The Geometry of Coupling:
Deriving the Fine Structure Constant (α
−1
≈ 137)
from Lattice Dilution Factors in a K = 12 Vacuum
Raghu Kulkarni
Independent Researcher, Calabasas, CA
February 13, 2026
Abstract
The Fine Structure Constant (α ≈ 1/137) determines the strength of the electromagnetic
interaction, yet its value remains one of the greatest mysteries in physics. In the Standard
Model, it is an arbitrary input parameter. We propose that α is a geometric invariant of a
discrete vacuum lattice. Using the Selection-Stitch Model (SSM), we model the vacuum
as a tensor network saturated at the Kepler packing limit (K = 12, Face-Centered Cubic). We
derive the inverse coupling α
−1
as the product of three geometric dilution factors: Topological
Selection (P
topo
= 1/2), Algebraic Projection (P
alg
= 1/4), and Resonant Coherence
(P
res
= 1/17). These factors represent the probability that a vacuum fluctuation successfully
couples to a vector boson channel. This yields a bare inverse coupling of α
−1
bare
= 136. Applying
the Self-Node Correction (+1), consistent with the counting principle used in prior SSM
derivations of the Hubble Tension (12 → 13) and Weak Mixing Angle (3 → 13), we derive
α
−1
= 137. This integer result suggests that the fundamental coupling of light is fixed by the
discrete geometry of the vacuum unit cell.
1 Introduction
The Fine Structure Constant α = e
2
/(4πϵ
0
ℏc) characterizes the strength of the electromagnetic
interaction. Its low-energy value is approximately 1/137.035999 [1, 2].
For a century, physicists have sought a derivation for the integer 137. Early attempts by
Eddington [3] and Wyler [4] sought to construct this number from algebraic ratios but lacked a
consistent physical framework, often failing to predict other observables. In continuum Quantum
Field Theory (QFT), this value is an empirical input. However, in discrete physics, coupling
constants often arise from Geometric Probability—the likelihood that a random fluctuation in
the medium aligns with a specific propagation channel.
In this work, we extend the Selection-Stitch Model (SSM) [5] to the electromagnetic sector.
We posit that the vacuum is a discrete graph with Cuboctahedral topology (K = 12). We
demonstrate that the value 137 arises naturally from the specific connectivity and algebra of this
unit cell, consistent with independent derivations of the mass spectrum and cosmological tension.
2 The Geometric Dilution Equation
We define the interaction strength g not as a force, but as a coupling efficiency. In a discrete lat-
tice, an interaction is a “handshake” between a node (fermion) and a link (boson). This handshake
is “diluted” by the geometry of the lattice.
1
We define the Inverse Coupling α
−1
as the ratio of the Total Vacuum State Space (Ω
total
)
to the Allowed Interaction Subspace (Ω
int
), plus a unitary correction for self-interaction:
α
−1
=
1
P
total
+ 1 (1)
The total geometric probability P
total
is the product of three independent dilution factors:
1. Topological Probability (P
topo
): Does the geometry allow flux screening?
2. Algebraic Probability (P
alg
): Does the algebra allow vector propagation?
3. Resonance Probability (P
res
): Does the scale allow coherent wave mechanics?
P
total
= P
topo
× P
alg
× P
res
(2)
3 Derivation of the Factors
3.1 Factor 1: Topological Selection (P
topo
= 1/2)
The electron resides on a vertex of the vacuum unit cell (Cuboctahedron). Interactions are local;
a node samples only the faces meeting at its vertex, not the global set of faces of the coordination
shell.
• Vertex Geometry: A vertex in a Cuboctahedron connects to exactly 4 faces: 2 Triangles
and 2 Squares [6].
• The Selection Rule: As established in the SSM framework, Triangular faces (C
3
sym-
metry) are non-bipartite (odd cycles). They cannot support alternating charge assignments,
leading to confinement (Strong Force). Square faces (C
4
symmetry) are bipartite (even
cycles), allowing for charge alternation and flux screening (Electroweak Force).
• The Probability: The chance that a local flux line couples to a screening face is:
P
topo
=
N
squares
N
total
=
2
2 + 2
=
1
2
(3)
3.2 Factor 2: Algebraic Projection (P
alg
= 1/4)
We define the vacuum state space using the spacetime algebra Cl(1, 3) rather than the spatial
algebra Cl(3, 0). This is necessary because the SSM vacuum is dynamical: every lattice node
carries both spatial connectivity (K = 12) and an intrinsic update clock τ (the stitching interval),
defining a full spacetime event.
• Total State Space: The dimension of Cl(1, 3) is 2
D
= 2
4
= 16. The basis elements are: 1
Scalar, 4 Vectors, 6 Bivectors, 4 Trivectors, 1 Pseudoscalar [7].
• Photon Sector: The photon is a vector boson (A
µ
). It can only occupy the Vector subspace
(D = 4).
• The Probability: The chance that a generic vacuum fluctuation projects onto the electro-
magnetic sector is:
2
P
alg
=
dim(V ector)
dim(Cl
1,3
)
=
4
16
=
1
4
(4)
3.3 Factor 3: Resonant Coherence (P
res
= 1/17)
A continuous wave packet can only propagate coherently through a discrete lattice if its wavelength
matches the integer harmonics of the grid. Mismatched waves suffer destructive interference and
decay.
