Micropolar Neutrinos: Deriving Mass Suppression and Oscillation from the
Cosserat Lagrangian of the Vacuum
Raghu Kulkarni
1, ∗
1
Independent Researcher, Calabasas, CA
(Dated: February 6, 2026)
We propose a dynamical origin for neutrino mass based on the mechanics of a structured
vacuum lattice. Modeling the vacuum as a Cosserat (micropolar) continuum, we decom-
pose the lattice field into two independent degrees of freedom: a translational displacement
vector u and a microrotational pseudo-vector ϕ. We identify charged fermions with the
translational modes (anchored defects) and neutrinos with the rotational modes (unanchored
twists). By constructing the linear Cosserat Lagrangian, we show that the mass terms for
these modes scale with distinct elastic moduli: the Bulk Modulus E
bulk
for charged species
and the Micropolar Twist Modulus γ for neutrinos. In a saturated K = 12 lattice, the
bulk modulus dominates, while the twist modulus appears only as a higher-order geometric
coupling. This creates a natural seesaw-like hierarchy m
ν
/m
e
∼ γ/E
bulk
≪ 1 purely from
classical lattice mechanics. We derive a specific scaling relation m
ν
≈ Φ
3
m
e
, where Φ is the
vacuum impedance derived from the lattice contact number and Clifford algebra dimension,
predicting m
ν
≈ 0.057 eV. Finally, we show that flavor oscillations arise as dispersive beating
patterns of the unpinned microrotation field.
I. INTRODUCTION
Narrative: The origin of neutrino mass is one of the central problems in particle physics.
The extreme hierarchy between the neutrino scale (< 0.1 eV) and the charged lepton scale (0.5
MeV) suggests they acquire mass through fundamentally different mechanisms. The discovery
that neutrinos oscillate between flavors [1, 2] further implies non-zero distinct masses. Standard
approaches, such as the See-Saw Mechanism [3], achieve this by introducing heavy sterile states.
We propose a low-energy alternative based on the geometry of the vacuum itself. We model the
vacuum as a discrete Face-Centered Cubic (FCC) lattice (K = 12). We posit that the neutrino is
not a point particle with a small Yukawa coupling, but a distinct *type* of lattice excitation—a
”floating twist”—described by Cosserat elasticity.
Formalism: We postulate that the vacuum manifold is a discrete tensor network saturated at
∗
raghu@idrive.com
2
the Kepler packing limit (K = 12) [4]. The fundamental lattice spacing ℓ
P
serves as the ultraviolet
cutoff. We treat the long-wavelength limit of this lattice as a continuum described by a Lagrangian
density L. Our goal is to derive the mass terms m
ψ
from the elastic moduli of this continuum.
II. THE COSSERAT VACUUM STRUCTURE
Narrative: In standard elasticity (Cauchy theory), a point in a medium has only one degree of
freedom: it can move (displacement). However, if the medium has microstructure (like a lattice of
finite-sized nodes), a point can also *spin* without moving. This second degree of freedom is called
”Microrotation.” We identify the two fundamental classes of fermions with these two independent
mechanical modes: 1. **Charged Leptons (Anchored):** Defects that displace the lattice nodes
(u). They stretch the strong bonds between nodes, acquiring large mass (Anchor Tension). 2.
**Neutrinos (Unanchored):** Defects that twist the lattice nodes without displacing them (ϕ).
They do not stretch bonds but only fight the residual rotational friction.
Formalism: We define the vacuum state vector at coordinate x by two independent fields,
following the notation of Eringen [5]:
1. **Displacement Vector u(x):** The translational shift of the node.
2. **Microrotation Pseudo-vector ϕ(x):** The rigid rotation of the node.
In a general medium, these modes couple. However, in the limit of a saturated lattice (K = 12), the
compression resistance (Bulk Modulus) becomes infinitely stiff compared to the rotation resistance.
This effectively decouples the modes into a heavy translational sector and a light rotational sector.
The linear isotropic Cosserat Lagrangian density L is given by [6]:
L = T − V (1)
where the potential energy density V is:
V =
λ
2
(∇ · u)
2
+ µ(∇u)
2
+ κ(∇ × u − 2ϕ)
2
+ γ(∇ϕ)
2
(2)
Here, the coefficients represent the elastic moduli of the vacuum:
• λ, µ: Lam´e coefficients (Compression/Shear).
• κ: **Coupling Modulus** (Links translation to rotation).
• γ: **Twist Modulus** (Resistance to pure microrotation).
3
III. DERIVATION OF THE MASS HIERARCHY
Narrative: Why is the neutrino so light? In our model, mass corresponds to the energy cost
of the defect. The electron is a ”push” on the lattice. Because the K = 12 lattice is saturated
(maximum packing), it is nearly incompressible. Pushing it requires enormous energy—hence, the
electron is heavy. The neutrino is a ”twist” on a node. Twisting a sphere in a packing does not
compress the neighbors; it only experiences surface friction. This friction is a higher-order effect,
much weaker than the compression resistance. The neutrino mass is suppressed because it couples
to the weak *Twist Modulus* rather than the strong *Bulk Modulus*.
