Photon–Gravity Coexistence on the FCC Vacuum

Photon–Gravity Coexistence on the FCC Vacuum:

Gauge Protection, a

√

3 Speed Obstruction,

and Conditional Domain-Boundary Universality

Raghu Kulkarni

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

raghu@idrive.com

Abstract

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

lattice raises two questions: whether coupling the sectors spoils the fragile Coulomb phase

supporting the photon, and whether they propagate at equal speeds, as GW170817 requires

to |c

− c

|/c ≲ 10

−15

. We address both in the face-centered-cubic vacuum of the Selection–

Stitch Model. The ⟨110⟩ network hosting the gravitational sector is non-bipartite and supports

only a gapped Z

gauge theory; the photon therefore lives on the bipartite ⟨111⟩ diamond–

pyrochlore sublattice, mapping onto the deconﬁned Coulomb phase of U(1) quantum spin ice.

The sublattices share sites but no bonds and carry opposite inversion symmetry (O

versus

), restricting their coupling to the gauge-invariant zincblende form ε

: the photon stays

exactly massless, the dielectric renormalization is isotropic, and a trapped defect sources the

emergent ﬂux with integer charge. The construction faces an obstruction: the sublattice speeds

diﬀer by the coordination invariant

√

3. We identify a conditional resolution in

the vacuum’s polycrystalline structure: when the grain-boundary crossing cost dominates intra-

grain traversal, the observed speed is set by the inter-grain stitch, a frame re-identiﬁcation we

prove acts on spin-1 and spin-2 as representations of the same rotation, without attenuation or

mixing. Under two stated dynamical assumptions, boundary dominance and delay universality,

= c

exactly, with residual corrections suppressed by (a/λ)

∼ 10

−75

. Collective re-encoding

between misaligned stabilizer frames supplies the natural mechanism class, with required factor

κN

≃ 35–40. GW170817 is thereby converted from a falsiﬁer into a constraint on the stitch

dynamics.

1 Background: the Selection–Stitch Model

We brieﬂy ﬁx the terms used below; full deﬁnitions are given in Refs. [1, 3, 2]. In the Selection–

Stitch Model (SSM), the early universe crystallizes from a frustrated K=4 tetrahedral foam into

a K=12 face-centered-cubic (FCC) lattice through a K=4 → K=12 phase transition [2]. The

FCC vacuum is a tensor network carrying a stabilizer code [1]; ordinary matter is modeled as a

region that failed to ﬁnish crystallizing—a trapped node in a tetrahedral void, whose geometry ﬁxes

the observed quark charges, color content, and baryon masses [2]. The FCC lattice is assembled

sheet by sheet from two elementary operations: a stitch, which welds a two-dimensional close-

packed hexagonal sheet into K=6 in-plane coordination, and a lift, which stacks successive sheets

in the ABC sequence to reach the K=12 cuboctahedral shell. The twelve nearest-neighbor bond

vectors {

}

j=1

of this shell deﬁne the rank-2 structure tensor S

µν

= 4 δ

µν

, an exact

consequence of the K=12 coordination in d=3 [2]; the isotropy of S

µν

and the vanishing of the odd-

rank tensor T

µνλ

= 0 (by inversion symmetry) are the origin of Lorentz-isotropic

dispersion in the model.

The emergent vacuum is polycrystalline: the crystallization nucleates in many regions and sat-

urates into FCC crystallites of diﬀering orientations, so the bulk consists of grains meeting at grain

boundaries. This is an established feature of the construction: the simulated bulk coordination dis-

tribution peaks at K=12 with a residual population at K=10, 11 concentrated at the interfaces be-

tween misoriented crystallites [2]. That residual population should be read as a transient signature

of incomplete healing, not the persistent structure of a mature boundary. The same reference es-

tablishes that under-coordination vanishes in the thermodynamic limit (f

K=12

= 1 −αN

−1/3

→ 1):

the saturation dynamics that builds the bulk keeps acting until every node reaches K=12 or hosts

a protected matter-type remnant. A mature boundary is therefore coordination-complete: it carries

misorientation without missing bonds. That this is geometrically possible is an elementary crys-

tallographic fact which we veriﬁed directly for the FCC lattice: at a coherent twin-type interface,

every node—including on the boundary plane itself—retains the full K=12 shell, while the local

bond-direction frames on the two sides diﬀer, so that crossing still requires a frame re-identiﬁcation.

Misﬁt that cannot be resolved into such a coordination-complete interface is trapped as localized

matter-type defects rather than persisting as extended under-coordination. Throughout this paper

we use grain-boundary transport (or, loosely, the inter-grain stitch) to mean the operation that

re-identiﬁes the coordinate frame of one crystallite with that of its misoriented neighbor across

such a coordination-complete boundary—distinct from the intra-sheet stitch operator above, which

acts within a single grain. It is this frame re-identiﬁcation between normal K=12 structure, not

any population of coordination defects, that plays the central role below.

2 The two-network superposition problem

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; Section 9 shows that this single-crystal anisotropy is removed in

the polycrystalline vacuum by orientation averaging. The coexistence result below uses only the

linearized strain sector, and is therefore conditional on that exact linear-order result alone. 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.

coordination 12 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.

3 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.

