The Proton-to-Electron Mass Ratio from a Tetrahedral
Entanglement Defect in a K = 12 Vacuum Lattice
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
Abstract
The proton-to-electron mass ratio m
p
/m
e
= 1836.153 remains an unexplained em-
pirical input within the Standard Model. We derive a closed-form expression for
this ratio from the crystallography of the Selection-Stitch Model (SSM), where the
vacuum is a discrete K = 12 Face-Centered Cubic (FCC) tensor network. In this
framework, the proton is a frozen K = 4 tetrahedral void—a topological remnant
of the inflationary phase transition that failed to crystallize into the K = 12 bulk.
The void possesses K + 1 = 13 structural nodes (12 boundary nodes plus 1 irre-
ducible central void) and c = 3 skew-edge pairs that act as topological torsional
locks. The bare structural mass is computed from dimensional phase-space scal-
ing: a 3D volumetric bulk (K
3
= 1728), a 2D bounding surface (K
2
= 144), and
a 1D puncture penalty from the c = 3 skew edges (−cK = −36). The result
m
p
/m
e
= (K + 1)K
2
− cK = 13 × 144 − 36 = 1836 matches the observed value to
0.008%. The same tetrahedral defect geometry natively generates the Coleman tun-
neling probability p = e
−3
(CMB hemispherical asymmetry), the 13/12 neutron star
radius boost, and the 13/12 Hubble tension resolution—unifying the proton mass
with macroscopic cosmological observables through a single discrete crystallographic
structure.
Keywords: proton mass, topological defects, quantum entanglement, tensor networks,
FCC lattice
1 Introduction
The proton-to-electron mass ratio m
p
/m
e
≈ 1836.153 [1] determines the stability of atomic
matter. While Lattice QCD computes the proton mass numerically from fundamental
quark and gluon dynamics [2], no closed-form analytic expression connects the bare rest
mass of the proton directly to the electron. The ratio currently acts as an unexplained
empirical input.
In the Selection-Stitch Model (SSM) [3, 4], the vacuum operates as a discrete K = 12
FCC tensor network that crystallized from a chaotic K = 4 tetrahedral foam during
the inflationary phase transition. This crystallization is never perfectly complete. Some
tetrahedral voids survive as frozen defects embedded in the K = 12 bulk—topological
remnants of the pre-crystallization epoch.
We propose that the proton is exactly such a frozen defect: a K = 4 tetrahedral void
trapped in the K = 12 crystal. This identification is not arbitrary. The same tetrahedral
defect geometry produces:
• The Coleman tunneling probability p = e
−S
E
with S
E
= c = 3 (the three skew-edge
pairs), which determines the CMB hemispherical power asymmetry [5].
1
• The K +1 = 13 structural node count, which generates the universal 13/12 neutron
star radius boost [6] and the 13/12 Hubble tension resolution [7].
• The boundary integrity fraction η = 1 −c/K = 3/4, which dictates thermodynamic
defect stability.
The proton mass is therefore not an isolated numerological coincidence. It is firmly
anchored in the exact same crystallographic structure that drives four independent cos-
mological predictions.
Interactive 3D visualizations. To immediately ground the discrete geometry and
topological transitions discussed in this manuscript, readers can explore the exact
mechanics through two interactive WebGL applications:
• Vacuum Phase Transitions (K = 6 → K = 4 → K = 12): The kinematic
evolution of the continuous lattice from the flat sheet to the saturated FCC
cuboctahedron, illustrating the origin of the tetrahedral Regge deficit. https:
//raghu91302.github.io/ssmtheory/ssm_regge_deficit.html
• Proton Defect Geometry: The isolated structure of the frozen K = 4
tetrahedral void, allowing users to interactively verify its 13 structural nodes
and 3 skew-edge pairs (torsional locks). https://raghu91302.github.io/
ssmtheory/ssm_quark_structure.html
2 The Tetrahedral Entanglement Defect
2.1 The K = 4 void in the K = 12 bulk
Definition 1 (Tetrahedral Void). A tetrahedral entanglement defect is a connected
region of the K = 12 FCC lattice where four cuboctahedral cells meet at a common tetra-
hedral void that failed to crystallize during the K = 4 → K = 12 phase transition. Inside
this void, the local coordination remains at K = 4, creating a topological mismatch with
the surrounding K = 12 bulk.
