Symmetry-Protected Masslessness of an Emergent U(1) Photon

Symmetry-Protected Masslessness of an Emergent U (1)

Photon

Coexisting with a Gravitational Network on the FCC

Vacuum

Raghu Kulkarni

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

raghu@idrive.com

June 2026

Abstract

Realizing both a continuous metric and a gapless U(1) gauge ﬁeld on a single emergent-spacetime

lattice raises a coexistence question: when the two sectors share degrees of freedom, does cou-

pling them spoil the fragile Coulomb phase that supports the photon? We address this within

the face-centered-cubic (FCC) vacuum of the Selection–Stitch Model (SSM). The primary ⟨110⟩

nearest-neighbor network hosts the model’s Regge-like gravitational strain sector and stabilizer

code, but its non-bipartite geometry supports only a gapped Z

gauge theory, not a photon. We

therefore place the electromagnetic sector on the bipartite ⟨111⟩ diamond–pyrochlore sublattice

of the same packing, where it maps onto the established deconﬁned Coulomb phase of U(1)

quantum spin ice. The two networks share the FCC sites but no bonds. Our result is a struc-

tural consistency statement: the shared-node coupling does not generate a photon mass. The

sublattices carry opposite inversion symmetry—centrosymmetric O

for the gravitational net-

work, tetrahedral T

for the gauge network—which restricts their magnetoelastic interaction [9]

to a gauge-invariant strain–ﬁeld-strength coupling of zincblende (d

) form, ε

rather than

. Because this coupling respects the emergent gauge symmetry, the standard Coulomb-

phase protection of the photon survives node-sharing: integrating out the strain isotropically

renormalizes the dielectric constant (slowing the emergent light) but cannot generate an A

mass. Placing a matter defect in a tetrahedral void puts it on a vertex of the charge lattice, so

its coupling to the emergent ﬂux is geometrically forced rather than inserted by hand, and the

defect carries integer emergent charge as in standard spin ice. We are explicit throughout that

the emergent photon, the diamond–pyrochlore geometry, and the gauge-invariance protection

are established physics located here within the FCC substrate; the coupling’s form is geometry-

ﬁxed but its magnitude is a free microscopic parameter, so the result is a structural consistency

proof rather than a parameter-free prediction.

1 The two-network superposition problem

The SSM treats the vacuum as an FCC tensor network carrying a stabilizer code, with matter

as defects in that network [1, 2]. Two low-energy sectors are assigned to the same crystal: a

gravitational sector, in which coarse-grained lattice strain plays the role of a Regge-like metric [3],

and an electromagnetic sector. The gravitational sector is posited here rather than re-derived: the

companion construction [3] shows that the intrinsic-lattice Regge operator reproduces linearized

Einstein (Fierz–Pauli) gravity exactly at quadratic order, while carrying a known obstruction (an

irreducible hypercubic anisotropy) at the cubic graviton vertex. The coexistence result below uses

only the linearized strain sector, and is therefore conditional on that exact linear-order result and

independent of the unresolved cubic-vertex question. The nearest-neighbor ⟨110⟩ graph is natural

for the metric and the code—it is isotropic at leading order and hosts the CSS stabilizers—but it

is non-bipartite: it contains the triangular odd cycles of the close-packed shell. A gapless compact-

U(1) photon lives only in a deconﬁned Coulomb phase, and that phase requires a bipartite charge

lattice admitting a height representation [7, 8]; on a non-bipartite lattice the compact gauge theory

is gapped (Z

) or conﬁned. The ⟨110⟩ network therefore cannot host the photon.

The same FCC packing, however, contains a second structure. The ⟨111⟩ bonds joining each

FCC site to its tetrahedral interstitial voids form a bipartite diamond lattice, and this is where we

place the electromagnetic sector. The two networks share the FCC sites but no bonds, and—as

we show—their opposite inversion symmetry is exactly what protects the photon when they are

coupled.

