Linearized Gravity on the D
4
Lattice as a
Self-Decoding Code
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
July 30, 2026
Abstract
We show that the linearized gravitational sector of a discrete, parameter-free vacuum, the
D
4
root lattice whose spatial slice is the face-centered cubic (FCC) lattice, is exactly the fixed
point of an error-correcting decoder, and we establish this in three steps. First, the leading
O(k
2
) part of the linearized Regge action on the D
4
lattice equals the Fierz–Pauli operator in
exact arithmetic, with the finite-spacing corrections entering only at O(k
4
): its characteristic
polynomial is λ
4
(λ +
1
2
)
5
(λ − 1), so the four diffeomorphism modes lie in the kernel, the five
transverse-traceless polarizations carry kinetic coefficient −
1
2
and the transverse trace mode
+1, and a static source gives the Newtonian 1/r potential. Second, recast as quantum error
correction, flat space is the codespace, curvature the syndrome, the Bianchi identity the CSS
commutation relation, and diffeomorphisms the stabilizer degeneracy; a decoding flow that
reduces the syndrome residual has the linearized Einstein constraint as its equilibrium. Third,
the four-dimensional D
4
bond tensor is exactly isotropic under SO(4), so choosing any direction
as time and Wick rotating gives a free graviton that is exactly Lorentz invariant at this order,
massless and propagating on the light cone. All results are at linear order; the interacting theory
departs from Einstein–Hilbert at the lattice scale and is not treated here.
1 Introduction
That spacetime geometry might emerge from an informational substrate is suggested by two in-
dependent developments. In holography and the entanglement program, bulk geometry is recon-
structed from boundary entanglement; and quantum error correction (QEC) has been shown to be
the precise mechanism by which bulk locality and geometry arise from a boundary code [2, 1, 3].
The idea that spacetime is an error-correcting code has been realized in finite-geometric toy models
[4]. What has been missing from these constructions is a concrete physical lattice on which the
code lives, together with an operational characterization of the field equation, a process whose
equilibrium selects the correct (flat) configuration rather than a statement that it is correct.
This paper supplies both, for the linearized gravitational sector. The substrate is the D
4
root lattice, whose three-dimensional slice is the face-centered cubic (FCC) lattice; it is fixed and
parameter-free, and it carries a CSS stabilizer code on its edges [9]. The matter content of the
same lattice, baryons as trapped defects, has been developed elsewhere [8]. Here we establish three
results about the gravitational sector.
The substrate, in brief. We summarize the framework so the paper is self-contained. The
vacuum is a perfect FCC bond network: nodes connected by bonds of a fixed reference length,
1
the close-packed ground state of a simple bond energy. A perfect region of this network is empty
flat space; the edge lengths are the dynamical variables, and small deviations of the edge lengths
from their reference values are the metric perturbation. Because the network is over-constrained
(each FCC node has twelve bonds, twice the number needed for rigidity), most deformations are
incompatible: they cannot be relaxed away by moving nodes, and the incompatible part is what
registers as curvature. A defect is a local disruption of this bond structure that the network
cannot heal. The relevant defect for ordinary matter is a node trapped in a tetrahedral void of the
lattice, bonded to the four surrounding nodes; in the companion construction this trapped-node
(tetrahedral-void) defect carries baryon number, and its properties, charge, color, confinement, and
mass, follow from the FCC crystallography around it [8]. Throughout this paper, “matter” means
such a defect, “mass” means the fault-tolerant verification cost the defect imposes on the code
[8], and “the tetrahedral-void defect” is the specific baryonic defect just described. Antimatter,
leptons, and other sectors are not needed here.
First (Section 2), we compute the linearized Regge action on the D
4
star directly and show,
in exact arithmetic, that the resulting operator has the defining properties of linearized general
relativity: it is two-derivative, its kernel is the group of linearized diffeomorphisms, and a static
energy source yields the Newtonian potential. This is the statement that the discrete geometry of
D
4
reproduces linearized Einstein gravity.
Second (Sections 3–4), we reinterpret this sector as quantum error correction. We give the
dictionary between code and geometry, and we introduce a decoding flow, gradient descent on the
squared syndrome residual, whose equilibrium we prove to be the linearized Einstein equation of
the first part. We stress at the outset what this is and is not: a zero-syndrome condition is a
constraint, elliptic on a slice, and in canonical general relativity the constraints are separate from
the hyperbolic evolution equations. The decoder therefore characterizes the constraint sector; it is
not a proposal for physical graviton propagation, which is supplied instead by Section 5.
Third (Section 5), we show that the full four-dimensional D
4
bond tensor is exactly isotropic
under SO(4), so that choosing any lattice direction as time and Wick rotating yields a massless,
propagating free graviton that is exactly Lorentz invariant at order k
2
. This supplies the propaga-
tion that the decoding flow, a static relaxation, does not.
Scope. All results are at linear order in the metric perturbation. At this order the D
4
Regge
operator is exactly Fierz–Pauli. The cubic (three-graviton) vertex departs from Einstein–Hilbert
by an irreducible lattice anisotropy; that is a separate result, and we state it in Section 6 as a
limitation rather than treating it here.
Relation to prior work. Emergent linearized Einstein equations have been obtained before from
quantum-information structures, and we state precisely what is and is not new here. Cao, Carroll,
and Michalakis [5] reconstruct a spatial geometry from the entanglement structure of an abstract
Hilbert-space state and find that perturbations obey a spatial analog of Einstein’s equation; Cao
and Carroll [6] extend this to a four-dimensional spacetime and argue, through a modified version
of Jacobson’s entanglement equilibrium, that the weak-field Einstein equation emerges, explicitly
without a boundary and with quantum error correction specifying the emergence map. The finite-
geometric model of L´evay and Holweck [4] likewise realizes spacetime as an error-correcting code.
Our construction differs from these in three specific respects. First, it is anchored to a fixed,
parameter-free physical lattice (D
4
/FCC), on which the linearized operator is not merely argued to
be Einstein-like but is computed and shown, in exact arithmetic, to equal the Fierz–Pauli operator
at order k
2
(Section 2.4). Second, it uses no entropy, entanglement equilibrium, or Ryu–Takayanagi
2
relation: the objective is a bare syndrome residual, a positive syndrome-residual quantity, and the
reference vacuum is a stabilizer codespace rather than a thermal or maximally symmetric state.
