Emergent Gravity from the Intrinsic D4 Lattice II: A Spherical-Design Obstruction to Einstein Gravity at the Cubic Graviton Vertex

Emergent Gravity from the Intrinsic D

4

Lattice II:

A Spherical-Design Obstruction to Einstein Gravity at

the Cubic Graviton Vertex

Raghu Kulkarni

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

raghu@idrive.com

June 26, 2026

Abstract

In the companion paper [1] we showed that the linearized Regge kinetic operator on the intrinsic

D

4

lattice equals the linearized Einstein operator exactly, the graviton arising as the incompatible

(Saint-Venant) edge-length mode, with the required Lorentz isotropy protected by the fact

that the 24 minimal vectors of D

4

form a spherical 4-design. The natural next question—

whether the emergent theory remains Einsteinian once gravitons interact with one another—

has been assumed answerable in the affirmative since Kleinert, who recovered linearized gravity

from a world crystal and asserted that the nonlinear generalization “presents no fundamental

problem” [5], without computing it. We compute it. We extract the cubic (three-graviton)

vertex at two-derivative order directly from the intrinsic Regge action, and we render it well

defined by two exact operations: a translation-invariant reformulation, and an exact average

over the full lattice automorphism group Aut(D

4

) (order 1152), which removes an otherwise

ambiguous dependence on the simplicial subdivision of the degenerate D

4

Delaunay cells. The

symmetrized vertex is exact (rational), lattice-symmetric to one part in 10

79

, and flat to one

part in 10

65

. It is not Einstein–Hilbert: it carries an irreducible hypercubic anisotropy of

≈ 12.7%, lying provably outside the complete nine-dimensional space of O(4)-invariant two-

derivative structures, and this fraction is stable from the signed-permutation subgroup (12.0%)

to the full automorphism group (12.7%)—ruling out a subgroup artifact. We trace the effect

to an exact design-theoretic fact: the 24-cell is a spherical 5-design but not a 6-design. The

linear theory probes degree-4 moments of the lattice directions, which are isotropic; the cubic

vertex probes degree-6 moments, which are not. Lorentz invariance therefore emerges order by

order, graded by the spherical-design strength of the lattice: D

4

’s strength secures the linear

theory exactly and fails first at the cubic vertex. The anisotropy is two-derivative (marginal, not

Planck-irrelevant), and roughly 56% of it is irreducible on shell; the precise Lorentzian-signature

reading requires Osterwalder–Schrader reconstruction, which we discuss but do not assume. A

corollary follows: because only the continuum is an ∞-design, no finite lattice reproduces general

relativity exactly to all orders.

1 Introduction

The proposal that gravity is emergent rather than fundamental—induced from a microscopic sub-

strate as elasticity is induced from a crystal—originates with Sakharov [2] and has been developed

most concretely in the world-crystal program, in which spacetime is a Planck-scale elastic lattice

and curvature is carried by its defects [4, 5, 6]. Within that program the gravitational field is the

1

incompatible part of the lattice strain: the deformation that cannot be relaxed by any reposition-

ing of lattice sites, the part orthogonal to the diffeomorphism (gauge) orbit. The complementary

identification of matter with lattice defects—developed for the face-centered-cubic lattice in three

dimensions, where a particle is a tetrahedral defect of incomplete crystallization [3]—supplies the

sources to which this field couples. Companion paper [1] placed the gravitational sector on an exact

footing for the D

4

root lattice in four dimensions, treated intrinsically (its data are edge lengths,

with no embedding): the linearized Regge kinetic operator was shown to equal the linearized Ein-

stein (Fierz–Pauli) operator term by term, an operator identity rather than an appeal to the spin-2

uniqueness theorem, with isotropy guaranteed by the spherical-4-design property of the D

4

minimal

vectors [9, 10].

That result is linear. The defining feature of general relativity, however, is its nonlinearity: the

gravitational field carries energy, and energy gravitates, so gravitons interact. Every claim that a

lattice “reproduces gravity” is therefore incomplete until the self-interaction is checked against the

Einstein–Hilbert vertex. In the thirty-five-year history of the world-crystal program this check was

never performed. Kleinert, having recovered the linearized theory, wrote that the extension to the

full nonlinear theory “presents no fundamental problem” [5] and proceeded as though the matter

were settled. The present paper carries out the computation he deferred, for the D

4

lattice, and

finds the assumption false.

