Quantum Entanglement as the Origin of the Gravitational Constant: Deriving the Planck Scale in the Selection-Stitch Model

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

) and the

Planck length (l

) 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 demonstrates

that G

is not a fundamental parameter. It is dynamically derived entirely from the topo-

logical 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 strictly

as G

2/3L

/(4 ln 2), where L is the physical lattice spacing. Equating this derived

to the standard Planck area ﬁxes the absolute physical lattice spacing of the universe at

L =

√

6 ln 2 l

≈ 1.84l

. We verify that the derived framework is consistent with all cur-

rent experimental constraints on gravitational physics, including the measured value of G

inverse-square law tests to 50 µm, equivalence principle bounds to 10

−15

, and gravitational

wave speed measurements. 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, Experimental Gravity

1. Introduction

A self-consistent theory of quantum gravity must explain the origin of its physical scales

rather than relying on them as empirical axioms. General Relativity mathematically couples

geometry to mass-energy via Newton’s gravitational constant (G

), while quantum mechan-

ics establishes the minimum limit of observability at the Planck length (l

ℏ/c

Standard physics treats these values as fundamental, measured inputs.

Holographic tensor networks oﬀer a discrete structural mechanism for spacetime geometry.

Within the Selection-Stitch Model (SSM) [1], the macroscopic 3D vacuum emerges as a

Face-Centered Cubic (FCC, K = 12) tensor network. This 3D bulk geometry projects

Email address: raghu@idrive.com (Raghu Kulkarni)

holographically from a 2D hexagonal (K = 6) boundary state [2]. The microscopic nodes

comprising this network do not possess inert spatial distances; their connections function

strictly as quantum entanglement bonds (maximally entangled Bell pairs).

Because bulk geometry is entirely determined by boundary entanglement [3], the macro-

scopic gravitational coupling must emerge directly from quantum information accounting.

We demonstrate that G

is not an independent physical parameter. By solving the Ryu-

Takayanagi (RT) holographic correspondence [4] exactly for the discrete geometry of the

SSM, we derive the explicit algebraic formula for G

. This geometric derivation dictates the

absolute physical length scale of the discrete vacuum, which we subsequently validate against

modern experimental astrophysical bounds.

2. The Holographic Entanglement Map

The derivation requires a strict geometric mapping between boundary quantum informa-

tion and bulk spatial area. The Stitch operator acts on the 2D hexagonal boundary as a

maximally entangled Bell pair projector [3].

Consider a connected, macroscopic bounded region A on the 2D boundary, enclosing a

perimeter P . Let L deﬁne the fundamental microscopic lattice spacing between adjacent

boundary nodes. The continuous boundary curve ∂A cuts across a discrete number of entan-

glement bonds. The exact number of severed bonds n

cut

evaluates to the perimeter divided

by the lattice spacing:

cut

(1)

Each severed bond represents a maximally entangled bipartite state. The reduced density

matrix ρ of half a Bell pair is maximally mixed, yielding a von Neumann entropy S =

−Tr(ρ ln ρ) = ln 2. The total entanglement entropy S

of region A strictly counts these

severed bonds:

= n

cut

ln 2 =





ln 2 (2)

The Lift operator projects this 2D boundary conﬁguration into the 3D FCC bulk [2].

Crystallographically, the continuous FCC lattice is mathematically constructed via the ABC

stacking sequence of 2D hexagonal layers.

To evaluate the resulting bulk area, we must determine the perpendicular vertical height

h between these stacked hexagonal planes. A tetrahedron formed by three adjacent nodes

in one layer and one nested node in the layer directly above possesses a uniform edge length

L. The centroid of the equilateral base lies at a horizontal distance of L/

√

3 from any base

vertex. Applying the Pythagorean theorem to this vertical cross-section extracts the exact

layer spacing:



√



= L

h =

L (3)

The minimal bulk surface γ

homologous to the boundary ∂A forms a straight vertical cur-

tain dropping exactly one unit layer deep into the bulk tensor network. The Ryu-Takayanagi

prescription selects the minimal-area bulk surface. Because penetrating additional lattice

layers multiplies the curtain area by an integer factor, the single-layer depth constitutes

the absolute geometric minimum. The area of this minimal bulk surface evaluates to the

boundary perimeter multiplied by the layer depth:

Area(γ

) = P × h = P

L (4)

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 con-

tinuous RT relation dictates that the entanglement entropy of the boundary region equals

the minimal bulk surface area divided by 4G

Area(γ

)

(5)

