Emergent Linearized Gravity from the Intrinsic D4
Lattice:
the Graviton as the Incompatible Edge-Length Mode
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
June 24, 2026
Abstract
We ask whether the D
4
root lattice, treated as a background-independent simplicial geometry
whose fundamental data are its edge lengths, supports an emergent theory of linearized grav-
ity. We show that it does, under one explicit foundational commitment: that the lattice is
intrinsic (Regge-like), not embedded in a flat background. We prove that the rank-four bond
tensor of D
4
is exactly isotropic and identify this with the strong spherical-design property
of the D
4
minimal vectors. We then show that a metric built from a displacement vector,
h
µν
= ∂
µ
u
ν
+ ∂
ν
u
µ
, is identically flat and cannot carry the transverse-traceless (TT) graviton
polarizations, whereas the intrinsic edge-length field can and does carry them with nonzero
curvature. The free dynamics of the intrinsic metric are computed explicitly on a concrete
D
4
simplicial subdivision: the leading continuum kinetic operator is a finite tensor (a lattice
sum over a single vertex’s star) that we prove equals the linearized Einstein operator term by
term: evaluating the dihedral-angle derivatives in closed form makes every entry of the kinetic
tensor rational, and the difference from the linearized-Einstein quadratic form expands to zero
identically in general polarization and wavevector. It is two-derivative, its gauge kernel is ex-
actly the diffeomorphisms with no extra modes, isotropic to a fractional spread below 10
−8
, and
carries two degenerate transverse-traceless polarizations with the ghost-free Fierz–Pauli trace
coefficient. The Fierz–Pauli identification is thus an exact operator identity on this subdivision,
not an invocation of the spin-2 uniqueness theorem. Under the further assumption of exactly
covariant induced matter coupling, localized lattice defects act as gauge-invariant stress-energy
sources; because all excitations propagate on a single emergent metric, the gravitational and
inertial masses coincide, and the static field is Newtonian with a Planck-scale coupling generated
as a Sakharov-induced term. Finally we note that the cosmological-constant catastrophe is, for
a self-bound vacuum of this type, dissolved at equilibrium by the thermodynamic argument of
Volovik. The construction is explicitly leading-order/linearized and weak-field, and con-
ditional on the intrinsic formulation. Two questions are left open and stated as such: whether
the induced matter coupling is fully and exactly general-covariant, and the magnitude of the
small residual cosmological constant. The matter sector of the broader program (fractional
charges, mass ratios) is deliberately not treated here. Every quantitative diagnostic used here
is reproduced by a self-contained verification script (Appendix A).
1 Introduction
The idea that gravity is not fundamental but emerges from a microscopic substrate, as hydrody-
namics emerges from molecular dynamics, dates to Sakharov [1] and has been pursued in lattice
1
elasticity [2], analogue-gravity [3], and induced-gravity [4] frameworks. In the SSMTheory program
the substrate is taken to be a specific physical lattice: the face-centered cubic (FCC) lattice in
three spatial dimensions and the D
4
root lattice in four spacetime dimensions [7, 8]. A compan-
ion analysis established that the rank-two structure tensor of D
4
is isotropic, giving an exactly
Lorentz-isotropic dispersion at long wavelengths.
This paper addresses the gravitational sector and makes one methodological commitment that
proves decisive. We treat the lattice as intrinsic: its fundamental dynamical data are the edge
lengths {ℓ
e
}, with no embedding into an external flat space. This is not an extra assumption grafted
onto the program; it is what a background-independent, relational lattice is. The distinction turns
out to be exactly the line between a theory with no graviton and a theory with one.
We establish four results and are explicit about the status of each. (i) The rank-four bond tensor
of D
4
is exactly isotropic (Theorem 1); this is the spherical-design property of the minimal vectors
[6] and guarantees Lorentz invariance of the emergent kinetic terms at leading order. (ii) The
displacement metric h = ∂u is flat (Proposition 1) and misses the TT polarizations, whereas
the intrinsic edge-length field carries them with genuine curvature (Section 4). (iii) The continuum
kinetic operator of the intrinsic metric, computed on a concrete D
4
subdivision, equals the linearized
Einstein operator term by term, an exact symbolic identity (Section 5); the Fierz–Pauli identification
is proven, not a uniqueness-theorem invocation. (iv) Defects source the emergent field as stress-
energy with the equivalence principle built in, giving a Planck-strength induced Newtonian limit
(Section 6). We close with the cosmological constant (Section 7) and an honest accounting of what
remains (Section 8).