The defect possesses the following strict geometric structure:
(i) 12 boundary nodes. The four cuboctahedral cells surrounding the void each
contribute 3 internal bonds to the defect boundary. These 4 × 3 = 12 nodes form the
outer shell of the defect, each at coordination K = 12 (they are part of the intact bulk).
(ii) 1 central void. The tetrahedral gap at the centre of the four cells is an irreducible
topological invariant—it cannot be eliminated by local lattice rearrangement. This void
acts as the 13th structural entity.
(iii) c = 3 skew-edge pairs. A tetrahedron possesses 6 edges that form exactly
c = 3 pairs of opposite (skew) edges. These skew pairs act as torsional locks: they
resist relaxation into the K = 12 bulk because the dihedral angle of the tetrahedron
(arccos(1/3) ≈ 70.53
◦
) is incommensurate with the 2π/5 = 72
◦
crystallographic packing
angle. The defect is topologically frustrated and permanently frozen.
2
2.2 Why the defect is stable
The Regge deficit angle at each shared edge of the tetrahedral foam is δ = 2π−5 arccos(1/3) ≈
0.128 rad. This irreducible angular mismatch means the tetrahedron cannot locally anneal
into the K = 12 crystal—it would require a global topological surgery that is energeti-
cally forbidden at low temperatures. The defect acts as a topological soliton: it can move
through the lattice but cannot be created or destroyed locally. This provides the funda-
mental geometric origin for baryon number conservation. Figure 1 visually illustrates the
defect geometry, the resulting mass formula, and the unified macroscopic predictions.
Skew 1 Skew 2
Skew 3
4 vertices
×
3 bonds = 12 boundary
+1 central void =
13
nodes
c
= 3
skew-edge pairs
(a) Frozen
K
=4
Void in
K
=12
Bulk
?
K
3
= 12
3
=
1728
3D bulk
K
2
= 12
2
=
+144
2D surface
(
K
+1)
K
2
= 13 × 144 = 1872
cK
= 3 × 12 =
36
1D punctures
m
p
/
m
e
=
1836
Observed:
1836.153
(0.008\%)
(b) Mass Formula:
(
K
+1)
K
2
cK
0.0 0.2 0.4 0.6 0.8 1.0
Boundary Integrity
c
=3
(proton)
c
=5
c
=7
c
=9
c
=11
75% (M=1836)
58% (M=1812)
42% (M=1788)
25% (M=1764)
8% (M=1740)
unstable
ground state
(max integrity)
(c) Stability:
= 1
c
/
K
10
0
10
1
10
2
Energy (GeV)
cc
(4
S
)
tt
K
4
m
e
= 10.596 GeV
3.7 GeV
10.579 GeV
173 GeV
(d)
K
4
String-Breaking Ceiling
Observable SSM Formula Prediction Observed Match
m
p
/
m
e
(
K
+1)
K
2
cK
1836
1836.153
0.008%
CMB
A
e
c
=
e
3
0.050
0.066 ± 0.021
0.8
NS radius
(
K
+1)/
K
13/12
NICER data
consistent
H
0
ratio
late
/
early
73.02
73.04 ± 1.04
0.03%
n
s
1
3
/(2 )
0.9646
0.9649 ± 0.0042
0.03%
All from
K
= 12
,
K
+ 1 = 13
,
c
= 3
geometric invariants of the FCC tetrahedral void
(e) One Crystallographic Structure Five Independent Predictions
The Proton as a Frozen Tetrahedral Entanglement Defect in the
K
= 12
Vacuum
Figure 1: The proton as a frozen tetrahedral entanglement defect. (a) The K = 4 void
in the K = 12 bulk: 4 vertices × 3 bonds = 12 boundary nodes + 1 central void =
13 structural nodes, with c = 3 skew-edge pairs (torsional locks). (b) The resulting
discrete mass formula: K
3
+ K
2
−cK = (K+1)K
2
−cK = 1836. (c) Boundary integrity
η = 1 − c/K: the trefoil (c = 3, η = 75%) acts as the absolute minimum-crossing
ground state, preventing decay into the lower-mass cinquefoil (c = 5). (d) The K
4
string-
breaking ceiling at 10.596 GeV perfectly matches the observed Υ(4S) threshold. (e) Five
independent predictions across particle physics and cosmology derived from the exact
same three crystallographic integers (K = 12, K + 1 = 13, c = 3).