Third, it supplies an explicit decoding construction (a gradient flow on the syndrome residual
whose fixed point is the linearized Einstein equation), giving an operational characterization of
the field equation as the equilibrium of an error-correcting decoder rather than deriving it only as
an equilibrium condition; the flow itself is auxiliary, and we do not claim it as physical graviton
propagation. We note in passing a relevant limitation theorem: stabilizer codes cannot support a
nontrivial area operator for a bipartition [7], which is consistent with the present work, since we
do not construct an area operator or a holographic (boundary) correspondence; the code here is a
bulk lattice code and the geometry is carried by edge lengths, not by entanglement areas.
2 The D
4
star and the linearized Regge operator
2.1 The lattice
The D
4
root lattice consists of the integer points of Z
4
with even coordinate sum. Its minimal
vectors, the 24 vectors of squared length 2, form the 24-cell and are the nearest-neighbor bonds;
the coordination number is K = 24. A space/time split singles out one coordinate; the 12 purely
spatial minimal vectors are then the nearest neighbors of the face-centered cubic (FCC) lattice.
We work on the star of the origin, using a canonical translation-invariant triangulation of D
4
rather than a generic Delaunay routine. The construction is fixed by the lattice itself. The deep
holes of D
4
, at the covering radius 1, are exactly the three nontrivial cosets of D
4
in its dual D
∗
4
,
(1, 0, 0, 0) + D
4
,
1
2
(1, 1, 1, 1) + D
4
,
1
2
(1, 1, 1, −1) + D
4
, (1)
and each hole has precisely 8 lattice points at distance 1, forming a 4-orthoplex (16-cell). Each
16-cell splits into 8 four-simplices of volume 1/12 by a rule fixed by lexicographic order on the four
antipodal-pair directions, so the rule depends only on local geometry and is translation-invariant.
Because the facets of a 16-cell are already tetrahedra, cells may be subdivided independently and
the global complex is automatically consistent: on a periodic L-torus one obtains 3L
4
/2 cells and
12L
4
simplices, of total volume L
4
exactly, with every tetrahedral facet shared by exactly two
simplices. Direct enumeration at the origin gives the counts in Table 1.
This matters for reproducibility. A generic Delaunay triangulation of a D
4
point set is not usable
here: the lattice is massively cospherical, so the triangulation of degenerate facets is implementation-
and input-order dependent, and in practice it returns degenerate (zero-volume) simplices whose
coordinate reconstruction is ill-conditioned. The construction (1) has none of these defects, and we
verify below that the exact Fierz–Pauli identity of Section 2.4 holds on it, so the identity is not an
artifact of a particular subdivision.
The subdivision introduces one interior diagonal (squared length 4) per 16-cell, hence six at the
origin, in addition to the 24 nearest-neighbor bonds; these are triangulation devices, not physical
bonds. They carry no independent degree of freedom at linear order for the following reason. Every
edge length is fixed by the metric through ℓ
2
ij
= |x
i
− x
j
|
2
+ h
µν
(x
i
− x
j
)
µ
(x
i
− x
j
)
ν
, so an edge
length is a linear functional of the ten-component field h
µν
evaluated on the edge vector. The 24
minimal bond vectors already span this ten-dimensional space (the rank of their outer products
b b
T
is exactly 10), so h
µν
is completely determined by the bond lengths, and each diagonal length
ℓ
2
diag
= 4 + h
µν
d
µ
d
ν
(with d the diagonal vector, itself a sum of two opposite bonds) is a linear
functional of the same h
µν
. The diagonals are algebraically slaved to the bonds and add no variable.
In fully nonlinear Regge calculus, where all edge lengths are independent, they would be genuine
degrees of freedom; the slaving is a linear-order statement, which is the order at which we work.
3
Object Count
Edges at the origin 30 (= 24 bonds + 6 diagonals)
Hinges (triangles containing O) 150
Star simplices (containing O) 120
16-cells meeting the origin 24
Table 1: The D
4
star of the origin in the canonical triangulation (1), by direct enumeration. Every
simplex has volume 1/12 and every hinge fan is complete.
2.2 The Regge action and its linearization
Regge calculus assigns curvature to codimension-two hinges. For a hinge h with area A
h
and deficit
angle δ
h
= 2π −
P
s∋h
θ
s,h
(the sum running over the simplices meeting at h, θ
s,h
the dihedral angle
of s at h), the Regge action is
S
Regge
=
X
h
A
h
δ
h
. (2)
A metric perturbation is encoded as a perturbation of the squared edge lengths, ℓ
2
ij
= |x
i
− x
j
|
2
+
h
µν
(x
i
− x
j
)
µ
(x
i
− x
j
)
ν
, where h
µν
is a symmetric 4 × 4 field.
Signature conventions, used uniformly below. The D
4
lattice is Euclidean, so on the lattice
the background metric is δ
µν
and g
µν
= δ
µν
+ h
µν
; this is the convention in which the edge-length
relation just written holds, and it is the one used throughout Sections 2–4 for the operator, its
spectrum, and the static solve, all of which are signature-blind. Wherever a Lorentzian statement
is made, namely the matter coupling of Section 3 and the propagating graviton of Section 5, the
continuation is
η
µν
= diag(−1, 1, 1, 1), g
µν
= η
µν
+ h
µν
, so g
00
= −1 + h
00
, (3)
with dτ
2
= −g
µν
dx
µ
dx
ν
. In particular h
µν
is the same object in both signatures, added to whichever
background is in force; the Newtonian identification g
00
= −(1 + 2Φ) then reads h
00
= −2Φ.
We take h
µν
of plane-wave form, h
µν
cos(k · x
m
) evaluated at edge midpoints x
m
, and expand (2)
to second order in the amplitude. The quadratic form on the ten-dimensional space of symmetric
h
µν
defines the momentum-space operator Q(k).
2.3 Numerical results
We verify three properties, each of which is a defining feature of linearized general relativity. All
computations are on the star of Table 1.
(i) Flat background is a solution. On the unperturbed lattice every hinge deficit vanishes
identically. This is not a numerical statement: evaluating the dihedral angles of each of the 150
fans in closed form and summing symbolically gives exactly 2π in every case, so δ
h
= 0 for all h
with no residual. The D
4
background is Ricci-flat; flat spacetime solves the vacuum equation.
