Emergent Gravity from the Intrinsic D
4
Lattice:
Exact Linearized Einstein Gravity, and its
Spherical-Design Obstruction
at the Cubic Graviton Vertex
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
June 26, 2026
Abstract
We ask whether the D
4
root lattice, treated intrinsically through its edge lengths, supports
emergent gravity—and how far that emergence extends. At linear order it succeeds exactly:
the graviton is the incompatible (Saint-Venant) edge-length mode, and the linearized Regge
kinetic operator equals the linearized Einstein (Fierz–Pauli) operator term by term, an ex-
act operator identity rather than an appeal to spin-2 uniqueness. The agreement is protected
by a spherical-design property—the 24 minimal vectors of D
4
(the 24-cell) form a 4-design,
forcing the degree-4 moment that builds the operator to be isotropic. Defects act as gauge-
invariant stress-energy sources, gravitational and inertial mass coincide, and the static field
is Newtonian with a Sakharov-induced coupling. We then test whether the exactness survives
self-interaction. Extracting the cubic (three-graviton) vertex at two-derivative order—made well
defined by a translation-invariant reformulation and an exact average over the full automorphism
group Aut(D
4
) (order 1152)—we find that it does not. The symmetrized vertex is exact and
rational but not Einstein–Hilbert: it carries an irreducible hypercubic anisotropy of ≈ 12.7%,
provably outside the complete nine-dimensional space of O(4)-invariant two-derivative struc-
tures, and stable from the signed-permutation subgroup (12.0%) to the full group. One design
fact explains both halves: the 24-cell is a spherical 5-design but not a 6-design, so the degree-4
linear theory is isotropic while the degree-6 cubic vertex is not. Lorentz invariance emerges or-
der by order, graded by design strength; the anisotropy is two-derivative (marginal) and ∼56%
irreducible on shell, with the precise Lorentzian reading awaiting Osterwalder–Schrader recon-
struction. A corollary follows: because only the continuum is an ∞-design, no finite lattice
reproduces general relativity exactly to all orders.
1 Introduction
The statement that gravity, at lowest order, is the unique Lorentz-invariant theory of a massless
spin-2 field is normally reached through the spin-2 uniqueness theorems: one posits a symmetric
tensor field, demands Lorentz invariance and the absence of ghosts, and finds the Fierz–Pauli
(linearized Einstein) action as the only consistent answer. This paper reaches the same statement
by a different route, and the route turns out to carry more information than the destination. We
show that on one specific discrete structure—the D
4
root lattice, treated intrinsically through its
edge lengths—the linearized Regge kinetic operator equals the linearized Einstein operator term
1
by term. This is an exact algebraic identity, not a continuum approximation and not an appeal to
uniqueness; the lattice does not approximate general relativity, it reproduces the linearized operator
on the nose. We then ask whether this exactness survives the first nonlinear correction, and find—
again exactly—that it does not, for a reason that turns out to explain both the success at linear
order and the failure beyond it. A single fact about the geometry of the 24-cell controls the entire
story.
The physical picture. The setting is the world-crystal program: the proposal, going back to
Sakharov’s induced gravity [1, 5] and developed concretely by Kleinert [2, 3] and Volovik [4], that
the gravitational field is not fundamental but is the long-wavelength elastic response of a Planck-
scale medium, with curvature carried by the medium’s defects rather than by an independent
field. The object that makes this precise is the classical decomposition of an elastic strain into
two parts. A compatible strain is one that can be removed entirely by repositioning the medium’s
constituents—it is a relabeling of points, physically empty, and in the gravitational dictionary it is
a diffeomorphism (a gauge transformation). An incompatible, or Saint-Venant, strain is one that
no repositioning can relax: it is locked into the medium, and it is precisely the part that registers
as intrinsic curvature. In Regge’s discretization of geometry [6], this incompatible content has an
exact and elementary representative—the deficit angle at a hinge, the failure of the cells around
a hinge to close up flat. The graviton, in this language, is the incompatible edge-length mode:
the fluctuation of edge lengths that produces deficit angles no relabeling of vertices can undo.
Diffeomorphism gauge invariance is the freedom to relabel; the physical graviton is what survives
that freedom. This identification is the conceptual core, and the operator identity above is its exact
realization on D
4
.
