Electromagnetism on the cubic sublattice of the FCC vacuum

Electromagnetism on the cubic sublattice of the FCC vacuum:

the correct photon polarization count, and

masslessness from centrosymmetry

Raghu Kulkarni

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

raghu@idrive.com

Abstract

A vacuum modeled as a face-centered-cubic lattice carrying a quantum error-correcting code,

with one qubit per nearest-neighbor bond, must nd room for a gapless photon alongside the

code and the gravitational sector. The nearest-neighbor

⟨110⟩

network cannot supply one. Its

one-form sector has only one propagating polarization per site, and it is non-bipartite, so a

compact

U(1)

theory on it is gapped.

A simpler structure in the same packing does the job. The FCC nodes and the octahedral

voids together form a simple cubic lattice, so the sector is Wilson's compact

U(1)

unmodied. A

one-form gauge eld on its

⟨100⟩

links has exactly one pure-gauge and two transverse modes per

site: the Maxwell count, and a property of the complex needing no dynamical assumption. The

sublattice is bipartite, so a Coulomb phase exists; its links are disjoint from the code's qubits,

so the two algebras commute trivially; and its bond star is centrosymmetric, so no piezoelectric

coupling exists at all, which protects masslessness without a gauge-invariance argument.

Matter couples to the sector without any added interaction, since a

⟨110⟩

hop is the com-

position of two

⟨100⟩

hops through a void, and the ambiguity between the two covering paths

is the ux through a cubic plaquette. Only this sector moves: non-bipartiteness is fatal for the

photon, neutral for the weak sector, and an asset for color connement and fermion doubling.

Putting qubits rather than a gauge eld on the same links gives the three-dimensional toric

code, which is the same gauge theory at

Z

2

rather than

U(1)

. The packing then carries two

CSS codes on identical qubit counts at opposite ends of the ratedistance trade-o. The code's

logical sector and the photon's zero modes are one topological invariant read over two coecient

groups. The construction also admits

α = 1/137

, which the pyrochlore route does not, though

the coupling remains a free parameter.

1 The setting

This section is self-contained. Readers who know Refs. [1, 3, 2, 5] can skip it.

The framework.

This paper contributes to a programme that models the vacuum as a face-

centered-cubic lattice carrying a quantum error-correcting code, and reads physical quantities o

its combinatorics. Reference [1] takes the rest mass of a particle to be the thermodynamic cost of

the syndrome measurements that keep it veried, and obtains the electron, muon, pion, proton and

neutron mass ratios from integer counts. Reference [3] builds the defects themselves from a single

extra node trapped in a tetrahedral void of the packing. Quark charges follow from the tetrahedral

bond angle. The eight triangular and six square faces of the coordination shell are assigned to

conning and screening sectors by whether they are odd or even cycles. Reference [5] constructs

1

the code. That third paper is the one this one uses; the rst two are cited so a reader knows what

the lattice is for.

The vacuum as a code.

The construction this paper uses is Ref. [5]. One qubit sits on each

nearest-neighbor bond. The stabilizer checks are uniform and of weight twelve: a

Z

-check on

the twelve bonds meeting at each lattice site, and an

X

-check on the twelve bonds bounding each

octahedral void. On an

L×L×L

periodic patch this is a

[[3L

3

, 2L

3

+2, 3]]

CSS code, so

[[192, 130, 3]]

at

L = 4

, with a rate above two thirds [5].

Particles and gravity on the same bonds.

Particles are topological defects of that lattice:

localized congurations the checks cannot remove.

The gravitational sector uses the same bonds for a dierent purpose. Reference [2] treats the

lattice as an intrinsic Regge geometry whose dynamical data are the squared edge lengths, with

the graviton identied as the incompatible part of the edge-length eld under the Saint-Venant

decomposition. Two points from that treatment matter here. The edge lengths are not code

degrees of freedom, so the code and the gravitational sector occupy the same bonds without sharing

variables. And the gravitational statements are four-dimensional: they hold on

D

4

, whose constant-

time slice is the FCC lattice, and the slice alone fails the rank-four isotropy that

D

4

satises exactly.

