Electromagnetism in the Selection

The Electromagnetic Sector of the Selection–Stitch Model:

A Compact U (1) Field beside a Stabilizer Code

on the FCC Lattice

Raghu Kulkarni

1,∗

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

∗

Correspondence: raghu@idrive.com

Keywords: lattice gauge theory; compact U(1); photon; Hodge decomposition; stabilizer

codes; toric code; close-packed structures; D

lattice and W (F

) symmetry; Selection–Stitch

Model

Abstract

The Selection–Stitch Model (SSM) treats the vacuum as a face-centered-cubic (FCC)

lattice whose ⟨110⟩ bonds carry a stabilizer code but no propagating electromagnetic ﬁeld.

We construct its electromagnetic sector by asking where a compact U (1) gauge ﬁeld can

live without sharing the code’s variables, comparing the Gaussian spectrum of the compact

U(1) Hamiltonian on three candidate complexes. With triangular plaquettes the ⟨110⟩

network supports a two-polarization photon, but only on the code’s bonds. On the code

complex, whose two-cells are the octahedral checks, four zero-frequency ﬂat bands appear,

with harmonic dimension equal to the code’s logical count, 2L

+ 2. The ⟨100⟩ links joining

FCC nodes to octahedral voids form a simple cubic lattice that shares no bond with the code

and has the Maxwell count: this is the proposed sector. Gauge invariance keeps its photon

massless even when the two site types diﬀer, and centrosymmetry removes the leading strain

coupling. Matter hopping along a ⟨110⟩ bond passes through an octahedral void, coupling

minimally to the gauge links. In four dimensions the cubic lattice is a slice of D

∗

, whose

point group W (F

) makes the Euclidean photon isotropic through quartic order. With

qubits instead of a gauge ﬁeld, the same links form the three-dimensional toric code.

1 Introduction

Lattice gauge theory places a gauge potential on the links of a lattice and its ﬁeld strength on

the plaquettes [1, 2]. The choice of lattice is usually a matter of convenience, and the hypercubic

lattice is almost universal. In some settings the lattice is given in advance and already carries

other structure, and then the question of where a gauge ﬁeld can live becomes a structural

one. Two familiar examples are frustrated magnets, where an emergent photon appears on the

pyrochlore lattice of a spin-ice material [3, 4], and topological quantum codes, where the toric

code is itself a Z

lattice gauge theory [5, 6].

This paper treats such a question for the face-centered-cubic packing. The setting is the

Selection–Stitch Model (SSM), a lattice model of the vacuum developed in earlier work [7, 8, 9,

10, 11], in which the nearest-neighbor ⟨110⟩ bonds of an FCC lattice carry a [[3L

, 2L

+2, 3]]

CSS stabilizer code [7]. The companion papers place a code, a gravitational sector and matter

defects on this lattice, but no propagating electromagnetic ﬁeld, and this paper constructs the

SSM’s electromagnetic sector. A model of this kind has to accommodate electromagnetism

alongside the code. We ask which sublattice of the packing can carry a compact U(1) gauge

ﬁeld with the Maxwell degree-of-freedom count, without sharing variables with the code, and

what protects its masslessness and isotropy. The results are statements about lattice complexes

and their spectra, and a reader who does not accept the wider programme can assess them on

those terms.

In plain terms, the question and its answer are as follows. The FCC packing is the densest

way to stack equal spheres. In the framework, the bonds between touching spheres already carry

a quantum code, so a gauge ﬁeld placed on those bonds would share its variables with the code.

Between the spheres lie small gaps, and joining every sphere to the six larger, octahedral gaps

around it gives an ordinary cubic grid that uses none of the code’s bonds. We show that this

grid carries a ﬁeld with the two polarizations of light, that gauge invariance keeps it massless,

that matter moving from sphere to sphere passes through the gaps and so couples to it, and

that the grid extends to a four-dimensional lattice whose symmetry keeps the ﬁeld isotropic to

high order in the Euclidean formulation. With qubits in place of the gauge ﬁeld, the same grid

is the three-dimensional toric code.

What is taken, what is standard, and what is new. Table 1 separates the ingredients.

From the earlier work we take only the placement of the code on the ⟨110⟩ bonds, with Z-checks

at nodes and X-checks on octahedral voids [7], and, in Sections 6 and 7, the placement of matter

defects in the interstitial voids [9, 11]; the methods are standard, and the new results are the

contributions listed below.

status content

taken from [7, 9, 11] the CSS code on ⟨110⟩ bonds; matter defects in interstitial voids (each

companion paper is summarized in Section 2)

standard method cochain complexes and Hodge decomposition [12]; compact U(1) and

its Gaussian limit [1, 2]; Coulomb phase [13, 14]; toric code as Z

gauge

theory [5, 15]; invariants of W (F

) [16]; the lattice D

∗

[17]

new structure the node–octahedral-void cubic sublattice as a gauge lattice beside the

code; one-form spectra of three FCC complexes; two-path matter cou-

pling; the D

∗

lift

established here Hodge splits and Bloch spectra (Sections 4, 7); the staggered-weight test

(Section 5); disjointness, moments and path coverings by enumeration

(Sections 5, 6); the parameters of the two codes (Section 8)

conditional that the vacuum realizes a compact U(1) theory on these links; the

value of β; the origin of the gauge variables; real-time dynamics; W (F

)

invariance of the matter actions (Section 10)

Table 1: Status of the ingredients of this paper.

Contributions.

1. The Gaussian Maxwell spectrum on three FCC complexes (Section 4). The close-packed

network with triangular plaquettes has two gapless transverse branches and three gapped

optical modes. The code complex has one gapless branch and four zero-frequency ﬂat bands,

whose harmonic dimension equals the code’s logical count. The cubic sublattice of nodes

and octahedral voids has exactly the Maxwell count.

2. The cubic sublattice shares no bond with the code, so a gauge ﬁeld on it coexists with

the code without sharing variables; its masslessness is protected by gauge invariance, is ro-

bust to inequivalent sublattices, and its leading strain coupling vanishes by centrosymmetry

(Section 5).

3. Motion along a ⟨110⟩ bond is the composition of two ⟨100⟩ links through an octahedral void,

so that minimal coupling to the gauge links is routed by the geometry (Section 6).

4. The cubic sublattice is the constant-time slice of D

∗

, on which the photon keeps the Maxwell

count, is isotropic through quartic order in the Euclidean formulation, and shares one

quadratic cone with every W (F

)-invariant ﬁeld (Section 7).

5. With qubits in place of the gauge ﬁeld, the same links give the three-dimensional toric code,

and the photon’s harmonic modes and the code’s logical qubits are the same cohomology

group over two coeﬃcient rings (Section 8).

Section 2 summarizes the framework and each of the companion papers it draws on. Sec-

tion 3 sets out the lattices, the complexes and the dynamics. Sections 4–8 give the results.

Section 9 relates them to standard lattice gauge theory, and Section 10 states what is and is

not established.

2 Background: the Selection–Stitch framework

This section summarizes the ﬁve companion papers on which the present work draws, so that

the paper can be read without them. The SSM models the physical vacuum as a discrete close-

packed lattice that carries a quantum error-correcting code, and reads particle properties oﬀ

the combinatorics of that lattice.