Independent computational support for the principle. The principle underneath this—
that a frozen geometric incompatibility produces a Regge deficit, and that the deficit drives real
physics—has independent and explicitly computational support in a different dimension and a
different sector. In the three-dimensional, face-centered-cubic vacuum studied in [11], the tetra-
hedrally coordinated intermediate phase is geometrically frustrated, because regular tetrahedra
do not tile Euclidean 3-space: five tetrahedra around a shared edge leave an irreducible deficit
δ = 2π − 5 arccos(1/3) ≈ 0.128 rad, and that residual wedge is shown, by direct simulation, to
drive the medium to crystallize into the close-packed lattice. A constituent that fails to crystallize
remains trapped as a strained, metastable defect—matter as incomplete crystallization. We stress
what this does and does not supply for the present work. It is a three-dimensional construction in
the matter sector, and its deficit is not the four-dimensional gravitational object computed below;
it does not stand in for any result of this paper. What it provides is independent evidence that the
mechanism we invoke is not an artifact of one lattice: the same frozen-incompatibility-produces-
deficit-produces-physics principle is realized, by simulation, in 3D where it makes matter, and—as
we now prove—exactly in 4D where it makes gravity. One principle, two sectors, two dimensions;
one demonstrated numerically, the other established as an identity.
Why D
4
, and what protects the linear theory. The exactness at linear order is not generic
to lattices; it is a property of D
4
specifically, and its origin is sharp. The 24 minimal vectors of D
4
—
the vertices of the 24-cell—form a spherical 4-design [8]: averaging any polynomial of degree ≤ 4
over these 24 directions reproduces the average over the full sphere. The linearized Regge operator
is built from a rank-four (degree-4) moment of the edge directions, so the 4-design property forces
that moment to be exactly isotropic, and isotropy at this order is exactly what promotes the discrete
2
operator to the rotationally invariant Einstein operator. The agreement is therefore protected, not
accidental: it is the design strength of the 24-cell, cashed out at the order the linear theory probes.
The obstruction, and the arc of the paper. This immediately raises the question that or-
ganizes the rest of the paper. The graviton self-interacts; gravity is nonlinear. Does the design
protection extend to the cubic (three-graviton) vertex? We extract that vertex at two-derivative
order directly from the intrinsic Regge action and find that it does not. The cubic vertex is built
from degree-6 moments of the edge directions, and the 24-cell, while a 5-design, is not a 6-design:
its degree-6 moments are anisotropic. The symmetrized vertex accordingly carries an irreducible
hypercubic anisotropy that we show lies provably outside the space of O(4)-invariant structures—it
is not Einstein–Hilbert. One spherical-design fact thus accounts for both halves of the story: the
4-design secures the linear theory exactly, and the failure to be a 6-design breaks the cubic vertex.
Lorentz invariance emerges order by order, graded by the spherical-design strength of the lattice,
and D
4
’s strength runs out at precisely the first nonlinear order. A corollary follows at once, since
only the continuous sphere is a t-design for every t: no finite lattice reproduces general relativity
exactly to all orders.
What is new. The world-crystal idea that linearized gravity can emerge from an elastic vacuum
with defects is not ours; it is Kleinert’s [2, 3], and the present paper is an extension of that program
rather than a departure from it. What is new is threefold: the exact operator identity for D
4
,
replacing an emergence argument with an algebraic equality; the identification of the spherical-
design property of the 24-cell as the precise mechanism protecting that identity; and the design-
theoretic obstruction at the cubic vertex, which locates exactly where, why, and at what order
emergent gravity ceases to be Einsteinian. The first two make the linear theory exact; the third
makes the nonlinear failure exact as well, and ties both to a single property of one polytope.
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 [12].
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.
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
3
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.
vectors form a spherical design of strength sufficient to render all moment tensors through rank
four isotropic [8]; 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 crystallization picture of matter. Regular
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)
4
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.
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 matter analysis of [11]. 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 frustration 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
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).
5
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.
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-
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.
6
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.
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.
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,
7
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.
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
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).
8
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.