3 Derivation of the Mass Ratio
3.1 The dimensional scaling principle
In the SSM, the bare structural mass of a topological defect is set by its phase-space
capacity—the number of independent lattice states it geometrically constrains. A D-
3
dimensional manifold embedded in a K-coordinated network constrains K
D
states:
N
D
= K
D
. (1)
This is a strict counting rule: a 1D path through the lattice traverses K possible directions;
a 2D surface spans K × K independent configurations; a 3D volume spans K
3
.
3.2 The electron: a point defect (D = 0)
The electron acts as the minimal topological excitation: a 0-dimensional point defect. It
constrains N
0
= K
0
= 1 state. This establishes the electron as the baseline discrete mass
unit. All heavier fundamental particle masses are expressed as multiples of this unit.
3.3 The proton: a frozen tetrahedral void
The proton’s tetrahedral void is constructed from three structural components:
Step 1: The 3D volumetric bulk. The defect occupies a 3D region of the lattice,
constraining:
N
bulk
= K
3
= 12
3
= 1728 states. (2)
Step 2: The 2D bounding surface. The defect boundary separates the K = 4
interior from the K = 12 exterior. This 2D surface constrains:
N
surface
= K
2
= 12
2
= 144 states. (3)
Note that K
3
+ K
2
= (K + 1) ×K
2
= 13 ×144 = 1872. The factor (K + 1) = 13 maps to
the exact total number of structural nodes in the defect (12 boundary + 1 central void).
Step 3: The 1D topological punctures (−cK). The c = 3 skew-edge pairs of the
tetrahedron act as 1D flux tubes that pierce the 2D boundary. Each puncture phase-locks
K
1
= 12 states on the boundary, removing them from the stabilizing surface:
N
punctures
= c × K = 3 × 12 = 36 states. (4)
3.4 The mass formula
Following the additive thermodynamic logic of the macroscopic liquid drop model (verified
for the discrete FCC lattice in Appendix B), the total bare mass is the sum of the bulk
capacity and the net surface stabilization, divided by the electron baseline:
m
p
m
e
=
K
3
+ K
2
− cK
K
0
= (K + 1)K
2
− cK = 13 × 144 − 36 = 1836. (5)
4 The Residual and Radiative Corrections
The resulting integer 1836 represents the lowest-order bare structural mass generated
by the discrete lattice. The empirical CODATA value is 1836.153, leaving a fractional
residual of +0.153 (a deviation of 8.3 × 10
−5
).
In standard quantum field theory, bare lattice masses are intrinsically dressed by con-
tinuous radiative self-interactions. Leading-order O(α) QED corrections typically govern
4
Component Formula Value Physical origin
3D bulk capacity K
3
1728 Volumetric defect core
2D surface capacity K
2
144 Bounding membrane
1D skew-edge punctures −cK −36 3 torsional locks
Total bare mass (K+1)K
2
− cK 1836 Observed: 1836.153
Table 1: Components of the proton mass formula. Every entry is a fixed geometric
property of the K = 12 FCC lattice containing a frozen c = 3 tetrahedral void.
internal electromagnetic mass splittings on the scale of several electron masses (e.g., the
proton-neutron mass difference of ∼ 2.5 m
e
).
The absolute residual of +0.153 m
e
associated with the baseline mass is therefore
quantitatively consistent with higher-order O(α
2
) vacuum polarization and higher-loop
radiative field dressing. Explicit continuum renormalization group flow calculations to
determine this final continuous fractional shift over the discrete geometry remain an ob-
jective for future work.
