The Two-Derivative Cubic Vertex of
D
4
Regge Calculus
Is Einstein–Hilbert
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
August 14, 2026
Abstract
The
D
4
root lattice, treated as an intrinsic simplicial geometry with squared edge lengths as the
dynamical data, reproduces linearized Einstein gravity exactly: its linearized Regge operator
equals the Fierz–Pauli operator term by term. Within a restricted constant-coefficient hinge
ansatz, flat stationarity selects weights proportional to the background hinge areas; the nonlinear
Regge area dependence remains an input. That result is established, and it leaves an obvious
question. Linearity is not where general relativity lives, and a lattice that reproduces free
gravitons has not yet been shown to reproduce their interaction. We compute the cubic (three-
graviton) vertex at two-derivative order and compare it against the continuum Einstein–Hilbert
vertex, obtained independently from the ΓΓ Lagrangian. On a closed periodic complex the two
agree, and the agreement is exact: carried out over the real quadratic field
Q
(
√
3
) in which
every algebraic number of the problem lies,
c
2
/N
v
=
V
EH
identically on all forty kinematic
configurations tested, with the
√
3
component of the answer vanishing throughout. Nothing is
fitted and no parameter is free, and the result is unchanged across all five subdivision assignments
tested, including a random cellwise assignment. In particular the vertex is exactly isotropic: the
lattice leaves no anisotropy at this order, despite the 24-cell failing to be a spherical 6-design.
Getting that answer requires the right domain, and we document what goes wrong on the
natural alternative. Applying the closed-complex Schl¨afli-reduced estimator to the origin-hinge
subset of an open vertex star does not define a consistent local cubic vertex: the result drifts
linearly with the subdivision, and at a typical subdivision the drift leaves a residue of some 13%
that would be easy to mistake for a real anisotropy. The cause is exact. Truncation breaks the
Schl¨afli identity
P
h
A
h
dδ
h
= 0, measured here at 2
.
5 against 6
.
1
×
10
−10
on the closed complex,
which makes the extracted quantity extensive in the star’s subdivision-dependent share of the
complex. At cubic order the intrinsic
D
4
lattice is therefore Einsteinian, and showing it requires
a complex without boundary.
1 Introduction
Recent work established the linearized half of the present question for the
D
4
root lattice. Taking
the edge lengths as the fundamental data, with no embedding into a surrounding flat space, its
linearized Regge kinetic operator equals the linearized Einstein (Fierz–Pauli) operator term by
term [
5
,
10
]. Section 5 re-establishes that result on the complex used here, in the conventions used
here, so that the linearized and cubic orders rest on a single computation.
The wider setting. Gravity may be emergent rather than fundamental, induced from a micro-
scopic substrate as elasticity is induced from a crystal. It originates with Sakharov [
1
] and has been
1
developed most concretely in the world-crystal program, in which spacetime is a Planck-scale elastic
lattice and curvature is carried by its defects [
2
,
3
,
4
]. The gravitational field is the incompatible
part of the lattice strain: the deformation that no repositioning of lattice sites can relax.
Why the nonlinear order matters. That result is linear, and linearity is not where general
relativity lives. The gravitational field carries energy, energy gravitates, and gravitons therefore
interact; the nonlinear structure is what distinguishes Einstein’s theory from a free spin-two field,
and the classic self-coupling arguments show that the nonlinearity is forced rather than optional [
11
].
A lattice that reproduces the linearized theory has not yet been shown to reproduce gravity. Within
the world-crystal program specifically, Kleinert, having recovered the linearized theory, wrote that
the extension to the full nonlinear theory “presents no fundamental problem” [
3
] and proceeded as
though the matter were settled; that assertion has not been tested.
What this paper does. This paper tests it, at the leading nonlinear order. We extract the
cubic graviton vertex directly from the intrinsic Regge action, at the two-derivative order where the
Einstein–Hilbert cubic vertex itself lives, and we compare it against a continuum Einstein–Hilbert
vertex computed independently. The two agree exactly, with no fitted normalization.
Relation to the selection–stitch framework. The
D
4
lattice used here is not an arbitrary
choice of substrate. It is the spacetime lattice of the selection–stitch model, in which the vacuum
is a close-packed bond network whose constant-time slice is the face-centered cubic lattice. Three
parts of that framework are already in the literature. The network carries a [[192
,
130
,
3]] CSS
stabilizer code on its edges [
6
]. Particles are localized defects of the network, with charges, color
and mass fixed by the crystallography around them [
7
]. And the linearized gravitational sector,
with the black-hole and cosmological consequences that follow from it, is developed in Ref. [
5
]. The
gravitational sector had been established only at linearized order. The present paper takes it to the
first nonlinear order, which is where a lattice claiming to reproduce gravity first has something to
fail at.
The result is independent of the framework. We stress that the computation does not
depend on accepting that framework. What is computed is a property of
D
4
Regge calculus: the
cubic vertex of the intrinsic action on a specific four-dimensional lattice, and the effect of truncating
the complex on which it is evaluated. Both statements are checkable by anyone working in discrete
gravity, whatever they take the vacuum to be.
Relation to previous work. The perturbative expansion of Regge calculus has a long history,
and the present computation belongs inside it. The four-dimensional linearized graviton propagator
was obtained by Roˇcek and Williams on a periodic simplicial lattice built by dividing a hypercubic
cell into 24 four-simplices [
9
], and the approach of the Regge action to the Einstein–Hilbert action
near the continuum was analyzed by Feinberg, Friedberg, Lee and Ren [
14
]. Hamber and Liu
developed the general formalism for diagrammatic expansions in lattice gravity, noting that it
applies in any dimension but working the Feynman rules explicitly in two, where they also checked
that two inequivalent triangulations give the same leading continuum answer [
15
]. More recently
Khatsymovsky has constructed an algorithmizable diagram technique on a periodic “hypercubic”
simplicial structure with part of the variables frozen, obtaining among other things additional
three-graviton vertices generated by the nonlinear parametrization that renders the functional
measure Lebesgue [16].