We compute the cubic graviton vertex—the leading graviton self-interaction—directly from the

intrinsic Regge action, to two-derivative order, the order at which the Einstein–Hilbert cubic vertex

itself lives. Two technical obstacles must be cleared first, and clearing them is half the content of

the paper. The vertex as naively extracted is (i) not translation invariant and (ii) dependent on how

the degenerate Delaunay cells of D

4

(regular 24-cells, which are cospherical) are subdivided into

simplices. Both are removed exactly: translation invariance by retaining the momentum-conserving

phase in full, and subdivision dependence by averaging over the full lattice automorphism group

Aut(D

4

)

∼

=

W (F

4

) of order 1152. The first is forced by momentum conservation; the second is

forced by the requirement that the physical vertex respect the lattice symmetry that an arbitrary

triangulation breaks.

The symmetrized vertex is then unambiguous, and we test it against general relativity. The

test is sharp because the space of admissible Einstein–Hilbert structures is finite and computable:

the O(4)-invariant two-derivative trilinear-polarization bilinear-momentum scalars form a nine-

dimensional space, which we determine exactly. The symmetrized D

4

vertex does not lie in it.

The shortfall is a genuine hypercubic anisotropy of about 12.7% of the vertex norm, and—the

decisive control—it does not diminish when the averaging group is enlarged from the 384 signed

permutations to the full 1152-element automorphism group. A subgroup artifact would have been

averaged away; this is not.

The origin is exact and independent of the vertex computation. The relevant moments of the

lattice directions are isotropic through degree 4 and anisotropic at degree 6, because the 24-cell

is a spherical 5-design but not a 6-design. The linear theory samples degree-4 moments and is

therefore exactly Einsteinian (Paper I); the cubic vertex samples degree-6 moments and therefore

is not. This single fact organizes both papers: Lorentz invariance is an emergent property graded

by the spherical-design strength of the lattice, exact up to the order the design controls and broken

beyond it.

We state plainly what is and is not established. The anisotropy is exact and robust. Its

magnitude is two-derivative—marginal in the renormalization-group sense, not a Planck-suppressed

irrelevant operator—and we show that roughly 56% of it survives projection onto the mass shell,

i.e. is not removable by field redefinition. Whether the surviving piece constitutes physical Lorentz

violation in the Lorentzian theory is a question the Euclidean lattice cannot answer directly, because

2

it has no null momenta; the rigorous statement requires Osterwalder–Schrader reconstruction [11],

which we frame but do not carry out. The result is thus a sharp obstruction theorem at the level

of the Euclidean vertex, with a clearly delimited open question about its continuation.

2 The intrinsic D

4

lattice and the cubic vertex

We recall only what is needed; see [1] for the construction. The substrate 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 [7, 8],

S =

X

h

A

h

δ

h

, δ

h

= 2π −

X

s⊃h

θ

h,s

, (1)

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 (1) 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

, (2)

and let S

3

( ·, ·, · ) be the (symmetric) third directional derivative of (1) about the flat background.

The vertex is computed on a concrete simplicial subdivision of a single vertex star.

3 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

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

, (3)

({i, j, l} = {1, 2, 3}). One verifies directly that under x 7→ x + a the variation of (3) 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

3

what makes the vertex translation invariant; without them the value depends on the (unphysical)

choice of origin.

Subdivision dependence. Even after (3), 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 (1) 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 (3) 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, (4)

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 (4) 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 (4)

is exact.)

4 The exact Aut(D

4

)-averaged vertex

We evaluate (3)–(4) 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 (1) demands. On a fixed triangulation this difference is O(1); the

averaging (4) 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.

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 (4) is a genuinely more complete

symmetrization than the signed-permutation average. This will matter in Section 5, where we find

that the anisotropy nonetheless persists.

4

5 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

. (5)

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 1 (Anisotropy). The exact Aut(D

4

)-symmetrized vertex c

sym

2

does not lie in the nine-

dimensional space spanned by (5). 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

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 (5) 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). (6)

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 1). This is the quantitative signature

distinguishing a genuine, lattice-symmetric anisotropy from a residual subgroup artifact.

6 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 [10, 9].

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 2):

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.

5

−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 1: 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).

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 1 (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.

7 Severity: marginality, the physical fraction, and continuation

Two questions determine the physical weight of Theorem 1: 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

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

6

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 (Paper I)

» degree 4 : EXACT

cubic vertex (Paper II)

» degree 6 : ANISOTROPIC

5-design

boundary

The spherical-design ladder of the 24-cell

Figure 2: 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.