Scope and limitations, stated up front. The construction is linearized and weak-field. We
do not derive the full nonlinear Einstein–Hilbert action, gravitational radiation, or strong-field
solutions. The central results are conditional on the intrinsic (Regge) formulation. Two physics
questions are genuinely open: the exact general covariance of the matter coupling (Section 8), which
is the keystone on which both the Newtonian limit and the cosmological-constant resolution rest;
and the magnitude of the residual cosmological constant. We make no claim to resolve either.
2 The D
4
lattice and the edge-length field
The D
4
root lattice is D
4
= {x ∈ Z
4
:
P
i
x
i
≡ 0 (mod 2)}, with 24 nearest-neighbor vectors
N = {±e
µ
± e
ν
: µ < ν}, each of squared length 2. The kissing number K = 24 saturates the
four-dimensional bound, and the spatial slice D
4
∩ {x
4
= 0} is the FCC lattice [8].
The fundamental field is the set of edge-length perturbations δℓ
e
about the flat ground state. By
the Saint-Venant decomposition this field splits into a compatible part — perturbations realizable
by displacing vertices in flat space, which carry no curvature — and an incompatible part, which
cannot be removed by any repositioning and is the seat of curvature. We will see that the compatible
sector contains the phonons (spin ≤ 1 matter excitations) and the incompatible sector contains the
graviton.
3 Rank-four isotropy
Theorem 1. The rank-four bond tensor T
µνρσ
=
P
n∈N
n
µ
n
ν
n
ρ
n
σ
of D
4
equals 4 (δ
µν
δ
ρσ
+δ
µρ
δ
νσ
+
δ
µσ
δ
νρ
) exactly. The hypercubic lattice Z
4
and the three-dimensional FCC sublattice both fail this
isotropy.
2
D
4
(24 nn)
4
(8 nn)
FCC slice
(12 nn)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
T
1111
/T
1122
3
.
00
degenerate
(
T
1122
= 0)
2
.
00
Rank-four bond-tensor isotropy
isotropic value
= 3
Figure 1: Rank-four isotropy diagnostic. D
4
attains the isotropic value exactly; the hypercubic and
FCC-slice lattices do not.
The diagnostic ratio T
1111
/T
1122
equals 3 for an isotropic tensor; direct enumeration (Fig. 1,
Appendix A) gives 3 for D
4
, a degenerate value for Z
4
(the cross-pair component T
1122
vanishes,
leaving cubic anisotropy), and 2 for the FCC slice. The structural reason is that the D
4
minimal
vectors form a spherical design of strength sufficient to render all moment tensors through rank
four isotropic [6]; this is what makes the isotropy exact rather than asymptotic. Isotropy of the
rank-four tensor is the condition that the leading long-wavelength operators of the emergent theory
carry no anisotropic correction, i.e. Lorentz invariance at leading order.
4 The graviton’s home: flat displacement metric vs. curved in-
trinsic metric
Proposition 1. For any smooth displacement field u
µ
, the linearized Riemann tensor of h
µν
=
∂
µ
u
ν
+ ∂
ν
u
µ
vanishes identically.
This is the statement that a metric built by displacing points in flat space is pure gauge: h
is a linearized diffeomorphism, R
µνρσ
[h] ≡ 0 (verified symbolically, Appendix A). Consequently
the displacement metric cannot represent the physical graviton. Concretely, the two transverse-
traceless polarizations of a wave along x
3
— the modes h
11
− h
22
and h
12
— are orthogonal to
the entire image of u 7→ ∂
(µ
u
ν)
: building h
µν
= k
(µ
ε
ν)
from any transverse polarization gives both
physical components identically zero. A metric built from a vector simply does not contain the
spin-2 degrees of freedom.