2
An open question answered. This paper also settles a question left open in Ref. [
5
], whose
Section 16.1 lists subdivision independence of the linearized identity as a check still to be made.
Sections 4 and 5 make it, and extend it to the cubic order.
What is new. What is added here is a direct classical statement rather than a diagram technique.
We work on the
D
4
root lattice, with no variables frozen and no input from the measure or from
gauge fixing. In that setting the two-derivative cubic vertex of the length-Regge action equals
the continuum Einstein–Hilbert cubic vertex exactly, with unit coefficient, nothing fitted, and no
dependence on the choice of subdivision. A second contribution is negative and of independent
methodological interest, and it concerns the domain of the computation. Section 3 documents a
failure mode that we believe will affect any similar calculation. Extracted from an open vertex
star—the obvious finite patch—the vertex is not a well-defined quantity. It drifts linearly with the
number of hinges in the chosen subdivision, by as much as the vertex itself, and the residue passes
every internal check one would naturally run: exact flatness, exact lattice symmetry, exact rationality.
It is large enough, and steady enough under those checks, to be mistaken for a hypercubic anisotropy
of the emergent theory. Section 3 traces it to a specific and measurable cause, the breakdown of the
Schl¨afli identity on a truncated complex, and Sections 4–6 show that the cause disappears, along
with the residue, once the complex is closed.
2 The intrinsic D
4
lattice and the cubic vertex
The lattice. The substrate is the
D
4
root lattice,
D
4
=
{x ∈ Z
4
:
P
i
x
i
even}
, whose 24 minimal
vectors are the vertices of the regular 24-cell. The dynamical data are the squared edge lengths
{ℓ
2
e
} of a simplicial complex built on the lattice, and the dynamics are Regge’s [8, 9],
S =
X
h
A
h
δ
h
, δ
h
= 2π −
X
s⊃h
θ
h,s
, (1)
where
A
h
is the area of the triangular hinge
h
and
θ
h,s
the dihedral angle subtended at
h
by
the four-simplex
s
. The dihedral angles are functions of the squared edge lengths through the
Cayley–Menger relations. That the curvature of a piecewise flat space is correctly captured by the
deficit angles in this way was placed on a rigorous footing by Cheeger, M¨uller and Schrader [
17
]; for
a broad review of lattice approaches built on (1) see Hamber [18].
The graviton. 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 perturbations realizable by displacing sites in flat space are pure
gauge, and the transverse-traceless remainder carries the curvature [
5
]. Throughout,
ε
i
(
i
= 1
,
2
,
3)
are symmetric traceless transverse polarization tensors and
k
i
the corresponding momenta, with
k
1
+ k
2
+ k
3
= 0.
The vertex we compute. The object of study is the off-shell cubic coefficient of the lattice
action, obtained by perturbing
(1)
with three Bloch waves and retaining the term linear in each.
Its small-momentum expansion begins at 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. We
denote the two-derivative coefficient c
2
.
3
The modes. With
p
i
(
e
) =
ε
i
:
d
e
⊗d
e
the polarization weight on edge
e
and
x
e
the edge midpoint,
define
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)
The physical weight is the square of the total momentum-conserving phase
ϕ
=
P
i
k
i
·x
e
i
, which
is invariant under a global shift because
P
i
k
i
= 0. Expanding
ϕ
2
and writing
S
3
for the third
directional derivative of (1) about the flat background,
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
, {i, j, l} = {1, 2, 3}. (3)
The cross terms are what make the vertex translation invariant; a form retaining only the squared
single-leg phases depends on the choice of origin.
2.1 What flat-space stationarity does and does not fix
Writing down
(1)
and then recovering Einstein–Hilbert from it would be close to circular, since
(1)
is the standard discretization of the Einstein–Hilbert action. We record here what can be said in
the model’s defence, and, more importantly, what cannot.
A constant-coefficient ansatz. Consider assigning a constant coefficient to each hinge according
to its congruence class at the flat background, of which the periodic
D
4
complex has four: the
triangles with squared edge lengths (2
,
2
,
2), and the three orientations of (2
,
2
,
4). Allowing an
independent hinge-area potential alongside the deficit term, the ansatz is
S(g, v) =
X
c
h
g
c
X
h∈c
δ
h
(ℓ) + v
c
X
h∈c
A
h
(ℓ)
i
, (4)
with eight constants and no area weighting put in by hand. Requiring
∂S/∂ℓ
2
e
= 0 at the flat
background for every edge
e
gives a linear system of rank 7, so its null space is one-dimensional,
and the surviving direction is
g
c
∝ A
(0)
c
, v
c
= 0, (5)
with
A
(0)
c
the background area of that class. The recovered coefficients are (0
.
86603
,
1
,
1
,
1) against
background areas (
√
3/
2
,
1
,
1
,
1), agreeing to 5
×
10
−11
, with the same rank and null direction on
every spine assignment and at
L
= 6. Within
(4)
, then, flat stationarity forces the deficit weights to
be proportional to the background hinge areas and forces the area potential to vanish. A version of
this selection on two congruence classes of an open star appears in Ref. [5].
This fixes background values, not the Regge action. The Regge action is
S
R
=
P
h
A
h
(
ℓ
)
δ
h
(
ℓ
),
in which the area varies with the edge lengths. Equation
(5)
determines only the value of the
coefficient at the background, and that is strictly weaker. Take any local, point-group-symmetric
f
h
(
ℓ
) with
f
h
(
ℓ
0
) =
A
h
(
ℓ
0
): every such action has the same first variation at flat space and passes
the same test, while different choices of
∂f
h
and
∂
2
f
h
give different quadratic and cubic vertices.
Since the third variation
(9)
is built from
∂A
h
and
∂
2
A
h
, this is exactly the freedom that matters
here, and flat stationarity cannot see it.
