Entropic Gravity as Regge Gravity on a Discrete Vacuum

Entropic Gravity as Regge Gravity on a Discrete

Vacuum:

A Selection–Stitch Realization of Gravity from Entropy

with a Fixed Dirac–K¨ahler Source

Raghu Kulkarni

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

raghu@idrive.com

Abstract

Gravity from Entropy (GfE) [Phys. Rev. D 111, 066001 (2025)] builds gravity from the quantum

relative entropy between the spacetime metric and an induced metric, and reduces at weak

coupling to Einstein–Hilbert gravity plus a Dirac–K¨ahler ﬁeld. We show that on the Selection–

Stitch Model (SSM), a discrete vacuum on the face-centered-cubic lattice, the reduction continues

one step further. The Einstein–Hilbert term becomes the Regge action on the lattice’s edges. The

Dirac–K¨ahler mass term, evaluated on the edges a defect adds to the complex at ﬁxed physical

amplitude, becomes a Regge point source. Stationarity of the entropic action is then a Regge

variational problem with a discrete source. The SSM’s own bond law, which gives a defect’s

edges a rest length, is one point on GfE’s continuous source family; it ﬁxes the source strength,

and at that point the two formulations agree on every edge to numerical precision. Along the

way the theory’s abstract objects acquire lattice meanings: the induced-metric deviation is the

edge strain, the

-ﬁeld is its inverse, the emergent cosmological constant is the strain energy of

the polycrystalline vacuum. We verify the correspondence on the SSM baryon, a node trapped

in a tetrahedral void. Two naive readings, compatible strain without curvature and incompatible

curvature without strain, each fail GfE’s ﬁeld equation; the family joining them satisﬁes it.

Across the SSM’s coupling range the defect is a curvature object, with deﬁcits of 86

◦

to 145

◦

its four edges. Its mass is its source term, equal on shell to the Regge sum over its edges, with

no free amplitude. The code’s harmonic sector, of dimension 2

+ 2 for every even

, is the

lattice’s own massless ﬁeld and carries 84% of the defect’s strain footprint. The correspondence

is to GfE’s Einstein sector; its higher-curvature terms, absent from the Regge action, rescale the

matched source by about 15% at the defect and leave the geometry unchanged.

Keywords: gravity from entropy; Regge calculus; quantum error-correcting codes; discrete spacetime;

Dirac–K¨ahler operator

1 Introduction

Gravity from Entropy (GfE) [

] is a modiﬁed theory of gravity with two metrics. The ﬁrst,

˜g

, is

dynamical. The second,

˜g

M − β

, is induced by matter and curvature. The action is the

quantum relative entropy between them,

Tr ˜g ln

−1

, with the metrics acting as operators on 0-,

1-, and 2-forms. An auxiliary

-ﬁeld

= (

G˜g

−1

)

−1

turns the action into a dressed Einstein–Hilbert

action with an emergent cosmological constant Λ

≥

0. At weak coupling the theory is Einstein

gravity plus a Dirac–K¨ahler matter ﬁeld.

The Selection–Stitch Model (SSM) [

] is a discrete vacuum: a network of unit entanglement

bonds that assembles into the face-centered-cubic (FCC) lattice in space and the

lattice in

spacetime, carries a [[192

130

3]] CSS code on its edges [

], has Regge gravity with an action the

lattice selects [6], and has matter as a defect, an extra node trapped in a tetrahedral void [4].

This paper establishes the following chain, and everything in it is computed.

1. GfE → Einstein–Hilbert. Bianconi’s weak-coupling limit: L

GfE

→ 3βR − αL

matter

Einstein–Hilbert

→

Regge on the SSM. On the lattice the curvature term is 3

ℓ

, the

Regge action in the edge labels. This is exact in the geometry; the correspondence is at the

order in

where GfE is Einstein, and Section 6.5 computes what GfE’s higher-curvature

terms add.

Dirac–K¨ahler matter

→

Regge point source. The ﬁeld lives on the edges a defect adds. At

ﬁxed physical amplitude its mass term is linear in those labels,

ℓ

, and with GfE’s own

sign it sets the deﬁcit on each ﬁeld edge: 3βδ

= µ.

The SSM bond law selects

∗

. GfE leaves the source strength free. The SSM’s rest-length

prescription for the defect’s bonds is one point on that family,

∗

= 3

βδ

SSM

, below the bound

µ < 3βπ a three-cell edge can carry.

The defect computation veriﬁes the correspondence. The trapped node is solved as a stationary

point of the discretized entropic action with edge labels as the only variables. Two naive

readings fail GfE’s ﬁeld equation in complementary places; the family joining them satisﬁes it

on every edge, and at µ

∗

the GfE form reproduces the SSM solution to 10

−15

In one line: the SSM provides a discrete Regge realization of the entropic gravitational action, and

the SSM bond law selects a deﬁnite member of its Dirac–K¨ahler matter family. The Regge side is

exact in the geometry. On the GfE side the correspondence is to the Einstein sector; GfE’s own

higher-curvature terms, which the Regge action does not contain, shift the matched source by about

15% at the defect and leave the geometry unchanged (Section 6.5).

The chain gives GfE’s abstract objects lattice meanings, collected in Section 3: the induced-

metric deviation is the edge strain, the

-ﬁeld its inverse, Λ

the strain energy of the polycrystalline

vacuum. The dictionary does not supply

˜g

, whose eigenvalues equal one by GfE’s deﬁnition of its

eigenvalue problem for any

˜g

[

], and it does not settle GfE’s matter content, which is an input

there as in any ﬁeld theory. Sections 4 and 5 supply two exact results the defect analysis uses: the

Hodge structure of the code complex, and the closed form of GfE’s relative entropy on it. Section 6

is the defect computation. Section 8 relates the result to the parent papers.

Status of the claims. (T) Exact: enumeration, diagonalization, or solution of a ﬁnite system,

with residuals given. (D) Derived: follows once the dictionary is accepted. (C) Consistency: order

of magnitude. (P) Premise: inherited from the parent papers. (S) Outlook.

2 The two frameworks

Gravity from entropy [

]. The eigenvalues of a rank-two tensor

are deﬁned by

µν

λV

These are the eigenvalues of the matrix

G˜g

−1

. So

˜g

has all eigenvalues equal to one and zero entropy.

