The Static Limit of D
4
Is Not the FCC Laplacian:
An Exact Cancellation, and What Gravity Sees
of a Bond Defect
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
September 4, 2026
Abstract
The D
4
root lattice has an exactly isotropic rank-four bond tensor, T = 4S, while its face-
centered cubic spatial slice does not, with T
1111
/T
1122
equal to 3 and 2 respectively. Both facts
are established in Ref. [1], along with a symbolic identity showing that the linearized Regge
operator on D
4
is the linearized Einstein operator. Together they raise a question. Static
physics lives on a slice, and the slice fails the isotropy test, so why is the static sector isotropic?
We show that the static sector is never the slice. Setting ω = 0 does not remove the cross-slice
bonds; it evaluates them at unit temporal phase, leaving their spatial projections ±e
i
with
weight two. The static operator is therefore a sum of two bond sets, each anisotropic by exactly
1/6 with opposite sign, cancelling identically. We then measure what that buys. The Green’s
function of the static operator is A/r with the analytically predicted A = −1/24π, from the
lattice expansion −6|k|
2
+
1
2
|k|
4
+O(k
6
) whose two coefficients are both moments of the bond set.
Its angular anisotropy is 3.4 × 10
−5
at nine lattice spacings, against a control series over simple
cubic, body-centered cubic and the bare FCC slice in which the anisotropy tracks the rank-four
deviation with a common constant. Finally, the monopole of that potential is the integrated
source exactly, however the source is distributed — so a localized bond defect couples to gravity
through
ρ alone. Within the framework that integral is the defect’s mass, which Ref. [2] gives
as its disrupted-bond count, so the far field sees the count alone. All results are at linear order
in the static scalar sector, on a fixed lattice with no continuum limit taken.
1 Introduction
A lattice singles out directions. If the lattice is a computational regulator this does no harm: the
spacing is sent to zero at the end, and the preferred directions go with it. Some approaches instead
treat a discrete structure as physical rather than as a regulator, among them causal set theory [15]
and causal dynamical triangulations [14]. The setting here is a fixed lattice with a spacing of order
the Planck length that is never taken to zero. The preferred directions then survive into every
prediction. Such a theory has to show two things: that what emerges is isotropic enough to match
the world, and that matter couples to it correctly.
This paper treats the simplest gravitational observable in that setting: the static potential of a
localized source on the D
4
root lattice.
The lattice. Readers unfamiliar with the setting may find a short orientation useful. The
selection–stitch model takes the vacuum to be a close-packed network of entanglement bonds.
1
Its geometry is the D
4
root lattice, read as physical four-dimensional spacetime. The lattice spac-
ing is of order the Planck length and is held fixed, not sent to zero. Constant-time slices are the
face-centered cubic lattice, in which every bulk site has coordination K = 12.
The code. The bond network carries a [[192, 130, 3]] CSS stabilizer code, with one qubit on each
edge [4]. We do not use the code below. We mention it because it is what makes the bonds physical
objects rather than bookkeeping.
Matter and mass. Matter is identified with defects of the network. A defect is an extra node
trapped in a tetrahedral void, or a site of anomalous coordination. The mass of a defect is taken
to be proportional to the number of bond states its presence disrupts. What Ref. [2] establishes is
a ratio: the proton, a node trapped in a tetrahedral void, disrupts 1836 times as many states as
the minimal single-node defect identified with the electron. Mass is therefore a count, up to one
overall scale.
Gravity. The gravitational sector is developed in Ref. [1], where the dynamics is Regge’s, with
the squared edge lengths as the variables, and the Bekenstein–Hawking entropy is obtained by
counting bonds severed by a horizon. That work counts bonds across a surface; the present paper
counts bonds disrupted in a volume, and asks what field the second count produces.
Why this matters here. Two features of the setting drive this paper. The spacing is fixed and
physical, so lattice anisotropy is a prediction, not an artifact to be removed. And mass is a count,
so the coupling question is sharp: a count is one number, while a gravitational source is in general
a field.