The section is organized around one principle. The results of Sections 4–8 are properties

of lattice complexes, that is, of sites, links and plaquettes, their incidence, and the spectra of

operators built on them, and they do not depend on accepting the physical claims summarized

below. Each subsection therefore separates three things. Under Construction it describes the

lattice object that the companion paper deﬁnes. Under Claims it reports what that paper

proposes about physics, in the paper’s own terms; these claims are context, and none of them is

assumed or tested here. Under Used here it lists the lattice facts the present paper takes from

it and where they are used. Where a later section refers to a physical object of the framework,

such as a baryon defect, it uses only the object’s location on the lattice, and the statement it

makes is a statement about that location. Section 2.6 collects these dependencies in one table.

Figure 1 shows the four lattice objects that recur throughout.

(a)

-check:

12 bonds at a node

(b)

-check:

12 bonds of an octahedron

dark: anchor bond

light: three valence bonds

node in a tetrahedral void

up/down: the two void classes;

every hop changes class

(d) tetrahedral-void

sublattice and one hop

Figure 1: The lattice objects of the framework, drawn from the deﬁning rules (FCC nodes at

even-sum integer points, octahedral voids at odd-sum points, tetrahedral voids at half-integer

points, cubic cell side a = 2). (a) A Z-check of the code: the twelve ⟨110⟩ bonds meeting at

a node, whose far ends form a cuboctahedron (grey) [7]. (b) An X-check: the twelve bonds

among the six nodes surrounding an octahedral void (square) [7]. (c) The baryon defect: an

extra node (star) at the center of a tetrahedral void, bonded to the four nodes of its tetrahedron,

one bond serving as anchor and three as valence bonds [9]. (d) The tetrahedral voids form a

simple cubic lattice of spacing a/2, two-colored by orientation; a defect migrates by hopping

between adjacent voids, and every hop changes the void class [11].

2.1 The vacuum as a stabilizer code

Construction. Reference [7] deﬁnes a CSS stabilizer code on the FCC lattice. One qubit

sits on each nearest-neighbor ⟨110⟩ bond. A Z-type check acts on the twelve bonds meeting at

each node (Figure 1a), and an X-type check on the twelve bonds joining the six nodes around

each octahedral void (Figure 1b). Each bond lies in two Z-checks and in the X-checks of the

two octahedra it borders, the checks commute, and every check has weight twelve. On an L

periodic patch the lattice has L

/2 nodes, L

/2 octahedral voids and 3L

bonds, and the code

has parameters [[3L

, 2L

+2, 3]]: [[192, 130, 3]] at L = 4 and [[648, 434, 3]] at L = 6. Each

family of checks has one global dependency, so only L

− 2 of the L

checks are independent,

which is the origin of the rate approaching two thirds. The distance is three at both sizes

computed, set by bond triangles, which are weight-three logical operators.

Claims. The framework reads this code as the vacuum itself, with particles as localized defects

that its checks detect. The paper also reports a matching decoder and a coding gain over

unencoded qubits at small error rates, conjectures that the logical qubits are associated with

the tetrahedral voids, and discusses hardware platforms with twelve-fold connectivity. The

paper notes that the distance stays at three, so the code trades scalable error suppression for

rate.

Used here. The placement of the qubits on the ⟨110⟩ bonds and the two families of checks.

These deﬁne the code complex of Section 3, whose one-form spectrum is computed in Section 4,

the disjointness argument of Section 5, and the comparison of codes in Section 8. All three are

statements about the complex and hold whether or not the vacuum is a code. In Section 4 we

ﬁnd that the harmonic sector of the code complex over the real numbers has dimension 2L

+ 2,

the code’s logical count.

2.2 Mass as veriﬁcation cost

Construction. Reference [8] works with the thirteen-node coordination cluster of an FCC

node, the node and its twelve neighbors, decomposed into one-, two- and three-sheet substruc-

tures with stated numbers of vertices, edges and faces. For each candidate defect geometry it

deﬁnes a veriﬁcation cost C

as the product of the number of bonds in the defect’s footprint

and the number of checks able to detect it.

Claims. The paper proposes that the rest mass of a particle is the thermodynamic cost of

verifying it as a defect of the code. The argument combines Landauer’s principle, that generating

one classical bit irreversibly costs at least kT ln 2, with E = mc

, and holds that a vacuum with

no other substrate has no other source of mass. Mass ratios are then ratios of veriﬁcation

costs, with the temperature canceling. Applying four stated axioms to twenty-ﬁve candidate

conﬁgurations, the paper retains ﬁve, with costs 1, 207, 273, 1836 and 1839, which it identiﬁes

with the electron, muon, charged pion, proton and neutron mass ratios.

Used here. Nothing. The paper is summarized because it is the reason the framework treats

the vacuum as a code, and hence the reason the ⟨110⟩ bonds are already occupied when a gauge

ﬁeld has to be placed. No mass assignment enters any result below.

2.3 Matter as incomplete crystallization

Construction. Reference [9] places an extra node at the center of a tetrahedral void of the

FCC lattice, bonded to the four nodes of its tetrahedron (Figure 1c). One of the four bonds is

designated the anchor and the other three the valence bonds. The paper also reports a graph-

growth simulation, built from two local operations, that reaches twelve-fold coordination in the

bulk by way of a four-coordinated tetrahedral stage.

Claims. The paper proposes that the FCC vacuum forms by a phase transition from the

tetrahedral phase, and that the trapped node is a baryon: matter as incomplete crystallization.

From the placement it derives quantum numbers. The projection of each valence bond on the

anchor is the tetrahedral cosine −1/3, identiﬁed with the down-quark charge; an integer winding

gives +2/3, so that Q = −1 + W with W ∈ {0, 1, 2, 3}. The three pairs of opposite tetrahedron

edges supply three color charges, the obstruction to extracting the node supplies conﬁnement,

and a counting of disrupted bonds gives m

= 1836. The paper also assigns the triangular

faces of the coordination shell, which are odd cycles, to the color sector, and its square faces,

which are even cycles, to electromagnetism.

Used here. Only the location of the defect: the tetrahedral voids, which in our coordinates

are the half-integer points. That location is used in Sections 6 and 7, where the statements made

are lattice statements: that the half-integer points are the cube centers of the cubic sublattice,

and that they are sites of D

∗

. They do not depend on what the defect is taken to be. The

charge, color, conﬁnement and mass claims are not used, and the face assignments are neither

used nor tested; Section 4 shows that the question of where a propagating photon can live is

separate from them.

2.4 Gravity on the four-dimensional lattice

Construction. Reference [10] takes the spacetime lattice to be the four-dimensional root

lattice D

, the even-sum points of Z

, whose constant-time slice is the FCC lattice, and treats

it as an intrinsic simplicial (Regge) geometry in Euclidean signature, whose variables are the

squared edge lengths. It shows that the rank-four bond tensor of D

is exactly isotropic while

that of the FCC slice is not, and computes the Regge kinetic operator on an explicit subdivision.

Claims. On that subdivision the long-wavelength kinetic operator is identiﬁed with the lin-

earized Einstein operator, with the graviton as the incompatible part of the edge-length ﬁeld,

and an induced Newton constant is matched to the observed one. The paper goes on to a black-

hole sector, with an area law whose coeﬃcient is ﬁxed by one calibration, and to a proposed

evaporation channel with consequences for primordial black holes.