The remaining statements in this section are conditional on the matter coupling being exactly
general-covariant, which we identify in Section 14.1 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
[4]. 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 The cubic graviton vertex
We now pass from the linearized theory established above to its first nonlinear correction. The
substrate, briefly recalled, is the D
4
root lattice, D
4
= {x ∈ Z
4
:
P
i
x
i
even}, with V
24
its 24
minimal vectors (norm-squared 2), the vertices of the regular 24-cell. The dynamical data are the
squared edge lengths {ℓ
2
e
} of a simplicial complex on the lattice; the dynamics are Regge’s [6, 7],
S =
X
h
A
h
δ
h
, δ
h
= 2π −
X
s⊃h
θ
h,s
, (4)
9
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.
with A
h
the area of hinge (triangle) h and δ
h
its deficit angle, the dihedral angles θ
h,s
being
functions of the squared edge lengths through the Cayley–Menger relations. The graviton is the
incompatible edge-length perturbation: writing δℓ
2
e
= ε
µν
d
µ
e
d
ν
e
with d
e
the edge vector, the per-
turbations realizable by site displacements are pure gauge (linearized diffeomorphisms), and the
transverse-traceless (TT) remainder carries the curvature. Throughout, ε
i
(i = 1, 2, 3) denote sym-
metric, traceless, transverse (k
i
·ε
i
= 0) polarization tensors and k
i
the corresponding momenta
with k
1
+ k
2
+ k
3
= 0.
The object of study is the connected three-graviton amplitude obtained by perturbing the
action (4) with a superposition of three Bloch waves and extracting the term linear in each. Its
small-momentum expansion begins at the two-derivative order: the O(k
0
) term vanishes identically
because a uniform strain is an affine map and an affine map of a flat simplicial complex is flat,
so δ
h
≡ 0. We confirm this numerically below to 10
−65
. The two-derivative coefficient, which
we denote c
2
, is the lattice analogue of the Einstein–Hilbert cubic vertex and is the quantity we
compute.
Concretely, c
2
is a third directional derivative of S along three TT edge-modes, weighted by the
momentum-conserving phase. With p
i
(e) = ε
i
:d
e
⊗d
e
the polarization weight on edge e and x
e
the
edge midpoint, define the uniform, linear, and quadratic phase modes
u
i
(e) = p
i
(e), y
i
(e) = p
i
(e) (k
i
·x
e
), w
i
(e) = p
i
(e) (k
i
·x
e
)
2
, (5)
and let S
3
( ·, ·, ·) be the (symmetric) third directional derivative of (4) about the flat background.
The vertex is computed on a concrete simplicial subdivision of a single vertex star.
9 Two obstacles and their exact resolution
Translation invariance. A first, naive extraction retains only a sum of squared single-leg phases,
c
2
?
= −
1
2
[S
3
(w
1
, u
2
, u
3
) + S
3
(u
1
, w
2
, u
3
) + S
3
(u
1
, u
2
, w
3
)]. This is incorrect: the physical weight is
the square of the total momentum-conserving phase ϕ = k
1
·x
e
+ k
2
·x
e
′
+ k
3
·x
e
′′
, which—because
10
P
i
k
i
= 0—is invariant under a global shift x 7→ x +a of all positions, whereas the single-leg phases
are not. Expanding ϕ
2
produces three cross terms that the naive form omits. Restoring them gives
c
2
= −
1
2
h
P
i
S
3
(w
i
, u
j
, u
l
) + 2
P
i<j
S
3
(y
i
, y
j
, u
l
)
i
, (6)
({i, j, l} = {1, 2, 3}). One verifies directly that under x 7→ x + a the variation of (6) cancels term
by term, using
P
i
k
i
= 0 and the flatness identity S
3
(u
1
, u
2
, u
3
) = 0. The cross terms are precisely
what makes the vertex translation invariant; without them the value depends on the (unphysical)
choice of origin.
Subdivision dependence. Even after (6), the single-subdivision vertex is not well defined: the
Delaunay cells of D
4
are regular 24-cells, which are cospherical (degenerate), and a Delaunay
triangulator breaks them into simplices in an orientation-dependent way that does not respect the
D
4
point group. Concretely, two kinematic configurations related by a lattice automorphism—
which the action (4) must map to equal values—come out unequal on a fixed triangulation, and a
single configuration spans an O(1) range across triangulations. The quadratic (linearized) operator
of Paper I does not suffer this, being protected by the rank-four bond-tensor isotropy; the cubic
order is not so protected.
The resolution is exact. The physical vertex is the average of (6) over the lattice automorphism
group G = Aut(D
4
) acting on the kinematics,
c
sym
2
(ε
i
, k
i
) =
1
|G|
X
g∈G
c
2
gε
i
g
⊤
, gk
i
, |G| = 1152, (7)
on a fixed subdivision. This is a finite exact sum, not a Monte Carlo estimate: it has no statistical
error, and configurations related by any g ∈ G are mapped to bit-identical values by construction.