Independence. Nothing below requires the reader to accept the framework. What is computed
are properties of D
4
and of the Regge action on it. They can be checked whatever one takes the
vacuum to be. The framework supplies the reason for asking about this lattice in particular.
The question this paper asks. Reference [1] establishes that D
4
is rank-four isotropic and
that its FCC slice is not. Static physics is slice physics, so the two statements together look like
a problem: the sector in which one would compute a Newtonian potential is the sector that fails
the isotropy test. Section 2 resolves it, and the remaining sections ask what the resolution buys
— how isotropic the resulting Green’s function is, against a control series, and what property of
a source its far field depends on. The answers compose into a chain, displayed as Eq. (11), which
takes a defect from a field on the bonds down to the one number its far field sees. Everything is at
linear order: the computations are second variations of the action about flat space, and the field
equation is the linearized one. Nonlinear orders are a separate problem and are not touched here.
The static limit is not the spatial slice. The first question has a non-obvious answer and it
drives the rest of the paper. Designate one axis, x
4
, as time. The 24 nearest neighbors of D
4
then
split by their x
4
component into 12 in-slice vectors ±e
i
± e
j
(1 ≤ i < j ≤ 3), which are exactly
the FCC kissing set, and 12 cross-slice vectors ±e
i
± e
4
, six with x
4
= +1 and six with x
4
= −1.
The split is elementary and the forward and backward counts are equal, so the cross-slice set is
symmetric under e
4
↦→ −e
4
. It is tempting to conclude that static physics sees only the FCC slice.
It does not. A time-independent field is not a field confined to one slice; the cross-slice bonds still
act on it, with their temporal factor evaluated at cos ω = 1. Their spatial shadow is a hop ±e
i
of
2
weight two. The static operator is therefore the sum of two bond sets, and, as Section 2 shows,
that sum is exactly isotropic at rank four while neither summand is.
Why this matters. The FCC lattice on its own is a poor substrate for isotropic physics: its
rank-four bond tensor has ratio T
1111
/T
1122
= 2 where isotropy requires 3. Had the static sector
been governed by FCC alone, the Newtonian potential would carry an angular anisotropy of order
10
−3
at ten lattice spacings. The measured value for the true static operator is 3.4 × 10
−5
and
falling as r
−2.6
. The difference is entirely due to the cross-slice bonds that a naive slicing argument
discards.
Relation to prior work. The gravitational sector of the framework was developed in Ref. [1]
as a supporting section of a paper about horizon thermodynamics, and we state at the outset
which results are taken from there and which are established here. Taken from Ref. [1]: the exact
rank-four isotropy of the D
4
bond tensor (its Theorem 1), the failure of the FCC slice and of Z
4
to satisfy it, the symbolic identity between the linearized D
4
Regge operator and the linearized
Einstein operator with sector ratios −1 : +3 : 0 (its §3.3), the selection of the Regge action within
the frustration-linear family (its Proposition 2), and the identification of a defect’s gravitational
source with its elastic self-energy rather than its misfit strain (its §3.4). Established here: that
the static limit is not the FCC Laplacian and why (§2); the reduction of the Regge action on
static conformal perturbations to that operator (§3); the analytic normalization A = −1/24π and
the lattice expansion behind it, the measured angular anisotropy of the Green’s function, and the
control series relating it to the rank-four deviation (§4); and the distribution-independence of the
monopole and what it implies for a bond defect (§5). No text is reused from Ref. [1], and its results
are restated only as far as needed to state ours.
2 How the isotropy survives the time slicing
The established result. The starting point is not new and we state it as such. Theorem 1 of
Ref. [1] establishes that the rank-four moment tensor of the 24 minimal vectors N of D
4
, T
µνρσ
=
n∈N
n
µ
n
ν
n
ρ
n
σ
, satisfies
T
µνρσ
= 4 (δ
µν
δ
ρσ
+ δ
µρ
δ
νσ
+ δ
µσ
δ
νρ
) (1)
exactly, in integer arithmetic; that the diagnostic ratio T
1111
/T
1122
is 3 for D
4
against 2 for the
FCC slice and a degenerate value for Z
4
; and, in its §3.3, that the linearized Regge operator on an
explicit D
4
subdivision equals the linearized Einstein operator as a symbolic identity, with sector
coefficients C
TT
: C
trace
: C
gauge
= −1 : +3 : 0. That D
4
is rank-four isotropic and its spatial slice
is not is therefore established. This paper takes those results as given and asks a question they
leave open.
