Three Routes to Regge Gravity on the FCC Vacuum Lattice

Three Routes to Regge Gravity on the FCC Vacuum

Lattice: Relative Entropy, Vacuum Fluctuations, and

Entanglement Equilibrium

Raghu Kulkarni

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

raghu@idrive.com

Abstract

The Selection–Stitch Model (SSM) is a discrete face-centered-cubic vacuum whose matter is

a node trapped in a lattice void. We derive its gravity by three routes: Bianconi’s Gravity

from Entropy (A), Sakharov’s gravity induced by vacuum ﬂuctuations (B), and Jacobson’s

entanglement equilibrium with the SSM’s bond area law (C). All three give the Regge constraint

with a rest-mass source on the defect’s four edges. On a time-symmetric slice, each spoke carries

the cosmic-string deﬁcit exactly, and the far ﬁeld matches Schwarzschild initial data to 0

14%.

As a consistency check, the solved slice satisﬁes the entanglement-equilibrium condition ball by

ball to within 4%. Gravitating ﬂuctuations must be regulated covariantly: lattice-regulated loops

destabilize the spin-2 mode, a result reproduced independently, while a randomly sprinkled ocean

restores covariance. Computed species by species on that ocean, route B’s Newton constant

turns on how fermions couple to curvature at the cutoﬀ. Route C ﬁxes Newton’s constant by the

area law, needs no sum over species, and keeps the vacuum energy out of the ﬁeld equations; we

adopt it as the preferred phenomenological framework. Its microscopic law, a bond entanglement

that follows area, speciﬁes the coupling between code and geometry that the SSM must supply.

Keywords: gravity from entropy; induced gravity; 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. It grew out of a

discrete precursor on higher-order networks [

]. The ﬁrst metric,

˜g

, is the usual dynamical metric.

The second,

˜g

M − β

, is induced by matter (

) and curvature (

). The action is the

geometric quantum relative entropy (GQRE) between the two metrics,

Tr ˜g ln

−1

. Here the

metrics act as matrices on 0-, 1-, and 2-forms. An auxiliary ﬁeld, the

-ﬁeld

= (

G˜g

−1

)

−1

, turns

the action into an Einstein–Hilbert action with an extra, emergent cosmological constant Λ

≥

When the couplings

and

are small (weak coupling), the theory is Einstein gravity plus a matter

ﬁeld of Dirac–K¨ahler type. GfE has since been applied to the Schwarzschild black hole, where its

relative entropy obeys an area law at large radius [

]. It has also been given a thermodynamic

reading, in which its emergent cosmological term is an energy density that obeys a ﬁrst law [3].

A second, older idea also reaches Einstein gravity without postulating it. In Sakharov’s induced

gravity [

], the vacuum ﬂuctuations of quantum ﬁelds, integrated out on a curved geometry,

produce the Einstein–Hilbert action, with Newton’s constant set by the ﬁeld content and the cutoﬀ.

Gravity is then a property of the vacuum.

The Selection–Stitch Model (SSM) [

] is a model of the vacuum as a discrete network.

Its bonds are entangled pairs of unit length. In space they assemble into the face-centered-cubic

(FCC) lattice, the densest packing of equal spheres [

]. In spacetime they form the

lattice.

The lattice edges carry a quantum error-correcting code, [[192

130

3]] on the smallest box [

Gravity on the lattice is Regge gravity [

], with an action the lattice selects [

]. Matter is a defect:

an extra node trapped in a tetrahedral void, one of the small empty cells of the lattice [

]. Section 3

describes the model, with its parent papers.

This paper keeps one gravitational sector ﬁxed, the Regge equations on the SSM lattice, reaches

them by three routes, and then adopts one of them. It builds the following chain. Step 1 collects

known results; the new content is steps 2 to 6. Each statement is either computed or marked as an

input (Table 1).

Three routes to Einstein gravity. In route A, Bianconi’s action at weak coupling becomes

GfE

→

βR − αL

matter

. In route B, the one-loop eﬀective action of covariantly regulated

vacuum ﬂuctuations contains

πG

ind

, with 1

ind

. In route C there is

no action: the entanglement across every surface is its area over 4

, and the vacuum’s

entanglement is in equilibrium with the matter it contains, which gives Einstein’s equation as

an equation of state (Section 4).

Einstein–Hilbert

→

Regge on the SSM. On the lattice the curvature term becomes the Regge

action,

over the hinges

of the four-dimensional lattice, with

the hinge area and

its deﬁcit angle. On a three-dimensional slice the hinges are edges and

= 2

ℓ

The variables are the edge lengths

ℓ

, which we call labels. The Regge form is exact for the

lattice geometry. The routes agree at ﬁrst order in the curvature; routes A and B diﬀer beyond

it, and route C, which has no action, is silent there (Section 6.9).

Matter

→

a rest-mass source. The defect adds four edges, the spokes. In route A the matter

is a Dirac–K¨ahler ﬁeld on the spokes. Held at ﬁxed physical amplitude, with ﬁxed dual areas,

its mass term is a ﬁxed mass per unit length,

ℓ

: the Regge coupling of a point particle.

This reduced one-form source is a stated prescription (Section 6.3). In routes B and C the

defect is a rest mass from the start. In every route the source is the defect’s rest mass, spread

over its spokes.

The slice obeys the Hamiltonian constraint. Three-dimensional Regge gravity on a slice has

no solution with a source on the spokes, because in three dimensions a line of curvature

cannot end (Section 6.2). The equation a spatial slice of four-dimensional gravity obeys is

the Hamiltonian constraint. We derive its Regge form, one equation per node, by varying the

four-dimensional Regge action with respect to the lapse, and solve it on a time-symmetric,

conformally ﬂat slice. At Einstein order the Regge action is the only gravitational term. The

source prescription and the choice of slice are stated inputs.

The defect is a cosmic-string-like source whose far ﬁeld agrees with Schwarzschild initial data.

Each spoke carries the deﬁcit 8

πGµ

exactly, as a cosmic string does. The far ﬁeld agrees with

Schwarzschild initial data in isotropic coordinates, with the source’s mass, and the lapse gives

the Newtonian potential with the correct normalization. The source is prescribed, not solved

for (it is oﬀ shell). For a proton the defect is ﬂat to a few parts in 10

A choice between the routes. The defect does not distinguish the routes, so we test what each

needs. Route B needs ﬂuctuations that induce a positive Einstein term. They cannot be the

lattice’s own modes: their loops would respond to a shear of the lattice, and for the SSM’s

content the gauge-constrained spin-2 spectrum is unstable at every lattice momentum. With

a covariant regulator in the proper-time scheme the induced term matches the area law within

a factor of 2, but an explicit covariant ocean, a random sprinkling with couplings in proper

distance, gives a negative term for the SSM’s content, with a Wilson fermion and with the

squared Dirac operator alike. Route C needs only the area law and equilibrium, and, as a

consistency check, the solved slice satisﬁes its equilibrium condition ball by ball. We adopt

route C as the preferred phenomenological framework. Its microscopic law, that a bond’s

entanglement follows its area, is a postulate, and a two-qubit bond whose coupling grows with

its dual area supplies it (Sections 6.5, 7, 8 and 9).

In short, the SSM lattice carries Einstein gravity as Regge initial data, and the trapped node

behaves as Einstein gravity predicts, whichever route leads to the Regge equations, at least at

physical masses. The defect computation therefore does not distinguish the routes. In route C,

Newton’s constant comes from the SSM’s area law, with no sum over species; a uniform vacuum

energy changes no entanglement and so does not gravitate; and the Hamiltonian constraint that

the defect solves is the condition of entanglement equilibrium, which the solved slice satisﬁes to

within 4% for balls whose radius is at least about twice the lattice spacing, and 1% for the largest.

If the vacuum’s bond density also maximizes its entanglement, 1

is the largest entanglement per

unit area the bonds can carry, and the bonds fall below their area at second order, so strong ﬁelds

require the lattice to restitch. Route C has no action, so it says nothing beyond Einstein order,

where routes A and B diﬀer (Section 6.9).

What kind of correspondence this is. We keep four levels apart. (i) Deﬁnitions. The dictionary

of Section 5.1 assigns lattice quantities to GfE’s objects. It is a choice, not a result. (ii) Internal

consistency. The lattice equations follow from the common Einstein limit of the routes, and the slice

choice and the source prescription are stated as such (Section 6.3). (iii) Numerical solution. The

defect solutions solve a ﬁnite set of equations, and we quote the residuals. (iv) Continuum limit.

We do not claim, and do not test, that the lattice solutions approach solutions of the continuum

Einstein equations as the lattice is reﬁned. In the SSM the lattice spacing is ﬁxed at

= 1

843

ℓ

with

ℓ

the Planck length, for a vacuum entanglement of

2 per bond (Section 9 gives the scaling),

so reﬁnement is not part of the model. Two mathematical results set the standard for level (iv).

Deﬁcit angles deﬁne the scalar curvature of a piecewise-ﬂat space in a precise sense [

]. This

supports reading

ℓ

. But a discrete Einstein tensor can be consistent without its

solutions converging to continuum solutions; convergence is a separate and stronger statement [

This paper works at levels (i) to (iii).

For route A, the chain gives GfE’s abstract objects a meaning on the lattice (Section 5.1).

The diﬀerence between the induced metric and the metric is the edge strain. (In discrete exterior

calculus [

], the metric on edge ﬁelds is a diagonal matrix built from edge lengths.) The

-ﬁeld is

the inverse strain. The emergent constant Λ

is a quadratic functional of the strain of the imperfect,

many-grained vacuum crystal. The dictionary does not supply the metric

˜g

. In GfE its eigenvalues

equal one for any metric, by the way the eigenvalue problem is deﬁned [

]. Nor does the dictionary

choose GfE’s matter content, which is an input there, as in any ﬁeld theory. Section 2 deﬁnes the

lattice and geometric objects. Section 3 describes the SSM, and Section 4 states the three routes

and what they share. Section 5 gives route A’s lattice form, with the exact structure of the code’s

zero modes and a closed form of GfE’s relative entropy. Section 6 is the defect computation, which

all routes share. Section 7 tests route B’s ﬂuctuations, Section 8 treats

and the cosmological term

in each route and chooses between them, and Section 9 asks what supplies route C’s microscopic

law. Section 10 states the limits of the work and relates it to the parent papers.

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

system, with residuals given. (D) Derived: follows from stated deﬁnitions, the dictionary, or the

shared Einstein limit. (S) SSM input: a modeling choice or postulate taken from the SSM. (C)

Consistency: an order-of-magnitude estimate. (P) Premise: taken from the parent papers. (L)

Literature: a standard result from the cited literature.

Claim Status

Postulates and inputs

The FCC complex and the code’s checks, which ﬁx

∂

and

∂

; the vacuum is the code and matter a

trapped node

Parent-paper results of Section 3, reported and not tested here P

Route A dictionary:

G˜g

−1

− I

= edge strain,

= inverse strain, Λ

= quadratic strain functional,

matter on the defect’s edges

Route A source: ﬁxed physical amplitude, a ﬁxed mass per unit length on the spokes; routes B and C:

rest mass

Route B: an ocean whose modes are counted per proper volume, with its cutoﬀ at the lattice spacing S

Route C: entanglement across a surface =

, carried by the bonds; the vacuum is in entanglement

equilibrium

Route C: the vacuum’s bond density maximizes its entanglement, Eq. (33) S

Time-symmetric, conformally ﬂat slice; constraint imposed inside a wall, ψ = 1 outside S

Exact results

Relation-preserving integer lifts form a 12-dimensional aﬃne family; lifts outside it lose a mode T

Harmonic counts 1

1 for all even

; torsion-free at

= 4

6; six localized zero modes per void

plus six global modes span them

Route A: relative entropy in closed form; its variation gives the Dirac–K¨ahler equation; zero modes are

its massless solutions

All defect labels at L

: +148.4

◦

× 4, −70.5

◦

× 6, a Planck-scale mass T

3D Regge gravity with the spoke source: no solution (residual 17–18% of the source at L = 4, 6) T

Vertex constraint reproduced by the lapse variation of an explicit 4D complex, to 5 × 10

−8

Spoke deﬁcit = 8πGµ exactly; source bounded by q < π T

Far ﬁeld ψ = 1 + GM/2r with M the source mass to 0.14%; lapse N = 1 − GM/r to 0.6%; γ = 0.998 T

Binding lowers the far-ﬁeld mass below the source mass; no horizon outside the void T

Defect adds three harmonic modes (b

→ 2L

+ 5) and binds no state through L = 10 T

Lattice modes as ﬂuctuations: non-covariant shear response; spin-2 unstable at every lattice momentum

(one loop; reproduced independently)

Sprinkled ocean passes the shear test (+0

005

004, against

−

059 on

and +0

24 on a ﬁxed

Delaunay complex)

Sprinkled ocean: C < 0 for the SSM’s content, with Wilson and with D

†

D fermions T

Solved slice satisﬁes equilibrium ball by ball, ratio 0.96–0.99 for R

≥ 2.5 (a consistency check) T

Bond model: slope one at the maximum of

s/a

;

h/J

= 0

691,

= 0

2 for

= 1; no maximum for

p = 1/2

Scalar ﬁeld across a plane: slope 0.75–1.05 with points carried, one when reﬁlled at ﬁxed density T

Derived

Route A at weak coupling: L

GfE

→ 3βR − αL

matter

Route B: 1/G

ind

= CΛ

/12π; the leading cone term is the Regge action L

Route B, SSM content (proper time): C = 4, n

− n

= −68 D

Route B: displacement of the ocean only rescales the local cutoﬀ, Eq. (14) D

Route C: equilibrium of every round ball = Hamiltonian constraint;

; Λ a constant of

integration

Vertex constraint

e∋v

ℓ

= 16

πGm

from the lapse variation on a 3+1 prism complex;

ℓδ

= 8

πGM D

Calibration: L

, L

∝ s

1/2

at ﬁxed G; results in lattice units unchanged (Section 9) D

Linearized D

Regge equations: Poisson’s equation with Newton’s normalization and γ = 1 D

Lattice corrections at order (ka)

; β

= γ = 1.000 ± 0.002 P

Estimates

Route B, proper time: G

ind

within a factor 2 of the area law at s

= ln 2; Λ

= 25.5Λ

Route A: β = ℓ

/48π if it gave all of G; correction ≲ 0.04 for q ≤ 2 C

Route A: Λ

from grain-boundary strain: right sign, 10

121

too large C

Near q = π: GM ≈ 0.85 L

, at Einstein order only C

Proton: q ≈ 4 × 10

−19

; defect ﬂat to that order C

Route C, strong ﬁelds: bonds fall below their area at second order; restitching above a model-dependent

stretch

Table 1: Claim ledger.

2 Background and deﬁnitions

The FCC lattice and its complex. We use half-cell units. The nodes are the integer points

x ∈ Z

with

even. Each node has twelve nearest neighbors, at the vectors

±e

± e

(

i < j

), all of length

√

. These bonds are the edges. The edges bound two kinds of empty cells.

A tetrahedral void has four nodes and six edges; its center is a point like (

). An octahedral

void has six nodes and twelve edges; its center is an odd-parity point like (1

0). Space is ﬁlled by

these cells, two tetrahedra for each octahedron. On a periodic box of side

(even) there are

nodes, 3L

edges, L

/2 octahedra, and L

tetrahedra.

Chain complex and Hodge Laplacians. Give every edge and cell an orientation. The boundary

map

∂

sends an edge to (head)

−

(tail); it is the

nodes

× N

edges

signed incidence matrix. The

map

∂

sends a 2-cell to the signed sum of its edges. The rule

∂

= 0 says that the boundary

of a boundary is empty. Functions on nodes, edges, and 2-cells are 0-, 1-, and 2-cochains, the

lattice versions of scalar ﬁelds, one-forms, and two-forms. The exterior derivative is

∂

and

the codiﬀerential is

∂

. (The codiﬀerential

appears only inside

; a

with an edge

subscript, δ

, is always a deﬁcit angle.) The Hodge Laplacians are

= ∂

∂

, L

= ∂

∂

+ ∂

∂

, L

= ∂

∂

. (1)

A cochain in the kernel of a Laplacian is harmonic; we also call it a zero mode. It has zero divergence

and zero curl at once. The dimension

of the kernel of

is the

-th Betti number. On a torus

= 1 (constants) and, for an ordinary cell complex,

= 3 (the three winding directions). The

Dirac–K¨ahler operator

acts on the sum of all three cochain spaces, and

is the direct

sum of the three Laplacians.

CSS codes. A quantum CSS code [

] on

qubits is deﬁned by two binary check matrices,

and

, with

= 0 (

mod

2). It encodes

n − rank H

− rank H

logical qubits. Its

distance

is the weight of the smallest undetectable error. The code is written [[

n, k, d

]]. The

condition

= 0 is the condition

∂

= 0 over the two-element ﬁeld. So a CSS code is

a chain complex: put qubits on edges,

-checks on nodes (

∂

mod 2), and

-checks on

2-cells (

∂

mod 2). The logical qubits are then the ﬁrst homology, and

over GF(2).

