Emergent Linearized Gravity from D4 Lattice Elasticity

Emergent Linearized Gravity from D

Lattice Elasticity:

Exact SO(4) Isotropy and Defects as Gravitational Sources

Raghu Kulkarni

∗

SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA

May 2026

Abstract

In the SSMTheory framework, the D

root lattice is physical four-dimensional spacetime

and the matter content arises from trapped lattice defects. We construct the elasticity theory

of small lattice deformations on D

and identify the resulting long-wavelength eﬀective action

with linearized gravity coupled to defect sources. The central technical result is that the rank-

four bond tensor T

µνρσ

n∈N

on the D

kissing set is exactly the fully-symmetric

isotropic tensor 4(δ

µν

ρσ

+ δ

µρ

νσ

+ δ

µσ

νρ

), with no anisotropic correction at any order. The

hypercubic Z

lattice fails this isotropy, exhibiting cubic anisotropy, and even the 3D FCC

sublattice contained as a spatial slice of D

fails rank-four isotropy; the missing pieces are

supplied exactly by the twelve cross-slice nearest neighbors. The elastic Lagrangian takes the

isotropic form L = (λ/2)(∇·u)

+µ ϵ

αβ

with equal Lamé parameters λ = µ = k, Poisson ratio

ν = 1/5, and bulk modulus K = 3k/2. Bare displacements correspond to gauge transformations

of the linearized metric; the gauge-invariant gravitational content is sourced by defects, with

the trapped tetrahedral defects identiﬁed in the matter paper as quarks playing the role of

localized gravitational sources. The static long-distance strain ﬁeld has Kelvin form u ∼ 1/r,

reproducing the Newton potential, and the Newton constant is identiﬁed as G

∼ ℓ

/ℏ,

equivalently G

∼ ℏc/M

, of Planck-scale magnitude. The construction is explicitly linearized

and weak-ﬁeld; full nonlinear General Relativity, the cosmological constant, and gravitational

radiation are not addressed here.

1 Introduction

1.1 D

as physical spacetime in the SSM framework

This paper continues the SSMTheory program in which the Face-Centered Cubic (FCC) lattice

is the substrate of three-dimensional physical space [1] and the D

root lattice is physical four-

dimensional spacetime [3]. The matter paper [1] establishes that trapped tetrahedral defects in

the FCC lattice carry quark quantum numbers (fractional charge, three color charges, linear con-

ﬁnement) and computes the proton-to-electron mass ratio m

= 1836 from a single-coupling

structural counting. The mass-energy-information paper [2] identiﬁes the ﬁve stable mass eigenval-

ues {1, 207, 273, 1836, 1839} with the electron, muon, pion, proton, and neutron via a [[192, 130, 3]]

CSS-code veriﬁcation cost. The D

paper [3] extends the spatial picture to four dimensions:

∩ {x

= 0} = FCC by the slicing theorem, the lattice is the densest 4D packing with kiss-

ing number K = 24, and the Wilson lattice gauge action on its 32 triangular plaquettes per unit

∗

raghu@idrive.com

cell gives an exactly SO(4)-symmetric Yang-Mills theory with no anisotropy correction at any order

in the lattice spacing.

The present paper addresses the third sector of physics: gravity. Under SSM the lattice spacing

a is a ﬁxed physical scale of order the Planck length ℓ

, not a continuum-limit regulator to be

sent to zero. The D

lattice is therefore not just a mathematical scaﬀold but an elastic medium

with ﬁnite bond stiﬀness, and small deformations of the lattice obey a well-deﬁned long-wavelength

elasticity theory. The idea that gravity emerges from underlying microscopic degrees of freedom, in

the same sense that hydrodynamics emerges from molecular dynamics, goes back to Sakharov [7]

and has been pursued in many forms since [8, 5, 6]. We work within the explicit dictionary between

lattice elasticity and linearized gravity developed in [5, 6], with the lattice geometry determining

the elastic constants.

1.2 Main results

We establish four results:

Rank-four isotropy of D

(Theorem 1). The bond tensor T

µνρσ

summed

over the 24 nearest-neighbor vectors of D

is exactly proportional to the unique fully-symmetric

isotropic rank-four tensor:

µνρσ

= 4 (δ

µν

ρσ

+ δ

µρ

νσ

+ δ

µσ

νρ

) .

This is a stronger isotropy property than the rank-two structure tensor

µν

= 6δ

µν

established

in [3]. The hypercubic lattice Z

and the 3D FCC sublattice both fail rank-four isotropy.

