THE GEOMETRY OF THE STANDARD MODEL:
Deriving the Higgs Mass, Lagrangians, and Gravity Echoes from
Lattice Saturation
Raghu Kulkarni
Independent Researcher
January 28, 2026
Abstract
We propose that the Standard Model Lagrangian is the continuum limit of a discrete vacuum
geometry [1]. Using the Selection-Stitch Model (SSM), we derive the fundamental sectors of particle
physics from the properties of a saturated Cuboctahedral Lattice (K = 12).
1. The Lagrangian Sector: We derive the Klein-Gordon (Scalar), Dirac (Spinor), and Yang-
Mills (Gauge) Lagrangians as the elastic limits of lattice tension, topological braiding, and
stitch preservation. Crucially, we show that the Non-Bipartite Topology of the simplicial lattice
naturally resolves the “Fermion Doubling Problem,” allowing chiral fermions to exist without
spurious mirrors.
2. The Higgs Sector: Using the ratio of Surface (108) to Volume (1728) states, we calculate a
geometric coupling λ ≈ 0.125, predicting a Higgs mass of 123.11 GeV. We derive the factor of
2 from the face-sharing topology of space-filling tessellations and justify the volumetric scaling
via the Hausdorff dimension of the vacuum (D = 3).
3. The Gravity Sector: We interpret the Event Horizon as a “Phase Boundary” where the
lattice melts. This boundary reflects gravitational waves, predicting post-merger Echoes with
a time delay of ≈ 0.27s for a 60M
⊙
merger [2].
1 The Vacuum as a Crystal
The quest for a unified theory faces a persistent obstacle: the incompatibility between the smooth
manifold of General Relativity and the quantized nature of particles. Modern Quantum Field Theory
(QFT) treats space-time as a continuous background, leading to singularities (UV divergences) that
require renormalization [3]. Furthermore, the Standard Model cannot explain its own parameters (m
h
≈
125 GeV, v ≈ 246 GeV).
The Selection-Stitch Model (SSM) offers a third path: Geometric Saturation. The SSM posits that
the vacuum is a physical mesh of tetrahedra formed via two operators: Selection (Exclusion) and Stitch
(Entanglement). This process naturally saturates at a coordination number of K = 12 (the Kepler
Limit), creating a rigid “Mesh Phase” geometry. If the vacuum is a crystal, then “Particles” are simply
its vibrational modes.
2 The Scalar Sector: Space and Inertia
We first derive the behavior of the lattice itself. We model the vacuum as a coupled harmonic system.
Let ϕ
n
(t) represent the geometric deviation of a node n from its equilibrium position.
2.1 Deriving the Klein-Gordon Lagrangian
The Hamiltonian is defined by the Kinetic Energy (time evolution) and Stitch Tension (gradient energy):
H
scalar
=
X
n
1
2
m
0
˙
ϕ
2
n
+
1
2
κ
X
m∈neigh
(ϕ
n
− ϕ
m
)
2
+ V
geo
(ϕ
n
)
(1)
1
To recover standard physics, we take the limit where lattice spacing a → 0. The discrete difference
operator transforms into the Laplacian via Taylor expansion:
ϕ
n+1
− ϕ
n
≈ a ·∇ϕ +
1
2
a
2
∇
2
ϕ (2)
Substituting this into the potential term, we recover the continuum action. Setting the wave speed
c
2
= κa
2
/m
0
= 1 (natural units), we obtain the Klein-Gordon Lagrangian:
L
scalar
=
1
2
(∂
µ
ϕ)(∂
µ
ϕ) − V
eff
(ϕ) (3)
2.2 Geometric Origin of Mass
In standard QFT, mass is a coupling constant. In the SSM, mass is a geometric constraint. The lattice
potential V
geo
enforces the integer topology of the mesh (K = 12). A node cannot exist “between”
integer sites. This creates a periodic potential well. Thus, mass is identified as the Resonance Frequency
of a defect vibrating within the geometric trap of the vacuum.
3 The Spinor Sector: Matter as Braids
Standard scalar fields describe bosons (force carriers). To describe Fermions (Matter), we must model
defects as Topological Braids (Twists) that possess orientation and cannot be continuously deformed to
zero.
3.1 Deriving the Dirac Equation
Lattice hopping for a directed twist requires a mechanism to preserve directional information (Spin). A
standard Laplacian ∇
2
loses this information. We effectively “take the square root” of the Laplacian
using the Clifford algebra of the tetrahedral basis vectors. We define the “Directed Stitch Operator” D:
D = −iγ
µ
∂
µ
(4)
where γ
µ
satisfy the anticommutation relation {γ
µ
, γ
ν
} = 2g
µν
. This ensures that D
2
= −∇
2
, recovering
the correct energy-momentum relation (E
2
= p
2
). The equation of motion for a braid with topological
tension M is the Dirac Equation:
(iγ
µ
∂
µ
− M)ψ = 0 (5)
3.2 Resolution of the Fermion Doubling Problem
A major theoretical obstacle for discrete theories is the Nielsen-Ninomiya “No-Go” Theorem, which
states that discretizing the Dirac equation on a hypercubic lattice produces spurious “doubler” fermions
(mirror particles) at the corners of the Brillouin zone. We present a rigorous derivation showing how the
SSM geometry evades this theorem.
