot
c}{a+2b};\overline{a+b}=\frac{\left(a+b\right)\cdot c}{a+2b}}.\end{array}$
(31)

In these equations, $\text{LCM}(x,y)$ represents the least common multiple of
$x$ and $y$. The bars over $a$, $b_{1}$, $b_{2}$, and $a+b$ denote the
adjusted values.

By employing these correlations, we can derive a system of interaction among
four particles in quasi-cyclic notation:

$H=\left(\begin{array}[]{cccc}{I_{\overline{a}}^{c}}&{I_{\overline{b_{1}}}^{c}}&{I_{\overline{b_{2}}}^{c}}&{I_{\overline{a+b}}^{c}}\end{array}\right).$
(32)

This matrix comprises four circulants, all standardized to the same size $c$,
aligning with the minimum field energy at points 1-3 and 2-4.

Numerical Example 2.

Consider an example with the following distances:

$\begin{array}[]{l}{r_{12}^{2}=2},\\\ {r_{23}^{2}=3},\\\
{r_{34}^{2}=5}.\end{array}$ (33)

Describing the impact of particles 1 and 3 on particle 2, as well as particles
2 and 4 on particle 3, we obtain the following matrix:

$H=\left(\begin{array}[]{cccc}{I_{2}^{5}}&{0}&{I_{3}^{5}}&{0}\\\
{0}&{I_{3}^{8}}&{0}&{I_{5}^{8}}\end{array}\right).$ (34)

This matrix comprises four circulants with sizes 5 and 8. The first circulant
represents the interaction between particles 1 and 2, the second circulant
represents the interaction between particles 2 and 3, the third circulant
represents the interaction between particles 3 and 4, and the fourth circulant
represents the interaction between particles 1 and 4.

Note that the circulants have different sizes due to the need for adjustment
to ensure they all have the same size. In this case, the adjusted sizes are 5
and 8. This matrix contains information about possible charge configurations
resulting in the minimum field energy at points 2-3.

To standardize the description to circulants of the same size, we can use the
following relationships:

$\begin{array}[]{l}{c=\text{LCM}(5,8)=40},\\\ {\overline{2}=\frac{2\cdot
c}{2+3}=16;\overline{3}=\frac{3\cdot c}{2+3}=24},\\\
{\overline{5}=\frac{5\cdot c}{3+5}=15;\overline{2+3}=\frac{(2+3)\cdot
c}{3+5}=25}.\end{array}$ (35)

Using these relationships, we can obtain a parity-check matrix with a
circulant size of 40 and cyclic shift values of 16, 15, 24, and 25:

$H=\left(\begin{array}[]{cccc}{I_{16}^{40}}&{I_{15}^{40}}&{I_{24}^{40}}&{I_{25}^{40}}\end{array}\right).$
(36)

In summary, we can apply a similar approach to the one-dimensional case to
find possible charge configurations resulting in the equilibrium state of the
system with the lowest field energy magnitude at a desired point. We can
construct a parity-check matrix describing charge interactions in the two-
dimensional case using distances between particles and insights gained from
the one-dimensional case.

Figure 12: (left) Interaction of 4 particles in 2d space. (right) Interaction
of 6 particles in 2d space

###  IV-B Interaction of Four Particles in Two-Dimensional Space Without
Projecting Onto One Axis

Now, let’s explore the interaction of four particles in two-dimensional space
without projecting them onto one axis (illustrated in Fig. 12, left).

For four particles in two-dimensional space, we can create a parity-check
matrix with two circulant rows: one for interaction along the X-axis and one
for interaction along the Y-axis. The order of each circulant is determined by
the sum of the projections of distances between particles onto the respective
axis. We define these values as:

$\begin{array}[]{l}{L_{x}=\sum_{i=1}^{4}L_{x_{i}x^{*}};L_{x_{i}x^{*}}=L_{x_{i}}},\\\
{L_{y}=\sum_{i=1}^{4}L_{y_{i}y^{*}};L_{y_{i}y^{*}}=L_{y_{i}}}.\end{array}$
(37)

We then define an interaction matrix as:

$H^{j}=\left(\begin{array}[]{cccc}{I_{L_{x1}^{j}}^{L_{x}^{j}}}&{I_{L_{x2}^{j}}^{L_{x}^{j}}}&{I_{L_{x3}^{j}}^{L_{x}^{j}}}&{I_{L_{x4}^{j}}^{L_{x}^{j}}}\\\
{I_{L_{y1}^{j}}^{L_{y}^{j}}}&{I_{L_{y2}^{j}}^{L_{y}^{j}}}&{I_{L_{y3}^{j}}^{L_{y}^{j}}}&{I_{L_{y4}^{j}}^{L_{y}^{j}}}\end{array}\right).$
(38)

However, we may need to adjust the sizes of the circulants in the interaction
matrix to ensure they are all the same. This can be achieved by finding the
least common multiple of $L_{x}^{j}$ and $L_{y}^{j}$ and scaling the sizes of
the circulants accordingly:

$\begin{array}[]{l}{L^{j}=LCM\left(L_{x}^{j},L_{y}^{j}\right)},\\\
{\overline{L_{xi}^{j}}=L_{xi}^{j}\cdot\frac{L^{j}}{L_{x}^{j}};\overline{L_{yi}^{j}}=L_{yi}^{j}\cdot\frac{L^{j}}{L_{y}^{j}}}.\end{array}$
(39)

The final form of the interaction matrix will be:

$H^{j}=\left(\begin{array}[]{cccc}{I_{\overline{L_{x1}^{j}}}^{L^{j}}}&{I_{\overline{L_{x2}^{j}}}^{L^{j}}}&{I_{\overline{L_{x3}^{j}}}^{L^{j}}}&{I_{\overline{L_{x4}^{j}}}^{L^{j}}}\\\
{I_{\overline{L_{y1}^{j}}}^{L^{j}}}&{I_{\overline{L_{y2}^{j}}}^{L^{j}}}&{I_{\overline{L_{y3}^{j}}}^{L^{j}}}&{I_{\overline{L_{y4}^{j}}}^{L^{j}}}\end{array}\right).$
(40)

This matrix describes the interaction between charges in two-dimensional space
and contains information about possible charge configurations resulting in the
lowest magnitude of the field energy at a desired point.

###  IV-C Interaction of Six Particles in Two-Dimensional Space Without
Projecting Onto One Axis

Let’s delve into the interaction scenario involving six particles in two-
dimensional space, where a pair of particles participates in both $j$ and
$j+1$ cells (as depicted in Fig. 12, b). The parity-check matrices for these
two adjacent cells can be defined as follows:

$H^{j}=\begin{pmatrix}I_{\overline{L_{x1}^{j}}}^{L^{j}}&I_{\overline{L_{x2}^{j}}}^{L^{j}}&I_{\overline{L_{x3}^{j}}}^{L^{j}}&I_{\overline{L_{x4}^{j}}}^{L^{j}}\\\
I_{\overline{L_{y1}^{j}}}^{L^{j}}&I_{\overline{L_{y2}^{j}}}^{L^{j}}&I_{\overline{L_{y3}^{j}}}^{L^{j}}&I_{\overline{L_{y4}^{j}}}^{L^{j}}\end{pmatrix},\\\
H^{j+1}=\begin{pmatrix}I_{\overline{L_{x2}^{j}}}^{L^{j}}&I_{\overline{L_{x2}^{j}}}^{L^{j+1}}&I_{\overline{L_{x4}^{j}}}^{L^{j}}&I_{\overline{L_{x4}^{j}}}^{L^{j+1}}\\\
I_{\overline{L_{y2}^{j}}}^{L^{j}}&I_{\overline{L_{y2}^{j+1}}}^{L^{j+1}}&I_{\overline{L_{y4}^{j}}}^{L^{j}}&I_{\overline{L_{y4}^{j+1}}}^{L^{j+1}}\end{pmatrix}.$
(41)

The columns of each matrix represent the contributions of particles when
listed with row priority. This approach can be extended to form quadrangular
or triangular cells in four directions using sets of particles. The spatial
configuration of particle locations is toroidal, enabling circular bypass of
the configuration both "from bottom to top" and "from left to right" from a
physical perspective on the boundary conditions.

######  Statement 3.

For a system of interacting particles in a two-dimensional space described by
the Ising model, with charges having the same sign and creating an electric
(magnetic) field, and distances between particles given by the sum of
projections on the X and Y axes, the interaction matrix will have two
circulant rows: interaction along the X axis and along the Y axis. The matrix
describing the interaction of particles can be defined as a pair of circulant
matrices that correspond to a set of pairs of circulants. To bring the orders
of the circulants to a common value, the least common multiple of the
distances is used to adjust the sizes.

Figure 13: Interaction of three particles on a plane. Example 2 from article
[36] Figure 14: Interaction of 4 particles in space. Example 3 from [36]
Figure 15: Interaction of five particles in space  Figure 16: Interaction of
five particles in a plane

Numerical Examples Using QC-LDPC Codes

In the paper [36], quasi-cyclic block and convolution LDPC codes based on
circulant matrices are explored. Their geometric representations can be
constructed using the proposed approach. The elementary codes from this paper,
based on 2x3, 3x4, and 3x5 circulants, can be depicted as a set of particles
in Euclidean space.

For instance, consider the code with the parity-check matrix:

$H=\begin{pmatrix}I_{1}^{7}&I_{2}^{7}&I_{4}^{7}\\\
I_{6}^{7}&I_{5}^{7}&I_{3}^{7}\end{pmatrix},$ (42)

representing the interaction of three particles on a plane (as shown in Fig.
13). Each row corresponds to a particle, and each column corresponds to a
check node. The identity matrices $I_{i}^{7}$ represent connections between
particles and check nodes, arranged in a circulant pattern. The position of
the center of mass of a triangular plate with particles at the vertices aligns
closely with the origin. In two-dimensional toric space, an analogue for such
a configuration is an isosceles right triangle with vertices equidistant from
the origin.

