Gravity from Stabilizer Defects on the FCC Lattice: 2+1D Conical Singularities & 3+1D Regge Curvature

Gravity from Stabilizer Defects on the FCC Lattice:

Conical Singularities in 2+1D and Regge Curvature

in 3+1D

Raghu Kulkarni

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

raghu@idrive.com

Abstract

We show that stabilizer defects on the FCC lattice produce Regge curvature in

both 2+1 and 3+1 dimensions. In 2+1D, removing one plaquette from a triad

sheet (

K = 4

rotated square lattice) creates a conical singularity with decit angle

δ = π/2

, matching the gravitational eld of a point mass

M = 1/(16G)

. Entropy

matching (

∆S = ln 2

per removed stabilizer equated to the Bekenstein-Hawking

formula) determines

G/a

= 1/(8 ln 2)

. Numerical verication conrms the conical

metric is exact at every radius. In 3+1D, the FCC lattice tiles space with regular

tetrahedra and regular octahedra; every edge has

2 arccos(1/3) + 2 arccos(−1/3) =

2π

, giving zero decit (at space). Removing one octahedral void (

-stabilizer)

produces decit angle

arccos(−1/3) ≈ 109.47

◦

at 12 edges spanning all three triad

sheets, coupling the gravitational degrees of freedom in all spatial directions. The

construction produces Regge curvature from code deformations in both 2+1D (exact,

topological) and 3+1D (veried at

L = 4

1 Introduction

In 2+1-dimensional general relativity, gravity is purely topological: there are no local

degrees of freedom, no gravitational waves, and the only eect of a point mass is to create

a conical singularity in the surrounding at space [1, 2]. A mass

removes a wedge of

angle

δ = 8πGM

from the plane, producing a cone.

The FCC triad sheet code

[[L

, 2L, L]]

[11] is a CSS stabilizer code dened on a

K = 4

rotated square lattice, with

independent layers per sheet. Each layer is a 2D toric code

[[L

, 2, L]]

[4]. The lattice is at: four plaquettes meet at each vertex, each subtending

π/2

, summing to

2π

We show that removing one plaquette (one stabilizer) at a vertex creates a conical singular-

ity with decit angle

π/2

, precisely matching 2+1D gravity. The thermodynamic match-

ing between the code's entropy (

∆S = ln 2

per removed stabilizer) and the Bekenstein-

Hawking formula determines the ratio

G/a

= 1/(8 ln 2)

. In 3+1D, the full FCC cell

complex (regular tetrahedra and octahedra) has zero decit at every edge; removing one

octahedral void produces Regge curvature coupling all three spatial directions.

2 The Sheet Lattice as Flat Spacetime

2.1 Geometry

Each triad sheet of the FCC lattice consists of

layers [11]. Within a single layer, the

vertices form a rotated square lattice with edge vectors

(±1, ±1)

(in units of

a/2

, where

is the FCC cubic lattice constant). Each vertex has

K = 4

edges and is surrounded by

4 plaquettes.

The edge length is

d = a/

√

, and each plaquette is a rhombus with area

2.2 Flatness

At every vertex, the 4 plaquettes contribute

π/2

each, giving total angle

θ = 2π

. In

Regge calculus [3], the decit angle is:

= 2π −

plaquettes

θ = 2π − 4 ×

= 0

(1)

The lattice is at. The vacuum Einstein equation

= 0

is satised at every vertex.

2.3 QEC structure

Each layer carries a 2D toric code

[[L

, 2, L]]

[4]. The

-stabilizers are the plaquette

checks (weight 4); the

-stabilizers are the vertex checks (weight 4). The code distance is

d = L

, set by the shortest non-contractible cycle on the torus. The entanglement entropy

across any cut equals (edges cut)

×ln 2

minus stabilizer corrections [6, 7].

3 The Disclination Defect

In the toric code, removing a plaquette means ceasing to measure one

-stabilizer. This

is equivalent to introducing a puncture [4]: the stabilizer constraint is erased, freeing one

logical degree of freedom. Geometrically, the plaquette is absent from the lattice, creating

a disclination.