The relative entropy between

˜g

and a positive-deﬁnite

reduces to

Tr ˜g ln

−1

−Tr ln

(

G˜g

−1

Matter is a Dirac–K¨ahler boson

⟩

ϕ ⊕ω ⊕ζ

on 0-, 1-, 2-forms, coupled through

. The

Claim Status

Dictionary:

G˜g

−1

−I

= edge strain,

= inverse strain, Λ

= strain

energy, matter = Dirac–K¨ahler ﬁeld on defect edges

D (deﬁnitions)

Unstrained FCC is the ﬂat solution of the one-form sector T (check, not supply)

Integer chain complex; relation-preserving lifts form a 12-dimensional

aﬃne family; lifts outside it lose a mode

Harmonic counts 1

1 for all even

; torsion-free at

= 4

permutation-similar to

; six CLS per void plus six global

modes span the band

Relative entropy in closed form; its variation gives the Dirac–K¨ahler

equation; harmonic sector survives back-reaction

Compatible defect: zero curvature, 20% negative shell, violates

M ⪰ 0

Incompatible defect: +148

◦

−

◦

6, zero elsewhere; violates

M ⪰ 0 on six edges

Intrinsic family solves GfE’s ﬁeld equation edge by edge;

sup-

ported on the ﬁeld edges only; both limits recovered

SSM rest-length prescription = one point

∗

of the Dirac–K¨ahler

source family; source bounded by δ

≤ π

Rigid source has no solution T

Exact-log GfE at the defect:

βR ≤

04; higher-curvature terms

shift matched source to 1.15 µ

∗

; geometry unchanged; dual-volume

dependent at second order

T/D

SSM coupling range

= 0

01–0

15: defect carries 86

◦

–145

◦

deﬁcits;

mass = source term = spoke Regge sum, 4

7–9

in units of 3

;

“locally ﬂat” is the λ → ∞ limit

D/C

84% of defect strain in the void’s harmonic states T

β = ℓ

/48π from Einstein–Hilbert matching C

from grain-boundary strain: right sign, 10

121

too large C

Vacuum is the code; matter is a defect of it P

Table 1: Claim ledger.

induced metric is

G = ˜g + α

M − β

M = D|Φ⟩⟨Φ|D + (m

+ ξR)|Φ⟩⟨Φ|,

R = R ⊕R

µν

⊕ R

µνρσ

. (1)

To ﬁrst order in α, β,

L = 3βR − α⟨Φ|D˜g

−1

D|Φ⟩ − α(m

+ ξR)⟨Φ|Φ⟩. (2)

This is Einstein–Hilbert with zero cosmological constant plus a Klein–Gordon-type ﬁeld. With the

-ﬁeld as a Lagrange multiplier for

G˜g

−1

, the ﬁeld equations are

−1

M˜g

−1

− β

R˜g

−1

a dressed Einstein equation, and

D ˜g

−1

D|Φ⟩ + ˜g

−1

+ ξR)|Φ⟩ = 0, ˜g

−1

˜g. (3)

The emergent constant is Λ

2β

(

G −

I − ln

)

≥

0. It is quadratic in

G −

. The couplings

α, β are positive and otherwise free. The matter content is an input.

The SSM vacuum [

]. Unit bonds assemble by two operators, a planar stitch and a rare

out-of-plane lift (

lift

−3

). The assembly passes through a frustrated tetrahedral foam and ends

in polycrystalline FCC [

]. The periodic complex of size

(half-cell units; bond length

√

) has

2 nodes, 3

edges,

2 octahedral voids, and

tetrahedral voids. The edges carry the code

[[3

3]]: weight-12

-checks on nodes, weight-12

-checks on octahedra [

]. Spacetime is

the

lattice. Its slice is FCC. Its rank-four bond tensor is exactly isotropic; the slice’s is not. The

linearized Regge operator on

equals the linearized Einstein operator as a symbolic identity. The

Regge action is the only member of the local frustration-linear family with a ﬂat vacuum [6].

Two facts about the static sector are used below. Both are elementary. First, at zero frequency

the

nearest-neighbor operator is not the FCC Laplacian. The twelve cross-slice bonds

±e

± e

survive at unit temporal phase as weight-two hops

±e

. The resulting 24-vector static operator

keeps the exact rank-four isotropy of

: the in-slice and projected cross-slice sets each deviate by

6, with opposite signs. In the static conformal sector the quadratic Regge action reduces to a

Poisson equation

stat

with an isotropic Laplacian,

∆

stat

(k) =

−

(

). Second,

summing a discrete Poisson equation over a region gives a boundary ﬂux. So the monopole of a

localized source is its total strength, however the source is spread. A defect couples to the static

ﬁeld through

ρ(x) alone. This is Gauss’s law on a graph. It needs no isotropy.

Matter is an extra node at the center of a tetrahedral void, bonded to the four bounding vertices

at distance

3/8 L

= 0

612

[

]. Throughout, a bond is a code degree of freedom, not a rod. A

defect changes the labels

ℓ

on the edges of a ﬁxed complex. Regge dynamics reads changed labels

as curvature. No node moves in a surrounding space. The only variable is the number each edge

carries. This is the reading used in every computation below.

3 The dictionary

Gravity from entropy SSM lattice status

˜g (dynamical metric) Regge geometry read from the labels {ℓ

} D

G (induced metric) the label ﬁeld itself, ℓ

= |d

(1 + ϵ

) D

G˜g

−1

− I, one-form block diag(ϵ

): the DEC edge-length Hodge star D (exact in DEC)

G-ﬁeld

G diag



1/(1 + ϵ

)



, the inverse strain D

≈

4β

Tr ϵ

elastic energy density of the strained lattice D

R Regge deﬁcits on hinges (edges in 3D) D

M αm

|ω⟩⟨ω| on the defect’s edges; zero on lattice edges T (Sec. 6)

αm

⟨Φ|Φ⟩ µ

ℓ

, µ = αm

|⋆s|: linear in the ﬁeld edges’ labels D (exact in DEC)

|Φ⟩ = ϕ ⊕ ω ⊕ ζ node, edge, and octahedral-cell cochains D

D = d + δ signed incidence operators of the covariant integer lift T

−Tr ln(

G˜g

−1

) −

ln(1 + ϵ

) on the one-form block D

β ℓ

/48π (Einstein–Hilbert matching) C

Table 2: The dictionary. Each left entry is an object GfE deﬁnes by algebra. Each right entry is a

quantity computable on the lattice. Nothing here supplies ˜g or its eigenvalue-one property.