Substituting the exact discrete geometric derivations (Eq. 2 and Eq. 4) into the RT

relation yields:





ln 2 =

2/3L

(6)

We solve this relation explicitly for Newton’s constant. First, the macroscopic bound-

ary perimeter P cancels identically from both sides, verifying the scale-independence of the

holographic mapping:

ln 2

2/3L

(7)

Multiplying both sides by 4G

L isolates the gravitational constant:

ln 2 =

(8)

2/3

4 ln 2

≈ 0.2946L

(9)

This algebraic reduction proves Newton’s gravitational constant G

is not a foundational

parameter of nature. It operates as an emergent proportionality scalar derived exclusively

from the K = 6 topology of the boundary entanglement network and the

2/3 geometric

layer spacing of the FCC bulk.

4. Fixing the Absolute Lattice Scale

Equation 9 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 l

is deﬁned such that the grav-

itational constant identically equals the Planck area: G

= l

. Enforcing this macroscopic

observational constraint onto our microscopic derivation establishes the absolute scale of the

discrete vacuum:

2/3

4 ln 2

(10)

We invert this relation to solve for the fundamental lattice spacing L as a function of the

Planck length. Multiplying both sides by the inverse geometric coeﬃcient gives:

4 ln 2

2/3

= 4

ln 2 l

= 2

√

6 ln 2 l

(11)

Taking the square root yields the exact algebraic scale of the universe’s minimum geom-

etry:

L =

√

6 ln 2 l

(12)

Evaluating the scalar components (

√

6 ≈ 2.4495 and ln 2 ≈ 0.6931) produces the numer-

ical spacing:

L =

√

3.3957 l

≈ 1.8427 l

(13)

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

in Eq. 12 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

) must be proportionally larger than the standard single-bit Planck area

) to physically accommodate the ln 2 entanglement contribution of all 6 bonds. The

prefactor

√

6 ln 2 explicitly encodes the topological saturation density of the K = 12 FCC

framework.

6. Consistency with Experimental Gravitational Physics

The derivation in Sections 3–4 establishes the algebraic structure of G

within the SSM

framework. We now verify that this structure is consistent with all current experimental con-

straints on gravitational physics. We also clarify the precise testable content of the derivation.

6.1. The Numerical Value of G

The CODATA 2018 recommended value is G

= 6.67430×10

−11

−1

−2

[6]. Equation

9 expresses G

2/3L

/(4 ln 2). Since the Planck length is deﬁned as l

ℏG

equating G

= l

is algebraically equivalent to adopting Planck units. The derivation

therefore does not provide an independent numerical prediction of G

in SI units—that

would require an independent determination of L in meters, which is not available.

What the derivation does provide is the dimensionless ratio L/l

√

6 ln 2 ≈ 1.8427.

This is a parameter-free prediction: the lattice spacing is 1.84 times the Planck length, not

1.0, not 10, not 0.1. This speciﬁc ratio is ﬁxed entirely by the K = 6 boundary topology and

the

2/3 FCC layer geometry. Any independent measurement of the lattice spacing (e.g.,

through gravitational scattering at Planck energies) would test this ratio directly.

6.2. The Inverse-Square Law

Torsion-balance experiments have veriﬁed the Newtonian 1/r

gravitational force law

down to length scales of approximately 50 µm [7]. The SSM lattice spacing is L ≈ 1.84l

≈

3.0 × 10

−35

m. The ratio of the smallest tested scale to the lattice scale is:

50 µm

≈ 1.7 × 10

(14)

The SSM predicts exact 1/r

behavior at all distances above the lattice scale. In the contin-

uum limit (r ≫ L), the FCC tensor network reproduces the Einstein-Hilbert action [3] and

therefore recovers Newtonian gravity exactly. The inverse-square law tests are satisﬁed by

30 orders of magnitude.

6.3. The Weak Equivalence Principle

The MICROSCOPE satellite experiment constrains violations of the weak equivalence

principle (WEP) to the Eötvös parameter η < 1.5×10

−15

[8]. In the SSM, both inertial mass

and gravitational mass originate from the same mechanism: the entanglement disruption

energy of a defect in the tensor network. The inertial mass (resistance to acceleration)

measures the total bond-state disruption required to translate the defect through the lattice.

The gravitational mass (coupling to curvature) measures the total bond-state disruption that

sources the Ricci tensor via the Einstein equation. Because both quantities count the same

disrupted bonds, inertial and gravitational mass are identically equal at the microscopic level.

No WEP violation is predicted.

