Quantum Entanglement as the Origin of the
Gravitational Constant: Deriving the Planck Scale from
Holographic Tensor Networks in the Selection-Stitch
Model
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
Abstract
Approaches to quantum gravity typically treat Newton’s gravitational constant (G
N
) and the
Planck length (l
P
) as fundamental, irreducible inputs. The Selection-Stitch Model (SSM) [1]
establishes the 3D Face-Centered Cubic (FCC) vacuum as an emergent structure projected
holographically from a 2D hexagonal boundary tensor network [2]. This Letter demon-
strates that G
N
is not a fundamental parameter. It is dynamically derived entirely from
the topological entanglement entropy of the boundary network. Solving the exact Ryu-
Takayanagi (RT) relation for the discrete geometry of the SSM [3, 4] yields Newton’s constant
as G
N
=
p
2/3L
2
/(4 ln 2), where L is the physical lattice spacing. Equating this derived G
N
to the standard Planck area fixes the absolute physical lattice spacing of the universe at
L =
p
2
√
6 ln 2 l
P
≈ 1.84l
P
. This result defines the scale of the discrete vacuum strictly from
quantum information mechanics.
Keywords: Quantum Gravity, Holographic Principle, Tensor Networks, Ryu-Takayanagi,
Planck Scale
1. Introduction
A self-consistent theory of quantum gravity must explain the origin of its physical scales.
General Relativity and the Standard Model treat Newton’s gravitational constant (G
N
) and
the Planck length (l
P
=
p
G
N
ℏ/c
3
) as empirical axioms. They are measured, not derived.
The Selection-Stitch Model (SSM) introduces a discrete structural mechanism for space-
time [1]. The 3D vacuum operates as an emergent Face-Centered Cubic (FCC, K = 12) tensor
network. This bulk is holographically projected from a 2D hexagonal (K = 6) boundary [2].
The connections between these microscopic nodes are not inert distances; they function as
physical quantum entanglement bonds (Bell pairs).
The geometry of the bulk is strictly determined by the entanglement structure of the
boundary [3]. Consequently, macroscopic gravitational dynamics must be derivable directly
from quantum information theory. We demonstrate that G
N
is not an independent pa-
rameter. Applying the Ryu-Takayanagi (RT) holographic correspondence [4], we derive the
Email address: raghu@idrive.com (Raghu Kulkarni)
exact algebraic formula for G
N
. This determines the absolute physical length scale of the
fundamental discrete vacuum.
2. The Holographic Entanglement Map
The derivation of the scale relies on tensor network holography. We review the geometric
mapping developed for the exact Lorentz invariance framework [3].
The Stitch operator on the 2D hexagonal boundary acts as a maximally entangled Bell
pair projector. For a connected macroscopic boundary region A with a perimeter P and
a microscopic lattice spacing L, the number of severed boundary stitches is n
cut
= P/L.
Each Bell pair contributes exactly ln 2 to the von Neumann entropy. The total entanglement
entropy of the region is determined by this bond-counting:
S
A
=
P
L
ln 2 (1)
The Lift operator projects this 2D boundary data into the 3D FCC bulk geometry. The
continuous FCC structure is mathematically constructed via the ABC stacking of hexagonal
layers. The perpendicular height between successive layers follows a strict geometric identity:
h =
p
2/3L.
The minimal bulk surface γ
A
homologous to the boundary ∂A forms a straight curtain
dropping exactly one layer deep into the bulk. The RT prescription selects the minimal-area
bulk surface. Since each additional layer multiplies the curtain area by an integer factor, the
single-layer surface at depth h =
p
2/3L constitutes the unique geometric minimum. The
area of this minimal bulk surface is therefore:
Area(γ
A
) = P ×
r
2
3
L (2)
3. Deriving Newton’s Constant
The Ryu-Takayanagi (RT) relation translates boundary quantum information into bulk
gravitational geometry [4]. The exact holographic mapping of the RT formula onto flat
isometric tensor networks is rigorously established by the HaPPY framework [5]. The RT
relation defines the entanglement entropy of the boundary region as proportional to the
minimal bulk surface area, scaled inversely by Newton’s constant:
S
A
=
Area(γ
A
)
4G
N
(3)
Substituting the discrete geometric identities (Eq. 1 and Eq. 2) into the RT relation
yields:
P
L
ln 2 =
P ×
p
2/3L
4G
N
(4)
The macroscopic boundary perimeter P cancels on both sides of the equation. Solving
the remaining expression for Newton’s constant yields:
G
N
=
p
2/3L
2
4 ln 2
≈ 0.2946L
2
(5)
2
Newton’s gravitational constant G
N
is not a foundational parameter of nature. It acts
as an emergent proportionality constant dictated exclusively by the K = 6 topology of the
boundary entanglement network and the
p
2/3 layer spacing of the emergent FCC bulk.