Another QC code with the parity-check matrix ([36]), Ex. 3):

$H=\left(\begin{array}[]{cccc}{I_{1}^{26}}&{I_{5}^{26}}&{I_{25}^{26}}&{I_{21}^{26}}\\\
{\begin{array}[]{c}{I_{9}^{26}}\\\
{I_{3}^{26}}\end{array}}&{\begin{array}[]{c}{I_{19}^{26}}\\\
{I_{15}^{26}}\end{array}}&{\begin{array}[]{c}{I_{17}^{26}}\\\
{I_{23}^{26}}\end{array}}&{\begin{array}[]{c}{I_{11}^{26}}\\\
{I_{11}^{26}}\end{array}}\end{array}\right),$ (43)

can be represented as a system of four particles in space (in Fig. 14).

Four particles form a tetrahedron with the center of mass coinciding with the
origin. In toric three-dimensional space, this configuration corresponds to
four points of the cube, forming a trihedral angle, equidistant from the
origin, with a parity-check matrix:

$H=\left(\begin{array}[]{cccc}{I_{1}^{31}}&{I_{2}^{31}}&{I_{4}^{31}}&{\begin{array}[]{cc}{I_{8}^{31}}&{I_{16}^{31}}\end{array}}\\\
{\begin{array}[]{c}{I_{5}^{31}}\\\
{I_{25}^{31}}\end{array}}&{\begin{array}[]{c}{I_{10}^{31}}\\\
{I_{19}^{31}}\end{array}}&{\begin{array}[]{c}{I_{20}^{31}}\\\
{I_{7}^{31}}\end{array}}&{\begin{array}[]{c}{\begin{array}[]{cc}{I_{9}^{31}}&{I_{18}^{31}}\end{array}}\\\
{\begin{array}[]{cc}{I_{14}^{31}}&{I_{28}^{31}}\end{array}}\end{array}}\end{array}\right),$
(44)

describing the interaction of five particles in space. The particle
configuration defining the code is shown in Fig. 15.

Consider the following parity-check matrix

$H=\left(\begin{array}[]{ccccc}{I_{10}^{11}}&{I_{9}^{11}}&{I_{8}^{11}}&{I_{7}^{11}}&{I_{6}^{11}}\\\
{I_{1}^{11}}&{I_{2}^{11}}&{I_{3}^{11}}&{I_{4}^{11}}&{I_{5}^{11}}\end{array}\right),$
(45)

describing the interaction of five particles in a plane. A polygon with
charges at the vertices is shown in (in Fig. 16).

###  IV-D Circular Hyperboloid

Now, let’s explore a system of particles arranged in two belts on a
cylindrical surface (Figure 4a). The connections between corresponding
particles on neighboring belts align with the generatrices of the cylinder:

$\left\\{\begin{array}[]{l}{\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}=1}\\\
{-\infty<z<\infty}\end{array},\right.$ (46)

forming a regular grid.

For example, if a disturbance is introduced into the upper belt (Figure 4b),
causing a displacement of particles at a certain angle $\alpha$, the particles
will maintain their regular grid formation. The system of connections between
corresponding particles remains linear; however, the connections no longer
form a cylinder but rather a single-sheet hyperboloid, corresponding to the
equation:

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}-\frac{z^{2}}{f(\alpha)}=1,$ (47)

where $f(\alpha)$ is a function dependent on the magnitude of the energy
disturbance introduced into the system.

Figure 17: Figure 4. Charges on a) cylindrical and b) hyperbolic surfaces.
Figure 18: Fermions are held in shifted lattices by a certain angle (left),
regular triangular shape made of wolfram disulfide and wolfram diselenide from
[48] Figure 19:  The 7 dipoles hexagonal prism  Figure 20:  Finite Element
Analysis local coordinate system grid from [49] Figure 21:  Grid
discretization of interaction

###  IV-E Example of Circular Hyperboloid: MET QC-LDPC Codes

The study in [48] investigates an experiment involving two lattices with
regular triangular shapes made of wolfram disulfide and wolfram diselenide,
each containing negatively charged electrons and positively charged electron
holes in the nodes. Fermions are held in lattice nodes through strong
interaction. When the lattices (Fig. 18) are shifted by a certain angle
$\alpha$ relative to the stationary center of rotation, and photon energy is
pumped, fermions enter into electromagnetic interaction, forming dipoles-
excitons. Due to their dipole nature, excitons act as dielectric insulators,
maintaining a strict lattice structure through strong interaction.

