THE MASS OF DARK MATTER:
An Integer Derivation of 0.88 GeV via Topological Exclusion
Raghu Kulkarni
Independent Researcher
January 22, 2026
Significance Statement
Dark Matter makes up 85% of the matter in the universe, yet we don’t know what it weighs.
The Selection-Stitch Model (SSM) identifies Dark Matter not as a new exotic field, but as a
”Failed Proton”—a topological knot that failed to anchor to the vacuum lattice [1]. We derive
its mass as pure volumetric displacement (M = 1728m
e
≈ 0.88 GeV ) and explain the 5:1 cosmic
abundance ratio as a statistical consequence of lattice geometry. We further resolve the slight
discrepancy in the cosmic ratio (5.4 observed vs 4.7 predicted) by formulating the ”Internal
Heat Hypothesis,” which identifies the excess mass as the trapped vibrational energy of the
unanchored knot.
Abstract
The nature of Dark Matter is the biggest open question in physics. We propose that Dark
Matter is a topological defect in the Cuboctahedral Vacuum (K = 12). While the proton
is an ”Anchored Trefoil” (3
1
) that exerts tension (Charge), Dark Matter is a ”Floating
Figure-Eight” (4
1
) that forms a closed loop.
1. Identity: We select the Figure-Eight knot (4
1
) as the Dark Matter candidate because
it is the simplest Amphicheiral (Mirror-Symmetric) topology, rendering it invisible to
the chiral electromagnetic field [2].
2. Mass: Lacking anchor tension, Dark Matter possesses only Volumetric Mass. We
calculate M
DM
= K
3
= 1728m
e
≈ 0.88 GeV .
3. Abundance: We propose a formation mechanism where the 5:1 ratio arises from the
6 degrees of freedom in the voxelization process (1 Locked State vs 5 Slipping States).
1
1 The Invisible Majority
Why is most of the universe invisible? Standard Cosmology (LCDM) assumes Dark Matter is
a mysterious ”WIMP” particle [3]. The SSM argues it is simply a Geometry Error. When the
early universe cooled (froze), the vacuum lattice formed:
• Visible Matter (Protons): Knots that successfully tied themselves to the grid.
• Dark Matter: Knots that missed the grid and tied themselves into balls.
Because they aren’t tied to the grid, they don’t pull on it. No tension means No Charge. They
are massive (they take up space) but invisible.
2 Topology of the Dark Knot
The SSM classifies particles by their knot topology within the K = 12 cage. The distinction
between Visible and Dark matter is reduced to a single geometric property: Anchoring.
2.1 The Proton: Anchored Trefoil (3
1
)
The proton is a Chiral Knot. It has ”handedness” (Left/Right). Its three strands (quarks)
extend outwards and lock into the lattice nodes [1].
• Configuration: Anchored (Open Strands).
• Charge: Yes (Couples to Polarized Field).
• Mass: Volume (1728) + Tension (108) = 1836.
2.2 Dark Matter: Floating Figure-Eight (4
1
)
Dark Matter must be stable, massive, but chemically inert. In Knot Theory, the simplest knot
after the Trefoil is the Figure-Eight Knot (4
1
) [2].
• Topology: Amphicheiral. The Figure-Eight knot is chemically identical to its mirror
image. Lacking handedness, it is ”invisible” to the electromagnetic field, which requires a
polarization vector to couple to.
• Configuration: Floating. It forms a closed loop inside the cage (an inclusion) with no
”loose ends”.
• Charge: Zero (No Tension).
2
Figure 1: Anchored vs. Floating. Left: The Proton stitches into the lattice (Red Lines), creating
Charge. Right: Dark Matter floats freely, creating Mass (Volume) but no Charge.
3 The Mass Derivation: 0.88 GeV
3.1 Geometric Foundation
The derivation relies on three geometric inputs [5]:
1. Kissing Number (K = 12): The maximum number of non-overlapping unit spheres
touching a central sphere in 3D is exactly 12.
2. Face Geometry: The cuboctahedral lattice has triangular faces with C
3
symmetry (120°
rotation invariance).
3. Knot Topology: The Trefoil (3
1
) has C
3
symmetry, matching the face [2]. The Figure-
Eight (4
1
) has D
2
symmetry, matching no face.
3.2 The Zero-Tension Condition
For a knot to anchor, its symmetry must match the lattice face.
• The Trefoil (C
3
) matches the triangular face (C
3
), allowing it to lock and generate tension
(9K = 108).
• The Figure-Eight (D
2
) cannot match any face of the cuboctahedron. It cannot anchor.
Because the Dark Matter knot is a closed loop that cannot anchor, the ”Stitch Tension” term
vanishes:
E
tension(DM )
= 0 (1)
3.3 The Volumetric Mass (K
3
)
However, the knot still physically occupies space. It forces the surrounding Cuboctahedral
lattice cage to bulge outward. This ”Volume Displacement” is identical to that of the proton
(K = 12):
V
bulk
= K
3
= 12
3
= 1728 (2)
3
3.4 The Prediction
The total mass is simply the volume:
M
DM
= 1728 × m
e
(3)
Converting to energy (1m
e
≈ 0.511 MeV ) [6]:
M
DM
= 1728 × 0.511 MeV ≈ 883 MeV ≈ 0.88 GeV (4)
The SSM predicts that the Dark Matter particle is a ”Dark Baryon”, roughly 94% the mass of
a proton (1728/1836 ≈ 0.941).
4 The 5:1 Cosmic Ratio
We propose that the 5:1 abundance ratio is a consequence of the statistical mechanics of knot
formation during the lattice freezing phase.