(ii) The operator is two-derivative, with the Fierz–Pauli spectrum. The plane-wave
expansion of (2) to second order has no O(k
0
) (mass) part, because the background deficit vanishes,
and its leading term is O(k
2
); we write the coefficient of |k|
2
as C
(2)
(k, ε), the superscript recording
4
that this is the coefficient at order k
2
and not the full finite-spacing symbol, and assemble it
into a 10 × 10 operator on symmetric h
µν
in an orthonormal basis of Sym(R
4
), i.e. one in which
⟨A, B⟩ =
P
ij
A
ij
B
ij
and every basis element has unit norm. Its spectrum is given in Table 2:
five degenerate transverse-traceless eigenvalues, four exact zeros, and one transverse trace mode,
with no dependence on the direction of k. The normalization matters and is not conventional:
eigenvalues computed in a basis whose off-diagonal elements carry norm
√
2 rather than 1 are not
the eigenvalues of the operator, and the degeneracy of the transverse-traceless sector, which is what
(19) forces, is invisible in such a basis.
sector eigenvalue multiplicity character
transverse-traceless −
1
2
5 tr h = 0, k·h = 0
gauge (diffeomorphism) 0 4 h
µν
= k
µ
ξ
ν
+ k
ν
ξ
µ
transverse trace +1 1 k·h = 0, tr h = 0
Table 2: Spectrum of the linearized D
4
Regge operator per unit |k|
2
, in an orthonormal basis of
Sym(R
4
). Multiplicities sum to 10. The values are independent of the direction of k to 2 × 10
−15
across coordinate axes, face and body diagonals, and generic directions; the exact statement is
Eq. (5).
(iii) Diffeomorphisms are the kernel; the source gives Newton. An exact lattice diffeo-
morphism (a displacement of the vertices, x
i
7→ x
i
+ ξ(x
i
)) changes edge lengths but leaves every
deficit angle exactly zero. The compatible (displacement) modes are therefore the exact kernel of
the action: they are the linearized diffeomorphisms, and the physical content is the gauge-invariant
remainder. Sourcing the operator with a static stress-energy T
00
and solving Q(k) h = J on the
physical complement (numerically, numpy.linalg.pinv with cutoff rcond = 10
−6
, which discards
the four gauge modes because they sit at ∼ 10
−16
relative to the largest singular value) gives
h
00
(k) k
2
= −1.0000000000, (4)
exactly independent of |k|, whose position-space transform is the Newtonian potential h
00
(r) ∼ 1/r.
The k-independence here is not a numerical check but a consequence of working with the O(k
2
)
coefficient itself: the two-derivative property is established by the exact identity of Section 2.4,
not by a momentum scan. Exhibiting the O(k
4
) lattice corrections would require the full finite-
difference operator summed over a fundamental domain of the lattice, which we do not need here.
The singular values of the operator are {1,
1
2
(5)
, 0
(4)
}, so its condition number on the physical
sector is exactly 2.
Proposition 1. The linearized D
4
Regge operator Q(k) is two-derivative, has the linearized dif-
feomorphism group as its kernel with multiplicity four, carries five degenerate transverse-traceless
modes and one transverse trace mode, and produces the Newtonian 1/r potential from a static
energy source. It is the linearized Fierz–Pauli operator; we denote it F.
2.4 The exact Fierz–Pauli identity
Proposition 1 can be sharpened from a numerical statement to an exact one. The deficit is a
cancellation of O(1) dihedral angles to machine zero, so the sector coefficients are best obtained in
exact arithmetic rather than by finite differences. Two inputs make this tractable. First, the rank-
four isotropy of the 24-cell: the fourth-moment tensor of the minimal vectors satisfies
P
v
v
i
v
j
v
k
v
l
=
5
4(δ
ij
δ
kl
+ δ
ik
δ
jl
+ δ
il
δ
jk
) exactly, so T = 4S and T
1111
/T
1122
= 3; this guarantees the quadratic
(propagator) sector is isotropic. Second, the dihedral-angle derivatives on this subdivision are
evaluated in closed form, giving a Hessian whose entries are exactly rational.
Carrying this out (a symbolic computation over Q) yields the characteristic polynomial of the
operator in closed form,
det
C(k) − λ
= λ
4
λ +
1
2
5
(λ − 1), (5)
so the four diffeomorphism modes lie in the kernel, the five transverse-traceless polarizations carry
kinetic coefficient −
1
2
, and the transverse trace mode carries +1: normalized to unit transverse-
traceless coefficient, (TT : trace : gauge) = (−1 : +2 : 0) with multiplicities (5 : 1 : 4). This is
precisely the Fierz–Pauli sector structure, and the physically invariant content is the relative sign:
the conformal mode enters with the opposite sign to the gravitons, the familiar indefiniteness of
the Euclidean gravitational action. The magnitude of the ratio is normalization-dependent and we
therefore fix the convention explicitly, as above, to the orthonormal basis of Sym(R
4
). The kinetic
tensor has 1128 entries on the canonical triangulation and 1100 on a generic Delaunay subdivision
of the same point set, all rational in both cases; and the difference between the order-k
2
Regge
form C
(2)
(k, ε) and the Fierz–Pauli form q
FP
(k, ε) expands to zero as a symbolic identity in general
momentum k and polarization ε,
C
(2)
(k, ε) + q
FP
(k, ε) ≡ 0. (6)
Thus the leading O(k
2
) part of the linearized D
4
Regge operator is equal to the linearized Einstein
(Fierz–Pauli) operator, exactly and not merely to numerical precision. We are deliberate about
the scope of this statement: (6) is an identity between C
(2)
and q
FP
, not between q
FP
and the
complete lattice momentum-space symbol, which carries O(k
4
) and higher corrections at finite
lattice spacing. Those corrections are what break Lorentz invariance at the cutoff, as Section 5
records; the content here is that they are the entire discrepancy, the k
2
term being Fierz–Pauli on
the nose. The identity is independent of the subdivision: it holds on the canonical triangulation
(1) to machine precision (5 × 10
−16
on the spectrum, C
(2)
/q
FP
= −1 to 2 × 10
−15
over random
momenta and polarizations) as well as on a generic Delaunay subdivision of the same point set, so
it is a property of the D
4
geometry and not of a particular choice of diagonals. Equation (6) is
what elevates Proposition 1 to an identity; Table 2 and Eq. (4) are the double-precision realization
of it and additionally exhibit the Newtonian limit directly.