Table 2 states the dictionary. Four remarks ﬁx its scope.

What is not claimed. In GfE,

˜g

has all eigenvalues equal to one because of how the tensor

eigenvalue problem is deﬁned. This holds for every metric, ﬂat or curved. A lattice cannot supply it

and does not need to. The lattice also has nothing to say about which matter ﬁelds GfE should

contain. That is an input in GfE, as in any ﬁeld theory. What the lattice supplies is a deﬁnition of

G − ˜g

, and with it of the

-ﬁeld and of Λ

. It also supplies a speciﬁc proposal for where matter

lives on this substrate: on the edges a defect adds to the complex. Section 6 tests that proposal

against GfE’s ﬁeld equation.

The ﬂat vacuum is a check. The twelve FCC bond vectors satisfy

µν

= 4

µν

[

Any bond set with cubic symmetry has an isotropic rank-two moment; simple cubic gives 2

body-centered cubic gives 8

δ/

3. The FCC value says only that the unstrained lattice is a ﬂat,

isotropic solution of the one-form sector:

= 0,

˜g

, Λ

= 0. That is the vacuum the rest of the

paper perturbs about. It is the only place

µν

enters. Rank-four isotropy, where FCC fails and

succeeds, is what matters for the gravity sector, and is treated in Ref. [6].

Euclidean signature. The lattice is Euclidean. GfE is Lorentzian. No ﬁnite bond set is boost-

invariant, because the Lorentz group is not compact. So there is no Lorentzian

, and none is

sought. Contact is through the

(4)-invariant continuum limit, continued to Minkowski signature as

in Euclidean ﬁeld theory. The rank-four identity of Ref. [

] continues to the invariant (

−ω

)

. Its

rank-six breaking continues to a Lorentz violation suppressed by (

E/M

)

. Everything computed

here is static or spatial and does not depend on signature. One consequence: in Euclidean signature

|∇ϕ|

≥

0, so GfE’s relative entropy on a matter conﬁguration is nonpositive. The two-signed action

of the Lorentzian theory, where timelike gradients are negative, is not available on a slice. The sign

remark in Section 5 is this restriction, not a defect.

Where matter lives. The most important entry is the one for

. In GfE the matter operator

⟩⟨

for a Dirac–K¨ahler ﬁeld. On the lattice the baryon is an extra node with

four extra edges. The ﬁeld is a one-form

on those four edges. Its one-form block

|ω⟩⟨ω|

nonzero on the four edges and zero on every lattice edge. Its 0-form block

|∂

ω⟩⟨∂

ω|

is nonzero

at the ﬁve nodes of the defect, since

∂

ω 

= 0 there; the ﬁeld is oﬀ-shell, as a localized massive

conﬁguration must be. Its 2-form block vanishes, since no octahedral cell contains a defect edge.

Two things follow. First, on every lattice edge the ﬁeld equation reads

−βR

: the metric

deviation there is curvature alone. Second, the ﬁeld’s norm depends on the labels. In DEC the

inner product on one-forms is

⟨ω|ω⟩

|⋆e|/ℓ

, with

ℓ

the integral of the ﬁeld

over the edge. At ﬁxed physical amplitude

this is

|⋆e|ℓ

, linear in the labels. The mass

term

αm

⟨

⟩

is then a Regge point-source term on the ﬁeld’s edges. That is a force on the labels.

It is the force the SSM assigns by hand when it gives the defect’s bonds a rest length. Section 6

makes both statements exact. The harmonic sector of the code complex is a diﬀerent object: it is

the lattice’s own massless ﬁeld, supported on lattice edges, and it is what a defect couples to.

4 The code complex and its harmonic sector

This section records the exact Hodge structure of the code complex. Section 6.6 uses it. All results

are enumerations or dense diagonalizations at

= 4

6 with residuals below 10

−14

. The Betti count

is proved for all even L.

4.1 Integer lift and its selection (T)

Over GF(2) the code is a chain complex because

= 0. A real ﬁeld needs an integer lift of

the

-checks. Each weight-12 octahedral stabilizer is, mod 2, the sum of the octahedron’s three

equatorial 4-cycles. Orienting the squares makes each column of

∂

a signed 1-cycle, and

∂

= 0

holds over

. The lift is not canonical. Reversing a whole generator multiplies a column by

−

1 and

0.0 0.2 0.4 0.6 0.8 1.0

eigenvalue index / sector dimension

eigenvalue

8(1 cos2 /

)=8

flat band,

=130

(nodes)

(edges)

(octahedra)

0.0 0.2 0.4 0.6 0.8 1.0

eigenvalue index / sector dimension

8(1 cos2 /

)=4

flat band,

=434

Figure 1: Spectra of the three Hodge Laplacians at

= 4 (left) and

= 6 (right). The one-form

sector has a ﬂat band at zero of dimension 2L

+ 2. All sectors share the gap 8(1 − cos 2π/L).

cannot change

rank ∂

. Changing the relative signs of the three squares within a generator gives a

diﬀerent integer lift of the same binary stabilizer. Requiring the uniform sum of all generators to

vanish over

is one linear condition per edge. At

= 4 this is 192 equations in 96 sign unknowns over

GF(2), of rank 84, and consistent. The relation-preserving lifts form a 12-dimensional aﬃne space,

of 2

. The translation-covariant lift is in it. Sixty random lifts outside it gave

rank ∂

−

zero times. A generic lift destroys a mode. The crystal’s translation symmetry selects one that does

not.

4.2 Harmonic count and localized basis (T)

With the covariant lift, L

= ∂

∂

, L

= ∂

∂

+ ∂

∂

, and L

= ∂

∂

have kernels of dimension

= 1, b

= 2L

+ 2, b

= 1. (4)

That is 130 and 434 one-form modes at

= 4

6, equal to the code dimension (Fig. 1). The count

holds for every even

. The node graph is connected, so

rank ∂

−

1. And

is the FCC

graph Laplacian on the octahedron sublattice (diagonal 12, oﬀ-diagonal

−

1, twelve per row), so

rank ∂

−

1 as well. The octahedron centers are the odd-parity points and the nodes are

the even-parity points. The translation

x 7→ x

+ (1

0) maps one FCC graph to the other. So

for a permutation

, and

spec L

as multisets. Equality of the

and real

dimensions rules out 2-torsion by universal coeﬃcients. A unit-determinant maximal minor of

∂

L = 4, 6 rules out the rest.

