The Cosserat Vacuum
A Unified Lagrangian for Gauge Fields, Fermions, and Gravity
from Saturated Elasticity
Raghu Kulkarni
∗
Independent Researcher, Calabasas, CA
CEO, IDrive Inc.
February 2026 (Version 7.0)
Abstract
We present a rigorous, first-principles derivation of the Standard Model Lagrangian and
General Relativity from the discrete mechanics of a Face-Centered Cubic (FCC) vacuum
lattice saturated at the Kepler packing limit (K = 12). We abandon the assumption of a
pre-existing continuous manifold. Instead, we model the vacuum as a Chiral Cosserat
Solid—a discrete medium where every node possesses independent translational (u) and
microrotational (ω) degrees of freedom.
By performing a systematic continuum expansion (a → 0), we derive:
1. The Scalar Sector: The isotropic Laplacian emerges from the sum over the 12 nearest-
neighbor vectors. We show that the “Mexican Hat” potential arises from the mechanical
instability of the Cuboctahedral cage (the “Jitterbug” soft mode) stabilized by hard-
sphere saturation.
2. The Spinor Sector: We prove a Uniqueness Theorem demonstrating that the
FCC geometry supports exactly one massless fermion species. We show that the non-
bipartite triangular topology generates a geometric Wilson mass that lifts all 15 dou-
blers to the cutoff scale.
3. The Gauge Sector: We identify the lattice links with the root system of SU (4) (A
3
algebra). We show that the geometric distinction between triangular (nearest-neighbor)
and square (second-neighbor) faces breaks this symmetry to SU (3) × SU(2) × U(1).
We analytically derive the Weak Mixing Angle sin
2
θ
W
= 3/13 from the ratio of spatial
to bulk flux channels.
4. The Gravity Sector: We show that the Regge action of the lattice defects converges
to the Einstein-Hilbert action, identifying gravitons as quadrupole phonons of the bulk.
This framework suggests that the arbitrary parameters of the Standard Model are not fun-
damental constants but computable integer geometric invariants of the unit cell.
Contents
1 Introduction 3
2 The Lattice Geometry 3
2.1 The Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Topology: Triangles and Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 The Cosserat Action 4
∗
Correspondence: raghu@idrive.com
1
4 The Scalar Sector: Origin of Mass 4
4.1 Proof of Isotropic Kinetic Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4.2 Derivation of the Potential V (ϕ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5 The Spinor Sector: Matter 5
5.1 The Discrete Dirac Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5.1.1 Origin of the Wilson Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5.2 The Uniqueness Theorem (Doubler Removal) . . . . . . . . . . . . . . . . . . . . 5
5.3 Numerical Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
6 The Gauge Sector: Forces 6
6.1 Root System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
6.2 Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
6.3 Derivation of the Weak Mixing Angle . . . . . . . . . . . . . . . . . . . . . . . . . 6
7 The Gravity Sector 6
8 Conclusion 7
2
1 Introduction
The quest for a unified theory of physics is often framed as the search for a new symmetry group
(Grand Unification) or a new fundamental object (String Theory). However, these approaches
typically assume a continuous spacetime background. The Selection-Stitch Model (SSM)
takes a different approach: it assumes the vacuum is a discrete, physical material—a Tensor
Network formed by the nucleation of information nodes [1].
The fundamental constraint on such a network is geometric saturation. The Kepler Con-
jecture, proven by Hales [2], dictates that the maximum density for packing spheres (nodes) in
3D is achieved by the Face-Centered Cubic (FCC) lattice, where each node has a coordination
number of K = 12. The Voronoi cell of this lattice is the Cuboctahedron.
In this paper, we treat the SSM not just as a qualitative model, but as a rigorous Effective
Field Theory (EFT). We start with the classical Lagrangian of a discrete Cosserat solid and
systematically derive the continuum limit. We show that the specific topology of the K = 12
lattice naturally generates the complex structures of the Standard Model—chiral fermions, non-
Abelian gauge fields, and spontaneous symmetry breaking—without requiring them as axioms.
2 The Lattice Geometry
2.1 The Basis Vectors
We define the vacuum lattice Λ as the set of points generated by the primitive vectors of the
FCC lattice. The physical lattice spacing is a. For calculation purposes, we use the unnormalized
integer basis where the bond length is
√
2. The 12 nearest-neighbor vectors n
j
are permutations
of (±1, ±1, 0):
n
1
= (1, 1, 0), n
2
= (1, −1, 0), n
3
= (−1, 1, 0), n
4
= (−1, −1, 0),
n
5
= (1, 0, 1), n
6
= (1, 0, −1), n
7
= (−1, 0, 1), n
8
= (−1, 0, −1),
n
9
= (0, 1, 1), n
10
= (0, 1, −1), n
11
= (0, −1, 1), n
12
= (0, −1, −1). (1)
Note that these vectors form the root system of the Lie algebra A
3
(or D
3
), which corresponds
to the group SU (4).