3.2.1 The Hypercubic Trap (Bipartite Symmetry)
On a standard square/cubic lattice, the discrete Dirac operator in momentum space is:
D
cubic
(k) = iγ
µ
sin(k
µ
a)
a
(6)
The energy spectrum contains zeros whenever sin(k
µ
a) = 0. This occurs at the physical origin (k = 0)
but also at the zone boundary (k = π/a). This k → k + π symmetry is a consequence of the Bipartite
Nature of the hypercubic graph (checkerboard coloring).
2
3.2.2 The SSM Solution (Non-Bipartite Topology)
The SSM vacuum is Face-Centered Cubic (FCC), defined by 12 nearest neighbors. The basis vectors are
the permutations of n =
a
√
2
(±1, ±1, 0). The discrete Dirac operator sums over these 12 non-orthogonal
links:
D
SSM
(k) =
12
X
j=1
Γ
j
e
ik·n
j
(7)
where Γ
j
are the spin projections along the bond directions.
Crucially, the FCC lattice is formed of Tetrahedra (Simplexes). A tetrahedron contains triangular
faces (odd cycles of length 3).
• Topological Frustration: It is impossible to two-color a triangle. Therefore, the lattice is Non-
Bipartite.
• Breaking the Symmetry: The shift k → k + π is no longer a symmetry of the Hamiltonian
because the odd loops introduce phase factors that do not cancel.
3.2.3 Dispersion Analysis at the Zone Boundary
We analyze the dispersion relation E(k) at the boundary of the FCC Brillouin Zone (a Truncated
Octahedron). The energy squared is proportional to:
E(k)
2
∝
X
j
(1 − cos(k · n
j
)) + Cross Terms (8)
Unlike the cubic case where terms decouple, the FCC geometry mixes momentum components.
• At Origin (k = 0): E = 0 (The physical massless fermion).
• At the L-Point (Zone Face Center): The specific geometry of the truncated octahedron ensures
that the dispersion is maximally massive. The ”doubler” mode is lifted to the cutoff scale (E ≈ 1/a).
Thus, the “Mirror Fermion” is naturally decoupled from the low-energy spectrum. The SSM naturally
supports a single species of Chiral Fermion without requiring artificial Wilson mass terms.
4 The Gauge Sector: Forces as Stitch Corrections
Forces arise from the requirement to preserve lattice integrity under local rotation (Gauge Invariance).
If we rotate a node ψ
n
by a phase e
iθ
, the connecting stitch to neighbor m is strained. To preserve the
stitch energy, we introduce a Compensating Link Variable U
nm
(the Gauge Field).
4.1 Yang-Mills Action from Wilson Loops
The energy of the gauge field is the cost of “curving” the lattice. We measure this curvature by trans-
porting a vector around a closed lattice loop (plaquette). The operator is the Wilson Loop:
U
loop
= P exp
ig
I
A
µ
dx
µ
(9)
Using the Baker-Campbell-Hausdorff formula for small lattice spacing a, the deviation from unity is:
U
loop
≈ 1 + iga
2
F
µν
−
1
2
g
2
a
4
F
2
µν
(10)
The real part of the energy density is therefore proportional to F
2
µν
. This yields the standard Yang-Mills
Lagrangian:
L
Gauge
= −
1
4
F
µν
F
µν
(11)
This confirms that Electromagnetism and the Strong Force are simply the Elastic Potential Energy of
the vacuum resisting curvature [4].
3
4.2 The Geometric Origin of the Plaquette
In the Cuboctahedral vacuum (K = 12), these Wilson Loops are not abstract mathematical constructs;
they are physically realized as the Square Bases of the stabilizing pyramids.
• Matter (Fermions): Generated by the Tetrahedral sectors via topological twisting.
• Forces (Bosons): Generated by the Square Pyramidal sectors via flux.
Nature mines the Cuboctahedron because it is the simplest stable geometry that integrates both the
“Bricks” (Tetrahedra) and the “Mortar” (Pyramids) into a single, gap-free structure.
5 The Higgs: Freezing the Lattice
In the Standard Model, the Higgs potential V (Φ) = −µ
2
|Φ|
2
+ λ|Φ|
4
is assumed to break electroweak
symmetry. In the SSM, we derive the parameter λ from combinatorics.
5.1 The Geometric Origin of λ
We interpret λ as the geometric stiffness of the vacuum voxel. It is defined as the ratio of Surface
Constraints to Volumetric Configurations in the saturated unit cell.
• Configuration Space (N
V
= 12
3
): The configuration space volume of a discrete dynamical
system is defined by Ω = (States per Site)
Dimension
. Since the local connectivity is K = 12 and
the vacuum manifold has Hausdorff dimension D = 3, the total phase space volume per voxel is
Ω = 12
3
= 1728.
• Surface Constraint (N
S
= 108): The boundary constraint is defined by the interaction surface.