Used here. Two points. The ﬁrst combines a framework choice with a lattice fact: the

framework’s four-dimensional lattice is D

, and the point group of D

, which is also that of

∗

, is W (F

) [17]. This motivates Section 7, whose results concern D

∗

and hold as statements

about it. The second is an input of the framework rather than a lattice fact: the edge lengths

of the gravitational sector are not code variables, so the code and the geometry already share

the ⟨110⟩ bonds without sharing degrees of freedom. It is recalled in Section 3.2 and, like the

code itself, motivates adding the gauge ﬁeld as new variables in Section 5; no computed result

depends on it. Nothing in the gravitational, black-hole or cosmological claims is used.

2.5 Defect migration

Construction. Reference [11] shows that the tetrahedral voids of the FCC slice form a simple

cubic lattice of spacing a/2, two-colored by the orientation of their tetrahedra (Figure 1d). Of

the ten ways out of a void, the four through faces are closed by the exclusion geometry of Ref. [9]

and the six across edges are open, so hops join adjacent voids with coordination six, and every

hop changes the void class. Each void class has a single bond-direction set, and the rotational

site symmetry of a void is the tetrahedral group.

Claims. Under a stated propagation rule, one D

bond per time step, each hop carries one

time step and a defect at rest oscillates between two adjacent voids. Because the void class

carries only one bit of orientation, the paper argues that the matter–antimatter distinction

cannot be the void class, as Ref. [9] had proposed, and assigns it to the time orientation of the

worldline. It also proposes a discrete-time quantum walk in which the defect’s mass enters as a

coin angle. Under the double cover of the site group, spin one-half is the only half-integer spin

that stays irreducible; the paper stresses that this selects a spin rather than determining one.

Used here. The void lattice and the hop geometry, which are lattice facts. Section 6 observes

that the void lattice is the dual of the photon’s cubic sublattice, so that on a single slice a hop

between voids crosses a plaquette, and Section 7 that the voids are sites of D

∗

. The propagation

rule, the matter–antimatter assignment, the spin selection and the quantum walk are not used.

2.6 What the results depend on

The inputs of the present paper are collected below, with their kind and where they enter.

A lattice fact is a statement about a complex that can be checked without reference to the

framework; a framework input is a choice of the framework that determines which complexes

are of interest or how a result is read.

input kind where used

qubits on ⟨110⟩ bonds; Z-checks at nodes, X-

checks on octahedra [7]

lattice fact (deﬁ-

nition)

Sections 4, 5, 8

tetrahedral voids at half-integer points; two

void classes; adjacent voids a/2 apart along an

axis [9, 11]

lattice fact Sections 6, 7

the point group of D

and of D

∗

is W (F

) [17,

16]

lattice fact Section 7

the vacuum is a code; the spacetime lattice is

; matter defects sit in tetrahedral voids and

hop between adjacent ones [7, 10, 9, 11]

framework input motivation; reading of Sec-

tions 6, 7

edge lengths are not code variables [10] framework input motivation in Sections 3.2, 5

mass, charge, color, conﬁnement, gravity, black-

hole and propagation claims [8, 9, 10, 11]

physical claims not used

No framework input enters a computed result. The Hodge splits, Bloch spectra and moments

of Sections 4–7 (Tables 2–4, Figure 3) are computed from the complexes deﬁned in Sections 3

and 7 alone, and the code comparison of Section 8 from the same complexes read over F

. The

framework inputs decide which complexes are worth comparing, and they give the results their

interpretation.

Against this background, the present paper ﬁlls a gap in the framework. The companion

papers place a code, a gravitational sector and matter defects on one close-packed lattice, but

none places a propagating electromagnetic ﬁeld. Reference [9] assigns electromagnetism to the

square faces of the coordination shell, which identiﬁes a sector but not a ﬁeld with a dispersion.

The present paper asks which sublattice can carry a compact U(1) gauge ﬁeld with the Maxwell

degree-of-freedom count, given that the ⟨110⟩ bonds already carry the code’s qubits, and answers

it with the Hodge decomposition and the Gaussian spectrum of the compact U(1) Hamiltonian,

set out in Section 3.

3 Lattices, complexes and dynamics

3.1 Three bond networks in one packing

We use coordinates in which the FCC nodes are the integer points of even coordinate sum, so

that the conventional cubic cell has side a = 2 and the nearest-neighbor bonds are the twelve

vectors (±1, ±1, 0) and permutations. The octahedral voids are then the integer points of odd

sum, and the tetrahedral voids are the half-integer points [17]. Wavevectors are measured in

units of 2/a, the inverse spacing of the cubic sublattice. Three bond networks join these sites

(Figure 2).

(a) ⟨110⟩ nearest neighbors

K = 12, non-bipartite:

close-packed gauge lattice

(b) ⟨111⟩, one void class

K = 4, bipartite:

diamond (spin-ice route)

K = 6, bipartite:

Maxwell count beside the code

odd cycle propagation path

Figure 2: Three structures in one FCC packing, drawn on equal axes. (a) The ⟨110⟩ network,

which carries the code; one triangular odd cycle of its cuboctahedral shell is shaded. With tri-

angular plaquettes it supports a two-polarization photon in the Gaussian weak-coupling sector,

but on the code’s own bonds (Section 4). (b) The ⟨111⟩ bonds to one class of tetrahedral voids:

the diamond lattice, the route through U(1) quantum spin ice. (c) The ⟨100⟩ bonds to the

octahedral voids: nodes (circles) and voids (squares) form the simple cubic lattice, with the

Maxwell degree-of-freedom count. The highlighted path alternates between the two site types.

All three structures contain the FCC nodes; (a) and (c) share no bonds.



The ⟨110⟩ network joins nodes to nodes. It is close-packed, with coordination twelve, and

its triangular faces are odd cycles, so it is not bipartite.



The ⟨111⟩ bonds join each node to its eight tetrahedral voids. Restricted to one of the two

void classes they form the diamond lattice, bipartite and non-centrosymmetric, whose bond

midpoints are the pyrochlore sites of spin ice [3, 4]. Including both classes gives the ﬂuorite

structure, which is centrosymmetric.



The ⟨100⟩ links join each node to its six octahedral voids. Nodes and voids together are all

integer points, so these links form the simple cubic lattice of spacing a/2, two-colored by

coordinate parity.

Direct enumeration conﬁrms the last statement: on periodic patches the 32 nodes and 32

voids at L = 4, and the 108 and 108 at L = 6, are exactly the L

integer points, and the link

network connects all of them.

3.2 The code on the ⟨110⟩ bonds

The code of Ref. [7], summarized in Section 2, places one qubit on each nearest-neighbor bond,

with a weight-twelve Z-check at each node and a weight-twelve X-check on the twelve bonds

bounding each octahedral void; on an L

patch it is a [[3L

, 2L

+2, 3]] CSS code. The octahedral

voids carry no degrees of freedom of their own. The edge lengths of the gravitational sector [10]

also live on these bonds but are not code variables; nothing below uses that sector.