2 The electromagnetic sector on the diamond–pyrochlore sublat-

tice

The FCC sites together with one tetrahedral-void sublattice, joined along ⟨111⟩, form the diamond

lattice: coordination 4 and bipartite (two-colorable). Its bonds’ midpoints form the pyrochlore

lattice of corner-sharing tetrahedra, coordination 6 (Fig. 1). This is the structure of U(1) quantum

spin ice. We import its established low-energy physics [4, 5]: the ice-rule (divergence-free) constraint

on the diamond vertices leaves an extensively degenerate ground manifold, the emergent ﬁeld is

transverse, and the structure factor exhibits the pinch-point singularities (Fig. 2) that are the

ﬁngerprint of an emergent gauge ﬁeld. On this sublattice the compact U (1) theory sits in its

deconﬁned Coulomb phase and supports a gapless emergent photon. We claim no novelty for this

mechanism; the content of this section is the location of the known quantum-spin-ice photon within

the FCC substrate.

Explicitly, place a compact phase (rotor) variable e

ℓ

on each pyrochlore site ℓ (a diamond bond-

midpoint), with conjugate integer electric ﬁeld E

ℓ

, [A

ℓ

, E

ℓ

′

] = iδ

ℓℓ

′

. The lattice gauge Hamiltonian

is that of Hermele, Fisher, and Balents [4],

H = U

diamond v



ℓ∈v

vℓ

ℓ



− g

hexagons p

cos



ℓ∈∂p

ℓ



, (1)

where η

vℓ

= ±1 orients the four bonds at diamond vertex v (the discrete divergence), and the

plaquette sum runs around the elementary hexagonal loops of the pyrochlore (the discrete curl).

In the limit U → ∞ the ﬁrst term projects onto the ice-rule (Gauss-law) subspace

ℓ∈v

vℓ

ℓ

= 0

at every vertex; the surviving ring-exchange term g cos(∇ × A) then supplies the photon dynamics

within that constrained manifold. This is the standard route by which a gapless U(1) photon

emerges on the pyrochlore, imported here unchanged.

3 Matter defects and geometrically forced coupling

In the SSM a matter defect is a node trapped in a tetrahedral void [2]. The tetrahedral void is a

vertex of the ⟨111⟩ diamond charge lattice, so the defect sits directly on the charge lattice of the

emergent gauge theory. Two consequences follow, at the level of rigor the geometry supports.

The coupling is geometrically forced. Because the defect occupies a charge-lattice vertex on

which the ice-rule (Gauss-law) constraint is deﬁned, it acts as a source or sink of the emergent ﬂux

coordination 8 · NON-bipartite

(red: an odd cycle) · gapped

, no photon

110 stabilizer bonds

coordination 4 · BIPARTITE (two colors)

charge lattice for the ice-rule Gauss law · hosts photon

111 diamond sublattice

Figure 1: Two structures in one FCC packing. Left: the ⟨110⟩ nearest-neighbor network (the code and

gravitational sector) is non-bipartite—it contains triangular odd cycles—and supports only a gapped Z

gauge theory. Right: the ⟨111⟩ bonds to the tetrahedral voids form a bipartite diamond charge lattice,

which hosts the deconﬁned U (1) photon. The networks share the FCC sites but no bonds.

carried by the incident pyrochlore bonds. The coupling of matter to the emergent gauge ﬁeld is

therefore ﬁxed by the graph connectivity—it need not be inserted as a hand-written A

term.

We are precise about what this does and does not give: the connectivity forces that the defect

couples to the ﬁeld; the form of the coupling and its strength are supplied by the emergent gauge

dynamics of the Coulomb phase, not by the graph alone.

The defect carries integer emergent charge. A defect that violates the ice rule at a coordination-

4 diamond vertex is an emergent monopole. Enumerating the 2

= 16 ﬂux conﬁgurations of the four

bonds (each + in or − out) and grouping by net divergence Q gives the complete emergent-charge

spectrum:

conﬁguration # states Q

4-in / 0-out 1 +4 monopole

3-in / 1-out 4 +2 monopole

2-in / 2-out 6 0 ice vacuum

1-in / 3-out 4 −2 monopole

0-in / 4-out 1 −4 monopole

The spectrum is {−4, −2, 0, +2, +4}—even integers spaced by two, i.e. integer monopole charge

in the standard normalization, exactly the integer emergent charge of quantum spin ice [4]. The

coordination-4 vertex does not partition ﬂux into thirds.

