THERMODYNAMIC EMERGENCE:
Deriving the Cuboctahedral Vacuum from Information Entropy
Raghu Kulkarni
Independent Researcher
January 28, 2026
Abstract
We present a thermodynamic derivation for the geometric ansatz of the Selection-Stitch Model
(SSM). We posit that the cooling vacuum evolves to maximize its Information Storage Capacity
(Entropy) subject to volumetric constraints. Using the Universality Hypothesis, we map this quan-
tum problem to the classical Sphere Packing Problem. The Kepler Conjecture (Hales, 2005) dictates
that the maximum density for 3D packing is strictly π/
√
18. While this density is shared by both
Face-Centered Cubic (FCC) and Hexagonal Close-Packed (HCP) lattices, we argue that the FCC
lattice is selected due to its superior isotropic symmetry (Point Group O
h
). To address the critique
of applying classical packing to quantum nodes, we cite the universality of geometric ground states in
quantum systems specifically Wigner Crystals, Abrikosov Lattices, and Coulomb Crystals. Finally,
we distinguish the crystalline ground state from Random Close Packing (RCP), arguing that Cosmic
Inflation acted as a thermodynamic annealing event that prevented a “glassy” vacuum. This sug-
gests the universe naturally establishes the Cuboctahedral (K = 12) boundary condition, effectively
setting the geometric baseline for the Hubble Tension resolution.
1 The Geometry Selection Problem
Imagine a box of oranges. If you pour them in randomly, they form a messy, amorphous pile (Random
Close Packing). However, if you shake the box—adding energy and allowing the system to explore phase
space—they spontaneously lock into an ordered pattern. They do this to minimize potential energy and
maximize density.
The early universe faced a similar problem. In the “Pre-Geometric” phase, quantum fluctuations
were stochastic. As the universe expanded and cooled, these fundamental “nodes” of information had to
pack together to form a stable space-time manifold.
The Selection-Stitch Model (SSM) assumes the universe settled into a Cuboctahedron (K = 12). A
fundamental critique of this model is: Why this shape? Why not a Tetrahedron (K = 4) or a disordered
glass? We argue the answer is Thermodynamic Efficiency.
2 The Holographic Packing Principle
2.1 Maximizing Information Density
The Holographic Principle suggests that information density is limited by surface area boundaries. To
build a universe capable of complex interactions (Unitary Stability), nature must maximize the number
of entanglement links (N ) per unit volume (V ). This optimization problem can be expressed as:
P
info
≈
N
links
V
−→ max (1)
This is physically identical to the Sphere Packing Problem: How to arrange nodes to waste the least
amount of empty space (voids).
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2.2 The Mathematical Proof (Hales, 2005)
In 2005, Thomas Hales provided the formal proof of the Kepler Conjecture, demonstrating that the
maximum density of spheres in 3D Euclidean space is strictly limited to:
ρ
max
=
π
√
18
≈ 0.7405 (2)
This rigorous proof eliminates all loose, random, or tetrahedral packings as energetically unfavorable. In
a cooling system seeking a ground state, only the close-packed lattices remain as viable configurations.
2.3 Breaking the Degeneracy: FCC vs. HCP
Critically, two lattices share this maximum density: the Face-Centered Cubic (FCC) and the Hexagonal
Close-Packed (HCP).
• HCP (D
6h
): Possesses a preferred vertical axis (anisotropic) and a “twist” in its stacking order
(ABAB).
• FCC (O
h
): Is centrally symmetric and isotropic (ABCABC).
Since the Big Bang was a highly isotropic event (the universe appears statistically identical in all
directions), we posit that Symmetry Breaking favored the isotropic FCC (Cuboctahedron) over the
twisted HCP. The Cuboctahedron is the unique geometry that satisfies both maximum density and
maximum symmetry.
3 First-Principles Derivation of the Vacuum State
Instead of assuming a phenomenological potential, we derive the vacuum energy landscape directly from
three fundamental physical axioms: Holographic Entanglement, Geometric Saturation, and Symmetry
Selection.
3.1 The Entanglement Drive (Deriving −αk)
We begin with the Unitary Stitch, defined as a single unit of quantum entanglement (a Bell pair)
connecting two domains.
• Axiom: The entanglement entropy (S) of the network is additive.