3.3 Cochain complexes and the Hodge decomposition

A two-dimensional cell complex on a lattice consists of sites, oriented links and oriented pla-

quettes. Write d

for the signed incidence of links on sites, so that (d

χ)

ℓ

= χ(head) − χ(tail)

for a zero-cochain χ, and d

for the signed incidence of plaquettes on links, so that (d

the circulation of a one-cochain A around p. The complex condition is d

= 0. Over the real

numbers the one-cochains split orthogonally as [12]

= im d

⊕ im d

⊕ H

, (1)

into exact (pure-gauge), coexact (transverse) and harmonic parts, with dim H

the ﬁrst Betti

number of the complex. We use three complexes:



the cubic sublattice, with its square plaquettes;



the ⟨110⟩ network with its triangular faces, eight per node;



the code complex: the ⟨110⟩ network with the octahedral X-checks as its two-cells.

The last needs a remark. The twelve edges of an octahedron are the one-skeleton of a solid,

not the boundary of a disk, so a lift of the code’s F

complex to real coeﬃcients needs a

sign convention. We take each check to be the sum of the three oriented great squares of

its octahedron, which partition its twelve edges. Every one of the eight sign choices satisﬁes

= 0 and gives the same ranks, at L = 4 and L = 6, so the conclusions below do not depend

on the choice.

3.4 Dynamics: the compact U (1) Hamiltonian

The gauge theory on a complex is the compact U(1) Hamiltonian [1, 2]

H =

ℓ

−

cos





, (d

= ρ

, (2)

with a compact phase A

ℓ

and its conjugate integer-valued electric ﬁeld E

ℓ

on each link, positive

plaquette weights w

, and Gauss’s law at every site s. In the Gaussian (weak-coupling) limit

the cosine is expanded to quadratic order, and the frequencies of the normal modes are

∈ spec



W d



, W = diag(w

), (3)

restricted by Gauss’s law to the coexact and harmonic subspaces. The ﬁeld is explicit: its gauge

structure is imposed by Gauss’s law in Eq. (2), not generated by hard-core constraints as in

spin ice, a distinction that matters in Section 4. Exact modes carry no energy and are removed

by the constraint. On a periodic lattice the operator is block-diagonal in crystal momentum q,

and its Bloch blocks are the matrices d

(q)

†

W d

(q).

A photon, in this language and in three spatial dimensions, is a pair of coexact branches

whose frequency vanishes linearly as q → 0, with no other coexact or harmonic branch at zero

frequency. Further coexact modes are allowed: if they are gapped at the lattice scale, they are

optical modes of the lattice and decouple at long wavelength. Zero-frequency ﬂat bands are not

allowed, since they make the Gaussian theory massively degenerate.

Gauge invariance makes the photon massless in the weak-coupling (Coulomb) phase: a

mass term A

ℓ

is not invariant, and the uniform harmonic modes are exact zero modes of Eq. (3)

for any positive weights, so the transverse branches connected to them are gapless. In 3 + 1

dimensions compact U(1) has such a phase at weak coupling, proved for the four-dimensional

Euclidean hypercubic lattice [13, 14]; in 2+1 dimensions it is always conﬁned by monopoles [18].

4 One-form spectra on three complexes

Result of this section. With triangular plaquettes the close-packed network supports a two-

polarization photon in the Gaussian sector, but only on the code’s own bonds; the code complex

supports none; the cubic sublattice has exactly the Maxwell count.

Table 2 gives the Hodge split of Eq. (1) per site at two sizes, and Table 3 the small-q Bloch

spectrum of Eq. (3) with unit weights. Figure 3 summarizes both.

complex L links/site exact/site coexact/site harmonic

cubic ⟨100⟩, squares 4 3 0.984 1.969 3

6 3 0.995 1.991 3

⟨110⟩, triangles 4 6 0.969 4.938 3

6 6 0.991 4.981 3

code complex, octahedra 4 6 0.969 0.969 130

6 6 0.991 0.991 434

Table 2: Hodge split of the one-cochains, Eq. (1), per site of the complex, on L

periodic

patches. The harmonic sector of the code complex has dimension 2L

+ 2, the logical count of

the code.

complex gauge gapless transverse gapped ﬂat, ω = 0

cubic ⟨100⟩ 1 2, ω

= q

— —

⟨110⟩, triangles 1 2, ω

= q

/2 3, at ω

= 8 —

code complex 1 1, ω

= 4q

— 4

Table 3: Bloch spectrum of d

(q)

†

(q) at small q, per site, with unit plaquette weights and q

in units of 2/a. The leading coeﬃcients are the same along [100], [110] and [111].

The cubic sublattice. The Hodge split is one gauge mode and two transverse modes per site,

approaching one and two as L grows, with a harmonic sector of dimension three, the winding

classes of H

). The Bloch spectrum is the familiar

(q) =

4 sin

/2) (4)

for both transverse branches, in units with link length one. This is the Maxwell ﬁeld.

The close-packed network: a Gaussian photon on the code’s bonds. With its trian-

gular faces as plaquettes, the ⟨110⟩ network has six link components per site: one pure gauge,

two transverse branches with ω

= q

/2, isotropic at leading order, and three optical modes

gapped at ω

= 8. Its harmonic sector is again three-dimensional. By the criterion of Section 3,

the close-packed network therefore supports a two-polarization photon in the Gaussian weak-

coupling sector; the optical modes play the same role as the gapped modes of D

∗

in Section 7.

It would require a compact phase on every ⟨110⟩ bond, and those bonds already carry the code’s

qubits.

The code complex does not carry one. With the octahedral checks as two-cells, rank d

is bounded by the number of octahedra, which equals the number of nodes, so at most one

transverse mode per node is coexact; this bound holds for any sign convention, and the ranks

0 2 4 6

one-form degrees of freedom per site

110

FCC, octahedral checks (code)

110

FCC, triangular plaquettes

100

cubic, square plaquettes

Maxwell (continuum)

(a)

pure gauge

gapless transverse

gapped (optical)

flat, zero frequency

0 1 2 3

along [100] (units of

)

(Gaussian limit)

(b)

100

cubic

110

triangles

110

code complex

code complex, 4 flat bands

Figure 3: (a) One-form degrees of freedom per site, classiﬁed by the Gaussian spectrum: pure

gauge, gapless transverse, gapped optical, and zero-frequency ﬂat. The continuum Maxwell

ﬁeld has one pure-gauge and two gapless transverse modes. The cubic sublattice reproduces

this count; the close-packed network with triangular plaquettes adds three optical modes; the

code complex has one gapless branch and four ﬂat bands. (b) The Gaussian spectra ω

(q) along

[100].

are the same for all eight. The Bloch spectrum has one gapless branch, ω

= 4q

, and four bands

at exactly zero frequency across the whole zone. Together with the six zero modes at q = 0,

these ﬂat bands make up the harmonic sector, whose dimension on the torus is 4 ·31 + 6 = 130

at L = 4 and 4 · 107 + 6 = 434 at L = 6, equal to 2L

+ 2, the number of logical qubits of

the code [7]. Over R as over F

, the octahedral checks constrain only about a third of the edge

variables (62 of 192 at L = 4), which is why the code has a high rate and why a gauge ﬁeld

on its complex has an extensive zero-mode degeneracy. A one-form on the code complex is

therefore not a photon.