What this paper needs from that.

Only two things. The qubits are on

⟨110⟩

bonds, and that

network is close-packed, so each site has twelve neighbors and its coordination shell is a cuboctahe-

dron.

Nothing below uses the mass spectrum of Ref. [1], the defect construction of Ref. [3], or the

gravitational sector of Ref. [2]. The results are properties of a lattice complex, so a reader who does

not accept the wider programme can still assess them.

2 The problem, and what changes

A vacuum built this way has to accommodate three things on one crystal: the code, the gravitational

sector, and electromagnetism. The rst two live on the

⟨110⟩

graph. The third does not, and the

reason is exact.

Why

⟨110⟩

fails, twice over.

There are two independent obstructions, and they need dierent

amounts of input.

The rst is a counting statement about the complex and assumes no dynamics. If the gauge

potential is a one-cochain and the eld strength is its coboundary, then the physical modes are the

coexact part of the one-form sector. On the code complex that part has dimension one per site, not

two (Section 3). A eld with one polarization is not a photon, whatever its equation of motion.

The second is the standard Coulomb-phase argument and does assume a compact gauge group.

The nearest-neighbor network is non-bipartite, since the close-packed shell contains triangular odd

cycles. A gapless compact

U(1)

photon exists only in a deconned Coulomb phase, and that phase

requires a bipartite charge lattice admitting a height representation [9, 10]; on a non-bipartite lattice

the compact theory is gapped or conned. A

non-compact

theory would evade it, so this argument

is not assumption-free. The counting obstruction is.

2

(a) ⟨110⟩ nearest neighbours

K = 12, non-bipartite: no photon

(b) ⟨111⟩ diamond

K = 4, bipartite: spin-ice photon

(c) ⟨100⟩ simple cubic

K = 6, bipartite: Maxwell

odd cycle propagation path

Figure 1: Three structures in one FCC packing, drawn on equal axes.

(a)

The

⟨110⟩

nearest-

neighbor network, which carries the code and the gravitational sector. Its coordination shell is a

cuboctahedron and contains triangular odd cycles (one shaded), so the graph is non-bipartite and a

compact

U(1)

theory on it is gapped.

(b)

The

⟨111⟩

bonds to the tetrahedral voids form a bipartite

diamond lattice of coordination four, the route through

U(1)

quantum spin ice.

(c)

The

⟨100⟩

bonds to the octahedral voids. Nodes (circles) and voids (squares) together are the simple cubic

lattice of spacing

a/2

, bipartite by parity, with the Maxwell degree-of-freedom count of Section 3.

The highlighted path shows how the eld propagates. It alternates between the two site types and

never breaks, because every even site has an odd neighbor one step along each axis. The photon

travels on the link network, not from node to node, and that network is connected: at

L = 6

every

one of the 216 sites is reachable from any other. The three structures share sites; (a) and (c) share

no bonds.

What the square faces do and do not give.

Reference [3] assigns electromagnetism to the

six square faces of the coordination shell, on the grounds that they are even cycles and so screen

rather than conne. That distinction is correct and we use it below. It identies a sector, not a

propagating eld: the squares are faces of the

⟨110⟩

complex, and Section 3 shows that complex has

one physical polarization per site, not two. The screening assignment survives; the photon has to

be found elsewhere.

The obvious alternative, and a third option.

The

⟨111⟩

bonds joining each FCC site to its

tetrahedral voids form a bipartite diamond lattice whose bond midpoints are the pyrochlore, and a

photon emerges there as the gauge eld of

U(1)

quantum spin ice [7, 8]. That route works, but it is

not the only one, and Section 3 shows it is not the one with the simplest degree-of-freedom count.

The cubic sublattice.

The FCC nodes are the even-parity sites of the cubic grid and the octa-

hedral voids are the odd-parity sites (Figure 1). Together they are the full simple cubic lattice of

spacing

a/2

, veried at two sizes in Section 3. Its

⟨100⟩

links join each node to an adjacent void.