Isotropic elastic Lagrangian (Theorem 2). The long-wavelength elastic energy density on

elastic

(∇ · u)

+ µ ϵ

αβ

, λ = µ = k,

where k is the bond spring constant. Both Lamé parameters are equal, the Poisson ratio is ν =

1/5, and the bulk modulus is K = 3k/2. Both longitudinal and transverse modes have isotropic

dispersion at every order in the wavevector.

Defects as gauge-invariant gravitational sources (Section 5). Bare displacement ﬁelds

(x) correspond to gauge transformations of the linearized metric perturbation h

µν

= ∂

∂

and carry no physical (gauge-invariant) content. The gauge-invariant gravitational content

is sourced by lattice defects. The trapped tetrahedral defects identiﬁed in the matter paper as

quarks [1] play the role of localized strain sources. The defect-induced strain ﬁeld has Kelvin form

at long distances with potential ∼ 1/r, reproducing the Newton law.

Planck-scale Newton constant (Section 5). Identifying the lattice spacing with the Planck

length and the bond spring constant with the Planck-scale stiﬀness k ∼ E

/ℓ

, the elastic Newton

constant works out to G

∼ ℏc/M

of the correct order of magnitude to match observed gravity.

The precise dimensionless prefactor depends on the topological strength of the defect source.

1.3 Scope and limitations

This paper is explicitly linearized and weak-ﬁeld. We do not derive the full nonlinear Einstein-

Hilbert action, do not compute Riemann curvature beyond the linearized order, and do not address

gravitational radiation, the cosmological constant, or the geometric structure of strong-ﬁeld solu-

tions. The construction shares the technical core of the elasticity-to-gravity programs of Kleinert

and others [5, 6]: the elastic Lagrangian on a regular lattice admits a linearized-gravity reinterpreta-

tion when topological defects are present, and the lattice geometry determines the elastic constants.

What is new here is the application to the D

lattice, where the rank-four isotropy theorem ensures

the construction is exactly Lorentz-invariant at every order in the lattice spacing, and where the

matter-as-defects identiﬁcation from [1] gives the gravitational source picture directly without an

additional matter Lagrangian.

2 Preliminaries on the D

Lattice

We recall the basic structural facts about D

established in [3].

The D

root lattice is deﬁned by

= {x ∈ Z

: x

+ x

≡ 0 (mod 2)}.

Its 24 nearest-neighbor vectors are

N = {±e

± e

: 1 ≤ µ < ν ≤ 4},

each of squared length |n|

= 2. The kissing number K = 24 saturates the four-dimensional

kissing-number bound. The slicing theorem [3] states that

∩ {x

= c}

∼

FCC for even c, D

∩ {x

= c}

∼

FCC (shifted) for odd c.

The 24 nearest-neighbor vectors partition naturally as 12 + 12:

• Spatial (in-slice) neighbors: the 12 vectors ±e

±e

with i, j ∈ {1, 2, 3}, i < j. These reproduce

the FCC kissing set in the spatial slice x

= 0.

• Cross-slice (cross-time) neighbors: the 12 vectors ±e

± e

for i ∈ {1, 2, 3}. These connect

adjacent x

slices.

The rank-two structure tensor of D

is [3]

µν

n∈N

= 12 δ

µν

or in unit-length normalization

µν

= 6δ

µν

. This rank-two isotropy is shared by FCC in three

dimensions, hypercubic in any dimension, and any lattice whose nearest-neighbor star has cubic or

higher symmetry.

3 The Rank-Four Isotropy Theorem

The novel content begins at rank four. For elasticity the relevant tensor is built from four powers

of nearest-neighbor vectors, and its symmetry properties determine the leading anisotropies of the

elastic action.

Deﬁnition 1. The rank-four bond tensor of a Bravais lattice with nearest-neighbor set N is

µνρσ

n∈N

This tensor is automatically fully symmetric in all four indices.

Theorem 1 (Rank-four isotropy of D

). The rank-four bond tensor of D

is exactly the fully-

symmetric isotropic rank-four tensor:

µνρσ

= 4



µν

ρσ

+ δ

µρ

νσ

+ δ

µσ

νρ



Proof. We give a symmetry argument followed by direct veriﬁcation.

Symmetry argument. The point group of D

is the Weyl group F

of order 1152, which acts on

the 24 nearest-neighbor vectors transitively. Any rank-four tensor T

µνρσ

deﬁned as a sum over an

-orbit is automatically F

-invariant. By construction T is also fully symmetric in all four indices.

