Geometric Renormalization of the Speed of Light and the
Origin of the Planck Scale in a Saturation-Stitch Vacuum
Raghu Kulkarni
1
1
Independent Researcher, Calabasas, CA
∗
(Dated: February 8, 2026)
Abstract
Attempts to reconcile General Relativity with Quantum Mechanics often falter on the prob-
lem of discretizing spacetime without violating Lorentz invariance. In this work, we propose
the Selection-Stitch Model (SSM), a discrete vacuum framework based on a saturated
Face-Centered Cubic (FCC) lattice (K = 12). We demonstrate two fundamental geometric
renormalizations. First, we show that the physical speed of light (c) emerges as a renormal-
ization of the lattice hopping speed, specifically c = 4v
lattice
, due to the constructive interfer-
ence of the 12 nearest-neighbor paths on the Cuboctahedron unit cell. Second, utilizing this
relation, we derive the lattice spacing a not as a free parameter, but as the inevitable geomet-
ric limit where the vacuum’s elastic energy density encounters the Schwarzschild constraint.
Fixed by the packing efficiency of the FCC unit cell (V = a
3
/
√
2), we derive a fundamental
lattice spacing of a ≈ 0.77l
P
. We explicitly derive the continuum limit, showing that Lorentz
invariance is restored for fermions up to O(a
2
) corrections due to the centrosymmetry of the
lattice. Furthermore, we demonstrate that this same lattice centrosymmetry enforces the
vanishing of the QCD topological charge density, naturally resolving the Strong CP problem
(θ
QCD
= 0) without requiring an axion. Finally, we discuss the geometric obstruction to
point-particle quantization of gravity in this framework.
I. INTRODUCTION
The unification of Quantum Field Theory (QFT) and General Relativity remains the
paramount challenge of modern physics. A central obstacle is the continuous nature of space-
time assumed in QFT, which leads to ultraviolet divergences when applied to gravity. Discrete
spacetime models offer a natural regularization mechanism [1], but they typically suffer from
the breakdown of Lorentz invariance (anisotropy) at the grid scale [2].
We present the Selection-Stitch Model (SSM), formalized within the Unified Geomet-
ric Lattice Theory (UGLT) framework [3]. This approach models the vacuum as a discrete,
saturated geometry. Unlike hypercubic models, the SSM utilizes a Face-Centered Cubic (FCC)
lattice (K = 12), the densest possible sphere packing [4], which possesses sufficient symmetry
to recover isotropy in the continuum limit.
In this paper, we derive:
1. The Geometric Renormalization of Light, deriving c = 4v
lattice
.
2. The Origin of the Planck Scale, deriving a ≈ 0.77l
P
from the geometry of the FCC
unit cell.
∗
raghu@idrive.com
2
3. The Restoration of Lorentz Invariance for fermions.
4. The Resolution of the Strong CP Problem via lattice centrosymmetry.
II. GEOMETRIC EMERGENCE OF SCALES
A. Geometric Boost of Light Speed (c = 4)
The propagation of a fermion ψ(x) is governed by a discrete hopping operator summed over
the 12 nearest-neighbor vectors ˆn
j
of the cuboctahedral unit cell.
We define the lattice vectors normalized by the spacing a:
n ∈ {(±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1)} (1)
The discrete Dirac operator D
SSM
is:
D
SSM
(k) =
1
τ
12
X
j=1
(γ
µ
ˆn
j,µ
)e
ik·n
j
(2)
where τ is the discrete time step.
Expanding for small k (long wavelength limit), the kinetic term depends on the tensor sum
S
µν
=
P
ˆn
µ
j
ˆn
ν
j
. For the FCC lattice, this sum is isotropic:
12
X
j=1
ˆn
µ
j
ˆn
ν
j
= 4δ
µν
(3)
This yields the effective speed of light:
c = 4
a
τ
= 4v
lattice
(4)
This indicates that physical light is a collective excitation moving 4 times faster than the raw
lattice updates (v
lattice
= a/τ ). This renormalization factor of 4 is a direct consequence of the
FCC coordination number and is a prerequisite for defining the energy scales of the vacuum.