The intrinsic edge-length field does. The 12 independent D
4
edge-direction dyads span the full
ten-dimensional space of symmetric rank-two tensors (rank 10, Appendix A), so the edge-strains
determine a general h
µν
, including the TT modes. Explicitly, the “+” TT mode maps to a definite,
nonzero pattern of edge-length changes, and that mode carries genuine curvature: its linearized
Riemann tensor is nonzero, R
0101
= ω
2
/2, R
0202
= −ω
2
/2 (Fig. 2), in contrast to the identically
vanishing curvature of the displacement metric. The graviton is the incompatible sector of the
edge-length field.
That the lattice supports genuine, non-removable intrinsic curvature is not abstract; the canon-
ical example is the tetrahedral frustration central to the SSM crystallization picture. Regular
3
0.0 0.5 1.0 1.5 2.0 2.5 3.0
wave frequency
ω
(lattice units)
0
1
2
3
4
linearized curvature
R
0101
The graviton lives in the intrinsic edge field
intrinsic TT mode:
R
0101
=
ω
2
/
2
displacement metric
h
=
u
:
R
≡
0
Figure 2: The displacement metric is flat (R ≡ 0); the intrinsic edge-length field carries the TT
graviton mode with nonzero curvature R
0101
= ω
2
/2.
tetrahedra cannot tile R
3
: five of them sharing a common edge subtend a total dihedral angle
5 arccos(1/3) = 352.64
◦
, leaving an angular deficit
δ = 2π − 5 arccos(1/3) = 0.128 rad = 7.36
◦
(1)
that no embedding into flat space can remove (Fig. 3). This is a concrete Regge curvature concen-
trated on the shared edge, and it is precisely the geometric frustration of the K=4 phase whose
relief drives crystallization to the flat K=12 ground state in the companion matter analysis [7]. It
demonstrates, with a single exact number, that the intrinsic edge-length geometry of this family of
lattices carries curvature inaccessible to any displacement description. (We stress that this frustra-
tion illustrates the lattice’s capacity for intrinsic curvature; it is a distinct question, addressed in
Section 6, whether a given localized defect carries any.)
5 Linearized dynamics are Fierz–Pauli
The dynamics of the intrinsic metric follow the Regge action S =
P
h
A
h
δ
h
, with A
h
the area of a tri-
angular hinge and δ
h
its deficit angle. On the flat background the Schl¨afli identity
P
h
V
(n−2)
h
dθ
h
=
0 holds (verified to machine precision in 3D and 4D, Appendix A), so the first variation of S is the
simplicial Einstein equation and the flat lattice is a stationary point. The physics is therefore in
the second variation — the graviton kinetic operator. Rather than assume the continuum limit has
the required structure, we compute that operator on a concrete D
4
subdivision and verify, term by
term, the hypotheses that identify it as Fierz–Pauli.
The subdivision. We take a finite patch of D
4
(all points with |x|
2
≤ 8; 169 vertices) and
triangulate it into 1696 four-simplices by the Delaunay construction, forced simplicial. The D
4
Delaunay cells are 16-cells (cross-polytopes), which the triangulation cuts into simplices; this intro-
duces a second, longer edge class beyond the 24 minimal vectors — a feature we return to below.
The triangulation is Qhull’s deterministic forced-simplicial (Qt) decomposition of the patch, so
the simplex list is fixed and exactly reproducible from the verification script; the results reported
here are for this subdivision. The deficit-map kernel count (the gauge structure) is topological
4
1
2
3
4
5
deficit
δ
= 7
.
36
◦
shared
edge
Five regular tetrahedra around an edge
δ
= 2
π
−
5arccos(1
/
3) = 0
.
128
rad
Figure 3: The tetrahedral frustration, viewed end-on down the shared edge: five regular tetrahedra
(dihedral angle arccos(1/3) = 70.53
◦
each) leave a 7.36
◦
deficit (δ = 0.128 rad) that no flat embed-
ding can close — a concrete unit of intrinsic Regge curvature.
and subdivision-independent, while the kinetic-operator match is evaluated on this decomposition;
whether the continuum operator is fully independent of the simplicial choice is a natural further
check. The background is flat: every interior hinge has |δ
h
| < 6 × 10
−15
(Appendix A).