Section 3 shows that a one-form gauge eld on those links has the Maxwell degrees of freedom,

which the code complex does not. That is a statement about the complex and holds for any such

eld. Compactness enters only in the phase argument and the coupling estimate, and is agged

where it is used; with it, the theory on these links is ordinary lattice quantum electrodynamics, and

Section 8 separates what that inherits from the standard treatment from what is new here.

3

0 1 2 3 4 5 6

one-form degrees of freedom per site

Maxwell

(continuum)

⟨100⟩ cubic

(this paper)

FCC code complex

(edge qubits)

3 dof

3 dof

6 dof

pure gauge transverse (physical) harmonic / flat

Figure 2: One-form degrees of freedom per site, split into pure gauge, physical transverse, and

harmonic. The continuum Maxwell eld has one of the rst and two of the second. The

⟨100⟩

cubic

sublattice reproduces that count exactly (

0.995

and

1.991

at

L = 6

, approaching one and two). The

FCC code complex, whose two-cells are the octahedral

X

-checks, has six edge degrees of freedom

per node but only one physical polarization, the remainder falling into a at band.

3 The polarization count

The cubic sublattice is simple cubic and bipartite.

Direct enumeration on a periodic patch:

FCC nodes octahedral voids union

L = 4

32 32 64

= L

3

L = 6

108 108 216

= L

3

The union is the full cubic grid, and parity two-colors it, so the graph is bipartite and carries no

odd cycles. The obstruction that rules out

⟨110⟩

therefore does not apply here.

Maxwell's degrees of freedom.

Write

d

1

for the signed incidence of links on sites and

d

2

for that

of square plaquettes on links;

d

1

d

2

= 0

exactly. The one-form sector splits as

C

1

= im d

1

⊕im d

2

⊕H

1

,

the three parts being pure gauge, physical (transverse), and harmonic.

L

sites links exact/site coexact/site harmonic

4 64 192 0.984 1.969 3

6 216 648 0.995 1.991 3

Per site this is one gauge mode and

two

transverse polarizations (Figure 2), with the harmonic

sector exactly the three winding classes of

H

1

(T

3

)

. That is the Maxwell eld.

The code complex does not have this.

The one-form sector of the

[[3L

3

, 2L

3

+2, 3]]

code

complex, whose two-cells are the octahedral

X

-checks, splits per node as approximately one exact,

4

one coexact and four harmonic out of six edge degrees of freedom. Only

one

propagating polarization

survives. The reason is structural rather than accidental: the complex has

L

3

/2

nodes and

L

3

/2

octahedral cells [5], and these are equinumerous, so

im d

1

and

im d

2

have equal dimension. A one-

form on the code complex is therefore not a photon, whatever its form degree suggests. The code's

edge qubits are the natural rst guess for where electromagnetism lives, and they do not work.

4 Coexistence with the code

Disjoint qubits.

A

⟨110⟩

code edge joins two even-parity sites; a

⟨100⟩

gauge link joins an even

site to an odd one. At

L = 4

there are 192 of each and they share none.

Two things follow, and they should be separated. The gauge eld is carried by new variables. The

published construction places one qubit on each

⟨110⟩

edge [5]. The octahedral voids are interstitial

positions where

X

-checks are dened, with no degrees of freedom of their own. A compact phase on

each

⟨100⟩

link is therefore an addition to the Hilbert space, not a reuse of code qubits. The same

is true of the pyrochlore route, which places a rotor on each pyrochlore site; neither construction is

carried by the code, and both add a gauge sector to it.

Given that, the disjointness is what makes the addition safe: the two operator algebras act on

dierent variables and commute trivially, so no stabilizer is violated by a gauge transformation and

no gauge transformation moves a code qubit.

Masslessness from centrosymmetry.