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.

4 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

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.

We state 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 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.

5 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 [9] 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 within a grain 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. This shift applies to the intra-grain propagation speed; Section 8 shows that in the

boundary-dominated polycrystalline vacuum it is washed out of the macroscopic speed along with

the coordination split itself.

6 The speed obstruction

The construction of Sections 2–5 is structurally consistent: the photon is massless and the coupling

between the sectors is gauge-invariant. It faces, however, a quantitative obstruction that the

masslessness argument does not touch: the two sublattices propagate their excitations at diﬀerent

speeds.

The separation forces a speed mismatch. In a hopping description the long-wavelength dis-

persion of a sector on a network with unit bond vectors {

}

j=1

is governed by the rank-2 struc-

ture tensor S

µν

. For any set of K unit vectors whose second moment is isotropic,

µν

= (K/d) δ

µν

, so the squared signal speed is proportional to the coordination-to-dimension ratio

K/d. The ⟨110⟩ graviton network has K = 12 (S = 4 I); the diamond photon sublattice has K = 4

(S =

I). Hence

4/3

√

≈ 0.577. (3)

This ratio is a graph invariant. It does not depend on the Euclidean bond lengths (which cancel

under a common signal rate v

lat

= ℓ/τ), on the microscopic time-step, or on any coupling constant;

distance measured as an optimized (geodesic) hop count leaves it unchanged, because the coordi-

nation number is a property of the graph prior to any metric. We record it as the obstruction

the construction must overcome: measured against GW170817 [11], an O(1) split of this size is

excluded (Fig. 3).

Antagonism. The obstruction is structural. Gauge protection of a massless U (1) requires a

sparse, bipartite (or otherwise frustration-free) host; exact-Einstein isotropy of the graviton requires

a dense, high-symmetry network. These conditions pull toward opposite coordination numbers, and

the signal speed

K/d then diﬀers. We veriﬁed that this antagonism recurs on the four-dimensional

lattice used for the gravitational sector: the exact-Fierz–Pauli graviton requires the full 24-cell

(K = 24, non-bipartite), while a gauge-protected photon requires a bipartite subnetwork (K = 8),

photon

111 diamond

graviton

110 network

structure eigenvalue

1.333

4.000

(a) single-crystal coordination

c c

0.0

0.5

1.0

1.5

2.0

signal speed

(units

lat

)

(b) speed split

= 1/

Figure 3: The two-sublattice speed obstruction. (a) The single-crystal structure-tensor eigenvalue K/d

for the two hosts: the graviton’s ⟨110⟩ network (K=12, K/d=4) and the photon’s ⟨111⟩ diamond sublat-

tice (K=4, K/d=4/3). (b) The resulting long-wavelength signal speeds

K/d diﬀer by the coordination-

invariant factor

√

3, giving c

= 1/

√

3 ≈ 0.577. The ratio is independent of bond length, time-step, and

coupling.

and the same

ratio reappears. Reshuﬄing the lattice within the class of regular networks

does not remove it.

7 Coarse-graining over grains

The vacuum is polycrystalline—an assembly of FCC grains of ﬁnite size, bound at misaligned

domain boundaries by the inter-grain frame transport deﬁned in Section 1 (the grain-boundary

“stitch”), a rotation that re-identiﬁes one grain’s coordinate frame with the next. We now ﬁx the

single length scale that the argument requires, and show that a propagating photon or graviton

necessarily spans an enormous number of grains as an observed fact rather than an assumption.

In the SSM each lattice bond is exactly one entanglement pair (a Bell pair) [2]. In emergent-

spacetime tensor-network vacua the entanglement bond is placed at the Planck length ℓ

≃ 1.6 ×

−35

m [18, 19]; we take the bond length L ≃ ℓ

accordingly. This is the only scale imported

from outside the SSM; grain sizes, the crystallization cascade, and the coordination statistics are

all ﬁxed internally in units of L [2, 1].

The propagating excitations we observe have wavelengths vastly larger than this bond length.

A photon of visible wavelength λ ∼ 5 × 10

−7

m corresponds to λ/L ∼ 3 × 10

bond lengths; a

gravitational wave in the LIGO band, λ ∼ 10

m, corresponds to λ/L ∼ 6 × 10

. Since a wave

of wavelength λ is a coherent object spread over a region of order λ, and cannot resolve structure

below λ, any observed photon or graviton is coherent across 10

–10

bonds, and therefore across

an enormous number of grains (each grain being a ﬁnite multi-bond crystallite). The coarse-graining

over grains is thus not a postulate about a hypothetical fundamental step: it is a direct consequence

of the observed wavelength-to-bond ratio, with the bond length ﬁxed at the Planck scale by the

entanglement identiﬁcation above. Equivalently, over any macroscopic path such a wave crosses an

enormous number of grain boundaries; we refer to this grain-boundary-dominated propagation as

the macro-hop regime.

The immediate consequence is that a propagating excitation is denied access to single-grain

(ultraviolet) physics: it exists only as a coarse-grained, macroscopic object averaged over many

grains. This is precisely the regime in which the two results below operate—the stitch bottleneck

(Section 8) and orientation averaging (Section 9)—and it is the same coarse-graining in both. The

one length taken from prior art is the Planck-scale entanglement bond; the wavelength ratio that

forces multi-grain spanning is measured.