Aut(D
4
)
∼
=
W (F
4
) is generated by the 384 signed permutations B
4
together with the order-three
triality; the 768 triality-coset elements have half-integer entries, so (7) is carried out in exact rational
arithmetic. (A Monte Carlo average over random subdivisions converges to the same object but,
with a per-subdivision spread of ±200 to ±400 in the natural lattice normalization, only as N
−1/2
;
reaching the precision needed to resolve the structure is computationally prohibitive, whereas (7)
is exact.)
10 The exact Aut(D
4
)-averaged vertex
We evaluate (6)–(7) for 40 independent integer kinematic configurations at 80-digit precision, with
the dihedral angles obtained in closed form and the third derivative by Richardson-extrapolated
finite differences. The computation is exact and passes three independent self-consistency tests
that would each fail on a contaminated vertex:
• Flatness. max
i
|c
(i)
0
| = 8.4 × 10
−65
: the O(k
0
) term vanishes on every configuration, con-
firming that uniform strain is exactly flat.
• Lattice symmetry. For a configuration and its image under a 90
◦
rotation in a coordinate
2-plane (a signed permutation R ∈ G), |c
sym
2
(cfg) − c
sym
2
(R · cfg)| = 4.0 × 10
−79
: exactly
equal, as the symmetry of (4) demands. On a fixed triangulation this difference is O(1); the
averaging (7) restores the symmetry to machine precision.
• Exact rationality. Every c
sym
2
is recognized as an exact rational with denominator in
{1, 3, 9}, as required of a finite average of rationals over the group.
11
The averaged values shift substantially from the B
4
-only average (mean change ≈ 13, max 91
in lattice units): the triality elements are not redundant, and (7) is a genuinely more complete
symmetrization than the signed-permutation average. This will matter in Section 11, where we
find that the anisotropy nonetheless persists.
Reproducibility of the cubic-vertex computation.
Simplicial data 169-point D
4
origin star; 100 simplices; 125 hinges
Symmetry group Aut(D
4
) = W (F
4
), |G| = 1152 (384 signed permuta-
tions + 768 half-integer entries); B
4
closed with the order-3
Hadamard triality
Kinematic configurations 40 (random TT polarizations ε
i
and momenta k
i
with k
1
+
k
2
+ k
3
= 0)
Derivative stencil 8-point sign-symmetric, 4 Richardson levels
Precision 80-digit; O(k
0
) flatness verified to 10
−65
Vertex arithmetic exact rational after the group average (orbit-deduplicated);
rationality self-check enforced before any value is reported
O(4)-invariant basis dimension 9 (9 nonzero singular values; remainder < 10
−13
relative)
Rank test rank(M) = 9 < 10 = rank([M |c
sym
2
]), exact rational arith-
metic over all 40 configurations (no floating-point tolerance)
Anisotropy 12.0% (B
4
, |G| = 384) −→ 12.7% (full Aut(D
4
), |G| =
1152): stable, not a subgroup artifact
Residual vs. Einstein–Hilbert irreducible at two-derivative order; after hypercubic com-
pletion the residual falls to 10
−12
, confirming the shortfall
is purely hypercubic
11 The vertex is not Einstein–Hilbert
We now test c
sym
2
against general relativity. The Einstein–Hilbert cubic vertex at two-derivative
order is some O(4)-invariant scalar that is trilinear in the polarizations ε
i
and bilinear in the
momenta k
i
. The space of all such scalars is finite. Enumerating the index contractions of three
symmetric tensors and two vectors into a scalar yields exactly three topological families:
(ε
i
:ε
j
) (k
a
·ε
l
·k
b
)
| {z }
trace + sandwich
, k
a
·(ε
i
ε
j
ε
l
)·k
b
|
{z }
chain
, tr(ε
1
ε
2
ε
3
) (k
a
·k
b
)
| {z }
triple trace
. (8)
Evaluated on generic (continuum) transverse-traceless kinematics, these span a space of dimension
exactly nine, which we verify numerically (nine nonzero singular values, the remainder vanishing).
This nine-dimensional space is the complete arena available to any O(4)-invariant two-derivative
cubic graviton vertex, Einstein–Hilbert included.