Gauge structure (no extra modes). Linearizing the deficit gives the map D
he
= ∂δ
h
/∂ℓ
2
e
from
edge-length perturbations to deficits. About a fully interior vertex (25 incident edges, 125 incident
hinges) this map has rank 21 and a kernel of dimension exactly 4, with a clean spectral gap (four
null values, then a gap to 0.81). The four null directions are exactly the vertex-translation modes
— the linearized diffeomorphisms — and they span the kernel (∥D G
gauge
∥ < 10
−8
). There are
no extra zero modes: the gauge group of the linearized theory is precisely the diffeomorphisms,
neither larger (which would signal a missing physical mode or accidental symmetry) nor smaller.
This is the discrete origin of δh
µν
= ∂
µ
ξ
ν
+ ∂
ν
ξ
µ
. Continuum diffeomorphisms reproduce it exactly:
a longitudinal mode h
µν
= k
µ
ξ
ν
+ ξ
µ
k
ν
produces identically zero deficit at every wavelength (Ap-
pendix A), whereas a physical transverse-traceless perturbation produces nonzero deficit scaling as
k
2
— the curvature is two-derivative and lives in the intrinsic, not the displacement, sector (Fig. 4).
The kinetic operator. The second variation is S
(2)
=
1
2
P
ee
′
M
ee
′
δℓ
2
e
δℓ
2
e
′
, with M
ee
′
=
P
h
(∂A
h
/∂ℓ
2
e
)(∂δ
h
/∂ℓ
2
e
′
).
Evaluating M on plane-wave metric perturbations h
µν
(x) = ε
µν
e
ik·x
and taking the long-wavelength
limit analytically (so the k
2
scaling is exact, free of finite-size artefacts) gives the kinetic co-
efficient C(
ˆ
k) = S
(2)
/|k|
2
. Three properties hold (Fig. 5): (i) two-derivative — S
(2)
∝ k
2
(S
(2)
/k
2
= 1.0004, 1.0001, 1.00003 at |k| = 0.2, 0.1, 0.05, converging with O(k
2
) corrections; uniform
strains are exactly null, which forces the leading term to be k
2
); (ii) isotropic — C(
ˆ
k) is direction-
independent to a fractional spread < 10
−8
across coordinate-axis, face-diagonal, body-diagonal,
and generic directions, so the continuum limit is Lorentz-invariant; (iii) massless spin-2 — the
two transverse-traceless polarizations are degenerate (C
+
= C
×
= −1, two propagating graviton
modes), the trace sector is distinct (C = +3, the conformal mode), and the longitudinal/gauge
sector is null (C ≈ 0).
A subtlety underlies (ii). The naive rank-four moment
P
e
n
⊗4
e
of the full edge set is not
isotropic, because the triangulation’s longer edges are anisotropic on their own. Isotropy is re-
5
0.575 0.600 0.625 0.650 0.675 0.700 0.725 0.750 0.775
spoke length
d
s
(units of cage edge
L
)
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
2.0
hinge deficit
2
π
−
3
θ
(rad)
perfect fit
(flat,
δ
= 0
)
A point defect is locally flat; curvature needs incompatibility
Figure 4: The hinge deficit (curvature) of a node bonded into a fixed cage, as a function of spoke
length. At the geometrically forced (“perfect-fit”) value the configuration is flat; deficits arise only
from intrinsic incompatibility, never from compatible strain. Vertex displacements (compatible
strain) are the linearized diffeomorphisms and leave every deficit zero.
stored by the physical Regge weights (∂A/∂ℓ)(∂δ/∂ℓ): weighted as the action demands, the edge
contributions combine so that the spherical-design isotropy of the minimal vectors (Theorem 1)
carries through to the kinetic term. Isotropy is thus a dynamical statement about the weighted
operator, and it holds.
The continuum limit is exactly linearized Einstein. The properties above already fix the
theory through the uniqueness theorem, but on this subdivision one can show more: the continuum
operator does not merely lie in the Fierz–Pauli universality class, it equals the linearized Einstein
operator term by term. Because the Regge Hessian is finite-range and constant metric perturbations
are null, the small-k action is analytic with leading order S
(2)
(k) =
1
2
k
α
k
β
ε
µν
ε
ρσ
G
µνρσαβ
+ O(k
4
),
where G is a finite lattice sum over the star of a single interior vertex — the two-derivative kinetic
tensor itself, not a continuum extrapolation. This tensor can be evaluated in closed form. The
derivative of each dihedral angle with respect to a squared edge length has a rational/Cayley–
Menger expression whose only irrationality is the simplex-volume surd
√
ST −R
2
(App. A); eval-
uated at the integer D
4
edge lengths, every ∂θ/∂ℓ
2
is a rational multiple of such a surd (e.g.