It is worth stating explicitly how the same tetrahedral-void defect can carry integer emergent

charge and, at the same time, fractional color. The two live in diﬀerent charge spaces of the same

cage. The emergent electromagnetic charge counts ﬂux lines: the net divergence of the ± ﬂuxes on

4 2 0 2 4

(

h h

0) [units of ]

(0 0

) [units of ]

pinch

point

Emergent-field structure factor

(

)

(hhl) plane pinch points at reciprocal points

0.0

0.2

0.4

0.6

0.8

1.0

Figure 2: Computed emergent-ﬁeld structure factor in the (hhl) plane on the ⟨111⟩ pyrochlore sublattice

(periodic cluster). The bowtie pinch-point singularities at the reciprocal points are the momentum-space

signature of a divergence-free emergent gauge ﬁeld—the Coulomb-phase ﬁngerprint underlying the gapless

photon.

the four ⟨111⟩ bonds, enumerated above, which is integer. The color charge counts perfect matchings

of the four-vertex cage K

: the three matchings furnish the color qutrit and the fractional (1/3)

charges of the strong sector. The ‘3’ of the thirds is the combinatorial 3 of the K

matchings, not

the coordination 4 of the ice vertex; the two counts are independent, so a single defect is an integer

emergent-electric monopole and a fractional-color object simultaneously, with no conﬂict and no

shared origin.

4 Coexistence without a photon mass

The two networks share the FCC nodes, so a node displacement—a gravitational strain ﬂuctuation—

perturbs the ⟨111⟩ gauge bonds. The question is whether integrating out these strain ﬂuctuations

generates a photon mass A

and conﬁnes the Coulomb phase. It does not, and the reason is a

symmetry mismatch between the two sublattices.

Opposite inversion symmetry. At an FCC site the twelve ⟨110⟩ bonds are centrosymmetric

): their odd moments vanish, and a displacement couples to them only as a rank-2, parity-

even strain ε. The four ⟨111⟩ bonds are tetrahedral and non-centrosymmetric (T

): their rank-3

moment is nonzero, with a single independent component—the zincblende d

form. The induced

magnetoelastic coupling is therefore g d

ijk

: the piezoelectric tensor contracts a symmetric

strain to a single vector index, and that vector is the gauge-invariant ﬁeld strength E, not the

potential A.

Gauge invariance protects the mass. A mass term A

is not invariant under the emergent

gauge transformation A → A + ∇λ, which is exact in the deconﬁned Coulomb phase. A term

forbidden by an exact symmetry cannot be radiatively generated by an interaction that respects

the symmetry. The coupling ε

is a product of two gauge-invariant objects (the strain is gauge-

neutral; E is the ﬁeld strength), so it cannot generate the non-invariant mass at any order. This is

the standard gauge-invariance protection of an emergent photon [6, 4]; the contribution here is that

the coupling induced by node-sharing is of the gauge-invariant εE type, so the protection survives

the superposition. A second, independent layer reinforces it: the centrosymmetric ⟨110⟩ network

has all odd moments vanishing and a uniform displacement sources no net gauge potential (the

four ⟨111⟩ bond vectors sum to zero), so the gravitational sector has no channel to source A even

before gauge invariance is invoked.

What the coupling does instead: renormalize the Maxwell term. Being gauge-invariant and mass-

less, the coupling can only renormalize the emergent dielectric constant. Integrating out the strain

gives the standard piezoelectric contribution to the permittivity, δϵ

= g

i,jk

−1

)

jk,mn

l,mn

with C the FCC ⟨110⟩ elastic stiﬀness. Direct evaluation ﬁxes its structure and sign, and proves

the isotropy explicitly. The four tetrahedral bond directions are

1,...,4

√



(1, 1, 1), (1, −1, −1), (−1, 1, −1), (−1, −1, 1)