The space of fully-symmetric rank-four tensors on R

is 35-dimensional. Imposing invariance

under B

(the signed-permutation subgroup of F

, of order 384) reduces this to a 2-dimensional

space spanned by

µνρσ

= δ

µν

ρσ

+ δ

µρ

νσ

+ δ

µσ

νρ

and δ

(4)

µνρσ

(

1 µ = ν = ρ = σ,

0 otherwise.

-invariance forces nonzero components only on patterns where each axis appears an even number

of times, leaving two pattern classes: (a, a, a, a) giving δ

(4)

, and (a, a, b, b) giving the cross-pair

pieces of S.) The full F

contains elements outside B

; speciﬁcally, the orthogonal transformation

H =



1 1 1 1

1 1 −1 −1

1 −1 1 −1

1 −1 −1 1



permutes the D

nearest neighbors (e.g. H(1, −1, 0, 0)

= (0, 0, 1, 1)

) and has

entries outside {0, ±1}, so H ∈ F

. Under H, the isotropic S is invariant (it is O(4)-invariant),

while δ

(4)

is not invariant. Hence F

-invariance reduces the B

-invariant 2-d space to the 1-d

subspace spanned by S, and T

µνρσ

= αS

µνρσ

for some α.

Normalization. Evaluating at (µ, ν, ρ, σ) = (1, 1, 1, 1): T

1111

. The D

neighbors with

= 0 are ±e

± e

for j ∈ {2, 3, 4}, giving 3 × 4 = 12 vectors each with n

= 1. So T

1111

= 12,

while S

1111

= 3. Hence α = 4.

Decomposition check. The decomposition N = N

⊔N

(12+12 neighbors) gives T = T

Neither summand alone is proportional to S — the spatial sum vanishes whenever any index equals

4, and the cross-slice sum has a speciﬁc bias toward indices that include 4. Their sum, however, is

exactly 4S. The component-by-component veriﬁcation and the explicit check that S is H-invariant

while δ

(4)

is not are carried out in the script verify_gravity.py [9] (§3, §3b, and §5).

Proposition 1 (Comparison with other lattices). For comparison:

1. Hypercubic Z

: T

µνρσ

= 2δ

(4)

µνρσ

, where δ

(4)

denotes the rank-four Kronecker symbol (which is

1 if all four indices are equal and 0 otherwise). This is anisotropic: it has no δ

µν

ρσ

cross-pair

structure.

2. 3D FCC sublattice of D

(alone): T

FCC

ijkl

with i, j, k, l ∈ {1, 2, 3} is not proportional to the 3D

isotropic tensor δ

+ δ

3. D

requires both spatial and cross-slice neighbors to achieve exact rank-four isotropy. Neither

subset alone is isotropic; the sum is.

The veriﬁcation of all three claims is in verify_gravity.py [9], §4 and §5.

The geometric meaning of Theorem 1: the cross-slice neighbors of D

carry exactly the anisotropy-

canceling weight that completes the spatial FCC’s rank-four anisotropy into the unique isotropic

rank-four tensor in 4D. The fact that D

is the lattice where this completion is exact, rather than

approximate, is the structural reason emergent gravity on D

inherits exact SO(4) Euclidean sym-

metry (equivalently, exact Lorentz invariance after Wick rotation) at every order in the lattice

spacing.

This parallels the gauge-sector result of [3]: the D

plaquette stiﬀness tensor (built from sums

of plaquette holonomies) is exactly proportional to the pair-antisymmetric isotropic tensor δ

µρ

νσ

−

µσ

νρ

at rank four, giving exact SO(4) Yang-Mills. The elastic stiﬀness tensor (built from bond

stretches) is exactly proportional to the fully-symmetric isotropic tensor S

µνρσ

at rank four, giving

exact SO(4) elasticity. Both rely on the same F

symmetry of D

; they are the unique O(4)-

singlets of their respective symmetry types (antisymmetric and symmetric pair structure). Gauge

and gravity inherit exact Lorentz invariance from the same lattice geometry.