The question. If the FCC slice fails the isotropy test, and static physics lives on a slice, why is
the static sector isotropic at all? The answer is that the static sector is never the slice. Designating
x
4
as time and setting ω = 0 does not remove the cross-slice bonds; it evaluates them at cos ω = 1.
Write the scalar operator as
∆(k) =
n∈N
(cos k·n − 1). At ω = 0,
∆
stat
(
k) = 4
i<j
cos k
i
cos k
j
+ 4
i
cos k
i
− 24, (2)
the first sum from the 12 in-slice bonds ±e
i
± e
j
and the second from the 12 cross-slice bonds
±e
i
± e
4
evaluated at unit temporal phase. In position space this is an operator on Z
3
, not on
3
the FCC sublattice: the in-slice bonds connect sites of equal coordinate parity, and the projected
cross-slice bonds connect opposite parity with weight two, because a time-independent field takes
the same value on the corresponding sites of adjacent slices.
An exact cancellation. The two bond sets are individually anisotropic and their anisotropies
cancel (Table 1). This is the spatial shadow of (1): T restricted to spatial indices is the same tensor
whether computed in four dimensions or from the projections. The decomposition is worth recording
because each summand deviates by exactly 1/6 in the normalized measure, and a treatment that
discards the cross-slice bonds retains one of them — which is precisely the failure Ref. [1] reports
for the bare slice.
bond set count T
1111
T
1122
ratio ∥T − αS∥
max
FCC in-slice, ±e
i
± e
j
12 8 4 2 1.333
cross-slice projections, ±e
i
(weight 2) 12 4 0 — 1.333
static D
4
, both 24 12 4 3 0.000
Table 1: Decomposition of the rank-four tensor under the time slicing, with S
ijkl
= δ
ij
δ
kl
+ δ
ik
δ
jl
+
δ
il
δ
jk
and α = T
1111
/3. Isotropy requires T
1111
= 3 T
1122
. Neither summand is isotropic; each
deviates by exactly 1.333, and the deviations cancel identically. The total is Theorem 1 of Ref. [1]
restricted to spatial indices.
The same condition appears elsewhere. That an emergent continuum equation is isotropic
exactly when the rank-four bond tensor is isotropic is not peculiar to gravity. It decides whether a
lattice gas reproduces the Navier–Stokes equations: the hexagonal lattice satisfies it and the square
lattice does not, which is why the FHP model works on the first and fails on the second [12, 13].
The construction is also standard in numerical analysis, where the isotropic 19-point stencil on a
cubic grid is isotropic for the same reason [6].
3 The Regge action reduces to this operator
Two geometries. Two different complexes are used in this paper, for reasons worth stating.
The reduction below is a bulk identity, and the Schl¨afli relation it rests on holds exactly only
without boundary, so it is checked on periodic complexes: four-tori of side L = 6, 8 and 10, with
L
4
/2 vertices and 6L
4
nearest-neighbor bonds. The simplicial complex carries further edges, the
triangulation diagonals, which are not bonds; at L = 6 there are 7776 of the former and 1944 of the
latter. The Newtonian potential of Section 4 is the opposite case: a source needs somewhere for its
field to fall off, which a torus does not provide, so that measurement uses balls with a boundary,
of radius R = 16, 20 and 24. Nothing is transferred between the two beyond the operator itself.
Setup. The gravitational sector is Regge’s, S =
h
A
h
δ
h
over triangular hinges of a simplicial
complex built on D
4
[7, 1]. That the deficit angles correctly capture the curvature of a piecewise
flat space was placed on a rigorous footing by Cheeger, M¨uller and Schrader [9]; for reviews see
Refs. [10, 11]. The perturbative expansion in four dimensions goes back to Roˇcek and Williams [8].