Theorem 2 (Anisotropy). The exact Aut(D
4
)-symmetrized vertex c
sym
2
does not lie in the nine-
dimensional space spanned by (8). Over the 40 configurations the exact rational rank test gives
rank(M) = 9 and rank([M |c
sym
2
]) = 10: the data are inconsistent with any O(4)-invariant two-
derivative structure. The fractional shortfall is ∥c
sym
2
− Π
O(4)
c
sym
2
∥/∥c
sym
2
∥ = 0.127.
Three points establish that this is physics, not an artifact of method.
First, the residual is genuinely outside a complete O(4) basis: the dimension of that basis is
nine on continuum kinematics, so no Einstein–Hilbert structure has been omitted. Second, because
12
−250 −200 −150 −100 −50 0 50 100
best Einstein–Hilbert [O(4)] fit
−250
−200
−150
−100
−50
0
50
100
exact c
sym
2
(Aut(D
4
))
irreducible off-diagonal
scatter = anisotropy
(12.7%)
Vertex vs. Einstein–Hilbert
B
4
(384)
Aut(D
4
)
(1152)
0
2
4
6
8
10
12
14
fractional anisotropy
degree-6 moment
defect (7.6%)
12.0%
12.7%
Stable under full symmetrization
Figure 8: The cubic vertex is not Einstein–Hilbert, and the anisotropy is not a subgroup
artifact. Left: the exact Aut(D
4
)-symmetrized vertex c
sym
2
for each configuration against its
best O(4)-invariant (Einstein–Hilbert) fit; the irreducible off-diagonal scatter is the hypercubic
anisotropy. Right: the fractional anisotropy is stable when the averaging group is enlarged from
the 384 signed permutations to the full 1152-element automorphism group, and sits at the scale set
by the degree-6 moment defect of the 24-cell (dashed).
c
sym
2
is by construction an exactly G-invariant multilinear form, its non-O(4) part is necessarily a
hypercubic (lattice-invariant but not rotation-invariant) structure; we confirm directly that aug-
menting (8) with the hypercubic “axis-diagonal” contractions reduces the residual to 1.4 × 10
−12
,
i.e. the shortfall is entirely hypercubic. Third, the fractional anisotropy is stable under enlargement
of the averaging group:
anisotropy = 12.0% (G = B
4
, |G| = 384) −→ 12.7% (G = Aut(D
4
), |G| = 1152). (9)
Had the anisotropy been an artifact of averaging over too small a symmetry group, enlarging the
group would have driven it toward zero. It does not (Figure 8). This is the quantitative signature
distinguishing a genuine, lattice-symmetric anisotropy from a residual subgroup artifact.
12 The spherical-design origin
The anisotropy is not numerical happenstance; it is forced by an exact property of the lattice
directions, independent of the Regge computation. A finite set V ⊂ S
d−1
is a spherical t-design
if the average over V of every polynomial of degree ≤ t equals its average over the sphere [9, 8].
Equivalently, the moment
P
v∈V
(v · u)
2m
is independent of the unit direction u for all 2m ≤ t.
For the 24 minimal vectors of D
4
we compute the relative variation of this moment over directions
(Figure 9):
degree d 2 4 6 8 10 12
rel. anisotropy 10
−16
10
−16
0.076 0.19 0.32 0.46
The 24-cell is isotropic through degree 4 (indeed it is a 5-design, the odd moments vanishing by
central symmetry) and anisotropic from degree 6 onward: it is a 5-design but not a 6-design.
13
2 4 6 8 10 12
polynomial degree d of the moment
X
v 2 V
24
(v ¢ u)
d
10
−17
10
−15
10
−13
10
−11
10
−9
10
−7
10
−5
10
−3
10
−1
relative anisotropy std
u
=mean
u
isotropic
(design holds)
anisotropic
(design fails)
linear theory
» degree 4 : EXACT
cubic vertex
» degree 6 : ANISOTROPIC
5-design
boundary
The spherical-design ladder of the 24-cell
Figure 9: The spherical-design ladder of the 24-cell. The relative anisotropy of the direction-
averaged moment
P
v∈V
24
(v · u)
d
is below 10
−15
through degree 4 and jumps to 7.6% at degree 6.
The linearized theory of Paper I probes degree-4 moments and is therefore exactly Einsteinian; the
cubic vertex of this paper probes degree-6 moments and is therefore anisotropic. The same property
of one fixed point set explains both the exactness of Paper I and the obstruction of Paper II.