The harmonic sector has a local basis. No mode lives on one void’s six edges: an oriented triangle

is closed but not coclosed. On the void’s 42-edge neighborhood

(its six edges plus the twelve of

each of its four face-adjacent octahedra) the structure is exact. Every tetrahedral void supports

exactly six compact localized states (CLS). Across all voids these span 2

voids

−

4 dimensions. The

remainder is exactly six: the uniform

1 cochains

of the six bond-direction classes, harmonic

for any lift. And (2

voids

−

4) + 6 = 2

+ 2 at both sizes. Three of the

are the torus winding

classes. The other three are zero-winding modes of the stabilizer-deﬁned 2-cell structure. They

disappear when the triangular faces are added as 2-cells; that completion collapses

to the ambient

value 3 at both sizes. The extensive harmonic sector is therefore a count of constraints the code

does not impose. It is anchored at the tetrahedral voids, where the defect sits.

5 The entropic action on the complex

On the unstrained lattice

˜g

and

= 0. GfE’s action on a matter conﬁguration is then the ﬁnite

functional

−Tr ln

(

) on the 4

-dimensional graded space. Because

is self-adjoint,

has rank at most two. For Φ homogeneous in one form degree, its two ranges are orthogonal.

Proposition 1 (T). Let Φ be a normalized eigenmode of

with eigenvalue

, homogeneous of

one degree. Then

(Φ) =

−ln

(1 +

αλ

)

− ln

(1 +

αm

) exactly. Proof: the nonzero eigenvalues of

are ∥DΦ∥

= λ and m

, on orthogonal lines. □

This agrees with the brute-force log-determinant to ten decimals. Its ﬁrst-order expansion is the

linearized Lagrangian

(2)

. On the dispersive branches (

up to 16 at

= 4) linearization fails once

αλ ∼

1: the relative error is 34% at

αλ

= 0

8 and 238% at

αλ

= 8. On the harmonic sector it is

controlled for every α.

Proposition 2 (T). For general graded Φ, let

= 1 +

α∥D

∥

= 1 +

αm

∥

, and

C = ⟨Φ, DΦ⟩ (real). Then L = −ln(AB − α

), and stationarity is

−

2αm

DΦ = 0. (5)

On single-degree ﬁelds

= 0. Then this is GfE’s matter equation

(3)

, with the dressing

˜g

−1

realized

as the scalars A

−1

, B

−1

. Checked against a central-diﬀerence gradient to 2 ×10

−8

Proposition 3 (T). A harmonic one-form solves Eq.

(5)

, back-reaction included, if and only if

= 0. The harmonic sector is exactly the massless solution set of the nonlinear theory. Residual

2 × 10

−15

at m = 0; 2 ×10

−1

at m

= 0.5.

Sign. The functional follows Ref. [

] and is not the normalized density-matrix relative entropy.

On the Euclidean slice it is nonpositive for

M ⪰ 0, for the signature reason given in Section 3.

These results are used in one way below. They show the linearized action is exact on the

harmonic sector and controlled at small

elsewhere. So the matter side of the defect problem can

be posed at linear order without loss. The curvature side cannot, at the deﬁcits the defect turns out

to carry; Section 8 states that limit.

6 The trapped-node defect as a solution of the entropic action

The SSM baryon is an extra node

at the center of a tetrahedral void with vertices

A, B, C, D

bonded to all four. In the label picture the complex gains one node and four edges, the spokes

cA, . . . , cD

. The Dirac–K¨ahler ﬁeld is a one-form

on the spokes. The ten edges of the resulting

are the only place anything has changed. GfE asks: what are the labels, and does the resulting

G satisfy

−1

= I + α

M˜g

−1

− β

R˜g

−1

Two readings of “the labels” exist before any dynamics. Both are computed.

6.1 The compatible reading: positions relax, curvature vanishes (T)

Give every node a position. Give every bond a harmonic energy

(

ℓ − L

)

with rest length

spokes included. Minimize. On periodic boxes

= 6

10 the result agrees to ﬁve ﬁgures. The

spokes relax from 0

612

to 0

6728

and store 0

254

κL

. The strain

ℓ

−

1 on the

original edges, by shell of edge midpoints from the void center (L = 8, half-cell units), is:

shell r edges ⟨ϵ⟩ share of ∥ϵ∥

0.50 (tetrahedron) 6 +0.207 76%

1.12 24 +0.003 0.1%

1.50 30 −0.028 17%

1.80 24 +0.017 2%

≥ 2 ∝ r

−3

Eighty percent of

∥ϵ∥

is positive and on the tetrahedron. Twenty percent is negative. It lies outside

the void’s 42-edge neighborhood and falls as

−3

, the ﬁeld of an Eshelby dilatation center [

]. The

pattern does not depend on the spoke stiﬀness: a factor 30 in

/κ

moves the cosines below by less

than 0.08.

The conﬁguration is realized by positions, so it is a ﬂat embedding. Every Regge deﬁcit vanishes.

This is the compatible sector in the Saint-Venant language of Ref. [

]. GfE then requires, on each

lattice edge,

− βR

with

= 0. With the ﬁeld on the spokes,

vanishes on lattice

edges and

must vanish there; it does not. Even if the ﬁeld were allowed to spread over the lattice,

M ⪰

0 could not produce the negative shell, which is 20% of the strain. The compatible reading

fails.

6.2 The incompatible reading: labels ﬁxed, curvature concentrated (T)

Now let every bond carry the label

, spokes included. This is the

with ten equal edges. It does

not embed in

. The curvature is the three-dimensional Regge deﬁcit on each edge,

= 2

π −

over the cells at that edge. The regular-cell dihedral angles are

tet

arccos

= 70

◦

and

oct

= 109.47

◦

. A bulk FCC edge has two tetrahedra and two octahedra, 360

◦

, ﬂat. At the defect:

edge class cells angle sum deﬁcit

4 spokes 3 inner tets 211.6

◦

+148.41

◦

6 outer tetrahedron edges 2 inner tets + 1 tet + 2 octs 430.5

◦

−70.53

◦

every other edge 2 tets + 2 octs 360

◦

The Regge sum is

ℓ

= 2

975

. The curvature is exactly compact. With labels unchanged,

, and GfE reduces to

on every edge. This works on the spokes, where the ﬁeld

lives and the curvature is positive. It fails on the six outer edges, where

= 0 and the curvature is

not. The incompatible reading also fails, in the complementary place.