Section 3.2 describes the SSM’s code, [[3L

, 2L

+ 2, 3]] on the FCC edges.

Compact localized states. A harmonic cochain supported on a ﬁnite set of edges is a compact

localized state (CLS). Flat bands in lattice models are often spanned by such states. Section 5.2

shows that the harmonic sector of the SSM complex is spanned by CLS anchored at the tetrahedral

voids, plus six global modes.

Regge calculus. Regge [

] describes a curved space by ﬂat simplices glued along shared faces.

The geometry is ﬁxed by the edge lengths

ℓ

alone. In three dimensions curvature lives on edges

(the hinges). Around edge

the cells that contain it each contribute an interior dihedral angle

e,σ

The deﬁcit angle is

= 2π −

σ∋e

e,σ

(ℓ). (2)

= 0 on every edge, the complex can be laid ﬂat in

near each edge. A positive deﬁcit is

a cone, like positive curvature; a negative deﬁcit is a saddle. The Regge action

ℓ

the lattice form of

√

g d

[

]. Its key property is the Schl¨aﬂi identity: on a closed

complex

ℓ

∂δ

/∂ℓ

= 0 for every f. So the variation of the Regge action is simple:

∂S

∂ℓ

= δ

. (3)

The vacuum equation of three-dimensional Regge gravity is therefore

= 0 on every edge. Now

add a source term

−µℓ

on one edge

. It describes a ﬁxed mass per unit length on that edge.

The equation on that edge becomes

∝ µ

. So the edge that carries the source carries a deﬁcit

proportional to the source. This is how three-dimensional Regge calculus describes a localized mass.

Section 6.2 shows why this three-dimensional equation is not the right one for a spatial slice of

four-dimensional gravity.

Discrete exterior calculus. A metric enters through the Hodge star. It maps each edge to its

dual face, the small face that the edge pierces in the dual lattice. In discrete exterior calculus [

]

the Hodge star is diagonal. The inner product of two one-cochains is

⟨ω, η⟩

|⋆e|/ℓ

, with

|⋆e|

the area of the face dual to

. A change of edge lengths changes this inner product. That is

how the metric acts on one-forms on the lattice.

The Dirac–K¨ahler ﬁeld. A Dirac–K¨ahler ﬁeld [

] is a sum of forms of all degrees, here

⟩

ϕ ⊕ ω ⊕ ζ

with

a 0-cochain,

a 1-cochain, and

a 2-cochain. Its kinetic operator is

. It is the natural matter ﬁeld on a lattice, because the operators it needs are the lattice’s

own boundary maps.

Terms used in the defect computation. A stationary point of an action is a set of edge lengths

at which its gradient vanishes. A source is oﬀ shell when it is prescribed by hand and does not solve

its own ﬁeld equation. Continuation means solving at one value of a parameter, then using that

solution as the starting guess at a nearby value. A spatial slice is at a moment of time symmetry

when its extrinsic curvature vanishes, as for a body momentarily at rest. It is conformally ﬂat when

its metric is a position-dependent factor times the ﬂat metric. The lapse

sets the rate of proper

time; in a weak static ﬁeld N ≈ 1 + Φ, with Φ the Newtonian potential.

3 The Selection–Stitch Model

The Selection–Stitch Model (SSM) is a program, developed in four papers, in which the vacuum is a

discrete network and matter is a defect in it. This section summarizes each paper as far as this work

needs it. The claims here are those of the parent papers. The exception is the last four paragraphs

of Section 3.5, which give this paper’s own convention and two facts about the static ﬁeld. The

ﬁrst of these facts is a parent result; the second follows from it. The parent results are reported,

not tested here, and carry status (P) in Table 1 unless a later section computes them. Section 3.5

lists exactly which of them this paper uses. We use units with

ℏ

= 1, so that

ℓ

with

ℓ

the Planck length. Lengths are given in terms of the bond length

, which is

√

in the half-cell

coordinates of Section 2. (The parent papers write the bond length as

; here

is the side of a

periodic box.)

3.1 Assembly of the vacuum

The primitive object of the SSM is a bond: a Bell pair that joins two nodes at unit distance

[

The network grows by two operators, and only two (Fig. 1a). A stitch adds a node at the apex of

stitch: new node

2 bonds, in plane

lift: new node, 3 bonds

out of plane,

√

2/3

(a) the two assembly operators

(b)

= 12 = 6 + 3 + 3

(FCC, ABC stacking)

6 in plane (stitch)

3 above (lift)

3 below (lift)

Bell pair

= 1

stitched sheet

= 6

tetrahedral foam

= 4

, frustrated

FCC crystal

= 12

, Kepler bound

stitch

lift,

lift

−

deficit

= 0.128

rad relieved

a node left in a tetrahedral void = matter

Figure 1: Assembly in the SSM [

]. (a) The two operators: a stitch adds a node in the plane

with two bonds; a lift adds a node above a triangle with three bonds, at height

2/3 L

. (b) The

resulting

= 12 coordination: six neighbors in the node’s own sheet and three in each adjacent

sheet. (c) The cascade from a single bond to the FCC crystal. The tetrahedral foam is frustrated

by the deﬁcit 0

128 rad; a node left behind in a tetrahedral void when the foam crystallizes is the

SSM’s matter (Section 3.3).

an equilateral triangle on an existing bond and bonds it to both ends. It grows a ﬂat triangulated

sheet in which every interior node has six neighbors (

= 6). A lift adds a node above the center of

an existing triangle, at height

2/3 L

, and bonds it to all three corners. It is the only way

out of the plane. No third operator exists at ﬁxed bond length, because four unit spheres in three

dimensions generically have no common point. A stitch is ﬁxed by two distance conditions and has

a one-parameter family of positions; a lift is ﬁxed by three and has none. Ref. [

] takes the lift to

be suppressed by the extra condition, with P

lift

= e

−3

≈ 5% relative to a stitch.

The two operators drive a cascade (Fig. 1c). A bond (

= 1) is stitched into sheets (

= 6).

Rare lifts build a foam of regular tetrahedra (

= 4). That foam cannot ﬁll space: ﬁve regular

tetrahedra around a shared edge leave a gap, the Regge deﬁcit

= 2

π −

arccos

= 0

128 rad

= 7

◦

. To relieve this strain, the foam turns into stacked sheets in the ABC order. That is the

face-centered-cubic (FCC) lattice. Each node has

= 12 neighbors: six in its own sheet and three

in each sheet above and below (Fig. 1b). Twelve is the kissing number in three dimensions, so the

cascade ends there; FCC is also the densest packing of equal spheres [

]. A random simulation

of the two operators in Ref. [

] supports this picture. The most common number of neighbors for

an interior node is 12, at every size from 250 to 1000 nodes. The fraction of nodes with all twelve

neighbors approaches one as 1

− a/N

1/3

, as expected when only the surface is incomplete. The

spacing between layers matches the FCC value 0

8165

to 0

01%. The result holds for exclusion

radii from 0

to 0

. It changes sharply at

√

, the radius of the circle through the

corners of a lattice triangle. The SSM calls this length the “metric wall.” Interior nodes with fewer

than 12 neighbors have 10 or 11; at the sizes simulated they are about 30% of the interior. They

mark grain boundaries, where crystals of diﬀerent orientation meet. The assembled vacuum is

therefore polycrystalline FCC with a bond-length tolerance of about

5%, set by the deﬁcit of the

foam. Section 8.1 uses this grain-boundary strain.

(a)

-check on a node (b)

-check on an octahedral void

Figure 2: Geometry of the SSM code and of its mass count. (a) A

-check: the twelve bonds of one

node. (b) An

-check: the twelve edges of one octahedral void [

]. (c) The 13-node coordination

cluster (a node, its twelve bonds, and the 24 edges among its neighbors), whose face counts set the

veriﬁcation costs of Ref. [7].

3.2 The vacuum as a quantum code

Ref. [

] puts one qubit on each edge of the FCC lattice and deﬁnes two kinds of check (Fig. 2a,b). A

-check acts on the twelve bonds of a node. An

-check acts on the twelve edges of an octahedral

void. Every check has weight twelve, and the two kinds commute,

= 0 over GF(2). A

periodic box of side

(half-cell units) has 3

edges and

2 checks of each kind. Each kind

obeys one global relation, so the number of independent checks is

−

2 and the code encodes

= 3

−

(

−

2) = 2

+ 2 logical qubits. That is [[192

130

3]] at

= 4 and [[648

434

3]] at

= 6; the rate tends to 2

3. The distance

= 3 is proved by exhaustive search over all errors

of weight one and two. For comparison, Ref. [

] quotes the cubic three-dimensional toric code at

distance four, [[108

4]], with a rate of 2

8%. The high rate has a simple cause. The FCC lattice

has only two kinds of void, so it has far fewer checks per edge than a cubic lattice. A matching

decoder gives a tenfold reduction of the block error rate at a physical error rate of 10

−3

. Because

the distance does not grow with size, the code does not give scalable error suppression; the SSM

uses it as the structure of the vacuum, not as a memory.

Ref. [

] conjectures that the logical qubits live at the tetrahedral voids, two per void plus two

global ones, which gives 2

+ 2. Section 5.2 of this paper veriﬁes a real-valued version of that

statement at

= 4 and 6. Every tetrahedral void carries six harmonic states of ﬁnite extent.

Together they span 2

voids

−

4 dimensions, and six global modes complete the set. The code is

what ﬁxes the boundary maps

∂

and

∂

of the complex used throughout (Section 2). Its

-checks

are the rows of ∂

and its X-checks the columns of ∂

, mod 2.

3.3 Matter as a trapped node

If crystallization is incomplete, a node of the

= 4 foam can be left at the center of a tetrahedral void

of the ﬁnished crystal [

]. It is bonded to the four corners of the void at distance

3/8 L

= 0

612

shorter than a lattice bond, so in the SSM’s reading the defect is under local strain. Together with

the six edges of the void it forms a complete graph

embedded in the

= 12 bulk (Fig. 4a).

This is the SSM baryon. It cannot leave the void. To pass through a face of the void, it would have

to come within

√

of all three corners of that face. That is the metric wall, below which the

lattice cannot hold a separate node. The surrounding bonds pull it back with a force that grows

with the displacement. Ref. [6] reads this as linear conﬁnement, V = σr.

Ref. [

] also derives charges and colors from the defect’s geometry, and Ref. [

] proposes that

a defect’s mass is the cost of verifying it, through Landauer’s principle, with mass ratios ﬁxed by

counts on the coordination cluster of Fig. 2c. The present paper uses the geometry of the defect, not

these identiﬁcations. Its mass enters only as the strength

of its gravitational source (Section 6.3).

3.4 Gravity and black holes

Ref. [

] supplies the SSM’s gravitational sector. Spacetime is the

lattice: the integer points

whose coordinates have an even sum. Its 24 shortest vectors are the most that can touch

one sphere in four dimensions. Its slice at ﬁxed time is the FCC lattice. The fourth-order moment

of the 24 vectors is exactly the same in every direction; that of the FCC slice is not. The lattice

is treated as a Regge geometry deﬁned by its edge lengths alone, without placing it in a larger

space. A change of edge lengths splits into two parts, as in classical elasticity. The compatible part

comes from moving vertices and carries no curvature. The incompatible part carries curvature; the

graviton is this part. On a subdivision of

into simplices, the second variation of the Regge action

equals the linearized Einstein operator term by term. Its coeﬃcients are those of the ghost-free

Fierz–Pauli theory, and the equality is an exact symbolic identity. The Regge action is also selected

rather than chosen: among local actions supported on hinges and linear in the deﬁcit, it is the only

one for which ﬂat space is a solution. The code does not ﬁx the geometry; edge lengths are not

code degrees of freedom. In that paper a matter defect sits at a geometrically forced position and

is described as locally ﬂat. Its gravitational eﬀect comes from its stress-energy, not from a local

curvature. That stress-energy is the elastic energy stored in its compressed bonds, which is also its

inertial mass. The Newtonian 1

ﬁeld is sourced by this stress-energy. In that paper’s reading,

then, gravity is the Regge (curvature) sector, and the bonds’ elastic energy is a source rather than

part of the gravitational action. The present paper assigns no energy to bonds (Section 10).

The same paper ﬁxes the lattice scale and treats black holes. Integrating out lattice modes gives

an induced Newton constant proportional to the squared spacing, which places the spacing at the

Planck scale (Section 7 shows that loops of lattice modes make the spin-2 mode unstable; the scale

used here rests on the calibration that follows). Requiring each bond, a Bell pair with entropy

to obey the Ryu–Takayanagi relation across the elementary (111) bilayer cell pins the coeﬃcient:

= 1

843

ℓ

. Applying the same relation at a horizon gives

ℓ

. The paper states that the

coeﬃcient comes from this postulate and a calibration. The lattice supplies a consistency check,

about 3

5 crossing bonds per cell. An SSM black hole is a region in which the lattice has crossed

its metric wall. Seen from outside it is a vacancy; seen from inside it is a saturated, inert part of

the code. Its boundary has a surface tension set by the bond energy

bond

ℏc/

. This gives

a non-thermal evaporation with lifetime

∝ M

. It also predicts a lower cutoﬀ in the masses of

surviving primordial black holes. One of these results enters below: L

3.5 What this paper takes from the SSM

This paper uses four things from the SSM. (i) The FCC complex and the code’s checks, which ﬁx

∂

and

∂

(Sections 2 and 5.2). (ii) The trapped-node defect: its position and its four spokes (Section 6).

(iii) The

setting, the selection of the Regge action, and the scale

= 1

843

ℓ

(Section 8.1). (iv)

For route B, the SSM’s ﬁeld content and its counting of the cutoﬀ (Section 4.2). These are inputs

of this paper, status S. It does not use the charge, color, or mass-ratio identiﬁcations of Refs. [

or the black-hole thermodynamics of Ref. [

] beyond the area relation that ﬁxes

. They show

what the SSM means by matter; nothing below depends on them. In particular, the paper does not

use the SSM’s assignment of a mass to the defect. The defect’s mass enters only as the strength

of its source.

Throughout, a bond is a code degree of freedom, not a rod. A defect changes the labels

ℓ

the edges of a ﬁxed complex. Regge dynamics reads changed labels as curvature. No node moves in

a surrounding space, and no bond has a rest length or a stiﬀness. The only variable is the number

each edge carries. The computations are on the spatial FCC slice, where curvature sits on edges.

The

spacetime lattice enters only through

and

(Section 8.1) and through the result

= 1

described next, which supports the choice of slice in Section 6.3.

Two facts about the static ﬁeld of a defect matter below. The ﬁrst is the Newtonian limit of

Regge gravity on

. Ref. [

] shows that the second variation of the Regge action on a subdivision

equals the linearized Einstein operator term by term. The linearized ﬁeld equation for a

metric change

µν

is therefore the continuum one, up to lattice corrections. For a static source with

energy density

, the only nonzero component of the stress tensor is

. In harmonic gauge the

trace-reversed ﬁeld

µν

−

µν

obeys

∇

µν

−

πG T

µν

, so only

is sourced. Undoing

the trace reversal gives

= −2U, h

= −2U δ

, ∇

U = 4πGρ, (4)

with

the Newtonian potential. The time–time part carries Newton’s law with Newton’s normaliza-

tion. The space–space part is the same potential times the ﬂat metric, which is the statement

= 1.

On the lattice the Laplacian is replaced by the

diﬀerence operator. Its symbol diﬀers from

at relative order (

)

, where

is the wavenumber and

the lattice spacing, and the diﬀerence

depends on direction. The SSM’s analysis of the linearized

equations with a static point source

gives the coeﬃcients of these corrections:

direction [100] [110] [111]

correction to U, in units of (ka)

0.028 0.070 0.084

correction to γ, in units of (ka)

0.028 0.007 < 10

−4

Beyond linear order, the same analysis ﬁnds that the exact Regge equations, projected onto smooth

metric variations and evaluated on the Schwarzschild exterior, give the post-Newtonian parameters

= 1

000

002 and

= 1

000

002. These results are for a smooth mass density. How a

lattice defect acts as a source is the question of Section 6.

The second fact follows from Eq.

(4)

. Since

−

U δ

, the spatial metric of a weak static

ﬁeld is a rescaling of the ﬂat metric by a position-dependent factor, with the same potential that

appears in

. A conformally ﬂat slice is therefore exact at linear order. Section 6.4 computes both

the spatial metric and h

(through the lapse) for the defect.

4 Three routes to the Regge equations

This paper keeps one gravitational sector ﬁxed: the Regge equations on the SSM lattice. It considers

three routes to them. Route A is Bianconi’s Gravity from Entropy. Route B is gravity induced

by the vacuum ﬂuctuations of quantum ﬁelds. Route C is Einstein’s equation as the condition of

entanglement equilibrium, with the SSM’s area law. Routes A and B give an action; route C gives

the ﬁeld equations without one. This section states the three and what they share.