About a flat background the Schl¨afli identity
h
A
h
dδ
h
= 0 reduces the second variation to
S
2
=
h
∂A ∂δ, symmetrized. We evaluate this on a static conformal perturbation
δℓ
2
e
= 2 ϕ(x
e
) |d
e
|
2
, ϕ(x) = cos(
k·x), (3)
4
with
k a spatial lattice momentum, and compare against the quadratic form of the static operator
(2).
What the perturbation does and does not change. Equation (3) rescales the squared lengths
and leaves the complex itself untouched. No node is added, no edge removed, no incidence altered:
at L = 6 the 648 nodes, 9720 edges, 32,400 hinges and 15,552 simplices are the same before and after,
and the only variable in the problem is the number ℓ
2
e
carried by each edge. Held at the baseline
value |d
e
|
2
the complex is exactly flat, with deficit angles below 10
−15
; curvature appears only when
those numbers depart from it. What varies is therefore the metric label a bond contributes, not a
distance between nodes moving in a surrounding space. On the reading natural to the framework,
where a bond is a code degree of freedom rather than a rod, a defect alters the state of the network
and the Regge dynamics reads the altered labels as curvature. Nothing below depends on that
reading; it is offered because the alternative picture, of a rigid lattice that somehow bends, is not
what is being computed.
The full spectrum, from Ref. [1]. We record what the operator looks like on all ten compo-
nents, because it fixes which mode (3) probes. Section 3.3 of Ref. [1] computes it on a D
4
subdivision
and finds the linearized Einstein form, in the sector ratios −1 : +3 : 0 for transverse-traceless, trace
and gauge. In an orthonormal basis of Sym(R
4
) and per unit |k|
2
, those ratios are the characteristic
polynomial
λ
4
λ +
1
2
5
(λ − 1), (4)
independent of the direction of k: four zeros, five eigenvalues at −
1
2
, one at +1, which we reproduce
here as a check on our own implementation. The four zeros are the linearized diffeomorphisms,
which change edge lengths but leave every deficit angle zero. The five degenerate modes are
transverse and traceless, and the isolated +1 is the transverse trace mode. This is the Fierz–Pauli
sector structure [5], and it places the conformal perturbation (3) in the distinguished trace sector
rather than at an arbitrary direction in field space; §6 states what that does and does not settle. The
direction-independence is not an accident of the momenta sampled: the quadratic form contracts
four bond vectors, so its angular dependence is carried entirely by the rank-four tensor of Table 1,
which is isotropic.
Result. The two agree up to sign:
S
Regge
2
= − ϕ
T
L
stat
ϕ as |k| → 0. (5)
Static means ω = 0, so the momentum is purely spatial and the admissible directions are those with
k
4
= 0; we use [1000], [1100] and [1110]. Along [1000] the ratio is −1 to 10
−10
at every lattice size
tested (L = 6, 8, 10). Off axis the deviation is nonzero at finite |k| and falls rapidly: 1.35 × 10
−2
,
4.14 × 10
−3
and 1.67 × 10
−3
along [1100] at L = 6, 8, 10, and 1.44 × 10
−2
, 4.28 × 10
−3
, 1.71 × 10
−3
along [1110]. These fall like the fourth power of the momentum, the leading anisotropic term that
rank-four isotropy permits, the rank-six sector being the first that can contribute. Three lattice
sizes do not determine an exponent and we quote none; Figure 1(a) shows the trend against a |k|
4
reference.
Equation (5) is what makes the rest of the paper about gravity rather than about a lattice
Laplacian. Varying the Regge action in the static conformal sector yields
L
stat
ϕ = ρ, (6)
a Poisson equation whose Laplacian is the exactly isotropic operator of Section 2.
5
4 The Newtonian potential
Conventions. Two points of formal setup. The complex is Euclidean throughout: every squared
edge length is positive and no Wick rotation is performed. Designating x
4
as time enters only
through the restriction ω = 0, which selects a spatial momentum; it does not change the signature.
