THE TOPOLOGY OF DARK MATTER:
Deriving the Cosmic Mass Abundance Ratio (Ω
DM
/Ω
b
≈ 5.4)
Raghu Kulkarni
Independent Researcher
raghu@idrive.com
February 16, 2026
Significance Statement
Cosmological observations indicate that Dark Matter is approximately 5.4 times more abundant
than baryonic matter. The Selection-Stitch Model (SSM) derives this ratio geometrically. Identi-
fying the proton as a chiral Trefoil Knot (3
1
) and Dark Matter as an amphicheiral Figure-Eight
Knot (4
1
), we calculate their formation probabilities (5:1) and effective masses (2016 vs 1836). The
resulting mass density ratio of 5.49 matches Planck satellite data with 1.6% precision, suggesting
Dark Matter is simply ”unanchored” knot topology in the vacuum lattice.
Abstract
We propose that Dark Matter consists of Figure-Eight Knots (4
1
) in the Cuboctahedral
Vacuum (K = 12). Unlike the proton (3
1
), which is chiral and topologically locked (generating
charge), the Figure-Eight knot is amphicheiral and unanchored (generating no charge).
1. Abundance (5:1): Combinatorial analysis of lattice pathways shows that 3-step loops
(Protons) and 4-step loops (Dark Matter) form in a strict 1:5 ratio during nucleosynthesis.
2. Mass (2016): We derive the Dark Matter mass as the sum of its static volume (1728)
and its chiral oscillation energy (288).
This yields a cosmic mass ratio of Ω
DM
/Ω
b
≈ 5.49, resolving the ”missing mass” problem
without invoking arbitrary non-baryonic particles.
1 The Geometric Identity of Dark Matter
In the SSM, the vacuum is a discrete tensor network with coordination number K = 12. Elementary
particles are topological defects (knots) in this lattice [1].
• The Proton (3
1
): The simplest non-trivial knot. It is Chiral (distinct Left/Right forms).
This chirality forces a geometric lock with the lattice, creating static tension (Electric Charge).
• Dark Matter (4
1
): The next simplest knot. It is Amphicheiral (topologically identical
to its mirror image). Because it is its own mirror image, it cannot lock into a chiral lat-
tice. It ”floats” through the vacuum, creating no static tension (Zero Charge) but retaining
volumetric inertia (Mass).
1
Figure 1: The Dark Matter Topology. The Figure-Eight Knot (4
1
) (Blue) is amphicheiral.
It continuously tunnels between Left and Right configurations inside the vacuum cage, creating
dynamic viscosity without static tension.
2 Mass as a Dynamic Path Integral
While the proton has a fixed mass of 1836 (1728 Volume + 108 Tension), Dark Matter has a
dynamic mass component due to its lack of anchoring.
2.1 The Base Volumetric Mass (K
3
)
Like the proton, the Dark Matter candidate is a strictly 3-dimensional topological defect. To define
its crossing number in the vacuum lattice, the defect must establish coherence along three principal
axes. Thus, its base rest mass is defined by the same 3-hop path integral sum derived for the
proton:
M
base
=
N=3
X
paths
≈ 12
3
= 1728 (1)
2.2 The Chiral Oscillation Correction (2K
2
)
Unlike the proton, which is topologically locked, the Figure-Eight knot is metastable. It continu-
ously tunnels between its Left-Handed (L) and Right-Handed (R) enantiomers [3].
2
• Mechanism: To flip chirality (L ↔ R), the knot must effectively invert its surface projection.
• Energy Cost: This tunneling is a vacuum fluctuation that energizes the 2nd Topological
Shell (Surface Area, K
2
) for both states simultaneously.
The energy cost of this chiral superposition is exactly two units of surface area:
E
osc
= 2 × (12 × 12) = 2 × 144 = 288 (2)
2.3 Distinguishing Tension from Viscosity
A critical distinction must be made:
• Proton (Tension): Static vector stress. Generates Charge (108).
• Dark Matter (Viscosity): Dynamic scalar stress. Generates Mass (288).
Because the oscillation is scalar (pressure) rather than vector (polarization), Dark Matter interacts
gravitationally but not electromagnetically.
2.4 Total Effective Mass
The total mass is the sum of the static volume and the dynamic oscillation energy:
M
DM
= 1728 + 288 = 2016 (3)
In physical units (1m
e
≈ 0.511 MeV), this corresponds to a particle mass of approximately 1.03
GeV.
3 Deriving the Cosmic Abundance Ratio
3.1 The Formation Ratio (5:1)
During the ”Geometric Nucleosynthesis” phase, the vacuum lattice crystallized. The probability of
forming a defect is driven by the combinatorics of closed loops in the K = 12 lattice [1].
• 3-Step Lo ops (Protons): There are 8 fundamental triangular faces on a Cuboctahedron.
• 4-Step Loops (Dark Matter): There are 6 fundamental square faces, but each square sup-
ports multiple crossing permutations (diagonals). Detailed path counting yields a formation
favorability of exactly 5:1 for the 4-step geometry.
Thus, for every 1 Proton formed, 5 Dark Matter knots are formed.
N
DM
/N
p
= 5 (4)
3.2 The Mass Density Ratio (Ω
DM
/Ω
b
)
We can now calculate the total mass density ratio of the universe:
Ratio =
N
DM
× M
DM
N
p
× M
p
(5)
Substituting our derived integers (M
p
= 1836, M
DM
= 2016, N = 5):
Ratio = 5 ×
2016
1836
= 5 × 1.098 ≈ 5.49 (6)
3
4 Conclusion
The Planck satellite measures the Dark Matter to Baryon ratio at Ω
c
h
2
/Ω
b
h
2
≈ 5.4 [2]. The SSM
derivation (5.49) matches this observation with remarkable accuracy.
• Why is it Dark? Because the knot is amphicheiral (4
1
), it cannot lock to the lattice to
generate static tension (Charge).
• Why is it Heavy? Because it possesses a ”Chiral Oscillation” term (288) that adds dynamic
viscosity to its base volume.
References
[1] Kulkarni, R. (2026). The Selection-Stitch Model (SSM): Emergent Gravity from Discrete Ge-
ometry. Zenodo.
[2] Planck Collaboration. (2020). P lanck 2018 results. VI. Cosmological parameters. Astronomy &
Astrophysics.
[3] Rolfsen, D. (1976). Knots and Links. Publish or Perish, Berkeley.
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