The freedom is not academic. Freezing the coefficient at its background value,
f
h
≡ A
h
(
ℓ
0
),
gives a quadratic form that is minus the Regge one. Differentiating the Schl¨afli identity once gives
P
h
[∂A ∂δ + A ∂
2
δ] = 0, so for any f with f(ℓ
0
) = A(ℓ
0
),
∂
2
S
f
= 2
X
h
∂f ∂δ +
X
h
A
(0)
∂
2
δ = ∂
2
S
R
+ 2
X
h
(∂f − ∂A) ∂δ, (6)
4
which for
∂f
= 0 returns
−∂
2
S
R
. We verify this numerically: in the same normalization the
frozen-coefficient action reproduces the Fierz–Pauli form with ratio +1
.
0000000 where Regge gives
−
1
.
0000000. Since the overall constant in
(5)
is itself unfixed, we draw no conclusion from the sign;
the point is only that agreement at first variation does not determine the kinetic operator, let alone
the cubic one.
Status. Within the eight-parameter constant-coefficient ansatz
(4)
, flat stationarity selects deficit
weights proportional to the background hinge areas and a vanishing area potential. It does not
determine the nonlinear edge-length dependence
A
h
(
ℓ
), and the Regge action remains the dynamical
assumption on which the rest of this paper rests. Deriving it from a microscopic lattice Hamiltonian,
free energy, or code dynamics is open, and we make no claim to have done so.
2.2 Variations of the action and the Schl¨afli identity
One identity, two reductions. Both the linearized operator and the cubic vertex follow from one
identity. The Schl¨afli relation holds simplex by simplex: for a single four-simplex,
P
h⊂s
A
h
dθ
h,s
= 0,
which we verify numerically to 3
.
7
×
10
−11
. Summing over all simplices of a complex without
boundary and using δ
h
= 2π −
P
s
θ
h,s
gives
X
h
A
h
dδ
h
= −
X
s
X
h⊂s
A
h
dθ
h,s
= 0, (7)
identically in the edge lengths. Differentiating
(7)
once and substituting into the second variation
of (1) about a flat background, where δ
h
= 0, collapses it to a single term,
S
ab
2
=
X
h
∂
a
A ∂
b
δ. (8)
Symmetry in (
a, b
) is not manifest but is automatic: differentiating
(7)
with respect to
ℓ
2
a
and to
ℓ
2
b
and subtracting the two results gives
P
h
(
∂
a
A ∂
b
δ − ∂
b
A ∂
a
δ
) = 0. Differentiating twice and
substituting into the third variation gives
S
abc
3
=
X
h
h
∂
ac
A ∂
b
δ + ∂
bc
A ∂
a
δ + ∂
c
A ∂
ab
δ
i
. (9)
Equation
(9)
is symmetric in (
a, b, c
) even though the elimination of one index is not manifest, which
supplies a sharp internal test: evaluating it with each slot in turn playing the role of
c
must give
the same number.
Both reductions require (7), and therefore a complex without boundary. Section 3 shows what
happens when that hypothesis fails.
3 The open vertex star, and why it fails
The obvious domain. The natural finite domain for
(3)
is the star of one lattice vertex: the
simplices containing it, and the hinges through it. It is the domain on which the linearized operator
of Ref. [
5
] is evaluated, and the obvious candidate at cubic order. It does not work there, and the
way it fails is instructive.
5
3.1 Subdivision dependence
Many subdivisions, many answers. The Delaunay cells of
D
4
are cospherical, so a triangulator
must break them into simplices, and the choice is not unique. Sampling 400 point orderings of the 169-
point neighborhood produces origin stars with hinge counts
N
h
∈ {
120
,
125
,
130
,
135
,
140
,
145
,
150
, . . . }
,
all of them legitimate Delaunay triangulations of the same point set. The total number of non-
degenerate four-simplices in that 169-point patch is fixed at 1536, independently of the ordering;
what varies is the origin star’s share of it, from 96 simplices upward in steps of four.
The drift is exactly linear. Evaluated on these subdivisions, the vertex drifts. Figure 2(a)
shows
c
2
for three kinematic configurations against
N
h
, after an exact average over the full lattice
automorphism group
Aut
(
D
4
)
∼
=
W
(
F
4
) of order 1152. The dependence is linear to the precision of
the computation,
c
sym
2
(N
h
) = a(K) + N
h
b(K), (10)
with fitted slopes
−
0
.
977778,
−
3
.
466666 and
−
2
.
311111 per hinge and residuals below 1
.
4
×
10
−6
.
Equation
(10)
predicts a subdivision not used in the fit (
N
h
= 145) to 6
.
3
×
10
−6
. Across the
observed range the drift is comparable to the vertex itself. It vanishes at
N
h
= 150, which Section 4
identifies as the star count of the chosen reference triangulation.
Averaging does not help. Group averaging does not repair this. It removes every part of
the subdivision artifact that is not invariant under the lattice point group, and the residue that
survives is invariant, so averaging leaves it untouched. Neither do the internal checks detect it: on
every subdivision the
O
(
k
0
) term vanishes, configurations related by a lattice automorphism agree
exactly, and the averaged values are exact rationals with small denominators. A residue extracted
at any fixed
N
h
will therefore look like a clean, reproducible number. At
N
h
= 130 it amounts to
12
.
7% of the vertex norm and lies outside the complete nine-dimensional space of
O
(4)-invariant
two-derivative structures—exactly the signature a genuine hypercubic anisotropy of the emergent
theory would produce.
3.2 The Schl¨afli identity on a truncated complex
Where the identity fails. The cause is exact, and it is the hypothesis of Section 2.2. An origin
star is not a complex without boundary. For each origin hinge it retains all of the simplices around
that hinge, but it discards the other hinges of those same simplices, so the cancellation in
(7)
misses
exactly those terms (Figure 1(b)). Measured on the star,
max
e
X
h
A
h
∂δ
h
/∂ℓ
2
e
= 2.5, (11)
nonzero on every one of the 156 edges. Figure 4(b) compares the residual edge by edge on the two
complexes.
The consequence at cubic order. Neither
(8)
nor
(9)
is then available. At cubic order the
consequence is immediate: the symmetry test below
(9)
fails, returning
−
1
×
10
−6
, 108
.
0 and 108
.