Second, L
stat
is the graph Laplacian of (2), negative semidefinite with L
stat
→ −6|
k|
2
at small
momentum, and the source enters through S ↦→ S −
x
ρ(x)ϕ(x), whose stationarity gives (6).
With these signs a positive source produces a negative potential.
Method. We solve (6) for a unit point source at the origin on balls of radius R with Dirichlet
conditions on the outer shell, by sparse LU factorization. The boundary contributes an image term,
so the solution has the form ϕ ≃ A/r + B; both constants are fitted per radial band. The angular
anisotropy is the normalized rms residual against that fit,
ε(r) =
rms
ϕ
i
− A/r
i
− B
⟨|ϕ
i
|⟩
, (7)
Equation (7) measures departure from any function of r alone and so is a purely angular diagnostic.
An analytic check. The normalization is fixed independently. Expanding the static operator in
lattice units,
∆
stat
(
k) = −6|
k|
2
+
1
2
|
k|
4
+ O(k
6
), (8)
where both coefficients follow from moments of the bond set; the same construction underlies the
isotropic finite-difference stencils used in numerical analysis [6], where a 19-point cubic stencil is
isotropic for the same reason. Explicitly,
(k·n)
2
= 12|k|
2
gives the first, and
(k·n)
4
= 12|k|
4
,
which is (1) again, gives the second. The O(k
4
) term is therefore isotropic as well, a further
consequence of rank-four isotropy. Since 1/k
2
and 1/4πr are Fourier conjugate, the leading term
of (8) gives, for a unit source,
A = −
1
24π
= −0.0132629 . . . (9)
Isotropy. The measured coefficient is A = −0.01326 at R = 24, constant to five decimal places
across all bands and matching (9). The anisotropy is 3.4 × 10
−5
at r = 9 and decays as r
−2.6
; the
rise beyond r ≃ 12 is Dirichlet contamination and recedes as R grows.
Size of the residual. The residual anisotropy is a calculable higher-order lattice correction, and
it decreases rapidly with distance in the numerical Green’s function. We make no claim here about
what it implies for macroscopic propagation: relating a dimensionless angular deviation at a few
lattice spacings to an observable dispersion coefficient requires a calculation we have not done. The
interest of the number is that it is calculable at all, and that Section 2 fixes what controls its size.
Control series. To calibrate what that number means we repeat the measurement on simple
cubic, body-centered cubic, and the bare FCC slice [17, 16], three lattices with the same cubic point
group but different rank-four deviations ∆ = ∥T − αS∥
max
/T
1111
. The anisotropy is proportional
to ∆ with a common constant (Table 2, Figure 1(c)):
ε(r ≃ 9) = c ∆, c = 0.0083 ± 0.0003. (10)
6
10
0
1.2 × 10
0
1.4 × 10
0
1.6 × 10
0
1.8 × 10
0
|
k
|
10
2
deviation of the ratio from 1
exact along [1000]
(a) Regge static sector =
static
D
4
Laplacian
k
[1100]
k
[1110]
|
k
|
4
10
1
6 × 10
0
r
(lattice units)
10
4
10
3
10
2
angular anisotropy of
(b) isotropy of the
Newtonian potential
FCC slice only
SC
BCC
static
D
4
0.0 0.2 0.4 0.6
rank-four deviation
0.000
0.001
0.002
0.003
0.004
0.005
0.006
anisotropy at
r
9
(c) =
c
, and
= 0 for static
D
4
FCC
SC
BCC
static
D
4
Figure 1: The static sector of D
4
, its isotropy, and the coupling. (a) Departure of
S
Regge
2
ϕ
T
L
stat
ϕ from −1, against momentum, at L = 6, 8, 10. The ratio is exact to 10
−10
along
[1000]; off axis it falls as |k|
4
(dashed), the leading correction permitted by rank-four isotropy. Only
static directions (k
4
= 0) are shown. (b) Angular anisotropy of the Newtonian potential. The static
D
4
operator is roughly forty times more isotropic than its own spatial slice would be, and decays
with nearly twice the exponent. (c) The anisotropy of the control lattices is proportional to the
rank-four deviation ∆ with slope 0.0083; the static D
4
operator has ∆ = 0 and lies at the origin.