Uniformity from structural homogeneity. Boundary-dominated transport makes the vacuum

strongly non-uniform at the grain scale: an excitation spends far longer at each boundary than

inside each grain interior. Macroscopic uniformity is nonetheless guaranteed, and—because the

mature boundaries are coordination-complete (Section 1)—it is guaranteed structurally, not just

statistically. Every grain interior is built of identical K=12 cells, and every mature boundary is a

frame re-identiﬁcation between such cells: the delay is a property of a standardized piece of structure

repeated throughout the vacuum, not of a random population of defects whose costs would have

to conspire. Statistical homogeneity remains as the secondary condition governing the orientation

and size distribution of the domains: every macroscopic path samples N

cross

≃ D/ℓ

grain

boundaries

drawn from the same distributions, with path-to-path ﬂuctuations ∼ 1/

√

cross

, negligible for

astrophysical baselines. Kibble-type nucleation at the crystallization transition naturally produces

homogeneous domain statistics, and the observed constancy of c across the sky and over cosmic

time is the corresponding consistency check; coordination-complete (twin-type) interfaces are also

the least mobile boundaries a crystal admits, strengthening the expectation that the frozen domain

structure and hence c do not drift.

8 The stitch bottleneck and its sector-independence

A photon or graviton originates in a local event and propagates by ordinary lattice steps; it is

not a single object the size of its wavelength, and no single cell emits a macroscopic excitation.

What is macroscopic is the path: over any distance D the excitation crosses N

cross

≃ D/ℓ

grain

grain boundaries, one per grain traversed. The propagation time is the sum of time spent moving

coherently within aligned grain interiors and time spent crossing the misaligned boundaries between

them:

t =

intra

+ N

cross

stitch

, c

macro

1/c

intra

+ τ

stitch

/ℓ

grain

−−−−−−−−−−−−−→

stitch

≫ℓ

grain

intra

ℓ

grain

stitch

(4)

When the per-grain boundary cost dominates the intra-grain traversal time, the macroscopic speed

is ℓ

grain

/τ

stitch

, independent of the intra-grain speed c

intra

(Fig. 4). The suppression is a per-grain

eﬀect, not a cumulative one. The arrival-time lag ∆t = N

cross

ℓ

grain

(1/c

intra

−1/c

intra

) accumulates

linearly with the number of grains and is never erased; but the fractional split

∆c

ℓ

grain

(1/c

intra

− 1/c

intra

)

ℓ

grain

intra

+ τ

stitch

is independent of N

cross

: each grain’s contribution to the lag is diluted by that same grain’s sector-

blind boundary cost, grain by grain, not cumulatively. The photon genuinely arrives later, by

(∆c/c) t

travel

; over an astrophysical baseline this is precisely the photon–gravitational-wave arrival-

time coincidence that GW170817 bounds. Taking the two intra-grain speeds in the ratio 1:

√

3 and

a common per-crossing cost, the macroscopic ratio c

rises monotonically to unity: 0.577 →

0.672 → 0.891 → 0.986 → 0.99985 as the per-path boundary cost τ

stitch

/ℓ

grain

grows through

0, 0.5, 5, 50, 5000 (units c

−1

). This is grain-boundary–dominated transport, as in the electrical

per-grain boundary cost

stitch

grain

(units

)

0.5

0.6

0.7

0.8

0.9

1.0

(intra-grain)

universality

boundary crossings wash out the split

Figure 4: The stitch bottleneck washes out the intra-grain speed split. As the sector-independent per-grain

boundary cost τ

stitch

/ℓ

grain

grows, the macroscopic ratio c

rises monotonically from the intra-grain value

√

3 (dashed) to unity (dotted): the sector-dependent micro-speeds are subleading to the shared boundary

crossings accumulated along the path.

resistance of ordinary polycrystals, where boundaries rather than grain interiors set the macroscopic

behavior.

Universality thus holds provided (i) the per-grain boundary cost dominates the intra-grain

traversal time, so that boundary crossings rather than grain interiors set the propagation time,

and (ii) the crossing cost τ

stitch

is the same for both sectors. Condition (i) is an assumption

about the stitch dynamics, not derived here, and it is quantitatively demanding: from Eq. (4),

1 − c

≈ (1/c

intra

− 1/c

intra

) ℓ

grain

/τ

stitch

, so satisfying the GW170817 bound of 10

−15

requires

stitch

≳ 10

ℓ

grain

: a boundary crossing must be slower than a grain traversal by roughly ﬁfteen

orders of magnitude. We state this explicitly in the scope ledger. Condition (ii) is the crux: if the

photon and graviton paid diﬀerent boundary costs, the split would merely relocate from the grain

interiors to the stitch, with c

→ τ

stitch

/τ

stitch

. We now establish the kinematic part of condition

(ii), and then state its remaining dynamical part explicitly as an assumption.

Proposition 1 (Kinematic sector-blindness of the stitch). Let the stitch across a domain boundary

be a ﬁxed rotation R ∈ SO(3) (the grain misalignment). Then R acts on the spin-1 photon ﬁeld

and the spin-2 graviton ﬁeld as two representations of the same rotation, generated by the same

misalignment angle; each action is a norm-preserving isometry; and, the two being inequivalent

irreducible representations, the crossing cannot mix them. The frame transport therefore attenuates

neither ﬁeld, converts neither into the other, and involves no representation-dependent angle.