The two sectors do share every site, so a node displace-

ment perturbs both its

⟨110⟩

bonds and its

⟨100⟩

links, and one must ask whether integrating out

the edge-length uctuations generates a photon mass. On

⟨111⟩

the answer required an argument:

the tetrahedral star is non-centrosymmetric, its rank-three moment is the zincblende

d

14

, and the

induced coupling is the gauge-invariant

ε

jk

E

i

rather than

ε

jk

A

i

, so gauge invariance forbids the

mass.

On

⟨100⟩

no argument is needed. The six-direction star is centrosymmetric (

O

h

), and its rank-

three moment vanishes identically,

max

ijk

|d

ijk

| = 0

by direct summation. There is no piezoelectric

channel at all: an edge-length perturbation cannot source the gauge potential at rank three, so the

leading coupling is rank four and quadratic in gauge-invariant objects. The mass term

A

µ

A

µ

is

forbidden a fortiori.

Two conditions the geometry does not supply.

Everything above is a property of the com-

plex. Two further requirements are conditions on the dynamics.

First: the two sublattices must be equivalent.

The links alternate between FCC nodes and octahedral voids, and those are physically dierent

objects: the nodes carry the code's qubits, the voids carry its

X

-checks. The gauge eld is indierent

to that, since it lives on links and the sites enter only through Gauss's law. But Gauss's law must

then be the

same

at both. On a uniform cubic lattice the dispersion is

ω

2

(k) =

X

µ

4 sin

2

(k

µ

a/4),

which vanishes as

k → 0

: the photon is massless. A staggered site term, diering between the

two parities, mixes

k

with

k + π

and opens a gap

ω

2

(0) ∼ ∆

2

; a ten percent asymmetry gives

ω

2

(0) = 0.01

in lattice units. Masslessness therefore requires the two sublattices to carry the same

constraint.

5

This is the sharpest form of the question of where the gauge variables come from: whatever

supplies them must supply them symmetrically across the two parities.

Second: the photon and matter cones must coincide.

Both are isotropic at leading order. The

photon's is isotropic to

O((ka)

2

)

in every direction, with

ω

2

/k

2

equal to

0.9992

,

0.9996

and

0.9997

along the axis, the face diagonal and the body diagonal at

ka = 0.1

; the matter cone is isotropic

because the

⟨110⟩

bond tensor is

8δ

µν

. But their

speeds

come from dierent couplings, the photon's

from the gauge action and matter's from the hopping amplitude on

⟨110⟩

, which Ref. [4] xes at

c = 4v

lat

from the kissing number. Nothing here ties the two. Equality is a condition, not a

consequence.

It is a weaker condition than the pyrochlore route faces. There the dielectric renormalization

gives

c

em

= c

0

/

√

1 + δε

with

δε > 0

, so the photon is

systematically

slower than the bare cone by an

amount set by a free coupling, and a two-cone structure is predicted rather than merely permitted.

On the cubic sublattice there is no dielectric background and no systematic shift; the two speeds

are simply independent until something relates them.

Anisotropy.

The rank-two moment is isotropic,

S

µν

= 2δ

µν

. The rank-four moment is not:

T

xxxx

= 2

and

T

xxyy

= 0

, whereas isotropy requires

T

xxxx

= 3T

xxyy

. The photon dispersion

therefore carries a cubic-anisotropic correction of relative order

(ka)

2

, far below current bounds on

Lorentz violation. It is present and should not be described as absent; Section 8 places it beside

the corresponding artifact in lattice QCD.

5 How matter couples to the sector

A gauge eld is only useful if charges can radiate into it. The defects live on the

⟨110⟩

network and

the eld lives on

⟨100⟩

links, which share no bonds, so the coupling looks as though it has to be

supplied by hand. It does not. The geometry forces it.

A

⟨110⟩

hop is two

⟨100⟩

hops.

