The Geometric Harmonics of Mass: Precise
Constraints on the Standard Model and
Dark Sector from Lattice Resonance
Raghu Kulkarni
Independent Researcher, Calabasas, CA
February 18, 2026
Abstract
Why is the Muon exactly 207 times heavier than the electron? The Standard
Model treats particle masses as arbitrary coupling constants, offering no funda-
mental explanation for the hierarchy of generations. We propose that the particle
spectrum is not arbitrary, but is a series of Geometric Harmonic s vibrating on
a saturated Face-Centered Cubic (FCC) 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 of the vacuum structure. 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 resolve the Higgs Boson mass (125 GeV) as the nodal information sat-
uration limit (K
5
), predicting a mass of 125.27 GeV with 99.8% accuracy. While
these integer relations are precise, they rely on sector-specific topological ansatzes.
Furthermore, the absence of corresponding signals in interim LHC Run 3 data
suggests that if these geometric resonances exist, they must be feebly interacting
or broad-width states. As Run 3 concludes in June 2026, we propose three specific
Dark Sector targets for the final dataset analysis: a 10.6 GeV flux resonance, a 30.1
GeV dark lepton, and a 136 GeV scalar.
1 Introduction: The Music of the Grid
Standard physics views elementary particles as point-like excitations floating in a contin-
uous void. The Selection-Stitch Model (SSM) takes a fundamentally different view: it
models particles as vibrations on a discrete, crystalline grid [1].
Consider the analogy of a guitar string. If you pluck the open string, you produce
a fundamental note. If you press down on a fret, you shorten the vibrating length and
produce a higher note—a harmonic. We argue that the Electron represents the funda-
mental note of the vacuum lattice. The heavy generations—the Muon and the Tau—are
not separate fundamental entities, but simply higher geometric harmonics of the same
string, vibrating in more complex topological shapes [4].
The fundamental ”tuning peg” of our universe is the Coordination Number (K =
12) of the Cuboctahedral unit cell. In this paper, we demonstrate how the entire Standard
Model mass spectrum emerges from simple integer resonances of this single geometric
parameter.
1
2 The Lepton Ladder: Geometric Symmetry Break-
ing
A central mystery of the Standard Model is the existence of three generations of matter.
In the SSM, these correspond to the three dimensions of lattice stress: Point (0D), Line
(1D), and Plane (2D).
Figure 1: The Geometric Origin of Generations. The Muon (Red) arises from linear
tension along the slip plane (λ). The Tau (Blue) arises from planar resonance of the
lattice face area (λ
2
).
2.1 The Geometric Key: The Slip Plane (λ ≈ 17)
The Cuboctahedron unit cell is composed of triangular and square faces. The structural
weakness of any square lattice is its Face Diagonal, known in crystallography as the ”slip
plane”. The length of this diagonal scales with the coordination number K = 12:
λ
raw
= K
√
2 = 12 × 1.414... ≈ 16.97 (1)
In a discrete lattice, a vibration cannot exist on a fractional node. The system must
”snap” to the nearest integer topology to maintain a standing wave. This forces the
quantization condition:
λ → ⌈16.97⌋ = 17 (2)
The integer 17 is the ”Topological Lock” of the lattice, representing the quantized diagonal
tension required to shear the unit cell [2].
2.2 Generation 1: The Electron (m = 1)
The Electron is the fundamental unit. It represents a Point Defect sitting on a single
node. We define its mass as Unity (1).
2
2.3 Generation 2: The Muon (Linear Resonance)
The Muon is a Linear Mode. It represents a vibration stretching across the face diagonal
of the unit cell (λ). Its mass is defined by the product of the slip plane (λ) and the lattice
connectivity (K):
µ
Muon
= λ × K = 17 × 12 = 204 (3)
Observation: The measured mass is 206.77m
e
. The slight discrepancy (1.3%) is at-
tributed to QED radiative corrections (vacuum polarization) added to this ”bare” geo-
metric mass [4].
2.4 Generation 3: The Tau (Planar Resonance)
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 (λ
2
):
µ
T au
= λ
2
× K = 17
2
× 12 = 289 × 12 = 3, 468 (4)
Observation: The measured mass is 3, 477m
e
. This implies an accuracy of 99.8% [4].
3 The Meson Sector: Volumetric Defects
We bridge the gap between leptons and baryons by modeling Mesons (quark-antiquark
pairs) as half-volume defects. 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), so it scales with the Face Diagonal
(λ = 17).
• Volume: V
meson
= V
bulk
/2 = K
3
/2 = 1728/2 = 864.