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 [11]: 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.

8 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

7

present paper shows that the long-assumed automatic extension to the nonlinear theory [5] 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 1 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 [5] 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 1 into a numerical waveform signature requires both the Lorentzian

continuation of Section 7 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.

9 Conclusion

We have computed the cubic graviton vertex of the intrinsic D

4

lattice exactly, by a translation-

invariant reformulation and an exact average over the full 1152-element lattice automorphism group,

and shown that it is not Einstein–Hilbert: it carries an irreducible 12.7% hypercubic anisotropy,

provably outside the complete nine-dimensional space of O(4)-invariant two-derivative structures,

stable under enlargement of the averaging group, and roughly half physical on shell. The cause is

exact and unifying: the 24-cell is a spherical 5-design but not a 6-design, isotropic at the degree-4

moments the linear theory probes and anisotropic at the degree-6 moments the cubic vertex probes.

Lorentz invariance is thus emergent order by order, graded by the spherical-design strength of the

lattice; D

4

’s strength makes the linearized theory exactly Einsteinian (Paper I) and the cubic vertex

the first to depart (Paper II). The nonlinear extension that the world-crystal program assumed

automatic is, for D

4

, obstructed—and the obstruction is calculable, with the corollary that no

finite lattice reproduces general relativity exactly to all orders. The one inference we defer is

the Lorentzian-signature reading of the physical fraction, which awaits an Osterwalder–Schrader

continuation argument.

A Methodology and reproducibility

The vertex (3) 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

8

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

(4) 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.

B The O(4) structure basis and the rank test

The candidate Einstein–Hilbert structures are the three families of (5), 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 7 uses the subset of these carrying an explicit k

a

·k

b

factor.

Data and code availability

All code and data used to produce the results of this paper are openly avail-

able as a single archive, ssmtheory paper2 scripts.zip. It contains the exact

Aut(D

4

)-averaged cubic-vertex computation (d4 cubic vertex autD4.py); the signed-

permutation (B

4

) average and the random-subdivision Monte Carlo cross-check

(d4 cubic vertex groupavg.py, d4 cubic vertex symmetrized.py); the structure-basis enu-

meration, exact rank test, anisotropy decomposition, on-shell projection, and spherical-design

ladder (d4 cubic anisotropy analysis.py); the figure generation (d4 cubic makefigs.py);

and the numerical output averaged over the full 1152-element automorphism group

(d4 cubic vertex autD4 data.json). Part I [1] is available at doi:10.5281/zenodo.20144486; the

code accompanying Part I is located as stated in that paper’s data availability section.

References

[1] R. Kulkarni, Emergent Linearized Gravity from the Intrinsic D

4

Lattice: the Graviton as the

Incompatible Edge-Length Mode (Part I, 2026), doi:10.5281/zenodo.20144486; under review at

Eur. Phys. J. C.

[2] A. D. Sakharov, Vacuum quantum fluctuations in curved space and the theory of gravitation,

Dokl. Akad. Nauk SSSR 177, 70 (1967).

[3] R. Kulkarni, Matter as Incomplete Crystallization, Physics Open 27, 100423 (2026),

doi:10.1016/j.physo.2026.100423.

[4] H. Kleinert, Gauge Fields in Condensed Matter, Vol. II: Stresses and Defects, World Scientific

(1989).

9

[5] H. Kleinert, Emerging gravity from defects in world crystal, Braz. J. Phys. 35, 359 (2005),

doi:10.1590/S0103-97332005000200022.

[6] G. E. Volovik, The Universe in a Helium Droplet, Oxford University Press (2003).

[7] T. Regge, General relativity without coordinates, Nuovo Cimento 19, 558 (1961),

doi:10.1007/BF02733251.

[8] M. Roˇcek and R. M. Williams, Quantum Regge calculus, Phys. Lett. B 104, 31 (1981).

[9] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer (1999).

[10] P. Delsarte, J. M. Goethals, and J. J. Seidel, Spherical codes and designs, Geom. Dedicata 6,

363 (1977), doi:10.1007/BF03187604.

[11] K. Osterwalder and R. Schrader, Axioms for Euclidean Green’s functions, Commun. Math.

Phys. 31, 83 (1973), doi:10.1007/BF01645738.

10