The mechanism is explicit. The linearized kinetic operator is built from the rank-four bond
tensor, a degree-4 moment of the edge directions, which the 4-design renders exactly isotropic,
giving Paper I’s exact Fierz–Pauli identity. The cubic vertex is built from degree-6 moments, which
the failure of the 6-design renders anisotropic; hence the ≈ 12.7% shortfall here. The moment defect
at degree 6 (7.6% in the raw moment) and the vertex anisotropy (12.7% in the specific contraction)
are the same phenomenon at the same scale. We record the organizing principle:
Proposition 2 (Order-by-order Lorentz emergence). On a lattice whose direction set is a spherical
t-design but not a (t + 1)-design, the emergent graviton dynamics are exactly O(4)-invariant for
every vertex whose momentum/polarization content samples moments of degree ≤ t, and generically
anisotropic at the first vertex sampling degree t + 1. For D
4
, t = 5: the quadratic (degree-4) sector
is exact and the cubic (degree-6) sector is the first to break.
Corollary 1 (No finite lattice is exactly Einsteinian). Since a finite point set has finite design
strength and only the full sphere is a t-design for all t, every finite lattice produces a graviton self-
interaction that departs from Einstein–Hilbert at some finite vertex order. Exact general relativity
to all orders is attained only in the continuum (∞-design) limit.
13 Severity: marginality, the physical fraction, and continuation
Two questions determine the physical weight of Theorem 2: at what derivative order the anisotropy
sits, and whether it survives on shell.
The anisotropy is two-derivative: c
2
is by definition the O(k
2
) coefficient, the same derivative
order as the Einstein–Hilbert cubic vertex. It is therefore marginal in the renormalization-group
14
sense—neither relevant nor irrelevant—and is not a Planck-suppressed higher-derivative correction
that flows away in the infrared. This is the consequential case rather than the benign one: a
higher-order (e.g. four-derivative) anisotropy would be an irrelevant operator washing out at low
energies, but a two-derivative anisotropy rides at the same order as gravity itself.
Whether the marginal anisotropy is physical or removable by field redefinition is a separate ques-
tion. A local field redefinition h 7→ h + O(h
2
) shifts the cubic vertex only by terms proportional to
the linearized equations of motion, i.e. by structures carrying an explicit k
2
i
(equivalently, vanishing
on the mass shell). Projecting the anisotropy onto the on-shell-vanishing subspace, we find that
after removing every such field-redefinition/off-shell structure, 56% of the anisotropy norm remains:
it cannot be redefined away and does not vanish on shell. A substantial part of the obstruction is
thus physical, not a gauge of the off-shell continuation.
We are deliberately careful about the final inference. The lattice is Euclidean and possesses no
null momenta, so the on-shell projection above is algebraic (identifying structures by their k
i
·k
j
factors) rather than an evaluation at k
2
= 0. The rigorous statement that the surviving anisotropy
is physical Lorentz violation in the Lorentzian theory requires Osterwalder–Schrader reconstruc-
tion [10]: the emergent theory is Euclidean and O(4)-broken at this order, and whether reflection
positivity continues that breaking to genuine SO(3, 1) violation is the precise open question. We
state the Euclidean obstruction as a theorem and the Lorentzian reading as a conjecture requiring
that continuation, rather than conflating the two.
14 Discussion
Relation to the world-crystal program. Our result refines rather than refutes the emergent-
gravity program. Paper I confirms its central claim at linear order with unusual exactness; the
present paper shows that the long-assumed automatic extension to the nonlinear theory [3] fails
for D
4
, and supplies the missing reason—a design-theoretic obstruction that earlier, continuum-
elasticity treatments could not see because they never resolved the lattice directions at degree 6.
What survives is a sharper and more honest claim: emergent gravity from a lattice is Einsteinian
up to the lattice’s design strength.
Routes forward. Proposition 2 is constructive. A direction set that is a spherical 6- or 7-design
would push the first anisotropy to the quartic vertex or beyond, recovering an exactly Einsteinian
cubic sector at the cost of more vertices (larger, less “minimal” configurations). This suggests
a hierarchy of emergent-gravity lattices graded by design strength, with D
4
the minimal 5-design
entry. Whether any four-dimensional structure of physical interest realizes higher design strength—
and whether the floppy, second-gradient elasticity of the world crystal [3] can suppress the degree-6
defect dynamically—are open and, we think, tractable questions.
