The Chiral Filter:
Deriving the Baryon Asymmetry (η ∼ 10
−10
) from
Geometric Frustration in the Vacuum Lattice
Raghu Kulkarni
Independent Researcher, Calabasas, CA
February 13, 2026
Abstract
The observed universe is dominated by matter, with a baryon-to-photon ratio of η ≈
6 × 10
−10
. The Standard Model requires an arbitrary CP-violation parameter to explain this
asymmetry, but known sources are insufficient by orders of magnitude. We propose that this
ratio is a Geometric Invariant of the vacuum’s crystallization. In the Selection-Stitch Model
(SSM), the vacuum is a Face-Centered Cubic (FCC) lattice (K = 12). Due to the Non-
Bipartite topology of the lattice (triangular faces), Right-Handed knots (Antimatter) are geo-
metrically frustrated and cannot anchor. This acts as a “Chiral Filter,” suppressing primordial
antimatter formation entirely. We derive the abundance of the surviving Matter (Protons) as
the statistical probability of a random fluctuation successfully locking into the lattice. Using the
proton’s geometric complexity (N
dof
= 9) and the lattice coordination (K = 12), we calculate
the formation probability as P ≈ 3 × (1/12)
9
≈ 5.8 × 10
−10
. This matches the Planck 2018
observation (6.10 ± 0.04 × 10
−10
) within 5%, suggesting that the baryon asymmetry is simply
the survival rate of knots during the Big Bang.
1 Introduction
One of the greatest mysteries in cosmology is the **Baryon Asymmetry**: the fact that the
universe contains matter but almost no antimatter. Standard Big Bang cosmology predicts equal
amounts of both (η = 0), requiring complete annihilation. The observed residual matter (η =
n
b
/n
γ
≈ 6 × 10
−10
) [1] implies that for every billion antimatter particles, one billion and one
matter particles were created.
We propose a geometric solution based on the **Selection-Stitch Model (SSM)** [2]. We posit
that the asymmetry arises from the **Crystallization** of the vacuum.
1. Total Suppression: The vacuum lattice (K = 12) is chiral (Non-Bipartite). It geometri-
cally allows Left-Handed knots (Matter) to anchor but physically rejects Right-Handed knots
(Antimatter) due to topological frustration.
2. Formation Rate: The abundance of matter is determined by the probability of a random
fluctuation satisfying the ”Locking Condition” of the lattice against the thermal photon bath.
2 The Chiral Filter Mechanism
In our companion paper Fermion Chirality [3], we established that the FCC lattice is composed of
**Triangular Faces**.
1
• Geometric Frustration: A triangle is a non-bipartite graph (odd cycle length). It cannot
support alternating charges (+/−) without conflict.
• Chiral Selection: This topology breaks the symmetry between Left and Right. A “Left-
Handed” twist can align with the lattice curvature (constructive interference), while a “Right-
Handed” twist fights against it (destructive interference).
• Result: During the high-temperature “Freezing” phase, Right-Handed knots (Antiprotons)
could not anchor. They slipped off the lattice and dissolved back into the thermal bath.
Left-Handed knots (Protons) anchored successfully. Consequently, η represents the total
production rate of matter, not a net excess over antimatter.
3 Deriving the Ratio (η)
The Baryon-to-Photon ratio η represents the efficiency of this locking process.
η =
n
b
n
γ
≈
Successful Knots
Total Fluctuations
(1)
3.1 The Lattice Probability (P
step
= 1/12)
During the high-temperature “Foam Phase” (pre-crystallization), the local geometry is not yet
fixed to a single face. A fluctuation explores the full coordination shell of the nascent lattice nodes.
• Coordination: The FCC lattice has K = 12 nearest neighbors.
• Selection: At each step of knot formation, the fluctuation must select the specific vector
that aligns with the final crystal structure out of the 12 available options. Thus, P
step
= 1/12.
3.2 The Knot Complexity (N
dof
= 9)
In our derivation of the Proton Mass [4], we established the geometric cost of a proton. A proton
is a 3-strand braid (quarks). To lock a strand into a triangular face (C
3
symmetry), it requires 3
discrete rotational steps (120
◦
each).
