THE GEOMETRIC HARMONICS OF MASS:
Deriving the Standard Model Spectrum from Lattice Resonance
Raghu Kulkarni
Independent Researcher
January 22, 2026
Significance Statement
Why is the Muon exactly 207 times heavier than the electron? The Standard Model offers
no answer. We propose that the particle spectrum is a series of Geometric Harmonics on
a Cuboctahedral Vacuum (K = 12). By identifying the face diagonal (λ ≈ 17) as the
fundamental slip plane, we derive the Muon (17K) and Tau (17
2
K) as resonant modes. We
further extend this logic to the Meson sector, deriving the Kaon and Pion masses with > 99%
accuracy, and identify the Square Pyramids within the lattice as the source of Gauge Bosons.
Finally, we derive the Higgs mass (125 GeV) as the nodal information limit (K
5
).
Abstract
We extend the Selection-Stitch Model (SSM) to the Lepton, Meson, and Boson sectors.
We model elementary particles not as independent entities, but as vibrational modes of the
Cuboctahedral Lattice (K = 12) [1].
1. Leptons: We identify a secondary geometric scaling factor, λ ≈ 17, corresponding to
the lattice face diagonal (
√
2K). This maps the generations: The Muon is a Linear
Mode (≈ 204m
e
) and the Tau is a Planar Mode (≈ 3468m
e
).
2. Mesons: We bridge the gap between leptons and baryons by modeling Mesons as half-
volume defects. We derive the Kaon (966m
e
) and Pion (272m
e
) with high precision.
3. Gauge Forces: We decompose the Cuboctahedron into 8 Tetrahedra (Fermions) and
6 Square Pyramids (Bosons). We identify the square faces as the Wilson Loops
(Plaquettes) that generate gauge forces. We further derive the W and Z boson masses
as vertex harmonics of the unit cell.
4. The Higgs: We resolve the Higgs Boson mass as the saturation limit of nodal infor-
mation density (K
5
), predicting a mass of 125.27 GeV (99.8% accuracy).
1
1 The Music of the Grid
Standard Physics views particles as points floating in a void. The SSM views them as vibrations
on a crystal grid. If you pluck a guitar string, you get a fundamental note. If you press a fret,
you get a higher note (a harmonic).
We argue that the Electron is the fundamental note of the vacuum. The Muon and Tau
are simply higher geometric harmonics of the same string, vibrating in a more complex shape.
The fundamental tuning peg of our universe is the Coordination Number (K = 12) [3].
2 The Lepton Ladder: 1, 17, 289
Why are there three generations of matter? In the SSM, these correspond to the three dimen-
sions of lattice stress: Point, Line, and Plane.
2.1 The Geometric Key: The Diagonal (λ)
The Cuboctahedron has square faces. The structural weakness of a square lattice is its Face
Diagonal (the slip plane).
λ = K
√
2 = 12 × 1.414... ≈ 16.97 → 17 (1)
The integer 17 is not random; it is the geometric ”Overtone” of the lattice.
2.2 Generation 1: The Electron (m = 1)
The Electron is a Point Defect. It sits on a single node. We define its mass as Unity (1).
Figure 1: Geometric Harmonics. The Muon (Red) is a linear vibration along the lattice diagonal
(17K). The Tau is the planar vibration of the face (17
2
K).
2.3 Generation 2: The Muon (m ≈ 204)
The Muon is a Linear Mode. It represents a vibration stretching across the face diagonal of
the unit cell [2].
µ
Muon
= λ × K = 17 × 12 = 204 (2)
Observation: The measured mass is 206.77m
e
. Accuracy: 98.7%.
2
2.4 Generation 3: The Tau (m ≈ 3468)
The Tau is a Planar Mode. It represents the vibration of the entire face area (A ”Drum Head”
resonance). In geometry, Area scales as Length squared.
µ
T au
= λ
2
× K = 17
2
× 12 = 289 × 12 = 3468 (3)
Observation: The measured mass is 3477m
e
. Accuracy: 99.8%.
3 The Meson Sector: Kaon and Pion
We extend the geometric logic to Mesons (Quark-Antiquark pairs). Since Baryons (3 quarks)
occupy the full unit cell volume (K
3
), Mesons (2 quarks) physically occupy half the volume
of the lattice cage.
3.1 The Kaon (K
±
): The Diagonal Meson
The Kaon contains a Strange quark (Generation 2). Thus, it scales with the Face Diagonal
(λ = 17) rather than the lattice side (K = 12).
• Volume: V
meson
= V
bulk
/2 = 1728/2 = 864.
• Tension: 2 strands × 3 steps = 6 degrees of freedom. Tension scales with λ.
m
Kaon
=
K
3
2
+ 6λ = 864 + (6 × 17) = 864 + 102 = 966 (4)
Observation: The measured mass is 966.1m
e
(493.7 MeV). Accuracy: 99.99%.
3.2 The Pion (π
±
): The Surface Resonance
The Pion is the lightest meson (Generation 1). It represents the relaxation of the planar mode
(λ
2
).
m
P ion
= λ
2
− λ = 289 − 17 = 272 (5)
Observation: The measured mass is 273.1m
e
(139.6 MeV). Accuracy: 99.6%.
