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 27, 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. The construction has two parts, each verified numerically in this
paper. First, we compute the linearized Regge action on the D
4
star and show that the resulting
momentum-space operator is two-derivative and gauge-invariant and, in exact arithmetic, equals
the Fierz–Pauli operator: the sector coefficients are (−1 : +3 : 0) for transverse-traceless, trace,
and gauge, exact lattice diffeomorphisms lie in its kernel to machine precision, and a static stress-
energy source produces a potential h
00
(k) ∼ 1/k
2
, i.e. the Newtonian 1/r law. Second, we recast
this sector in the language of quantum error correction. The FCC edge network carries a CSS
stabilizer code; flat spacetime is its zero-syndrome codespace; curvature is the incompatibility
syndrome of an edge-length error; the discrete Bianchi identity is the CSS commutation relation
∂
1
∂
2
= 0; and linearized diffeomorphisms are stabilizer degeneracies. Within this dictionary
we introduce a single dynamical ingredient: the vacuum actively reduces its own syndrome
residual, and the minimal such flow is gradient descent on the squared residual. Its operator
is the square of the Regge/Fierz–Pauli operator of the first part, so the flow is an auxiliary
dissipative relaxation, not the physical graviton propagation; its equilibrium, however, is exactly
the linearized Einstein equation F h = J, with the correct sign on every sector. The physical
claim is this fixed point, not the time dependence of the approach to it. The picture is compact
and self-contained: gravity is the equilibrium of a vacuum that corrects its own geometric errors,
and the Einstein equation is the decoder’s fixed point. We work strictly at linear order; the
known departure of the cubic vertex from Einstein–Hilbert is noted as a limitation and 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 a dynamics, an account of not merely what the correct (flat) configuration
is but why the vacuum evolves toward it.
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
1
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 two
things and connect them.
First (Section 2), we compute the linearized Regge action on the D
4
star directly and show,
numerically and to machine precision, that the resulting operator has the defining properties of lin-
earized 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
elastic geometry of D
4
reproduces linearized Einstein gravity.
Second (Sections 3–4), we reinterpret this sector as quantum error correction and add the
missing dynamics. 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. The flow is an auxiliary relaxation used to characterize
that fixed point, not a proposal for physical graviton propagation.
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 5 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 it is important to 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 (Section 2.4). Second, it uses no entropy, entanglement equilibrium, or
Ryu–Takayanagi 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
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: all lattice points within squared distance 8, Delaunay-
triangulated. Direct enumeration gives the counts in Table 1. Note that the Delaunay triangulation
introduces one diagonal edge (squared length 4) at the origin in addition to the 24 nearest-neighbor
bonds; this is a triangulation device, not a physical bond. It carries 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 the 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
. It is algebraically
slaved to the bonds and adds no variable. In fully nonlinear Regge calculus, where all edge lengths
are independent, the diagonal would be a genuine degree of freedom; the slaving is a linear-order
statement, which is the order at which we work.
Object Count
Vertices in star region 169
Edges at the origin 25 (= 24 bonds + 1 diagonal)
Hinges (triangles containing O) 125
Star simplices (containing O) 100
Table 1: The D
4
star of the origin, by direct enumeration.
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
. (1)
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. We take h
µν
of plane-wave form,
h
µν
cos(k · x
m
) evaluated at edge midpoints x
m
, and expand (1) 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.
3
(i) Flat background is a solution. On the unperturbed lattice every hinge deficit vanishes: the
total deficit over the 125 hinges is −4.6 × 10
−14
, i.e. zero to machine precision. The D
4
background
is Ricci-flat; flat spacetime solves the vacuum equation.
(ii) The operator is two-derivative. Table 2 lists the largest eigenvalue of Q(k) divided by k
2
for several momenta. The ratio is constant to better than 1% over the tested range: Q(k) = O(k
2
),
with no O(k
0
) (mass) or O(k
4
) (higher-derivative) leading part. A two-derivative operator on a
symmetric rank-two tensor, gauge-invariant (below) and isotropic at this order, is the Fierz–Pauli
operator.
|k| max |eig Q(k)|/k
2
0.05 11.004
0.10 10.983
0.20 10.899
Table 2: The Regge operator scales as k
2
: the ratio is constant, so the operator is two-derivative
(Fierz–Pauli structure). The residual ∼ 1% drift is the expected O(k
2
) lattice correction.