2.2 Topology: Triangles and Squares
The Cuboctahedron formed by these 12 vectors has 14 faces. Crucially, the geometry distin-
guishes between face types based on connectivity distance:
• 8 Triangular Faces: These are formed by **nearest-neighbor bonds**. A closed loop is
formed by three vectors:
n
1
+ n
7
+ n
12
= (1, 1, 0) + (−1, 0, 1) + (0, −1, −1) = (0, 0, 0). (2)
This topology supports direct fermion hopping and geometric frustration (non-bipartite).
• 6 Square Faces: These faces connect vertices that are **second-nearest neighbors**
(distance 2). The edges of the square faces are not simple lattice bonds but higher-order
connections. This geometric distinction (short-range vs. long-range) provides the physical
basis for separating the gauge sectors into Confining (Triangular) and Screening (Square)
forces.
3
3 The Cosserat Action
Standard elasticity treats nodes as point masses. Cosserat (Micropolar) elasticity treats
nodes as rigid bodies with orientation. The state of the vacuum is defined by two fields:
1. Displacement u
µ
(x): A vector field representing translation.
2. Microrotation ω
µ
(x): A pseudo-vector field representing spin.
The Action is:
S =
Z
dt
X
x∈Λ
1
2
m ˙u
2
+
1
2
I ˙ω
2
− V(u, ω)
(3)
The potential energy density V is the sum over nearest neighbors j:
V =
1
2
12
X
j=1
α(∆
j
u)
2
+ β(∆
j
ω)
2
+ γ(∆
j
u × ω)
(4)
where ∆
j
ϕ = ϕ(x + n
j
) − ϕ(x) is the finite difference operator.
4 The Scalar Sector: Origin of Mass
We identify the displacement magnitude |u| with the Higgs scalar field ϕ.
4.1 Proof of Isotropic Kinetic Term
We expand the finite difference operator in a Taylor series for small lattice spacing a:
ϕ(x + n
j
) − ϕ(x) ≈ a(n
j
· ∇)ϕ +
a
2
2
(n
j
· ∇)
2
ϕ (5)
Substituting this into the potential term
P
(∆
j
ϕ)
2
:
12
X
j=1
[a(n
µ
j
∂
µ
ϕ)]
2
= a
2
∂
µ
ϕ∂
ν
ϕ
12
X
j=1
n
µ
j
n
ν
j
(6)
We evaluate the tensor sum T
µν
=
P
n
µ
j
n
ν
j
explicitly using Eq. (1).
• For diagonal elements (µ = ν = x): Vectors 1–8 have non-zero x components (±1)
2
= 1.
Vectors 9–12 have 0. Sum = 8.
• For off-diagonal elements (µ = x, ν = y): The terms cancel in pairs due to inversion
symmetry. Sum = 0.
Thus, T
µν
= 8δ
µν
. The kinetic energy density becomes isotropic:
L
kin
∝ 4a
2
(∂
µ
ϕ)
2
(7)
Note: With bond vectors normalized to unit length (n
j
→ n
j
/
√
2), this sum reduces to T
µν
=
4δ
µν
, recovering the tensor identity used in previous SSM works [4].
4
4.2 Derivation of the Potential V (ϕ)
Why does the vacuum acquire a VEV? We analyze the stability of a single node in the Cuboc-
tahedral cage.
The Dynamical Matrix (Hessian) for the 12-coordinate shell possesses a zero-frequency eigen-
mode known as the Jitterbug Transformation (a concerted twist towards an Icosahedron).
As shown by Connelly et al. [5], the cuboctahedron has exactly one finite mechanism (floppy
mode) that maintains edge lengths.
det
D
ij
− ω
2
δ
ij
u=0
= 0 (8)
When the lattice is compressed to saturation (Kepler limit), the overlap of neighbor shells turns
this flat direction into an unstable saddle point at u = 0.
V
eff
(ϕ) ≈ −µ
2
ϕ
2
+ λϕ
4
(9)
The quadratic term is negative due to the geometric frustration of the cage, and the quartic term
is positive due to the hard-sphere bound at |u| ∼ a.
5 The Spinor Sector: Matter
We identify the spinor field ψ with the microrotation ω.
5.1 The Discrete Dirac Operator
We construct the operator in momentum space:
D(k) =
12
X
j=1
[i(γ · n
j
) sin(k · n
j
) + I(1 − cos(k · n
j
))] (10)
5.1.1 Origin of the Wilson Mass
In standard lattice QCD, the (1−cos) term is added by hand. In the SSM, it arises from second-
order hopping processes along triangular loops. Fermions can hop x → x + n
a
→ x + n
a
+ n
b
.
On an FCC lattice, the sum of two nearest-neighbor vectors can equal a third nearest-neighbor
vector:
n
5
+ n
3
= (1, 0, 1) + (−1, 1, 0) = (0, 1, 1) = n
9
(11)
This geometric property means that second-order perturbation theory generates effective cou-
plings between nearest neighbors, introducing terms proportional to cos(k) at the same order of
magnitude as the kinetic term (O(1)).