The step

(1, 1, 0)

is the composition of

(1, 0, 0)

and

(0, 1, 0)

, and

the intermediate site has odd parity, so it is an octahedral void. This holds for every edge: enumer-

ating 2400 of them, each is covered by exactly two two-step

⟨100⟩

paths, and every intermediate is

a void, with no exceptions. A defect moving along a

⟨110⟩

edge therefore passes through a void and

accumulates the product of two

⟨100⟩

link phases. That product is the coupling.

The path ambiguity is the eld strength.

The two covering paths of the edge from

(0, 0, 0)

to

(1, 1, 0)

run through

(1, 0, 0)

and through

(0, 1, 0)

. Traversing one and reversing the other gives

the closed loop

(0, 0, 0) → (0, 1, 0) → (1, 1, 0) → (1, 0, 0) → (0, 0, 0),

a unit square of the cubic sublattice: a plaquette. The induced

⟨110⟩

phase is therefore well dened

only up to the ux through that plaquette. That is not a defect of the construction. It is what a

gauge coupling is, and the ambiguity is the eld strength the defect radiates into.

This removes the pyrochlore route's one advantage over this one. There the ice rule forces the

matter coupling because defects sit on the charge lattice; here it is forced because a close-packed

hop cannot avoid passing through a gauge site.

6

The same construction in four dimensions.

Nothing above is special to the spatial slice. The

D

4

lattice is the even-sum subset of

Z

4

, and adding its odd-sum coset gives

Z

4

itself:

128+128 = 256

at

L = 4

. The joining links are the eight axis vectors, four per site, which is the hypercubic lattice

on which ordinary lattice gauge theory is done. And the coupling argument lifts verbatim: every one

of the 24

D

4

bonds splits into two axis steps in exactly two ways, through an interstitial site, with

the two routes diering by a hypercubic plaquette. The three-dimensional statement is therefore

the spatial section of a four-dimensional one, not an accident of the slice.

6 Why only electromagnetism moves

The cubic sublattice is convenient enough to raise an obvious objection: if ordinary lattice technique

applies there, why not put every sector on it? The answer is that non-bipartiteness is a liability

in exactly one place. Section 7 records separately what the same links carry if one puts qubits on

them instead of a gauge eld.

sector structure why status

code

⟨110⟩

the qubits are dened there settled

gravity

⟨110⟩

the graviton is the incompatible edge-length mode [2] settled

matter defects

⟨110⟩

voids the trapped node sits in a void settled

electromagnetism

⟨100⟩

needs a Coulomb phase and two polarizations this work

color

⟨110⟩

triangles odd cycles conne open dynamics

weak

⟨110⟩

squares three antipodal pairs give

su(2)

open, see below

fermions

⟨110⟩

non-bipartiteness suppresses doubling open

Two checks conrm that the other sectors gain nothing by moving.

First, chirality. The weak sector's handedness is statically obstructed by centrosymmetry. The

⟨100⟩

star is centrosymmetric too: its rank-three moment vanishes identically, as the

⟨110⟩

star's

does. So the obstruction travels with the sector. Only the

⟨111⟩

diamond is non-centrosymmetric,

with rank-three moment

4/3

√

3 = 0.770

. It is the one structure here that can carry a handedness.

Whether that makes it the home of a chiral sector is not settled. We record the geometric property

and claim nothing about the weak interaction.

Second, doubling. The NielsenNinomiya theorem forces the zeros of a local Hermitian lattice

Dirac operator to come in pairs, and on a bipartite lattice they sit at the zone corners,

2

d

of them,

sixteen in four dimensions. That is why lattice QCD needs Wilson, staggered, domain wall or

overlap fermions. Non-bipartiteness is what Creutz-type constructions exploit to reduce the count,

so moving fermions to the cubic sublattice would reinstate the standard problem. Section 8 takes

this up.

The principle is therefore narrow. Non-bipartiteness is fatal for the photon, neutral for the weak

sector, and an asset for color connement and for fermion doubling. Electromagnetism is the one

sector for which close packing cannot work, and the interstitial sublattice is the one place it can go.