(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 (verified: maximum deficit 3.6 × 10
−15
under a plane-wave vertex dis-
placement). 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 by the Moore–Penrose
pseudoinverse (numerically, numpy.linalg.pinv with singular-value cutoff rcond = 10
−6
, which
discards the four near-zero gauge modes and inverts on the physical complement) gives the results
of Table 3: h
00
(k) k
2
is constant, so h
00
(k) ∼ 1/k
2
, whose position-space transform is the New-
tonian potential h
00
(r) ∼ 1/r. The small residual drift is again an O(k
2
) lattice correction and
extrapolates to a constant as k → 0.
|k| h
00
(k) k
2
0.06 −0.11104
0.10 −0.11134
0.16 −0.11220
0.24 −0.11410
Table 3: Static source T
00
: h
00
(k) k
2
is constant (2.7% spread across a factor of four in momentum),
so h
00
∼ 1/k
2
and the potential is Newtonian, 1/r.
Proposition 1. The linearized D
4
Regge operator Q(k) is two-derivative, has the linearized dif-
feomorphism group as its kernel, and produces the Newtonian 1/r potential from a static energy
source. It is the linearized Fierz–Pauli operator; we denote it F.
4
0.00 0.05 0.10 0.15 0.20
|
k
|
10.6
10.7
10.8
10.9
11.0
11.1
11.2
max|eig
Q
(
k
)| /
k
2
constant
Q
(
k
) =
O
(
k
2
)
(a) two-derivative scaling
0.00 0.05 0.10 0.15 0.20 0.25
|
k
|
0.118
0.116
0.114
0.112
0.110
0.108
h
00
(
k
)
k
2
constant
h
00
1/
k
2
1/
r
(b) Newtonian limit
Figure 1: Double-precision diagnostics of the linearized D
4
Regge operator. (a) The largest eigen-
value of Q(k) divided by k
2
is constant across momenta, so the operator is two-derivative (Fierz–
Pauli structure), with a residual ∼ 1% O(k
2
) lattice drift. (b) For a static T
00
source, h
00
(k) k
2
is
constant, so h
00
∼ 1/k
2
and the position-space potential is Newtonian, 1/r. The values are those
of Tables 2 and 3.
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
=
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 exact sector coefficients: the four
transverse-traceless polarizations each carry kinetic coefficient −1, the trace mode carries +3, and
the gauge sector carries 0,
(TT : trace : gauge) = (−1 : +3 : 0), (2)
which is precisely the Fierz–Pauli sector structure. The full kinetic tensor has 1100 entries, all
rational; and the difference between the Regge kinetic form C(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(k, ε) + q
FP
(k, ε) ≡ 0. (3)
Thus the linearized D
4
Regge operator is equal to the linearized Einstein (Fierz–Pauli) operator,
exactly and not merely to numerical precision, on this subdivision. Equation (3) is what elevates
Proposition 1 to an identity; the double-precision diagnostics of Tables 2–3, plotted in Figure 1,
are consistent with 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
5
than from the double-precision Q(k); the latter suffices to exhibit the k
2
scaling, 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, X-checks on nodes and Z-checks on
octahedral cells, both weight 12, satisfying the CSS condition
H
X
H
T
Z
= 0 ⇐⇒ ∂
1
∂
2
= 0, (4)
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 Table 3 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 (4) 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.
6
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 = defect syndrome source. A trapped-node defect [8] occupies edges that no com-
patible field can bring to reference length; it sources a fixed, localized syndrome J ∈ C
1
. By the
Bianchi/CSS identity it is automatically conserved, ∂
1
J = 0.
Remark 1. Gravity is sourced here through the defect/energy channel (a syndrome source J), 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 a dynamics: it says what flat
spacetime is, not why the vacuum moves toward it. Rather than import a thermodynamic relax-
ation, we use the mechanism QEC supplies in its own terms, an active decoder, and promote it
from a discrete correction to a continuous flow.
Syndrome residual. A decoder does not minimize the action; it drives the syndrome toward the
value the source demands. We must be careful about what plays the role of the syndrome here. The
binary FCC stabilizer code produces integer check violations through an incidence map; the contin-
uous 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 con-
traction with the hinge weights (area times deficit, linearized). Define the sourced gravitational
syndrome residual
r(h) = F h − J, (5)
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
, (6)
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 (6)
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 negative kinetic coefficient, the −1 in the
−1 : +3 : 0 structure of Eq. (2)), so the action is not bounded below and cannot be the decoder’s
objective. The residual is. We emphasize 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.