4.1 Route A: Gravity from Entropy

Gravity from Entropy [

] is stated here in the form used below. The eigenvalues of a rank-two

tensor

are deﬁned by

µν

νV

(we write

for the eigenvalue). These are the eigenvalues of

the matrix

G˜g

−1

. So

˜g

has all eigenvalues equal to one and zero entropy. The GQRE between

˜g

and a positive-deﬁnite

reduces to

Tr ˜g ln

−1

−Tr ln

(

G˜g

−1

). Matter is a Dirac–K¨ahler

boson [19] |Φ⟩ = ϕ ⊕ ω ⊕ ζ on 0-, 1-, 2-forms, coupled through D = d + δ. The induced metric is

G = ˜g + α

M − β

M = D|Φ⟩⟨Φ|D + (m

+ ξR)|Φ⟩⟨Φ|,

R = R ⊕ R

µν

⊕ R

µνρσ

. (5)

To ﬁrst order in α, β,

L = 3βR − α⟨Φ|D˜g

−1

D|Φ⟩ − α(m

+ ξR)⟨Φ|Φ⟩. (6)

This is the Einstein–Hilbert action with zero cosmological constant, plus the kinetic and mass

terms of the Dirac–K¨ahler ﬁeld. In the full theory the

-ﬁeld

enters as a Lagrange multiplier

for the induced metric

(5)

. The ﬁeld equations are then a modiﬁed (“dressed”) Einstein equation,

−1

= I + α

M˜g

−1

− β

R˜g

−1

, and

D ˜g

−1

D|Φ⟩ + ˜g

−1

+ ξR)|Φ⟩ = 0, ˜g

−1

˜g. (7)

The emergent constant is Λ

2β

(

G − I − ln

)

≥

0. It is quadratic in

G − I

. The couplings

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

4.2 Route B: vacuum ﬂuctuations

The second route treats gravity as the response of the quantum vacuum, in Sakharov’s sense [

The quantum ﬁelds of the vacuum ﬂuctuate even in their ground state. In route B these vacuum

ﬂuctuations are integrated out on a given geometry, and the gravitational action is what remains.

The ocean. In the SSM the ﬂuctuating degrees of freedom are not the lattice’s own modes. Route

B postulates a background of quantum degrees of freedom present everywhere, which we call the

ocean. Its constituents are two-state systems with no geometry of their own. The lattice is the

arrangement into which they bind: each bond is a Bell pair drawn from the ocean, and the densest

packing is the ocean’s ground state. Route B takes the ocean’s ﬂuctuations as the ﬁelds that are

integrated out. These are ﬂuctuations of the ocean as a whole, not ﬂips of the code qubits on the

lattice’s bonds; this is an assumption of route B (Section 8, item (d)). Three properties of the

ocean are used below. Its modes are counted per unit of proper volume, not per lattice site, so its

regulator is covariant: the cutoﬀ is a proper length, and it does not deform when the lattice strains.

And its cutoﬀ sits at the lattice spacing, because the ocean built the lattice: the bond length is set

by the ocean’s own correlation length, with about one ocean mode per

site, so Λ

≈

4 (with

the area-law length deﬁned below). And matter displaces the ocean. A trapped node occupies

a region that the ocean would otherwise ﬁll, as a boat displaces water, so the ocean’s density is

lower around it:

(

) =

−

∆

[

(

)], with

the defect’s occupancy. All three are postulates of

the SSM’s route B, not results of this paper. Section 7 shows why route B needs the ﬁrst, and the

paragraph on displacement below what the third does and does not do.

The induced Einstein–Hilbert term. Take a real scalar ﬁeld with kinetic operator

−∇

ξR

on a Euclidean geometry with metric

. Integrating it out gives the one-loop eﬀective

action Γ =

ln det D. With a proper-time cutoﬀ at s = Λ

−2

Γ = −

∞

−2

Tr e

−sD

. (8)

At small s the trace has the heat-kernel expansion [26]

Tr e

−sD

(4πs)

√

g d

1 + s



− ξ



R + O(s

)

. (9)

The ﬁrst term gives a cosmological term of order Λ

. The second gives

Γ ⊃ −

32π



− ξ



√

g R d

x. (10)

This has the form of the Euclidean Einstein–Hilbert action,

−

16πG

√

g R

, with 1

πG

= (1

−

)Λ

192

, that is 1

= (1

−

)Λ

. Each species adds its own term, with the weight set

by its spin. Fermions enter with the opposite sign of the determinant, and gauge ﬁelds with their

ghosts. Summing over species,

ind

CΛ

12π

, C = N

(1 − 6ξ) + N

Weyl

+ 2N

Dirac

− 4N

, (11)

counting real scalars (with curvature coupling

), Weyl and Dirac fermions, and gauge ﬁelds. The

numerical coeﬃcient is for a proper-time cutoﬀ and depends on the regulator scheme. So does the

gauge-ﬁeld weight

−

4, which includes a contact term at the surface [

]; the count

, and every

ratio built from it, inherits that dependence. Minimally coupled scalars and fermions make 1

ind

positive; gauge ﬁelds reduce it; conformally coupled scalars (

= 1

6) contribute nothing. The

(

) terms give curvature-squared terms with a logarithm of Λ. No gravitational action has been

postulated; it is a property of the vacuum.

The induced action at a hinge. On a piecewise-ﬂat geometry the curvature sits on the hinges,

and the induced action can be computed hinge by hinge. Near a hinge of area

and deﬁcit

the space is a two-dimensional cone of angle 2

πϑ

, with

= 1

− δ

, times the hinge. The heat

kernel of a cone carries, besides the ﬂat-space term, a vertex term

(

−1

− ϑ

) [

]. Expanding in

the deﬁcit,



− ϑ



12π



1 +

4π

+ O(δ

)



. (12)

For the whole hinge this term multiplies

πs

. Inserting it into Γ and integrating over

gives,

for one minimally coupled scalar in Euclidean signature,

= −

96π



1 +

4π

+ O(δ

)



= −

8πG

ind



1 +

4π

+ O(δ

)



, (13)

using 1

ind

= Λ

for this ﬁeld. The leading term is exactly the Regge action, with the

same Newton constant as Eq.

(11)

. So on the lattice, route B gives the Regge action directly; the

correction at order

is not used here. Matter enters as a rest mass,

−M

dτ

along the defect’s

worldline.

Why the regulator must be covariant. Route B carries one requirement that route A does not:

the vacuum term of the ﬂuctuating ﬁelds must be covariant. Their vacuum energy is

vac

Under a uniform strain of the lattice, a ﬁeld whose modes are set by the lattice can change its

frequencies with the shape of the cells, not only with their volume. The part of

vac

that depends on

the shear of the cells is a term quadratic in the traceless metric change, with a coeﬃcient of order Λ

That is a mass term for the graviton, with a mass of order the cutoﬀ. A covariant vacuum energy

can depend on the volume only, as a cosmological term. So route B needs a covariant regulator. In

route B it is the ocean’s; Section 7 tests whether the lattice’s own modes could take its place. This

also distinguishes route B from the induced constant of Ref. [

], which integrates out lattice modes.

The value of

. The SSM’s area law gives each bond, with entropy

2, the area

= 4

G ln

2 and

√

4 ln 2 ℓ

= 1

665

ℓ

. For the (111) bilayer cell of Section 3.4 this is

= (2

1/4

, with

= 1

843

ℓ

. These values take

2; Section 9 gives them for the

bond model’s

. Route B ties this law to the induced action. The vacuum ﬂuctuations carry

entanglement across any surface, and for minimally coupled scalars and fermions, with a common

cutoﬀ, that entanglement entropy equals

ind

species by species: the two come from the same

ﬂuctuations [

]. Gauge ﬁelds diﬀer by a contact term at the surface [

]. We identify the

entanglement of the lattice’s bonds with the entanglement of the ocean’s ﬂuctuations across the

same surface. This identiﬁcation is a postulate of route B. With it, the area law ﬁxes

, and the

induced action supplies its form, the Regge action. The value of Eq.

(11)

then has to be consistent

with the area law, and this is a condition on the ﬁeld content and the scheme.

What displacement does. In route B, let the ocean’s free energy be

[

;

g, d

]: its density

has

a preferred value

and a restoring stiﬀness, it depends on the geometry

, and the defects displace

it through their occupancy

. The ocean relaxes to equilibrium,

∗

[

g, d

], and the eﬀective action

is Γ

eﬀ

[

g, d

] =

[

∗

;

g, d

]. Because

∗

minimizes

, the ﬁrst-order change of Γ

eﬀ

with the curvature

is the change of F at ﬁxed density:

δΓ

eﬀ

δg

∂F

∂g



n=n

∗

. (14)

The density’s re-adjustment to the curvature costs nothing at ﬁrst order. So the Einstein term of

an ocean in equilibrium is its curvature response at ﬁxed density, which is Eq.

(11)

evaluated at the

local density

∗

, equal to

away from defects. The explicit dependence on

∗

only rescales the

local cutoﬀ, Λ

∝ n

1/2

∗

, which changes the size of the induced term near a defect but not the sign of

. Everything else that displacement does enters at second order. It shifts each defect’s energy,

adds curvature-squared terms, and, through the gradient energy of the depleted region, gives two

defects an attraction that depends on the square of the displacement coupling, as two holes in a

medium attract. If that restoring stiﬀness vanishes, the attraction falls as 1

; it is then a scalar

force, and it can pass the light-bending bound only as the Newtonian part of the lattice’s tensor

ﬁeld, which needs a positive Einstein term. In route B, displacement would therefore explain how

defects pull together, but it does not set the sign of

. A density that followed the curvature, rather

than the defects, would add an Einstein term

∝ n

times its coupling to

, which is positive only if

curvature draws the ocean in.

Matching the area law. The two values of

agree when

= 48

πs

(Λ

)

. The cutoﬀ is ﬁxed

by matching the number of covariant ﬁeld modes per unit four-volume to one mode per

site,

whose four-volume is

2. The proper-time scheme’s own count is the heat-kernel trace at

= Λ

−2

modes per unit four-volume; a sharp cutoﬀ on

|k| <

Λ gives Λ

. The two counts

give Λ

= (32

)

1/4

= 4

216 and (64

)

1/4

= 5

013, that is Λ

= 3

81 and 4

53, so at

agreement requires

= 7

20 or 5

09 (Section 9 gives other values of

). The ﬁeld content gives

directly. Three generations of Standard-Model fermions, each in one 16-component multiplet of

Spin(10), are 48 Weyl fermions. The Standard-Model gauge group has 12 gauge ﬁelds. A minimal

Higgs doublet is 4 real scalars with

= 0. So

= 48

−

12 + 4 = 4. With a covariant regulator

and the proper-time weights, the induced and area-law values of

then agree, at

2, to

within a factor of 2:

ind

area

= 1

80 for the heat-kernel count, which belongs to the scheme, and

27 for the sharp one. Other ﬁeld contents fare worse. The Standard Model without right-handed

neutrinos has

= 1; a fourth generation gives

= 20; and grand-uniﬁed groups with many gauge

ﬁelds give a negative

. These statements hold in the proper-time scheme. On an explicit ocean

the weights change, and Section 7 computes them.

The vacuum energy. The ﬁrst heat-kernel term gives a cosmological term,

ρ = −

− n

)Λ

64π

, (15)

where

−n

+ 2

−

Weyl

−

Dirac

counts bosonic minus fermionic degrees of freedom.

For the ﬁeld content above it is 4 + 24

−

96 =

−

68, with no cancellation, so

= 17Λ

; its

sign depends on the regulator, and Section 8.1 gives its size. The next terms of the expansion are

curvature-squared terms with a logarithm of Λ, which we do not use.

4.3 Route C: entanglement equilibrium

Route C does not start from an action. It starts from the SSM’s area law and asks which geometries

the vacuum’s entanglement allows. The idea is Jacobson’s: Einstein’s equation is an equation of state

of the vacuum’s entropy [

], and in its later form the condition that the vacuum’s entanglement be

in equilibrium [

]. In holography the linearized Einstein equation follows in the same way from the

ﬁrst law of entanglement [

], and geometry and Einstein’s equation have been sought directly

from the entanglement between abstract quantum degrees of freedom [

]. Route C applies this

to the SSM. The degrees of freedom that carry the entanglement are the lattice’s bonds, and the

area law is the SSM’s own. Two postulates deﬁne it.

Entanglement follows area. The entanglement entropy across any surface Σ in a slice is

(Σ) =

(Σ)

, with

the area in the lattice’s geometry. In the ﬂat vacuum this is the area

law of Section 3.4: each bond that Σ cuts carries the entropy

and a share

of its area, so

. When the lattice strains, a bond’s share of area changes with its labels, and its

entanglement must change with it,

(0)

, with

the area of the bond’s dual face. A

Bell pair carries at most

2. So a bond can follow its area only up to a maximum stretch, and

the vacuum value

must lie below

2 by at least the largest stretch the lattice has to absorb.

On a conformally ﬂat slice a bond’s share of area grows by

. Far from a mass

that is about

1 + 2

GM/r

, a stretch below 10

−18

for a proton at the lattice scale. But the requirement is set by

the strongest ﬁelds, not by a proton: at the surface of a neutron star the stretch is about 50% (for

4 solar masses within 12 km), and at a black hole’s horizon on such a slice it is a factor of 16. So

either

lies below

2 by enough to absorb the strongest ﬁelds, or, above some stretch, the lattice

restitches, inserting bonds rather than stretching them. The bond model of Section 9 ﬁxes

and,

under stated assumptions, requires restitching.

The vacuum is in entanglement equilibrium. For every small ball at ﬁxed volume, the change

of the entanglement across its boundary from the geometry balances the change from the matter

inside:

δA|

+ 2π δ⟨K⟩ = 0, K =

− r

dV, (16)

with

the modular Hamiltonian of a ball of radius

in ﬂat space [

]. On a time-symmetric

slice, at linear order in the curvature, the area deﬁcit at ﬁxed volume is

δA|

−

(

−

)

R dV

. Equilibrium for every ball then requires

= 16

πGρ

at every point. This is

the Hamiltonian constraint. On the lattice

R dV

is 2

ℓ

on the edges, so equilibrium is the

vertex constraint

e∋v

ℓ

= 16

πGm

of Section 6.3. Two checks ﬁx the normalization. For

a point mass, Schwarzschild initial data give

δA|

−

πGMR

and 2

πδ⟨K⟩

πMR

. For a

straight cosmic string through the ball,

δA|

−

ℓδ

, which balances the string’s 2

πδ⟨K⟩

when

ℓδ = 8πGµℓ. Section 6.5 checks Eq. (16) on the solved defect slices, ball by ball.

With boosts, the same equilibrium for every small causal diamond gives the full Einstein

equation [

]. We use only the time-symmetric slice. In route C,

is ﬁxed by the area law,

with no sum over species. A uniform vacuum energy changes no entanglement, so it does not enter,

and the cosmological constant appears only as a constant of integration [

]. Route C has no action.

It therefore says nothing beyond Einstein order, where routes A and B diﬀer (Section 6.9).

4.4 What the routes share

All three routes give, at leading order in the curvature, the same thing on the lattice: the Regge

equations with a Newton constant

, plus a rest-mass source for the defect. The constraint solutions

of Section 6 use only this common content. They therefore hold for every route wherever the

Einstein order is accurate, which includes every physical mass. The subsections on the full matter

action and the dictionary’s objects at the defect are speciﬁc to route A. The routes diﬀer in four

places, which Table 2 collects: where

comes from, how matter is described, the terms beyond

Einstein order, and the cosmological term. Sections 7 and 8 test route B’s regulator and choose

between the routes.

Route A: Gravity from En-

tropy

Route B: vacuum ﬂuctuations

Route C: entanglement

equilibrium

Origin of the

action

relative entropy between

˜g

and

G, Eq. (6)

one-loop eﬀective action of co-

variantly regulated ﬁelds

no action; area law and

equilibrium, Eq. (16)

Newton con-

stant

πG

= 3

β/ℓ

, with

free coupling (Section 8.1)

ind

, set by

the ﬁeld content and the cut-

oﬀ

, from the

area law

Matter

Dirac–K¨ahler ﬁeld on the

spokes; rest-mass coupling at

ﬁxed amplitude (S)

rest mass −M

dτ rest mass, through T

Beyond Ein-

stein

(

R˜g

−1

)

, relative size

βR

one-loop curvature-squared

terms, times ln Λ

none given

Regulator none needed

the ocean: covariant, cutoﬀ

at the lattice spacing; the lat-

tice’s own modes fail (Sec-

tion 7)

none needed

Cosmological

term

2β

(

G −I −ln

)

≥

= 17Λ

(proper time),

Eq.

(15)

; needs a separate can-

cellation

a constant of integration;

uniform vacuum energy

does not enter

Table 2: Three routes to the Regge equations. Section 6 uses only what they share; Section 8 adopts

route C.

The dictionary of Section 5.1 and the closed-form relative entropy of Section 5.3 belong to route

A. The code’s zero modes (Section 5.2) belong to the lattice and to no route.

5 Route A on the lattice and the code’s zero modes

Route A has more lattice structure than the other routes: a dictionary for GfE’s objects and a

closed form of its relative entropy on the complex. Both rest on the code’s chain complex, whose

zero modes belong to the lattice and to no route.