Consider a hexagonal prism containing 7 dipoles, one of which serves as the
center of rotation for the grid (Fig. 19). By introducing a Cartesian
coordinate system centered at the grid’s rotation center, we can derive a
matrix describing the effect of grid displacement on particle interaction at a
fixed center. Let $R$ be the step of the triangular grid, and $L_{ij}$
represent the distance between two nodes. Then,

$\begin{array}[]{l}{\Delta R_{jx}=L_{1jx}-L_{1jx}\cos\alpha},\\\ {\Delta
R_{jy}=L_{1jy}-L_{1jy}\sin\alpha}.\end{array}$ (48)

Introduce quantization:

${dR=\left\lfloor\frac{R}{q}\right\rfloor},{qdR=\overline{R}},{\overline{\Delta
R}=\left\lfloor\frac{\Delta R}{dR}\right\rfloor}.$ (49)

Then, we can describe the interaction matrix $H$ as follows:

$\left(\begin{array}[]{ccccccc}{C_{0}^{q}}&{C_{\overline{\Delta
R_{2x}}}^{q}}&{C_{\overline{\Delta R_{3x}}}^{q}}&{C_{\overline{\Delta
R_{4x}}}^{q}}&{C_{\overline{\Delta R_{5x}}}^{q}}&{C_{\overline{\Delta
R_{6x}}}^{q}}&{C_{\overline{\Delta R_{7x}}}^{q}}\\\
{C_{0}^{q}}&{C_{\overline{\Delta R_{3y}}}^{q}}&{C_{\overline{\Delta
R_{3y}}}^{q}}&{C_{\overline{\Delta R_{4y}}}^{q}}&{C_{\overline{\Delta
R_{5y}}}^{q}}&{C_{\overline{\Delta R_{6y}}}^{q}}&{C_{\overline{\Delta
R_{7y}}}^{q}}\end{array}\right).$ (50)

At a sufficient distance from the center of rotation, the bevel of the dipoles
may exceed the value at which the connection between the i-th electrons and
the holes of the corresponding layers is preserved. In this case, the
influence passes to neighboring nodes, changing the direction of the circulant
shift. Additionally, three main directions can be distinguished on such a
grid, and a local non-orthogonal system of three coordinates L1-3 can be
introduced. Coordinates vary from 0 to 1 (exceeding 1 when the grid is
beveled), [49].

The local coordinate system is connected by perpendiculars, projecting from a
point to a side. Each coordinate expresses the ratio of the area of the
triangle constructed on the corresponding side and point to the area of the
original triangle (Fig. 20). The coordinate of point B in the L-system will
have the value:

$\begin{array}[]{l}{B_{L}=(L_{1},L_{2},L_{3})=\left(\frac{S_{\Delta
Bjk}}{S_{\Delta ijk}},\frac{S_{\Delta Bki}}{S_{\Delta ijk}},\frac{S_{\Delta
Bij}}{S_{\Delta ijk}}\right)}\\\
{=\left(\frac{h_{B1}}{h_{1}},\frac{h_{B2}}{h_{2}},\frac{h_{B3}}{h_{3}}\right)}.\end{array}$
(51)

Each displaced lattice node participates in the formation of six grid cells
and can be represented, according to the proposed hypothesis, by six sets of
columns of three circulants describing the shift of the node relative to the
local coordinates of each of the 6 cells. By the properties of additivity, the
columns describing the interaction in each cell will be the sum of three
columns describing the shifts of each of the cell nodes relative to the local
coordinate system.

##  V Implementing the Proposed Approach in Finite Geometry QC-LDPC Codes

Examine the QC-LDPC code introduced in [50], featuring mixed circulant sizes
with various automorphism structures:

$H=\left(\begin{array}[]{c}{\begin{array}[]{ccccc}{I_{17}}&{I_{17}}&{I_{17}}&{I_{17}}&{I_{17}}\end{array}}\\\
{H_{1}}\\\ {H_{2}}\\\ {H_{3}}\\\ {H_{4}}\end{array}\right)=\\\
\left(\begin{array}[]{c}{\begin{array}[]{ccccc}{I_{17}}&{I_{17}}&{I_{17}}&{I_{17}}&{I_{17}}\end{array}}\\\
{C_{85}^{0}+C_{85}^{24}+C_{85}^{40}+C_{85}^{71}+C_{85}^{84}}\\\
{C_{85}^{1}+C_{85}^{49}+C_{85}^{58}+C_{85}^{81}+C_{85}^{84}}\\\
{C_{85}^{3}+C_{85}^{14}+C_{85}^{32}+C_{85}^{78}+C_{85}^{84}}\\\
{C_{85}^{16}+C_{85}^{33}+C_{85}^{50}+C_{85}^{67}+C_{85}^{84}}\end{array}\right).$
(52)

Here, $H_{i}$ represents circulant size 85 with weight 5, determined by the
number of circulant terms with weight 1. The suitability of this parity-check
matrix as a representation of Nitrogen is readily confirmed using the
following logic. For ease of geometric analogies, let’s focus on the first 3
rows of the matrix $H$:

$H=\left(\begin{array}[]{c}{\begin{array}[]{ccccc}{I_{17}}&{I_{17}}&{I_{17}}&{I_{17}}&{I_{17}}\end{array}}\\\
{H_{1}}\\\ {H_{2}}\end{array}\right)=\\\
\left(\begin{array}[]{c}{\begin{array}[]{ccccc}{I_{17}}&{I_{17}}&{I_{17}}&{I_{17}}&{I_{17}}\end{array}}\\\
{C_{85}^{0}+C_{85}^{24}+C_{85}^{40}+C_{85}^{71}+C_{85}^{84}}\\\
{C_{85}^{1}+C_{85}^{49}+C_{85}^{58}+C_{85}^{81}+C_{85}^{84}}\end{array}\right).$
(53)

In the context of the considered model, this matrix can be obtained by
collapsing the matrix along one of the coordinates:

$\left(\begin{array}[]{c}{\begin{array}[]{ccccc}{C_{85}^{k}}&{C_{85}^{k}}&{C_{85}^{k}}&{C_{85}^{k}}&{C_{85}^{k}}\end{array}}\\\
{\begin{array}[]{ccccc}{C_{85}^{0}}&{C_{85}^{24}}&{C_{85}^{40}}&{C_{85}^{71}}&{C_{85}^{84}}\end{array}}\\\
{\begin{array}[]{ccccc}{C_{85}^{1}}&{C_{85}^{49}}&{C_{85}^{58}}&{C_{85}^{81}}&{C_{85}^{84}}\end{array}}\end{array}\right)\to\\\
\left(\begin{array}[]{c}{\begin{array}[]{ccccc}{I_{17}}&{I_{17}}&{I_{17}}&{I_{17}}&{I_{17}}\end{array}}\\\
{C_{85}^{0}+C_{85}^{24}+C_{85}^{40}+C_{85}^{71}+C_{85}^{84}}\\\
{C_{85}^{1}+C_{85}^{49}+C_{85}^{58}+C_{85}^{81}+C_{85}^{84}}\end{array}\right).$
(54)

Consider a scenario where five indistinguishable particles are evenly
positioned from the center, described by spherical coordinates
$\left(r,\varphi,\theta\right)$ within a matrix. Now, let’s explore the
electron clouds of a carbon atom within the proposed model. Carbon carries a
charge of 6, distributed as 2 electrons in the first energy level and 4
electrons in the second. A code matrix, presented in a general form, is
constructed as follows:

$\left\\{\begin{array}[]{c}{r}\\\ {\varphi}\\\
{\theta}\end{array}\right\\}\left(\begin{array}[]{cccccc}{C_{k}^{r_{1}}}&{C_{k}^{r_{1}}}&{C_{k}^{r_{2}}}&{C_{k}^{r_{2}}}&{C_{k}^{r_{2}}}&{C_{k}^{r_{2}}}\\\
{C_{k}^{\varphi_{1}}}&{C_{k}^{\varphi_{2}}}&{C_{k}^{\varphi_{3}}}&{C_{k}^{\varphi_{4}}}&{C_{k}^{\varphi_{5}}}&{C_{k}^{\varphi_{6}}}\\\
{C_{k}^{\theta_{1}}}&{C_{k}^{\theta_{2}}}&{C_{k}^{\theta_{3}}}&{C_{k}^{\theta_{4}}}&{C_{k}^{\theta_{5}}}&{C_{k}^{\theta_{6}}}\end{array}\right).$
(55)

To determine the numerical characteristics of the circulants, specific
conditions are defined. Specifically, for collapsing the matrix along the
radii, it is necessary to observe the multiplicity of the order of the
circulant to the radii and their sum:

$\begin{array}[]{l}{k\,\mathrm{mod}\,{r_{1}}=0},\\\
{k\,\mathrm{mod}\,{r_{2}=0}},\\\
{k\,\mathrm{mod}\,{\left(r_{1}+r_{2}\right)}=0}.\end{array}$ (56)

In a more general case, expressed as:

$\begin{array}[]{l}{k\,\mathrm{mod}\,{r_{i}=0}},\\\
{k\,\mathrm{mod}\,{\sum_{i=1}^{N}r_{i}=0}}.\end{array}$ (57)

Or, in the most general case (e.g., when the radii are prime numbers),
${a,r_{i}\in{\mathbb{N}}}$:

$\begin{array}[]{l}{k\,\mathrm{mod}\,{\left(a\left(\sum_{i=1}^{N}r_{i}\right)\prod_{i=1}^{n}r_{i}\right)}}.\end{array}$
(58)

The presented mathematical framework introduces a cycle-based gauge, denoted
as TS(a,0) or the Schrödinger-Heisenberg-Bohr-Fossorier Electron Cloud Gauge
(SHBF Cycle Gauge), represented by the equation [51]:

$\left(\sum_{i=1}^{N-1}\Delta_{ji,ji+1}\left(l_{i}\right)\right)\,\mathrm{mod}\,{k}=0.$
(59)

This equation ensures that the sum of shifts of the circulant along the row
(or column) is a multiple of the dimension of the circulant. A specific matrix
meeting this criterion is given by:

$\left\\{\begin{array}[]{c}{r}\\\ {\varphi}\\\
{\theta}\end{array}\right\\}\left(\begin{array}[]{cccccc}{C_{48}^{24}}&{C_{48}^{24}}&{C_{48}^{36}}&{C_{48}^{36}}&{C_{48}^{36}}&{C_{48}^{36}}\\\
{C_{48}^{1}}&{C_{48}^{7}}&{C_{48}^{13}}&{C_{48}^{19}}&{C_{48}^{25}}&{C_{48}^{31}}\\\
{C_{48}^{23}}&{C_{48}^{17}}&{C_{48}^{47}}&{C_{48}^{41}}&{C_{48}^{35}}&{C_{48}^{29}}\end{array}\right).$
(60)

It is crucial that the condition along the row is satisfied when the shift of
all circulants of the row changes simultaneously by a multiple of:

$S=k/N.$ (61)

Following this, the matrix is collapsed along the radii, resulting in a
structure denoted as $H$:

$H=\left(\begin{array}[]{c}{\begin{array}[]{cccccc}{C_{8}^{0}}&{C_{8}^{0}}&{C_{8}^{2}}&{C_{8}^{2}}&{C_{8}^{2}}&{C_{8}^{2}}\end{array}}\\\
{C_{48}^{1}+C_{48}^{7}+C_{48}^{13}+C_{48}^{19}+C_{48}^{25}+C_{48}^{31}}\\\
{C_{48}^{23}+C_{48}^{17}+C_{48}^{47}+C_{48}^{41}+C_{48}^{35}+C_{48}^{29}}\end{array}\right)\to\\\
\left(\begin{array}[]{c}{\begin{array}[]{cccccc}{I_{8}}&{I_{8}}&{I_{8}}&{I_{8}}&{I_{8}}&{I_{8}}\end{array}}\\\
{C_{48}^{1}+C_{48}^{7}+C_{48}^{13}+C_{48}^{19}+C_{48}^{25}+C_{48}^{31}}\\\
{C_{48}^{23}+C_{48}^{17}+C_{48}^{47}+C_{48}^{41}+C_{48}^{35}+C_{48}^{29}}\end{array}\right).$
(62)

The first row, where the collapse was performed, consists of 6 circulants of
size 8 and weight 1. Two shift 0 circulants correspond to the first energy
level, and four shift 2 circulants correspond to the second energy level,
representing the electron configuration $1s^{2};2s^{2}2p^{2}$.

Lines 2 and 3 are circulants of weight 6 and size 48. The parity-check matrix
of the quasi-cyclic LDPC code from Fossorier-Shu Lin’s work corresponds to
electron clouds in the ground state of a Nitrogen and Carbon atom under
Spherical Topology and SHBF Cycle Gauge.

######  Statement 4.

The carbon and nitrogen state diagram can be interpreted as a transition
between states of matter under different temperatures (SNR). Such QC-LDPC
codes representation allows greatly reducing the complexity of physical and
chemical properties of matter. The existence of matter under different
environmental properties is equivalent to the existence of a code on graphs
with the required capacity properties. Moreover, Bethe-permanent allows
simplifying it to linear complexity.

Figure 22: Repeat Accumulate Code on the graph non-linear low dimensional
toric and hyperbolic circular topology embedding as combination of symmetry
and asymmetry(loss surface lattice/Hessian), fast features avoid non-linear
transformation Figure 23: Illustration of the exponential moving average (EMA)
approach, which averages the input values X with weights decaying
exponentially over timesteps. Equivalent to GeIRA QC-LDPC codes

##  VI  Demonstrating Equivalence: Generalized Repeat Accumulate Codes and
Cage-Graph QC-LDPC Codes in NLP Transformer DNN

Let’s examine the application of an LDPC code parity-check matrix, denoted as
$H$, in the context of encoding (embedding) a message (Tensor) $m$ for error
correction. It’s noteworthy that the nonlinear mapping and encoding can be
viewed as embedding in machine learning (ML).

$x=H\times m,$ (63)

Here, $m$ represents the transmitted message, $H$ is the parity-check matrix,
$G$ is the generator matrix, and $x$ is the encoded message.