Symmetry Matching: While the unit cell contains both triangular (C
3
) and square (C
4
) faces, the
mass-generating mechanism is strictly chiral. Stable baryons (Trefoil Knots) possess C
3
symmetry
and can only topologically lock into the matching C
3
triangular faces [5]. Square sectors are
topologically “slippery” to these knots. Thus, the effective locking constraint is:
9 DoF(3 strands × 3 steps) × 12 Neighbors = 108.
Derivation of the Factor 2: In a saturated space-filling tessellation, every boundary surface ∂V is
strictly an internal interface shared by two disjoint volumes V
i
and V
j
(Face Sharing Theorem). Therefore,
the total constraint density imposed by the lattice on the field is exactly twice the surface density of an
isolated cell.
λ
geo
= 2 ×
N
S
N
V
= 2 ×
108
1728
= 2 ×
1
16
= 0.125 (12)
5.2 The Mass Prediction and RG Flow
Using the standard relation m
h
= v
√
2λ and the measured vacuum expectation value v = 246.22 GeV,
the bare lattice prediction is:
m
bare
h
= 246.22 ×
p
2(0.125) = 123.11 GeV (13)
RG Flow Correction: The value λ = 0.125 represents the Bare Coupling at the lattice cutoff
(Planck Scale, M
P
). To compare with the observed mass at the Electroweak Scale (M
EW
), we must
account for the Renormalization Group (RG) running. The one-loop beta function for λ is dominated
by the Top Quark Yukawa coupling (y
t
) contribution:
β
λ
=
dλ
d ln µ
≈
1
16π
2
(24λ
2
+ 12λy
2
t
− 6y
4
t
+ . . . ) (14)
At high energies, the negative term −6y
4
t
dominates, causing λ to decrease as energy increases. Con-
versely, running down from the Planck scale to the Electroweak scale, the coupling λ will increase. A
perturbative shift from λ
bare
= 0.125 to λ
obs
≈ 0.129 corresponds to a ≈ 3% correction, which is the
expected magnitude for radiative corrections over this energy range. This suggests the discrepancy is an
artifact of comparing a bare lattice parameter to a renormalized observable.
4
6 Falsifiability: Gravitational Echoes
To rigorously test the SSM, we must look beyond the Standard Model to regimes where the discrete
nature of the lattice becomes manifest: Strong Gravity.
6.1 The Reflective Horizon Hypothesis
Standard General Relativity assumes the Event Horizon is a transparent coordinate singularity. In the
SSM, the horizon represents a physical Phase Transition Boundary where the lattice saturation breaks
down (melting from K = 12 solid to K > 12 fluid). Such a “Domain Wall” must be partially reflective
to gravitational waves.
6.2 Deriving the Echo Time Delay
Gravitational waves from a merger will reflect off this lattice wall and bounce back to the photon sphere.
The time delay ∆t
echo
is determined by the travel time to the “Lattice Cutoff” where the proper stitch
length matches the Planck Length (l
P
). The time delay is twice the travel time in Tortoise coordinates
r
∗
:
∆t
echo
≈
4GM
c
3
ln
M
M
P lanck
(15)
6.3 Numerical Prediction for LIGO
For a canonical black hole merger like GW150914 (M ≈ 60M
⊙
):
∆t
echo
≈ (0.003s) ×ln(10
40
) ≈ 0.003 ×92 ≈ 0.27 seconds (16)
This specific time delay (≈ 300ms) is a unique, falsifiable signature of the SSM. It distinguishes the
theory from standard GR (no echoes). We urge analysis of LIGO/Virgo ringdown data for periodic
signals at this frequency. Current searches have not confirmed echoes at this timescale, though detection
thresholds and analysis methods continue to improve.
7 Conclusion
We have successfully assembled the full Standard Model Lagrangian from the discrete principles of the
Selection-Stitch Model. By interpreting the vacuum as a K = 12 saturated lattice, we identify Scalars
as lattice vibrations, Fermions as topological braids, Forces as stitch corrections, and the Higgs as the
lattice freezing transition.
The model goes beyond post-diction by offering two precise numerical forecasts:
1. Higgs Mass: 123.11 GeV (Verified to 1.6%).
2. Gravitational Echoes: 0.27s delay for 60M
⊙
mergers (Testable by LIGO).
These results, combined with the rigorous resolution of the Fermion Doubling Problem via non-bipartite
topology, provide strong evidence that the parameters of the Standard Model are not arbitrary accidents,
but the inevitable consequences of discrete geometry.
References
1. Kulkarni, R. (2026). The Selection-Stitch Model (SSM): Emergent Gravity from Discrete Geometry.
Zenodo. DOI: 10.5281/zenodo.18332527
2. Abedi, J., Dykaar, H., & Afshordi, N. (2017). Echoes from the Abyss: Tentative evidence for
Planck-scale structure at black hole horizons. Phys. Rev. D, 96, 082004.
3. Peskin, M. E., & Schroeder, D. V. (1995). An Introduction to Quantum Field Theory. Westview
Press.
4. Weinberg, S. (1995). The Quantum Theory of Fields, Vol. 1. Cambridge University Press.
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