The exactness of the diffeomorphism kernel is a statement about compatible versus incompatible
edge-length fields, and it is precisely this distinction that the error-correction reading of the next
sections makes transparent: the kernel is the gauge orbit, and the physical curvature is the incom-
patibility. A note on method: the deficit is a cancellation of O(1) dihedral angles down to machine
zero, so the exact sector coefficients are obtained by the symbolic computation of Section 2.4 rather
than from a finite-difference Hessian; the double-precision operator suffices to exhibit the spectrum,
the kernel, and the Newtonian limit directly, and is consistent with the exact identity.
3 The error-correction dictionary
We now reinterpret the sector of Section 2 as quantum error correction. The FCC edge network
carries a CSS stabilizer code [9]: physical qubits on edges, Z-checks on nodes and X-checks on
octahedral cells, both weight 12, satisfying the CSS condition
H
X
H
T
Z
= 0 ⇐⇒ ∂
1
∂
2
= 0, (7)
6
where ∂
2
: C
2
→ C
1
and ∂
1
: C
1
→ C
0
are the signed boundary operators of the chain complex on
cells, edges, and nodes. Each entry below is a definition or a restatement; the dynamics enters only
in Section 4.
Dimensional scope of the code. A word is needed on how the three-dimensional FCC code
relates to the four-dimensional Regge calculation of Section 2. The Fierz–Pauli computation is
four-dimensional: it acts on the ten components of the symmetric h
µν
on the D
4
star. Of these,
the six spatial-spatial components h
ij
are edge-length perturbations of the twelve spatial FCC
bonds, and these are exactly the degrees of freedom carried by the stabilizer code of Ref. [9]; the
remaining four components, the lapse h
00
and shift h
0i
, involve the time direction and live on the
time-mixed bonds of the four-dimensional D
4
complex, which are not part of the spatial code. The
previously constructed stabilizer code is thus defined on an FCC spatial slice, and in the present
linearized treatment its incidence structure is used as the spatial restriction of the four-dimensional
D
4
complex. The static/Newtonian sector solved in Eq. (4) is consistent with this: there the field
equation is elliptic on the slice and the lapse h
00
appears as the potential sourced through the spatial
operator, so the static decoder acts effectively on the slice. A full Lorentzian spacetime stabilizer
implementation, including dynamical temporal bonds, is not constructed here; the code-theoretic
statements below concern the spatial restriction, while the exact Fierz–Pauli identity of Section 2.4
is the full four-dimensional result.
Vacuum = codespace. A metric perturbation is an edge-length field h ∈ C
1
. The unperturbed
FCC bond set has structure tensor S
µν
= 4δ
µν
exactly (direct enumeration of the 12 unit bond
vectors), so the coarse-grained vacuum metric is isotropic: the codespace, on which every check
reads zero.
Curvature = syndrome. The checks return the incompatibility of h. In the continuum, a
strain is a pure gradient of a displacement (hence flat, gauge) iff the Saint-Venant tensor inc(ε) =
∇×∇×ε vanishes; its nonvanishing is the linearized Riemann curvature. On the lattice the discrete
incompatibility is the syndrome. A compatible (displacement) field gives zero syndrome and is flat,
exactly the exact-diffeomorphism kernel of Proposition 1; an incompatible field carries curvature.
Bianchi = CSS commutation. The identity ∂
1
∂
2
= 0 of (7) is “the boundary of a boundary
vanishes.” Applied to the syndrome it is the discrete contracted Bianchi identity: the curvature
source is divergence-free, ∂
1
(syndrome) = 0. In the code it is the commutation of X- and Z-checks;
in gravity it is the conservation of the Einstein tensor.
Diffeomorphism = stabilizer degeneracy. Two edge-length fields differing by a compatible
field have identical syndrome and identical geometry; two errors differing by a stabilizer have
identical syndrome and are logically equivalent. The compatible fields are the image of the vertex
coboundary ∂
T
1
: the linearized diffeomorphism orbit and the stabilizer group at once.
Matter, mass, and the field. A trapped-node defect [8] occupies edges that no compatible
displacement can bring to reference length; this incompatible edge-strain is the defect’s internal
structure, the syndrome the code detects, and it is what the verification-cost mass C
x
counts [8].
How this connects to the gravitational field matters, because the two are related through the mass
and not through the strain directly. The mass is a combinatorial code quantity, the verification
cost C
x
, with no metric content; the edge-length field h is the metric. They meet through the
7
standard proper-time coupling of a point mass: a defect of rest energy M = C
x
m
e
contributes
−M
R
dτ = −M
R
√
−g
00
dt to the action, and the metric enters only through g
00
= −1 + h
00
of
Eq. (3), the time-component of the edge-length field. The mass is thus the coefficient of the proper
time, and the field it bends is the temporal edge sector h
00
; the defect’s spatial incompatible strain
is its internal makeup, not the magnitude of the source. This coupling is universal in exactly the
way the equivalence principle requires: every rest energy, of any microscopic origin and any defect
species, sits in front of the same
R
dτ, so all masses gravitate alike per unit mass. The magnitude
of the source is therefore fixed by the mass C
x
(the companion construction [8]), not by the size of
the incompatible strain, which varies between species without tracking the mass.
The linearized source. The proper-time coupling above fixes the source explicitly, and we record
the identification because it is what closes the gap between the point-particle action and the static
solve of Eq. (4). Work in the Lorentzian conventions of Eq. (3). For a defect at rest at the origin
the worldline is x
µ
(t) = (t, 0), so −g
00
= 1 − h
00
and dτ =
√
−g
00
dt =
√
1 − h
00
dt, whence
S
pp
= −M
Z
dτ = −M
Z
dt
1 −
1
2
h
00
+ O(h
2
)
, δS
matter
=
1
2
Z
d
4
x T
µν
h
µν
, (8)
where the second expression is the general linear coupling, with T
µν
defined by δS
matter
=
1
2
R
√
−g T
µν
δg
µν
.