7 The same sector, read as a code

Putting qubits on the

⟨100⟩

links rather than a gauge eld gives the standard three-dimensional

toric code. That is not a second, unrelated construction. The toric code

is

a

Z

2

gauge theory on

those same links: the vertex check is the

Z

2

Gauss law and the plaquette check the

Z

2

magnetic

7

term. Compact

U(1)

Higgsed by a charge-two condensate breaks to

Z

2

, and the deconned phase

of the resulting theory is the toric code phase. The photon sector and the code sector are one gauge

theory on one complex at two points of its phase diagram, diering in gauge group rather than in

substrate.

Read as a code, it gives the FCC packing two CSS codes on the same number of qubits:

code

n k d

rate check weights

FCC

⟨110⟩

,

L = 4

192 130 3 67.7 % 12, 12

cubic

⟨100⟩

,

L = 4

192 3 4 1.6 % 6, 4

FCC

⟨110⟩

,

L = 6

648 434 3 67.0 % 12, 12

cubic

⟨100⟩

,

L = 6

648 3 6 0.5 % 6, 4

The

k

values are computed here. The distances are not: three is the published value for the FCC

code [5], and

L

is the standard result for the three-dimensional toric code [6]. We verify only the

upper bound, exhibiting a straight winding line of weight

L

that commutes with every vertex check,

so

d ≤ L

.

The two sit at opposite ends of the ratedistance trade-o on the same substrate. The

⟨110⟩

code maximizes rate at xed distance three. The

⟨100⟩

code has distance growing with

L

at xed

k = 3

.

The phase-diagram reading explains something the counts would otherwise leave as a coinci-

dence. The cubic code has

k = 3

and the

⟨100⟩

gauge eld has a three-dimensional harmonic sector,

and these are the same three objects: the rst cohomology of one complex, taken over

F

2

in one

case and over

R

in the other. Both equal

b

1

(T

3

) = 3

, computed here at

L = 4

and

L = 6

. The

topological sector is xed by the complex and does not care which gauge group sits on it, so the

photon's zero modes and the code's logical qubits are one invariant read twice.

8 Relation to standard lattice gauge theory

The electromagnetic sector proposed here is Wilson's compact

U(1)

on a simple cubic lattice, un-

modied. Most of what follows from that is standard.

What transfers unchanged.

The gauge potential sits on links and the eld strength on square

plaquettes. The Hodge count of Section 3 is the textbook statement that a cubic lattice carries three

link variables per site, one pure gauge and two transverse. The connement transition at

β

c

≃ 1.01

,

the Coulomb phase above it, the relation

α = 1/4πβ

, and the requirement of a bipartite charge

lattice for a deconned phase are all standard [9, 10], as is the toric code obtained by putting qubits

on the same links [6]. So the whole apparatus of the eld applies without adaptation: Symanzik

improvement, the standard fermion formulations, scale setting, and the accumulated knowledge of

where compact

U(1)

does and does not have a photon.

Doubling, and what the close-packed network buys.

The Nielsen Ninomiya theorem for-

bids a local, Hermitian, translation-invariant lattice Dirac operator on a torus from having a single

Weyl zero: the zeros come in pairs. On a bipartite lattice they sit at the zone corners, where every

bond phase is

±1

, and there are

2

d

of them, sixteen in four dimensions. Lattice QCD lives with

this by construction, and the remedies are expensive: Wilson fermions give up chiral symmetry,

staggered fermions reduce sixteen to four, and domain wall and overlap fermions recover a modied

chiral symmetry at substantial computational cost.

8

That cost is incurred because the matter elds are on a hypercubic grid. The present ar-

rangement does not put them there. Matter stays on the close-packed

⟨110⟩

network, which is

non-bipartite, and non-bipartiteness is precisely the structural feature that Creutz-type and hy-

perdiamond constructions exploit to reduce the doubler count. Whether a satisfactory operator

exists on this particular network is a separate question and not settled here. What the arrangement

supplies is the geometry those constructions need, at no cost, and it supplies it for the same reason

the photon had to move elsewhere.

Anisotropy, in context.