6.3 The intrinsic family (T)

The physical conﬁguration is neither reading. Treat every edge length as a variable. Use no

embedding. Subdivide each octahedron about its center into eight tetrahedra, with six auxiliary

spokes of ﬂat length

√

and no elastic cost. Replace the defected void by its four inner tetrahedra.

The discretized action has three terms.

The lattice bonds are the vacuum. They carry the bond energy

(

ℓ

− L

)

. This term is not

in GfE. It is the SSM’s own, part of its gravitational sector: Ref. [

] obtains the Regge action from

an elastic expansion of the lattice and Newton’s constant from integrating out lattice modes. It

enters with the same sign as the Regge term, as an energy of the medium. The opposite sign also

gives a solvable family with the same two endpoints; it diﬀers in the interior, where it compresses

the tetrahedron edges while the void expands and makes the compatible limit an energy maximum.

The sign used is the physical one, and it is an SSM input.

= 3 /

0.6

0.7

0.8

0.9

1.0

spoke label

compatible limit (B)

incompatible limit (C)

(a)

spoke label /

150

100

150

deficit angle (deg)

(b)

spoke deficit

tetrahedron-edge deficit

shell (

= 1.5) deficit ×10

0.0

0.5

1.0

1.5

2.0

2.5

3.0

energy / action

(c)

elastic energy [

]

Regge sum

e e

[

]

Figure 2: The intrinsic family of trapped-node solutions at

= 4, against the coupling

= 3

β/κL

(a) Spoke label, from the incompatible limit (

ℓ

) to the compatible limit (0

673

). (b) Deﬁcit

angles on the spokes, the tetrahedron edges, and the

= 1

5 shell. The shell’s curvature changes

sign near

λ ≈

2 and is positive wherever its strain is negative. (c) Elastic energy and Regge sum.

The defect’s energy is elastic at large λ and curvature at small λ, where the physical coupling lies.

The Regge term is 3β

ℓ

The four spokes carry the ﬁeld. Its mass term is

αm

⟨ω|ω⟩

αm

|⋆s|/ℓ

, where

is the cochain. In the continuum, GfE’s stress tensor is the variation of the matter action with

respect to the metric at ﬁxed ﬁeld (Ref. [

], Eq. 65). On the lattice one must say what is held ﬁxed

when a label changes, and the answer is not a free choice.

For the 0-form component there is nothing to decide. Its mass term is

|⋆v|

, the dual volume

grows with the labels, and ﬁxed component and ﬁxed amplitude agree: the term rises with the size

of the cell, and with the sign of Eq.

(2)

it gives positive deﬁcit. For the 1-form there are two readings.

Holding the cochain

ﬁxed is holding the covariant component ﬁxed. Then

|ω|

∝

/ℓ

against a

measure

∝ ℓ

, the term falls as 1

/ℓ

, and with GfE’s sign the deﬁcit on the edge is negative. That is

the stress of a massive vector ﬁeld, whose content scales with the edge it lives on, and it is a diﬀerent

object from a ﬁxed excitation. Holding the physical amplitude

ﬁxed, with

wℓ

, gives

ℓ

with

αm

|⋆s|

, linear in the label, and with GfE’s sign it gives positive deﬁcit. This is the

reading forced by what a bond is. In the SSM a bond is a Bell pair. Its label is its size. Its content

does not change with its size. A ﬁeld on that bond has a deﬁnite frame-measured amplitude, and its

cochain is that amplitude times the length. Holding

ﬁxed is the discrete form of Regge’s coupling

of any localized matter, mass ﬁxed and length varied; holding

ﬁxed would hold the product of

mass and length ﬁxed, which is not a variation at ﬁxed matter. A check that does not depend on

the prescription: the

with all labels at

has +148

◦

on its spokes (Section 6.2), and only

the ﬁxed-amplitude reading reproduces that sign with the sign of Eq.

(2)

. The linear form is the

Regge point-source term. The kinetic term

α∥δω∥

(with

dω

= 0, since no octahedral cell contains

a spoke) has (

δω

)

|⋆v|

−1

e∋v

±w

|⋆e|

at ﬁxed amplitude, with no label dependence at ﬁxed

dual volumes; it does not enter the stationarity conditions. With the sign of Eq. (2) for the ﬁeld,

E(ℓ) =

e ∈ bonds

(ℓ

− L

)

+ 3β

ℓ

(ℓ) − µ

s ∈ spokes

ℓ

. (6)

The complex is a closed periodic torus, so the three-dimensional Schl¨aﬂi identity

ℓ

dδ

= 0

holds exactly (it fails on an open star, where the hinge sum is incomplete), and the stationarity

conditions are

κ(ℓ

− L

) + 3β δ

= 0 (bonds), 3β δ

= µ (spokes), δ

= 0 (auxiliary edges), (7)

with two dimensionless parameters,

= 3

β/κL

and

ˆµ

µ/κL

. The spoke equation is GfE’s

-ﬁeld equation on the ﬁeld’s support, and it is the Regge equation for a point source: the deﬁcit

on each ﬁeld edge equals the source strength, in units of 3

. A spoke is shared by three inner

tetrahedra whose dihedral angles at it cannot fall below 60

◦

, so δ

< π and the source is bounded,

µ < 3βπ. (8)

The SSM’s own prescription, that the spokes are bonds of rest length

, replaces the spoke equation

by the bond equation. It is one point of the ﬁeld family. Matching the two equations at that point

gives

∗

= 3β δ

SSM

, ˆµ

∗

= λ δ

SSM

, (9)

evaluated on the SSM solution. Eq.

(9)

ﬁxes GfE’s source strength

αm

|⋆s|

by the SSM bond

law. Eq.

