THE MASS OF DARK MATTER:
An Integer Derivation of 1.03 GeV via Chiral Oscillation in a Discrete Vacuum
Raghu Kulkarni
Independent Researcher
raghu@idrive.com
February 16, 2026
Significance Statement
Dark Matter constitutes 85% of the matter in the universe, yet its mass remains unknown. 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. We derive its base mass
as pure volumetric displacement (M
vol
= 1728m
e
≈ 0.88 GeV). Crucially, we resolve the cosmic
abundance ratio by deriving a specific ”Chiral Oscillation Correction.” By identifying the
Figure-Eight knot (4
1
) as an amphicheiral structure that tunnels between two mirror states, we
calculate the oscillation cost as exactly two units of surface energy (2K
2
). This yields a corrected
mass of M
tot
= 2016m
e
, predicting a cosmic ratio of Ω
DM
/Ω
b
≈ 5.49, matching Planck observations
to within 1.6%.
Abstract
The nature of Dark Matter is the central puzzle of modern cosmology. 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, unanchored 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. Cold Mass (M
v ol
): Lacking anchor tension, Dark Matter possesses only Volumetric Mass.
We calculate M
v ol
= K
3
= 1728m
e
≈ 0.88 GeV.
3. Corrected Mass (M
tot
): Accounting for the vacuum energy required for chiral tunneling,
we add a surface term 2K
2
, yielding M
tot
= 2016m
e
≈ 1.03 GeV.
4. Abundance: We derive the 5:1 formation ratio from the 6 degrees of freedom in the
voxelization process (1 Locked State vs 5 Slipping States).
1 The Invisible Majority
Why is most of the universe invisible? Standard Cosmology (ΛCDM) assumes Dark Matter is a
mysterious ”WIMP” particle [1]. The SSM argues it is simply a Geometry Error. When the early
universe cooled (”froze”), the vacuum lattice formed [2]:
• Visible Matter (Protons): Knots that successfully tied themselves to the grid.
1
• Dark Matter: Knots that missed the grid and tied themselves into closed loops.
Because they are not anchored to the grid, they do not exert tension. 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 [3].
• Configuration: Anchored (Open Strands).
• Charge: Yes (Couples to Polarized Field via Tension).
• 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
) [4].
• Topology: Amphicheiral. The Figure-Eight knot is topologically 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).
3 The Cold 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 touch-
ing 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. The Figure-Eight
(4
1
) has D
2
symmetry, matching no face.
2
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)
This is the ”Cold” mass, simply the volume of displacement. Converting to energy (1m
e
≈ 0.511
MeV) [6]:
M
cold
= 1728 × 0.511 MeV ≈ 883 MeV ≈ 0.88 GeV (3)
4 The 5:1 Formation 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 and 3 Rotational.
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?
The Figure-8 (4
1
) is the Topological Ground State for unanchored loops. Simulations 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
form
=
Misses
Hits
=
5
1
(4)
3
5 The Corrected Mass: Chiral Oscillation Hypothesis
While the formation ratio is 5:1, the Mass Density ratio depends on the weights of the particles.
Using only the cold mass (1728), the predicted ratio is ≈ 4.7. However, Planck observations show
Ω
DM
/Ω
b
≈ 5.4 [1]. We resolve this by accounting for the Chiral Oscillation Energy.
5.1 The Cost of Amphicheirality
The Figure-Eight knot is unique because it is amphicheiral—it is its own mirror image. Unlike the
chiral Proton, which is locked in one state, the Dark Matter knot can ”tunnel” between Left-Handed
(L) and Right-Handed (R) configurations. This tunneling is a continuous oscillation.
5.2 Deriving the Oscillation Energy (2K
2
)
To flip chirality, the knot must invert its surface projection. In the SSM lattice (K = 12), energy
scales with geometry:
• Volumetric Energy (Bulk): Scales as K
3
= 1728.
• Surface Energy (Boundary): Scales as K
2
= 144.
For a knot to exist as a superposition of two chiral states (L ↔ R), it must energize the
boundary surface of both configurations simultaneously. The energy cost of this duality is exactly
two units of surface area:
E
osc
= 2 × K
2
= 2 × 144 = 288 (5)
5.3 The Total Corrected Mass (M
tot
)
We add this oscillation energy to the base volumetric mass:
M
tot
= V
bulk
+ E
osc
= 1728 + 288 = 2016 (6)
Converting to physical units (1m
e
≈ 0.511 MeV):
M
phys
≈ 2016 × 0.511 MeV ≈ 1.03 GeV (7)
5.4 Verification against Cosmic Ratio
We now compare the predicted mass density ratio to the observed Planck value.
Ratio
pred
= Count Ratio ×
M
DM
M
proton
(8)
Substituting the derived integers (M
proton
= 1836, M
DM
= 2016):
Ratio
pred
= 5 ×
2016
1836
= 5 × 1.098 ≈ 5.49 (9)
Result: This derived value of 5.49 matches the Planck observation (≈ 5.4) with an error of only
1.6%. This suggests that Dark Matter is slightly heavier than the proton due to its internal chiral
oscillation.
4
6 Numerical Verification
To test the topological stability, we performed a simplified Monte Carlo simulation [7].
6.1 Simulation Results
The simulation applies a Hamiltonian H = J
vol
L + J
bend
K 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, while the Figure-Eight
(4
1
) finds a deep ground state.
• Thermal Jitter: The simulation reveals significant Kinetic Jitter in the relaxed 4
1
state.
We identify this jitter with the Chiral Oscillation (2K
2
) derived in Section 5.
7 Conclusion
The Selection-Stitch Model provides a complete identity for the ”Missing Mass”:
1. Identity: Dark Matter is a Figure-Eight Knot (4
1
), selected for its amphicheiral neutrality.
2. Cold Mass: 1728m
e
(0.88 GeV), derived from volumetric displacement (K
3
).
3. Corrected Mass: 2016m
e
(1.03 GeV), derived by adding the Chiral Oscillation cost (2K
2
).
4. Abundance: It outnumbers visible matter by 5 to 1 due to the statistical likelihood of
”Missed Stitches” (6 DoF).
5. Agreement: The resulting cosmic mass ratio of 5.49 naturally explains the Planck obser-
vations without requiring arbitrary dark energy parameters.
References
[1] Planck Collaboration. (2020). Planck 2018 results. VI. Cosmological parameters. Astronomy &
Astrophysics, 641, A6.
[2] Kulkarni, R. (2026). The Selection-Stitch Model (SSM): Emergent Gravity from Discrete Ge-
ometry. Zenodo.
[3] Kulkarni, R. (2026). The Geometric Origin of Mass: A Topological Derivation of the Proton-
Electron Ratio. Zenodo.
[4] Sossinsky, A. (2002). Knots: Mathematics with a Twist. Harvard University Press.
[5] Conway, J. H., & Sloane, N. J. A. (1999). Sphere Packings, Lattices and Groups. Springer New
York.
[6] Tiesinga, E., et al. (2021). CODATA recommended values of the fundamental physical constants:
2022. Rev. Mod. Phys., 93, 025010.
[7] Kulkarni, R. (2026). SSM Theory Simulation Scripts. GitHub Repository.
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