The rank-four moment of the

⟨100⟩

star is cubic rather than isotropic,

so the photon dispersion carries a correction of relative order

(ka)

2

. This is the same class of artifact

as the hypercubic breaking in lattice QCD gluon and quark propagators, and it is handled the same

way: the Symanzik improvement programme removes

O(a

2

)

terms systematically by adding higher-

dimension operators to the action. Nothing here is worse than what a lattice practitioner routinely

corrects, and the correction is available o the shelf.

Two grids, and what that costs.

One feature departs from the standard setup. In lattice QCD

matter and gauge elds share a single grid, matter on sites and gauge on the links joining them,

which guarantees a common continuum cone. Here they do not share one: matter is on

⟨110⟩

and

the gauge eld on

⟨100⟩

, joined by the two-step decomposition of Section 5. Their propagation

speeds are therefore set by dierent couplings and are not automatically equal.

That is the trade. A single grid buys an automatic common speed limit and pays for it in

doubling and in a photon the close-packed network cannot host. Two grids buy a photon with the

right polarization count, masslessness protected by centrosymmetry, and a non-bipartite home for

fermions; the price is that the two cones must be matched rather than inherited. Which is preferable

is not something this paper settles, but the choice is structural and the cost is one condition, not a

family of them.

9 Scope

Established here.

Each of the following is a direct enumeration at

L = 4

and

L = 6

.



The FCC nodes and octahedral voids form a simple cubic lattice.



That lattice is bipartite.



Its one-form sector carries one gauge and two transverse modes per site.



The code complex's one-form sector carries only one.



The

⟨100⟩

links are disjoint from the code qubits.



The

⟨100⟩

star's rank-three moment vanishes, as the

⟨110⟩

star's does. Only the

⟨111⟩

diamond

is non-centrosymmetric.



α = 1/137

lies inside the Coulomb phase.



Qubits on the same links give the toric code,

[[192, 3, 4]]

against

[[192, 130, 3]]

. Its logical sector

and the gauge eld's harmonic sector are both

H

1

(T

3

)

, of dimension three over

F

2

and over

R

.



Every

⟨110⟩

edge is covered by exactly two two-step

⟨100⟩

paths, each through an octahedral

void, and the two dier by one cubic plaquette.



The same holds in four dimensions:

D

4

plus its odd-sum coset is

Z

4

, and each of the 24

D

4

bonds splits into two axis steps in exactly two ways.

9

The sector assignment of Section 6 follows from these properties and from nothing else.

Against the pyrochlore route, for the photon.

The

⟨111⟩

construction forces the matter

coupling through the ice rule, since a defect in a tetrahedral void sits on the charge lattice. That

looked like an advantage over this one until Section 5, which shows the cubic sublattice forces it

too, by a dierent route. Set against that, the pyrochlore needs an ice rule, a ring-exchange term

and a collective mechanism to produce a photon, where the cubic sublattice has the Maxwell count

outright. And its emergent coupling is bounded below, excluding the physical value of

α

.

On the pyrochlore the coupling goes as

p

U/g

and the Coulomb phase requires

g ≪ U

, so

α

is bounded

below

and

1/137

is not reachable while the photon exists. On the cubic sublattice the

theory is ordinary compact

U(1)

, with

α = 1/4πβ

. The Coulomb phase

β > β

c

≃ 1.01

bounds

α

above

at

1/12.7

, and the physical value corresponds to

β = 10.9

. The coupling is admitted, not

derived:

β

is an input here exactly as in ordinary lattice quantum electrodynamics.

On the photon the comparison therefore favors the cubic sublattice. That is a statement about

the photon only. The diamond's distinguishing property is that it is non-centrosymmetric, and

selecting one of the two tetrahedral sublattices is a binary choice between inversion partners. Those

are the features a parity-violating sector would need, and both the

⟨110⟩

and the

⟨100⟩

networks

lack them. The diamond may have a role in this packing; it is not the photon's.

Not established.

The following are open.