Proposition 1

(Conical singularity from stabilizer removal)

Removing one plaquette

(one

-stabilizer) at a vertex

in the sheet lattice produces:

1. Coordination reduction:

K = 4 → K = 3

2. Decit angle:

δ = π/2

3. Code parameter change:

∆k = +1

(one new logical qubit).

4. Entropy change:

∆S = ln 2

Proof.

(1) Removing one plaquette removes one edge and reduces the coordination at

from 4 to 3.

(2) The remaining 3 plaquettes each subtend

π/2

, giving total angle

3π/2

. The decit

δ = 2π − 3π/2 = π/2

(3) The removed plaquette was one

-stabilizer. Removing it frees one degree of freedom

from the code space. Computed exactly at

L = 4

∆k = +1

(4) Each logical qubit contributes

ln 2

to the entanglement entropy [6]. Therefore

∆S =

ln 2

In Regge calculus, a decit angle

δ > 0

at a vertex corresponds to positive curvature (the

vertex is the apex of a cone). The discrete Einstein equation for a point mass in 2+1D [1]

is:

δ = 8πGM

(2)

The defect mass is therefore:

M =

8πG

π/2

8πG

16G

(3)

Figure 1 illustrates the at lattice, the defect, and the resulting cone.

4 stabilizers

4 plaquettes

Flat:

= 4

= 2

3 stabilizers

1 removed

Defect:

= 3

Conical singularity

(2+1D gravity)

Figure 1: Left: at vertex with

K = 4

, four plaquettes, total angle

2π

. Center: disclina-

tion defect with

K = 3

, one plaquette removed, decit angle

δ = π/2

(red wedge). Right:

the resulting conical singularity, the gravitational eld of a point mass

M = 1/(16G)

2+1D.

4 The Gravitational Coupling

The defect creates both a geometric change (decit angle, plaquette area removed) and

an information change (one stabilizer removed,

∆S = ln 2

). Matching these determines

4.1 The matching condition

The Bekenstein-Hawking entropy formula [5] relates the entropy change to the area

change:

∆S =

∆A

(4)

This formula was derived for black hole horizons; its application to a single lattice pla-

quette is the central physical assumption of this paper (see Section 9). The removed

plaquette has area

plaq

= a

. The entropy change is

∆S = ln 2

. Substituting:

ln 2 =

(5)

Solving for

G =

8 ln 2

≈ 0.1803 a

(6)

4.2 The Planck scale

Setting

G = ℓ

(the denition of the Planck length in natural units) gives the lattice

spacing in Planck units:

ℓ

√

8 ln 2 ≈ 2.355

(7)

This is a unit conversion, not an independent prediction: it expresses Eq. (6) in Planck

units.

4.3 The mass quantum

From Eq. (3) and (6):

M =

16G

8 ln 2

16 a

ln 2

(8)

In 2+1D, mass has dimensions of inverse length (in natural units

c = ℏ = 1

). The

minimum mass is set by a single removed plaquette. Masses are quantized:

removed

plaquettes give

= N ln 2/(2a

)

and decit

= Nπ/2

, valid for

< 2π

(i.e.,

N ≤ 3

;

removing all 4 plaquettes annihilates the vertex).

5 Multiple Defects and Superposition

In 2+1D Regge calculus, the metric between two conical singularities is at [2]. The total

decit angle is the sum of individual decits. For

defects at vertices

, . . . , v

total

i=1

Nπ

(9)

Each defect removes one stabilizer, adding

ln 2

to the total entropy. The total code

modication is

∆k = N

∆S = N ln 2

For well-separated defects, the gravitational potential between two masses separated by

distance

on the 2D sheet is logarithmic [1]:

Φ(r) = −4GM ln(r/r

)

. This is the exact

2+1D Newtonian gravity.

6 Gravitational Collapse

The Bekenstein-Hawking bound constrains the total number of independent degrees of

freedom in a region of area

[5]:

total

≤

4G ln 2

(10)

(see also [12] for a detailed analysis on the FCC lattice). For one sheet at lattice size

the at code has

= 2L

logical qubits, and the sheet occupies area

A = 2L

(two faces

of the cubic domain). Each defect adds

∆k = 1

. With

defects:

+ N ≤

4G ln 2

(11)

Substituting

G = a

/(8 ln 2)

2L + N ≤ 4L

(12)

The maximum number of defects per sheet before collapse is:

max

= 4L

− 2L

(13)

L = 4

max

= 56

. At

L = 10

max

= 380

When

N > N

max

, the information content exceeds the holographic bound. The stabilizer

code can no longer maintain error correction. In gravitational language, the total decit

angle exceeds

2π

per fundamental domain, and the geometry closes  a discrete analog

of black hole formation.