• Tension: The defect consists of 2 strands (quark-antiquark), each stepping through
3 dimensions. 2 ×3 = 6 degrees of freedom. This tension scales with the slip plane
λ.
m
Kaon
=
K
3
2
+ 6λ = 864 + (6 × 17) = 864 + 102 = 966 (5)
Observation: The measured mass is 966.1m
e
(493.7 MeV). The accuracy is 99.99% [2].
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
) minus the linear tension (λ).
m
P ion
= λ
2
− λ = 289 − 17 = 272 (6)
Observation: The measured mass is 273.1m
e
(139.6 MeV). The accuracy is 99.6%.
3
4 The Baryon Sector: Proton and Neutron
4.1 The Proton (K
3
+ 9K)
As derived in our foundational work on the SSM, the proton represents a volumetric knot.
Its mass is the lattice volume (K
3
) stabilized by a surface term (9K) [2].
m
p
= K
3
+ 9K = 1728 + 108 = 1836 (7)
Observation: 1836.15m
e
. Accuracy: 99.99%.
4.2 The Neutron (Composite)
The neutron is dynamically modeled as a Proton fused with an Electron to achieve charge
neutrality. Binding them requires overcoming the Unitary Lattice Gap (δ = 1).
m
n
= m
p
+ m
e
+ δ = 1836 + 1 + 1 = 1838 (8)
Observation: 1838.68m
e
. Accuracy: 99.96% [2].
5 The Gauge Sector: Vertex Harmonics
We model the heavy weak bosons as vibrational modes of the unit cell’s vertices. A cubic
node has 8 vertices.
5.1 W Boson (Vacancy Mode)
The W Boson represents a vibration on 7 vertices (8−1), scaled by the proton mass (M
p
).
m
W
= (7K + 1)M
p
= 85 × 1836 = 156, 060 (9)
Observation: 157, 300m
e
(80.37 GeV). Accuracy: 99.2% [4].
5.2 Z Boson (Full Mode)
The Z Boson represents a vibration on the full 8 vertices, plus the unitary carrier.
m
Z
= (8K + 1)M
p
= 97 × 1836 = 178, 092 (10)
Observation: 178, 450m
e
(91.18 GeV). Accuracy: 99.8% [4].
5.3 Geometric Stabilization (Square Pyramids)
Pure tetrahedral lattices (K = 4) are unstable due to geometric frustration. Nature
stabilizes the vacuum by inserting Square Pyramids between the tetrahedra. In Lattice
Gauge Theory, forces are defined by the flux through a closed loop called a Plaquette.
We identify the square base of these pyramids as the physical manifestation of the Wilson
Loop Plaquette [5].
4
6 The Higgs Boson: Nodal Saturation (K
5
)
The Higgs Boson is the heaviest fundamental scalar. In the SSM, it represents the Max-
imum Information Capacity of a single lattice node.
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
(11)
The mass is the full capacity of the node (K
5
) minus the vacuum polarization ”tax” of a
baryon pair (2M
p
) required to isolate the node from the lattice.
m
H
= K
5
− 2M
p
= 12
5
− 3672 = 245, 160 (12)
Converting to GeV (1m
e
≈ 0.000511 GeV):
m
H
≈ 245, 160 × 0.000511 ≈ 125.27 GeV (13)
Observation: 125.25 ± 0.17 GeV. The match is nearly exact (> 99.8%) [3].
7 Discussion: Theoretical Constraints and Interpre-
tive Gaps
Our model achieves high precision, but we must address the interpretive choices that
bridge the gap between the raw lattice parameter (K = 12) and the observed spectrum.
7.1 The Interpretive Gap: Ansatz vs. Derivation
A valid critique of this model is that the specific algebraic combinations (e.g., 6λ for the
Kaon, 7K for the W boson) act as ”hidden parameters.” While K = 12 is the anchor, the
selection rules constitute a Phenomenological Ansatz.
• Lowest-Order Harmonics: The Lepton sector follows the simplest possible di-
mensional scaling: Point (λ
0
), Line (λ
1
), and Plane (λ
2
). There are no other lower-
order geometric objects available in the unit cell.
• Euler Constraints: We interpret the ”6” in the Kaon mass (2×3) as the minimum
Euler path for a 2-strand defect in 3D. A defect in a 3D lattice requires a minimum
of 3 spatial steps to define a closed loop or knot.
• Fundamental Defect Types: We interpret the ”7” in the W mass (8 − 1) as a
Vacancy Mode. In crystallography, vacancies are fundamental, inevitable modes of
any lattice.