6.4. Gravitational Wave Speed and Lorentz Invariance

Treating spacetime as a discrete lattice fundamentally challenges established experimental

physics due to the threat of Lorentz Invariance Violation (LIV). In standard discrete spaces,

the propagation speed of waves theoretically depends on their energy, as shorter wavelengths

interact more directly with the lattice grain. These eﬀects have been rigorously tested by

modern astrophysics. Time-of-ﬂight measurements of high-energy photons from Gamma-Ray

Bursts, most notably the observations of GRB 090510 by the Fermi Large Area Telescope

[9], constrain linear Lorentz invariance violation to energy scales exceeding the Planck mass.

This eﬀectively rules out any naive discrete spacetime lattice with a spacing larger than

∼ 0.8l

Furthermore, the joint detection of GW170817 (gravitational waves) and GRB 170817A

(gamma-ray burst) constrains the relative speed of gravitational waves to |v

−c|/c < 10

−15

[10].

In a standard Newtonian or Minkowskian lattice framework, a grain size of L ≈ 1.84l

would produce massive, observable energy-dependent delays, severely violating these bounds.

However, the SSM survives these observational constraints because the 3D bulk is an isomet-

ric holographic projection of the 2D boundary [3]. Because the boundary state maintains

exact continuous rotational and translational symmetry, the projected 3D bulk inherits exact

macroscopic Lorentz invariance at all energy scales.

In the SSM, gravitational waves are transverse strain oscillations of the FCC tensor net-

work. The maximum propagation speed of any disturbance is set by the bond transmission

rate, which is identical to the maximum speed of electromagnetic disturbances. Both gravita-

tional and electromagnetic waves propagate on the same lattice at the same maximum bond

speed. The SSM therefore predicts exactly zero energy-dependent dispersion and strictly en-

forces v

= c, perfectly satisfying the Fermi LAT and LIGO/Virgo observational bounds.

6.5. Bekenstein-Hawking Entropy

The Bekenstein-Hawking entropy S

= A/(4l

) [11] is automatically satisﬁed by the

SSM construction. Equation 2 counts the entanglement entropy as the number of severed

bonds times ln 2. The RT relation (Eq. 5) maps this to S = Area/(4G

). Since G

= l

in natural units, this recovers S = A/(4l

) identically. The bond-counting origin of entropy

provides a microscopic derivation of the Bekenstein-Hawking formula: the entropy counts the

number of entanglement bonds severed by the horizon surface, with each bond contributing

exactly ln 2.

6.6. The Anomalous Spread in G

Measurements

Laboratory measurements of G

exhibit an anomalous spread of approximately ±500 ppm

across diﬀerent experimental groups [12], far exceeding the quoted individual uncertainties.

The SSM does not currently explain this discrepancy, as it predicts a single, exact value of

for a given L. The anomalous spread likely originates in systematic errors in the torsion-

balance experiments (material properties, metrology, vibration isolation) rather than in new

physics. We note this as an open experimental question that the SSM framework does not

address.

Table 1: Summary of experimental consistency checks. The SSM derivation strictly satisﬁes modern gravi-

tational and astrophysical constraints.

Experimental Test Constraint SSM Prediction Status

numerical value CODATA 2018 [6] G

2/3L

/(4 ln 2) Consistent

Inverse-square law 1/r

to 50 µm [7] 1/r

for r ≫ L Consistent

Equivalence principle η < 1.5 × 10

−15

[8] η = 0 (same bonds) Consistent

Lorentz invariance E

> 1.22E

P lanck

[9] Exact symmetry inherited Consistent

GW speed |v

− c|/c < 10

−15

[10] v

= c exactly Consistent

BH entropy S = A/4l

[11] Bond counting recovers S Consistent

spread ±500 ppm [12] Single exact scalar Open

7. 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

naturally precipitates from the foundational bond-counting

mathematics of the Ryu-Takayanagi relation.

This quantum information approach locks the absolute physical scale of the discrete vac-

uum at exactly L =

√

6 ln 2 l

≈ 1.84l

. This derivation relies exclusively on topological

integers and strict geometric identities, establishing the SSM as a parameter-free geometric

theory of quantum gravity. As demonstrated in Section 6, the derived framework is fully con-

sistent with all current experimental constraints on gravitational physics: the inverse-square

law to 50 µm, the weak equivalence principle, identical gravitational and electromagnetic

wave speeds, and the stringent Fermi Space Telescope bounds on Lorentz Invariance Viola-

tion. The testable content of the derivation is the dimensionless ratio L/l

= 1.8427, which

is a rigid, parameter-free prediction of the boundary topology.

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 =

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].

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