5.1 The dictionary of route A

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 change of the DEC one-form metric D (exact in DEC)

G-ﬁeld

G diag



1/(1 + ϵ

)



, the inverse strain D

≈

4β

Tr ϵ

quadratic in the edge strain 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| (ﬁxed amplitude) S (Sec. 6.3)

|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 if route A gave all of G C

Table 3: 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 3 states the dictionary, and Table 4 collects the symbols used from here on. The rows for

˜g

and

use the same edge labels in two roles. As geometry, the labels have a curvature, the Regge

deﬁcit

; that is

˜g

and

. As strain, the labels have a deviation

ℓ

/ℓ

(0)2

−

1 from their ﬂat

values

ℓ

(0)

(

for a lattice bond); that is

G − ˜g

. In the defect computation of Section 6, the strain

is set by the Hamiltonian constraint through the conformal factor of the slice. The dictionary is a

set of deﬁnitions, tagged D. Section 6 uses only its route-A parts (the source in Section 6.3 and

Sections 6.6–6.7). 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 the lattice: on the edges a defect adds. In route A, Section 6 uses that proposal as the

source of the gravitational constraint.

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|

symbol meaning

nearest-neighbor bond length (

√

2 in half-cell units; 1.843 ℓ

for s

= ln 2)

ℓ

label of edge e (its length)

= ℓ

/ℓ

(0)2

− 1 edge strain relative to the ﬂat label ℓ

(0)

; the one-form block of

G˜g

−1

− I

Regge deﬁcit angle at edge e (hinges are edges in 3D)

|⋆e|, |⋆v| dual face area of an edge; dual volume of a node

= ℓ

edge curvature (V

: dual volume; mean Ricci scalar 2R

)

β = ℓ

/48π GfE curvature coupling, if route A gave all of G (Section 8.1)

µ source mass per unit spoke length (µ = αm

|⋆s| in route A)

G, q = 8πGµ Newton’s constant (1/16πG = 3β/ℓ

in A, G

ind

in B, L

/4s

in C); the source

C, Λ ﬁeld-content count and covariant cutoﬀ of route B, Eq. (11)

conformal factor of the slice at node v; ℓ

= ψ

ℓ

(0)

M mass of the defect: µ

ℓ

(source), or read from ψ = 1 + GM/2r (far ﬁeld)

, L

vacuum entanglement per bond; area-law length, L

= 4Gs

(route C)

dimension of the harmonic one-form sector (2L

+ 2 in vacuum)

Table 4: Notation.

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. So the ﬁeld does not solve its own equation; it

is oﬀ shell, as any localized massive ﬁeld held in place 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 has no matter term: the metric deviation there is set by 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

, which is linear in the labels when the dual areas

|⋆e|

are held ﬁxed. The mass term

αm

⟨

⟩

is then a ﬁxed mass per unit length on the ﬁeld’s edges, the Regge form of a point particle.

Holding the amplitude ﬁxed is a prescription, an SSM input. Section 6.3 states it, compares it with

the alternative, and uses it as the source of the Hamiltonian constraint. The harmonic sector of the

code is a diﬀerent object. It is the set of zero modes of

on the lattice edges, the lattice’s own

static massless sector. A defect couples to it but is not part of it.

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

µν

= 4

µν

[

Any set of bonds with cubic symmetry has this property; simple cubic gives 2

and body-centered

cubic 8

δ/

3. So the FCC value says only that the unstrained lattice is a ﬂat solution that looks the

same in every direction:

= 0,

˜g

, Λ

= 0. The rest of the paper perturbs about this vacuum.

This is the only place

µν

enters. For gravity the fourth-order moment matters, and there FCC

fails and D

succeeds (Ref. [8]).

Euclidean signature. The lattice is Euclidean. GfE is Lorentzian. No ﬁnite set of bonds is

unchanged by Lorentz boosts, because the Lorentz group is not compact. So there is no Lorentzian

lattice, and we do not seek one. The link to Lorentzian physics is through the continuum limit,

which is rotation invariant in four dimensions, continued to Minkowski signature as in Euclidean ﬁeld

theory. The fourth-order identity of Ref. [

] continues to the invariant (

− ω

)

. Its sixth-order

breaking continues to a Lorentz violation suppressed by (

E/M

)

, with

the Planck mass.

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 of a matter conﬁguration is never

positive. In the Lorentzian theory, time derivatives enter with the opposite sign, and that is not

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 3: 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).

available on a spatial slice. The sign remark in Section 5.3 reﬂects this restriction.

5.2 The code complex and its harmonic sector

This section records the exact zero modes of the code’s Laplacians. Sections 5.3 and 6.8 use them.

All results are exact counts or full matrix diagonalizations at

= 4

6, with residuals below 10

−14

The count of zero modes (the Betti numbers) is proved for all even L.

5.2.1 Integer lift and its selection (T)

Over GF(2), the ﬁeld with two elements, the code is a chain complex because

= 0. A

real-valued ﬁeld needs an integer version of the

-checks, which we call a lift. Each octahedral check

acts on twelve edges. Mod 2, it is the sum of the octahedron’s three equatorial squares. Giving

each square an orientation makes each column of

∂

a signed loop, and then

∂

= 0 holds over

the integers. This lift is not unique. Reversing a whole generator multiplies a column by

−

1 and

cannot change

rank ∂

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

a diﬀerent integer lift of the same binary stabilizer. The code has one global relation among its

-checks: their sum is zero mod 2. We require the same relation over the integers. This gives one

linear condition per edge. It is what makes

rank ∂

−

1, and hence

= 1 in Section 5.2.2.

= 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, 2

of 2

. The translation-covariant lift

is in it. Sixty random lifts outside it gave

rank ∂

−

1 zero times. A generic lift destroys a

mode. The crystal’s translation symmetry selects one that does not.

5.2.2 Harmonic count and localized basis (T)

With the covariant lift, L

= ∂

∂

, L

= ∂

∂

+ ∂

∂

, and L

= ∂

∂

have kernels of dimension

= 1, b

= 2L

+ 2, b

= 1. (17)

That is 130 and 434 one-form modes at

= 4

6, equal to the code dimension (Fig. 3). 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. The counts over GF(2)

and over the reals agree, which rules out torsion of order 2 by the universal coeﬃcient theorem. A

maximal minor of ∂

with determinant one, found at L = 4, 6, rules out all other torsion.

The zero modes have a local basis. No mode lives on the six edges of one void alone: an oriented

triangle has zero curl but not zero divergence. Each void has a neighborhood

of 42 edges: the

edges of the four octahedra that share a face with it (4

12 = 48 edges, less 6 because each void

edge lies in two of the octahedra). On this neighborhood the structure is exact. Every tetrahedral

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

voids

−

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

wind

around the torus. The other three do not wind; they exist because the code uses only octahedral

squares as 2-cells. If the triangular faces are added as 2-cells, these modes disappear and

drops

to 3 at both sizes, the value for an ordinary torus. So the large number of zero modes counts the

constraints the code does not impose. These modes sit at the tetrahedral voids, where the defect

sits.

5.3 The entropic action on the complex

This section evaluates GfE’s action for a Dirac–K¨ahler ﬁeld on the lattice, including the zero modes

of Section 5.2. That ﬁeld is the lattice’s static massless sector, not the defect of Section 6. The aim

is to ﬁnd the exact form of the action on the lattice. This form justiﬁes treating the defect’s ﬁeld at

ﬁrst order later. On the unstrained lattice

˜g

and

= 0. GfE’s action of a matter conﬁguration

is then

−Tr ln

(

), a function on a space of dimension 4

(nodes, edges, and 2-cells

together). Since

is a sum of two terms of rank one, it has rank at most two. If Φ has only one

form degree, the two directions in its range are orthogonal.

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

with eigenvalue

, homogeneous of

one degree. Then

(Φ) =

−ln

(1 +

αν

)

− ln

(1 +

αm

) exactly. Proof: the nonzero eigenvalues of

are ∥DΦ∥

= ν and m

, on orthogonal lines. □

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

linearized Lagrangian (6). For modes with nonzero ν (up to 16 at L = 4) the ﬁrst-order form fails

once

αν ∼

1. With

= 0

3, the relative error is 34% at

αν

= 0

8 and 238% at

αν

= 8. On the

harmonic sector the kinetic term vanishes identically, and the only linearization error is

(1 +

αm

)

against αm

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

= 1 +

α∥D

∥

= 1 +

αm

∥

, and

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

), and stationarity is

−

2αm

DΦ = 0. (18)

For ﬁelds of a single degree,

= 0. Then this is GfE’s matter equation

(7)

, with the factor

˜g

−1

given by the numbers A

−1

and B

−1

. A ﬁnite-diﬀerence gradient agrees to 2 × 10

−8

Proposition 3 (T). A zero mode on the edges solves Eq.

(18)

, with its eﬀect on the metric

included, if and only if

= 0. So the zero modes are exactly the massless solutions of the full

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 5.1.

These results are used in one way below, in route A. On the zero modes, the only error of the

ﬁrst-order form is in the mass term. Elsewhere the error is small when

is small. So the matter

side of the defect problem can be posed at ﬁrst order with a known error. The curvature side is

controlled by βR

, which Section 6.9 evaluates at the defect.

6 The trapped-node defect as Regge initial data

The SSM baryon is an extra node

at the center of a tetrahedral void with corners

A, B, C, D

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

cA, . . . , cD

. In route A the matter is a Dirac–K¨ahler one-form

on the spokes; in routes B and C

the defect is a rest mass. All three routes give Einstein gravity, in Regge form, plus a rest-mass

source on the defect. So the question is: what geometry does Einstein gravity give the lattice around

such a source? This section answers it on a spatial slice. Figure 4a shows the defect.

6.1 Two ﬁxed geometries (T)

Two geometries can be written down before any ﬁeld equation. If the spokes take the length the

void gives them,

3/8 L

= 0

612

, the defect sits in ﬂat space and every deﬁcit is zero. If instead

every edge, spokes included, keeps the bond length

, the ten edges of the resulting

cannot be

built in ﬂat space. The deﬁcit on each edge is then

= 2

π −

over the cells at that edge. The

dihedral angles of the regular cells are

tet

arccos

= 70

◦

and

oct

= 109

◦

. A bulk FCC

edge has two tetrahedra and two octahedra, 360

◦

, and is 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

. Below we ﬁnd that a slice’s Regge sum measures its mass,

ℓδ

= 8

πGM

(Eq.

(22)

). So holding the spoke labels at

would put a mass of order

, the

Planck mass, into every defect. That is 10

times the proton mass. The defect’s labels cannot be

ﬁxed by hand; they must be set by the ﬁeld equations.

6.2 Why the slice needs the constraint, not 3D gravity (T)

The direct approach is three-dimensional Regge gravity on the slice, with the rest-mass source on

the spokes. Its stationary points satisfy

= 0 on every edge except the spokes, and a ﬁxed deﬁcit

on each spoke. We solved these equations by least squares at

= 4 and 6. They have no solution

for any nonzero source. At

= 4, asking for spoke deﬁcits of 0

◦

, 5

◦

, and 20

◦

leaves residual

curvature of 0

087

◦

, 0

◦

, and 3

◦

on other edges. The residual is 17–18% of the target and does

not shrink at

= 6 (0

◦

at 5

◦

), so the obstruction is local. The reason is the Bianchi identity of

Regge calculus. In three dimensions curvature sits on lines, and such a line cannot end, just as a

cosmic string cannot end. The four spokes end at the corners of the void. In continuum terms, the

three-dimensional Einstein equation needs a conserved source, and a source on four line segments

is not conserved. An elastic energy for the bonds would hide the problem by letting lattice strain

carry the missing stress. No route has such a term, and we add none.

(a) trapped node: spokes (red), cage (blue)

0.6 1 2 3 4 6

distance of edge midpoint from void centre (half-cells)

−

largest

| /

spoke

−

(b) curvature off the spokes (

= 0.1

)

Figure 4: (a) The trapped-node defect: a node

at the center of a tetrahedral void, joined by four

spokes (red) to the corners

A, B, C, D

of the void (blue). The source lives on the spokes; in route A

it is a Dirac–K¨ahler ﬁeld. (b) Curvature oﬀ the spokes in the initial-data solution at

= 0

= 16:

the largest

|δ

among edges whose midpoints lie in each distance shell, in units of the spoke deﬁcit

q. The dotted line shows r

−3

for comparison. The ﬁxed wall at r = 7.6 steepens the last points.

Three-dimensional Einstein gravity is also not the gravity of a spatial slice. A slice of a four-

dimensional spacetime obeys the constraint equations of general relativity. At a moment of time

symmetry, when the slice’s extrinsic curvature vanishes, the only nontrivial one is the Hamiltonian

constraint,

R = 16πGρ, (19)

with

the energy density. It is a single scalar equation, so it carries no line-ending obstruction. We

use it from here on. In route A, 1

πG

corresponds to the coeﬃcient 3

of the curvature term in

Eq.

(6)

, with each lattice term carrying the volume

ℓ

: 3

β/ℓ

= 1

πG

(Section 8.1). In route B,

G = G

ind

of Eq. (11). In route C, G = L

/4s

from the area law.

6.3 The equations (D, S)

The constraint from the four-dimensional action. On a piecewise-ﬂat slice,

√

h d

x =

ℓ

[

]. This ﬁxes the integrated constraint, but not how it divides among nodes. We

obtain the division from the four-dimensional Regge action, by varying it with respect to the lapse,

as in the 3+1 form of Regge calculus [23].

Extend the slice to a static four-dimensional complex. Each node

gets a vertical edge of length

, the lapse times a time step. Each tetrahedron of the slice, times the interval, becomes a

prism, which we split into four 4-simplices. The hinges are now triangles. A spatial edge

and

its copy one step later bound a trapezoid of area

ℓ

(

)

dt/

2. At a moment of time symmetry,

and to leading order in

, the four-dimensional geometry is the slice times the interval. So the

deﬁcit around this trapezoid is the slice’s deﬁcit

, and all other hinges are ﬂat; when

varies

from node to node, the other hinges acquire deﬁcits that enter the variation only at higher order in

. The source is a rest mass, with action

−

M dτ

and

dτ

N dt

. A spoke

sweeps out a

strip of area

ℓ

(

)

dt/

2 and carries mass

per unit length. The action of one time step is

therefore

S =

8πG

ℓ

+ N

dt −

dt, m

s∋v

µℓ

. (20)

So each spoke gives half of its mass

µℓ

to each of its two end nodes. We write

with the Lorentzian

signs. The Euclidean action of the check complex below is

−S

, which has the same stationary

points. By the four-dimensional Schl¨aﬂi identity,

∂δ

/∂N

= 0, so varying

changes only

the hinge areas. Setting ∂S/∂N

= 0 gives one equation per node,

e∋v

ℓ

= 16πG m

. (21)

This is the vertex form of the Hamiltonian constraint. It is the discrete counterpart of the continuum

statement that varying the lapse gives the Hamiltonian constraint. Summing over all nodes counts

each edge twice, so

ℓ

= 8πGM, (22)

with M the total mass: the curvature of a time-symmetric slice measures its mass.

We checked this derivation on an explicit four-dimensional complex (T). We took a solved slice

= 8, three periodic time steps of

= 0

02, and the staircase split of each prism into four

4-simplices, with diagonal edges of length

ℓ

+ (dt (N

+ N

)/2)

. The complex has about 3

simplices. At

= 1 the four-dimensional Regge sum equals 3

ℓ

. The trapezoid hinges

carry exactly the slice’s deﬁcits, and every other hinge has a deﬁcit below 10

−11

degrees. A central

diﬀerence of the four-dimensional Regge sum with respect to

equals

e∋v

ℓ

at every node

tested, at

= 0

1 and at

= 1 (spoke deﬁcits of 57

◦

). The largest diﬀerence is 5

−8

, the level

of rounding in the ﬁnite diﬀerence, against values up to 0

13. With a lapse that varies by about

10% across the box, the absolute diﬀerence is larger but falls by a factor of 8 when

is halved, as

expected for a correction of relative order

. What the derivation assumes is the 3+1 complex

itself: a lapse on each node and a vertical edge joining each node to its own copy. The

lattice of

Ref. [

] joins slices diﬀerently, so this is a derivation from the standard 3+1 Regge complex built on

the slice, not from the D

complex. A derivation from the D

complex is left open.

The source (route A). The ﬁeld’s mass term is

αm

⟨ω|ω⟩

αm

|⋆s|/ℓ

, where

is the value of the ﬁeld on spoke

. On the lattice we must say what is held ﬁxed when a label

changes. This is a choice, and there are two natural ones. They match two diﬀerent variations in

the continuum.

Fixed covariant components. In the continuum, the mass term of a one-form is

µν

√

Varying the metric at ﬁxed

gives the stress tensor

(

−

µν

) of a massive vector ﬁeld.

On the lattice, ﬁxed

means a ﬁxed edge value

. Then the term falls as 1

/ℓ

. This is the direct

lattice version of GfE’s variation at ﬁxed ﬁeld (Ref. [1], Eq. 65).

Fixed physical amplitude. Instead, we can hold ﬁxed the components measured in a local

orthonormal frame. Then

µν

is ﬁxed and only the volume element varies. The mass term then

varies like the rest mass of a localized object. Its stress is

−

µν

, the same in all directions.