4 Elastic Lagrangian on D

4.1 Bond-stretch energy in the long-wavelength limit

Let u

(x) denote a small displacement ﬁeld deﬁned on D

sites. For a nearest-neighbor pair

(x, x + n) the bond length changes by

∆L

= |x + n + u(x + n) − x −u(x)| − |n| ≈ ˆn · [u(x + n) − u(x)],

where ˆn = n/|n| is the unit vector along the bond. Expanding u(x + n) − u(x) ≈ n

∂

for

slowly-varying u:

∆L

≈

|n|

∂

The Hookean bond energy per bond is

k(∆L

)

, where k is the bond spring constant (assumed

uniform on the lattice). Summing over bonds touching site x and dividing by 2 to avoid double-

counting:

elastic

(x) =

|n|

n∈N

∂

4|n|

αβγδ

∂

For D

, |n|

= 2 and T

αβγδ

= 4S

αβγδ

where S denotes the fully-symmetric isotropic tensor, so

elastic

(x) =

αβγδ

∂

4.2 Expansion in strain and rotation

Decompose ∂

= ϵ

αβ

+ ω

αβ

into symmetric strain and antisymmetric rotation,

αβ

(∂

+ ∂

), ω

αβ

(∂

− ∂

Expanding S

αβγδ

= δ

αβ

γδ

+ δ

αγ

βδ

+ δ

αδ

βγ

αβγδ

∂

= (∇ · u)

+ (∂

)(∂

) + (∂

)(∂

Substituting ∂

= ϵ

αβ

+ ω

αβ

and using ϵ

αβ

= 0:

(∂

)(∂

) = ϵ

αβ

+ ω

αβ

, (∂

)(∂

) = ϵ

αβ

− ω

αβ

The rotational pieces cancel, leaving

αβγδ

∂

= (∇ · u)

+ 2 ϵ

αβ

Theorem 2 (D

elastic Lagrangian). The long-wavelength elastic Lagrangian on D

elastic

(∇ · u)

+ µ ϵ

αβ

, λ = µ = k.

The Lamé parameters are equal, both given by the bond spring constant.

Corollary 1 (Elastic moduli in 4D). With λ = µ = k:

• Bulk modulus K = λ + 2µ/D = 3k/2 in D = 4.

• Poisson ratio ν = λ/[(D − 1)λ + 2µ] = 1/5 in D = 4.

• Longitudinal wave speed c

= (λ + 2µ)/ρ = 3k/ρ.

• Transverse wave speed c

= µ/ρ = k/ρ.

• Ratio c

√

Here ρ is the lattice mass density per unit cell.

The equality λ = µ is geometrically forced by the rank-four isotropy theorem: any lattice with

µνρσ

proportional to the unique fully-symmetric isotropic tensor automatically gives equal Lamé

parameters at the bond-stretch level. The D

lattice realizes this exactly; hypercubic and FCC

lattices do not.

5 Defects as Gravitational Sources

5.1 Pure-gauge nature of bare displacements

The displacement ﬁeld u

has four components in 4D; the symmetric metric perturbation h

µν

has

ten components. The two cannot be in one-to-one correspondence. The standard observation is

that under the linearized identiﬁcation

µν

= ∂

+ ∂

= 2 ϵ

µν

the metric perturbation h has exactly the form of a linearized coordinate transformation, which is

pure gauge in linearized gravity:

δh

µν

= ∂

+ ∂

(gauge transformation, ξ arbitrary).

A pristine D

lattice with smooth displacement u

therefore carries no gauge-invariant gravitational

content. The bond-stretch elastic energy is real and positive, but in the gravitational interpretation

it corresponds to gauge degrees of freedom.

5.2 Defects supply the gauge-invariant content

Following Kleinert [5] and the elasticity-to-gravity dictionary, the gauge-invariant gravitational con-

tent of the lattice is carried by topological defects: dislocations (which break translational continuity

of u) and disclinations (which break rotational continuity). The linearized Riemann curvature ten-

sor of the emergent metric is built from second derivatives of h

µν

, and the defect density gives a

gauge-invariant version of these second derivatives that does not vanish for a pristine lattice with

smooth deformations.

The trapped tetrahedral defect identiﬁed in the matter paper [1] is a localized conﬁguration

where the lattice’s regular tetrahedral structure is broken by an extra node bonded to four FCC

sites. Such a defect is a source for the displacement ﬁeld u in the surrounding lattice. The defect

is also the carrier of quark quantum numbers in the matter paper: charge, color, and conﬁnement

properties. The same defect therefore plays two roles: as the matter content of the theory (via the

matter paper) and as the gravitational source (via the present paper).

5.3 Static defect strain: Kelvin solution

We consider a defect localized in the x

= 0 slice, with spatial location (x

, x

) at the origin and

with the displacement ﬁeld u

, x

) independent of x

. The x

direction here plays the role

of Euclidean time, and the static restriction projects onto time-translation-invariant conﬁgurations.