−
√
3/8), and on each hinge this surd cancels against the matching surd in the triangle-area deriva-
tive ∂A/∂ℓ
2
. The result is that all 1100 entries of G are exactly rational. The linearized Einstein
tensor gives the quadratic form
q
FP
(ε,
ˆ
k) =
1
2
|ε|
2
− |
ˆ
k·ε|
2
+ (tr ε) (
ˆ
k·ε·
ˆ
k) −
1
2
(tr ε)
2
. (2)
Contracting the rational tensor G against a general symmetric ε and general k, the difference
C(k, ε)+q
FP
(k, ε) expands to exactly zero — a symbolic polynomial identity in the four components
of k and the ten of ε, not a numerical match. The canonical sectors give C
TT
: C
trace
: C
gauge
=
−1 : +3 : 0, the linearized-Einstein values +1 : −3 : 0 (Fig. 5b): the trace coefficient is the ghost-
free Fierz–Pauli value, the gauge sector is null by the exact vertex-move invariance, and the spin-2
sector is isotropic by Theorem 1. On this subdivision the continuum kinetic operator of linearized
D
4
Regge dynamics is the linearized Einstein operator, identically.
6
0 25 50 75
angle of
k
:
(0001)
→
(1110)
[deg]
1.1
1.0
0.9
C
(
ˆ
k
)
/
|
k
|
2
spread
<
10
−
8
(a) isotropy
TT
+
TT
×
TT trace
gauge
2
1
0
1
2
3
C/
|
k
|
2
(b) polarization sectors
D
4
Regge
lin. Einstein
20 0 20
−
q
FP
(
ε,
ˆ
k
)
(lin. Einstein)
20
0
20
C
Regge
(
ε,
ˆ
k
)
matches lin.
Einstein (all
ε
)
(c) generic polarizations
Figure 5: The graviton kinetic operator on the D
4
subdivision matches the linearized Einstein
operator. (a) The kinetic coefficient C(
ˆ
k)/|k|
2
for both transverse-traceless polarizations is flat
as k sweeps from a coordinate axis to a body diagonal: isotropic to a fractional spread < 10
−8
.
(b) Sector coefficients (TT, trace, gauge) coincide with the linearized-Einstein values, the trace
term carrying the ghost-free Fierz–Pauli coefficient. (c) For 60 random symmetric polarizations
and directions, the Regge coefficient agrees with −q
FP
(the linearized-Einstein quadratic form) to
a part in 10
9
— the numerical shadow of the exact symbolic identity proved in the text.
Conclusion of the section. The continuum kinetic operator of linearized D
4
Regge dynamics
equals the linearized Einstein operator, established here not by invoking the uniqueness theorem
but by an exact symbolic identity of the two quadratic forms for all polarizations. (Equivalently,
the four hypotheses of the uniqueness theorem — locality/two-derivative form, diffeomorphism
gauge invariance, isotropy, and spin-2 content — are all verified, and the overall normalization is
fixed.) The action is therefore linearized Einstein–Hilbert with the Fierz–Pauli relative coefficients,
ghost-free by construction: a single-cone, two-polarization graviton. Because the leading kinetic
tensor is a finite, exactly rational sum rather than a continuum extrapolation, and the identity
holds symbolically rather than to finite precision, this is a proven continuum-limit result on the
given subdivision, not a numerical approximation. The two-speed structure of the elastic matter
sector does not afflict the gravitational sector, which is single-cone by construction.
6 Defects as sources and the induced Newtonian limit
A localized lattice defect — in the matter program, a point node trapped in an interstitial void
— sits at a geometrically forced position and is locally flat: its hinge deficit is exactly zero (Ap-
pendix A). The forced spoke length, d
s
=
p
3/8 L ≃ 0.612 L, lies just above the hard-core “metric
wall” L/
√
3 ≃ 0.577 L and is exactly the value at which the four sub-tetrahedra tile flatly; a point
at the centroid of a fixed regular void is a flat 1 → 4 subdivision and carries no curvature. Its
gravitational influence is therefore not a local curvature but its stress-energy: the defect’s bonds
are strongly compressed relative to the natural lattice length, storing elastic self-energy. This
self-energy is the defect’s inertial mass.