On the lattice this is

wℓ

with

ﬁxed. With the dual areas held ﬁxed, the mass term is

ℓ

with µ = αm

|⋆s|, a ﬁxed mass per unit length.

In route A we adopt the ﬁxed-amplitude choice. It is a stated prescription, an SSM input (S in

Table 1), not a result of GfE. In routes B and C no choice is needed: the defect is a rest mass from

the start, and the same line source is the natural way to spread it over the defect’s bonds. It is the

standard Regge coupling of a point particle: a ﬁxed mass per unit length on the edges it occupies [

How to place matter ﬁelds on a simplicial lattice is a known open part of simplicial gravity [

], and

the other choice deﬁnes a diﬀerent lattice theory. The defect’s mass is then

ℓ

, divided

among the nodes as in Eq. (20). We write the source strength as the dimensionless number

q = 8πGµ. (23)

We call this model the reduced one-form source. Section 6.6 tests it against the full matter action.

The full matter operator (route A). In route A the mass term is not the whole matter operator.

Two other parts need attention. First,

|ω⟩⟨ω|

has oﬀ-diagonal entries

′

that link diﬀerent

spokes. Second, the 0-form block

|δω⟩⟨δω|

is nonzero at the ﬁve defect nodes. No octahedral cell

contains a spoke, so

dω

= 0 and

Dω

δω

. Then

|δω⟩⟨δω|

|ω⟩⟨ω|

. This is a sum of two

rank-one matrices acting on diﬀerent form degrees, so their ranges are orthogonal. The determinant

plus a rank-one matrix is a single number (the matrix determinant lemma, as in Proposition 1).

−Tr ln



I + α



= −ln



1 + α∥δω∥



− ln



1 + αm

∥ω∥



(24)

exactly, to all orders in

. The whole matter action, including the oﬀ-diagonal entries and

the 0-form block, depends on the labels only through two numbers:

∥ω∥

|⋆e|/ℓ

and

∥δω∥

|⋆v|

(

δω

)

, with (

δω

)

|⋆v|

−1

e∋v

±|⋆e|ω

/ℓ

. The labels enter these through the

Hodge stars. The reduced source keeps the ﬁrst number with ﬁxed dual areas and drops the

second. One limitation is built into the lattice. On a lattice of labels the metric on edge ﬁelds is

a diagonal matrix, so GfE’s matrix ﬁeld equation is represented only through its eﬀect on label

changes. Oﬀ-diagonal parts of the metric have no lattice variable.

The slice. Eq.

(21)

gives one equation per node, but there are about six edges per node. The

extra freedom is the freedom to choose the slice, and it must be ﬁxed. We use a conformally ﬂat

slice. Each node v carries a factor ψ

, and each edge has length

ℓ

= ψ

ℓ

(0)

, (25)

with

ℓ

(0)

its ﬂat length. The system is then square: one unknown and one equation per node.

This is an ansatz for the initial data, not a gauge choice: it restricts the intrinsic geometries

the slice can have, and we state it as such. It is exact for Schwarzschild initial data in isotropic

coordinates,

= 1 +

GM/

, and it is exact at linear order in static general relativity whenever

the post-Newtonian parameter

= 1, which holds on the

lattice (Section 3.5). Nodes include

the octahedron centers and the trapped node. The complex is a periodic box, but the constraint

is imposed only at nodes with

r < R

from the void center; nodes with

r ≥ R

are held at

= 1

and act as a wall. Some wall is needed. With

ρ ≥

0 everywhere and positive at the defect, Eq.

(19)

asks for a metric whose scalar curvature is nonnegative everywhere and positive somewhere. A

closed three-torus has no such metric, by a theorem of Schoen–Yau and Gromov–Lawson [

so the constraint cannot be imposed everywhere on a torus. The edges near the wall carry the

compensating curvature, and Eq. (22) holds for the sum over the free nodes.

The spoke deﬁcit. At the trapped node only the four spokes meet, and the node holds half of

each spoke’s mass. Eq. (21) at c reads

ℓ

= q

ℓ

. By the symmetry of the void,

= q = 8πGµ (26)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

source

= 8

spoke length

(a) spoke length

0.0 0.5 1.0 1.5 2.0 2.5 3.0

source

= 8

0.0

0.2

0.4

0.6

0.8

1.0

far-field mass / source mass

(b) binding

0.0 0.5 1.0 1.5 2.0 2.5 3.0

source

= 8

0.0

0.2

0.4

0.6

0.8

far-field

Figure 5: The trapped node as Regge initial data against the source

= 8

πGµ

(

= 20). (a) Spoke

length. (b) Far-ﬁeld mass over source mass; the diﬀerence is the gravitational binding energy. (c)

Far-ﬁeld mass GM in units of L

. The dotted line marks the bound q = π.

exactly, at every source strength. This is the deﬁcit angle of a cosmic string of mass per unit length

. A spoke is shared by three inner tetrahedra. In the symmetric conﬁguration their angles at the

spoke cannot fall below 60

◦

, so δ

< π, and the source is bounded,

q < π. (27)

6.4 Solutions (T)

We solve Eq.

(21)

under Eq.

(25)

by sparse least squares, by continuation in

from the ﬂat lattice.

Every solution reported has residual below 3

−13

, except the closest approaches to the bound

(

q >

13), where it is below 10

−10

. Table 5 and Fig. 5 give the results at

= 20 with

= 9

half-cells. At

= 16 (

= 7

6) the entries agree to within 0

5% for

q ≤

1 and to within 3% near

the bound, where the long spokes feel the wall.

q = 8πGµ spoke deﬁcit spoke length/L

far-ﬁeld mass / source mass

0.001 0.057

◦

0.6125 1.0001 1.001

0.1 5.73

◦

0.626 1.014 0.991

0.5 28.6

◦

0.689 1.076 0.944

1 57.3

◦

0.794 1.179 0.875

2 114.6

◦

1.221 1.588 0.683

2.5 143.2

◦

1.816 2.138 0.531

3 171.9

◦

5.01 4.92 0.254

3.13 179.3

◦

22.4 19.1 0.071

Table 5: The trapped node as Regge initial data (

= 20). The spoke deﬁcit equals

exactly

(Eq.

(26)

is the conformal factor at the trapped node. The far-ﬁeld mass is read from

ψ = 1 + GM/2r ﬁtted on 3 < r < 7.6; the source mass is µ

ℓ

on the solution.

The far ﬁeld. Away from the defect the conformal factor is

= 1 +

GM/

. This agrees with

Schwarzschild initial data in isotropic coordinates. At small

the mass read from the ﬁt equals

the source mass

ℓ

to 0

14% at

= 20 (0

36% at

= 16). On shells at

r ≈

6 (

= 16) the

quantity (

ψ −

is the same at every node to

4%. Eq.

(22)

holds by construction; the ﬁt is an

independent check that the lattice reproduces the continuum ﬁeld and its normalization.

The Newtonian potential and

. The time–time part of the metric is ﬁxed by the lapse

which obeys ∆

= 4

πGρN

in the static, weak-ﬁeld limit, with ∆

the Laplacian of the slice. We

solve it on the solved slice, with the cotangent Laplacian of the slice’s own geometry and

= 1 on

the wall. At

= 0

001,

= 16, it gives

= 1

−GM/r

with the coeﬃcient equal to the source mass

to 0.6%, and the same at every node of a shell to ±0.4%. The ratio of the spatial to the temporal

coeﬃcient is

= 0

998. On a conformally ﬂat slice

= 1 is expected once both equations are solved

correctly, so this is a check that the two lattice operators have the right normalization. The defect’s

far ﬁeld is Newtonian with the correct strength. At large

a defect held at rest is not in static

equilibrium; that would need internal pressure, which the source does not have. We therefore quote

γ only in the weak-ﬁeld regime.

Curvature near the defect. The spokes carry the largest deﬁcits. Edges near the void carry

deﬁcits up to about a quarter of the spoke value, and the largest deﬁcit per shell falls oﬀ roughly as

−3

, the tidal falloﬀ, out to the wall (Fig. 4b). Individual deﬁcits away from the defect are nonzero,

but at every node away from the source they sum to zero by Eq.

(21)

. At the trapped node the

constraint makes the four spokes’ terms

ℓ

equal to 8πGµ

ℓ

, the source mass times 8πG,

exactly. The six edges of the void then carry negative curvature, about

−

65% of 8

πGM

= 0

and edges within about three half-cells of the void center carry a matching positive amount.

How the source is spread. The line source spreads the rest mass over the spokes. We also

placed the whole rest mass on the trapped node (

= 16). For the same total mass the far ﬁeld

again agrees with Schwarzschild initial data, with the source mass to 0

35% at small mass, while

the spoke deﬁcit doubles (0

115

◦

against 0

057

◦

). The near ﬁeld depends on how the mass is spread;

the far ﬁeld does not.

Binding near the bound. As

grows, the far-ﬁeld mass falls below the source mass (Fig. 5b).

The diﬀerence is the gravitational binding energy, as for any self-gravitating body. As

q → π

the

spokes stretch into a long throat and the source’s own mass grows without bound. The far-ﬁeld

mass grows much more slowly. At

= 16, as the spoke length grows from 22 to 130

(

= 3

to 3

1412), 8

πGM

rises only from 27

8 to 30

3 half-cells, and each doubling of the spoke length

adds less than the one before. At the last point computed the far-ﬁeld mass is

GM ≈

, or

ℓ

for

2, close to the Planck mass, and still rising slowly; whether it reaches a ﬁnite limit

cannot be decided at this resolution. These statements near the bound are about the Einstein-order

constraint. Route A’s correction there grows to order one (0

25 at

= 3, about 1 at

= 3

13,

Section 6.9), so it does not control this regime; route C, which has no action, does not address it.

No horizon forms outside the void: the area radius

grows steadily with

outside the corners.

= 3

13 the defect’s corners sit at 2

GM/R ≈

91 in area radius, close to but outside their own

Schwarzschild radius. Any narrow neck would lie inside the void, where a single node cannot resolve

it. So the geometry near

is resolved only at the lattice scale, and we do not claim more than

this.

6.5 Entanglement equilibrium of the solved slice (T)

Route C requires every small ball of the slice to satisfy Eq.

(16)

. This is an internal consistency

check, not an independent test. The slice was built to satisfy the Regge constraint, and at linear

order equilibrium of every round ball is equivalent to that constraint (Section 4.3). What the check

tests is that the equivalence survives the discretization: that the areas and volumes of round balls,

measured through the piecewise-ﬂat geometry, reproduce the equilibrium condition on a lattice whose

curvature sits on edges, for balls that cut the spokes and for balls with no source. We check this on

a solved slice (

= 16, wall radius 7

= 0

01). Each tetrahedron carries the constant metric ﬁxed

by its six edge lengths. Through these metrics we integrate the area of a coordinate sphere of radius

and the volume inside it, and compare the area deﬁcit at ﬁxed volume,

δA|

A − A

ﬂat

(

with the equilibrium value

−

πG δ⟨K⟩

−q

s∩B

(

− r

)

dℓ

of the spoke source. At

linear order a coordinate sphere serves as well as a geodesic one, because a change of shape at ﬁxed

volume does not change the area of a sphere.

(half-cells; L

√

2) 1.5 2.5 3.5 4.5 5.5

δA|

−0.0195 −0.0398 −0.0583 −0.0761 −0.0937

−8πG δ⟨K⟩ −0.0231 −0.0417 −0.0595 −0.0771 −0.0947

ratio 0.84 0.96 0.98 0.99 0.99

The ratio approaches one once the radius is about twice the lattice spacing; at

= 1

5 the radius

is about one spacing. Balls of radius 2 that contain no source have

|δA|

between 10

−5

and 10

−4

while their area changes by 1

−2

: they are in equilibrium to about 0

5%. Regions built from

whole tetrahedra do not pass the test, even without a source, because their jagged boundaries pick

up the individual edge strains; equilibrium is a statement about round balls. So the lattice form of

the constraint and the equilibrium condition agree on the solved slices, to this accuracy. This does

not test route C’s postulates, which the check assumes.

6.6 The full matter action of route A (T)

We now drop the two simpliﬁcations of the reduced source, at

= 12 with

= 5

6. In model

M the spokes’ dual areas are barycentric and computed from the labels, so each spoke’s mass is

(

|⋆s|/|⋆s|

(0)

)

ℓ

. In model F we add the 0-form block, with the 0-form energy at the trapped node

and at each corner. Its weight η is set so that at the ﬂat lattice the 0-form energy is 16.45 η times

the mass term. We take

= 0

1. For each model we follow the solution by continuation in

. At

each solution we also nudge the conformal factor at one corner by ±5% and solve again.

The reduced source has a locally unique solution at every q tried: after either nudge it returns

to the same solution to 10

−15

. Model M behaves the same way, with symmetric and locally unique

solutions, up to

= 0

81. At

= 0

82 the solver ﬁnds no solution. The reason is feedback: longer

spokes have larger dual areas, so the source grows with the geometry it produces, and the branch of

solutions turns back (a fold) while the spoke deﬁcit, about 1

6 rad at

= 0

81, is still well below

. (In model M the spoke’s mass per unit length is

µ|⋆s|/|⋆s|

(0)

, so its deﬁcit is

q|⋆s|/|⋆s|

(0)

, larger

than

.) Model F is also symmetric and locally unique where it is found, up to

= 0

60; at 0

65 the

solver ﬁnds none. In neither model does the trapped node move toward a corner, and in neither

does it approach the metric wall: with a positive mass the spokes lengthen, away from the wall. A

largest

of this kind is known in the conformal method of numerical relativity when the physical

density is held ﬁxed [

], so we do not read it as a physical maximum mass. It depends on how the

Hodge star is discretized. The results of Sections 6.3 and 6.4 use the reduced source.

6.7 Physical scale (C)

For a proton,

is spread over spokes of total length 4

612

= 4

ℓ

, so

πGm

ℓ

≈

−19

. The spoke deﬁcit is then about 2

−17

degrees, and every strain is

of order 10

−19

. The defect is locally ﬂat to this order, which is the description of Ref. [

]. In route A,

the dictionary’s other objects are equally small at this scale. The strain

ℓ

/ℓ

(0)2

−

1 is positive

everywhere. Inside the

= 16 box

≈

; the sum grows with the box radius, as the square

of a 1

ﬁeld does. For a proton this is a few times 10

−36

per box, so the defect’s contribution to

∝

/β is negligible. The defect therefore leaves both GfE’s emergent cosmological constant

and its G-ﬁeld at their vacuum values to this order.

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

The code’s zero modes sit at the same voids as the defect. Let

be the projector onto the void’s

six localized zero modes. For the strain of the initial-data solution at small

(

= 16), the cosine

between

and

diag P

is 0

90 on the void’s 42-edge neighborhood and 0

51 over all lattice edges.

(The strain is the conformal factor, which falls oﬀ as 1

, so it is less concentrated than a footprint.)

A local positive term placed on the defect’s footprint lifts exactly the zero-mode components that

overlap it (38 of 130 at

= 4, whatever its strength). The zero modes are what the defect couples

to; they are not the defect.

The coupling does not remove those modes. The defect adds one node, four spokes, and no 2-cells

(no octahedral cell contains a spoke). So the Euler characteristic changes by 1

−

4 + 0 =

−

3. With

= 1 and b

= 1 unchanged, b

→ 2L

+ 5. Direct diagonalization of L

on the defected complex

conﬁrms

= 133 at

= 4 and 437 at

= 6, three above the vacuum values, with kernel residuals

below 3 × 10

−15

. The number of zero modes is ﬁxed by topology. The 38 overlapping components

are moved among the nonzero modes, not out of the zero modes. Nor does the defect bind a state.

The lowest nonzero level with the defect lies below the lowest vacuum level 8(1

− cos

π/L

) and

follows it to zero: 3

603, 3

083, 2

219, 1

499 at

= 4

10 against 8

000, 4

000, 2

343, 1

528. Its

weight on the four spokes falls from 81% at

= 4 to 1

3% at

= 10. The number of edges it

eﬀectively spreads over (its participation ratio) grows from 6 to 1300. It looks localized at small

only because the box is small. At large

it is the lowest vacuum mode, slightly perturbed and

spread out. So the defect changes the number of zero modes but neither removes nor binds them.

Whatever excited states a defect has, they do not come from the code’s zero modes.

6.9 Beyond the Einstein sector: where the routes diﬀer (C)

Everything above uses the Einstein limit that all three routes share. Beyond it routes A and B

diﬀer; route C has no action. In route A, GfE’s curvature term is

−Tr ln

(

I − β

R˜g

−1

), which at

second order adds

(

R˜g

−1

)

. Its size relative to the Einstein term is

βR

. With

ℓ

√

ℓ

, and

= 1

843

ℓ

(

2; Section 9), this is

βR

≈

017 (

ℓ

)

On the spokes, where

, it is 0

013 at

= 1, 0

04 at

= 2, and 4

−21

for a proton. Near

the bound it grows with the spoke length, to 0

08 at

= 2

5 and 0

25 at

= 3, although the dual

volume used for it is a poor estimate for such long spokes. (This value is unrelated to the limit

= 0

60 of model F in Section 6.6.) It applies only if a route-A term is present, and it depends

on how curvature-squared terms are put on a lattice, which is not unique [

]. Route B’s one-loop

action adds curvature-squared terms with a logarithm of the cutoﬀ; we do not evaluate them on the

lattice. The strong-source solutions of Table 5 solve the Regge constraint that the routes share at

Einstein order.

