  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.