0
for the three slot assignments on a representative configuration. A well-defined truncated action of
course has a symmetric third variation; what the test shows is that the Schl¨afli-reduced estimator
(9)
,
derived using the complete cancellation, is not valid on a restricted hinge sum. We also note that the
origin-hinge restriction is not the full Regge action of a finite region: it omits the non-origin hinges
6
spine
4 tetrahedra
(a) coning a cross-polytope
3D analogue of the 16-cell
O
kept: the
O
term
of every triangle
discarded: the other
two terms of the same
triangles
(b) what the vertex star discards
2D analogue: hinges are vertices
Figure 1: The construction, and what truncating it discards. Both panels are drawn one
dimension below the computation and show the mechanism, not the
D
4
geometry itself. (a) A 16-cell
is coned from one antipodal vertex pair, the spine, into eight four-simplices. The three-dimensional
analogue drawn here is an octahedron coned into four tetrahedra. The construction leaves the
boundary faces whole, which is why neighboring cells may choose their spines independently and
still agree across shared faces. (b) In two-dimensional Regge calculus the hinges are vertices and
the simplices are triangles, and the Schl¨afli identity holds triangle by triangle, summing over all
three of its vertices. The star of the origin (shaded) keeps every triangle touching
O
but contributes
only the
O
term of each; the other two terms of those same triangles are dropped. That incomplete
cancellation is what Eq. (11) measures.
of its own simplices and the boundary (exterior-angle) term that a Regge region with boundary
requires.
Why the linear theory escapes. At quadratic order the star survives, but only because of a
compensating restriction. Ref. [
5
] evaluates not
(8)
but the contraction
P
h
(
∂A
h
/∂ℓ
2
e
)(
∂δ
h
/∂ℓ
2
e
′
)
with the area-derivative index restricted to the edges incident on the origin, which assigns each
hinge a fixed per-vertex share and yields an intensive quantity. We have checked directly that the
restricted form is subdivision independent: run on six subdivisions with
N
h
from 120 to 150, the
ratio of that kinetic form to the Fierz–Pauli quadratic form is
−
1
.
000000 in every case, with a spread
across polarizations below 10
−8
. On a closed complex no such restriction is needed, because
(8)
holds outright; Section 5 verifies this.
4 The closed periodic complex
Why a torus, and why an explicit triangulation. Equation
(7)
is exact on a complex without
boundary. We therefore build
D
4
on a periodic four-torus and triangulate it explicitly, without
delegating the subdivision to a general-purpose Delaunay routine.
7
Cells. The cell structure is fixed by the lattice, not chosen. The dual
D
∗
4
contains
D
4
with index
four, and the three nontrivial cosets are represented by (1
,
0
,
0
,
0),
1
2
(1
,
1
,
1
,
1) and
1
2
(1
,
1
,
1
, −
1).
These representatives are the deep holes of
D
4
, the points at the covering radius 1 [
12
]; each
has exactly eight lattice points at unit distance, and those eight are the vertices of a 16-cell, the
four-dimensional cross-polytope. The Delaunay tessellation of
D
4
is therefore a tiling by 16-cells,
one per deep hole, with 3
L
4
/
2 of them on an
L
-torus. Half are centered at integer points
c
with
P
i
c
i
odd, with vertices
c ±e
i
. The rest are centered at half-integer points, with vertices
c
+
s/
2 for
the eight sign patterns
s ∈ {±
1
}
4
that place the vertex in
D
4
. Each cell has volume 2
/
3, and three
cells per lattice point reproduce the covolume 2.
Triangulation. A 16-cell has four antipodal vertex pairs. Coning from a chosen pair, which we
call the spine, produces eight four-simplices
{a, a
′
, p
1
, p
2
, p
3
}
with
p
i
one vertex from each remaining
pair. Coning leaves the boundary tetrahedra whole, so any assignment of spines to cells fits together
globally (Figure 1(a)). The subdivision ambiguity is thereby reduced to an explicit finite label—four
choices per cell—instead of an uncontrolled consequence of point ordering, and we exploit this below
by varying it deliberately. Fixing the spine by a rule that refers only to the local geometry, for
instance lexicographic order on the four antipodal-pair directions, makes the assignment translation
invariant. We call this the chosen spine assignment rather than a canonical one: the rule is coordinate
dependent and we have not shown it invariant under the full automorphism group of
D
4
. Nothing
below depends on the choice, since the result is unchanged across the five assignments tested. Direct
enumeration on that complex gives, at every vertex, 30 incident edges (24 nearest-neighbor bonds
and 6 cell diagonals), 150 hinges and 120 simplices, with 12
L
4
simplices of volume 1
/
12 and total
volume L
4
on the torus.
Where
N
h
= 150 comes from. This identifies the value singled out in Section 3. The drift
(10)
vanishes at
N
h
= 150, and 150 is exactly the number of hinges through a vertex of the complex built
above. A general-purpose Delaunay routine returns subdivisions spanning
N
h
from 120 upward, all
of them legitimate triangulations of the same point set. The coincidence at
N
h
= 150 identifies the
star count of the chosen reference triangulation; we assign no intrinsic significance to that value,
and in particular we do not claim that
D
4
selects it. The complex built here is the one on which
the vertex is computed below.
Validation. Table 1 collects the checks at
L
= 4. The volume is exact, no simplex is degenerate,
every facet is shared by exactly two simplices, and the background is flat to machine precision. The
property that matters is the last: the Schl¨afli identity is restored to 6
.
1
×
10
−10
, the accuracy of the
finite differences used to evaluate it, from 2.5 on the open star.
Translation invariance on a torus. An absolute edge midpoint is not well defined on a torus,
but the vertex never requires one. The momentum-conserving phase
ϕ
=
P
i
k
i
·x
e
i
depends only on
relative offsets when
P
i
k
i
= 0, so all phases in
(2)
are measured from a local origin attached to
each hinge. The per-hinge contributions are then individually translation invariant and their sum is
unambiguous.