The encoding process satisfies the zero syndrome condition:

$G=H^{-1},\quad H\times x=0.$ (64)

If the generator matrix of the code, by definition, satisfies $G=H^{-1}$,
then:

$H\times G^{T}=0.$ (65)

The parity-check matrix of QC-LDPC codes for Repeat Accumulated (RA) codes can
be expressed as:

$H=[\,H_{1}|H_{2}].$ (66)

Here, $H_{1}$ denotes the parity-check part, which, when multiplied by
systematic bits (analogous to features in DNN), is directly embedded through
fast connections. Simultaneously, $H_{2}$ represents the special structures of
the RA submatrix.For instance, if the RA parity-check matrix is defined as
follows:

$H_{2}=\left[\begin{array}[]{cccccc}{I}&{I}&{0}&{...}&{0}&{0}\\\
{0}&{I}&{I}&{...}&{0}&{0}\\\ {0}&{0}&{I}&{...}&{0}&{0}\\\
{\vdots}&{0}&{0}&{...}&{0}&{0}\\\
{\vdots}&{\vdots}&{\vdots}&{...}&{\vdots}&{\vdots}\\\
{\vdots}&{\vdots}&{\vdots}&{...}&{I}&{0}\\\ {0}&{0}&{0}&{...}&{I}&{I}\\\
{0}&{0}&{0}&{...}&{0}&{I}\end{array}\right],$ (67)

where $I$ represents the identity circulant matrix of size $L\times L$. The
inverse matrix $H_{2}^{-1}$ becomes triangular, and the encoding transforms
into $G\times x$, as illustrated through the use of a convolutional
accumulator (Fig. 28, [60, 17, 61]), or within DNN embedding accomplished
through dynamic convolution (involution and convolution features) (Fig. 29,
[62, 63, 64]).

The Mega and Mega-chunk Attention models are built upon a Generalized
Irregular Repeat Accumulate (GeIRA) protograph, as outlined in [17, 55]. In
these models, the Exponential Averaging Alpha filter is employed to propel
features forward without transformation, ensuring robustness to high
temperatures and a quicker recovery. Additionally, we demonstrate that such
Repeat Accumulate (RA) codes on graph structures can be generalized to
irregular RA codes (GeIRA). This is exemplified through the Transformer MEGA
architecture, which is represented by GeIRA codes on the graph, as depicted in
[17, 55], Fig. 23.

Figure 24: Chord protocol weight 4 cage-graph. Distance graph and related LDPC
code parity-check matrix. ChordMixer Transformer DNN Architecture

A cutting-edge attention architecture (according LRA challenge [21]) in the
realm of long-range interactions is introduced in the work by [23]. This
architecture is built upon the P2P Chord protocol, as outlined in [47] (refer
to 24, left). ChordMixer leverages Cage graphs as distance graphs to formulate
its attention mechanism, as elucidated in the study by [56]. The utilization
of Cage graphs enables ChordMixer to design the attention mechanism in a
manner that is analogous to the LDPC codes.

Figure 25: The architecture of symmetric dilated convolutions. Computation
graph of regular weight 3 codes. CDIL Transformer DNN Architecture [58]

Finally, we examine the dilated Convolutional model ("CDIL") introduced in the
publication by [58]. This convolutional code is founded on the 2P Chord
protocol and is categorized as a column weight 3 convolutional code, as
illustrated in Fig. 25. The theoretical characterization of the trapping set
for such a code was comprehensively provided by [27], proposing an efficient
decoding method based on [59].

Figure 26:  Left side: Kitaev’s Toric 2D code [53]. Right side: Natural image
Topology First layer, CIFAR-10,[57] Figure 27: Klein bottle and Zigangirov’s
Spatially-Coupled Convolutional QC-LDPC code [35] as half-twisted Hyperbolic
Torical Topology

An intriguing observation lies in the link to Torical quantum codes under
Ising models, established through the Calderbank-Shor-Steane (CSS) stabilizer
code construction [52, 53, 54], as illustrated in Figure 26 on the left. This
connection with the ground states of the Ising model has been substantiated in
previous subsections. The study [57] utilized topological data analysis to
gain insights into the computations performed by convolutional deep neural
networks. Analyzing the internal states of these networks, they modified
computations to enhance speed and generalization across various datasets.

Their analysis also resulted in a new geometry for sets of features on image
datasets (MNIST, CIFAR-10, SVHN, ImageNet, etc.). This allowed them to
construct analogues of CNNs for various geometries, including graph structures
derived from topological data analysis.

Building on previous research, such as Carlsson et al.’s study [57], which
demonstrated that local patches in images concentrate around a primary circle
and can be modeled by algebraically defined functions (Fig. 26 on the right),
their work led to the construction of features for images admitting a Klein
bottle Topology.

The natural generalization of Torical topology under infinite-size circulant
(automorphism) is the half-twisted Klein bottle topology equivalent to
convolutional codes: Spatially-coupled Tailbitted QC-LDPC codes, Polar
Spatially-Coupled codes, and Serial Turbo codes (Fig. 27). This connection
elucidates the effectiveness of convolutional codes in computer vision tasks.
Moreover, it demonstrates the link between topological analysis and the
calculation of topological invariants, such as Betti numbers, indicating the
number of multidimensional voids formed by code and pseudo-code words of
trapping sets (TS enumerators).