Proof. The photon ﬁeld is a vector A

, carrying the deﬁning (spin-1) representation A 7→ RA. The

graviton ﬁeld is a symmetric traceless tensor h

, carrying the spin-2 representation h 7→ R h R

⊤

Both are representations of the single group element R. If R has rotation angle θ, its spin-1

eigen-phases are {−θ, 0, +θ} and its spin-2 eigen-phases are {−2θ, −θ, 0, +θ, +2θ}: the higher-spin

phases are integer multiples of the same fundamental angle θ, veriﬁed numerically (e.g. θ = 0.3

gives spin-1 phases {−0.3, 0, 0.3} and spin-2 phases {−0.6, −0.3, 0, 0.3, 0.6}). No independent angle

enters. The map h 7→ R h R

⊤

preserves the Frobenius norm, ∥R h R

⊤

∥ = ∥h∥, since R is orthogonal;

the spin-1 map A 7→ RA likewise preserves ∥A∥. Each crossing is thus a unitary parallel transport

that re-expresses the ﬁeld in the neighboring frame without attenuation. Finally, spin-1 and spin-2

are inequivalent irreducible representations of SO(3), and a group element acts block-diagonally

on inequivalent irreducible representations: the crossing cannot convert a photon into a graviton

or vice versa.

The physical content is that the stitch acts on the frame, not on the internal dynamics of a sector.

A single rotation transports every tensor ﬁeld simultaneously; it cannot “know” whether it carries

a vector or a tensor, because both are rotated by the same R. We are explicit, however, about what

the Proposition does and does not establish. It proves that the stitch is kinematically sector-blind

at the level of frame transport: no attenuation, no photon–graviton conversion, no representation-

dependent angle. Equality of the dynamical crossing time, τ

stitch

= τ

stitch

, is a stronger statement:

in ordinary wave physics, boundary delays can depend on mode-matching conditions, impedance

mismatch, or vector-versus-tensor scattering phase shifts even when the bulk transport does not.

Equality of the crossing times therefore requires the additional assumption that the boundary

delay depends only on the frame re-identiﬁcation and not on representation-dependent boundary

scattering. We state this as the construction’s second dynamical assumption; the kinematic result

makes it natural—the transport itself supplies no representation-dependent structure on which a

diﬀerential delay could depend—but does not by itself derive it.

Constraints on the stitch dynamics. The two dynamical assumptions are not unconstrained:

any microscopic stitch dynamics that realizes them must also leave the vacuum transparent and

dispersionless, and both requirements are quantitatively mild for this geometry. Elastic scattering

at boundaries is in the extreme Rayleigh regime—the wave is enormously longer than the grains—

with per-crossing scattering ∼ (2πℓ

grain

/λ)

∼ 10

−99

for 10

-bond grains at visible wavelengths,

giving an optical depth ∼ 10

−41

over a Hubble distance: the polycrystal is transparent for the

same reason nanograined ceramics are, with roughly forty orders of magnitude to spare. Dispersion

from homogenizing the periodic boundary delays enters at (2πℓ

grain

/λ)

∼ 10

−50

; and the frame

rotation itself is achromatic, acting identically on every frequency component, so the proven part of

the mechanism introduces no frequency dependence. Any hidden chromaticity of τ

stitch

is bounded

empirically by gamma-ray-burst arrival-time measurements, which constrain the energy dependence

of the photon speed over Gpc baselines at the 10

−15

level [15]; the natural expectation for a

geometric transport—achromatic—is consistent, and a strongly chromatic stitch would already be

excluded. The one loss channel not closed by the Proposition is inelastic excitation of internal

boundary modes; the observed transparency of the universe bounds its per-crossing cross-section

to be extraordinarily small, and its derivation belongs with the other open items of the stitch

dynamics.

Nature of the crossing: an active process, not a defect barrier. The transparency

requirement also identiﬁes what kind of process the crossing can and cannot be. A passive

boundary—a static defect layer, barrier, or region of anomalous local response—obeys a delay–

bandwidth constraint: a weak link of relative strength t

′

delays only by reﬂecting, with trans-

mission T = 4(t

′

/t)

/[1 + (t

′

/t)

]

collapsing as the link weakens, and the one passive route to

unit transmission, resonant tunneling, is transparent only within a window of width ∆ω ∼ 1/τ,

so a passive delay of 10

lattice ticks would transmit only a ∼ 10

−15

fraction of the band. Over

∼ 10

crossings per Hubble path, either alternative renders the universe opaque to broadband

light. The observed transparency therefore excludes any passive origin for a dominant boundary

delay: the thin under-coordinated interface layer (the K=10, 11 ﬁngerprint of Ref. [2]) marks where

the misaligned frames meet, but scattering from that imperfection is not, and cannot be, the delay

mechanism. Nor are the sparse accidental proximity bonds that bridge a misaligned interface a

crossing route: a Haar misalignment with generic registration decorrelates the two grains’ sublattice

colorings, so such incidental bonds do not continue the photon’s bipartite gauge structure across

the boundary and cannot carry the sector. What remains is the crossing the construction actually

posits: the active frame re-identiﬁcation itself—capture into the boundary transport, translation

between the two internally perfect K=12 frames, release into the far grain—a process that is trans-

parent, achromatic, and sector-blind by its nature (the Proposition), and whose duration τ

stitch

a property of that re-identiﬁcation between intact grains, not of boundary damage. No current

analysis bounds this duration in either direction; the dominance assumption is a statement about

it, and it is the model’s honest unknown.