(7)

is solved by least squares, ﬁrst in the SSM form (spokes as bonds) to give the

-family

of Table 3, then in the GfE form at µ = f µ

∗

to give Table 4. The residual is below 10

−13

at every

entry of both tables. The undefected complex is ﬂat to 10

−15

. The

λ →

0 conﬁguration reproduces

the deﬁcits of Section 6.2 exactly. Table 3 and Fig. 2 give the result. The family does not depend

on size:

= 4 and

= 6 agree to 1% at

= 1. It joins the two readings. At

λ → ∞

it reproduces

the position relaxation of Section 6.1 (spoke 0

673

, elastic energy 0

253

κL

, tetrahedron strain

+0.207, shell −0.023). At λ → 0 it reproduces the incompatible K

λ spoke ℓ/L

spoke

tet

shell

/κL

ℓδ/L

→ 0 1 +148.4

◦

−70.5

◦

0 0 0 0 2.975

0.003 0.992 +147.5

◦

−69.9

◦

+0.007 0.000 −0.12

◦

0.0002 2.87

0.01 0.975 +145.2

◦

−68.5

◦

+0.024 +0.0001 −0.41

◦

0.0017 2.63

0.1 0.818 +104.3

◦

−59.0

◦

+0.217 +0.060 −16.8

◦

0.135 0.233

1 0.683 +18.1

◦

−5.6

◦

+0.204 −0.028 +0.83

◦

0.239 0.0069

10 0.674 +1.9

◦

−0.6

◦

+0.208 −0.024 +0.07

◦

0.252 0.0001

→ ∞ 0.673 0 0 +0.207 −0.028 0 0.254 0

Table 3: The intrinsic family at

= 4. “Shell” is the 30-edge orbit at

= 1

5 that carries the

negative strain of the compatible limit. On every bond, at every

−

λδ

to the stated

residual.

The structural result. On every lattice bond, at every

−

λ δ

edge by edge. This

includes the shell that broke the compatible reading. At

= 1 it has

−

028 and

= +0

◦

. It

is positive curvature, not negative matter. In GfE’s terms the ﬁeld equation

−1

− I

M − β

holds on the whole complex with

M = αm

|ω⟩⟨ω| on the four spokes,

M = 0 on every lattice edge. (10)

The ﬁeld acts on the lattice through curvature alone. On lattice bonds Eq.

(7)

is the stationarity of

Eq.

(6)

, so the proportionality of

and

there follows from adopting GfE’s linear structure; it is not

a test of it. What is established is this. GfE’s structure, with its Dirac–K¨ahler ﬁeld on the defect’s

edges and its own relative sign between curvature and matter, has a solution at every coupling. The

solution reproduces two limits computed by diﬀerent methods. Neither limit alone is consistent

with the theory. The family is.

λ f = µ/µ

∗

spoke ℓ/L

spoke

tet

ℓ

0.01 0.25 0.638 +36.3

◦

−15.1

◦

+0.005 1.62

0.01 0.5 0.680 +72.6

◦

−30.9

◦

+0.011 3.45

0.01 0.75 0.762 +108.9

◦

−48.2

◦

+0.017 5.79

0.01 1 (ˆµ

∗

= 0.0253) 0.975 +145.2

◦

−68.5

◦

+0.024 9.88

0.01 1.1 1.214 +159.7

◦

−78.8

◦

+0.028 13.5

0.01 1.2 2.156 +174.2

◦

−93.4

◦

+0.033 26.2

0.01 1.24 bound of Eq. (8): no solution

1 0.5 0.647 +9.1

◦

−2.8

◦

+0.100 0.41

1 1 (ˆµ

∗

= 0.317) 0.683 +18.1

◦

−5.6

◦

+0.204 0.87

1 2 0.761 +36.3

◦

−11.2

◦

+0.429 1.93

1 4 0.945 +72.6

◦

−22.7

◦

+0.950 4.79

1 8 1.799 +145.1

◦

−49.9

◦

+2.50 18.2

Table 4: The Dirac–K¨ahler source family at

= 4. Spokes carry the ﬁeld; lattice bonds carry the

vacuum. At

= 1 the GfE form reproduces the SSM solution of Table 3 on every edge to 10

−15

The last column is the spoke Regge sum, equal on shell to the source term

ℓ

in units of 3

βL

The spoke deﬁcit is

f δ

SSM

exactly, by Eq.

(7)

; the spoke length and the rest of the lattice adjust to

it.

The source strength. Table 4 shows the family. At

= 1 the GfE form and the SSM form agree

on every bond to 10

−15

. The spoke deﬁcit is proportional to the source exactly; the spoke length

responds nonlinearly and diverges at the bound

(8)

, which at

= 0

01 is

= 1

24. At

= 0

the SSM point sits at 81% of the bound; at

= 0

15, at 48%. In GfE the source strength is a free

input, so the mass is a free input. In the SSM it is not: the defect’s bonds are code bonds of rest

length

, and that ﬁxes

∗

. The two statements are compatible. GfE supplies the form of the

matter term. The SSM supplies its strength. Eq. (9) is the exchange rate.

A rigid source is inadmissible (T). Hold the four spoke labels at

and relax everything

else. There is no solution. The residual stalls at 10

−2

to 10

−3

, with strains of order one on the

tetrahedron edges at

= 1. The lattice cannot hold a rigid incompatible

. The source must be

elastic. This settles the hard-core versus soft-core question of Ref. [

]’s exclusion-radius analysis

from the gravity side. With binary hard-core bonds the embedded defect costs nothing and, as

found here, has no solution. Only the soft-core defect exists.

6.4 The physical regime and the mass (D/C)

Take

ℓ

(Section 7) and

= 1

843

ℓ

[

]. Then

= 3

β/κL

≈

−3

/ˆκ

with

ˆκ

κℓ

the bond stiﬀness in Planck units, up to the measure convention of Section 7. Two estimates bracket

ˆκ

. A Planck-scale stiﬀness,

ˆκ ∼

1, gives

λ ∼

01. The SSM’s bond energy

bond

ℏc/

[

] with

κ ∼ E

bond

gives ˆκ ≈ 0.04 and λ ≈ 0.15. Table 5 gives the SSM point across that range.

Across the range the defect is a curvature object: deﬁcits of 86

◦

to 145

◦

on its four edges, lattice

strain from 2% to 22%, elastic energy from 0

4% to 22% of the source term. The gravitating mass

is the source term. By the Gauss-law argument of Section 2 the far ﬁeld sees the integrated source,

and on shell the source equals the Regge sum over the ﬁeld’s own edges, since 3βδ

= µ there:

defect

∝ µ

ℓ

= 3β

s ∈ spokes

ℓ

, (11)

λ spoke ℓ/L

spoke

tet

4ℓ

(mass, units 3β) elastic / source

0.01 0.975 145

◦

+0.02 9.9 0.004

0.05 0.887 130

◦

+0.11 8.0 0.02

0.10 0.818 104

◦

+0.22 6.0 0.12

0.15 0.773 86

◦

+0.22 4.7 0.22

Table 5: The SSM point across the plausible coupling range. The last column is the lattice’s elastic

energy over the source term µ

ℓ

between 4

7 and 9

in units of 3

over the range, and 4

590

= 10

λ →

0, where it

is ﬁxed by the single dihedral angle 2

π −

arccos

. The six outer edges carry a negative Regge

sum (

−

λ →

0); that is the geometry’s response, not the source, and the far ﬁeld does not

see it. The mass has no free amplitude once the SSM bond law is imposed: it is a function of the

one coupling λ. In GfE alone it scales with the source (Table 4).