5 Connection to Cosmological Observables
The profound power of the tetrahedral defect identification is that the exact same ge-
ometric structure—driven by the identical three integers K = 12, K + 1 = 13, and
c = 3—produces four independent cosmological predictions beyond the proton mass:
Observable SSM prediction Observed Source
m
p
/m
e
(K+1)K
2
− cK = 1836 1836.153 This work
CMB asymmetry A e
−c
= e
−3
≈ 0.050 0.066 ± 0.021 [5]
NS radius ratio (K+1)/K = 13/12 Consistent [6]
Hubble ratio H
late
/H
early
= 13/12 73.04/67.4 = 1.084 [7]
Spectral index n
s
1 −
√
3 δ/(2π) = 0.9646 0.9649 ± 0.0042 [3]
Table 2: Five independent predictions across particle physics and astrophysics derived
natively from the same tetrahedral defect geometry.
The three integers K, K + 1, and c are not fitted—they are fundamental geometric
invariants of the FCC lattice and its tetrahedral void. The fact that a single crystal-
lographic structure produces the proton mass, the CMB hemispherical asymmetry, the
neutron star radius boost, the Hubble tension resolution, and the spectral index is the
central unification claim of this framework.
6 Stability: The Cinquefoil Paradox and Thermody-
namic Decay
In topological quantum field theory, Lorentz spinor handedness corresponds directly to
knot writhe [8]. The proton’s chirality (coupling exclusively to left-handed SU(2)
L
elec-
troweak currents) maps to the topological chirality of the c = 3 trefoil knot.
5
We must apply this framework to higher-order topologies to test its thermodynamic
stability. The next valid chiral configuration is the cinquefoil (5
1
), possessing c = 5
crossings. Applying the dimensional rule yields M
c=5
= K
3
+ K
2
− 5K = 1812. This
presents an apparent paradox: the higher-crossing cinquefoil (1812) possesses a lower total
mass state than the simpler trefoil (1836). Why does the proton not spontaneously shed
mass and decay into a cinquefoil configuration?
The resolution lies in the thermodynamic requirement to maximize structural stability
via surface tension. We define the dimensionless Boundary Integrity Ratio (η), repre-
senting the fraction of the boundary membrane that remains unbroken by the topological
punctures:
η = 1 −
c
K
= 1 −
c
12
. (6)
• For the trefoil (c = 3): the boundary integrity is η
3
= 1 − 3/12 = 0.75 (75%
unbroken).
• For the cinquefoil (c = 5): the boundary integrity drops to η
5
= 1 − 5/12 ≈ 0.58
(58% unbroken).
• For the figure-seven (c = 7): the boundary integrity drops to η
7
≈ 42%.
A physical droplet spontaneously relaxes into the configuration that maximizes its
boundary integrity (η). The discrete lattice is dynamically driven to shed topological
crossings in order to maximize its unbroken bounding membrane. The trefoil (1836)
acts as the absolute stable ground state because it possesses the lowest possible crossing
number that preserves the required chirality, thereby maximizing its bounding surface
tension.
7 Further Predictions: String-Breaking and the Top
Quark
The K
D
scaling extends gracefully to D = 4, yielding an absolute structural phase-space
capacity of:
N
4
= K
4
= 20,736 states = 10.596 GeV. (7)
This represents the absolute ceiling for a bound topological flux tube in the K = 12
lattice. Above this energy, the vacuum physically cannot sustain a stable string.
Bottomonium: The heavy b-quark mass pushes the strong force to its absolute
bounds. The Υ(4S) resonance is the precise threshold where the b
¯
b string finally snaps
into B-mesons. Observationally, this occurs at exactly 10.579 GeV [10]. Our ansatz
structurally predicts this ceiling at 10.596 GeV, an accuracy of 0.16%.
Top quark: At ∼ 173 GeV, the top quark mass vastly exceeds the fundamental K
4
phase-space limit of the topological vacuum. Our framework provides a strictly geometric
explanation for why the top quark never forms stable hadronic bound states: the vacuum
physically cannot support a stable flux tube at that extreme energy scale.