7 Testing route B: lattice modes and the sprinkled ocean

Route B needs the ﬂuctuating ﬁelds to be regulated covariantly, and any route needs the same of

quantum ﬁelds that live on the lattice. If they were the lattice’s own modes, the regulator would be

the lattice itself: each ﬁeld would have one mode per site, with momenta in the Brillouin zone of

. We test whether such a regulator could replace the ocean’s.

The test is a uniform strain. Change the metric of the ﬂat

lattice from

µν

lattice coordinates, with

constant. This moves the vertices without creating curvature: every

deﬁcit stays zero, and the Regge action does not change. It is the lattice form of a linear change of

coordinates. A covariant vacuum term can therefore respond to it only through the volume, that is,

through

det g

. A response to a shear at ﬁxed volume breaks this invariance. In the continuum it is

a mass term for the graviton.

For one degree of freedom, the one-loop vacuum term per site is

(

) for a boson and

−

(

)

for a fermion, where

f(g) =



ln Q(k; g)



(28)

is the average over the Brillouin zone of the logarithm of the ﬁeld’s lattice operator

. Expand

(

) +

tr h

(

tr h

)

tr h

. . .

. If

depends on

only through

det g

, then 2

−κ

Every ﬁeld below has

−

4, because its operator scales as 1

/λ

when

g → λg

. Covariance

therefore requires the shear response

∂

, for

with

traceless and

tr H

= 1, to equal

1/4.

We computed

for scalars and Wilson–Dirac fermions on

, with 16

momenta. The scalar

operator is

− cos k · e

) over the 24 nearest-neighbor vectors

, with bond weights

chosen so that

−1

; the fermion has a Wilson term of strength

, and its hopping

couples to the metric through

−1

, through the scalar’s bond weights, or through a vierbein with

the spin connection. No lattice action tested is covariant. The scalar gives

= 0

464 with nearest

neighbors and 0

30–0

48 with next-nearest-neighbor couplings; an ideal isotropic cutoﬀ with a ﬁxed

number of modes gives 5

12. The fermion gives 0

50–1

98 over

= 0

1–4, and 0

752 with vierbein

and spin connection at

= 1. The response is the same for every shear direction, as the symmetry

requires, and no value of

makes the fermion match the scalar, so bosons and fermions do not

cancel. A dispersion that is ﬂat at the cutoﬀ could remove the excess over 1

4, so covariance with a

ﬁxed number of modes is not excluded in principle, but it needs every species tuned at the cutoﬀ.

Fermions enter with the opposite sign and outnumber bosons 96 to 28 in the SSM’s content, so the

net shear term is a tachyonic spin-2 mass term, with

between 18 and 69 (Appendix A): the

lattice-regulated vacuum resists shear with the wrong sign, at the Planck scale.

The spin-2 spectrum. A coeﬃcient at zero momentum does not yet show that the spin-2 mode is

unstable. That needs the quadratic action of the spin-2 mode at every lattice momentum, including

the kinetic terms from the Regge action and from the same loops. We compute it. Take the metric

wave

+ 2

ϵH cos

(

p ·x

), with

transverse to

, traceless, and normalized. For

along a lattice

axis or a root, these polarizations are orthogonal to the lattice gauge modes, so they carry the

gauge-invariant spin-2 content. For each ﬁeld we expand

ln det

to second order in

and average over

the Brillouin zone. The vertices come from ﬁnite diﬀerences of the full nonlinear lattice operators,

so no expansion is done by hand. The fermion now has the lattice vierbein,

1/2

, and

the spin connection on every bond,

exp

(

µab

), with

built from the Levi-Civita

connection. Four checks pass. The spinor curvature built from the code’s spin connection equals the

Riemann tensor to 10

−7

at ﬁnite amplitude; with the opposite sign it fails by 5

−2

. As

p →

the scalar and fermion values return to 0

464 and 0

752, the uniform-strain values of the scalar and

of the fermion with vierbein and spin connection at

= 1. The scalar agrees with an independent

closed-form bubble formula at every momentum. And exact real-space determinants on an 8

lattice

reproduce the momentum-space values to 10

−4

. For the SSM’s content, with Wilson parameter

= 1, the spin-2 quadratic action, averaged over two orthogonal transverse-traceless polarizations,

is, in units where m

< 0 means unstable:

|p| along an axis (lattice units) 0 1.05 2.09 3.14 (zone edge)

loops, m

−1249 −1217 −1125 −979

Regge kinetic term, 2ˆp

0 +2 +6 +8

Along a root direction the loop term rises to

−

716 at the zone edge (

π, π,

0), where the Regge

term is +11. For

= 0

5 and 2 the loop term is

−

1054 and

−

1040 at the axis edge, and

−

443

and

−

990 at the root edge. Over all ﬁve transverse-traceless polarizations the loop term varies by

about

6 along an axis and

20 at the root edge. The values are converged in the momentum

grid. An independent implementation, written from this description alone and sharing no code

with ours, reproduces every value in this paragraph within the polarization spread, with the same

normalization; it also reproduces the uniform-strain values above and agrees with exact real-space

determinants on a 6

lattice to 10

−6

. The spin-2 action is negative at every lattice momentum, by

about forty times the Regge kinetic term or more. So the spin-2 mode is unstable across the whole

Brillouin zone, with or without a bare Regge term. The loops’ own kinetic term has the sign that

makes 1

positive, but it is far too small to help. It is also the least certain part: ordering the

fermion’s spin transporter symmetrically about the bond makes the action more negative, by about

150 in

at the axis edge and 310 at the root edge, which does not change the sign. The scalar’s

kinetic term is ten times smaller per degree of freedom than the fermion’s, the reverse of what a

covariant regulator gives; this is one more sign that the lattice regulator is not covariant.

The instability has a deﬁnite origin. At

p →

0 the spin-2 wave is a uniform shear, a linear change

of coordinates, so the negative term is the non-covariant graviton mass of the lattice regulator. It

could be removed only by a counterterm that breaks diﬀeomorphism invariance, a bare term

tr h

on every bond, tuned against the loops. That counterterm is a bond stiﬀness. This paper adds no

such term, so with the actions tested the instability stands. For the lattice actions tested, at one

loop, induced gravity in the SSM therefore cannot come from the lattice’s own modes. The test

isolates why: counting modes per site stiﬀens the vacuum against shear with the wrong sign, while

counting them per proper volume, as the ocean does, gives a covariant vacuum term. The test is

one loop, for free ﬁelds, in Euclidean signature, and it counts gauge ﬁelds as two scalar-like degrees

of freedom. A lattice gauge ﬁeld on the 12 links per site has more modes near the cutoﬀ. This could

change the size by a factor of order one, and it could change the sign only if the gauge ﬁelds’ shear

response exceeded the fermions’ by about a factor of four.

A covariant ocean can be constructed. The test also shows what a covariant ocean needs: its

modes must not be tied to a regular arrangement of sites. A random sprinkling meets this. Place

the ocean’s degrees of freedom at the points of a Poisson process, about one per lattice cell, and

let their couplings depend only on proper distance. A uniform strain maps a Poisson process onto

another Poisson process, changed only in density. So the vacuum term, averaged over sprinklings,

depends on the metric only through its volume, and its shear response equals the covariant value.

A regular lattice has no such property. This is the Euclidean counterpart of the theorem that a

Poisson sprinkling is discrete without breaking Lorentz symmetry [

]. We checked it numerically.

The same operators were placed on sprinkled points and on the

lattice points, at equal density,

with a Gaussian kernel in proper distance; for the fermion a constant mass removes near-zero modes.

Covariance requires K + κ

= 0:

points, ﬁeld box side K + κ

lattice, scalar 8, 10 −0.059, −0.059

sprinkled, scalar (200 and 16 samples) 8, 10 +0.005 ± 0.004, −0.006 ±0.009

lattice, fermion 6 −0.045 and −0.097 (two shear directions)

sprinkled, fermion (60 samples) 6 −0.019 ± 0.014

sprinkled, Delaunay held ﬁxed (2 samples) 5 +0.240 ± 0.001

The sprinkled ocean passes, within the sampling error, and the lattice fails with the same operators.

In the smallest box both ﬁelds show a small oﬀset of the same size, which for the scalar disappears

in the larger boxes. A single sprinkling ﬂuctuates about the covariant value, but the ﬂuctuation

averages out over a region of many cells. Randomness alone is not enough. The last row uses the

same sprinkled points, joined by their Delaunay triangulation, with the ﬁnite-element Laplacian

built from the edge lengths, as in Regge calculus. When the metric changes, the triangulation is

held ﬁxed and only the edge lengths change. This regulator is four times less covariant than the

lattice, with the opposite sign. A ﬁxed complex, random or regular, remembers the metric in which

it was built. Rebuilding the triangulation in each metric restores covariance by construction, but

then

ln det

jumps at every change of the triangulation, and in a curved metric the triangulation

itself is not available in four dimensions. The couplings of a covariant ocean must therefore be

functions of the proper distance in the current metric, as the Gaussian kernel’s are. This gives the

ocean’s ﬁrst postulate a construction, in Euclidean signature and at one loop. It leaves open the

Lorentzian case, where sprinklings are causal sets and their nonlocality is harder, and how the ocean

assembles the lattice and responds to its geometry.

The sprinkled ocean’s Newton constant. Covariance does not ﬁx the value of

. We computed

it directly for the sprinkled ocean, species by species. For a transverse-traceless wave, the induced

Einstein term appears as the

coeﬃcient of the one-loop spin-2 response, 1

πG

per species. We

obtain it by the exact second-order expansion of

ln det

on a ﬁxed sprinkling, at several momenta,

and check it against exact determinants to six digits. The scalar is the graph Laplacian, whose bond

weights fall oﬀ with the distance between the two points in the curved metric. The fermion is the

graph Dirac operator with vierbein, spin connection, and a Wilson term. The gauge ﬁeld is the

vector graph Laplacian with Levi-Civita transport, minus two ghost scalars; the Ricci term of the

Feynman-gauge operator has no graph form of its own, so we add it at each point, normalized with

the graph Laplacian’s continuum constant. The result depends on choices at the cutoﬀ, so we ﬁrst

made every ingredient covariant. The bond weight uses the geodesic distance, not the metric at the

bond’s midpoint; the two diﬀer by terms of order (

∂h

)

and

∂∂h

for a bond

, the same order

as the Einstein term. The kernel is a Gaussian of width

times the smooth cut (1

− d

)

proper distance

, with

= 3

, so no bond is selected by a coordinate distance. The fermion’s

vierbein is averaged along the bond, and its spin transporter is the path-ordered product along it.

The gauge ﬁeld is computed with and without its Ricci term. In units of the proper-time value for

one minimal scalar with Λ set by the sharp count, Λ

modes per unit volume equal to the

sprinkling density, 2.3 × 10

−3

, the weights are (with the heat-kernel count they are larger by

√

2):

species σ = 0.55 σ = 0.70 σ = 0.80 proper time

real scalar +2.1 +2.8 +3.5 +1

Weyl fermion (Wilson r = 1) −6.1 −9.4 −12.0 +1

Weyl fermion (Wilson r = 0.5) — −8.4 — +1

Weyl fermion (D

†

D, no Wilson term) — −0.6 to −3.5 — +1

gauge ﬁeld with ghosts, with Ricci term −9.2 −10.6 — −4

gauge ﬁeld with ghosts, no Ricci term +4.9 +7.7 — +2

C for the SSM’s content, with Ricci term −396 −565 < −400 4

C for the SSM’s content, no Ricci term −227 −346 < −400

= 0

80 the scalars and fermions alone give

−

560, and no gauge weight in the range found lifts

above

−

400. Cruder versions give a wide range: with the metric at the bond’s midpoint and a

hard cut in coordinate distance, the scalar’s weight runs from

−

5 to +4 and the gauge ﬁeld’s from

−

20 to +10, depending on the choice. With every ingredient covariant, three things are robust. The

weights grow with

at ﬁxed density, so they are set by the kernel’s shape rather than by a count of

modes. The scalar and the gauge ﬁeld keep the signs of the proper-time scheme, enlarged by factors

of 2 to 4. The fermion does not: its weight is negative and large, and with 48 Weyl fermions it makes

negative at every width. So, with these operators, the sprinkled ocean induces a negative Newton

constant for the SSM’s content. Since this ﬁxed-density term is the whole ﬁrst-order response of an

ocean in equilibrium, Eq. (14), displacement of the ocean by defects cannot change its sign.

The fermion’s sign is the part to doubt. A fermion’s entanglement entropy across a surface is

positive, and with a common cutoﬀ it equals

ind

for that fermion [

]. So we tested the

fermion operator. Halving the Wilson parameter changes the weight by about 10%. A fermion

with no Wilson term at all follows from

det D

det

(

†

)

1/2

, with

†

−∇

4: on the

sprinkling it is the spinor graph Laplacian with path-ordered spin transport plus the Lichnerowicz

term

4, normalized with the graph Laplacian’s constant, regulated exactly as the gauge ﬁeld is.

Its weight is

−

6 from a linear ﬁt and

−

5 from a quadratic one; the response is curved in

. Its

two parts explain the sign. The kinetic part alone gives

−

9, the proper-time value

−

2 enlarged

by the same factor of about 3 that every bosonic piece shows (scalar 2

8; vector kinetic part 3

vector Ricci part 3

1). The Lichnerowicz part gives +5

3, its proper-time value +3 enlarged by

only 1

8. With all parts enlarged alike the fermion would give about +3 and

about +12. On

the sprinkled ocean the fermion’s coupling to curvature is too weak to beat its kinetic term, and

with either fermion regulator

is negative for the SSM’s content: about

−

145 with the squared

operator, using the linear ﬁt. The proper-time weights, and with them

= 4 and the agreement of

within a factor of 2, remain a result of the proper-time scheme that no covariant ocean we could

construct reproduces.

8 Choosing a route

8.1 Newton’s constant and the cosmological term in each route (C)

in route A. Match the curvature term 3

βR

of Eq.

(6)

to the Einstein–Hilbert normalization

at the Planck length, with each lattice term carrying the four-dimensional volume

ℓ

. This gives

ℓ

π ≈

−3

ℓ

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

]

shows that the lattice selects the Regge action, so the match ﬁxes

rather than testing it. This

value assumes that route A supplies all of

. In route C, which this paper adopts,

comes from

the area law and no β is needed.

from the polycrystal (route A). In the dictionary, Λ

≈ Tr ϵ

per site. Ref. [

] identiﬁes

GfE’s energy density with its emergent dark-energy term. On the lattice that density is the quadratic

strain functional, so our identiﬁcation is the lattice form of hers. The assembled vacuum is made of

many small crystals. Ref. [

] reports a bond-length spread of

5%, set by the deﬁcit of the strained

foam. It also reports that about 30% of interior nodes have 10 or 11 neighbors, the mark of grain

boundaries. Take

rms

∼

05 on the bonds of that fraction of sites. Each site owns six of its twelve

bonds, since each bond is shared by two sites. Then

Tr ϵ

∼

05)

≈

−3

per site

and

∼

4.5 × 10

−3

× 48π

4 ℓ

≈ 0.2 ℓ

−2

, (29)

against the observed

∼

−122

ℓ

−2

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

121

, the usual

problem. The two inputs are the grain-boundary fraction and the leftover strain. Their product

rms

is 7

−4

here; it would have to be

∼

−125

, a single crystal with no leftover strain.

The dictionary makes this a deﬁned quantity.

and the cosmological term in route B. In route B in its area-law form,

is ﬁxed by the

area law, and Eq.

(11)

must be consistent with it. The cosmological term of route B follows from

Eq.

(15)

: Λ

= 8

πG

ind

= 102Λ

= 25

5 Λ

, or Λ

≈

370–520 for the two cutoﬀs. This is

the vacuum-energy problem in its usual form, with no cancellation among species and a sign that

depends on the regulator. Left uncanceled, it would curve the vacuum at the lattice scale. The ﬂat

background assumed by every calculation in this paper therefore requires, in route B, the vacuum

energy to be canceled or renormalized separately, which the induced-gravity argument does not

supply. So routes A and B face the cosmological constant problem in diﬀerent forms: route A as a

strain functional that is positive but 10

121

too large, route B as a vacuum energy that must cancel.

and the cosmological term in route C. In route C,

is the area law’s constant,

with

the vacuum’s entanglement per bond, below

2, and 0

2 in the bond model of Section 9.

A uniform vacuum energy changes no entanglement across any surface, so it does not enter the

equilibrium condition. The cosmological constant then appears only as a constant of integration, as

in unimodular gravity [

]; route C removes the vacuum-energy problem from the ﬁeld equations

but does not ﬁx the constant that remains.