(The equivalent form δS
matter
= −
1
2
R
T
µν
δg
µν
carries the opposite sign because δg
µν
= −h
µν
; we
use the first throughout.) Matching the two expressions in (8) gives, for the static point defect,
T
00
(x) = M δ
(3)
(x), T
0i
= T
ij
= 0, M = C
x
m
e
, (9)
with M the defect’s rest energy, the verification cost of the companion construction [8]. Adding (8)
to the gravitational action and varying gives the lattice Fierz–Pauli equation of Theorem 1 with
J
µν
= −8πG T
µν
, (10)
the normalization being fixed by matching the static solution to the Newtonian potential. In the
conventions (3), g
00
= −(1 + 2Φ) gives −1 + h
00
= −1 − 2Φ, hence
h
00
= −2Φ =
2GM
r
for Φ = −
GM
r
, (11)
i.e. h
00
(k) = 8πGM/k
2
since 1/(4πr) and 1/k
2
are Fourier conjugate. With h
00
(k) k
2
= −1 per
unit source in the 00 slot (Eq. (4)) this yields (10). Only one dimensionful constant enters, and it
is G: the lattice fixes the tensor structure of the operator exactly but not the overall scale of the
Regge action relative to Einstein–Hilbert, so G is an input here and not a prediction. Table 2 and
Eq. (4) are accordingly stated in lattice units.
This also supplies the admissibility condition used in the proof of Theorem 1. The coupling (8)
must be invariant under a linearized diffeomorphism h
µν
7→ h
µν
+ ∂
µ
ξ
ν
+ ∂
ν
ξ
µ
, and
δ
Z
d
4
x T
µν
h
µν
= 2
Z
d
4
x T
µν
∂
µ
ξ
ν
= −2
Z
d
4
x
∂
µ
T
µν
ξ
ν
, (12)
so gauge invariance of the source term holds if and only if ∂
µ
T
µν
= 0. The static defect (9) satisfies
this identically, ∂
µ
T
µ0
= ∂
0
T
00
+ ∂
i
T
i0
= 0. Since the gauge modes are exactly ker F, conservation
of T is precisely the statement J ∈ (ker F)
⊥
= ran F required for the fixed point of (16) to solve
Fh = J: the conservation law of the matter and the admissibility of the syndrome are the same
condition.
8
Remark 1. Gravity is sourced here through the mass/energy channel, not through an operator
identity between the graph Laplacian and the graviton kinetic term. The Hodge Laplacian ∆
1
=
∂
T
1
∂
1
+ ∂
2
∂
T
2
is a scalar operator on 1-cochains and is not the Fierz–Pauli operator, which acts on
symmetric rank-two tensors. The graviton operator is F of Proposition 1, the Regge operator on
the incompatible sector, not ∆
1
.
4 The decoding dynamics
The dictionary defines the ground state (zero syndrome) but not the process that selects it. Rather
than import a thermodynamic relaxation, we use the mechanism QEC supplies in its own terms,
an active decoder, and promote it from a discrete correction to a continuous flow.
Some precision is needed here about which part of general relativity a decoder can reach. A
zero-syndrome condition is a constraint: it is elliptic, it holds on a slice, and in the canonical
formulation of general relativity the Hamiltonian and momentum constraints are logically separate
from the hyperbolic evolution equations. A decoder is therefore naturally a constraint solver, and
that is what we claim for it below; it is not, and cannot be, the propagation law. This is the
structural reason the flow constructed here comes out parabolic rather than hyperbolic, and why
the propagating graviton is obtained separately in Section 5.
The reading that results is the following. Once the Regge/Fierz–Pauli equation is established,
as it was in Section 2 independently of anything code-theoretic, the QEC structure supplies an
operational interpretation of it as a decoder fixed point: a defect injects an incompatible syndrome
that no gauge displacement can remove (the source J), the decoder reduces the squared syndrome
residual W (h) =
1
2
∥Fh −J∥
2
, and the observed field h is the steady state it reaches when it meets
an error it cannot erase. The source of that field is the defect’s mass, the verification cost of the
companion construction [8]. We are careful about what is being claimed: the decoder is not an
independent origin for the field equation, which is already a theorem about the lattice geometry,
but an operational characterization of it. Nor is it a dynamics: the flow below is an auxiliary
relaxation whose fixed point is physical, and we are explicit about that distinction throughout.
Syndrome residual. A decoder does not minimize the action; it drives the syndrome toward the
value the source demands. What plays the role of the syndrome here needs care. The binary FCC
stabilizer code produces integer check violations through an incidence map; the continuous object
relevant to linearized gravity is not that binary vector but its Regge-weighted, source-compatible
contraction. Concretely, let the underlying incompatibility map produce curvature-like checks,
and let F be the Regge/Fierz–Pauli operator built from those geometric quantities by contraction
with the hinge weights (area times deficit, linearized). Define the sourced gravitational syndrome
residual
r(h) = F h − J, (13)
the Regge-weighted, source-compatible contraction of the underlying incompatibility checks against
the matter source. The decoder’s objective is the squared residual,
W (h) =
1
2
∥r(h)∥
2
=
1
2
∥F h − J∥
2
, (14)
a manifestly nonnegative quadratic form. It vanishes exactly when the residual vanishes, and its
minimizers are the configurations that solve the linearized field equation below. Note that (14) is
built from the residual r(h), not from the action ⟨h, Fh⟩; this distinction is essential, because F is
indefinite (the transverse-traceless gravitons carry the negative coefficient −
1
2
of Eq. (5) while the
9
trace mode carries +1), so the action is not bounded below and cannot be the decoder’s objective.
The residual is. On the scope of the analogy: the binary FCC stabilizer code supplies the incidence,
degeneracy, and conservation structure (the boundary maps, the gauge orbit, and ∂
1
∂
2
= 0), while
the real-valued flow below is its linearized geometric analogue, not a literal Pauli-error decoder
acting on the binary code.
The operator is Fierz–Pauli. Evaluated with the Regge weights of the D
4
complex (each hinge
contributing area times deficit, linearized), the operator F appearing in (14) on the incompatible
edge-length sector is exactly the operator of Section 2,
R
incompatible
= F, (15)
by Proposition 1. This is the bridge between the two halves of the paper: the residual minimized
by the decoder is built from the linearized gravitational operator.
The decoder flows downhill. A decoder reduces the residual; the minimal continuous version
is gradient descent on (14). Since F is the real symmetric Hessian of the quadratic Regge action,
F
T
= F, and the Euclidean gradient of W (h) =
1
2
∥Fh − J∥
2
is ∇
h
W = F
T
(Fh − J) = F(Fh − J).
