THE GEOMETRIC ORIGIN OF MASS:
A Topological Derivation of the Proton-Electron Ratio (µ ≈ 1836)
Raghu Kulkarni
Independent Researcher
raghu@idrive.com
January 21, 2026 (Revised Feb 16, 2026)
Significance Statement
In the Standard Model, the mass ratio between the proton and electron (µ ≈ 1836.15) is an
empirical parameter without a theoretical origin [1]. We propose a solution using the Selection-
Stitch Model (SSM). By defining mass as ”Topological Impedance” against a Cuboctahedral
Vacuum Lattice (K = 12), we derive the integer 1836 from two geometric principles: Volumetric
Phase Space (K
3
) and Rotational Locking Symmetry (9K). This suggests the proton’s mass is an
eigenvalue of the vacuum geometry itself.
Abstract
Why is the proton roughly 2000 times heavier than the electron? We propose that this
hierarchy is a direct consequence of the Cuboctahedral Vacuum Geometry (K = 12).
Modeling the electron as a surface defect (L
2
) and the proton as a volumetric knot (L
3
), we
derive the proton’s mass from first principles.
1. Base Mass (1728): We rigorously derive the volumetric mass factor 12
3
= 1728 using
three distinct physical frameworks: Tensor Network Capacity, Mechanical Green’s Func-
tions, and Feynman Path Integrals.
2. Tension Energy (108): We calculate the locking energy of a 3-strand braid (Trefoil
Knot) as 9 × 12 = 108.
The sum yields µ = 1728 + 108 = 1836, matching CODATA observations to within 0.008% [2].
We further address the relation between the derived ”9 degrees of freedom” and the Standard
Model’s gluon octet.
1 The Mystery of 1836
The ratio of the proton mass to the electron mass is one of the most stable constants in nature [2]:
µ =
m
p
m
e
≈ 1836.15267... (1)
Standard physics treats this as a random number. The SSM treats it as a geometric necessity. If
the vacuum is a discrete crystal defined by the Kissing Number (K = 12), then ”Mass” is simply
the aggregate number of lattice nodes (or constraints) disturbed by a particle.
1
2 Mass as Topological Impedance
2.1 The Electron: Surface Impedance (m = 1)
The electron is a point-like lepton. In our lattice model, it represents a Planar Defect—a distur-
bance restricted to the 2D surface of the lattice domains. We normalize the mass of this unitary
surface defect to m
e
= 1.
2.2 The Proton: Volumetric Impedance (m = 1836)
The proton is a composite baryon (3 quarks). Unlike the electron, it cannot exist on a surface; it
requires 3D depth to form a stable knot. Thus, defining a proton requires locking a Volumetric
State of the vacuum lattice.
Figure 1: The Geometric Model. The proton is modeled as a Trefoil Knot (Red) trapped inside
a Cuboctahedral Cage (Gray) [3].
3 Deriving the Bulk Mass: The Proof of 1728
A common critique of integer-based physics is ”numerology.” Why must the mass scale as K
3
=
1728? Why not count the atoms in the unit cell (N = 4)?
We present three distinct physical derivations that all converge on the integer 1728 as the
fundamental Topological Impedance of a 3D defect in a K = 12 vacuum.
2
3.1 Proof 1: The Tensor Network Capacity (Information Theory)
In a quantum tensor network (e.g., PEPS), the complexity of a state is defined by its Bond
Dimension.
• Connectivity (K = 12): In the Cuboctahedral vacuum, every node has a coordination
number of 12. This is the ”bandwidth” of the vacuum at any point.
• The Tensor Product: A knot is a strictly 3-dimensional topological object. To uniquely
define a crossing (distinguishing ’over’ from ’under’), the defect must constrain degrees of
freedom along three independent principal axes (X, Y, Z).
• The Voxel: The total phase space volume (Ω) required to encode this 3D information is the
tensor product of the linear resolutions:
Ω = K
x
⊗ K
y
⊗ K
z
= 12 × 12 × 12 = 1728 (2)
Thus, 1728 is the size of the Information Voxel required to write a baryon into the vacuum.