Their rank-3 moment d

ijk

ˆn

has, by direct summation, only the fully-mixed component

nonzero: d

xyz

= d

xzy

= · · · =

√

≡ d

, while every d

xxx

, d

xxy

vanishes (each contains an odd

power of some component, which cancels over the four sign-balanced vectors). This is exactly

the zincblende (

43m) piezoelectric tensor [10]. The FCC ⟨110⟩ central-force stiﬀness is cubic,

with Voigt compliance S

, S

and shear S

(and S

= S

= 0). The induced permittivity is

δϵ

= g

jk,mn

i,jk

jk,mn

l,mn

. Because d

i,jk

is nonzero only when {i, j, k} = {x, y, z}, the ﬁeld

index i ﬁxes the strain to the single shear with the other two axes: i = x selects ε

(Voigt 4), i = y

selects ε

(Voigt 5), i = z selects ε

(Voigt 6). Hence

δϵ

= g

, δϵ

= g

, δϵ

= g

, δϵ

x=l

= g

4,5-type

= 0, (2)

and cubic symmetry forces S

= S

with the oﬀ-diagonal shear compliances zero. Therefore

δϵ

= g

: the two anisotropic inputs contract, through the cubic symmetry shared by

both sublattices, to an exactly isotropic scalar. Numerically (unit spring constant, a = 2) the

shear compliance is S

and δϵ = 0.593 g

I, with all oﬀ-diagonal entries identically zero—the

renormalization shifts the emergent speed of light without inducing birefringence, a non-trivial

consistency requirement the construction meets. Integrating out a stable elastic mode can only

add to the E

coeﬃcient, so δϵ > 0 and the emergent speed of light is always reduced, c

1 + δϵ/ϵ

, controlled by the shear compliance of the ⟨110⟩ bonds. The magnitude is δϵ ∼ g

carrying the free microscopic coupling and the bare bond stiﬀness; it is not ﬁxed by geometry.

5 Scope and status

Reached. Within one FCC packing, the electromagnetic sector is placed on the bipartite ⟨111⟩

diamond–pyrochlore sublattice, where it maps to the deconﬁned Coulomb phase of U (1) quantum

spin ice (imported); the coexisting gravitational strain sector on the centrosymmetric ⟨110⟩ network

couples to it only through the gauge-invariant zincblende εE magnetoelastic interaction, so the

photon stays exactly massless and the induced dielectric renormalization is isotropic and speed-

reducing. A matter defect in a tetrahedral void is forced by connectivity to source the emergent

ﬂux and carries integer emergent charge.

Open. The overall magnitude of the coupling (∼ g

) is a free microscopic parameter, so the

result is a structural consistency proof, not a parameter-free quantitative prediction. Whether the

physical FCC Hamiltonian realizes the deconﬁned phase for the speciﬁc ⟨111⟩ model is inherited

from the quantum-spin-ice mapping and not independently established here. The relation to the

electroweak sector is open in a speciﬁc way: an electroweak-shaped SU(2)×U(1) structure has been

identiﬁed on the centrosymmetric ⟨110⟩ square faces, whereas the propagating photon requires the

non-centrosymmetric ⟨111⟩ diamond. Since the physical photon is the unbroken combination that

survives electroweak symmetry breaking rather than the electroweak U(1) itself, the map from the

⟨110⟩ electroweak structure to the ⟨111⟩ propagating photon is a well-posed open problem, not a

contradiction; the two sectors are consistent precisely because they occupy sublattices of opposite

inversion symmetry.

Not claimed. No new condensed-matter physics. The emergent photon, the diamond–pyrochlore

geometry, the gauge-invariance protection of masslessness, and the piezoelectric renormalization of

permittivity are all established. The content of this paper is the demonstration that these known

results can be embedded in the SSM gravitational model without the two sectors spoiling one

another, with computed geometric and consistency evidence.

Statement Status

⟨110⟩ non-bipartite ⇒ gapped Z

, no photon derived / known

⟨111⟩ diamond bipartite (coord. 4); midpoints pyrochlore (co-

ord. 6)

derived / computed

Deconﬁned U(1) photon on the ⟨111⟩ sublattice imported [4, 5]

Defect on a void couples to the ﬂux by connectivity (form from

dynamics)

derived

Defect carries integer emergent charge ({−4, −2, 0, 2, 4}; no

thirds)

derived / computed

vs T

: coupling is εE (gauge-invariant), not εA derived / computed

Photon stays massless under node-sharing derived (standard protection)