B. Geometric Origin of the Planck Scale (a ≈ 0.77l
P
)
With the physical speed of light established (c = 4a/τ ), we can now determine the absolute
scale of the lattice spacing a.
In the SSM, the vacuum energy is distributed across discrete nodes. The maximum energy
density the lattice can support is constrained by the volume of the Wigner-Seitz cell. For an
FCC lattice with bond length a, the effective volume per node is:
V
node
=
a
3
√
2
(5)
The maximum energy a single node can transmit is limited by the lattice update rate τ. Using
the derived speed of light relation c = 4a/τ , the maximum energy per mode is:
E
max
=
ℏ
τ
=
ℏc
4a
(6)
3
The condition for gravitational stability is that the lattice energy density ρ
lattice
must not exceed
the critical Planck energy density ρ
P
. We identify the physical vacuum with the maximally
packed configuration, where the energy density saturates this gravitational bound; any smaller
spacing would produce nodes whose energy density exceeds the Schwarzschild limit, triggering
collapse.
ρ
lattice
=
E
max
V
node
=
ℏc/4a
a
3
/
√
2
=
√
2ℏc
4a
4
(7)
Setting this equal to the Planck Energy Density ρ
P
= c
7
/(ℏG
2
):
√
2ℏc
4a
4
=
c
7
ℏG
2
=⇒ a
4
=
√
2
4
ℏ
2
G
2
c
6
(8)
Solving for a:
a =
√
2
4
!
1/4
r
ℏG
c
3
≈ 0.77 l
P
(9)
Substituting the values of fundamental constants, we obtain a theoretical lattice spacing of a ≈
1.25×10
−35
m, which is approximately 77% of the conventional Planck length (l
P
≈ 1.62×10
−35
m). This geometric correction factor arises directly from the specific packing efficiency of the
K = 12 lattice . It demonstrates that the Planck scale is not an arbitrary regulator, but
the specific geometric limit where the vacuum packing density encounters the Schwarzschild
constraint. Furthermore, the precise factor a ≈ 0.77l
P
constitutes a falsifiable prediction of the
SSM, distinguishing it from generic Planck-scale discreteness models which typically assume a
coefficient of unity.
III. THE CONTINUUM LIMIT AND LORENTZ INVARIANCE
To validate the model as a physical theory, we must show that it recovers the continuous
Dirac equation as a → 0. We perform a Taylor expansion of the finite difference operator acting
on a smooth field ψ(x):
ψ(x + n) =
∞
X
m=0
1
m!
(n · ∇)
m
ψ(x) (10)
A. First Order (Mass Term)
The m = 0 term involves the sum of direction vectors:
12
X
j=1
ˆn
j
= 0 (11)
This vanishes due to inversion symmetry (n → −n), ensuring no lattice-induced mass generation
(doubling problem resolution) [5].
4
B. Second Order (Kinetic Term)
The m = 1 term gives the derivative ∂
µ
:
1
τ
X
j
(γ · ˆn
j
)(n
ν
j
∂
ν
)ψ =
a
τ
γ
µ
X
j
ˆn
µ
j
ˆn
ν
j
∂
ν
ψ (12)
Using the identity
P
ˆn
µ
ˆn
ν
= 4δ
µν
, we recover:
4
a
τ
γ
µ
∂
µ
ψ (13)
This is the standard relativistic Dirac operator, identifying c
phys
= 4(a/τ).
C. Third Order (Lorentz Violation)
Crucially, the next order term (m = 2, proportional to a
2
∂
2
) involves the tensor
P
n
µ
n
ν
n
λ
.
Because the lattice is centrosymmetric (invariant under x → −x), all odd moments of the
distribution vanish:
12
X
j=1
n
µ
j
n
ν
j
n
λ
j
= 0 (14)
This implies that the leading-order Lorentz violating terms are automatically zero. The first
deviations appear at O(a
2
), suppressed by the square of the Planck scale. This is consistent
with current experimental bounds on Lorentz violation [6, 7].