Bipartiteness and the Coulomb phase. A bipartite charge lattice is sometimes taken to

be necessary for a gapless compact U(1) photon. That requirement belongs to emergent gauge

ﬁelds whose gauge structure comes from hard-core constraints, as in dimer and spin-ice models,

where a bipartite lattice is needed to orient the height or ﬂux variables [19, 20]. An explicit

compact gauge ﬁeld has a weak-coupling Coulomb phase without it [13, 14]; the Gaussian

spectrum above is its weak-coupling limit, although a proof of the phase on the close-packed

network is not attempted here. Bipartiteness therefore matters here only for the pyrochlore

route; on the cubic sublattice it is convenient but not required.

5 Coexistence with the code

Result of this section. The cubic sublattice shares no bond with the code, and its photon is

massless by gauge invariance, stays massless when the two kinds of site diﬀer, and has no leading

strain coupling.

Disjoint variables. 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.

The gauge ﬁeld is carried by new variables. The published construction places one qubit

on each ⟨110⟩ edge [7], and the octahedral voids carry 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 spin or rotor on each

pyrochlore site, and of the Gaussian close-packed photon of Section 4, which would place a

second variable on each code bond. Given that, the disjointness makes the addition safe at

the level of the operator algebras: the two act on diﬀerent variables and commute trivially, so

no stabilizer is violated by a gauge transformation and no gauge transformation moves a code

qubit. How the two sectors couple through matter is a separate question, taken up in Section 6.

Of the complexes in Section 4, the cubic sublattice is the one that carries the Maxwell count

on bonds the code does not use.

Masslessness with inequivalent sublattices. The links of the cubic sublattice alternate

between FCC nodes and octahedral voids, which are physically diﬀerent objects, and one might

expect a diﬀerence between them to open a gap. It does not. Every link joins one node to one

void, so no site-parity diﬀerence can make links inequivalent, and a staggered background charge

produces only a static ﬁeld. More generally, unequal positive plaquette weights leave Eq. (3)

gauge invariant, and its uniform harmonic modes remain exact zero modes, so the transverse

branches stay gapless. A numerical check conﬁrms this: with plaquette weights 1 ±ε staggered

by parity, the lowest transverse eigenvalue on an L

torus scales as (2 sin(π/L))

for ε = 0.1 and

0.3 at L = 6 and 10, with coeﬃcient 0.99 and 0.90 respectively, and the three harmonic zero

modes persist. What would gap the photon is the loss of gauge invariance on one sublattice,

for instance a Gauss law imposed at nodes but not at voids, under which mass terms built

from links that share a node become invariant, or a Higgs condensate. The condition on the

dynamics is therefore that the Gauss law hold at every site; the couplings need not be equal.

Centrosymmetry and the strain coupling. The two sectors share the FCC nodes, so a

node displacement perturbs both its ⟨110⟩ bonds and its ⟨100⟩ links, and one may ask how

lattice strain u

couples to the gauge ﬁeld. A coupling linear in the potential, of the form

, is excluded on any lattice by gauge invariance. On a non-centrosymmetric star, such as

the tetrahedral star of the diamond with its zincblende moment d

, a gauge-invariant coupling

is allowed at rank three. It renormalizes the dielectric response but does not generate a

mass. The ⟨100⟩ star is centrosymmetric (O

), and its rank-three moment vanishes identically,

max

ijk

| = 0 by direct summation, so this piezoelectric channel is absent and the leading

coupling is rank four and quadratic in gauge-invariant objects. Centrosymmetry therefore re-

moves a coupling channel; it does not by itself protect the masslessness, which follows from

gauge invariance as above. The ⟨110⟩ star is centrosymmetric as well, so this property does not

distinguish the two networks.

Anisotropy on the slice. The rank-two moment of the ⟨100⟩ star is isotropic, S

µν

= 2δ

µν

The rank-four moment is not: T

xxxx

= 2 and T

xxyy

= 0, whereas isotropy requires T

xxxx

xxyy

. The photon dispersion, Eq. (4), therefore carries a cubic-anisotropic correction of rela-

tive order (qa)

, with ω

equal to 0.9992, 0.9996 and 0.9997 along the axis, the face diagonal

and the body diagonal at q = 0.1. Section 7 shows that in four dimensions, in the Euclidean

formulation, this correction is absent by symmetry.

6 How matter couples to the gauge sector

Result of this section. A defect moving along a ⟨110⟩ bond passes through an octahedral void,

so minimal coupling attaches it to the gauge links; defects in tetrahedral voids are taken up in

Section 7.

A gauge ﬁeld is only useful if charges can radiate into it. Consider a charged defect that

moves between FCC nodes along ⟨110⟩ bonds; the argument concerns that motion and does not

depend on what the defect is taken to be. The ﬁeld lives on ⟨100⟩ links, which share no bonds

with the ⟨110⟩ network, so the coupling looks as though it has to be supplied by hand. Minimal

coupling is still an input, but the geometry determines which links it runs through.

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:

enumerating all 192 edges at L = 4 and all 648 at L = 6 shows that each is covered by exactly

two two-step ⟨100⟩ paths, and every intermediate is a void. A defect moving along a ⟨110⟩ edge

therefore passes through a void, and minimal coupling assigns it the product of two ⟨100⟩ link

phases.

The path ambiguity is the ﬁeld 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 deﬁned

only up to the ﬂux through that plaquette. Either path is a gauge-covariant transporter, and

the two diﬀer by the plaquette holonomy e

i(d

, which is the lattice ﬁeld strength; this is the

term through which the defect radiates.

Defects in tetrahedral voids. In the companion papers [9, 11] baryonic defects sit in tetra-

hedral voids and migrate between adjacent voids. The tetrahedral voids are the half-integer

points, the cube centers of the cubic sublattice, so on a constant-time slice they are sites of

the dual lattice, and a hop between adjacent voids pierces a plaquette. In compact U (1) that

is how a magnetic rather than an electric charge couples. On the slice alone, baryonic defects

would therefore couple as magnetic charges, although the framework treats them as electrically

charged. A resolution is available on the four-dimensional lattice D

∗

of Section 7, where the

tetrahedral voids are sites of the photon lattice itself.

The same construction in four dimensions. The D

lattice is the even-sum subset of Z

and adding its odd-sum coset gives Z

itself: 128 + 128 = 256 at L = 4. The joining links are

the eight axis vectors, the hypercubic lattice of ordinary lattice gauge theory. The coupling

argument lifts verbatim: every one of the 24 D

bonds splits into two axis steps in exactly two

ways, through an odd-sum site, with the two routes diﬀering by a hypercubic plaquette. But

is not the only four-dimensional lift, and not the one whose point group is that of D

7 The four-dimensional lattice D

∗

Result of this section. On the four-dimensional lattice D

∗

the photon keeps the Maxwell count,

is isotropic through quartic order in the Euclidean formulation, and shares one quadratic cone

with any ﬁeld whose action has the full W (F

) symmetry.