The crossing as collective stabilizer re-synchronization. The natural mechanism class for

that unknown can nonetheless be identiﬁed, and it changes the character of the dominance as-

sumption. A grain boundary is more than a geometric interface: in a vacuum that is a stabilizer

code [1], it is a mismatch between two stabilizer-frame conventions—the two grains deﬁne their

check operators on rotated, mutually incommensurate lattices. To move a photon or graviton across

it, the excitation must be re-encoded from one grain’s local stabilizer frame into the neighboring

grain’s rotated frame: a collective syndrome-matching process over a boundary patch, not a nearest-

neighbor hop. The relevant timescale is therefore not the microscopic a/c

intra

of a lattice step but a

logical re-stitching time, and transition amplitudes for processes that require N

boundary degrees

of freedom to act coherently are generically exponentially suppressed in N

: τ

stitch

∼ t

lat

κN

for

order-unity κ. On this scaling, the required factor τ

stitch

≳ 10

ℓ

grain

corresponds to κN

≃ 35–

40, i.e. a modest collective action or entropy cost. For κ = O(1), this would involve a boundary

patch of order tens of coherently re-encoded degrees of freedom (comparable, for orientation, to

the bond count of a single coordination cell). We emphasize that this is an order-of-magnitude

plausibility estimate, not a computed value. Two features of this identiﬁcation deserve comment.

First, it closes the shortcut worry structurally: a logical re-encoding is gated by its completion

over the excitation’s code support—fault-tolerant translation of logical content is not divisible into

cheap partial steps—so incidental interface bonds, which carry no logical content, cannot bypass

it. Second, we are explicit about its status: this identiﬁes the mechanism class and its scaling,

not the values of κ or N

. Known fault-tolerant costings do not settle them—standard lattice-

surgery merges between aligned patches complete in O(d) syndrome rounds [16], but that result

does not transfer to re-synchronization between incommensurate frames, a protocol that has not

been constructed. The dominance assumption is thereby restated in its physical form: the inter-

grain re-encoding is a high-order collective process over a patch of a few tens of degrees of freedom.

Its derivation reduces to constructing the frame-translation protocol and computing its exponent.

A further consequence ties back to Section 5: the piezoelectric dielectric renormalization δϵ ∼

shifts the photon’s intra-grain speed by an amount carrying the free microscopic coupling g.

In the boundary-dominated regime this shift is washed out along with the coordination split itself:

the macroscopic speed is insensitive to every intra-grain detail, including the unknown magnitude

of the magnetoelastic coupling. The same mechanism that removes the

√

3 split removes the g-

dependence of the emergent light speed.

9 Universality and the residual

Combining the two conditions: when boundary crossings dominate the propagation time (con-

dition i) and the crossing cost is sector-independent (condition ii—kinematically established by

Proposition 1, dynamically assumed as stated above),

= c

ℓ

grain

stitch

, (5)

a single macroscopic speed ﬁxed by the domain-boundary network’s transition rate, common to

all sectors. The emergent light-cone is a property of the vacuum’s topological construction, not

of any particle. The interpretation is that the photon and graviton are distinct excitations that

diﬀer at the single-crystal level—this is precisely the

√

3 split of Eq. (3)—but coincide at the

polycrystalline level, because no excitation ever propagates within a single crystal: every macro-

hop crosses grains, and the shared boundary transport sets one speed. Universality here is a

property of the propagation, not of a shared origin; the two sectors remain diﬀerent objects, and it

is the grain-boundary bottleneck, not a common source, that renders their observed speeds equal.

The leading intra-grain split is not suppressed but removed: it does not enter the stitch-

dominated speed at all. The surviving corrections are subleading. The residual direction-dependence

(from the graviton’s rank-4 lattice anisotropy) and any residual sector-dependence enter the dis-

persion at order (ka)

= (2πa/λ)

beyond the isotropic term. With a Planck-scale spacing a ∼

1.6 × 10

−35

m and a gravitational-wave wavelength λ ∼ 10

m, this suppression is (a/λ)

∼ 10

−75

so the residual |c

−c

|/c is of order 10

−75

, far below the GW170817 bound of 10

−15

. The coarse-

graining enforced by the large wavelength-to-bond ratio is what guarantees the wave never resolves

the lattice-scale structure that would otherwise distinguish the sectors.