• Thermodynamics: Systems evolve to maximize entropy (S → max), which minimizes the Free
Energy (F = U − T S).
• The Hamiltonian Map: This linear drive corresponds to the Vertex Coupling Constant (J
e
)
in the Stabilizer Hamiltonian H = −J
e
P
A
v
.
Since each neighbor (k) represents a valid unitary link, the energy benefit scales linearly with con-
nectivity:
E
bind
= −J
e
· k = −αk (3)
Where α is the energy equivalent of one bit of entanglement (dS).
3.2 The Metric Wall (Deriving βe
k−12
)
Unlike chemical bonds, geometric exclusion is not a soft force but a hard metric limit.
• Axiom: The Kepler Conjecture dictates that the maximum local density in 3D is strictly limited
to the ”Kissing Number” of k = 12.
• The Singularity: As k → 12, the available phase space volume (V
free
) for a new node approaches
zero. Since pressure P ∝ 1/V
free
, the resistance to a 13th node diverges asymptotically.
• Simulation Constraint: Our constructive simulation demonstrates this by enforcing a ”Hard
Shell” exclusion radius of 0.95L. This 5% tolerance defines the ”stiffness” of the metric.
The repulsive potential is therefore an asymptotic wall, modeled effectively as:
E
excl
= βe
k−12
(4)
Where β is the Bulk Modulus of the vacuum lattice, derived from the metric jitter tolerance (J
m
).
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3.3 Breaking the Degeneracy: The Symmetry Term (−T S
sym
)
A purely density-driven model cannot distinguish between Face-Centered Cubic (FCC) and Hexag-
onal Close-Packed (HCP), as both satisfy k = 12 and ρ = π/
√
18.
• Observation: The early universe (Big Bang) was a highly isotropic event.
• Entropic Penalty: The HCP lattice (D
6h
) possesses a preferred vertical axis (anisotropy). Ori-
enting this axis in a hot, isotropic plasma entails an entropic cost. The FCC lattice (O
h
) is isotropic
and carries no such penalty.
• Selection: The system minimizes free energy by maximizing Symmetry Entropy (S
sy m
).
F
select
= −T · S
sy m
(5)
3.4 The Combined Free Energy Functional
Combining these three terms yields the complete First-Principles Equation for Vacuum Selection:
F (k) = −αk
|{z}
Entanglement
(Vertex A
v
)
+ βe
k−12
| {z }
Metric
(Plaquette B
p
)
− T S
sy m
| {z }
Isotropy
Selection
(6)
This equation dictates that the vacuum MUST settle into the Cuboctahedral (K = 12, FCC)
geometry, as it is the unique state that simultaneously maximizes connectivity, respects metric saturation,
and maximizes symmetry.
4 Micro-Foundation & Hamiltonian Mapping
We formally map the macroscopic parameters (α, β) to the microscopic constants of the Selection-Stitch
Model’s Toric Code Hamiltonian derived in the primary theory (H
SSM
).
Phenomenological Fundamental Physical Origin
α (Binding) J
e
(Vertex Coupling) The Information Gap. Cost of breaking a unitary
link. Derived from Ryu-Takayanagi (S ∝ Area).
β (Metric Stiffness) J
m
(Plaquette Coupling) The Geometric Gap. Cost of flux violation (overlap).
Derived from ”Metric Jitter” tolerance (0.95L).
Transition (K = 12) A
v
(Star Constraint) The Kepler Limit. Mathematical limit of 3D
sphere packing (π/
√
18).
Table 1: Mapping Effective Field Theory to Quantum Stabilizers
4.1 The Annealing Constraint
For the universe to find this global minimum (Crystal) rather than getting trapped in a local minimum
(Glass/Random Close Packing, K ≈ 12, ρ ≈ 0.64), the system required a high-temperature ”shaking”
event.
• Identification: We identify Cosmic Inflation as this thermodynamic annealing process.
• Requirement: k
B
T
inflation
> ∆E
g lass−crystal
.
This confirms that the rigid geometric background required for the 13/12 Hubble Tension boost
is not an accident, but a thermodynamic necessity.