5 The linearized theory on the same complex
Why we redo the linear order here. Before the cubic vertex we record the linearized operator
on the periodic complex, so that the two orders are established on one footing and the comparison
8
quantity value expected
vertices 128 L
4
/2
16-cells 384 3 per vertex
four-simplices 3072 24 per vertex
edges / hinges 1920 / 6400 —
total volume 256.000000 L
4
= 256
simplex volumes all 1/12 (|det | = 2) none degenerate
facet sharing all 7680 facets shared by 2 closed complex
flat-background deficit max |δ
h
| = 1.3 × 10
−15
0
Schl¨afli residual 6.1 × 10
−10
0
slot symmetry of (9) 1 × 10
−6
relative 0
Table 1: The periodic
D
4
complex at
L
= 4. The Schl¨afli residual is
max
e
|
P
h
A
h
∂
e
δ
h
|
; on the open
vertex star the same quantity is 2
.
5. The final row reports the spread among the three evaluations
of (9) obtained by letting each slot play the role of c; on the open star that test fails outright.
against the continuum is made in a single set of conventions. The rank-four isotropy of Eq.
(12)
and the identification of the graviton as the incompatible edge-length mode are results of Ref. [
5
];
they are restated here, with the isotropy re-derived by direct enumeration, because both enter
the argument below. The verification of the Fierz–Pauli identity on the periodic complex, and its
independence of the subdivision, are new.
Two ingredients. Two things enter. The first is a moment condition on the lattice directions.
Direct enumeration over the 24 minimal vectors of D
4
gives the rank-four bond tensor exactly,
T
µνρσ
=
X
n∈V
24
n
µ
n
ν
n
ρ
n
σ
= 4
δ
µν
δ
ρσ
+ δ
µρ
δ
νσ
+ δ
µσ
δ
νρ
, (12)
the isotropic value, so the leading long-wavelength operators carry no anisotropic correction. Neither
Z
4
nor the three-dimensional FCC sublattice satisfies
(12)
, as the same enumeration shows [
5
]. The
second is the identification of the graviton: edge-length perturbations split, by the Saint-Venant
decomposition, into a compatible part realizable by displacing sites in flat space, which carries no
curvature, and an incompatible remainder, which does. The 12 independent
D
4
edge-direction dyads
span the full ten-dimensional space of symmetric rank-two tensors, so the edge strains determine a
general h
µν
including the transverse-traceless modes [5].
The result. On the closed complex the second variation is
(8)
, with no restriction imposed by
hand. We contract it with the modes
(2)
, in the two-mode form obtained by setting
k
2
=
−k
1
, and
compare against the Fierz–Pauli quadratic form
q
FP
(k, ε) =
1
2
k
2
|ε|
2
− |k·ε|
2
+ (tr ε)(k·ε·k) −
1
2
k
2
(tr ε)
2
. (13)
Over 60 random generic (not transverse-traceless) polarizations and directions this gives
S
2
/N
v
q
FP
= −1.00000000, spread 2.1 × 10
−9
. (14)
On transverse-traceless configurations
q
FP
reduces to +
1
2
k
2
|ε|
2
, so the same statement reads: the
ratio is 1
.
0000000 against
−
1
2
k
2
(
ε
1
:
ε
2
) for every direction tested, including (1
,
1
,
0
,
0), (1
,
0
,
1
,
0),
(0
,
1
,
1
,
0), (1
, −
1
,
0
,
0), (1
,
1
,
1
,
1), (1
,
1
, −
1
, −
1) and (2
,
1
,
1
,
0). The five spine assignments of
9
120 128 136 144 152
origin-star hinge count
N
h
125
100
75
50
25
0
25
50
c
sym
2
(open star)
N
h
= 150
(a) open star: extensive drift
1
2
3
0 10 20 30 40
kinematic configuration
1.0
0.5
0.0
0.5
1.0
c
2
/
N
v
V
EH
1e 4
(b) closed torus: subdivision independent
spine 0
spine 1
spine 2
spine 3
random spine
Figure 2: The star extraction drifts; the periodic extraction does not. (a) The
Aut
(
D
4
)-
averaged vertex on an open origin star, for three kinematic configurations
K
1
, K
2
, K
3
, against the
star’s hinge count
N
h
. The dependence is linear to 1
.
4
×
10
−6
; stars mark
N
h
= 150. Over the
accessible range the drift is comparable to the vertex. (b) On the closed periodic complex, the
same vertex minus the Einstein–Hilbert value, for all forty configurations and for five independent
assignments of 16-cell spines, including one drawn at random per cell. The residuals sit at 10
−5
, the
accuracy of the underlying finite differences, with no systematic dependence on the subdivision.
Section 4 return
−
287
.
9999999,
−
287
.
9999999,
−
288
.
0000000,
−
288
.
0000000 and
−
287
.
9999999
for the (1
,
1
,
1
,
1) mode against a Fierz–Pauli value of 288: the linearized operator is subdivision
independent, as it must be.
Same conventions throughout. The coefficient is unity in the same conventions in which the
cubic vertex is reported below. The linearized theory on the periodic
D
4
complex is therefore
Einstein’s, exactly, and the question of the next section is whether the same is true of the graviton
self-interaction.
6 Results
What is compared. We evaluate
(3)
through the factorized form
(9)
on the periodic complex,
for the same forty integer kinematic configurations used throughout, and compare against the
continuum Einstein–Hilbert cubic vertex
V
EH
of Section 7. All quantities are reported per lattice
vertex.
The vertex. In double precision, over the forty configurations,
max
c
2
/N
v
− V
EH
= 4.8 × 10
−5
, corr
c
2
/N
v
, V
EH
= 1.0000000000, (15)
with no fitted normalization: the coefficient relating the two is unity in the edge-strain conventions
of Section 2, not a number obtained by matching. Figure 3(a) shows the comparison.
10
The identity in exact arithmetic. The statement is in fact exact, and we establish it as such.
At the flat background the dihedral cosines satisfy
cos
2
θ ∈ {
0
,
1
4
}
and the hinge Heron discriminants
are
H ∈ {
12
,
16
}
, so every algebraic number entering
(9)
lies in the real quadratic field
Q
(
√
3
).