Classically, a self-equilibrated localized source produces only a short-range (Eshelby) elastic
field ∼ 1/r
2
; two such defects have no long-range interaction. The Newtonian 1/r law is instead
an induced (Sakharov) effect: integrating out the lattice (phonon) modes generates the Fierz–Pauli
kinetic term with coupling 1/(16πG
ind
), and the defect stress-energy sources it. For the N
b
= 3
minimally-coupled transverse modes with a Brillouin-zone-edge cutoff Λ = π/a and a ∼ ℓ
P
, the
7
10
0
10
1
10
2
10
3
distance
r
(lattice units)
10
6
10
5
10
4
10
3
10
2
10
1
10
0
field strength (arb.)
lattice scale
The
1
/r
law is induced, not classical
induced gravity
∝
1
/r
(Newton)
classical elastic
∝
1
/r
2
(Eshelby)
Figure 6: The classical elastic field of a defect is short-range (1/r
2
); the induced gravitational field
is 1/r and dominates beyond the lattice scale. The Newtonian law is induced, not classical.
induced constant is
1
16πG
ind
=
N
b
(1/6) Λ
2
32π
2
⇒ G
ind
≃
1.3
M
2
P
, (3)
of Planck magnitude (the O(1) prefactor is cutoff-dependent). The static limit gives ∇
2
Φ = 4πG
ind
ρ
and Φ = −G
ind
M/r (Fig. 6).
The remaining statements in this section are conditional on the matter coupling being exactly
general-covariant, which we identify in Section 8 as the keystone open question; we present them
here under that assumption rather than as established. Granting it, because every excitation prop-
agates on the same emergent metric, matter couples universally via
1
2
h
µν
T
µν
; the source strength is
the total stress-energy, so the gravitational and inertial masses coincide, M
grav
= M
inert
, to leading
order — the equivalence principle is geometric, not imposed.
7 The cosmological constant for a self-bound vacuum
The induced vacuum energy is naively ∼ Λ
4
∼ M
4
P
, the familiar 10
123
catastrophe, and the usual
remedies are unavailable here: the phonon sector is purely bosonic (no supersymmetric cancella-
tion), and residing at the energy minimum does not suppress the quantum zero-point sum.
The one mechanism specific to this class of theory is Volovik’s self-sustained-vacuum argument
[3]. The D
4
vacuum is self-bound: held together by its own bonds, background-independent, subject
to no external pressure. A self-bound medium relaxes to the density where its pressure vanishes,
P = 0, and there the thermodynamically relevant (gravitating) vacuum energy is ˜ε = ε − µn =
−P = 0, even though the bare ε remains Planck-scale (Fig. 7). The Planck-scale zero-point energy
is cancelled by the Planck-scale binding energy through self-tuning — the density relaxes to P = 0
— rather than fine-tuning. This dissolves the equilibrium cosmological constant. The small nonzero
observed value must then arise from departures from equilibrium (expansion, matter); its magnitude
is model-dependent and not fixed here.
8
0.4 0.6 0.8 1.0 1.2 1.4 1.6
lattice density
n
3.0
2.5
2.0
1.5
1.0
0.5
0.0
energy density (arb.)
self-bound
P
= 0
,
˜
ε
= 0
Self-bound vacuum: gravitating energy self-tunes to zero
bare
ε
(
n
)
(Planck scale)
gravitating
˜
ε
=
ε
−
µn
=
−
P
Figure 7: Self-bound vacuum. The bare energy density ε(n) is Planck-scale, but the gravitating
combination ˜ε = ε − µn = −P vanishes at the self-bound equilibrium where P = 0.
8 What remains open
We state the open questions plainly. (1) The keystone: whether the induced matter coupling is
exactly general-covariant. The Newtonian limit and the Volovik cosmological-constant resolution
both rest on the induced action coupling to matter in the genuinely covariant (Einstein) way —
equivalently, on whether the emergent gravity sees the thermodynamic vacuum energy ˜ε rather
than the bare ε. These are the same structural question, and it is not settled for D
4
; it is the
four-dimensional dual-gravity frontier. (2) The magnitude of the residual cosmological constant.