Two lifts. The point group of Z

is W (B

), of order 384, whose invariant polynomials have

degrees 2, 4, 6 and 8 and include the quartic

. The point group of D

is W (F

), of

order 1152, whose basic invariants have degrees 2, 6, 8 and 12 [16]. Its only quartic invariant

is therefore (q

)

, and the ﬁrst direction-dependent term in a W (F

)-invariant dispersion is of

sixth order. The lift that carries the cubic sublattice and has this point group is the dual lattice

∗

: any orthogonal map that preserves a lattice also preserves its dual, so D

and D

∗

have the

same point group, and matter on D

and a gauge ﬁeld on D

∗

are acted on by the same W (F

Structure of D

∗

. D

∗

is D

together with its three glue classes, the cosets of (0, 0, 0, 1),

(

) and (

, −

) [17]; equivalently it is the Hurwitz lattice Z

∪ (Z +

)

, which is

itself a rotated and rescaled copy of D

. Taking the fourth coordinate as Euclidean time, its

constant-time slices alternate. At integer times they contain D

and its (0, 0, 0, 1) coset, which

are the FCC nodes and the octahedral voids of that slice, exactly the cubic sublattice; since the

points have odd spatial sum at odd times, nodes and voids exchange coordinates from one

integer slice to the next. At half-integer times they contain the two spinor classes, which are

the half-integer spatial points, the tetrahedral voids. Its 24 shortest vectors are the spatial ±e

the ⟨100⟩ links of the slice; the temporal ±e

; and the sixteen (±

, ±

), which join

each site to the eight tetrahedral voids around it at each adjacent half-integer time. In space

these last have length a

√

3/4, the node-to-void distance of a trapped baryon [9].

Polarizations. We take the Gaussian sector of a compact U(1) ﬁeld on the links of D

∗

, with

triangular plaquettes formed by three shortest vectors summing to zero. There are 12 link classes

and 32 triangles per site. At small q the spectrum of d

(q)

†

(q) has one pure-gauge mode, three

gapless modes with eigenvalue q

/3, and eight coexact modes gapped at exactly 12 (Table 4).

Here and in the rest of this section lengths are in units of the shortest D

∗

vector, the node–void

link of length a/2, so that q is measured in units of 2/a. Three gapless transverse modes in

four Euclidean dimensions are Maxwell’s count, two physical polarizations after continuation

to Minkowski signature. The eight gapped modes are optical modes of the lattice, as in the

close-packed case of Section 4. The link star is centrosymmetric, so the remarks of Section 5

carry over.

hypercubic Z

∗

link classes, plaquettes per site 4, 6 squares 12, 32 triangles

gauge modes 1 1

gapless transverse modes, eigenvalue 3, q

3, q

gapped coexact modes none 8, at 12

rank-three moment of the link star 0 0

quartic moment

(ˆq · ˆv)

0.5 to 2 3, every direction

directional spread of ω

/ω

lead

, q = 0.1 5.8 × 10

−4

1.8 × 10

−8

directional spread, q = 0.05 1.5 × 10

−4

1.2 × 10

−9

Table 4: The photon on the two four-dimensional lifts of the cubic sublattice, Gaussian sector.

Halving q reduces the spread by a factor of four on Z

, as q

, and by a factor of about sixteen

on D

∗

, as q

, with q in units of 2/a. Spreads are over 100 random directions in four Euclidean

dimensions.

Isotropy through quartic order. The sixteen half-integer vectors and the eight axis vectors

together form a spherical ﬁve-design, so the quartic moment of the link star is 3 in every

direction, against a range of 0.5 to 2 for the hypercubic star. The photon branch on D

∗

expands as

λ(q) =



1 −

+ . . .



, (5)

with the coeﬃcient 1/72 the same in every direction to 3 × 10

−7

. The correction depends on

alone, so under continuation to Lorentzian signature it becomes a term in (k

−ω

)

which is Lorentz invariant and vanishes on the light cone. Centrosymmetry excludes odd powers.

In the Euclidean formulation the photon on D

∗

therefore has no direction-dependent correction

below sixth order, where the cubic slice and the hypercubic lift have one at second relative

order.

One quadratic cone. Because W (F

) acts irreducibly on R

, its only invariant quadratic

form is

, up to normalization (Schur’s lemma). Any ﬁeld with a W (F

)-invariant lattice

action therefore has the Euclidean kinetic operator Z

+ m

+ . . . , with a ﬁeld-dependent

normalization Z and mass m but a common quadratic form, and after continuation a common

limiting speed. The hypercubic group W (B

) is also irreducible, so this leading-order statement

holds there too. The diﬀerence lies at quartic order: W (B

) admits

with a ﬁeld-dependent

coeﬃcient, and with it ﬁeld-dependent speeds at relative order (qa)

, which improved actions [21]

reduce order by order; W (F

) admits only (q

)

. The argument applies to a matter ﬁeld only

if its lattice action is invariant under the whole of W (F

), including the elements that mix

time and space. Whether the defect actions of the companion papers have that invariance is

not established here. The argument also concerns Euclidean actions continued to real time; a

real-time formulation with discrete updates, such as that of Ref. [11], is not covered by it.

Where charge lives. On the cubic slice the Gauss law holds at nodes and octahedral voids,

and the tetrahedral voids are dual sites (Section 6). On D

∗

the tetrahedral voids are sites of the

photon lattice at half-integer times, so every interstitial position carries a Gauss law. A defect

in a tetrahedral void then couples to the gauge ﬁeld at its own site, through the spokes whose

spatial projections are its node-to-void bonds, and the dual-lattice issue of Section 6 does not

arise. The condition of Section 5 becomes that the Gauss law hold at all three kinds of site.

8 The same sector, read as a code

Result of this section. With qubits in place of the gauge ﬁeld the cubic sublattice is the three-

dimensional toric code, and on both complexes the code’s logical qubits and the gauge ﬁeld’s

zero modes are one cohomology group.

Putting qubits on the ⟨100⟩ links rather than a gauge ﬁeld gives the standard three-

dimensional toric code [22]. The toric code is a Z

gauge theory on those same links: the

vertex check is the Z

Gauss law and the plaquette check the Z

magnetic term. Compact U(1)

Higgsed by a charge-two condensate breaks to Z

[15], and the deconﬁned phase of the resulting

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

one complex, at two points of its phase diagram with diﬀerent gauge groups.

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

(Table 5).

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

Table 5: The two CSS codes carried by the FCC packing on the same number of qubits: the

code of Ref. [7] on the ⟨110⟩ bonds and the three-dimensional toric code on the ⟨100⟩ links.

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

FCC code [7], and L is the standard result for the three-dimensional toric code [5]. We verify

only the upper bound, exhibiting a straight line of weight L that commutes with every check

and, because it winds around the torus, is not a product of checks, so d ≤ L. The two codes sit

at opposite ends of the rate–distance trade-oﬀ on the same substrate.

The cubic code has k = 3 and the ⟨100⟩ gauge ﬁeld has a three-dimensional harmonic sector,

and both count the ﬁrst cohomology of one complex, computed over F

in one case and over R

in the other; here the two have the same dimension, b

) = 3 [6]. Section 4 found the same

correspondence on the code complex, where the real harmonic sector has dimension 2L

+ 2,

the logical count of the FCC code. On both complexes, then, the code’s logical qubits and the

gauge ﬁeld’s zero modes count the same cohomology, over two coeﬃcient rings. On the cubic

complex it is three-dimensional, so no ﬂat bands arise and the spectrum of Section 4 is that of

a photon; on the code complex it grows with the volume, which gives the code its rate and ﬁlls

the spectrum with zero-frequency modes.