The ten local squared edge lengths take only seven distinct values across all 30 720 hinge–simplex
incidences, and the hinge triangles only four, so the derivative tensors are computed symbolically
once per congruence class, cleared to integers over the common denominator 288, and the contraction
is performed in exact integer arithmetic. On every one of the forty configurations,
c
2
/N
v
= V
EH
exactly,
c
2
√
3
= 0, (16)
the vanishing of the
√
3
component being an independent check, since a contaminated vertex would
generically retain it. Twenty-five of the forty values are nonzero; representative exact values are
−30, −40, −8, −16, +36 and +16.
Subdivision independence. Repeating the computation with each of the four axis spines and
with a spine drawn at random for every cell independently leaves the result unchanged, as Figure 2(b)
shows. The exact computation run on each of the five subdivisions returns
−
30,
−
40 and
−
8 for
the three configurations of Figure 2(a) in every case, with vanishing
√
3
component throughout, so
the independence is exact and not merely numerical. Enlarging the torus to
L
= 6 (648 vertices,
15552 simplices, 32400 hinges) reproduces the same values to 4 × 10
−5
.
O
(4) structure. The
O
(4)-invariant scalars that are trilinear in the polarizations and bilinear in
the momenta form three topological families,
(ε
i
:ε
j
) (k
a
·ε
l
·k
b
), k
a
·(ε
i
ε
j
ε
l
)·k
b
, tr(ε
1
ε
2
ε
3
) (k
a
·k
b
), (17)
spanning a space of dimension exactly nine on generic continuum kinematics. This is the complete
arena available to any
O
(4)-invariant two-derivative cubic graviton vertex. The periodic lattice
vertex lies inside it: the exact rational rank test returns
rank(M) = 9, rank
[ M |c
2
]
= 9, (18)
and the least-squares residual against the nine-dimensional space is, in double precision, 4
.
0
×
10
−7
of the vertex norm.
From finite tests to an identity of structures. The forty configurations are a finite sample,
so it is worth stating why exact agreement on them settles the coefficients. Both objects lie in the
nine-dimensional space spanned by
(17)
: the continuum vertex by the rank test of Section 7, the
lattice vertex by the test just quoted. The evaluation matrix
M
of those structures on the forty
configurations has rank nine, so the configurations span the dual space and a structure vanishing on
all of them vanishes identically. Exact agreement of the two vertices on a spanning set therefore
forces their nine coefficients to agree, and the identity holds for all kinematics, not only those tested.
The statement is about the two-derivative
O
(4)-invariant sector; it does not extend to
O
(
k
4
) terms,
which the extraction does not probe. The same test applied to the star extraction at
N
h
= 130
returns ranks 9 and 10 with a residual of 0.127 (Figure 3(b)).
11
40 20 0 20 40
Einstein--Hilbert vertex
V
EH
40
30
20
10
0
10
20
30
40
lattice vertex
c
2
/
N
v
max|
c
2
/
N
v
V
EH
| = 4.8 × 10
5
(a) 40 kinematic configurations
open star
N
h
= 130
open star
N
h
= 120
periodic
torus
10
8
10
7
10
6
10
5
10
4
10
3
10
2
10
1
10
0
anisotropy
c
2
O
(4)
c
2
/
c
2
0.127
0.130
4 × 10
7
(b) departure from
O
(4) invariance
Figure 3: The periodic cubic vertex is Einstein–Hilbert. (a) Lattice vertex per lattice
site against the independently computed continuum Einstein–Hilbert vertex, for forty kinematic
configurations; the line is the identity, not a fit. (b) Departure from
O
(4) invariance, measured as the
least-squares residual against the complete nine-dimensional space of
O
(4)-invariant two-derivative
structures. The open-star extraction yields 0
.
127 at
N
h
= 130 and 0
.
130 at
N
h
= 120; the periodic
extraction yields 4 × 10
−7
.
7 The Einstein–Hilbert comparison
Method. The continuum vertex is computed independently of the lattice, from the first-order
form of the Einstein–Hilbert Lagrangian,
L =
√
g g
µν
Γ
α
µβ
Γ
β
να
− Γ
α
µν
Γ
β
αβ
, (19)
which differs from
√
gR
by a total derivative. At cubic order a total derivative contributes a factor
k
1
+
k
2
+
k
3
= 0 and drops out, so
(19)
is exactly the two-derivative object required. Setting
g
=
δ
+
H
with
H
µν
(
x
) =
P
i
t
i
ε
i
µν
e
ik
i
·x
and retaining the coefficient of
t
1
t
2
t
3
, all series may be
truncated to squarefree monomials in the three modes. With traceless
ε
i
one has
tr H
= 0, so
√
g = 1 −
1
4
trH
2
+
1
6
trH
3
and g
−1
= 1 − H + H
2
− H
3
to the order needed.
Two validation checks. Two checks fix confidence in the implementation. Applied at quadratic
order the same machinery returns the Fierz–Pauli form exactly,
L
(2)
/
[
−
1
2
k
2
(
ε
1
:
ε
2
)] = 1
.
000000
±
3
.
1
×
10
−16
over random transverse-traceless pairs. And the cubic vertex it produces lies inside the
nine-dimensional space
(17)
, with residual 1
.
5
×
10
−13
and exact rational ranks 9 and 9, as any
O
(4)-invariant two-derivative structure must. The second check also confirms, independently, that
the enumeration (17) is complete.
8 Relation to the spherical-design argument
The design defect is real. The 24 minimal vectors of
D
4
form a spherical 5-design but not
a 6-design [
13
,
12
]. The set is antipodal, so all odd moments vanish identically and 4-design
strength already implies 5; the failure at degree six is exact rather than marginal. The moment
12
0 30 60 90
direction
u
: (0001) (1111) [deg]
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
moment
M
d
(
u
) =
v
(
v u
)
d
d
= 2, 4 coincide, constant to 1.5 × 10
15
M
6
= 44/3
M
6
= 12
(a) the 24-cell is a 5-design, not a 6-design
d
= 2
d
= 4
d
= 6
0.0 0.2 0.4 0.6 0.8 1.0
fraction of edges (sorted)
10
13
10
11
10
9
10
7
10
5
10
3
10
1
10
1
|
h
A
h h
/
2
e
|
2.5
6.1 × 10
10
(b) the Schläfli identity, edge by edge
open vertex star
closed periodic complex
Figure 4: The design defect is real, and the truncation is what breaks. (a) The moment
M
d
(
u
) =
P
v
(
v·u
)
d
over the 24 minimal vectors of
D
4
, as
u
sweeps from a coordinate axis to a
body-type diagonal. At
d
= 2 and
d
= 4 the two curves coincide and are constant to 1
.