These interpretations are geometrically consistent but not yet derived from a first-principles
lattice Lagrangian. Future work must demonstrate that these specific integer modes are
indeed the lowest-energy eigenstates of the K = 12 Hamiltonian.
7.2 Bare vs. Renormalized Mass
We contend that the SSM derives the Bare Geometric Mass (m
0
). The small dis-
crepancies (e.g., 1.3% for the Muon) likely correspond to QFT radiative corrections
(m
obs
= m
0
+ δm). This implies the geometric contribution is dominant.
5
8 Targets for the Final Run 3 Analysis (2026-2027)
The SSM predicts three resonances at 10.6 GeV, 30.1 GeV, and 136 GeV. As the LHC
completes its Run 3 data collection in June 2026, we note that interim analysis has not yet
confirmed these states. This suggests they are likely Dark Sector or feebly interacting
candidates.
1. 10.6 GeV: The Hidden Flux Resonance (K
4
): m ≈ 10.6 GeV. This lies deep
within the Bottomonium (Υ) family. The lack of a sharp *new* peak suggests this
state is not a distinct hadron but a Broad Flux Resonance of the vacuum itself,
possibly manifesting as an anomalous width or mixing parameter in the Υ sector
data currently being analyzed by **LHCb**.
m
K4
= 12
4
= 20, 736m
e
≈ 10.6 GeV (14)
2. 30.1 GeV: The Dark Lepton (λ
3
): m ≈ 30.1 GeV. Since a charged lepton at
this mass is ruled out, this geometric mode must be a **Heavy Neutral Lepton
(HNL)**. The absence of signal in early Run 3 implies a mixing angle |U|
2
below
current thresholds, placing it in the specific ”gap” region targeted by the full Run
3 dataset of **FASER** and **SND@LHC**.
m
L4
= λ
3
× K = 17
3
× 12 = 58, 956m
e
≈ 30.1 GeV (15)
3. 136 GeV: The Shadow Scalar (12K + 1): m ≈ 136 GeV. This state lies in the
high-mass tail of the Higgs (125 GeV). Current resolution may obscure a secondary
scalar this close to the main peak. We predict that the full integrated luminosity of
Run 3 may reveal this not as background noise but as a distinct BSM scalar.
m
Max
= (12K + 1)M
p
= 145 × 1836 = 266, 220m
e
≈ 136 GeV (16)
9 Summary of Results
Table 1: Comparison of SSM Geometric Predictions vs. Experimental Data
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%
We have derived the fundamental scales of physics from the geometry of a saturated
geometric crystal (K = 12). While interpretive gaps remain regarding the selection rules,
6
Figure 2: Geometric Predictions vs. Observation. The SSM integer predictions
(Blue) track with the Standard Model observations (Gray) across 5 orders of magnitude
with > 99% accuracy.
the model’s ability to unify the mass spectrum with a single geometric input warrants
serious investigation, particularly in the Dark Sector searches of the upcoming HL-LHC
era.
References
[1] Kulkarni, R. ”The Selection–Stitch Model (SSM): Space-Time Emergence via Evo-
lutionary Nucleation in a Polycrystalline Tensor Network.” Zenodo (2026). https:
//doi.org/10.5281/zenodo.18138227
[2] Kulkarni, R. ”The Geometric Origin of Mass: A Topological Derivation of the Proton-
Electron Ratio (µ ≈ 1836).” Zenodo (2026). https://doi.org/10.5281/zenodo.
18253326
[3] Kulkarni, R. ”The Geometry of the Standard Model: Deriving the Higgs Mass, La-
grangians, and Gravity Echoes from Lattice Saturation.” Zenodo (2026). https:
//doi.org/10.5281/zenodo.18292757
[4] Kulkarni, R. ”Unified Geometric Lattice Theory (UGLT): Deriving Gauge Couplings,
Mass Spectra, and Gravity from a K = 12 Vacuum.” Zenodo (2026). https://doi.
org/10.5281/zenodo.18520623
[5] Kulkarni, R. ”Structural Correspondence between the Standard Model and Vacuum
Geometry: SU(3) × SU(2) × U(1) from the Cuboctahedron.” Zenodo (2026). https:
//doi.org/10.5281/zenodo.18503168
7
[6] Kulkarni, R. ”The Cosserat Vacuum: A Unified Lagrangian for Gauge Fields,
Fermions, and Gravity from Saturated Elasticity.” Zenodo (2026). https://doi.
org/10.5281/zenodo.18645620
[7] Tiesinga, E., et al. ”CODATA recommended values of the fundamental physical
constants: 2018.” Rev. Mod. Phys. 93, 025010 (2021).
8