8.2 Weighing the routes

Nature has one origin of gravity, so we ask which route the SSM supports. If routes A and B both

acted, their Einstein coeﬃcients would add,

48πβ

ℓ

CΛ

12π

, (30)

the ﬁrst term from route A (Eq.

(6)

) and the second from route B (Eq.

(11)

), in the same convention.

Route B with the ocean. With the ocean as the ﬂuctuating ﬁelds, the regulator is covariant and

the cutoﬀ sits at the lattice spacing. The cutoﬀ is physical, so the induced term cannot be absorbed

into a bare coupling. In the proper-time scheme it would supply 0

56 or 0

79 of 1

for the SSM’s

content at

2 (Section 4.2), and a route-A term could supply the rest. On the sprinkled ocean,

with the graph operators we can build, the induced term is negative, so it would not supply 1

all (Section 7).

A covariant regulator is required. With the lattice as their regulator, the loops that would give

make the spin-2 mode unstable (Section 7). So induced gravity in the SSM must come from

ﬂuctuations whose modes are counted per proper volume, as the ocean’s are postulated to be. This

makes a covariant regulator necessary, for the lattice actions tested and at one loop; it does not

show that the ocean exists.

The choice. We adopt route C as the preferred phenomenological framework. Route C makes no

claim about the vacuum’s stiﬀness, so the regulator test does not apply to it. Its two postulates

give the Hamiltonian constraint, with

ﬁxed by the area law and no sum over species, and the

defect solutions satisfy its equilibrium condition ball by ball (Section 6.5), a consistency check, since

they were built to satisfy the constraint. Route B needs ﬂuctuations that induce a positive Einstein

term, and no covariant ocean we could construct gives one for the SSM’s content; displacement of

the ocean cannot change the sign, Eq.

(14)

. Route A is a consistent action with exact lattice results,

but it takes

as an input, and its cosmological term is 10

121

too large. Route C also ﬁts the SSM

for reasons of its own. Entanglement equilibrium needs something to carry the entanglement, and

the SSM supplies it: the bonds, each a Bell pair. Its area law is the one that already ﬁxes the lattice

spacing (Section 3.4), so route C adds no new constant. The usual weakness of deriving gravity from

entanglement is that the entanglement of quantum ﬁelds depends on the species and on the cutoﬀ;

in route C the entanglement sits on the bonds, not on ﬁelds, so that problem does not arise, and it

is the problem that defeats route B. Nor does route C need the boundary of anti-de Sitter space, on

which the holographic derivations rest. Its microscopic law, that a bond’s entanglement follows its

area, is a postulate: the code as deﬁned cannot supply it, and the bond model of Section 9 gives

a coupling that can. Gravity in the SSM is then the condition that the vacuum’s entanglement,

carried by the lattice’s bonds, be in equilibrium with the matter it contains. Routes A and B remain

as the two actions whose Einstein limit route C shares, and the ocean computation remains as the

test that route B does not pass.

What the choice leaves open. (a) Route C’s postulates are not derived from the code’s

dynamics, and the stabilizer code cannot supply the ﬁrst; Section 9 gives a minimal coupling that

can and states what it assumes. (b) Only the time-symmetric slice is treated here. The full Einstein

equation in route C needs the equilibrium of small causal diamonds in Lorentzian signature [

]. (c)

Route C has no action, so it gives no corrections beyond Einstein order. (d) Any quantum ﬁeld

whose modes live on the lattice would generate the shear term of Section 7 and give the graviton a

mass, in any route. Such ﬁelds must carry no loops at the cutoﬀ scale, or their bosons and fermions

must cancel. The test is one loop, for free ﬁelds, in Euclidean signature, and does not cover the

code qubits on the bonds. (e) The cosmological constant is a constant of integration in route C;

its value is not ﬁxed. (f) Route B could be revived by a covariant ocean whose fermions couple to

curvature as strongly as its bosons do; we did not ﬁnd one.

9 Route C’s microscopic law

Route C’s ﬁrst postulate asks each bond’s entanglement to follow its share of area. This section

asks what can supply that rule: a count of bonds, the code as deﬁned, or a coupling between the

code and the labels.

Counting bonds is not enough. One could take the entanglement across a surface to be

times

the number of bonds it cuts. When the labels strain, that number does not change while the

area does. In a conformally ﬂat slice with a uniform conformal factor, which has no curvature, a

count-based entropy still changes at ﬁxed volume, in proportion to the conformal factor. A count

measures geometry only if curvature comes from restitching at ﬁxed bond length, not from strained

labels. In that reading a proton’s gravity corresponds to about one missing bond crossing on a

sphere of radius 0

05 fm around it and about 18 at 0

84 fm, for

2 (the count scales as 1

The defect solutions of this paper strain their labels, so route C uses entanglement that follows area.

What the code can and cannot supply. The SSM’s vacuum is the code space of a stabilizer code,

and its labels are not code degrees of freedom (Section 3.4). Two consequences follow. In any

stabilizer state the entanglement across a cut is a whole number of ebits, ﬁxed by the independent

checks that straddle the cut [

]; it cannot vary continuously. And because no check involves the

labels, no code state changes when the labels strain. So the code, as deﬁned, supplies the count rule

and not the rule that entanglement follows area. Route C needs a bond whose entanglement varies

with its label. For a pair of qubits this means a state cos θ

|00⟩ + sin θ

|11⟩, with entropy

s(θ

) = −cos

ln cos

− sin

ln sin

, (31)

which rises to

2 as

→ π/

4. The rule

(0)

then ﬁxes

as a function of the label,

with the vacuum at

(

) =

< ln

2. The code as deﬁned contains no coupling that produces it, so

route C’s ﬁrst postulate is phenomenological here. The growth simulation of Ref. [

] assumes that

bonds form only near the bond length. If error correction shared that tolerance, the code would

depend on the labels, but not in the way route C needs: its performance would be best at equal

lengths and fall with strain of either sign, a response of second order that favors equal lengths. Such

a tolerance would instead be a natural origin of the stretch at which the lattice restitches.

A minimal coupling between code and labels. We tested whether a simple label-dependent

Hamiltonian gives the required response without imposing it. With

set by the vacuum’s own

area law, the condition

∂S

/∂a

= 1

reads

d ln s

/d ln a

= 1 at the vacuum point. Take each

bond’s two qubits with

= −J(a

) (X

+ Z

) − h (Z

+ Z

), J(a

) = J



(0)



, (32)

so the coupling grows with the dual area. The ground state is

cos θ

⟩

sin θ

⟩

, and Table 6

gives its entanglement and response for

= 1. Stretching a bond raises its entanglement at ﬁrst

order, with the sign route C needs. The slope equals one only at

h/J

= 0

69, and an equilibrium

condition selects that point.

Table 6: Bond model of Eq. (32): vacuum entanglement and its response to the dual area.

h/J

0.10 0.30 0.50 0.69 1.00 1.50 2.00

/ ln 2 0.97 0.80 0.60 0.45 0.30 0.17 0.11

d ln s

/d ln a

0.06 0.39 0.75 1.00 1.25 1.45 1.54

Maximal entanglement ﬁxes the slope. The condition that selects this point is a second use of

maximal vacuum entanglement, and it diﬀers from the equilibrium condition, Eq.

(16)

, in what it

varies. Jacobson’s condition varies the geometry of a small ball at ﬁxed volume, with the vacuum’s

microscopic state held ﬁxed, and balances the change of area against the change of matter energy [

It takes the entanglement per unit area, 1

, as given. Here the geometry is held ﬁxed and the

microscopic state varies. The ball, its volume and its boundary area

stay the same, while the

density of bonds that cross the boundary changes. The boundary is crossed by

A/a

bonds, where

is the vacuum’s dual area per bond, so the entanglement across it is

= (

A/a

)

(

). We postulate

that the vacuum’s bond density is the one that maximizes this entanglement. Then

, and with it

s/a, is stationary at the vacuum:

s(a)



= 0 ⇐⇒

d ln s

d ln a



= 1 ⇐⇒

∂S

∂a



. (33)

The condition holds for any bond model in which

s/a

has a maximum, and it identiﬁes 1

as the

largest entanglement per unit area the bonds can carry. In the bond model it ﬁxes the ratio: for

= 1 the maximum of

s/a

lies at

h/J

= 0

691, with

= 0

2. A coupling that grows faster

with the dual area gives a larger

(Table 7). A coupling that follows the bond length,

= 1

has no maximum and fails. So the FCC geometry ﬁxes the ratio once the coupling is tied to the

dual area. The stabilizer structure cannot ﬁx it, since it gives whole ebits.

Table 7: Bond model of Eq.

(32)

at the maximum of

s/a

, Eq.

(33)

, for three exponents

; the slope

is one in each case.

loc

is the local Newton constant at a 10% and a 49% stretch of the dual

area, under assumptions (i)–(iii) of the text; the last column is the stretch at which

loc

= 1

08.

For p = 1/2, s/a has no maximum.

p h/J

/ ln 2 G

loc

/G (10%) G

loc

/G (49%) stretch at 1.08

1 0.691 0.45 1.08 1.54 9.8%

1.5 0.450 0.65 1.20 2.62 4.4%

2 0.358 0.74 1.33 4.33 2.8%

The variation is admissible for two reasons. First, the number of bonds is not conserved in

the SSM. Bonds form by selection and stitching, so a vacuum with a diﬀerent bond density is an

allowed state of the same geometry, not a diﬀerent geometry. Second, the geometry ﬁxes the area,

not the count: the labels give each bond its dual area, and the bond count through a ﬁxed surface

follows from them. The variation compares uniform vacua. Changing the density in one region

alone requires lattice defects, which is the restitching discussed under strong ﬁelds. The postulate is

an addition to route C. It ﬁxes the entanglement per unit area that Jacobson’s condition takes as

given, and Eq. (16) does not imply it.

Calibration. The area law ﬁxes the bond’s area through

= 4

. The values of

and

used in this paper, from Section 3.4, take

2. With the bond model’s

= 0

2 both

shrink by

√

0.45

= 0

67, and every scale set by

in Planck units changes with them (Table 8).

Newton’s constant is held ﬁxed, since it is measured. So are all results stated in lattice units: the

deﬁcits, the constraint solutions labeled by

, the far-ﬁeld agreement, the equilibrium ratios, the

cutoﬀ per site ΛL

, and the ﬁeld content’s C.

Strong ﬁelds. At a maximum,

s/a

curves downward. So in any such model each bond follows its

area exactly at ﬁrst order and falls below it at second order. Reading this shortfall as a change of

Newton’s constant takes three assumptions. (i) The bonds hold the stretch, with no restitching to

relieve it. (ii) The stretch is uniform over a ball small enough for the equilibrium condition to apply

about the stretched state, with the matter side unchanged. The area-law coeﬃcient there is then

′

(

) per unit dual area, so

loc

G s

′

(

)

′

(

). (iii) The stretch is measured from the vacuum

labels. On a conformally ﬂat slice we estimate it by

, which depends on the slicing and on how

the lattice reached its strained state. Under these assumptions, for

= 1,

loc

grows by 1

5% at

a stretch of 2%, by 8% at 10%, and by 54% at the 49% stretch of a neutron star’s surface. The

existence of two-solar-mass neutron stars bounds

inside them to at most 8% above its laboratory

value [

]. That bound assumes a uniform rescaling of

in the stellar structure equations and the

equations of state considered there. Applied to

loc

, it says that bonds do not hold neutron-star

Table 8: Quantities that depend on the area-law calibration, at

2 (used in the text) and at

the bond model’s s

= 0.45 ln 2, with G held ﬁxed.

quantity scales as s

= ln 2 s

= 0.45 ln 2

/ℓ

1/2

1.665 1.121

/ℓ

1/2

1.843 1.240

cutoﬀ Λℓ

, heat kernel / sharp s

−1/2

2.29 / 2.72 3.40 / 4.04

route B: C required, heat kernel / sharp s

7.20 / 5.09 3.26 / 2.31

route B: G

ind

/G at C = 4 s

1.80 / 1.27 0.82 / 0.58

route A: βR

/[(ℓ

) δ

] s

−1

0.017 0.038

proton source q s

−1/2

4 × 10

−19

6 × 10

−19

missing bond crossings, proton, 0.84 fm s

−1

18 40

far-ﬁeld GM at the largest q computed s

1/2

1.6 ℓ

1.05 ℓ

stretches, and that the lattice restitches once the stretch reaches about 10% for

= 1, or about 4%

and 3% for

= 1

5 and 2. These thresholds are model-dependent. They rest on the bond model

and its exponent, on assumptions (i)–(iii), and on comparing a

loc

that varies with depth to a

bound derived for a uniform

. Route C’s departure from general relativity near compact objects is

then set by how fully restitching relieves the stretch, which is not computed here.

For comparison, a scalar ﬁeld’s entanglement across a plane responds to a transverse stretch

with slope between 0

75 and 1

05 when its lattice points are carried along, depending on the lattice’s

aspect ratio and the mass, and with slope one, to 0

2%, when the ﬁeld is reﬁlled at ﬁxed density

per proper area. That slope then comes with an entanglement per unit area that is a sum over

species [

], which is route B’s problem. The bond model avoids it: one bond, one coupling, and

a maximum that ﬁxes its parameters.

What the model assumes. The bond model ﬁxes its parameters by maximal entanglement, but

two of its inputs are assumed rather than derived from the code: the coupling’s dependence on

the dual area,

J ∝ a

, and the maximization itself, which is a postulate added to route C. The

calibration of L

and L

follows s

(Table 8).

10 Scope

What the defect tests. The constraint solutions of Section 6 use only the Einstein limit that the

routes share; the full matter action and the dictionary’s objects at the defect belong to route A.

They show that the SSM lattice carries Einstein gravity consistently around a defect, They do

not distinguish the three routes, and they are not evidence that gravity comes from entropy, from

vacuum ﬂuctuations, or from entanglement. The choice comes from elsewhere: from the regulator

and sprinkled-ocean tests and from the equilibrium check (Sections 7, 8 and 6.5).

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

the constraint is imposed inside a ﬁxed wall. Interactions between defects, the ∆ multiplet of

Ref. [6], and the proton–neutron distinction are not addressed.

A moment of time symmetry. The slice is a moment of time symmetry, with zero extrinsic

curvature. The lattice is Euclidean. The far ﬁeld is that of a static body, and the lapse gives

its Newtonian potential in the weak-ﬁeld limit. Nothing here shows that the construction gives

Lorentzian dynamics, gravitational waves, or moving matter. Section 5.1 discusses the Lorentzian

case only through the continuum limit. At large source strength the defect is not in static equilibrium,

which would need internal pressure. Where we call the zero modes a massless sector, we mean zero

modes of the static Laplacian, not a ﬁeld shown to propagate.

Oﬀ-shell source. In route A the defect’s Dirac–K¨ahler ﬁeld is prescribed, not solved for. Its

amplitude is ﬁxed, and it does not satisfy the Dirac–K¨ahler equation; a localized massive ﬁeld held

in place cannot. The computation gives the geometry that a prescribed source produces. We do not

solve the coupled problem, in which the ﬁeld and the labels are found together.

The choice of slice. The slice is conformally ﬂat. This is an initial-data ansatz, not a gauge

choice, and it restricts the geometries allowed. It is exact at linear order when

= 1 and exact for

Schwarzschild initial data, but near a strong source a general slice would have more freedom than

one factor per node. The vertex form of the constraint follows from varying the four-dimensional

Regge action with respect to the lapse on a 3+1 prism complex built on the slice (Section 6.3).

That complex is itself a choice; a derivation from the D

complex is not given.

Reduced source. In route A the main results use the reduced one-form source, with the dual

areas in the mass term held ﬁxed and the 0-form block dropped. Section 6.6 shows that relaxing

either one keeps the defect symmetric and locally unique but lowers the largest source for which a

solution exists.

No continuum limit. The lattice spacing is ﬁxed at

= 1

843

ℓ

for

2, or 1

240

ℓ

for

= 0

2. Convergence of lattice solutions to continuum solutions under reﬁnement is not

claimed (Section 1; Ref. [13]).

Beyond Einstein gravity. Section 6.9 estimates route A’s second-order term at the defect; route

B’s one-loop curvature-squared terms are not evaluated on the lattice, and route C has none. A

lattice form of curvature-squared terms free of convention choices is the general problem of putting

higher-curvature actions on a lattice [14], and it is not special to this lattice.

The regulator test. The test of Section 7 is one loop, for free ﬁelds, in Euclidean signature,

at a uniform strain and for the spin-2 spectrum along an axis and a root. It uses nearest- and

next-nearest-neighbor scalar actions and two Wilson-fermion forms, and it models a gauge ﬁeld as

two scalar-like degrees of freedom. An idealized dispersion that is ﬂat at the cutoﬀ can be covariant,

so lattice actions tuned separately for each species could perhaps reach the covariant value; we

found none that does. The lattice spacing

rests on the area-law calibration, not on an induced

constant from lattice modes.

Relation to the parent papers. Ref. [

] describes the defect as locally ﬂat. Section 6.7 ﬁnds this

to hold for any physical mass: for a proton the deﬁcits are of order 10

−19

. Ref. [

] also measures the

defect’s mass by the elastic energy of its compressed bonds. The present construction has no bond

energy. The defect’s mass enters only as the strength

of its source, and the far ﬁeld measures that

mass, so the construction is compatible with any assignment of

. The code’s zero modes are not

the matter. They are the lattice’s static massless sector, and the defect couples to them.