Formalism: We identify the mass scales with the respective elastic moduli:
m
e
∼ (λ + 2µ) ≡ E
bulk
(3)
m
ν
∼ γ ≡ E
twist
(4)
The ratio E
twist
/E
bulk
is determined by the lattice microstructure. For granular packings, this ratio
is calculable in principle from the contact geometry [7]. Here, we propose a **Geometric Ansatz**
for this ratio based on the concept of Virtual Anchoring. The microrotation (ϕ) is a surface effect
that must couple virtually to the 3D bulk to acquire mass. This coupling is suppressed by the
**Vacuum Impedance** Φ.
We explicitly define Φ via the phase space of the K = 12 unit cell. The total phase space is
the tensor product of the kinematic coordination shell (N
kin
= 13, from the contact number of the
FCC unit cell [8]) and the internal spinor algebra (N
alg
= 16, from the Clifford algebra dimension
[9]).
Φ =
1
N
kin
× N
alg
=
1
13 × 16
=
1
208
(5)
Since mass is a bulk (3D) property, the suppression acts in each spatial dimension (D = 3):
γ ≈ Φ
3
× E
bulk
(6)
Substituting the mass identifications:
m
ν
≈ m
e
× Φ
3
= m
e
×
1
208
3
≈ 0.057 eV (7)
This matches experimental upper bounds from Planck and KATRIN [10, 11]. While first-principles
calculations of γ for FCC packings are in progress, this ansatz provides a phenomenological bridge
consistent with the virtual coupling interpretation.
4
IV. DYNAMICS OF OSCILLATION
Narrative: Why do neutrinos oscillate? A charged lepton is ”anchored” or pinned to the
lattice by the strong bulk tension, forcing it into a specific geometric state (Point, Line, or Plane).
It behaves like a localized particle. A neutrino, being unanchored, is not pinned. It behaves like a
wave in a dispersive medium. When a neutrino is created, it is a wave packet composed of different
polarization modes (geometric harmonics). Because the lattice has microstructure, these modes
travel at different speeds. As they propagate, they drift out of phase, creating the beating pattern
we measure as oscillation.
Formalism: The equation of motion for the free microrotation field ϕ is derived from the
variation of Eq. (2):
ρJ
¨
ϕ − γ∇
2
ϕ + 4κϕ = 0 (8)
where J ∼ ℓ
2
P
is the micro-inertia density, representing the geometric size of the rotating nodes.
This is a wave equation with a dispersive mass term 4κ. The dispersion relation ω(k) depends on
the twist modulus γ. Crucially, in the FCC lattice, the twist modulus is anisotropic for different
geometric modes (Point/Line/Plane harmonics).
∆m
2
ij
∼
γ
i
− γ
j
J
(9)
The flavor state |ν
α
⟩ is a coherent superposition of these mass eigenstates. The oscillation proba-
bility is governed by the beat frequency determined by the anisotropy of γ.
V. CHIRALITY AND DIRAC NATURE
Narrative: Why are neutrinos left-handed? In our model, weak interactions happen at
the ”corners” of the vacuum structure (triangular faces). These corners are geometrically frus-
trated—they have a built-in twist (C
3
symmetry). When a neutrino is created here, the lattice
forces it to twist in the same direction as the local geometry. A ”right-handed” neutrino would be
a twist against the grain, which the lattice forbids. Also, a twist is topologically different from an
anti-twist (like a screw thread). Therefore, the neutrino and anti-neutrino are distinct particles.
Formalism: The coupling term in the Lagrangian is κ(∇ × u − 2ϕ)
2
. This term converts
orbital angular momentum (∇ × u) into spin (ϕ). On a triangular lattice face, the curl operator
∇×u breaks parity due to the C
3
frustration. This parity breaking selects a single chirality for the
5
generated field ϕ
L
. Since the field ϕ is a vector quantity in the Cosserat continuum, ϕ (particle)
and −ϕ (antiparticle) are distinct states. Thus, we predict the neutrino is a **Dirac Fermion**:
ψ
ν
=
¯
ψ
ν
(10)
This precludes neutrinoless double beta decay [12].
VI. CONCLUSION
Narrative: We have reformulated the neutrino problem using the rigorous mechanics of
Cosserat lattice elasticity. By identifying the neutrino with the microrotation field ϕ and the
electron with the displacement field u, we explain the mass hierarchy not as an arbitrary param-
eter choice, but as the natural ratio of the Twist Modulus to the Bulk Modulus in a saturated
solid.
Formalism: The SSM identifies the neutrino as the Unanchored Surface Defect. We have
shown this corresponds to the rotational mode of a Cosserat continuum. The mass suppression
m
ν
≈ Φ
3
m
e
and the Dirac nature of the neutrino follow directly from the Lagrangian stability
conditions of the K = 12 vacuum.
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