The static D
4
operator has ∆ = 0 and sits at the origin of that line, an order of magnitude below
the others. That is as it must be: its leading anisotropy is a rank-six effect, while the control
lattices’ is rank four. Note that ∆, not coordination number, orders the lattices: SC has the lowest
coordination and is more isotropic than BCC.
operator K ∆ ε(r ≃ 9) ε/∆ decay
BCC 8 2/3 5.46 × 10
−3
0.0082 r
−0.90
SC 6 1/3 2.86 × 10
−3
0.0086 r
−0.76
FCC slice only 12 1/6 1.34 × 10
−3
0.0081 r
−1.43
static D
4
24 0 3.4 × 10
−5
— r
−2.6
Table 2: Angular anisotropy of the discrete Newtonian potential against the rank-four deviation.
Control lattices at R = 20, static D
4
at R = 24.
5 Coupling to a source
The monopole is the integrated source. Summing (6) over a region telescopes to a boundary
flux, so the far field of a localized source depends on that source only through its total. We verify
this directly on the same operator L
stat
used in Section 4. A total weight 1836 was placed at a
single site, spread evenly over four neighboring sites, and spread unevenly as 1000/500/300/36; the
three give A = −24.3493, −24.3465 and −24.3476, agreeing to one part in 10
4
and matching the
unit-source coefficient scaled by 1836. The residual spread is higher-multipole contamination from
the finite separation of the sources relative to the fit radius, not a failure of the identity, which is
exact. This is Gauss’s law on a graph; it requires no isotropy and holds exactly.
Consequence for a lattice defect. In the selection–stitch model a localized defect is charac-
terized by the set of bonds it disrupts, and its mass is that count up to a fixed scale [2, 1]. The
7
three results now compose. A defect is a disruption pattern on the bonds. That is one number per
disrupted bond, so the source is extensive: it grows with the size of the defect. Each step below
discards part of that information, until only a single number is left:
{δℓ
2
e
}
a field on the bonds
Fierz–Pauli; S
2
=−ϕ
T
L
stat
ϕ
−−−−−−−−−−−−−−−−−→
§3
L
stat
ϕ = ρ
T =αS exactly
−−−−−−−−−→
§2
ϕ =
A
r
A∝
ρ
−−−−−−−→
Gauss
M
one number
.
(11)
The first arrow says the gravitational sector reduces in the static limit to a Poisson equation. The
second says the Laplacian of that equation is exactly rank-four isotropic, so its Green’s function
carries no directional memory of the pattern, to a part in 3 ×10
4
at nine lattice spacings. The third
says the coefficient of the resulting 1/r is the integrated source and nothing else. The far field of a
defect is therefore
ϕ(r) −→ −
GM
r
, M ∝ N
dis
, (12)
with no dependence on the defect’s internal arrangement. The constant of proportionality involves
the bond energy and Newton’s constant, neither of which is fixed here; what the computation
establishes is that the far field scales with the count and with nothing else about the defect. Two
defects with the same count and different geometry are distinguishable near the source, where the
field carries the full structure of the disruption, and indistinguishable at distance, where only the
monopole survives.
Which field is being sourced. One point needs stating, because the classical elastic field of a
lattice defect is not the gravitational one. Reference [1], §3.4, notes that a self-equilibrated localized
defect produces only a short-range Eshelby field falling as 1/r
2
; its gravitational influence is instead
its stress-energy, which sources an induced Poisson equation. The ρ of (6) is that stress-energy,
not the misfit strain, and the 1/r law below is the induced field rather than the elastic one; the
paragraph after next gives the framework’s value for it. We do not rederive the induced coupling;
we take the identification from Ref. [1] and ask what the lattice operator does with it.