7
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 (6) on the incompatible
edge-length sector is exactly the operator of Section 2,
R
incompatible
= F, (7)
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 (6). 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). (8)
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 (8) are exactly the
solutions of
F h = J, (9)
modulo the gauge kernel. By (7) and Proposition 1, F is the Fierz–Pauli operator, so (9) 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 Table 3.
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 (4). The identification of F as Fierz–Pauli and the Newtonian reduction are
Proposition 1.
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 (8), whose operator F
2
is
positive semidefinite, started from h = 0 converges to the solution of Fh = J (Figure 2): the residual
∥Fh−J∥ falls to ∼ 10
−6
and the iterate matches the exact Fh = J solution, with the correct sign on
the transverse-traceless sector, to one part in 10
4
. (Because F
2
squares the condition number, plain
gradient flow approaches the ill-conditioned modes slowly; a preconditioned solve reaches the same
fixed point faster. The fixed point itself is exact.) The decoder relaxes to the linearized Einstein
solution, as Theorem 1 requires.
Remark 2 (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 (8) 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
8
0 1 2 3 4 5 6 7 8
decoding-flow iteration
1e6
10
6
10
5
10
4
10
3
10
2
10
1
10
0
residual
Fh J
/
J
decoder converges to
Fh
=
J
Figure 2: Convergence of the decoding flow
˙
h = −F(Fh − J) at a representative momentum. The
relative residual ∥Fh − J∥/∥J∥ decreases monotonically to ∼10
−6
; 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.
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 mobil-
ity constant multiplying (8) would rescale the relaxation time without moving the fixed point, and
deriving it from the microscopic lattice dynamics is left open. Recovering the true second-order,
Lorentzian graviton dynamics requires a Hamiltonian (wave) completion that we do not construct
here; the present results are the static/Newtonian fixed point and its code-theoretic reading.
5 Scope and limitations
Established here. (i) The linearized D
4
Regge operator is two-derivative, gauge-invariant, and
reproduces the Newtonian limit: the Fierz–Pauli operator (Proposition 1), verified numerically on
the D
4
star. (ii) The linearized gravitational sector admits a complete restatement in QEC vocab-
ulary, 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 Einstein equation as its fixed point
(Theorem 1).
Limitations. (a) All results are at linear order. 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. (b) 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. (c) The
matter source J is taken as given; deriving its magnitude, the defect mass, is a separate problem
[8] not treated here. (d) A Lorentzian continuation is not required for the static/Newtonian sector
(which is signature-blind) but is required for propagation; we do not address it. (e) The stabilizer
9
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 (6) and a relative-entropy
action are the same quadratic form read two ways: as − log likelihood under a Gaussian syndrome
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.
6 Conclusion
The linearized gravity of the D
4
lattice is the fixed point of an error-correcting decoder. We com-
puted the linearized Regge operator on the D
4
star and showed it to be Fierz–Pauli: two-derivative,
with the diffeomorphism group as its exact kernel, and the Newtonian 1/r potential as its static
response. We then recast the sector as quantum error correction: flat spacetime the codespace,
curvature the syndrome, Bianchi the CSS commutation, diffeomorphism the stabilizer degeneracy,
and added a decoding dynamics whose fixed point is exactly the linearized Einstein equation. Grav-
ity, in this reading, is the vacuum correcting its own geometric errors. The coupling that makes
this correction long-ranged is the shared-edge structure of the code: every edge lies in exactly two
octahedral cells, so each local correction constrains its neighbors, and the cumulative effect of this
overlap is the propagation of a localized matter syndrome into the long-range gravitational field.
The construction is self-contained at linear order; the nonlinear completion, the decoding rate, and
the origin of the matter source remain open, and we have marked each as such.
Data availability
All numerical results in this paper are reproduced by two open scripts. The exact-arithmetic
Fierz–Pauli identity of Section 2.4 (the rank-four isotropy T = 4S, the (−1 : +3 : 0) sec-
tor coefficients, the rational Hessian, and the symbolic identity C + q
FP
≡ 0) is verified by
linearized gravity verify.py. The double-precision diagnostics (Table 1, the flat-background
deficit, Table 2, the diffeomorphism kernel, the Newtonian solve of Table 3, and the decoding-flow
convergence of Section 4) are reproduced by decoding gravity diagnostics.py. Both require
only numpy and scipy and run in seconds.
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