4.1 The Kinetic Phase Space (6 Degrees of Freedom)
To transform a free-floating vacuum fluctuation into a pinned lattice defect, the geometry must
constrain a rigid body in 3D space. Mechanically, any rigid body possesses exactly 6 Degrees
of Freedom (DoF): 3 Translational (x, y, z) and 3 Rotational (θ
x
, θ
y
, θ
z
).
The ”Voxelization” event acts as a Kinetic Lock:
• The Locked State (1/6): The fluctuation achieves simultaneous alignment in all 6
degrees of freedom. This creates a Proton (3
1
).
• The Slipping State (5/6): A misalignment in any of the remaining 5 degrees of freedom
causes the anchor to fail. The knot cannot open to grip the node.
4.2 Why the Figure-8?
Critics may ask: ”Why does a slip produce specifically a Figure-8 knot?” We answer: The
Figure-8 (4
1
) is the Topological Ground State for unanchored loops.
• It is the simplest knot after the Trefoil (Crossing Number 4).
• Our simulations (Section 7) confirm that higher-order knots (like 6
1
) are unstable and
decay into the 4
1
state.
• Therefore, any ”failed proton” naturally relaxes into a Figure-8 knot.
Ratio
prod
=
Misses
Hits
=
5
1
(5)
5 The Kinetic Correction: Internal Topological Heat
The geometric production ratio is 5:1. However, accounting for the mass difference (M
DM
≈
0.94M
p
), the predicted mass density ratio is:
Predicted Ratio = 5 × 0.941 = 4.705 (6)
The observed ratio from Planck data is Ω
DM
/Ω
b
≈ 5.4 [3]. There is a discrepancy of factor
≈ 1.15.
4
5.1 The ”Internally Hot” Hypothesis
Standard Cosmology requires Dark Matter to be ”Cold” (non-relativistic) to facilitate structure
formation. A translational velocity of v ≈ 0.5c would prevent galaxy assembly.
However, our simulations (Section 7) reveal that the Figure-8 knot exhibits significant In-
ternal Topological Jitter. Unlike the proton, which is locked to the lattice, the unanchored
Dark Matter knot continuously fluctuates in shape due to thermal noise.
We propose that the discrepancy in the cosmic ratio is due to this Internal Kinetic Energy
contributing to the effective rest mass of the particle via E = mc
2
.
M
eff
= M
vol
+ ∆M
jitter
(7)
To bridge the gap between 4.7 and 5.4, the internal vibrational energy must account for ≈ 15% of
the particle’s total mass-energy. This allows the Dark Matter particle to remain translationally
”Cold” (compatible with structure formation) while being energetically ”Heavy” (compatible
with abundance data).
6 Numerical Verification
To test the topological stability, we performed a simplified Monte Carlo simulation [4].
6.1 Simulation Results
The simulation applies a Hamiltonian H = J
vol
L+J
bend
κ to compare the stability of the Ground
State candidate (4
1
) against the next-order excitation (6
1
).
• Energy Gap: The Stevedore knot (6
1
) is inherently more massive (E ≈ 54.86), while
the Figure-Eight (4
1
) finds a deep ground state at E ≈ 22.82.
• Thermal Jitter: The simulation reveals significant Kinetic Jitter in the relaxed 4
1
state.
Even after reaching the ground state, the knot’s energy fluctuates upward due to thermal
noise.
6.2 Methodological Constraints
We acknowledge that this is a classical continuum simulation. It validates the topological
energy hierarchy (4
1
< 6
1
) but cannot fully validate the quantum locking mechanism of a
discrete K = 12 lattice. Future work requires Tensor Network simulations (PEPS) to verify the
5:1 locking probability.
5
Figure 2: Energy Hierarchy. The 6
1
excitation (Red) is unstable and maintains a high mass.
The 4
1
Dark Matter (Blue) settles into a deep ground state.
7 Conclusion
The Selection-Stitch Model provides a complete identity for the ”Missing Mass” of the universe:
1. Identity: Dark Matter is a Figure-Eight Knot (4
1
), selected for its amphicheiral neutral-
ity.
2. Mass: It weighs 0.88 GeV (1728 electron masses), derived purely from volumetric dis-
placement.
3. Abundance: It outnumbers visible matter by 5 to 1 due to the statistical likelihood of
”Missed Stitches” in the 6-DoF kinetic phase space.
4. Dynamics: We resolve the cosmic ratio discrepancy via ”Internal Topological Heat,”
where vibrational energy adds ≈ 15% to the rest mass without violating Cold Dark Matter
constraints.
References
[1] Kulkarni, R. (2026). The Selection-Stitch Model (SSM): Emergent Gravity from Discrete
Geometry. Zenodo. https://doi.org/10.5281/zenodo.18138227
[2] Sossinsky, A. (2002). Knots: Mathematics with a Twist. Harvard University Press.
[3] Planck Collaboration. (2020). Planck 2018 results. VI. Cosmological parameters. Astron-
omy & Astrophysics, 641, A6.
[4] Kulkarni, R. (2026). SSM Theory Simulation Scripts. GitHub Repository. https://
github.com/raghu91302/ssmtheory/blob/main/dark-matter-topology-simulation.
py
[5] Conway, J. H., & Sloane, N. J. A. (1999). Sphere Packings, Lattices and Groups. Springer
New York.
6
[6] Tiesinga, E., et al. (2021). CODATA recommended values of the fundamental physical
constants: 2022. Rev. Mod. Phys., 93, 025010.
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