What is demonstrated, and what is assumed. It is worth separating the two. What the
calculation establishes is that the far-field gravitational charge is the integrated source,
far-field charge =
x
ρ(x), (13)
and nothing else about how that source is arranged. Equation (13) is the gravitational content of
this section, and it is a statement about ρ alone, independent of what ρ is taken to represent.
The framework supplies the interpretation. Reference [2] defines the mass of a defect as the
total count of crystalline bond states it disrupts, each contributing one unit of m
e
, and gives
(K + 1)K
2
− c
skew
K = 1836 for the trapped tetrahedral-void defect; Ref. [3] obtains the same
integer independently as the fault-tolerant verification cost C
x
= E
s
× C
s
= 36 × 51 in the FCC
stabilizer code. Identifying
ρ with that count N
dis
, and hence with the mass, turns (13) into
(12): ϕ → −GM/r, the Newtonian coupling of mass to gravity, with the far field seeing the count
and nothing else about the defect’s internal arrangement. Reference [3] relates the count to an
energy by Landauer’s principle,
m
x
= C
x
kT ln 2
c
2
, (14)
in which the temperature cancels from any ratio. Equation (14) fixes the form of the relation
but not its scale: mass ratios are ratios of integers, while the absolute scale depends on T and
8
is not determined by the framework or here. The two counts agree arithmetically, 36 × 51 =
13 × 144 − 36, because K = 4c
skew
in the cuboctahedral geometry. The source density of (6) is
therefore proportional to N
dis
, with an undetermined constant; we take that identification from
Refs. [2, 3] rather than rederiving it, as we take Theorem 1 from Ref. [1]. Being blind to everything
except the count is the same as being blind to everything except the mass, which is what one
requires of a gravitational source. That is not automatic on a lattice. A count is a single number,
while the disruption pattern that produces it is a field on the bonds. The reduction from the second
to the first is carried out by the two results that precede it — the isotropy of L
stat
, which removes
any directional memory of the pattern, and Gauss’s law, which removes everything but its integral.
6 Scope
We state the limits plainly.
Which sector has been tested. The reduction (5) is verified for one family of perturbations:
static, conformal, of the form (3). The spectrum (4) identifies that family with the distinguished
trace sector of the Fierz–Pauli Hessian rather than an arbitrary direction in field space. Establishing
its complete constraint-reduced interpretation as the physical Newtonian scalar lies beyond the
present calculation: the lapse and shift sector, and the separation into scalar, vector and tensor
parts, are not worked out here, nor is the transverse-traceless sector, so the graviton propagator on
this complex is not computed. The designation of x
4
as the time axis is an input. The D
4
point
group acts transitively on the 24 roots and mixes all four axes, so nothing in the geometry singles
one out; a static sector exists only once a direction has been chosen, and we choose one. Everything
below is conditional on that choice and independent of which of the four is taken. We do not derive
the identification of ρ with an energy density, nor the value of Newton’s constant, which belongs
to the constants sector of the program.
What the source is. The identification of ρ with the disrupted-bond count, and its conversion
to an energy, are taken from Refs. [2, 3] and not rederived here. The gravitational statements of
§5 hold for any source density: the far field scales with
ρ, whatever that integral counts. The
anisotropy constant c in (10) is measured at one radius on three lattices and should not be assumed
universal. All results are classical and linearized, and all measurements are at fixed lattice spacing
with no continuum limit taken.
7 Conclusion
Setting ω = 0 in the D
4
nearest-neighbor operator leaves a rank-four bond tensor that is isotropic on
the nose. Neither the spatial slice nor the cross-slice bonds have that property alone; it comes from
an exact cancellation between them. The quadratic Regge action reduces, in the static conformal
sector, to the quadratic form of that operator, with corrections entering at O(k
4
) as rank-four
isotropy requires. The resulting Newtonian potential is isotropic to 3.4 × 10
−5
at nine lattice
spacings, forty times better than the spatial slice alone would give, and its monopole is the integrated
source exactly. A localized defect therefore couples to gravity through the integrated source it
presents and through nothing else about its structure. That integral is the framework’s mass, so
the result is the Newtonian coupling of mass to gravity on a fixed lattice. Equation (11) collects
the three steps that take a defect from a field on the bonds to that single number.