9 Relation to standard lattice gauge theory

On each constant-time slice the electromagnetic sector proposed here is compact U (1) on a sim-

ple cubic lattice in its Hamiltonian (Kogut–Susskind) form [2], unmodiﬁed; its four-dimensional

form is compact U(1) on D

∗

with triangular plaquettes. Most of what follows from that is

standard.

What transfers unchanged. The gauge potential sits on links and the ﬁeld strength on

square plaquettes. The Hodge count of Section 4 is the textbook statement that a cubic lattice

carries three link variables per site, one pure gauge and two transverse. The Coulomb phase of

four-dimensional compact U(1) [13, 14], its conﬁnement transition at β

≃ 1.01 for the Wilson

action [23, 24], and the relation α = 1/4πβ are standard, as is the toric code obtained by

putting qubits on the same links [5, 22]. The apparatus of lattice gauge theory therefore applies

to the gauge sector without adaptation, including Symanzik improvement [21]. On D

∗

, whose

triangular plaquettes make it non-bipartite, the existence of the Coulomb phase is expected

from the weak-coupling arguments but is not veriﬁed here, and neither is reﬂection positivity,

which a transfer matrix in the fourth direction requires.

Doubling. The Nielsen–Ninomiya theorem [25] forbids a local, Hermitian, translation-invariant

lattice Dirac operator from having a single Weyl zero. For the naive operator on the hypercubic

lattice the zeros sit at the 2

zone corners, sixteen in four dimensions. The standard reme-

dies remove or reinterpret the extra species at a price: explicit chiral-symmetry breaking for

Wilson fermions, taste mixing for staggered fermions, and computational cost for domain-wall

and overlap fermions. Minimally doubled constructions of Karsten–Wilczek and Bori¸ci–Creutz

type [26, 27, 28] reduce the count to two, at the cost of breaking part of the hypercubic symme-

try. Matter in the present arrangement is not placed on the cubic sublattice: in the companion

papers its defects sit on the close-packed network or in the interstitial voids [9, 11]. Whether a

satisfactory Dirac operator exists for them is a separate question, and we claim no advantage

here.

Anisotropy, in context. The rank-four moment of the ⟨100⟩ star is cubic rather than

isotropic, so the photon on the slice carries a correction of relative order (qa)

. This is the

same class of artifact as the hypercubic breaking in lattice QCD propagators, and Symanzik

improvement reduces it order by order [21]. On D

∗

, in the Euclidean formulation, no improve-

ment is needed at this order, since W (F

) admits no quartic anisotropy (Section 7).

Two grids. In lattice QCD matter and gauge ﬁelds share a single grid, matter on sites and

gauge on the links joining them. Here they do not: matter defects move on the close-packed

network or between interstitial voids, while the gauge ﬁeld lives on ⟨100⟩, and on D

∗

also on

the spokes that join each slice to the tetrahedral voids at the adjacent half-integer times. The

two are joined by the two-step decomposition of Section 6 and, on D

∗

, by the coincidence of the

defect sites with sites of the photon lattice (Section 7). On a constant-time slice with continuous

time, their propagation speeds are set by diﬀerent couplings and must be matched. In the four-

dimensional Euclidean formulation, W (F

) ﬁxes one quadratic cone for both, provided both

actions are W (F

)-invariant (Section 7).

10 Discussion

Comparison with the pyrochlore route. The two routes reach a photon diﬀerently. In the

⟨111⟩ construction the photon emerges from spins, at the price of an ice rule, a ring-exchange

term and a collective mechanism [3, 4], and a defect in a tetrahedral void sits on the charge

lattice, its charge a violation of the ice rule. On the cubic sublattice the gauge ﬁeld is explicit: it

is assumed rather than derived, and in return it has the Maxwell count directly, while the two-

step decomposition of Section 6 determines which links minimal coupling runs through. The two

routes do not diﬀer in whether the physical coupling can be reached. Exact diagonalization of

the microscopic quantum spin-ice model gives an emergent ﬁne-structure constant of order 0.1

for realistic parameters, more than an order of magnitude above 1/137, but it can in principle

be tuned from zero up to the conﬁnement limit by engineering the microscopic Hamiltonian [29].

On the pyrochlore the coupling is an output of the spin model; on the cubic sublattice it is a

parameter of the action. On the cubic sublattice the theory is ordinary compact U(1), with

α = 1/4πβ. The Coulomb phase β > β

≃ 1.01 [24] 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.

The other sublattices. The diamond sublattice is the only non-centrosymmetric structure

considered here, with rank-three moment 4/(3

√

3) ≈ 0.770; with both void classes the ﬂuo-

rite structure is centrosymmetric. Its site group T

contains mirrors, so it supports a rank-

three (piezoelectric) response but no handedness. We record this and claim nothing about a

parity-violating sector. Assignments of other interactions to the close-packed network, made in

companion papers [9], are not used or tested here.

Established in this paper. By enumeration at L = 4 and L = 6:



The FCC nodes and octahedral voids form a simple cubic lattice, which is bipartite, and its

links are disjoint from the code qubits.



The Hodge splits of Table 2: one gauge and two transverse modes per site on the cubic

sublattice; about ﬁve transverse modes per site on the close-packed network with triangles;

at most one on the code complex, whose harmonic dimension is 2L

+ 2.



The ⟨100⟩ and ⟨110⟩ stars have vanishing rank-three moment; the single-class ⟨111⟩ diamond

star does not.



Every ⟨110⟩ edge is covered by exactly two two-step ⟨100⟩ paths through octahedral voids,

diﬀering by one plaquette; the same holds for the 24 D

bonds in Z



Qubits on the cubic links give the toric code, with k = 3 at n = 192 against k = 130 for the

FCC code, and a weight-L logical operator bounding its distance by L.

By Bloch analysis of the Gaussian sector:



The spectra of Table 3: two gapless transverse branches on the cubic sublattice and on the

close-packed network with triangles, the latter with three optical modes; one gapless branch

and four ﬂat zero bands on the code complex.



Staggered plaquette weights do not gap the cubic-sublattice photon.



On D

∗

, in the Euclidean formulation: three gapless modes and eight gapped at 12, in units

of (2/a)

; isotropy through quartic order, with the direction-independent coeﬃcient 1/72.

From the literature: the Coulomb phase of four-dimensional compact U (1) [13, 14] and β

≃

1.01 [23, 24]; the toric-code distance [5, 22]; the invariants of W (F

) [16]; the emergent coupling

of quantum spin ice and its tunability [29].

Not established.



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 the Gauss law holds

at every kind of site; the photon is massless only if it does.



Whether the matter actions are invariant under the whole of W (F

), which is the condition

for a common light cone, and whether that survives a real-time formulation.



The Coulomb phase beyond the Gaussian sector, on D

∗

and on the close-packed network

with triangular plaquettes, and reﬂection positivity on D

∗



Any role for the ⟨111⟩ diamond beyond the property recorded here.