Orientation averaging. The graviton’s single-crystal anisotropy is, in addition, removed by

ensemble averaging over grain orientations, a second and independent mechanism. In a single FCC

grain the elastic (rank-4) response is only cubically symmetric, so a shear (graviton) wave carries

a direction-dependence—the irreducible rank-4 lattice anisotropy, of relative magnitude ≈ 0.20

for the ⟨110⟩ network. In the polycrystalline vacuum the macro-hop wave samples an ensemble

of randomly oriented grains and experiences their orientation average. The Haar (orientation)

average of any rank-4 tensor equals its isotropic projection: the cubic cross-terms cancel exactly,

leaving a strictly isotropic tensor characterized by two directionless (Lam´e) parameters. This is

the Voigt–Reuss–Hill result of continuum elasticity. We conﬁrm it directly: averaging the FCC

cubic structure tensor over N random SO(3) orientations drives the anisotropy to zero as 1/

√

(0.054, 0.019, 0.0052, 0.0012 at N = 10, 10

, 10

), with the ensemble mean exactly isotropic

(Fig. 5).

For a wave of wavelength λ the number of grains sampled is N

grain

∼ λ/ℓ

grain

, smaller than the

bond count λ/L but still enormous for any reasonable grain size (e.g. N

grain

∼ 10

for grains of 10

bonds at visible wavelengths, and far larger in the gravitational-wave band); the residual orientation

ﬂuctuation ∼ 1/

grain

is then negligible, and the (ka)

suppression multiplies it further. The

single-crystal 0.20 anisotropy is thus not a macroscopic feature of the emergent spacetime at all: it

is a microscopic artifact of a single grain, washed out both by orientation averaging (mean exactly

isotropic) and by coarse-graining (residual (a/λ)

). The emergent vacuum is isotropic.

The nonlinear cubic-vertex obstruction. The same mechanism bears on the residual anisotropy

of the gravitational sector at nonlinear order. The single-crystal construction of Ref. [3] reproduces

linearized Einstein gravity exactly, but carries a residual cubic-vertex anisotropy, reported there

number of grain orientations

rank-4 anisotropy

single-crystal 0.20

orientation averaging isotropy

FCC 110 average

1/2

Figure 5: Orientation averaging drives the single-crystal graviton anisotropy to zero. The rank-4 anisotropy

of the FCC structure tensor, averaged over N random grain orientations, falls as N

−1/2

from the single-

crystal value 0.20 (dotted) toward zero; the ensemble mean is exactly isotropic (Voigt–Reuss–Hill). For a

wave sampling N

grain

∼ λ/ℓ

grain

grains the residual is negligible.

as a 12.7% deviation under its normalization—a single-crystal feature of the same cubic-symmetry

origin as the linear shear anisotropy above. In the rank-6 moment measure used here the same ob-

struction has single-crystal magnitude 0.44; the two ﬁgures quantify one anisotropy under diﬀerent

normalizations. Orientation averaging removes it by the identical Haar-projection argument: the

rank-6 moment’s anisotropic part averages to zero (numerically, from the single-crystal 0.44 toward

0.011 at N = 3000 orientations, vanishing as 1/

√

N), while the exact linear operator, being already

isotropic, is a ﬁxed point of averaging and is left strictly unchanged (the orientation average of an

isotropic tensor equals itself, veriﬁed to machine precision). The polycrystalline vacuum therefore

renders the gravitational sector isotropic at both linear and cubic order, without perturbing the

exact linear result. This establishes isotropy of the eﬀective-medium tensor mean; a full nonlinear

Lorentzian eﬀective action, including disorder-induced scattering and nonlocal corrections, remains

open. We stress that this resolution of the cubic-vertex anisotropy is conditional on the polycrys-

talline structure of the vacuum: on a single crystal the anisotropy is a genuine feature, as reported

in Ref. [3]; it is the ensemble of misoriented grains that renders it macroscopically absent.

10 Relation to common-origin universality

Media generically detune the speeds of distinct emergent ﬁelds; this is the origin of the strin-

gent constraints on scalar–tensor and Horndeski gravity from GW170817 [12], and it is the reason

emergent-ﬁeld velocities do not in general coincide [6]. Two routes to universality are established.

The ﬁrst is common origin: when all low-energy excitations descend from a single underlying

structure, they inherit a single limiting velocity by construction, as for the emergent photon, gravi-

ton, and chiral fermions near a Fermi point in superﬂuid

He-A [14], or for photon and graviton

coupled identically to a common ﬁeld in Kaluza–Klein-type constructions (see the discussion in

Ref. [6]). The second is renormalization-group ﬂow: in a theory whose species start with unequal

velocities, interactions can drive the velocities to a common infrared value, making Lorentz in-

variance a low-energy attractor [17]. The stitch bottleneck constitutes a third, transport-limited

route: the commonality lives neither in the source nor in the ﬂow but in the path—every sector

traverses the same domain-boundary network by the same frame transport, so the observed speed

is a property of the shared boundaries rather than of any sector. The mechanism does not tune the

intra-lattice couplings into agreement, nor wait for them to ﬂow together; it buries them beneath

a sector-independent boundary cost.