9
Declarations
Funding
No funding was received for conducting this study.
Availability of data and material
All data and code supporting the findings of this study are openly available at github .
com/raghu91302/ssmtheory, in the archive d4 static code.zip. No experimental data were
used. Every number, table and figure in this paper is regenerated by those scripts, which
require only numpy, scipy and matplotlib and run from a single flat directory. The cor-
respondence is one script per section: static 01 tensor.py for the rank-four bond tensors
of Section 2 and Table 1; static 02 regge.py for the reduction (5) of Section 3 and its
O(k
4
) power law; static 03 potential.py for the Newtonian potential and the control series
of Section 4; static 04 gauss.py for the distribution-independence of the monopole in Sec-
tion 5; and static 05 figure.py for Figure 1. The simplicial complex and its Regge variations
are in static d4 complex.py and static d4 engine.py; the control-lattice construction is in
static lattices.py.
Competing interests
The author declares no competing interests.
Authors’ contributions
Not applicable. This is a single-author manuscript.
References
[1] R. Kulkarni, Black holes in the FCC selection–stitch model: Bekenstein–Hawking entropy, ge-
ometric evaporation, and primordial-black-hole signatures, Eur. Phys. J. Plus 141, 916 (2026),
doi:10.1140/epjp/s13360-026-08148-9.
[2] R. Kulkarni, Matter as incomplete crystallization: quark charges, color confinement, and the
proton mass from a single extra node in the vacuum lattice, Phys. Open 27, 100423 (2026),
doi:10.1016/j.physo.2026.100423.
[3] R. Kulkarni, The mass-energy-information equivalence: a bottom-up identification of the
particle spectrum via FCC lattice error correction, Phys. Open 27, 100414 (2026),
doi:10.1016/j.physo.2026.100414.
[4] R. Kulkarni, A 67%-rate CSS code on the FCC lattice: [[192, 130, 3]] from weight-12 stabilizers
(2026), arXiv:2603.20294.
[5] M. Fierz and W. Pauli, On relativistic wave equations for particles of arbitrary spin in an
electromagnetic field, Proc. R. Soc. Lond. A 173, 211 (1939), doi:10.1098/rspa.1939.0140.
[6] P. K. Patra and M. Karttunen, Stencils with isotropic discretization error for differential oper-
ators, Numer. Methods Partial Differ. Equ. 22, 936 (2006), doi:10.1002/num.20129.
10
[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),
doi:10.1016/0370-2693(81)90848-0.
[9] J. Cheeger, W. M¨uller and R. Schrader, On the curvature of piecewise flat spaces, Commun.
Math. Phys. 92, 405 (1984), doi:10.1007/BF01210729.
[10] T. Regge and R. M. Williams, Discrete structures in gravity, J. Math. Phys. 41, 3964 (2000),
doi:10.1063/1.533333.
[11] H. W. Hamber, Quantum Gravitation: The Feynman Path Integral Approach, Springer (2009),
doi:10.1007/978-3-540-85293-3.
[12] U. Frisch, B. Hasslacher and Y. Pomeau, Lattice-gas automata for the Navier–Stokes equation,
Phys. Rev. Lett. 56, 1505 (1986), doi:10.1103/PhysRevLett.56.1505.
[13] S. Wolfram, Cellular automaton fluids 1: Basic theory, J. Stat. Phys. 45, 471 (1986),
doi:10.1007/BF01021083.
[14] J. Ambjørn, J. Jurkiewicz and R. Loll, Reconstructing the universe, Phys. Rev. D 72, 064014
(2005), doi:10.1103/PhysRevD.72.064014.
[15] L. Bombelli, J. Lee, D. Meyer and R. D. Sorkin, Space-time as a causal set, Phys. Rev. Lett.
59, 521 (1987), doi:10.1103/PhysRevLett.59.521.
[16] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Springer
(1999), doi:10.1007/978-1-4757-6568-7.
[17] N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston (1976).
11