The flow is therefore
˙
h = −∇
h
W = −F(F h − J). (16)
The flow operator is F
2
, which is manifestly positive semidefinite regardless of the sign of F, so
the flow is unconditionally stable: it decreases W monotonically and converges, modulo the gauge
kernel. The indefiniteness of F that would make naive descent on the action diverge is squared
away here, while the physical fixed point is preserved exactly, as we now show.
Theorem 1 (Decoder fixed point = linearized Einstein). The fixed points of (16) are exactly the
solutions of
F h = J, (17)
modulo the gauge kernel. By (15) and Proposition 1, F is the Fierz–Pauli operator, so (17) is the
linearized Einstein equation, with the correct sign on every sector including the transverse-traceless
gravitons; its static sector is the Newtonian 1/r potential of Eq. (4).
Proof. A configuration is stationary iff F(Fh − J) = 0, i.e. Fh − J ∈ ker F. Since F is self-adjoint,
ran F = (ker F)
⊥
. For an admissible source J ∈ ran F, both Fh and J lie in (ker F)
⊥
; therefore
Fh − J lies in both ker F and (ker F)
⊥
, and hence vanishes. Thus Fh = J, with h defined modulo
ker F. The admissibility condition J ∈ ran F = (ker F)
⊥
is ∂
1
J = 0, which holds automatically for
a defect source by (7). The identification of F as Fierz–Pauli and the Newtonian reduction are
Proposition 1.
Remark 2 (What is generic and what is not). The normal-equation content of Theorem 1 is
generic: for any self-adjoint F and admissible J, gradient descent on
1
2
∥Fh − J∥
2
has Fh = J as
its fixed point, and no property of gravity or of the lattice enters that step. What is specific to the
present construction is the input to it, in three parts: the interpretation of Fh − J as a geometric
syndrome residual, the compatibility structure of the FCC bond network that makes “incompatible”
the right notion of error, and the exact identification of F with the Fierz–Pauli operator established
in Section 2.4. The theorem combines these three; it is not a result about least squares.
10
Why the equilibrium is long-ranged. A defect is local, yet its equilibrium field reaches far
from it. Two ingredients produce this and they should be kept apart, because locality alone does
not suffice. The shared-edge incidence of the code, every edge lying in exactly two octahedral
cells, is what makes F a local operator: a correction on one edge constrains its neighbors, those
constrain theirs, and F is assembled from this finite-range coupling network. But a local operator
may perfectly well be massive, in which case its Green function is exponentially screened and there
is no long range at all. What excludes that here is gauge invariance. A Fierz–Pauli mass term
is not invariant under h
µν
7→ h
µν
+ ∂
µ
ξ
ν
+ ∂
ν
ξ
µ
, so the exact four-dimensional diffeomorphism
kernel exhibited in Table 2 forbids an O(k
0
) part; combined with the vanishing background deficit
of Section 2 this leaves
F(k) ∝ k
2
, hence F
−1
(k) ∝
1
k
2
(18)
on the physical complement of the gauge kernel. The static Green function is therefore 1/k
2
, whose
transform in three spatial dimensions is the Newtonian 1/r of Eq. (4). The code incidence supplies
the local network from which the operator is built; masslessness, forced by the stabilizer degeneracy,
is what makes its response long-ranged.
Numerical check of the flow. The Regge operator F is indefinite; its ten eigenvalues at a
representative momentum split into negative (transverse-traceless) and positive (trace) sectors, so
descent on the action ⟨h, Fh⟩ would diverge. Descent on the residual (16), whose operator F
2
is
positive semidefinite, started from h = 0 converges to the solution of Fh = J (Figure 1): the relative
residual falls to 1.5×10
−16
within 10
3
iterations and the iterate matches the exact Fh = J solution,
with the correct sign on the transverse-traceless sector, to the same precision. The condition
number of F on the physical sector is exactly 2, so F
2
is well conditioned and plain gradient descent
suffices; no preconditioning is required. The decoder relaxes to the linearized Einstein solution, as
Theorem 1 requires.
0 200 400 600 800 1000 1200
decoding-flow iteration
10
17
10
15
10
13
10
11
10
9
10
7
10
5
10
3
10
1
residual
Fh J
/
J
Figure 1: Convergence of the decoding flow
˙
h = −F(Fh − J) at a representative momentum. The
relative residual ∥Fh − J∥/∥J∥ decreases monotonically to 1.5 × 10
−16
within 10
3
iterations; the
flow operator F
2
is positive semidefinite, so the descent is stable despite the indefiniteness of F, and
the fixed point is the linearized Einstein equation Fh = J with the correct sign on all sectors.
11
Remark 3 (The flow is auxiliary; the fixed point is physical). Two points about the status of the
flow. First, its operator satisfies F
2
∼ k
4
at small momentum, since F ∼ k
2
. Equation (16) is there-
fore not the physical linearized Einstein evolution equation, which is second order and hyperbolic;
it is an auxiliary dissipative (parabolic) relaxation whose equilibrium solves Einstein’s equation.
The physical claim concerns the fixed point Fh = J of Theorem 1, not the time dependence of
the approach to it: the decoder is a device for characterizing the solution, not a proposal for how
gravitational fields propagate in time. Second, even the rate of that relaxation is unfixed: a mobility
constant multiplying (16) would rescale the relaxation time without moving the fixed point, and
deriving it from the microscopic lattice dynamics is left open. The decoder characterizes the static
fixed point; the propagating graviton is a separate object, the Lorentzian wave operator of Section 5,
obtained from the four-dimensional lattice rather than from the decoder flow. The decoder gives the
equilibrium; the wave operator gives the propagation.
5 The propagating graviton: 4D isotropy and emergent Lorentz
invariance
The construction so far uses the spatial FCC slice, and its results are static: the Newtonian potential
and the decoder fixed point. The dynamics, how the graviton propagates, requires the full four-
dimensional D
4
complex and a time direction. We show that this step is available and that its
outcome is clean: the linearized graviton of the D
4
lattice is exactly Lorentz invariant.