Proposition 3: The research paper [36] establishes a connection between QC
block codes and spatially coupled QC convolutional codes with limited memory
termination. This connection, observed under the Ising model, is rooted in the
klein bottle twisted topology present in block QC codes featuring a toric
(hyperbolic) topology. Exploiting this relationship enables the creation of a
dynamic convolution aligned with the mentioned topology, incorporating low-
complexity non-linear multiplicative correlation. This dynamic convolution
involves fast and slow features, systematic bits (fast feature, pass forward
without changes), and coded bits in channel codes, utilizing highly efficient
convolutional accumulator processing commonly employed in wireless encoders,
such as QC-LDPC codes [55, 17, 60, 61]. This aligns with the TDA analysis
findings by Carlsson [57], as originally proposed by Schmidhuber (1991) and
described in papers [62, 63, 64].

The integration of QC-LDPC block and convolutional codes, along with their
relationship to Polar and Turbo codes, through serial processing on Klein
bottle and Torus Topology, significantly contributes to enhancing the quality
of structured codes on graph models with high automorphism. This integration
plays a crucial role in various applications, spanning material science,
complexity theory, and efficient implementations of deep neural networks. A
key metric for assessing the gauge quality of any dynamic system is the
Trapping sets enumerator, encompassing codewords and pseudocodewords. It is
asserted that the Trapping sets enumerator serves as a fundamental metric for
evaluating the gauge, defining the loss surface (energy landscape) of
spherical and toroidal dynamic systems, including Ising models, DNNs, control
systems, differential equations, social networks, and other related topology
graph models and tensor networks.

Proposition 4: Spatially-Coupled Convolutional codes resembling a Klein bottle
topology can be constructed by adopting a suitable graph structure, achieved
by twisting opposite edges on a torus. The link between block and
convolutional QC-LDPC codes is evident in [36], while the connection between
Polar codes and LDPC codes, post-sequential processing with a successive
cancellation scheduler, is discussed in [67, 68]. A similar connection is
showcased between Turbo codes and QC-LDPC codes under serial processing [37],
providing a potential approach for hardware-efficient optimal embedding using
Codes on the graphs TS enumerators and related Topology Invariant from
Topology Data Analysis.

Non-linear sparse factorization serves as a low-dimensional embedding
capturing the structure of high-dimensional data. Leveraging topology data
analysis and information geometry, we can analyze data curvature, facilitating
efficient and accurate tasks like classification, interpolation, and
regression. Stochastic dynamic systems accommodate inherent data uncertainty,
ensuring robust predictions in real-world scenarios. The spectral gap, tied to
topological properties like Bethe tree or lattice, Bethe permanent
(pseudocodewords TS(a,b)) [31, 73], and permanent (codewords TS(a,0)) [74,
75], along with the graph spectral bound [76, 77, 78], are crucial. Neural
networks with minimal clustering coefficients, particularly entangled nets
(optimized Trapping set enumerator), exhibit promising traits for constructing
high-capacity and high-performance artificial neural networks [78]. Improving
spectral properties, especially bethe permanent and permanent, enhances
Trapping set enumerators in graph models, leading to increased capacity and
improved performance in DNNs.

Figure 28: Repeat Accumulate Code on the graph structures, fast part
systematic bits(features for DNN) and repeat accumulated coded bits, [55, 60,
17, 61] Figure 29: The dynamic approach combines: convolutional and block
codes by incorporating attention and convolution within a deep neural network
(DNN) proposed by Schmidhube at 1991, [62]. RA codes example of such approach

##  VII CONCLUSION

This paper introduces a pioneering approach to achieve equilibrium in the
ISING Hamiltonian when dealing with unevenly distributed charges on an
irregular grid. By leveraging QC-LDPC codes and the Boltzmann machine, our
method involves expanding the system dimensionally, replacing charges with
circulants, and expressing distances through circulant shifts. This systematic
mapping of the charge system onto a space transforms the irregular grid into a
uniform configuration, applicable to Torical and Circular Hyperboloid
Topologies. The fundamental definitions and notations related to QC-LDPC
Codes, Multi-Edge QC-LDPC codes, and the Boltzmann machine are covered. The
paper delves into the marginalization problem in code on the graph
probabilistic models for evaluating the partition function, exploring exact
and approximate estimation techniques. Rigorous proof is provided for the
attainability of equilibrium states for the Boltzmann machine under Torical
and Circular Hyperboloid, laying the foundation for the application of our
methodology. Practical applications of our approach are investigated in Finite
Geometry QC-LDPC Codes, particularly in Material Science. The paper further
explores its effectiveness in the domain of Natural Language Processing
Transformer Deep Neural Networks, examining Generalized Repeat Accumulate
Codes and Cage-Graph QC-LDPC Codes, Convolutional codes. The versatile and
impactful nature of our topology-aware equilibrium method is demonstrated
across diverse scientific domains, transcending specific section delineations.

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