Relation to holographic codes. This construction diﬀers from holographic codes [

]. Here

the code lives on the bulk lattice, and geometry comes from how the lattice is assembled. We claim

no duality between a bulk and its boundary.

11 Conclusion

The SSM lattice carries one gravitational sector, the Regge equations, and this paper reaches it by

three routes. In route A, Gravity from Entropy reduces at weak coupling to Einstein gravity plus a

Dirac–K¨ahler ﬁeld. In route B, the vacuum ﬂuctuations of covariantly regulated ﬁelds induce the

Einstein–Hilbert action, as Sakharov proposed. In route C, Einstein’s equation is the condition that

the vacuum’s entanglement, with the SSM’s area law, be in equilibrium with matter. All three give

the Regge equations plus a rest-mass source for the trapped node.

A spatial slice must obey the Hamiltonian constraint, not three-dimensional Einstein gravity.

Three-dimensional Regge gravity has no solution with a source on the spokes, because a line of

curvature cannot end. The constraint’s lattice form, one equation per node, follows from varying

the four-dimensional Regge action with respect to the lapse, and we checked this on an explicit

four-dimensional complex. Solved on a conformally ﬂat, time-symmetric slice, it has a solution at

every source strength tried below its bound.

The solutions are what Einstein gravity predicts. Each spoke carries the cosmic-string deﬁcit

πGµ

exactly. The far ﬁeld agrees with Schwarzschild initial data in isotropic coordinates, with the

source’s mass, and the lapse gives the Newtonian potential with the correct strength. As the source

grows, gravitational binding lowers the mass seen from far away. For a proton the defect is ﬂat to a

few parts in 10

, which is the locally ﬂat defect of the SSM. These results hold for every route at

physical masses, and, as a consistency check, the solved slices satisfy the equilibrium condition ball

by ball, to within 4% for balls whose radius is at least about twice the lattice spacing. At strong

sources they are solutions of the shared Einstein-order constraint, and how far they can be trusted

depends on the route. Route A’s correction, if such a term is present, stays below a few percent up

to q = 2, with the reduced source, and becomes of order one only near the bound.

The routes diﬀer elsewhere. Route A gives GfE’s abstract objects lattice deﬁnitions (edge strain,

inverse strain, and a strain functional for Λ

), has a closed-form relative entropy on the lattice, and

leaves

free. In route B,

is induced by the ﬂuctuations and the cosmological term is a vacuum

energy to be canceled. In route C,

is the area law’s constant and a uniform vacuum energy does

not gravitate. Beyond Einstein order, route A, if present, adds corrections of relative size

βR

negligible for real matter. The code’s zero modes, 2

+ 2 for every even

, belong to no route.

The defect adds three and neither removes nor binds any.

We adopt route C as the preferred phenomenological framework. Route B needs ﬂuctuations

that induce a positive Einstein term. In the proper-time scheme they do, within a factor of 2 of

the area law at

2, but on an explicit covariant ocean the fermions’ term is negative and

C <

0 for the SSM’s content, and displacement of the ocean cannot change this. The ﬂuctuations

also cannot be the lattice’s own modes, whose loops make the spin-2 mode unstable, a result an

independent implementation reproduces. Route A remains a consistent action with exact lattice

results, but it takes

as an input. Route C needs no induced term. The entanglement across any

surface is its area over 4

, carried by the bonds; the vacuum is in entanglement equilibrium; and

these two postulates give the Hamiltonian constraint the defect solves, with

from the area law,

no sum over species, and a vacuum energy that does not gravitate. Its condition is that each bond’s

entanglement follow its share of area. A two-qubit bond whose coupling grows with its dual area

meets it, and, if the vacuum’s bond density also maximizes its entanglement, that maximum ﬁxes

the bond’s parameters, so 1

is the largest entanglement per unit area the bonds can carry. At

second order the bonds fall below their area, so in strong ﬁelds, in the bond model, the lattice must

restitch, and how fully it does is where route C could depart from general relativity. Route C has

no action, so it says nothing beyond Einstein order.

Several questions remain open: how a bond’s coupling comes to follow its dual area, which

ﬁxes

, and how fully restitching relieves the stretch near compact objects and horizons, route C’s

candidate departure from general relativity; route C on general slices and in Lorentzian signature,

with small causal diamonds; corrections beyond Einstein order, which route C does not supply, and

a lattice form of the curvature-squared terms free of convention choices; fermion operators on a

covariant ocean whose induced 1

is positive, which route B would need, and the sprinkled ocean

in Lorentzian signature, as a causal set; how the ocean assembles the lattice and sets its cutoﬀ; the

condition on ﬁelds that live on the lattice, with lattice gauge ﬁelds and Dirac–K¨ahler fermions as

cross-checks of the spin-2 spectrum; a matter action that solves for the ﬁeld instead of prescribing

it; the defect in static equilibrium, with its internal stress; the continuum limit; the value of the

cosmological constant, a constant of integration in route C; and a comparison of GfE’s relative

entropy for the Schwarzschild geometry [2] with the bond-count area law of Ref. [8].

A Numerical methods

All computations are ﬁnite and use NumPy and SciPy.

Building the complex. Nodes are the even-parity points of a periodic box of side

. Edges join

nodes at the twelve vectors

±e

± e

, with minimum-image wrapping. Octahedra are indexed by

their odd-parity centers, each with its six vertices at

±e

. The boundary matrices

∂

(node–edge)

and

∂

(edge–octahedron, through the three equatorial squares of the integer lift of Section 5.2) are

built as integer matrices. The check ∂

∂

= 0 is exact.

Spectra. The Hodge Laplacians

(1)

are diagonalized densely (

numpy.linalg.eigh

) at

= 4 (192

edges) and

= 6 (648 edges), and at

= 8

10 for the defected complex. The number of zero modes

is the number of eigenvalues below 10

−9

. The next eigenvalue is above 1

4 in every case, so the

count is clear. Integer ranks are conﬁrmed over GF(2) and by a unit-determinant minor.

Regge geometry from labels. For the defect computations every edge length is a variable. Each

octahedron is split into eight tetrahedra about its center, adding six auxiliary edges. The defected

void is split into four tetrahedra about the trapped node, adding the four spokes. The complex is

then a simplicial complex of tetrahedra. For each tetrahedron the six lengths ﬁx its shape. We place

its vertices in

: one at the origin, one on an axis, one in a plane, and the last from the remaining

distances. We then compute the six dihedral angles from the face normals. The deﬁcit of each edge

is 2

minus the sum of the dihedral angles at that edge over all tetrahedra containing it, Eq.

(2)

On the ﬂat lattice every deﬁcit is zero to 10

−15

The 3D obstruction. For Section 6.2 the unknowns are all edge lengths of a periodic box. The

equations are

= 0 on every edge except the spokes and

target

on the spokes. We minimize the

sum of squared residuals with a trust-region least-squares method (

scipy.optimize.least squares

method

trf

) with a sparse ﬁnite-diﬀerence Jacobian; two edges interact only if they share a

tetrahedron. The same method reaches 10

−14

on the solvable systems of this paper; here the smallest

residual stays at 17–18% of the target.

The lapse variation. For the check in Section 6.3 we build a four-dimensional complex from a

solved slice: three periodic time steps, each prism split into four 4-simplices by the staircase rule

with a global order of nodes, so that neighboring prisms agree on shared faces. Each 4-simplex

is placed in

from its ten edge lengths through its Gram matrix, and its ten dihedral angles at

triangles are computed as for tetrahedra. The deﬁcit of a triangle is 2

minus the sum of its dihedral

angles. We take a central diﬀerence of

with step 10

−5

in one N

at a time.

Solving the constraint. For Sections 6.3–6.6 the unknowns are the conformal factors

at the

nodes, octahedron centers, and trapped node with

r < R

; nodes with

r ≥ R

are held at

= 1.

Edge lengths are

ℓ

(0)

. The residual at each free node is

e∋v

ℓ

−

πGm

, Eq.

(21)

We solve with the same least-squares method, with a sparse Jacobian whose pattern links two nodes

when an edge at one shares a tetrahedron with an edge at the other. We continue in

from the ﬂat

lattice and accept a solution when the largest residual is below 10

−10

; most reach a few times 10

−14

The far-ﬁeld mass comes from a linear least-squares ﬁt of

ψ −

1 =

over lattice nodes with

3 < r < R

− 2; the constant b

absorbs the eﬀect of the wall, and GM = 2a

The lapse. On a solved slice we build the cotangent Laplacian of the slice’s own geometry.

The weight of edge

ℓ

opp

cot θ

opp

, summed over the tetrahedra

that contain

the edge, where

ℓ

opp

is the length of the opposite edge and

opp

its dihedral angle. We solve

(

− N

) = 4

πGm

with

= 1 on the wall by a sparse direct solver and ﬁt 1

− N

/r + b

; then γ = 2a

The full matter action. We compute barycentric dual areas and volumes from the labels of every

tetrahedron that contains a defect node. In model M each spoke’s mass is

(

|⋆s|/|⋆s|

(0)

)

ℓ

. In model

F the 0-form energy

η µ

(

|⋆s|

)

(

|⋆s|

(0)

|⋆c|

) is added at the trapped node and

η µ |⋆s|

(

|⋆s|

(0)

|⋆A|

)

at each corner

. The nudge test multiplies

at one corner by 1

05 and solves again from there.

The lattice regulator. For Section 7 we average

ln Q

over

midpoint momenta in [

−π, π

)

which covers two Brillouin zones of

exactly;

= 16 and 20 agree to four digits. The bond

weights solve

−1

with the smallest change from the ﬂat weights. The Wilson term

(

det g

)

1/8

times the scalar operator.

is a central second diﬀerence with step 0

02; exact

derivatives give the same values. For the ﬁeld content, the noncovariant shear energy per site is

tr h

with

dof

(

K −

1 for bosons and fermions. With the Fierz–Pauli term

/64πG) tr h

and a site volume L

/2, m

= 128πκ

G/L

The spin-2 spectrum. For each ﬁeld we build the lattice operator at the origin for a background

: a site term and one hopping term per root, with bond quantities at the midpoint. The

fermion’s vierbein is

1/2

, and its spin connection uses the exact metric and its derivatives at

the midpoint, with

∂

−1/2

by central diﬀerences. First-order vertices for

±ip·x

come from waves

ϵH cos

and 2

ϵH sin

with

= 10

−3

; the second-order zero-momentum vertex averages four phases

of the wave, which removes the

parts. The second-order term of

ln det

is then the tadpole

minus the bubble, averaged over

momenta;

= 12 and 16 agree to four digits. As checks,

the spinor curvature from the spin connection is compared with the Riemann tensor, and the

momentum-space result is compared with exact real-space determinants on an 8

periodic lattice

with 8192

8192 fermion matrices. For the ﬁeld content, the loops give

= 128

πκ

(

)

G/L

with κ

(p) =

dof

σ(K(p) −

), and the Regge term gives 2ˆp

with ˆp

(1 − cos p ·e

The sprinkled ocean. Points are placed uniformly at random in a periodic box, at density

2 per coordinate volume, and move with the strain. Pairs within proper distance 3

couple

with weight

−d

/2σ

= 1. The scalar operator is the graph Laplacian; the fermion operator is

w γ ·

(

1/2

∆

)

/σ

between pairs plus a mass 0

3. We compute

ln det

by dense diagonalization,

take

and

by central diﬀerences with step 0

02, and average over sprinklings and two randomly

oriented shears each. The control uses the same operators on the D

points.

Entanglement equilibrium. For Section 6.5 each tetrahedron within the wall radius is unwrapped

to its minimum image, and its metric is the solution of the six equations

g e

ℓ

for its edges.

Points on a sphere (30000 Fibonacci directions) and inside it (32 Gauss–Legendre radii) are located

by barycentric coordinates. The area element is

√

det g

−1

and the volume element

√

det g

;

the ﬂat area and volume use the same quadrature, so its error cancels in

δA|

. The spoke integrals

use 2000 points per spoke.

The Delaunay complex. The periodic Delaunay triangulation of 625 points in a box of side

5 is built from copies of the points within 2

5 of the box. One copy of each simplex is kept,

and the simplex volumes sum to the box volume to machine precision. The operator is the

ﬁnite-element Laplacian

with the consistent mass matrix

, both from edge lengths through

each simplex’s Gram matrix, and

= [

ln det

(

µM

)

− ln det M

]

with

= 0

02, from sparse

LU factors. The circumcentric (Voronoi) Laplacian agrees with

in two dimensions but not on

general four-dimensional simplices, so we use K. The volume response is κ

= −0.249.

The sprinkled ocean’s

. On a periodic box of side 6 (gauge ﬁeld, fermion) or 8 (scalar) at

density 1, with Gaussian weights of width

= 0

55, 0

70, or 0

80 times the smooth cut (1

−d

)

= 3

in proper distance, we invert each operator once (dense, up to 5184 rows) and evaluate the

tadpole minus the bubble for transverse-traceless waves with

|p|

= 2

πn/L

|n|

≤

4. The

slope of

a linear ﬁt is 1

πG

; quadratic ﬁts are also reported. The geodesic distance is the straight-segment

integral of the metric plus its second-order correction, obtained from the Green function of the

path’s deviation. Evaluated with ten-point Gauss quadrature, as in the runs, it is within 2% of

the energy drop along the corrected path; with forty points, within 0

2%. The fermion’s vierbein

is averaged over six sub-segments of the bond, and its spin transporter is the ordered product of

the six sub-segment transporters. The fermion has mass 0

1 and Wilson parameter 1, and the

scalar and vector have mass 0

02, all covariant. The cruder variants quoted in Section 7 use the

metric at the bond’s midpoint or the straight-segment integral, and a hard cut at 2

or 3

coordinate distance. The gauge ﬁeld’s Ricci tensor is computed from the metric by ﬁnite diﬀerences

and checked against

µν

. Scalar values are for side 8; at side 6 and 9 and over sprinklings the

hard-cut values agree to about 10% (geodesic) or within noise around zero (midpoint). Gauge and

fermion values are for side 6 and one sprinkling.

Proﬁles and footprint. The curvature proﬁle groups edges by the distance of their midpoints

from the void center and reports the largest

|δ

in each group. The footprint test uses the strain

ℓ

/ℓ

(0)2

−

1 on lattice edges of the small-

solution and the six localized zero modes of the void,

found as the null space of L

restricted to the void’s 42-edge neighborhood.

Data availability

All results reproduce from the Python scripts in the archive

ssmtheory entropy scripts.zip

the repository github.com/raghu91302/ssmtheory. They need only NumPy, SciPy, and Matplotlib.



Section 3, Figs. 1 and 2: entropy fig ssm.py (schematics).



Section 5.2:

entropy fcc real harmonic.py

and

entropy void localization.py

;

ﬁgures and geometric completion:

entropy make figures.py

and

entropy complex completion.py.



Section 5.3: entropy action backreaction.py.



Section 4.2: entropy induced action.py (induced coeﬃcients and the mode count).



Section 7:

entropy d4 regulator.py

(shear response and spin-2 mass) and

entropy d4 regulator checks.py

(other lattice actions, ideal cutoﬀ); spin-2 spec-

trum:

entropy spin2 spectrum.py

, with checks in

entropy spin2 checks.py

;

sprinkled ocean:

entropy ocean sprinkling.py

; its Newton constant:

entropy ocean G scalar.py

entropy ocean G fermion.py

entropy ocean G gauge.py

;

geodesic-distance check:

entropy ocean geodesic check.py

; Delaunay

complex:

entropy ocean delaunay shear.py

; squared-operator fermion:

entropy ocean G spinor.py

; an independent implementation of the

uniform-strain and

spin-2 computations, sharing no code with the scripts above: folder independent spin2.



Section 6.1:

entropy defect deficit.py

. The complex and the deﬁcit routine used by all

defect scripts are in entropy defect intrinsic.py.



Section 6.2: entropy 3d obstruction.py.



Section 6.3, lapse variation of the four-dimensional action: entropy lapse variation.py.



Sections 6.3 and 6.4, Table 5, Fig. 5:

entropy id solve.py

(constraint solver, far-

ﬁeld ﬁt),

entropy id lapse.py

(lapse and

entropy id profile.py

(curvature proﬁle),

entropy fig initialdata.py and entropy fig defect.py (Figs. 4 and 5).



Section 9, Tables 6–8:

entropy routeC bond coupling.py

(bond model) and

entropy routeC field cut.py (scalar ﬁeld across a plane).



Section 6.5: entropy routeC equilibrium.py.



Section 6.6: entropy id fullmatter.py.



Section 6.8:

entropy id footprint.py

(overlap and the rank 38 at

= 4); hybridization:

entropy defect hybridization.py.

The constraint solves take seconds per source value at

= 16 and 20 after a setup of 40 seconds

(

= 16) to 3 minutes (

= 20). All random numbers (gradient-check vectors, random lifts,

sprinklings) come from ﬁxed seeds.

Declarations

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

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