4 The Proton and Neutron
4.1 The Proton (K
3
+ 9K)
As derived in Part I, the proton is a volumetric knot.
m
p
= K
3
+ 9K = 1728 + 108 = 1836 (6)
Observation: 1836.15m
e
. Accuracy: 99.99%.
4.2 The Neutron (Composite)
The neutron is a Proton fused with an Electron to achieve neutrality. Binding them requires
overcoming the Unitary Lattice Gap (δ = 1).
m
n
= m
p
+ m
e
+ δ = 1836 + 1 + 1 = 1838 (7)
Observation: 1838.68m
e
. Accuracy: 99.96%.
3
5 The Gauge Sector: Weak Force and Stabilization
We model the heavy gauge bosons as large-scale harmonics of the proton mass (M
p
) and the
lattice capacity (K
5
).
5.1 The Weak Force (W
±
, Z
0
): Vertex Harmonics
We model the heavy weak bosons as vibrational modes of the unit cell’s vertices. A cubic node
has 8 vertices.
• W Boson (Vacancy Mode): Represents a vibration on 7 vertices (8−1), plus a unitary
charge carrier.
m
W
= (7K + 1)M
p
= 85 × 1836 = 156, 060 (8)
Observation: 157, 300 m
e
(80.37 GeV). Accuracy: 99.2%.
• Z Boson (Full Mode): Represents a vibration on the full 8 vertices (8), plus the unitary
carrier.
m
Z
= (8K + 1)M
p
= 97 × 1836 = 178, 092 (9)
Observation: 178, 450 m
e
(91.18 GeV). Accuracy: 99.8%.
5.2 Geometric Stabilization (Square Pyramids)
Pure tetrahedral lattices (K = 4) are unstable due to geometric frustration (gaps). Nature
stabilizes the vacuum by inserting Square Pyramids between the tetrahedra.
In Lattice Gauge Theory (LGT), forces are defined by the flux through a closed loop called
a Plaquette [4]. In the SSM, the Square Base of the pyramid is the physical manifestation
of this Plaquette. While Tetrahedra (Matter) create mass via twisting, the Square Pyramids
(Forces) act as Stabilizers, transmitting tension (Force) to glue the lattice together.
6 The Higgs: Nodal Saturation
The Higgs Boson is the heaviest fundamental particle. In the SSM, it represents the Maximum
Capacity of a lattice node.
6.1 Derivation
A Cuboctahedral node possesses both Volumetric (L
3
) and Surface (L
2
) complexity. The total
information capacity is the product:
I
total
= L
3
× L
2
= L
5
(10)
The mass is the full capacity of the node (K
5
) minus the vacuum polarization of a baryon pair
(2M
p
):
m
H
= K
5
− 2M
p
= 12
5
− 3672 = 248, 832 − 3672 = 245, 160 (11)
Converting to GeV (1m
e
≈ 0.000511 GeV):
m
H
≈ 245, 160 × 0.000511 ≈ 125.27 GeV (12)
Observation: 125.25 ± 0.17 GeV. The match is nearly exact.
4
7 Summary of Results
Figure 2: The Proof is in the Prediction. SSM integer predictions (Blue) track with the Standard
Model observations (Gray) across 5 orders of magnitude.
Particle SSM Formula Prediction (m
e
) Observed (m
e
) Accuracy
Electron (e
−
) 1 (Unitary) 1 1 100%
Muon (µ
−
) 17 × K 204 206.77 98.7%
Pion (π
±
) λ
2
− λ 272 273.1 99.6%
Kaon (K
±
) K
3
/2 + 6λ 966 966.1 99.99%
Tau (τ
−
) 17
2
× K 3,468 3,477 99.8%
W Boson (W
±
) (7K + 1)M
p
156,060 157,300 99.2%
Z Boson (Z
0
) (8K + 1)M
p
178,092 178,450 99.8%
Proton (p
+
) K
3
+ 9K 1,836 1,836.15 99.99%
Neutron (n
0
) p
+
+ e
−
+ 1 1,838 1,838.68 99.96%
Higgs (H
0
) K
5
− 2M
p
245,160 244,814 99.8%
Table 1: Comparison of Integer Geometric Predictions vs. Experimental Data (Complete Spec-
trum)
References
[1] Kulkarni, R. (2026). The Selection-Stitch Model (SSM): Emergent Gravity from Discrete
Geometry. Zenodo. https://doi.org/10.5281/zenodo.18138227
[2] Tiesinga, E., Mohr, P. J., Newell, D. B., & Taylor, B. N. (2021). CODATA recommended
values of the fundamental physical constants: 2018. Reviews of Modern Physics, 93(2),
025010. https://doi.org/10.1103/RevModPhys.93.025010
[3] Conway, J. H., & Sloane, N. J. A. (1999). Sphere Packings, Lattices and Groups. Springer
New York. https://link.springer.com/book/10.1007/978-1-4757-2016-7
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