Relation to the Weinberg–Witten theorem. A remark is in order on the no-go theorem most

often cited against lattice gravitons. Weinberg and Witten [13] forbid massless helicity-2 particles

in any theory possessing an exactly Lorentz-covariant, conserved energy–momentum tensor under

which those particles are charged. The present construction does not satisfy the hypotheses, at

either level. At the substrate level, the lattice deﬁnes a preferred frame and conserves momentum

only modulo reciprocal-lattice vectors, so no exactly conserved covariant T

µν

exists—the standard

escape of the condensed-matter-analog family [14, 6]. At the emergent level, the low-energy theory

is linearized Einstein gravity with its gauge invariance intact [3], and a diﬀeomorphism-invariant

theory admits no local, gauge-invariant, covariant energy–momentum tensor for the gravitational

ﬁeld: the emergent graviton evades the theorem the same way general relativity itself does. The

theorem’s real content for a lattice model is the burden it imposes: Lorentz invariance can only

be approximate, so every residual violation—anisotropy, dispersion, a sector speed split—must be

shown unobservable. Sections 7–9 are the discharge of exactly that burden.

11 Scope and status

The central result of this paper is conditional, and we state its structure plainly before itemiz-

ing. What is established unconditionally: the two-sublattice construction and its masslessness

protection (Sections 2–5); the

√

3 speed obstruction as a coordination invariant (Section 6); the

kinematic sector-blindness of the stitch (Proposition 1); and the two washout results—that the

boundary-dominated speed is independent of every intra-grain detail, and that orientation aver-

aging removes the lattice anisotropies. What is conditional: the observed equality c

= c

itself,

which follows exactly given the two stated dynamical assumptions (dominance and delay universal-

ity) and not otherwise. The value of the conditional is that it converts GW170817 from a falsiﬁer

of the construction into a quantitative constraint on a single object, the stitch: if the microscopic

stitch dynamics fails to deliver the two assumed properties, the obstruction of Section 6 stands

and the construction is excluded; if it delivers them, universality is automatic. The ledger below

itemizes each claim’s status.

Reached (construction). 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 inter-

action, so the photon stays exactly massless and the induced dielectric renormalization is isotropic

and speed-reducing (intra-grain; washed out of the macroscopic speed, Section 8). A matter de-

fect in a tetrahedral void is forced by connectivity to source the emergent ﬂux and carries integer

emergent charge.

Proven (speed mechanism). (a) The two-sublattice construction splits the sector speeds by

the coordination-invariant factor

√

3, independent of length, time-step, and coupling,

and this split recurs on D

; the naive coexistence construction therefore does not satisfy GW170817.

(b) The stitch, as a frame rotation, acts on spin-1 and spin-2 ﬁelds as representations of the same

rotation generated by the same angle; each action is a norm-preserving isometry; and the two

irreducible representations cannot mix under the crossing (kinematic sector-blindness). (c) In the

stitch-dominated limit the macroscopic speed is independent of the intra-grain speed, so the sector

split washes out and c

= c

. (d) The orientation (Haar) average of the FCC rank-4 elastic tensor

is exactly isotropic, so the single-crystal graviton anisotropy has zero macroscopic mean, with a

ﬁnite-grain residual ∼ 1/

grain

Assumed. Two properties of the stitch dynamics are stated as assumptions rather than derived.

(i) Dominance: the per-grain boundary-crossing cost dominates the intra-grain traversal time,

stitch

≫ ℓ

grain

intra

; meeting the GW170817 bound requires τ

stitch

≳ 10

ℓ

grain

. The cost

concerned is that of the active frame re-identiﬁcation between internally perfect grains; a passive

(defect-scattering) origin for a dominant delay is excluded by the delay–bandwidth argument of

Section 8, and the natural mechanism class—collective stabilizer re-synchronization, exponentially

suppressed in the number of coherently re-encoded boundary degrees of freedom—renders the

required magnitude a modest collective exponent (κN

≃ 35–40, an order-of-magnitude plausibility

estimate) rather than an arbitrary large number. (ii) Delay universality: the boundary delay

depends only on the frame re-identiﬁcation, not on representation-dependent boundary scattering,

so τ

stitch

= τ

stitch

; Proposition 1 establishes the kinematic part of this statement but not the

dynamical part.

Grounded in observation. That an observed photon or graviton spans many grains (the macro-

hop) follows from the measured wavelength-to-bond ratio λ/L ∼ 10

–10

, with the bond length

ﬁxed at the Planck scale by the entanglement-pair identiﬁcation [2, 18, 19]. The remaining mod-

eling input is that the vacuum is polycrystalline with ﬁnite, misoriented domains whose mature

boundaries are coordination-complete (Section 1); the polycrystalline structure is established in

Ref. [2], and the healing of transient boundary under-coordination is the asymptotic saturation of

the same reference’s ﬁnite-size scaling.

Imported. The emergent U(1) photon and its gauge-invariance protection [4, 5]; the linearized

gravitational sector [3]; the GW170817 bound [11]; the generic-detuning problem and its common-

origin resolution [12, 14, 6].