Observability. The effect should not be overstated. It is a marginal, hypercubic, gravity-of-
gravity anisotropy, a faint directional “grain” in the graviton self-interaction, sourced only where
the gravitational field is strong (compact-object mergers, the early universe) and tied to a Planck-
scale spacing. The natural channel, if any, is the strong-field vacuum probed by gravitational-wave
emission. Turning Theorem 2 into a numerical waveform signature requires both the Lorentzian
continuation of Section 13 and a propagation calculation we have not done; we flag the category as
the one in which astrophysical and cosmological Lorentz-invariance tests are in principle sensitive,
without claiming a measurement.
15
14.1 Open questions
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.
15 Conclusion
The D
4
root lattice, treated as an intrinsic simplicial geometry whose data are its edge lengths,
reproduces linearized general relativity exactly: the incompatible edge-length mode is the graviton,
and the linearized Regge operator equals the Fierz–Pauli operator term by term, an algebraic
identity protected by the 4-design property of the 24-cell. That same property, pursued one order
further, sets the boundary of the agreement. The cubic graviton vertex, computed exactly as an
average over the full automorphism group Aut(D
4
), is provably not Einstein–Hilbert; it carries a
≈ 12.7% hypercubic anisotropy whose origin is the single fact that the 24-cell is a spherical 5-design
but not a 6-design. The linear theory samples degree-4 moments and is isotropic; the cubic vertex
samples degree-6 moments and is not. Emergent Lorentz invariance is therefore graded, order by
order, by the spherical-design strength of the underlying lattice—exact where the design protects
it, anisotropic at the first order it does not. Because only the continuum is a t-design for every
t, the construction also shows that no finite lattice can be exactly Einsteinian to all orders: the
continuum limit is not a convenience but a necessity. What is gained is a setting in which both the
success and the failure of emergent gravity are exact, computable, and traceable to one geometric
property of one polytope.
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.
16
B Methodology and reproducibility
The vertex (6) is evaluated on a concrete simplicial subdivision of the D
4
origin star (the 169-point
neighborhood; 125 origin hinges, 100 simplices), with dihedral angles in closed form via the Cayley–
Menger determinant and the third directional derivative obtained by an 8-point sign-symmetric
stencil with four levels of Richardson extrapolation at 80-digit precision. The automorphism group
Aut(D
4
) is generated by closure of the axis transposition, the 4-cycle, a single sign flip (these
three generate B
4
), and the order-three Hadamard triality H =
1
2
1 1 1 1
1 −1 1 −1
1 1 −1 −1
1 −1 −1 1
; closure yields 1152
elements, of which 384 are signed permutations and 768 carry half-integer entries. The average
(7) rotates each polarization ε
i
7→ gε
i
g
⊤
and momentum k
i
7→ gk
i
and is performed in exact
rational arithmetic, deduplicated over the orbit. Self-checks (flatness, the lattice-symmetry pair
identity, exact rationality) are required to pass before any value is reported. All scripts are provided
(d4 cubic vertex autD4.py and the analysis routines); the run averaged over 1152 group elements
×40 configurations ×6 third-derivative evaluations.
C The O(4) structure basis and the rank test
The candidate Einstein–Hilbert structures are the three families of (8), generated over all mo-
mentum assignments and all chain orderings. Their span has dimension nine, established by the
numerical rank of the structure matrix on generic continuum kinematics (nine nonzero singular val-
ues; the rest zero to 10
−13
relative). The obstruction is the exact rational rank test rank(M) = 9 <
10 = rank([M |c
sym
2
]) over the 40 configurations, performed in exact arithmetic (no floating-point
tolerance). The hypercubic completion—axis-diagonal sums
P
µ
(ε
1
)
µµ
(ε
2
)
µµ
(ε
3
)
µµ
k
aµ
k
bµ
and their
relatives—reduces the residual to 10
−12
, confirming the shortfall is purely hypercubic; the on-shell
projection of Section 13 uses the subset of these carrying an explicit k
a
·k
b
factor.
Data and code availability
All code reproducing the results of this paper is openly available. The linearized-theory diagnostics
(Appendix A) are reproduced by linearized gravity verify.py; the cubic-vertex computation,
the exact Aut(D
4
) average, the rank test, the anisotropy decomposition, and the spherical-design
ladder (Appendices B–C) are reproduced by the archive ssmtheory gravity scripts.zip, to-
gether with the numerical output averaged over the full 1152-element automorphism group.
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] H. Kleinert, Emerging gravity from defects in world crystal, Braz. J. Phys. 35, 359 (2005),
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17
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