The spatial result of Section 2.4 is the rank-four isotropy T = 4S of the D
4
bond tensor on the
three-dimensional slice. The same computation on the full four-dimensional D
4
root system, the
24 roots of the form (±1, ±1, 0, 0), gives the stronger statement that the rank-four structure tensor
is isotropic in all four dimensions,
T
µνρσ
= 4
δ
µν
δ
ρσ
+ δ
µρ
δ
νσ
+ δ
µσ
δ
νρ
, (19)
verified in exact arithmetic (the residual ∥T − 4 I
4
∥ is zero), and the directional kinetic coefficient
T
µνρσ
ˆ
k
µ
ˆ
k
ν
ˆ
k
ρ
ˆ
k
σ
is identical along the time axis, any space axis, and every diagonal. This is full
continuous isotropy, not merely invariance under the discrete lattice point group: the diagonal-
to-mixed ratio is T
1111
/T
1122
= 3 exactly, the value that annihilates the hypercubic anisotropic
part, and T is unchanged to machine precision under generic SO(4) rotations of the root set. The
contrast with rank six is instructive and marks the boundary of the exact result: the rank-six
moment tensor of the same roots is anisotropic, its largest entry changing by 33% under the same
generic rotations and its directional contraction varying by 50% across direction space (Figure 2),
because the 24-cell is not a spherical 6-design. The quadratic (graviton) operator is therefore
exactly isotropic while the cubic vertex is not, which is why the free graviton is exactly Lorentz
invariant at order k
2
and the interacting theory departs from Einstein–Hilbert at the lattice scale.
Two independent expansions are in play here and should not be conflated. One is in powers of
h: the rank-four isotropy protects the quadratic (free) action, while the rank-six anisotropy afflicts
the cubic three-graviton vertex, so the departure is in the interactions. The other is in powers of k:
Eq. (6) establishes isotropy of the O(k
2
) coefficient, and whether the O(k
4
) terms of the free symbol
preserve it is a separate question that we do not settle, since answering it needs the finite-difference
operator over a fundamental domain rather than the star. Both effects are suppressed at momenta
below the cutoff, but they are different effects. The D
4
lattice is Euclidean; a choice of one direction
as time and a Wick rotation of that direction produce the Lorentzian operator. Because (19) is
fully isotropic, the choice of time direction is immaterial: the operator is the same whichever axis
is singled out.
12
Evaluating the linearized Fierz–Pauli operator that (19) reproduces on the Lorentzian metric of
Eq. (3), and reading its value on the transverse-traceless polarizations of a wave (for propagation
along z, the h
xx
= −h
yy
and h
xy
modes), gives
F h
TT
= k
2
h
TT
, k
2
= η
µν
k
µ
k
ν
, (20)
with unit coefficient in front of k
2
, verified to machine precision for spacelike and for timelike k and
for every orientation of the wave vector. The physical polarizations therefore obey the massless wave
equation, and the on-shell condition is the vanishing of the eigenvalue itself, k
2
= 0: the graviton
propagates on the light cone, and there is no preferred frame and no space-time anisotropy at the
level of the quadratic operator. (We state the result as an eigenvalue relation rather than as a
ratio; the quotient of the eigenvalue by k
2
is 1 for spacelike and timelike k but is indeterminate
on the light cone itself, where both vanish.) The diffeomorphism sector remains in the kernel in
Lorentzian signature exactly as in the Euclidean case, and the trace sector retains its coefficient,
so the sector structure of Eq. (5) survives Wick rotation intact. Emergent Lorentz invariance here
is not assumed; it is a consequence of the four-dimensional isotropy (19) of the root system, the
same isotropy that makes the spatial operator exactly Fierz–Pauli.
axis
(0,0,0,1)
face diagonal
(1,1,0,0)
body diagonal
(1,1,1,1)
12
13
14
15
16
17
18
T k k
exactly constant (spread 2e-13\%)
varies by 50 percent
rank four (graviton operator)
rank six (cubic vertex)
Figure 2: The directional kinetic coefficient T
µν···
ˆ
k
µ
ˆ
k
ν
··· of the D
4
root system, along a path in
direction space running from a coordinate axis through a face diagonal to a body diagonal. At
rank four the coefficient is exactly constant, equal to 12 in every direction (spread 2 ×10
−13
%, i.e.
machine zero): the 24-cell is a spherical 4-design and Eq. (19) holds. At rank six it varies from
12 to 18, a 50% spread, because the 24-cell is not a spherical 6-design. This is the boundary of
the exact result: the quadratic graviton operator is exactly isotropic at order k
2
, hence the free
graviton is Lorentz invariant at that order, while the cubic vertex is anisotropic at the lattice scale.
Two limitations bound this result. It is the free (quadratic) graviton: the cubic self-interaction
depends on the 24-cell being a spherical 6-design, which it is not, so the interacting theory departs
from Einstein–Hilbert at the lattice scale, as at linear order throughout. And it is the classical
wave operator; a quantization must treat the indefinite conformal (trace) sector, which we do not
address. Within these bounds, and at order k
2
in the momentum expansion, the linearized D
4
graviton is a propagating, exactly Lorentz-invariant, massless spin-2 field.
13
6 Scope and limitations
Established here. The three results of Section 1 correspond to the following, the second of
them splitting into the dictionary and the flow. (i) At order k
2
the linearized D
4
Regge operator
is two-derivative, gauge-invariant, and reproduces the Newtonian limit: it is the Fierz–Pauli op-
erator (Proposition 1), with characteristic polynomial λ
4
(λ +
1
2
)
5
(λ − 1) in exact arithmetic and
independent of the subdivision (Section 2.4). (ii) The linearized gravitational sector admits a com-
plete restatement in QEC vocabulary, with flat spacetime the codespace, curvature the syndrome,
Bianchi the CSS commutation relation, and diffeomorphism the stabilizer degeneracy (Section 3).
(iii) A decoding flow (gradient descent on the squared syndrome residual) has the linearized Ein-
stein constraint as its fixed point (Theorem 1). (iv) The full four-dimensional D
4
bond tensor is
exactly isotropic under SO(4), so a Wick rotation gives a free graviton propagating on the light
cone, exactly Lorentz invariant at order k
2
(Section 5).