11 Conclusions

To give the SSM an electromagnetic sector, three complexes on its FCC packing were compared

as carriers of a compact U(1) gauge ﬁeld. The close-packed network, with its triangular faces

as plaquettes, supports a photon in the Gaussian weak-coupling sector, but only by placing

gauge variables on the bonds that carry the code’s qubits. The code complex does not: its

harmonic sector is the code’s logical sector, and a gauge ﬁeld on it has an extensive set of

zero-frequency modes. The cubic sublattice of nodes and octahedral voids carries the Maxwell

count on bonds the code does not use. Its photon is massless by gauge invariance, robust to

inequivalent sublattices, free of the leading strain coupling by centrosymmetry, and coupled to

matter through links that the geometry of the packing selects. In four dimensions it is the slice

of D

∗

, where, in the Euclidean formulation, the point group W (F

) protects its isotropy through

quartic order and ﬁxes a common quadratic cone for every W (F

)-invariant action. Read with

qubits instead of a gauge ﬁeld, the same complex is the toric code, and on both complexes

the code’s logical qubits and the gauge ﬁeld’s zero modes count the same cohomology. What

remains open is whether the vacuum realizes such a ﬁeld, the value of its coupling, and whether

the matter dynamics share its light cone (Section 10).

Data Availability Statement

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

archives in the repository github.com/raghu91302/ssmtheory. The scripts added in this re-

vision, marked below, are in photon scripts revision1 additions.zip; the others are in

photon scripts.zip. They require only numpy, apart from the ﬁgure generators, which also

need matplotlib, and each runs in under a minute. Nothing is hard-coded: the lattices are

built from their deﬁning rules and the ranks and spectra are computed by the scripts.



photon revision verify.py (revision archive): the Hodge splits of the three complexes, the

Bloch spectra, the staggered-weight test, the two-path covering, the moments of all bond

stars, and the D

∗

spectrum, isotropy and quartic coeﬃcient. It prints a pass/fail line for

each claim.



photon revision figure.py (revision archive): Figure 3, computed from the spectra.



photon framework figure.py (revision archive): Figure 1, drawn from the lattice rules.



photon verify 100.py: the cubic sublattice, its bipartiteness, the disjointness from the

code, and the four-dimensional lift.



photon two codes.py: the two CSS codes on the same qubit count, and the weight-L wind-

ing operator bounding the distance.



photon make figures.py: Figure 2; in the revision its panel titles are removed from the

graphic and set in the manuscript instead.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Conﬂicts of Interest

The author is employed by SSMTheory Group, IDrive Inc. The author declares that the re-

search was conducted in the absence of any commercial or ﬁnancial relationships that could be

construed as a potential conﬂict of interest.

References

[1] K. G. Wilson, Conﬁnement of quarks, Phys. Rev. D 10, 2445 (1974). doi.org/10.1103/

PhysRevD.10.2445

[2] J. B. Kogut, An introduction to lattice gauge theory and spin systems, Rev. Mod. Phys.

51, 659 (1979). doi.org/10.1103/RevModPhys.51.659

[3] M. Hermele, M. P. A. Fisher and L. Balents, Pyrochlore photons: the U(1) spin liquid in

a S =

three-dimensional frustrated magnet, Phys. Rev. B 69, 064404 (2004). doi.org/

10.1103/PhysRevB.69.064404

[4] 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

[5] A. Yu. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003).

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

[6] H. Bombin and M. A. Martin-Delgado, Homological error correction: classical and quantum

codes, J. Math. Phys. 48, 052105 (2007). doi.org/10.1063/1.2731356

[7] R. Kulkarni, A 67%-rate CSS code on the FCC lattice: [[192, 130, 3]] from weight-12 stabi-

lizers, arXiv:2603.20294 (2026). arxiv.org/abs/2603.20294

[8] R. Kulkarni, The Mass–Energy–Information Equivalence: a bottom-up identiﬁcation of

the particle spectrum via FCC lattice error correction, Phys. Open 27, 100414 (2026).

doi.org/10.1016/j.physo.2026.100414

[9] R. Kulkarni, Matter as incomplete crystallization: quark charges, color conﬁnement, and

the proton mass from a single extra node in the vacuum lattice, Phys. Open 27, 100423

(2026). doi.org/10.1016/j.physo.2026.100423

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

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

(2026). doi.org/10.1140/epjp/s13360-026-08148-9

[11] R. Kulkarni, Defect migration in the D

vacuum lattice: hop parity, site symmetry, and

soliton kinematics, Symmetry 18, 1571 (2026). doi.org/10.3390/sym18091571

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

arXiv:math/0508341 (2005). arxiv.org/abs/math/0508341

[13] A. H. Guth, Existence proof of a nonconﬁning phase in four-dimensional U(1) lattice gauge

theory, Phys. Rev. D 21, 2291 (1980). doi.org/10.1103/PhysRevD.21.2291

[14] J. Fr¨ohlich and T. Spencer, Massless phases and symmetry restoration in abelian gauge

theories and spin systems, Commun. Math. Phys. 83, 411 (1982). doi.org/10.1007/

BF01213610

[15] E. Fradkin and S. H. Shenker, Phase diagrams of lattice gauge theories with Higgs ﬁelds,

Phys. Rev. D 19, 3682 (1979). doi.org/10.1103/PhysRevD.19.3682

[16] J. E. Humphreys, Reﬂection Groups and Coxeter Groups (Cambridge University Press,

Cambridge, 1990). doi.org/10.1017/CBO9780511623646

[17] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed. (Springer,

New York, 1999). doi.org/10.1007/978-1-4757-6568-7

[18] A. M. Polyakov, Quark conﬁnement and topology of gauge theories, Nucl. Phys. B 120, 429

(1977). doi.org/10.1016/0550-3213(77)90086-4

[19] 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

[20] 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

[21] K. Symanzik, Continuum limit and improved action in lattice theories, Nucl. Phys. B 226,

187 (1983). doi.org/10.1016/0550-3213(83)90468-6

[22] A. Hamma, P. Zanardi and X.-G. Wen, String and membrane condensation on three-

dimensional lattices, Phys. Rev. B 72, 035307 (2005). doi.org/10.1103/PhysRevB.72.

035307

[23] M. Creutz, L. Jacobs and C. Rebbi, Monte Carlo study of Abelian lattice gauge theories,

Phys. Rev. D 20, 1915 (1979). doi.org/10.1103/PhysRevD.20.1915

[24] G. Arnold, B. Bunk, Th. Lippert and K. Schilling, Compact QED under scrutiny: it’s ﬁrst

order, Nucl. Phys. B Proc. Suppl. 119, 864 (2003), arXiv:hep-lat/0210010. doi.org/10.

1016/S0920-5632(03)01704-3

[25] H. B. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice (I): proof by homotopy

theory, Nucl. Phys. B 185, 20 (1981). doi.org/10.1016/0550-3213(81)90361-8

[26] L. H. Karsten, Lattice fermions in Euclidean space-time, Phys. Lett. B 104, 315 (1981).

doi.org/10.1016/0370-2693(81)90133-7

[27] M. Creutz, Four-dimensional graphene and chiral fermions, JHEP 04 (2008) 017. doi.

org/10.1088/1126-6708/2008/04/017

[28] A. Bori¸ci, Creutz fermions on an orthogonal lattice, Phys. Rev. D 78, 074504 (2008).

doi.org/10.1103/PhysRevD.78.074504

[29] S. D. Pace, S. C. Morampudi, R. Moessner and C. R. Laumann, Emergent ﬁne structure

constant of quantum spin ice is large, Phys. Rev. Lett. 127, 117205 (2021). doi.org/10.

1103/PhysRevLett.127.117205

Electromagnetism on the cubic sublattice of the FCC vacuum