5
×
10
−15
.
Because the root set is antipodal all odd moments vanish identically, so the 24-cell is a spherical
5-design, which is what protects the linearized operator. At
d
= 6 the moment runs between the
exact values 12 and 44
/
3 along this sweep, and between 12 and 18 over the full sphere. The cubic
vertex is built from degree-six moments, so an anisotropy might have been expected; Section 8 shows
it does not appear. (b) The Schl¨afli residual
|
P
h
A
h
∂δ
h
/∂ℓ
2
e
|
, sorted by magnitude over every edge
of each complex. On the closed periodic complex it vanishes to 6
.
1
×
10
−10
on all 1920 edges, the
accuracy of the finite differences. On the open origin star it is nonzero on all 156, reaching 2
.
5.
That failure, not the design defect, is what produces the spurious anisotropy of Section 3.
M
d
(
u
) =
P
v
(
v · u
)
d
over unit directions
u
equals 12 identically for
d
= 2 and
d
= 4 (a dense scan
bounds the variation by 2.7 × 10
−15
), while at d = 6 it takes the values
M
6
(1, 0, 0, 0) = 12, M
6
(1, 1, 1, 1) = 12, M
6
(1, 1, 1, 0) =
44
3
, M
6
(1, 1, 0, 0) = 18, (20)
spanning 40% of its mean, with a root-mean-square variation of 7.6% over the sphere. Figure 4(a)
shows the moment along a great-circle sweep. Since the linearized kinetic operator is built from
a degree-four moment of the edge directions and the cubic vertex from degree-six moments, it is
natural to expect the cubic order to be the first to feel the anisotropy. A numerical near-coincidence
reinforces the expectation: the root-mean-square degree-six defect, 7
.
6%, and the star residue of
Section 3, 12.7%, are of the same order.
But it does not obstruct the vertex. The design defect at degree six is a true property of
the 24-cell, but the cubic vertex is Einstein–Hilbert regardless: the moment anisotropy does not
survive into the physical contraction. The agreement in magnitude is accidental, and the residue is
accounted for entirely by the truncation analyzed in Section 3. Whether design strength obstructs
some higher vertex remains open; the periodic machinery developed here is what such a question
would require.
13
9 Discussion
What this establishes. At cubic order and two-derivative order, the intrinsic D
4
Regge action
reproduces the Einstein–Hilbert graviton self-interaction, with no free normalization and no depen-
dence on the simplicial subdivision. Together with the linearized result of Section 5, established
here on the same complex, the emergent theory is Einsteinian through the first nonlinear order,
which is the check the world-crystal program has always assumed and never performed.
A caveat on off-shell comparison. The quantity compared is the cubic coefficient of the action,
not an on-shell amplitude, and an off-shell cubic vertex is not invariant under field redefinition. The
comparison here is made in a fixed parametrization: the edge strain
δℓ
2
e
=
ε
µν
d
µ
e
d
ν
e
on the lattice
side, and
g
=
δ
+
H
on the continuum side, with no intermediate redefinition. What is established
is that the two agree in that parametrization, which is the natural one for a length-Regge action.
Nonlinear field redefinitions can shift off-shell structures proportional to the linear equations of
motion; we have not computed the image of that freedom in the nine-dimensional space. The
equality reported here is coefficient by coefficient in one fixed, explicitly stated parametrization.
Newton’s constant. The lattice fixes the tensor structure of the vertex and its coefficient relative
to the linearized operator, both being unity in the conventions of Section 2. It does not fix the
overall scale of the Regge action against Einstein–Hilbert, so Newton’s constant is an input here
and not a prediction; all results are quoted in lattice units.
What it does not. The selection argument of Section 2.1 narrows the choice of action but does
not derive it from a microscopic dynamics: locality, hinge support and linearity in the deficit are
inputs, and a derivation of
(1)
from a lattice Hamiltonian, free energy, or code dynamics remains
open. The computation is Euclidean, classical, and truncated at cubic order. It says nothing
about the quartic and higher vertices. Nor does it address the nonlinear theory as a whole, or the
Lorentzian continuation. The lattice spacing is physical, not a regulator to be removed, so the
statement concerns a fixed lattice and not a continuum limit taken along a renormalization-group
trajectory. Nothing here bears on the general covariance of the induced matter coupling, which
Ref. [
5
] leaves open. That gap has a concrete form in the present framework: the same lattice
carries a matter sector in which particles are localized defects of the bond network, with masses and
charges fixed by the crystallography around them [
7
]. Whether the vertex established here couples
to those defects as Einstein gravity couples to stress-energy is a well-posed question, and it is not
settled by anything in this paper.
A methodological remark. The failure documented in Section 3 is not specific to this lattice.
Restricting the hinge sum of the bulk Schl¨afli-reduced expression to an incomplete local subset
breaks the cancellation that
(8)
and
(9)
require, and any cubic coefficient extracted with that
reduced estimator inherits a truncation-dependent contribution. The warning is about the estimator,
not about every finite Regge calculation: a complete Regge region with its boundary term has a
perfectly good variational principle. Perturbative Regge calculations have used periodic lattices
since Roˇcek and Williams [
9
], so the hazard is avoided there by construction. What we add is the
quantitative statement. On an open star the violation of
(7)
is 2
.
5, against 6
×
10
−10
on a closed
complex. The induced error in the cubic vertex is linear in the hinge count, with slopes of order
unity. It survives averaging over the full lattice automorphism group and passes flatness, symmetry,
and rationality checks unscathed, so it is large enough and stable enough to be mistaken for physics.
14
Both diagnostics are cheap—vary the subdivision, and measure
P
h
A
h
dδ
h
—and are worth running
whenever a finite patch is used.