Under this restriction the 4D Navier equation

(λ + µ) ∂

(∂ ·u) + µ □ u

= j

(where □ =

∂

is the 4D Laplacian) splits into independent equations for the spatial components

(i = 1, 2, 3) and the time component u

. Since ∂

u = 0, the operator □ reduces to the 3D

Laplacian ∇

, and ∂ · u = ∂

(the 3D divergence). The time-component equation becomes

µ∇

= j

, which is decoupled and not relevant for gravitational physics (it corresponds to a

longitudinal lattice translation in the time direction). The spatial equations

(λ + µ) ∂

(∂

) + µ ∇

= j

with λ = µ = k become

2k ∂

(∂

) + k ∇

= j

which is the standard Navier equation of 3D isotropic elasticity with the Lamé parameters extracted

in Theorem 2.

For a point defect at the origin with isotropic source strength F

, j

= F

(r), the standard

Kelvin solution gives

(r) =

16πµ(1 − ν)



(3 − 4ν)



With ν = 1/5 and µ = k (Corollary 1):

(r) =

16πk(4/5)





16πk





The leading long-distance behavior is u ∼ 1/r, equivalent to a Newton potential. Figure 1 visualizes

the strain ﬁeld around the defect.

2 1 0 1 2

(lattice units)

(A) Kelvin displacement field

(

) around a defect

tetrahedral

defect

def

(

(lattice units)

(

(B) Long-distance falloff: |

| 1/

matches Newton

Kelvin solution

(Newton same slope)

(steeper, for contrast)

Figure 1: Defect-induced strain ﬁeld on D

. (A) Two-dimensional cross-section of the Kelvin

displacement ﬁeld u(r) in the (x

, x

) plane around a static tetrahedral defect (cyan) sourcing a

body-force



def

(yellow). The background colormap shows the magnitude |u(r)| on a logarithmic

scale; white arrows show the displacement direction. The strain ﬁeld is most intense near the

defect and decays radially. (B) Angle-averaged ⟨|u(r)|⟩ versus r on a log–log scale. The D

Kelvin

solution (solid blue) is parallel to the ∝ 1/r reference (dashed black), conﬁrming the Newton-

potential scaling; a ∝ 1/r

reference (dotted red) is shown for contrast to make the slope visibly

distinct from a steeper power law. The Planck-scale lattice cutoﬀ at r ∼ ℓ

is marked by the

gray dotted vertical. The defect (matter) and the strain ﬁeld (gravity) come from the same lattice

conﬁguration.

5.4 Identiﬁcation with linearized Newton gravity

The map from elastic Kelvin solution to gravitational potential requires care, because for a smooth

the metric perturbation h

µν

= 2ϵ

µν

is pure gauge (Section 5.1). The defect-induced Kelvin

displacement u

∼ 1/r, however, is singular at the defect location, and this singularity is what

makes it physical: ∇

(1/r) = −4πδ

(r), so the strain ﬁeld has a delta-function distributional

source at the defect location. The defect location is the support of gauge-invariant content; in the

Kleinert framework [5] this localized source is precisely a dislocation/disclination density that does

not vanish for the singular displacement.

We make the gravitational identiﬁcation precise via the static Poisson equation. The trace of

the linearized Einstein equation in the Newtonian gauge gives

∇

Φ = 4πG

where Φ = −h

/2 is the gravitational potential and ρ

is the matter density. In the elasticity

picture, the static defect density ρ

def

acts as the source. Both equations have the same Green’s

function structure: Φ(r) ∼ 1/r in 3D.

The Kelvin solution displacement u ∼ F/(8πµ r) has strain ϵ ∼ F/(µr

), and its trace ∂ · u ∼

F/(µr

). The defect-induced potential (built gauge-invariantly from second derivatives of h via the

linearized Ricci scalar) inherits the 1/r long-distance behavior, with proportionality

Φ(r) ∼

def

µ r

where F

def

is the topological strength of the defect (units of force in elasticity language).

Matching F

def

/(µr) to G

M/r gives the order-of-magnitude identiﬁcation (in natural units

ℏ = c = 1):

∼

def

µ M

For a defect with topological strength F

def

∼ k ·ℓ

(one bond-stretch quantum at the lattice scale)

and a Planck-scale mass density at the lattice site, and using µ = k in SSM-natural units, this gives

∼ ℓ

(since ℓ

= 1/M

in natural units), recovering

∼ ℓ

∼

This is the correct order of magnitude for Newton’s constant. Gravity is naturally weak in the SSM

framework for the simple reason that the substrate is stiﬀ at the Planck scale: µ ∼ M

in natural

units.