N
dof
= (3 Strands) × (3 Steps/Strand) = 9 (2)
Independence Argument: The relevant timescale for locking is the fundamental lattice update
time τ (Planck time). Since τ is the minimum quantum of time, there is no sub-tick sequential
process. All 9 geometric constraints must be satisfied simultaneously within a single τ. Therefore,
the probabilities multiply.
3.3 The Calculation
The probability of a random fluctuation simultaneously satisfying 9 specific geometric constraints
on a K = 12 grid is:
P
lock
=
1
K
N
dof
=
1
12
9
≈ 1.93 × 10
−10
(3)
2
3.4 The Generation Factor (N
g
= 3)
The vacuum supports **3 Generations** of fermions (Electron, Muon, Tau families) [5]. During
the high-energy freeze-out, knots form in all three channels. Although heavy baryons (Charm, Top,
Bottom) are unstable today, they were produced during the Big Bang and subsequently decayed
into the ground state (Protons/Neutrons). Therefore, all three generation channels contribute to
the final baryon yield.
η
pred
= N
g
× P
lock
= 3 ×
1
12
9
(4)
η
pred
= 3 × 1.93 × 10
−10
≈ 5.81 × 10
−10
(5)
4 Consistency Checks
4.1 Comparison with Dark Matter
In our companion paper on Dark Matter [7], we derived that for every stable baryon (Locked),
there are 5 unanchored knots (Dark Matter). This implies n
DM
≈ 5 × n
b
. Comparing this to the
photon density:
n
DM
n
γ
= 5 × η ≈ 3 × 10
−9
(6)
This confirms that n
γ
≫ n
DM
≫ n
b
. The photon bath dominates the number density by 9 orders
of magnitude, validating our approximation that the denominator is effectively the photon count.
4.2 Comparison with Observation
We compare the prediction with the Planck 2018 data derived from Ω
b
h
2
= 0.0224 [1].
Parameter Symbol SSM Derivation Predicted Planck 2018 Error
Baryon Ratio η 3 × (1/12)
9
5.81 × 10
−10
6.10 ± 0.04 × 10
−10
4.8%
Table 1: The Geometric Derivation of the Baryon Asymmetry. The predicted value matches
observation to within 5%.
5 Discussion: Symmetry of Laws vs. Inventory
This result clarifies a crucial distinction in the SSM framework regarding CP Violation. In our Speed
of Light paper [6], we proved that the Laws of Physics (Strong Force) conserve CP (θ
QCD
≈ 0)
because the Centrosymmetry of the unit cell forces odd field moments to vanish. Here, we show
that the Inventory of the Universe (Matter vs. Antimatter) violates CP (η = 0) because the
Non-Bipartite Topology of the faces acts as a chiral filter. The universe’s rules are symmetric,
but its contents are filtered.
6 Conclusion
We have derived the Baryon-to-Photon ratio from the geometry of the vacuum. The Non-Bipartite
lattice acts as a Chiral Filter, preventing antimatter formation. The abundance of matter (≈
3
6 × 10
−10
) is determined by the statistical likelihood of forming a 9-step knot on a 12-neighbor
grid across 3 generations. This unifies the origin of the Baryon Asymmetry with the origin of the
Proton Mass (N = 9) and the Mass Spectrum (N
g
= 3).
References
1. Planck Collaboration, “Planck 2018 results. VI.,” Astron. Astrophys. 641, A6 (2020).
2. R. Kulkarni, “The Selection-Stitch Model (SSM),” Zenodo (2026).
3. R. Kulkarni, “Fermion Chirality from Non-Bipartite Topology,” Zenodo (2026).
4. R. Kulkarni, “The Geometric Origin of Mass,” Zenodo (2026).
5. R. Kulkarni, “The Geometric Harmonics of Mass,” Zenodo (2026).
6. R. Kulkarni, “Geometric Renormalization... and Strong CP,” Zenodo (2026).
7. R. Kulkarni, “The Mass of Dark Matter,” Zenodo (2026).
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