3.2 Proof 2: The Mechanical Green’s Function (Elasticity)
Alternatively, we can view the mass as the total mechanical strain energy in the lattice.
• Propagation: A central twist propagates stress to its 12 neighbors. These 12 neighbors
must brace against their 12 neighbors to hold the tension.
• The Screening Hori zon: Because the knot is 3-dimensional, the strain field must extend to
exactly 3 lattice steps to stabilize the volumetric topology. Beyond 3 steps, the stress dilutes
into the continuum (1/r field).
• Cumulative Strain: The total number of active stress vectors (N
stress
) is the branching
product of the cascade:
N
stress
=
3
X
Steps=1
K
branching
→ 12
3
= 1728 (3)
Even if lattice paths converge on the same nodes (reducing the unique node count), the Super-
position Principle dictates that the total energy stored is proportional to the sum of the stress
vectors. 1728 is the total Action of the strain field.
3.3 Proof 3: The Path Integral Sum (Quantum Mechanics)
In the Feynman Path Integral formulation, mass is related to the number of histories a particle
explores.
• Winding Numbers: For a topological defect, the path taken matters. A path that winds
”over” is distinct from a path that winds ”under,” even if they end at the same node [3].
• The Sum Over Histories: The number of unique quantum trajectories of length 3 in a
K = 12 lattice is exactly 12
3
.
• Result: The integer 1728 represents the Partition Function (Z) of the proton’s wavefunc-
tion over its coherence length.
Conclusion: Whether viewed as Information (12
3
bits), Mechanics (12
3
stress units), or Prob-
ability (12
3
paths), the bulk mass of a 3D defect in this vacuum is invariant at 1728.
3
4 The Tension Term: 9 Degrees of Freedom
The bulk mass (1728) represents the volume occupied by the knot. However, the proton is not just
a void; it is a Tensioned Braid. We must add the energy required to lock the knot to the lattice.
4.1 Rotational Locking Symmetry
The proton consists of 3 quarks (strands) braided together [3].
• Symmetry Constraint: The faces of the Cuboctahedron are triangular (C
3
symmetry).
• The Locking Condition: To lock a strand into a lattice node, it must perform a full rotation
(360
◦
). On a triangular face, this requires 3 discrete sub-steps (120
◦
per step).
• Total Degrees of Freedom:
N
dof
= (3 Strands) × (3 Steps/Strand) = 9 (4)
4.2 The Tension Calculation
Each of these 9 twist-steps pulls on the surrounding K = 12 neighbors. The total binding energy
is:
E
tension
= 9 × 12 = 108 (5)
4.3 Connection to the Gluon Octet (8 vs 9)
The Standard Model describes 8 gluons. Our geometric derivation yields 9 degrees of freedom [1].
We propose that the 8 massless gluons correspond to the 9 − 1 degrees of freedom remaining after
a global gauge constraint is applied (similar to how a photon polarization removes 1 DOF). The
9th degree corresponds to the ”Massive Mode” that gives the proton its weight.
5 The Total Mass Prediction
Combining the Bulk Volume (Section 3) and the Locking Tension:
µ
pred
= V
bulk
+ E
tension
= 1728 + 108 = 1836 (6)
This integer derivation matches the observed CODATA value (1836.15) with a precision of 99.992%
(Error ≈ 0.008%).
6 Conclusion
The derived ratio of 1836 is robust. It relies only on the geometry of the Cuboctahedron (K = 12)
and the topology of the Trefoil Knot [3].
• 1728 (12
3
): The cost of occupying 3D space.
• 108 (9 × 12): The cost of topological stability.
This suggests that the ”Hierarchy Problem” is an artifact of treating particles as points. Once
treated as geometric knots, their masses emerge naturally from the lattice background [1].
4
References
[1] Kulkarni, R. (2026). The Selection-Stitch Model (SSM): Emergent Gravity from Discrete Ge-
ometry. Zenodo.
[2] Tiesinga, E., et al. (2021). CODATA recommended values of the fundamental physical constants:
2022. Rev. Mod. Phys., 93, 025010.
[3] Rolfsen, D. (1976). Knots and Links. Publish or Perish, Berkeley.
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