10 Conclusion
The cubic graviton vertex of the intrinsic
D
4
lattice, computed on a closed periodic complex, equals
the Einstein–Hilbert cubic vertex. Per lattice site the two agree exactly, over the field
Q
(
√
3
), on
all forty kinematic configurations. The coefficient is unity, the exact rational ranks are 9 and 9,
and the answer is the same on every subdivision tested. The nonlinear extension Kleinert deferred
works, at least at this order. An apparent obstruction at this order is an artifact of extracting the
vertex from a complex with boundary, and its cause—the breakdown of the Schl¨afli identity under
truncation—is exact, measurable, and now quantified.
A Methodology
Dihedral angles are evaluated in closed form from the squared edge lengths through the Gram
construction underlying the Cayley–Menger relation. First and second derivatives of
θ
with respect
to squared edge lengths are obtained by central differences at step 10
−4
, vectorized over all hinge–
simplex incidences (30 720 at
L
= 4); areas and their first two derivatives are analytic. The
third variation is assembled through
(9)
. The double-precision computations use IEEE arithmetic,
with accuracy 10
−5
to 10
−7
set by the finite differences rather than by conditioning. The central
identity is independently established in exact arithmetic: derivatives of
θ
=
arccos c
follow from
∂
i
θ
=
−c
i
/
√
1 − c
2
and
∂
ij
θ
=
−c
ij
/
√
1 − c
2
− c
i
c
j
c
(1
− c
2
)
−3/2
with
c
a rational function of the
local squared lengths, so
c
i
and
c
ij
are exact rationals obtained symbolically; at the background
1 − c
2
∈ {1,
3
4
} and H ∈ {12, 16}, placing every quantity in Q(
√
3).
The open-star computations of Section 3 are carried out in double precision and cross-checked at
80-bit extended precision, the two agreeing to 3
×
10
−8
on the configurations tested. The
Aut
(
D
4
)
averages are exact rationals with denominators dividing 9, which the double-precision values identify
unambiguously.
The exact rational rank tests are performed over
Q
with no floating-point tolerance. For the
periodic complex the vertex values are integers to within 5
×
10
−5
and are rounded before the test;
the rounding is unambiguous, the nearest competing rational being further away by more than three
orders of magnitude.
B An explicit basis of the structure space
Equation
(17)
gives the three topological families; the completeness and rank of the resulting span
is what the identity of Section 6 rests on, so we record an explicit independent basis here.
The families are generated with
k
3
=
−k
1
− k
2
imposed at the outset, so only
k
1
and
k
2
appear
as independent momenta, and with
ε
i
symmetric, traceless and transverse to
k
i
. Permutation
symmetry is not imposed by hand: every assignment of the labels is generated, and the redundancy
is removed by the rank test rather than by a symmetry argument. That produces 93 structures in
all (27 from family (i), 54 from (ii), 12 from (iii)), whose span has dimension exactly 9 on generic
continuum kinematics. Greedy selection over 120 random transverse-traceless configurations returns
the independent set
15
family structure
1 (i) (ε
2
:ε
3
) (k
2
·ε
1
·k
2
)
2 (i) (ε
1
:ε
3
) (k
1
·ε
2
·k
1
)
3 (i) (ε
1
:ε
2
) (k
1
·ε
3
·k
1
)
4 (ii) k
2
·(ε
1
ε
2
ε
3
)·k
1
5 (ii) k
2
·(ε
1
ε
3
ε
2
)·k
1
6 (ii) k
1
·(ε
2
ε
1
ε
3
)·k
1
7 (iii) tr(ε
1
ε
2
ε
3
) (k
1
·k
1
)
8 (iii) tr(ε
1
ε
2
ε
3
) (k
1
·k
2
)
9 (iii) tr(ε
1
ε
2
ε
3
) (k
2
·k
2
)
Three elements come from each family, and the momentum labels reduce to
k
1
and
k
2
throughout.
Any other independent set of nine spans the same space;
emergent o4 analysis.py
regenerates
this one and verifies the dimension.
C Reproducibility
Every quantitative result in this paper is reproduced by the scripts in
emergent scripts.zip
, at
github.com/raghu91302/ssmtheory
. They require only
numpy
,
scipy
,
sympy
and
matplotlib
,
and run from a single flat directory.
The periodic complex of Section 4, with its cells, coned triangulation and validation counts, is
built by
emergent periodic d4.py
; the Regge variations on it, including the Schl¨afli residual and
the plane-wave modes with per-hinge local phases, by
emergent periodic engine.py
. The open
star used in Section 3, together with the
Aut
(
D
4
) average, is
emergent star.py
. The continuum
Einstein–Hilbert cubic vertex of Section 7, which uses no lattice input, is
emergent eh vertex.py
.
The results follow one script per section:
emergent action selection.py
for the selection
of the Regge action in Section 2.1;
emergent linearized.py
for the rank-four isotropy and the
Fierz–Pauli identity of Section 5;
emergent cubic vertex.py
for the double-precision comparison,
subdivision independence and
L
= 6 convergence of Section 6;
emergent exact tensors.py
and
emergent exact vertex.py
for the exact identity
(16)
over
Q
(
√
3
);
emergent o4 analysis.py
for the structure basis
(17)
and the exact rational rank tests;
emergent star drift.py
for the
drift law
(10)
and its out-of-sample prediction;
emergent schlaefli.py
for the Schl¨afli diagnos-
tic; and
emergent make figs.py
for Figures 2 and 3, written as
emergent fig1 drift.pdf
and
emergent fig2 result.pdf.
Timings on one core: the exact tensors take about 40 s to build and the exact vertex a few
seconds thereafter; Section 6 at
L
= 4 runs in under a minute,
L
= 6 in about half a minute plus a few
seconds per configuration; the linearized, structure-basis and Schl¨afli checks are seconds each. Only
the open-star scan is slow, the
Aut
(
D
4
) average costing roughly half a minute per configuration, so
the two open-star datasets are included in the archive and regenerated by
emergent star drift.py
full.
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17