(3) The O(1) prefactor in G. (The continuum kinetic operator is shown in Section 5 to equal the
linearized Einstein operator exactly, by closed-form evaluation of the dihedral-angle derivatives,
for this subdivision; establishing the same identity subdivision-independently is a natural further
step.) (4) The matter sector of the broader program (fractional charges, mass ratios) is outside the
scope of this paper and is not relied upon by any result above.
9 Conclusion
Treated as an intrinsic, background-independent simplicial geometry, the D
4
lattice supports emer-
gent linearized gravity. The graviton is the incompatible (curvature-carrying) sector of the edge-
length field — a degree of freedom absent from the displacement description, which is flat. Its
leading continuum kinetic operator, computed on an explicit D
4
subdivision as a finite lattice ten-
sor, equals the linearized Einstein operator term by term (an exact symbolic identity) — gauge
group exactly the diffeomorphisms, isotropic, two-derivative, spin-2, with the ghost-free Fierz–Pauli
coefficients. Under the further assumption of exactly covariant induced matter coupling, defects
source it as stress-energy with the equivalence principle built in, and the static limit is Planck-
strength Newtonian gravity, induced rather than classical. The cosmological-constant catastrophe
is, for a self-bound vacuum, dissolved at equilibrium by Volovik’s mechanism. The construction is
leading-order/linearized, weak-field, and conditional on the intrinsic formulation, and it reduces the
remaining difficulties to a single structural question — the exact covariance of the induced coupling
9
— together with the model-dependent residual cosmological constant.
Data availability. Every quantitative diagnostic used in the paper is reproduced by the self-
contained verification script linearized gravity verify.py, available at
github.com/raghu91302/ssmtheory/blob/main/linearized gravity verify.py.
References
[1] A. D. Sakharov, Vacuum quantum fluctuations in curved space and the theory of gravita-
tion, Dokl. Akad. Nauk SSSR 177, 70 (1967); reprinted in Gen. Rel. Grav. 32, 365 (2000),
doi:10.1023/A:1001947813563.
[2] H. Kleinert, Gauge Fields in Condensed Matter, Vol. II: Stresses and Defects, World Scientific
(1989), doi:10.1142/0356.
[3] G. E. Volovik, The Universe in a Helium Droplet, Oxford University Press (2003),
doi:10.1093/acprof:oso/9780199564842.001.0001.
[4] M. Visser, Sakharov’s induced gravity: a modern perspective, Mod. Phys. Lett. A 17, 977 (2002),
arXiv:gr-qc/0204062.
[5] M. Roˇcek and R. M. Williams, Quantum Regge calculus, Phys. Lett. B 104, 31 (1981),
doi:10.1016/0370-2693(81)90848-0; The quantization of Regge calculus, Z. Phys. C 21, 371
(1984), doi:10.1007/BF01581603.
[6] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Springer
(1999), doi:10.1007/978-1-4757-6568-7.
[7] R. Kulkarni, Matter as Incomplete Crystallization, Physics Open 27, 100423 (2026),
doi:10.1016/j.physo.2026.100423.
[8] R. Kulkarni, An FCC-Lattice CSS Quantum Code (2026), arXiv:2603.20294.
A Verification script
The script also includes the exact-symbolic proof of the Fierz–Pauli identity: it forms the closed-
form dihedral-angle derivatives, verifies that all 1100 kinetic-tensor entries are rational, and checks
that C(k, ε) + q
FP
(k, ε) expands to zero in general k and ε. The diagnostics reproduced by the
verification script (numpy, scipy, sympy; runs in a few seconds) are: rank-four isotropy (Theorem 1),
the vanishing of R[∂u] (Proposition 1), the edge-dyad span and the TT-mode curvature (Section 4),
the Schl¨afli identities and the embedded-vs-intrinsic deficit (Section 5), the local flatness of the point
defect (Section 6), the induced G estimate, and the self-bound vacuum cancellation (Section 7); and
the explicit D
4
Regge→Fierz–Pauli computation of Section 5 (flat background, deficit-map kernel
= the four diffeomorphisms with no extra modes, isotropy of the kinetic coefficient to a fractional
spread below 10
−8
, two-derivative scaling, and the spin-2 polarization content of Fig. 5). The script
is linked in the Data-availability statement above.
10