Figure 2 summarizes the QEC-gravity matching and the collapse criterion.

7 Extension to 3+1 Dimensions

The per-sheet 2+1D construction of Sections 26 treats the three triad sheets as in-

dependent. Three independent 2D conical metrics produce the correct Ricci scalar

(

R = R

+ R

for a diagonal metric) but the wrong Newtonian potential (

ln r

per sheet rather than

1/r

). The correct 3D gravitational eld requires the full FCC cell

complex, where all three sheets are coupled through the octahedral void structure.

7.1 The FCC cell complex

The FCC lattice tiles three-dimensional space with two types of cells: regular tetrahedra

at tetrahedral void positions and regular octahedra at octahedral void positions. At lattice

QEC

Remove 1 stabilizer

= +1

= ln 2

Remove 1 plaquette (

/2)

= /2

ln 2 =

8ln 2

2.35

Thermodynamic Matching

2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0

Lattice size

250

500

750

1000

1250

1500

1750

2000

Max defects before collapse

Holographic Collapse Criterion (per sheet)

max

defects per sheet

Stable (flat spacetime)

Collapse (black hole)

Figure 2: Left: the thermodynamic matching between QEC (stabilizer removal,

∆S =

ln 2

) and GR (plaquette removal,

∆S

= a

/(8G)

), determining

G = a

/(8 ln 2)

. Right:

maximum number of defects before holographic collapse, as a function of lattice size

size

L = 4

(periodic), the complex consists of 32 vertices, 192 edges, 64 tetrahedra, and

32 octahedra. The known dihedral angles are:

tet

= arccos(1/3) ≈ 70.53

◦

, θ

oct

= arccos(−1/3) ≈ 109.47

◦

(14)

7.2 Flat-space verication

Every FCC edge is shared by exactly 2 tetrahedra and 2 octahedra. The dihedral angle

sum is:

2θ

tet

+ 2θ

oct

= 2 arccos(1/3) + 2 arccos(−1/3) = 2π

(15)

since

arccos(1/3) + arccos(−1/3) = π

. The decit angle

= 2π −

θ = 0

at every

edge. The 3D Regge action vanishes:

S = (1/8πG)

ℓ

= 0

. The at-space vacuum

Einstein equation is satised on the full FCC cell complex. This is veried numerically

L = 4

: all 192 edges have

= 0

to machine precision.

7.3 The 3D defect: octahedral void removal

The

-stabilizers of the FCC code are octahedral void checks [10]. Removing one

stabilizer means ceasing to measure one octahedral void check, erasing the constraint

and freeing one logical degree of freedom  the 3D analog of the puncture described in

Section 3. In the Regge picture, this is equivalent to removing the octahedron's dihedral

contributions. The 12 edges of the removed octahedron each lose one dihedral contribution

oct

= θ

oct

= arccos(−1/3) ≈ 109.47

◦

(

12 aected edges

)

(16)

All other edges retain

= 0

The 12 aected edges are distributed equally across the three triad sheets: 4 edges in

4 in

, 4 in

. The defect couples all three sheets through the octahedral geometry.

This is the 3D analog of the 2D conical singularity: curvature localized on codimension-2

hinges (edges in 3D, vertices in 2D).