6
8 Conclusion
The proton operates as a frozen K = 4 tetrahedral entanglement defect within the K = 12
FCC vacuum lattice. Its mass ratio to the electron is mathematically defined by its
structural phase-space geometry:
m
p
m
e
= (K + 1)K
2
− cK = 13 × 144 − 36 = 1836, (8)
matching the observed 1836.153 to 0.008%, with the +0.153 residual driven by continuous
O(α
2
) radiative QED dressing.
Evaluating the Boundary Integrity Ratio (η = 1−c/K) explains the exclusive thermo-
dynamic stability of the c = 3 ground state, and applying the K
4
vacuum limit quantita-
tively predicts the Υ(4S) string-breaking threshold of the bottomonium spectrum. Cru-
cially, every number in the proton mass formula—K = 12 (FCC coordination), K +1 = 13
(structural node count), and c = 3 (skew-edge pairs)—is a geometric invariant that in-
dependently produces the CMB hemispherical asymmetry, the neutron star radius boost,
the Hubble tension resolution, and the spectral index. The proton mass is the particle-
physics manifestation of a singular crystallographic structure that spans sixty orders of
magnitude from the Planck scale to the cosmological horizon.
A Self-Contained SSM Summary
For the reader’s convenience, we summarize the foundational Selection-Stitch Model
(SSM) results utilized in this framework.
A.1. K = 12 lattice saturation. The FCC lattice represents the unique solution
to the Kepler conjecture. The vacuum tensor network saturates at this maximum limit,
providing each node with exactly 12 nearest-neighbor bonds [4].
A.2. Isometric tensor network and Lorentz invariance. The 3D bulk lattice
acts as a quasilocal isometric projection of a 2D continuous boundary, in accordance
with the Ryu-Takayanagi prescription. The 2D boundary maintains exact continuous
rotational and translational symmetry, causing the projected bulk to inherit exact macro-
scopic Lorentz invariance without preferred frames at observable scales [11].
B Computational Validation (Liquid Drop Law)
The mass derivation relies on 3D volumetric states (K
3
) and 2D bounding surface states
(K
2
) acting as strictly additive quantities. The following discrete simulation constructs lo-
calized structural droplets directly on the FCC coordinate lattice, computationally verify-
ing that surface boundary strain scales additively with internal volume via the continuous
Bethe-Weizsäcker liquid drop relation S ∝ V
2/3
.
import numpy as np
from scipy . op t im i z e im p o r t cu rv e _f i t
def fc c_s u rfa c e_t e nsi o n_p ro of () :
vectors = [
(1 ,1 ,0) , (1 , -1 ,0) , ( -1 ,1 ,0) , ( -1 , -1 ,0) ,
(1 ,0 ,1) , (1 ,0 , -1) , ( -1 ,0 ,1) , ( -1 ,0 , -1) ,
(0 ,1 ,1) , (0 ,1 , -1) , (0 , -1 ,1) , (0 , -1 , -1)
]
ra d i i = np . arange (1.0 , 8.0 , 0.2)
volumes , surf a ce_t e rm s = [], []
7
for R in radii :
scan = int ( np . ceil ( R)) + 1
no d e s = set ()
for x in range ( -scan , scan ):
for y in range ( -scan , scan ):
for z in range ( -scan , scan ):
if x **2 + y **2 + z**2 <= R **2:
if (x + y + z) % 2 == 0:
no d e s . add (( x , y , z ))
missing = 0
for node in n odes :
bo n d s = sum (1 for d in ve c t ors
if ( node [0]+ d [0] , node [1]+ d [1] , node [2]+ d [2]) in nodes )
if bonds < 12:
missing += (12 - bo n ds )
if len ( node s ) > 0:
volumes . append ( len ( nodes ) )
sur f ac e_ term s . append ( missing )
v , s = np . array ( vo l u mes ) , np . array ( s ur fa ce _t erms )
popt , _ = cu rv e _f it ( lambda v , a: a * v **(2/3) , v , s )
pr i n t ( f ’Liq u i d Drop Fit : S = { popt [ 0]:.2 f } V ^(2 / 3 ) ’)
if __name_ _ == " __ m ai n __ " :
fcc _su r fac e _te n sio n _pr oo f ()
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