The precise dimensionless prefactor in this identiﬁcation requires ﬁxing the defect-source-to-

mass conversion: how much body-force density F

def

does a baryon-defect source in the elastic

equilibrium equation, and how does this relate to the baryon’s rest mass. This is in principle

calculable from the matter paper’s structural counting [1], but a careful numerical determination

is left to follow-up work. The order-of-magnitude estimate is robust: G

comes out at the Planck

scale because the lattice’s only intrinsic scale is the Planck scale.

6 Distinguishing Features of the D

Construction

Three features of the D

construction distinguish the emergent-gravity picture here from related

approaches.

Exact SO(4) at every order. The rank-four isotropy theorem (Theorem 1) is exact, not asymp-

totic. Hypercubic lattice elasticity has cubic anisotropy at O(a

) (Proposition 1); D

has none.

The corresponding gravitational dispersion is exactly Lorentz-invariant at the linearized level after

Wick rotation x

→ ix

of the 4D Euclidean theory, with no expected (E/M

)

Lorentz-violation

signatures in gravitational-wave dispersion or graviton propagation. This parallels the gauge-sector

null Lorentz-violation prediction of [3].

Matter as defects, gravity as strain. The trapped tetrahedral defect of the matter paper [1]

plays simultaneously the role of (a) the matter content (carrying quark quantum numbers, mass,

and color charge) and (b) the source of long-distance strain (acting as a localized mass-energy

source for emergent gravity). The two roles are uniﬁed: matter and gravitational source are not

separately postulated. The same combinatorial structure that gives a baryon its mass-energy via

the matter paper’s counting also determines how much strain it produces in the surrounding D

lattice.

Planck-scale Newton constant. The lattice has a single intrinsic scale, the lattice spacing

a ∼ ℓ

. The bond stiﬀness k is Planck-scale by dimensional analysis. The emergent Newton

constant is therefore G

∼ ℏc/M

(equivalently G

∼ ℓ

/ℏ), matching the observed value.

This is not a derivation of the precise observed value, but a parameter-free order-of-magnitude

consistency: gravity is weak in the SSM framework because the substrate is stiﬀ at the Planck

scale.

7 Conclusion

We have shown that the D

root lattice, established as physical four-dimensional spacetime in the

SSM framework [3], has an elastic structure that supports a linearized emergent-gravity interpreta-

tion. The rank-four bond tensor is exactly the fully-symmetric isotropic tensor, with no anisotropy

at any order in the lattice spacing. The long-wavelength elastic Lagrangian takes the standard

isotropic form with equal Lamé parameters λ = µ = k and Poisson ratio ν = 1/5. Bare displace-

ments correspond to gauge transformations of the linearized metric and carry no physical content;

the gauge-invariant gravitational content is sourced by lattice defects. The trapped tetrahedral

defects identiﬁed in the matter paper [1] as quarks act simultaneously as the matter content and as

localized gravitational sources, giving a Kelvin-form 1/r static strain ﬁeld that matches the Newton

potential. The Newton constant works out to G

∼ 1/M

of the correct order of magnitude.

The construction is explicitly linearized and weak-ﬁeld. What remains for further work: the

derivation of the full nonlinear Einstein-Hilbert action from the lattice (Sakharov-style integration

over bulk ﬁelds [7, 8] or a geometric reformulation), the precise dimensionless prefactor relating

defect strength to gravitational mass, gravitational radiation on D

, the cosmological-constant

problem, and the connection between strong-ﬁeld strain conﬁgurations and the Schwarzschild-

like solutions of general relativity. These belong to the broader SSM program and to companion

analytical and numerical investigations.

Acknowledgments. This work builds on the FCC lattice studies of [1, 3, 4] and the broader

SSMTheory program.

Data availability. A Python script that veriﬁes all numerical claims of this paper — the rank-

four isotropy theorem, the comparison with hypercubic and FCC, the spatial/cross-slice decompo-

sition, the Lamé parameter extraction, the Poisson ratio and bulk modulus, and the dispersion

relations — is publicly available at https://github.com/raghu91302/ssmtheory/blob/main/

verify_gravity.py. No other data were generated or analyzed in this study.

References

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[2] R. Kulkarni, The mass-energy-information equivalence: A bottom-up identiﬁcation of the particle spectrum via

FCC lattice error correction, Phys. Open 27, 100414 (2026), doi:10.1016/j.physo.2026.100414.

[3] R. Kulkarni, D

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[9] Veriﬁcation script: #.