• The Fundamental Scale (K): The lattice coordination number is K = 12.
• The Coherence Path: The “Path of Least Resistance” for a shear wave in a cubic crystal
is the Face Diagonal of the unit cell [8].
• The Calculation:
L
diag
= K ×
√
2 = 12 × 1.4142... = 16.9705... (5)
• Integer Quantization: Because the lattice is discrete, the coherence length must snap to
the nearest integer resonance. This defines the Lattice Harmonic (λ):
λ = ⌈16.97⌉ = 17 (6)
• Cross-Validation: We emphasize that λ = 17 is not chosen ad hoc. This same integer
independently derives the Geometric Harmonics of Mass [9], specifically predicting the
Muon mass (17K) and Tau mass (17
2
K).
• The Probability: The wave interaction is diluted by the path length:
P
res
=
1
λ
=
1
17
(7)
4 The Calculation
We substitute these three geometric invariants into the dilution equation to find the Bare Inverse
Coupling (α
−1
bare
):
α
−1
bare
=
1
(1/2)(1/4)(1/17)
= 136 (8)
This counts the geometric channels surrounding the node. However, the physical observable
must account for the node itself.
The Self-Node Correction (+1): Consistent with the SSM framework, physical values scale
with the Total Cluster Count (N
neighbors
+ N
center
).
• Hubble Tension: Expansion boost is derived as (12 + 1)/12.
• Weak Mixing Angle: Derived as 3/(12 + 1).
• Fine Structure: The inverse coupling counts the 136 vacuum channels plus the 1 self-
interaction channel (the particle itself).
α
−1
= 136 + 1 = 137 (9)
3
5 Discussion
The derived value α
−1
= 137 matches the integer part of the experimental Fine Structure Con-
stant (137.035999...). The residual 0.036 (0.026%) remains unexplained within the current integer
framework and may arise from polycrystalline defects or higher-order symmetry breaking.
It is worth noting that the factorization 136 = 2
3
× 17 is unique among integers near 137;
no other integer in this range factors purely into small primes and geometric harmonics (K, λ) of
the Cuboctahedron. This suggests the result is not a numerological coincidence but a structural
necessity.
Conversely, employing the static spatial algebra Cl(3, 0) yields dim = 8 and P
alg
= 3/8, resulting
in α
−1
bare
≈ 91, which firmly rules out a static lattice model in favor of the dynamic spacetime
formulation.
5.1 Comparison with Prior Art
Unlike the historical numerology of Eddington or the algebraic construction of Wyler, the SSM
derivation is not an isolated formula. It is part of a unified geometric framework. We present below
a set of independent predictions derived from the same K = 12 geometry without free parameters.
5.2 Zero-Parameter Prediction Table
Observable Symbol SSM Derivation Predicted Observed Error
Inv. Fine Structure α
−1
2 · 4 · 17 + 1 137 137.036 0.026%
Hubble Boost H
L
/H
E
13/12 1.0833 1.0836 0.03%
Weak Mixing Angle sin
2
θ
W
3/13 0.2307 0.2312 0.2%
Proton Mass Ratio µ 12
3
+ 108 1836 1836.15 0.008%
Muon Mass Ratio m
µ
/m
e
12 × 17 204 206.8 1.4%
Tau Mass Ratio m
τ
/m
e
12 × 17
2
3468 3477 0.3%
Table 1: The consistency of the SSM framework. Six distinct physical constants are derived from
the same integers (12, 13, 17) inherent to the Cuboctahedron unit cell. Hubble Boost observed ratio
uses SH0ES (73.04) and Planck (67.4).
5.3 Falsifiability
This model is explicitly falsifiable. The derivation collapses if:
1. A fourth generation of fermions is discovered (violating the 3D spatial basis of the lattice).
2. The Hubble Tension is resolved to a value significantly different from the 13/12 prediction.
3. The mass spectrum is found to deviate from the λ = 17 harmonics.
6 Conclusion
We have derived α
−1
= 137 using the same geometric principles used to resolve the Hubble Tension
and derive the Proton mass. By treating the coupling as a geometric probability, we identified it
4
as the product of the lattice’s topology (1/2), algebra (1/4), and scale (1/17), corrected for the
self-node (+1). This result reinforces the core hypothesis of the SSM: the fundamental constants
of nature are Geometric Invariants of a K = 12 Face-Centered Cubic vacuum.
References
1. L. Morel et al., “Determination of the fine-structure constant with an accuracy of 81 parts
per trillion,” Nature 588, 61–65 (2020).
2. R. H. Parker et al., “Measurement of the fine-structure constant as a test of the Standard
Model,” Science 360, 191 (2018).
3. A. S. Eddington, Fundamental Theory, Cambridge Univ. Press (1946).
4. A. Wyler, “L’espace sym´etrique du groupe des ´equations de Maxwell,” C. R. Acad. Sci.
Paris 269, 743 (1969).
5. R. Kulkarni, “The Selection-Stitch Model (SSM),” Zenodo (2026).
6. H. S. M. Coxeter, Regular Polytopes, Dover Publications (1973).
7. P. Lounesto, Clifford Algebras and Spinors, Cambridge Univ. Press (2001).
8. D. Hull and D. J. Bacon, Introduction to Dislocations, Butterworth-Heinemann (2011).
9. R. Kulkarni, “The Geometric Harmonics of Mass,” Zenodo (2026).
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