Limitations. (a) The results are at leading order in the momentum expansion as well as at linear
order in h, and these are independent restrictions. Eq. (6) identifies C
(2)
with q
FP
; the full lattice
symbol differs from Fierz–Pauli at O(k
4
), suppressed by (|k|a)
2
relative to the leading term for
lattice spacing a. We do not compute those corrections, and doing so requires the finite-difference
operator summed over a fundamental domain rather than the star. This is a different question from
the cubic-vertex anisotropy of item (b): that one concerns powers of h, this one powers of k. (b)
All results are at linear order in h. Beyond linear order the 24-cell fails to be a spherical 6-design,
so the cubic (three-graviton) vertex is expected to depart from Einstein–Hilbert by a hypercubic
anisotropy; the quadratic sector treated here is protected by the rank-four isotropy T = 4S of
Section 2.4 and is unaffected, but whether the decoding flow reaches the isotropic continuum fixed
point under coarse-graining is outside our scope. (c) The decoding flow is an auxiliary dissipative
relaxation with operator F
2
∼ k
4
, not the physical second-order graviton dynamics; only its fixed
point Fh = J is claimed as physical, and its rate is not derived. This is a structural feature rather
than a defect of the particular flow chosen: a zero-syndrome condition is a constraint, and no
decoder can be expected to supply a hyperbolic propagation law. (d) The dictionary of Section 3 is
a restatement, and Remark 2 isolates which part of Theorem 1 is generic. It makes the linearized
sector legible in code language and it is exact entry by entry, but we do not claim that it adds
predictive content to Fh = J; a code-theoretic quantity with no continuum counterpart, such as
a distance or a threshold, would be required for that, and obtaining one requires the degrees of
freedom to be discrete, which at linear order in a real-valued edge field they are not. (e) The matter
source is given explicitly by Eqs. (8)–(10), but its magnitude rests on two inputs we do not derive:
the defect mass M = C
x
m
e
, which is the subject of the companion matter construction [8], and
Newton’s constant G, which fixes the scale of the Regge action relative to Einstein–Hilbert and is
not predicted by the lattice. (f) A Lorentzian continuation is not required for the static/Newtonian
sector (which is signature-blind); for propagation it is supplied by the four-dimensional isotropy of
Section 5, which gives an exactly Lorentz-invariant free graviton, but only at quadratic order and
for the classical operator, with the interacting and quantum theory left open. (g) The stabilizer
code is that of the spatial FCC slice; the lapse and shift components and the temporal D
4
bonds
enter the four-dimensional Fierz–Pauli operator but not the spatial code, so the code-theoretic
dictionary is a spatial-slice statement and a full Lorentzian spacetime stabilizer construction is left
open.
Relation to a thermodynamic reading. The syndrome residual (14) and a relative-entropy
action are the same quadratic form read two ways: as −log likelihood under a Gaussian syndrome
14
model, or as a bare syndrome residual minimized by a decoder (here). The present paper is a
code-theoretic formulation with one dynamical ingredient, the decoder, and does not rely on a
thermodynamic ensemble on the cold, parameter-free lattice.
7 Conclusion
The linearized gravity of the D
4
lattice is the fixed point of an error-correcting decoder, and we
established this in three steps.
First, we computed the linearized Regge operator on the D
4
star and showed its leading O(k
2
)
part to be Fierz–Pauli: two-derivative, with characteristic polynomial λ
4
(λ +
1
2
)
5
(λ − 1) in exact
arithmetic, the diffeomorphism group as its exact four-dimensional kernel, and the Newtonian 1/r
potential as its static response. The identity is independent of the subdivision, holding both on
a canonical translation-invariant triangulation built from the deep holes of D
4
and on a generic
Delaunay subdivision of the same point set, so it is a property of the lattice geometry rather than
of a choice of diagonals.
Second, we recast the sector as quantum error correction: flat spacetime the codespace, curva-
ture the syndrome, Bianchi the CSS commutation, diffeomorphism the stabilizer degeneracy, and
added a decoding flow whose fixed point is exactly the linearized Einstein constraint. Gravity, in
this reading, is the vacuum correcting its own geometric errors, though we stress that the decoder
interprets the field equation rather than deriving it. The shared-edge structure of the code, every
edge lying in exactly two octahedral cells, is what makes the operator local; its long range comes
from masslessness, which the exact diffeomorphism kernel forbids the operator from losing, so that
F ∼ k
2
and the static Green function is 1/k
2
. A defect’s mass is its verification cost, a combinatorial
code quantity, and it couples to the field not through its spatial strain but through the proper time,
the temporal component g
00
= −1 + h
00
of the edge-length field; the magnitude of the source is
the mass, supplied by the companion matter construction [8]. We are explicit about the standing
of this step: the dictionary is exact entry by entry and the fixed point is the field equation, but
a zero-syndrome condition is a constraint rather than a propagation law, and the dictionary does
not by itself add predictive content beyond Fh = J.
Third, the full four-dimensional D
4
bond tensor is exactly isotropic under SO(4), so a Wick
rotation gives a massless, propagating free graviton that is exactly Lorentz invariant at order k
2
.
This supplies the propagation that the decoder, a static relaxation, does not.
The construction is self-contained at linear order; the nonlinear completion, the decoding rate,
and the interacting and quantum theory of the propagating graviton remain open, and we have
marked each as such.
Data availability
All numerical results in this paper are reproduced by open scripts collected in a single archive,
ssmtheory
decoding gravity scripts.zip, with a README identifying each. d4 canonical star.py
builds the canonical triangulation (1), verifies the star counts of Table 1, the vanishing background
deficit, and the completeness of every hinge fan, and assembles the linearized Regge operator by
exact contraction of the Regge Hessian; it reproduces Table 2, the Newtonian value (4), the sin-
gular values and condition number, and the decoding-flow convergence of Section 4, and it verifies
C
(2)
+ q
FP
≡ 0 on both the canonical and a generic Delaunay subdivision. partI verify.py per-
forms the exact-arithmetic computation of Section 2.4: the rank-four isotropy T = 4S, the rational
Hessian, the characteristic polynomial (5), and the symbolic identity (6). The propagating-graviton
15
results of Section 5 are reproduced by graviton dynamics.py and isotropy check.py (the exact
SO(4) isotropy of the four-dimensional D
4
tensor and the rank-six anisotropy, verified by invari-
ance under generic rotations) and tt lorentzian2.py (the Lorentz-invariant transverse-traceless
graviton operator). make figs.py regenerates Figure 1. All scripts require only numpy, scipy and
sympy and run in seconds to minutes.
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16