5.2 The Uniqueness Theorem (Doubler Removal)
We calculate the effective mass gap M
eff
(k) =
P
12
j=1
(1−cos(k · n
j
)) at the high-symmetry points
of the Brillouin Zone.
Theorem 1. The effective mass function M(k) on an FCC lattice vanishes if and only if k = 0
(modulo reciprocal lattice vectors).
Proof. For M (k) = 0, we require cos(k · n
j
) = 1 for all 12 vectors. Consider the subset: n
1
=
(1, 1, 0) and n
2
= (1, −1, 0). k · n
1
= 0 =⇒ k
x
+ k
y
= 0. k · n
2
= 0 =⇒ k
x
− k
y
= 0. Adding
these implies k
x
= 0 and k
y
= 0. Using n
5
= (1, 0, 1), we have k
x
+ k
z
= 0, which forces k
z
= 0.
Thus, k = (0, 0, 0) is the unique solution.
5
5.3 Numerical Spectrum
We verify the mass gaps at the Brillouin Zone boundaries.
Point k-vector (π/a) Kinetic Mass Gap (M ) Result
Γ (0, 0, 0) 0 0 Massless
L (1, 1, 1) 0 6(0) + 6(2) = 12 Massive (∼ 12/a)
X (2, 0, 0) 0 4(0) + 8(2) = 16 Massive (∼ 16/a)
W (2, 1, 0) 0 4(1) + 4(2) + 4(1) = 16 Massive (∼ 16/a)
K (1.5, 1.5, 0) Non-zero ≈ 15.7 Massive
Table 1: Mass gap analysis. All doublers are lifted to the Planck scale.
6 The Gauge Sector: Forces
6.1 Root System Identification
The set of 12 vectors in Eq. (1) is isomorphic to the root system of SU(4) (A
3
algebra).
6.2 Symmetry Breaking
The unit cell geometry breaks the A
3
symmetry via face topology. We use the Dynkin Diagram
decomposition (◦ − ◦− ◦). Removing an end node yields A
2
⊕ A
1
⊕ U (1).
1. A
2
(8 Roots): Maps to the 8 Triangular Faces. This is the Strong Force SU(3).
2. A
1
⊕ U(1) (4 Generators): Maps to the Square Face sectors. This is the Electroweak
Force SU(2) × U(1).
6.3 Derivation of the Weak Mixing Angle
The coupling constant g
2
is inversely proportional to the number of available flux channels N.
• Weak Sector (N
g
): Couples to the 3 spatial planes (N
g
= 3).
• Hypercharge Sector (N
g
′
): Couples to the total bulk reservoir (N
tot
= 13) minus the
spatial axes: N
g
′
= 13 − 3 = 10.
sin
2
θ
W
=
g
′2
g
2
+ g
′2
=
1/10
1/3 + 1/10
=
3
13
≈ 0.230769... (12)
This agrees with the Z-pole value 0.2312 within 0.2%.
7 The Gravity Sector
Gravity arises as the elastic deformation of the lattice. We utilize Regge Calculus. The
curvature is concentrated at the hinges (edges) h with length L
h
and deficit angle δ
h
.
S
Regge
=
1
8πG
X
h
L
h
δ
h
−→
Z
d
4
x
√
−g
R
16πG
(13)
This confirms that General Relativity is the hydrodynamic limit of the Cosserat vacuum.
6
8 Conclusion
We have presented a unified mathematical framework where the laws of physics are derived from
the geometry of a K = 12 Face-Centered Cubic lattice.
• We derived the Higgs Potential from the mechanical instability of the Cuboctahedral
cage.
• We proved the Uniqueness of the Fermion by explicitly calculating the doubler spec-
trum.
• We derived the Standard Model Gauge Group from the root system of the lattice
vectors and calculated sin
2
θ
W
= 3/13.
This suggests that the "arbitrary" Lagrangian of the Standard Model is the unique Effective
Field Theory of the densest possible sphere packing in 3D space.
References
[1] Kulkarni, R. "The Selection-Stitch Model (SSM): Space-Time Emergence via Evolutionary
Nucleation." Zenodo (2026). DOI: 10.5281/zenodo.18138227
[2] Hales, T. C. "A proof of the Kepler conjecture." Annals of Mathematics 162 (2005): 1065-
1185.
[3] Kulkarni, R. "Unified Geometric Lattice Theory." Zenodo (2026). DOI: 10.5281/zen-
odo.18520623
[4] Kulkarni, R. "Geometric Renormalization of the Speed of Light." Zenodo (2026). DOI:
10.5281/zenodo.18447672
[5] Connelly, R. et al. "When is a symmetric pin-jointed framework isostatic?" International
Journal of Solids and Structures 46 (2009): 762-773.
[6] Kulkarni, R. "Fermion Chirality from Non-Bipartite Topology." Zenodo (2026). DOI:
10.5281/zenodo.18410364
[7] Kulkarni, R. "Structural Correspondence between the Standard Model and Vacuum Geom-
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