Topological Selection Rules:
Deriving the Cosmic Abundance Ratio (Ω
DM
/Ω
b
≈ 5.4)
via Lattice Symmetry Breaking and Doubler Heat
Raghu Kulkarni
Independent Researcher, Calabasas, CA
February 13, 2026
Abstract
The cosmic abundance of Dark Matter outweighs baryonic matter by a mass ratio of Ω
DM
/Ω
b
≈
5.4. In the Standard Model, this ratio is an empirical parameter. We propose that this ratio is
a Topological Selection Rule governed by the symmetry group of the vacuum’s nucleation
sites. Building on the Selection-Stitch Model (SSM), we identify the baryon-generating interface
as the triangular face of the K = 12 vacuum unit cell. The symmetry group of this interface
is D
3
(Order 6). We demonstrate that for a Chiral Knot (Proton) to lock into the lattice, it
must align with the unique Identity element (P = 1/6). The remaining 5 symmetries repre-
sent phase misalignments, producing unanchored “Floating” knots (Dark Matter). This yields
a fundamental number ratio of exactly 5:1. Furthermore, we resolve the mass discrepancy by
accounting for the Lattice Doubler Modes. While the anchored Proton couples to the lattice
geometry that lifts the 15 spurious fermion doublers to the cutoff, the unanchored Dark Matter
knot traps these modes. Assigning the fundamental lattice harmonic energy (λ = 17m
e
) to
each of the 15 trapped ghosts yields an internal heat correction that predicts a total abundance
ratio of Ω
DM
/Ω
b
= 5.400. This matches the Planck 2018 observation (5.36 ±0.06) to within 1σ
without fine-tuning.
1 Introduction
Precision cosmology has established that the universe is dominated by Dark Matter. Despite
decades of searching, the particle identity of Dark Matter remains unknown. Most models (WIMPs,
Axions) assume a new fundamental field and treat the abundance ratio as a result of thermal freeze-
out dynamics.
We propose an alternative: Dark Matter is a **Topological Defect** in the same vacuum lattice
that generates visible matter. In the **Selection-Stitch Model (SSM)** [1], elementary particles
are modeled as knots in a K = 12 Face-Centered Cubic (FCC) vacuum lattice.
• Visible Matter (Proton): A Trefoil Knot (3
1
). It is Chiral (handed) and Anchored to
the lattice, generating tension (Charge).
• Dark Matter: A Figure-Eight Knot (4
1
). It is Amphicheiral (mirror-symmetric) and
Floating, generating volume (Mass) but no tension (Neutral).
In this work, we derive the abundance of these two species from the **Symmetry Breaking** of
the vacuum crystallization. We show that the 5 : 1 number ratio is determined by the D
3
symmetry
of the lattice interface, while the specific mass ratio of 5.4 arises from the conservation of vacuum
fluctuation energy (Doubler Modes) within the unanchored defects.
1
2 Topological Identity of the Species
2.1 The Proton: Chiral Locking
The vacuum lattice is composed of Cuboctahedra. As derived in our companion work on the Fine
Structure Constant [2], the lattice possesses frustrated **Triangular Faces**. A stable particle
must “stitch” into this face to anchor itself. The triangular face possesses C
3
rotational symmetry.
To lock into this geometry, a knot must share this chiral symmetry.
• The **Trefoil Knot (3
1
)** is the unique prime knot with C
3
symmetry. It is chiral (distinct
from its mirror image).
• Mechanism: Its chirality allows it to thread the lattice chirality, locking it in place. The
energy cost of this lock is the mass term 9K, representing 3 strands × 3 rotational steps ×
12 neighbors [3].
2.2 Dark Matter: Relaxation to the Ground State
If a knot forms but fails to lock, it relaxes into the lowest-energy state allowed for an unanchored
loop.
• The **Figure-Eight Knot (4
1
)** is the simplest knot after the Trefoil. Crucially, it is **Am-
phicheiral** (chemically identical to its mirror image).
• Mechanism: Lacking chirality, it cannot couple to the chiral lattice face. It slips off the
anchor points and becomes a “Floating” inclusion.
3 Derivation of the Number Ratio (5:1)
We calculate the probability of a random vacuum fluctuation becoming a Locked Proton versus a
Floating Dark Matter particle. This is a problem of **Geometric Probability**.
3.1 The Interface Symmetry Group (D
3
)
The nucleation site is the Triangular Face. Its symmetry group is the **Dihedral Group D
3
**
(Order 6). The elements are:
• **1 Identity (e):** The perfect alignment.
• **2 Rotations (r, r
2
):** Rotations by 120
◦
, 240
◦
.
• **3 Reflections (s
1
, s
2
, s
3
):** Mirror flips.
3.2 The Locking Condition
Because the lattice nodes are time-ordered by the stitching sequence (t
1
, t
2
, t
3
), they are distin-
guishable. A physical knot is a directed path.
• **Identity (e):** Maps knot segment 1 to node t
1
. **Phase Match (LOCK).**
• **Rotations (r, r
2
):** Map knot segment 1 to node t
2
. While the shape matches, the causal
phase is mismatched. **Phase Mismatch (SLIP).**
2
• **Reflections (s):** The knot is Chiral. Reflections create a chirality mismatch. **SLIP.**
Result: Only 1 out of 6 configurations allows locking.