Dielectric renormalization isotropic (no birefringence), sign-

ﬁxed (slows light)

derived / computed

Coupling magnitude (∼ g

) open / microscopic

Map to the ⟨110⟩ electroweak structure through EW breaking open

5.1 The electroweak map and a shortcut that fails

The ⟨110⟩ nearest-neighbor shell is a cuboctahedron whose fourteen faces divide, by direct enu-

meration, into eight triangles oriented along ⟨111⟩ and six squares oriented along ⟨100⟩. The eight

triangles are the close-packed planes that carry the color sector; the six squares pair antipodally

into three ⟨100⟩ axes whose rotation generators J

, J

close as so(3)

∼

su(2), and together with

the diagonal bond-phase U(1) the rank matches SU(2) × U(1). We take this as a group-theoretic

structure carried by the geometry—a necessary condition, not a dynamical gauge theory—and note

that FCC centrosymmetry forbids a static chiral assignment on it: as established in Section 4, the

interlayer up- and down-environments of any bulk node are inversion partners, so a handedness

cancels under parity. An electroweak-shaped structure is therefore present on the ⟨110⟩ squares

but its chirality is statically obstructed by the same centrosymmetry that protects the photon’s

masslessness.

The relation between this ⟨110⟩ structure and the ⟨111⟩ propagating photon deserves a precise

framing, because it is the natural next problem and it contains a tempting but incorrect shortcut.

Geometrically, embedding the diamond charge lattice requires selecting one of the two tetrahedral

interstitial sublattices to host the photon; that choice reduces the site symmetry from O

to T

This O

→ T

reduction is a spontaneous selection of one orientation over its inversion partner,

structurally analogous to a Higgs-like choice of vacuum, and it is what makes the gauge sublattice

non-centrosymmetric (hence piezoelectrically active). The open question is dynamical: how does

an unbroken U(1) subgroup of an SU(2) × U(1) structure carried on the ⟨110⟩ square faces project,

after symmetry breaking, onto the ⟨111⟩ bonds that carry the propagating photon?

We ﬂag one identiﬁcation that does not work, so that it is not mistaken for a lead. The ﬁxed

angle between the ⟨110⟩ and ⟨111⟩ directions is arccos

2/3 ≈ 35.26

◦

, a crystallographic constant

of any cubic lattice. Identifying it with the weak mixing angle fails on two counts: it would give

sin

= cos

(35.26

◦

) = 2/3 ≈ 0.667, whereas the measured value is sin

≈ 0.231; and θ

is a running coupling that varies with energy scale, so a ﬁxed geometric angle cannot represent

it even in principle. Any geometric derivation of θ

in this framework must therefore produce it

dynamically, from the coupling that maps the two sublattices, not kinematically from a bond angle.

We record the shortcut here only to close it.

A Methods

Structures are built on a periodic FCC packing (conventional cell a = 2; sites at integer coordinates

of even sum). Bipartiteness is tested by two-coloring; coordination by minimum-image neighbor

counting. The ⟨111⟩ diamond is the FCC sites with the interstitial sublattice shifted by (

)a/2,

bonded at squared distance

; pyrochlore sites are the diamond bond-midpoints. The emergent-

ﬁeld structure factor uses the divergence-free projector on the ice-rule manifold. The elastic stiﬀness

is the central-force rank-4 bond tensor of the ⟨110⟩ network; the piezoelectric tensor is the rank-3

moment of the four ⟨111⟩ bonds; the dielectric renormalization is their contraction δϵ = g

d C

−1

evaluated in an orthonormal symmetric-tensor basis. The integer-charge enumeration counts the

ﬂux conﬁgurations of a coordination-4 vertex.

Code availability

This work uses no external or measured data; all reported quantities are computed from the FCC

lattice geometry. The Python scripts that reproduce every reported number and ﬁgure are avail-

able at github.com/raghu91302/ssmtheory/blob/main/photon verification.zip, and run on

a standard scientiﬁc Python environment (numpy, scipy, matplotlib). The magnetoelastic cou-

pling magnitude is a free microscopic parameter (g); the geometry ﬁxes the tensor structure and

sign of the induced dielectric renormalization, not its overall scale.

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