4. Fixing the Absolute Lattice Scale
Equation 5 expresses the macroscopic gravitational constant in terms of the microscopic
lattice spacing L. We use this algebraic relationship to determine the physical size of the
vacuum lattice relative to the standard Planck scale.
In standard natural units (c = ℏ = 1), the Planck length is defined such that the gravita-
tional constant equals the Planck area, G
N
= l
2
P
. Enforcing this macroscopic observational
constraint establishes the absolute scale of the discrete vacuum:
l
2
P
=
p
2/3L
2
4 ln 2
(6)
Solving for L yields the fundamental lattice spacing of the universe:
L =
q
2
√
6 ln 2 l
P
(7)
Evaluating this algebraic constant gives the numerical value:
L ≈ 1.8427 l
P
(8)
This length represents the exact nearest-neighbor distance between nodes in the 3D FCC
bulk spacetime. This is not an arbitrary UV cutoff introduced to regularize mathematical
divergences. It is the rigid physical scale required by the exact holographic equivalence
between quantum entanglement and bulk geometry.
5. Physical Interpretation of the Geometric Factor
The Bekenstein-Hawking entropy formula implies a saturation limit of exactly 1 bit of
information (or 1/4 of the area in Planck units) per Planck area on a holographic screen. The
algebraic coefficient connecting L to l
P
in Eq. 7 characterizes the true degrees of freedom
housed within the SSM vacuum.
The hexagonal boundary lattice does not provide a single bit of information per node. It
provides K/2 = 6 independent entanglement channels (Bell pairs) per node. Because each
microscopic node acts as a multi-channel entanglement hub, the physical area corresponding
to one node (L
2
) must be proportionally larger than the standard single-bit Planck area (l
2
P
)
to physically accommodate the ln 2 entanglement contribution of all 6 bonds. The prefactor
p
2
√
6 ln 2 encodes the topological saturation density of the K = 12 FCC framework.
6. Conclusion
The Selection-Stitch Model does not assume the existence of gravity; it explicitly derives
it. By formalizing the discrete vacuum as an emergent holographic tensor network, New-
ton’s gravitational constant G
N
naturally precipitates from the foundational bond-counting
mathematics of the Ryu-Takayanagi relation.
3
This quantum information approach locks the absolute physical scale of the discrete vac-
uum at exactly L =
p
2
√
6 ln 2 l
P
≈ 1.84l
P
. This derivation relies exclusively on topological
integers and strict geometric identities, establishing the SSM as a parameter-free geometric
theory of quantum gravity.
Appendix A. Self-Contained SSM Summary
For peer-review context, we summarize the foundational Selection-Stitch Model (SSM)
kinematics utilized in this framework. Detailed derivations are available in the linked preprints.
A.1. K = 12 Lattice Saturation. The FCC lattice represents the unique solution to
the Kepler conjecture; the densest packing of identical spheres in 3D possesses a coordination
of K = 12. The vacuum tensor network saturates at this maximum limit, providing each
node with exactly 12 nearest-neighbor bonds of length L, where L is the fundamental lattice
constant [2].
A.2. The Hexagonal Boundary and the Lift Operator. The continuous FCC
structure is mathematically constructed via the ABC sequence stacking of 2D hexagonal
layers (K = 6). The Lift operator in the SSM maps states from the 2D hexagonal boundary
into the 3D FCC bulk. Because the geometric height between these stacked hexagonal planes
is mathematically fixed at h =
p
2/3L, discrete bulk surfaces inherit exact volumetric scaling
rules [2].
A.3. Isometric Tensor Network and Lorentz Invariance. The 3D bulk lattice acts
as a quasilocal isometric projection of the 2D boundary, in strict accordance with the Ryu-
Takayanagi prescription. The isometry mathematically maps boundary entanglement entropy
to bulk geodesic area. Because the 2D boundary maintains exact continuous rotational and
translational symmetry, the projected bulk inherits exact macroscopic Lorentz invariance [3].
References
[1] R. Kulkarni, “Geometric Phase Transitions in a Discrete Vacuum,” Zenodo: 10.5281/zen-
odo.18727238 (In Review) (2026).
[2] R. Kulkarni, “Constructive Verification of K = 12 Lattice Saturation: Exploring Kine-
matic Consistency in the Selection-Stitch Model,” Zenodo: 10.5281/zenodo.18294925
(In Review) (2026).
[3] R. Kulkarni, “Exact Lorentz Invariance from Holographic Projection: Explicit RT Ver-
ification and the Boundary Origin of Bulk Symmetry in the Selection-Stitch Model,”
Zenodo: 10.5281/zenodo.18856415 (In Review) (2026).
[4] S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from the
anti-de Sitter space/conformal field theory correspondence,” Phys. Rev. Lett. 96, 181602
(2006).
[5] F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, “Holographic quantum error-
correcting codes: Toy models for the bulk/boundary correspondence,” JHEP 06, 149
(2015).
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