IV. COROLLARY: RESOLUTION OF THE STRONG CP PROBLEM
The vanishing of odd lattice moments derived in Section III.C has profound implications for
the Strong CP problem. The QCD θ-term is given by:
L
θ
=
θg
2
32π
2
G
a
µν
˜
G
aµν
(15)
This term violates CP symmetry because the operator G
˜
G ∝ E ·B is a pseudoscalar (odd under
Parity).
Under the lattice inversion operation I : x → −x (which is a symmetry of the FCC vacuum,
P
latt
= +1), the pseudoscalar density transforms as:
P (G
˜
G) = −(G
˜
G) (16)
Since the vacuum state is invariant under Parity (P |0⟩ = |0⟩), the expectation value of any
P-odd operator summed over the centrosymmetric unit cell must vanish:
⟨θ
QCD
⟩ ∝
X
cell
⟨G
˜
G⟩ = 0 (17)
Thus, the ”fine-tuning” θ ≈ 0 is actually a geometric identity forced by the O
h
point group
symmetry of the vacuum. This resolves the Strong CP problem without requiring an axion.
5
V. DISCUSSION: OBSTRUCTION TO GRAVITON QUANTIZATION
The discrete nature of the SSM creates a geometric distinction between gauge fields and
gravity.
• Fermions (Topological): Modeled as ”braids” or knots in the lattice links. These are
topological defects characterized by a winding number, which can be strictly localized
within a single unit cell.
• Gravity (Metric): Modeled as the elastic deformation of the lattice spacing a(x). Cur-
vature is a relation between multiple unit cells; one cannot define the curvature of a single
point.
This creates a geometric obstruction to quantization. While fermions can be treated as point-like
particles, gravitational excitations are inherently non-localizable below the scale of the lattice
patch. This implies that a ”graviton” (a point-particle of gravity) is an ill-defined concept in
the SSM, suggesting that gravity must remain semi-classical in this framework [8].
VI. CONCLUSION
We have derived the fundamental scales of physics from the geometry of a saturated K = 12
vacuum lattice. We showed that the speed of light is a geometric renormalization (c = 4a/τ)
arising from the constructive interference of nearest-neighbor paths. Using this relation, we
derived the lattice spacing a ≈ 0.77l
P
, demonstrating that the fundamental pixel of the universe
is fixed by the packing efficiency of the Cuboctahedron unit cell against the Schwarzschild
constraint.
Crucially, the centrosymmetry of the FCC lattice eliminates leading-order Lorentz violations
and simultaneously forces the vanishing of the QCD θ parameter, resolving the Strong CP
problem. These results suggest that the fundamental symmetries of the Standard Model are
emergent properties of a discrete, centrosymmetric vacuum.
[1] K. G. Wilson, ”Confinement of quarks,” Phys. Rev. D 10, 2445 (1974).
[2] J. B. Kogut and L. Susskind, ”Hamiltonian formulation of Wilson’s lattice gauge theories,” Phys.
Rev. D 11, 395 (1975).
[3] R. Kulkarni, ”Unified Geometric Lattice Theory (UGLT): Deriving Gauge Couplings, Mass Spectra,
and Gravity from a K=12 Vacuum,” Zenodo, doi:10.5281/zenodo.18520623 (2026).
[4] T. C. Hales, ”A proof of the Kepler conjecture,” Annals of Mathematics 162, 1065 (2005).
[5] R. Kulkarni, ”Fermion Chirality from Non-Bipartite Topology,” Zenodo, doi:10.5281/zenodo.18410364
(2026).
[6] V. A. Kosteleck´y and S. Samuel, ”Spontaneous breaking of Lorentz symmetry in string theory,” Phys.
Rev. D 39, 683 (1989).
[7] A. A. Abdo et al. (Fermi LAT Collaboration), ”A limit on the variation of the speed of light arising
from quantum gravity effects,” Nature 462, 331 (2009).
[8] R. Kulkarni, ”The Geometry of the Standard Model: Deriving the Higgs Mass,” Zenodo,
doi:10.5281/zenodo.18292757 (2026).