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5 Discussion: The Universality of Geometric Ground States
5.1 Precedents for Quantum Geometric Packing
A valid critique of this model is the application of classical sphere packing logic (Kepler) to quantum
objects (vacuum nodes). However, we invoke the principle of Universality: in statistical mechanics,
phase transitions often depend only on symmetry and dimensionality, not on the microscopic details of
the constituents. We observe distinct precedents in condensed matter physics where purely quantum
constituents spontaneously organize into classical lattices to minimize energy:
5.1.1 Wigner Crystallization (Electrons → BCC)
When the potential energy of an electron gas dominates its kinetic energy (at low densities and tempera-
tures), the electrons—pure quantum wavefunctions—“freeze” into a classical crystal lattice to minimize
Coulomb repulsion. As shown by Wigner (1934), the ground state is a Body-Centered Cubic (BCC)
lattice. This demonstrates that quantum repulsion naturally leads to classical geometric ordering.
5.1.2 Abrikosov Lattices (Flux → Hexagonal)
In Type-II superconductors, magnetic field lines penetrate the material as quantized flux tubes (vortices).
These vortices are topological defects. Remarkably, they do not arrange randomly; they self-assemble
into a perfect Triangular (Hexagonal) Lattice (Abrikosov, 1957). This is mathematically identical to 2D
circle packing (ρ = π/
√
12), proving that topological defects naturally seek maximum packing density.
5.1.3 Coulomb Crystals (Ions → FCC Shells)
When ions are laser-cooled in a Paul trap, they form “Coulomb Crystals”. Despite being governed by the
Heisenberg Uncertainty Principle, the ions arrange themselves into concentric shells that mimic classical
close-packing geometry (often FCC-like structures in the bulk limit) (Birkl et al., 1992).
5.2 Why Not a Glass? (The Annealing Hypothesis)
A related objection considers Random Close Packing (RCP), where spheres form a disordered glass with
local coordination K ≈ 12 but significantly lower density (ρ ≈ 0.64). Why does the vacuum not freeze
into this amorphous state?
We argue that the energy difference between the local minimum (Glass, ρ ≈ 0.64) and the global
minimum (Crystal, ρ ≈ 0.74) is substantial. The violent expansion of Cosmic Inflation acted as a
thermodynamic annealing process—effectively “shaking the box” with sufficient energy to overcome the
glassy energy barriers. This allowed the vacuum nodes to settle into the true, densest ground state
(FCC) rather than getting trapped in a disordered local minimum.
5.3 Implications for the Vacuum
These examples confirm that “packing efficiency” is a universal physical constraint.
• Electrons minimize repulsion via BCC packing.
• Flux lines minimize tension via Hexagonal packing.
• Vacuum Nodes minimize information entropy via Cuboctahedral (K = 12) packing.
Just as fluid dynamics applies to both water molecules and quark-gluon plasmas, we propose that geo-
metric packing efficiency applies to both marbles and Planck volumes.
6 Conclusion
The Cuboctahedral geometry (K = 12) of the Selection-Stitch Model is not an arbitrary parameter. It
is the necessary result of two fundamental principles:
1. Thermodynamics: The vacuum seeks maximum density (The Kepler Solution, π/
√
18).
2. Symmetry: The vacuum seeks isotropy (Selecting FCC over HCP).
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Therefore, the “13/12” Hubble boost (≈ 8.3%) derived in the primary SSM paper is a direct consequence
of the universe settling into its most efficient ground state.
References
1. Hales, T. C. (2005). A proof of the Kepler conjecture. Annals of Mathematics, 162(3), 1065-1185.
2. Kulkarni, R. (2026). The Selection-Stitch Model (SSM): Emergent Gravity, Dark Energy, and the
Hubble Tension. Preprint. DOI: 10.5281/zenodo.18332527
3. Wigner, E. (1934). On the Interaction of Electrons in Metals. Physical Review, 46(11), 1002.
4. Abrikosov, A. A. (1957). On the Magnetic Properties of Superconductors of the Second Group.
Soviet Physics JETP, 5, 1174.
5. Birkl, G., Kassner, S., & Walther, H. (1992). Multiple-Shell Structures of Laser-Cooled
24
Mg
+
Ions in a Quadrupole Storage Ring. Nature, 357, 310-313.
6. Conway, J. H., & Sloane, N. J. A. (1999). Sphere Packings, Lattices and Groups. Springer New
York.
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