6.5 Beyond the Einstein sector (T/D)

Everything above uses GfE at ﬁrst order in

, where it is Einstein gravity. The Regge action is

exact in the geometry, so the order-one deﬁcits of the defect are not a problem for the lattice side.

The question is what GfE’s own nonlinearity does. Its curvature term is

−Tr ln

(

I −β

R˜g

−1

), and at

second order this is

(

R˜g

−1

)

: terms quadratic in the curvature. Beyond ﬁrst order GfE is

a higher-curvature theory. The Regge action contains no such term at any order, so beyond the

Einstein sector the two theories diﬀer. We quantify the diﬀerence at the defect.

On the lattice, take the curvature density on an edge to be

ℓ

with

the edge’s dual

volume, and the exact-log action

E =

bonds

(ℓ

− L

)

− 3



1 − βR



−

ˆκ



1 + ˆκˆµ

ℓ



, (12)

whose ﬁrst-order expansion is Eq.

(6)

. At ﬁrst order the dual volumes cancel by Schl¨aﬂi; at second

order they do not, which is the standard ambiguity of any lattice

term, and we take the uniform

honeycomb value

√

. The expansion parameters at

= 0

01 are small:

βR

= 0

041 on

the spokes, 0

020 on the lattice, and

ˆκˆµ

ℓ

= 0

058. The stationary point of Eq.

(12)

is found by

a ﬁxed-point iteration on the non-uniform Schl¨aﬂi remainder and checked against the full numerical

gradient (residual 3 × 10

−5

against 3 × 10

−3

at the ﬁrst-order solution).

= 0

01 and the ﬁrst-order source

∗

, the exact-log solution moves the spoke label from

975 to 0

836

, the spoke deﬁcit from 145

◦

to 127

◦

, and the spoke Regge sum by

−

25%; lattice

bonds move by 10

−3

. The corrections themselves are small, +4% on the curvature weight and

−

6% on the source, but they act in the same direction and the spoke response is steep near the

bound at this

(Table 4:

= 1

→

1 moves

ℓ

by 24%). Re-matching the source recovers the

SSM geometry: at

= 1

∗

the exact-log stationary point has the SSM spoke label, lattice labels

within 4

−4

, and deﬁcits within 2

◦

of the ﬁrst-order solution. With the spoke dual volume

taken as a quarter of the void instead, the shift at ﬁxed

∗

−

24% in

ℓ

; the second-order result

depends on the dual-volume assignment, as a lattice R

term must.

The reading is this. The SSM realizes GfE’s Einstein sector exactly. GfE’s higher-curvature

sector is what the SSM’s selection of the frustration-linear family, and hence of the Regge action,

leaves out. At a defect the diﬀerence is a

∼

15% rescaling of the source strength that the SSM

bond law must be matched to, with the geometry and the mass at the SSM point unchanged. This

is a stated point of diﬀerence between the two theories, not an error in the correspondence; the

correspondence is between GfE’s Einstein limit and the lattice.

6.6 The defect’s footprint in the harmonic sector (T)

The code’s harmonic modes sit at the same voids as the defect. Take the compatible strain of

Section 6.1 (the family’s strain is concentrated on the tetrahedron edges at every

, more so at

small

, so this is the least favorable case) and

, the projector onto the void’s six CLS. The cosine

between

and

diag P

is 0

84 on the 42-edge neighborhood and 0

74 over all edges. A local positive

operator supported on the defect’s footprint lifts exactly the harmonic components that overlap it

(38 of 130 at

= 4, independent of its strength), and the defect’s footprint has 84% of its weight

in that span. The harmonic ﬁeld is what the defect couples to; it is not the defect. Whether the

coupling gaps those modes is the hybridization problem of the defected complex, not solved here.

7 Couplings and the cosmological constant (C)

. Match the curvature term 3

βR

of Eq.

(2)

to the Einstein–Hilbert normalization at

ℓ

, with

the lattice sum carrying the four-dimensional measure

ℓ

. This gives

ℓ

π ≈

−3

ℓ

The coeﬃcient 3 is the trace over the three form degrees. Ref. [

] shows the Regge action is selected

within its family rather than chosen, so this is a match between two derived coeﬃcients. If a

Sakharov-type induced term adds to the same coeﬃcient [6], this is an upper bound.

from the polycrystal. In the dictionary, Λ

≈ Tr ϵ

per site. The assembled vacuum is

polycrystalline. Ref. [

] reports a bond-length tolerance of

5%, set by the Regge deﬁcit of the

frustrated intermediate, and a bulk fraction near 30% of nodes at

= 10

11, the signature of grain

boundaries. Take

rms

∼

05 on the twelve bonds of that fraction of sites. Then

Tr ϵ

∼

−3

per site and

∼

9 × 10

−3

× 48π

4 ℓ

≈ 0.3 ℓ

−2

, (13)

against the observed

∼

−122

ℓ

−2

. The sign is automatic. The magnitude is oﬀ by 10

121

, the usual

problem. The only lever in the dictionary is the grain-boundary fraction, which would have to

∼

−121

: a single crystal. The number is reported because the mechanism is now a deﬁned

quantity on a deﬁned ensemble, and any future claim about it can be checked against it.

8 Scope

Beyond the Einstein sector. Section 6.5 computes GfE’s second-order terms at the defect. What is

not done is a lattice deﬁnition of GfE’s

sector free of the dual-volume choice; that is the general

problem of discretizing higher-curvature actions and is not speciﬁc to this substrate.

Relation to the parent papers. Ref. [

] prices the defect by the elastic self-energy of its compressed

bonds and describes it as locally ﬂat; that is the

λ → ∞

member of the family of Section 6.3, where

bonds are soft relative to gravity. The SSM’s inputs place

between 0

01 and 0

15 (Table 5), where

the mass is the source term and the defect carries deﬁcits of 86

◦

to 145

◦

on its edges. The hard-core

exclusion window of Ref. [

] is moot: a rigid source has no solution. The static-sector results of

Section 2, compact curvature and a Newtonian exterior sourced by the integrated strength, hold at

every

. The code’s harmonic sector is not the matter sector; it is the lattice’s massless ﬁeld, and

the defect couples to it.