The 3D Regge action of a single defect is:

defect

8πG

e∈oct

ℓ

12 (a/

√

2) arccos(−1/3)

8πG

(17)

The value of

in Eq. (17) is taken from the 2D entropy matching (Eq. 6). This is

the fundamental determination: the holographic capacity of the code is set by the 2D

sheet structure, where each plaquette carries exactly one stabilizer constraint and

ln 2

entanglement entropy. The octahedral void is a composite geometric object spanning all

three sheets; its 12-edge decit is the sum of contributions from three independent sheet

sectors, each governed by the same

7.4 Comparison of 2D and 3D defects

Per-sheet (2+1D) Full cell complex (3+1D)

Defect Remove plaquette Remove octahedron

Curvature locus Vertex (0D) Edges (1D)

Decit angle

π/2

per vertex

arccos(−1/3)

per edge

Sheets aected 1 All 3 (4 edges each)

Gravitational potential

ln r

(2D)

1/r

(continuum limit)

The per-sheet construction gives the correct curvature (Ricci scalar) but the wrong po-

tential because the sheets do not communicate. The full 3D construction couples all three

sheets through the octahedral void. In the continuum limit, this coupling is expected to

produce the correct

1/r

fallo, because the 3D Laplacian acts on all three spatial direc-

tions through the 12-edge octahedral structure; explicit computation of the lattice Green's

function at nite spacing remains open (Section 9).

8 Numerical Verication

We verify the 2+1D conical metric on a nite lattice patch (

N = 10

, 145 vertices, 240

triangles on the rotated square lattice). Removing one plaquette (2 adjacent triangles) at

the origin gives:

Quantity Computed Expected Error

at defect 1.5708

π/2 = 1.5708 < 10

−6

at all other interior vertices 0 0 exact

C(r)/C

ﬂat

(r)

r = 1

0.7500

3/4

exact

C(r)/C

ﬂat

(r)

r = 2

0.7500

3/4

exact

C(r)/C

ﬂat

(r)

r = 4

0.7500

3/4

exact

The circumference ratio

C/C

ﬂat

= α = 1 −δ/(2π) = 3/4

is exact at every radius, conrm-

ing that the discrete lattice with one removed plaquette is an exact conical metric. The

uniform-edge-length solution automatically satises the Regge equations: no numerical

optimization is required. This is the lattice realization of the fact that 2+1D gravity is

topological  the conical metric is the unique solution for a point mass.

An edge-length perturbation test conrms locality: perturbing one edge by

ϵ = 0.001

produces decit changes only at the two vertices adjacent to that edge (

∆δ = ±0.0014

with all other decits unchanged to machine precision.

9 Assumptions and Scope

1. Regge calculus.

The identication of stabilizer removal with a Regge decit angle is

exact for a lattice with square plaquettes. This is not an approximation.

2. The

k ↔ S

identication.

The entropy

∆S = ln 2

per removed stabilizer is exact

for stabilizer codes [6]. The identication of this with Bekenstein-Hawking entropy is the

physical assumption; it is standard in the QEC-spacetime literature [8, 9] but not derived

from rst principles.

3. Scope.

The 2+1D construction (conical singularities, per-sheet) is exact and veried

numerically. The 3+1D construction (octahedral void removal on the full cell complex)

produces the correct decit angles and couples all three sheets, veried at

L = 4

. What

remains open is the explicit computation of the lattice Green's function to conrm the

1/r

potential fallo at nite lattice spacing, and time evolution (the 4D Regge action).

The quantized mass spectrum (

= N/(16G)

N = 1, 2, 3

in 2+1D) is a prediction of

the discrete framework.

10 Conclusion

Stabilizer defects on the FCC lattice produce Regge curvature at both 2+1 and 3+1

dimensions. In 2+1D, removing one plaquette from a triad sheet creates a conical singu-

larity with

δ = π/2

, matching the gravitational eld of a point mass. Entropy matching

determines

G/a

= 1/(8 ln 2)

. The conical metric is exact:

C/C

ﬂat

= 3/4

at every radius

without optimization.

In 3+1D, the FCC lattice tiles space with regular tetrahedra and octahedra, with zero

decit at every edge (at space). Removing one octahedral void  one

-stabilizer 

creates decit

arccos(−1/3) ≈ 109.47

◦

at 12 edges spanning all three triad sheets. The

curvature lives on codimension-2 hinges (edges), as required by 3D Regge calculus, and

couples the gravitational degrees of freedom in all three spatial directions through the

octahedral geometry.

The construction maps the QEC vocabulary directly onto Regge gravity: a stabilizer is

a cell, removing it creates curvature, and the code's entropy determines the gravitational

coupling.

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