That the vacuum realizes a compact

U(1)

theory on these links.



The value of

β

.



Where the gauge variables on the

⟨100⟩

links come from, and whether whatever supplies them

treats the two sublattices alike. The photon is massless only if it does.



Whether the photon and matter cones have the same speed. Both are isotropic; their normal-

izations come from dierent couplings.



Any role for the

⟨111⟩

diamond beyond the negative one established here.



Any relation to an electroweak structure on the

⟨110⟩

square faces.

Data availability

Every number in this paper is produced by the scripts below, collected in

photon_scripts.zip

at

github.com/raghu91302/ssmtheory

. They require only

numpy

, apart from the gure generator

which also needs

matplotlib

, and each runs in seconds. Nothing is hard-coded: the lattices are

built from their dening rules and the ranks are computed rather than quoted.



photon_verify_100.py

: the cubic sublattice, its bipartiteness, the Hodge split, the disjointness

from the code, the moments of the

⟨100⟩

star, the dispersion and its isotropy, and the four-

dimensional lift.



photon_emission.py

: the two-step covering of every

⟨110⟩

edge and the plaquette relating the

two paths.



photon_two_codes.py

: the two CSS codes on the same qubit count, and the weight-

L

winding

operator bounding the distance.



photon_sector_table.py

: the structural properties behind the sector assignment.

10



photon_diamond_weak.py

: the rank-three moments, showing the

⟨111⟩

diamond is the only

non-centrosymmetric structure.



photon_make_figures.py

: both gures. It recomputes the geometry and the Hodge ranks rather

than drawing them by hand.

References

[1] R. Kulkarni,

The MassEnergyInformation Equivalence: a bottom-up identication of the par-

ticle spectrum via FCC lattice error correction

, Phys. Open

27

, 100414 (2026).

doi.org/10.

1016/j.physo.2026.100414

[2] R. Kulkarni,

Black holes in the FCC selectionstitch model: BekensteinHawking entropy, ge-

ometric evaporation, and primordial-black-hole signatures

, Eur. Phys. J. Plus

141

, 916 (2026).

doi.org/10.1140/epjp/s13360-026-08148-9

[3] R. Kulkarni,

Matter as incomplete crystallization: quark charges, color connement, and the

proton mass from a single extra node in the vacuum lattice

, Phys. Open

27

, 100423 (2026).

doi.org/10.1016/j.physo.2026.100423

[4] R. Kulkarni,

Emergent quantum mechanics on the FCC lattice

, Quantum Rep.

8

, 78 (2026).

doi.org/10.3390/quantum8030078

[5] R. Kulkarni,

A 67%-rate CSS code on the FCC lattice:

[[192, 130, 3]]

from weight-12 stabilizers

,

arXiv:2603.20294 (2026).

arxiv.org/abs/2603.20294

[6] A. Yu. Kitaev,

Fault-tolerant quantum computation by anyons

, Ann. Phys.

303

, 2 (2003).

doi.

org/10.1016/S0003-4916(02)00018-0

[7] M. Hermele, M. P. A. Fisher and L. Balents,

Pyrochlore photons: the

U(1)

spin liquid in a

S =

1

2

three-dimensional frustrated magnet

, Phys. Rev. B

69

, 064404 (2004).

doi.org/10.

1103/PhysRevB.69.064404

[8] M. J. P. Gingras and P. A. McClarty,

Quantum spin ice

, Rep. Prog. Phys.

77

, 056501 (2014).

doi.org/10.1088/0034-4885/77/5/056501

[9] D. A. Huse, W. Krauth, R. Moessner and S. L. Sondhi,

Coulomb and liquid dimer models in three

dimensions

, Phys. Rev. Lett.

91

, 167004 (2003).

doi.org/10.1103/PhysRevLett.91.167004

[10] C. L. Henley,

The Coulomb phase in frustrated systems

, Annu. Rev. Condens. Matter Phys.

1

,

179 (2010).

doi.org/10.1146/annurev-conmatphys-070909-104138

11