R
count
=
N
DM
N
b
=
P (Slip)
P (Lock)
=
5/6
1/6
= 5 (1)
4 Derivation of the Mass Ratio
Cosmology measures mass density Ω = N × M. The static geometric mass ratio is determined by
volume displacement (1728) vs volume + tension (1836).
M
static
=
1728
1836
≈ 0.941 (2)
Using only this static mass, the predicted abundance is 5 × 0.941 = 4.71. This is lower than the
observed 5.36. We now derive the missing energy.
4.1 The Ghost Heat Hypothesis
The **Nielsen-Ninomiya Theorem** [4] states that a lattice discretization naturally produces 2
D
fermion species.
• **Spacetime Dimension:** Consistent with the dynamical Cl(1, 3) algebra used in the SSM
[2], the vacuum is 4-dimensional.
• **Doubler Count:** The theorem predicts 2
4
= 16 species: 1 Physical Mode + 15 Doubler
Modes.
**The Selection Mechanism:**
• **Proton (Anchored):** The anchoring condition couples the particle to the specific lattice
geometry that lifts the doublers to the cutoff scale [5]. The state is purified; the ghosts are
decoupled.
• **Dark Matter (Unanchored):** The floating knot cannot resolve the lattice geometry re-
quired to cancel the doublers. It effectively “sees” the raw vacuum state. Consequently, it
**traps the vacuum fluctuation energy** of the 15 doubler modes.
4.2 Calculating the Internal Energy
What is the energy of a trapped lattice mode? The energy scale of the lattice is defined by the
**Brillouin Zone Boundary**. In an FCC lattice, the zone boundary (X-point) lies along the face
diagonal. As derived in our companion work [3], the resonant length scale along this diagonal is
the integer **Lattice Harmonic**:
λ = ⌈K
√
2⌉ = 17 (3)
(This same λ predicts the Muon mass 12 × 17m
e
and the α
−1
resonance factor 1/17). We assign
one quantum of lattice harmonic energy (λ × m
e
) to each trapped doubler mode.
E
heat
= N
doublers
× λ = 15 × 17 = 255 m
e
(4)
3
4.3 The Total Dark Mass
Adding this trapped vacuum energy to the static volume displacement:
M
DM
= V
static
+ E
heat
= 1728 + 255 = 1983 m
e
(5)
Note: This increases the effective mass of the Dark Matter particle from the geometric rest mass
(0.88 GeV) to an effective mass of ≈ 1.01 GeV.
4.4 The Final Ratio
The total abundance ratio becomes:
R
Ω
= 5 ×
M
DM
M
b
= 5 ×
1983
1836
(6)
R
Ω
= 5 × 1.080065... = 5.4003 (7)
5 Results and Comparison
We compare the derived value against the Planck 2018 data [6].
Parameter SSM Derivation Predicted Planck 2018 Match
Number Ratio |D
3
| − 1 5.00 – –
Doubler Count 2
4
− 1 15 – –
Ghost Heat 15 × 17 255 – –
Mass Ratio 1983/1836 1.080 – –
Abundance 5 × 1.080 5.400 5.36 ± 0.06 ¡ 1σ
Table 1: The SSM dark sector prediction. By accounting for the energy of the 15 trapped doubler
modes (15 ×17), the predicted abundance ratio matches the observed cosmological value with high
precision.
6 Falsifiability
This model makes specific, testable claims distinct from WIMP/Axion models:
1. **Mass:** The Dark Matter particle must have an effective mass of ≈ 1.01 GeV (1983m
e
).
2. **Internal Structure:** The particle should exhibit an internal excitation spectrum corre-
sponding to the 15 trapped doubler modes.
3. **Charge:** It must be strictly neutral (no lattice tension).
4. **Abundance:** The number density ratio of Dark Matter to Baryons must be exactly 5:1.
4
7 Conclusion
We have derived the cosmic abundance of Dark Matter (Ω
DM
/Ω
b
= 5.40) without free parameters.
The result unifies three distinct features of the vacuum: the **D
3
Symmetry** of the nucleation
site (5:1 count), the **K = 12 Geometry** of the unit cell (static mass), and the **Topology** of
the lattice fermions (15 trapped doublers). This suggests that Dark Matter acts as the energetic
reservoir for the “missing” mirror fermions of the Standard Model.
References
1. R. Kulkarni, “The Selection-Stitch Model (SSM),” Zenodo (2026).
2. R. Kulkarni, “The Geometry of Coupling: Deriving α
−1
≈ 137,” Zenodo (2026).
3. R. Kulkarni, “The Geometric Origin of Mass,” Zenodo (2026).
4. H. B. Nielsen and M. Ninomiya, “Absence of neutrinos on a lattice,” Nucl. Phys. B 185, 20
(1981).
5. R. Kulkarni, “Fermion Chirality from Non-Bipartite Topology,” Zenodo (2026).
6. Planck Collaboration, “Planck 2018 results. VI.,” Astron. Astrophys. 641, A6 (2020).
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