One defect, one void. The computation is for one trapped node in a periodic box. Interactions

between defects, the ∆ multiplet of Ref. [

], and the proton–neutron distinction are not addressed.

Dual volumes. The spoke dual area

|⋆s|

and the dual volumes in the kinetic term are held

at their ﬂat-embedding values. In an intrinsic geometry they depend on the labels. For the mass

term this changes

by a factor that is itself a function of

ℓ

, and would be absorbed into

∗

at the

SSM point; for the kinetic term it would introduce a weak label dependence not present here.

The vacuum bond sign. The sign with which the bond energy enters Eq.

(6)

relative to the

Regge term is an SSM input, chosen by the physical criteria stated in Section 6.3. It is not ﬁxed by

GfE, which has no vacuum stiﬀness term.

No dynamics. Everything is static. The hybridization of the defect with the gapless harmonic

band would decide whether the modes overlapping its footprint are gapped. That is a deﬁned ﬁnite

computation, not done here.

The measure convention.

depends on how the three-dimensional Regge sum and the elastic

energy are given a common measure. The value 6

−3

/ˆκ

is an estimate. The bond stiﬀness

ˆκ

not ﬁxed by the program. The qualitative conclusion, that the defect carries curvature, needs only

λ ≲ 1. That holds unless ˆκ ≲ 10

−3

Relation to holographic codes. This construction diﬀers from holographic code-subspace pro-

grams [

]. The code lives on the bulk lattice. Geometry arises from assembly. No subsystem

duality is claimed.

9 Conclusion

On the Selection–Stitch complex, Gravity from Entropy continues one step past its own weak-

coupling limit. Einstein–Hilbert plus a Dirac–K¨ahler ﬁeld becomes a Regge action plus a discrete

point source, and stationarity of the entropic action is a Regge variational problem. The SSM

supplies the discrete Regge realization of the entropic gravitational action. Its bond law selects one

member of the entropic theory’s continuous matter family, and at that member the two formulations

agree on every edge to numerical precision. The theory’s abstract objects acquire lattice meanings:

edge strain, inverse strain, strain energy of the polycrystalline vacuum. The SSM baryon, solved

with no embedding, is a curvature object across the SSM’s coupling range, with the locally-ﬂat

description as its soft-bond limit; its mass is its source term, equal on shell to the Regge sum over

its own edges, with no free amplitude; a rigid source has no solution. The code’s harmonic sector,

of dimension 2

+ 2 for every even

, is the lattice’s massless ﬁeld and carries 84% of the defect’s

strain footprint. What remains open is a lattice deﬁnition of GfE’s higher-curvature sector free

of the dual-volume choice, the Lorentzian continuation, and the size of Λ; each is now a ﬁnite

computation on a deﬁnite complex.

Data availability

All results reproduce from the Python scripts archived at Zenodo, doi:10.5281/zenodo.22652351.

They need only NumPy, SciPy, and Matplotlib:

entropy fcc real harmonic.py

and

entropy void localization.py

(Section 4);

entropy make figures.py

and

entropy complex completion.py

(Section 4 ﬁgures

and geometric completion);

entropy action backreaction v1.py

(Section 5);

entropy defect strain.py

(Sections 6.1 and 6.6);

entropy defect deficit.py

(Section 6.2); and

entropy defect intrinsic.py

(Section 6.3, Tables 3 and 4, Fig. 2; variant 1 is the SSM form, variant 4 the Dirac–K¨ahler source

form). The exact-log computation of Section 6.5 is

entropy defect exactlog.py

, and the re-

matched source is

rematch.py

. The intrinsic script runs in minutes at

= 4 and

= 6; the

exact-log solve takes about ten minutes at

= 4; the position relaxation runs in minutes at

= 8

and L = 10. The only random numbers are the ﬁxed-seed gradient-check vectors of Section 5.

Declarations

Conﬂict of interest: The author declares no conﬂict of interest.

References

[1]

G. Bianconi, Gravity from entropy, Phys. Rev. D 111, 066001 (2025).

doi:10.1103/PhysRevD.111.066001; arXiv:2408.14391

[2]

G. Bianconi, Quantum entropy couples matter with geometry, J. Phys. A: Math. Theor. 57,

365002 (2024). doi:10.1088/1751-8121/ad6f7e

[3]

R. Kulkarni, A 67%-rate CSS code on the FCC lattice: [[192

130

3]] from weight-12 stabilizers,

arXiv:2603.20294 (2026).

[4]

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

[5]

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

[6]

R. Kulkarni, Black holes in the FCC selection–stitch model: Bekenstein–Hawking entropy,

geometric evaporation, and primordial-black-hole signatures, Eur. Phys. J. Plus 141, 916 (2026).

doi:10.1140/epjp/s13360-026-08148-9

[7]

M. Desbrun, A. N. Hirani, M. Leok, and J. E. Marsden, Discrete exterior calculus,

arXiv:math/0508341 (2005); A. N. Hirani, Discrete Exterior Calculus, Ph.D. thesis, Caltech

(2003), doi:10.7907/ZHY8-V329

[8]

J. D. Eshelby, The determination of the elastic ﬁeld of an ellipsoidal inclusion, and related

problems, Proc. R. Soc. Lond. A 241, 376 (1957). doi:10.1098/rspa.1957.0133

[9]

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

doi:10.1007/BF02733251

[10]

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

doi:10.1016/0370-2693(81)90848-0

[11]

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

[12]

T. C. Hales, A proof of the Kepler conjecture, Ann. Math. 162, 1065 (2005).

doi:10.4007/annals.2005.162.1065

[13]

A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A

54, 1098 (1996). doi:10.1103/PhysRevA.54.1098

[14]

E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math.

Phys. 43, 4452 (2002). doi:10.1063/1.1499754

[15]

F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, Holographic quantum error-correcting

codes, JHEP 2015, 149 (2015). doi:10.1007/JHEP06(2015)149

[16]

P. Becher and H. Joos, The Dirac–K¨ahler equation and fermions on the lattice, Z. Phys. C 15,

343 (1982). doi:10.1007/BF01614426