Open. (i) The absolute value of the emergent speed c = ℓ

grain

/τ

stitch

, which is set by the stitch

network’s transition rate and is a parameter of the construction rather than a prediction. (ii)

A derivation of the two dynamical assumptions (dominance and delay universality) from the mi-

croscopic stitch dynamics—concretely, the construction of the inter-grain frame-re-synchronization

protocol and the computation of its exponent κN

; any such dynamics is constrained to be elastic

and achromatic by the transparency and dispersion estimates of Section 8, and acts at coordination-

complete interfaces rather than at coordination defects. (ii

′

) The healing dynamics itself: the relax-

ation of transient boundary under-coordination into coordination-complete interfaces and trapped

matter-type remnants is argued from the saturation drive of Ref. [2], not derived here. (iii) The

overall magnitude of the magnetoelastic coupling (∼ g

), a free microscopic parameter. (iv)

Whether the physical FCC Hamiltonian realizes the deconﬁned phase for the speciﬁc ⟨111⟩ model,

inherited from the quantum-spin-ice mapping. (v) The map from the ⟨110⟩ electroweak structure

to the ⟨111⟩ propagating photon (Section 11.1).

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 construction sections demonstrate that these known results

embed in the SSM gravitational model without the two sectors spoiling one another. We do

not derive the numerical value of c; we do not claim a chiral gauge theory; and we take the

polycrystalline (ﬁnite-grain) structure of the vacuum from Ref. [2] rather than re-deriving it. The

result is a structural mechanism—that a shared, sector-independent domain-boundary transport

forces a common propagation speed—assembled from the SSM stitch construction and the common-

origin principle, and conditional on the stated dominance assumption.

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 intra-grain light)

derived / computed

Coupling magnitude (∼ g

) open / microscopic

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

11.1 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 {111} planes, the natural home of the color structure whose qutrit

content at a matter defect is ﬁxed by the K

cage of Section 4; 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 5, 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.

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.

All reported quantities are computed from the lattice geometry and the rotation-group repre-

sentation theory. The structure-tensor eigenvalues use unit bond vectors of the ⟨110⟩ (K = 12) and

⟨111⟩-diamond (K = 4) networks. The stitch sector-independence uses the spin-1 (3×3) and spin-2

(h 7→ RhR

⊤

on symmetric traceless tensors) representations of R = exp(θ

a ·J ) ∈ SO(3); eigen-

phases and Frobenius-norm preservation are evaluated directly. The stitch-washout uses Eq. (4)

with intra-grain speeds in the ratio 1:

√

3. The residual suppression uses (2πa/λ)

with a at the

Planck scale. The passive-boundary exclusion uses the standard one-dimensional results: weak-link

transmission T = 4(t

′

/t)

/[1 + (t

′

/t)

]

at band center, and the resonant-level delay–bandwidth re-

lation (Wigner delay ∼ 2/Γ against transmitting width Γ), veriﬁed numerically. The orientation

average uses Haar-random SO(3) rotations of the rank-4 structure tensor, with the anisotropy

measured as the residual after isotropic projection.

Code availability

All reported quantities are computed from the FCC lattice geometry; no external or measured data

are used. Two script packages reproduce every number and ﬁgure. The construction results (Sec-

tions 2–5) are reproduced by https://github.com/raghu91302/ssmtheory/blob/main/photon_

verification.zip. The speed-universality results are reproduced by the scripts below, available

https://github.com/raghu91302/ssmtheory/blob/main/uniformity_scripts.zip



uniformity_structure_tensors.py — Eq. (3), the two-sublattice speed split;



uniformity_stitch_isometry.py — Proposition 1, shared eigen-angle and isometry;



uniformity_washout.py — Eq. (4), the N -independence of the fractional split, and the 10

dominance requirement;



uniformity_vrh_averaging.py — the orientation-averaging, ﬁxed-point, and cubic-vertex

results of Section 9;



uniformity_residual_suppression.py — the wavelength-to-bond ratios and the (ka)

resid-

ual;



uniformity_figures.py — Figs. 3–5.

Each analysis script asserts agreement with the quoted values and runs on a standard numpy/scipy

environment with no external or measured data.

References

[1] R. Kulkarni, A face-centered-cubic stabilizer code for the vacuum lattice, arXiv:2603.20294

(2026), under review at Quantum. arXiv:2603.20294.

[2] 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:10.1016/j.physo.2026.100423.

[3] R. Kulkarni, Emergent Gravity from the Intrinsic D4 Lattice: Exact Linearized Einstein Grav-

ity, and its Spherical-Design Obstruction at the Cubic Graviton Vertex (2026), under review

at Eur. Phys. J. C. doi:10.5281/zenodo.20952333.

[4] M. Hermele, M. P. A. Fisher, and L. Balents, Pyrochlore photons: the U (1) spin liq-

uid in a S =

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

doi:10.1103/PhysRevB.69.064404.

[5] M. J. P. Gingras and P. A. McClarty, Quantum spin ice, Rep. Prog. Phys. 77, 056501 (2014).

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

[6] X.-G. Wen, Artiﬁcial light and quantum order, Phys. Rev. B 68, 115413 (2003).

doi:10.1103/PhysRevB.68.115413.

[7] 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:10.1103/PhysRevLett.91.167004.

[8] C. L. Henley, The Coulomb phase in frustrated systems, Annu. Rev. Condens. Matter Phys.

1, 179 (2010). doi:10.1146/annurev-conmatphys-070909-104138.

[9] S. Lee and E.-G. Moon, Spin–lattice coupling in U(1) quantum spin liquids, arXiv:1802.05280.

arXiv:1802.05280.

[10] J. F. Nye, Physical Properties of Crystals (Oxford Univ. Press, 1985): piezoelectric tensor of

the

43m (T

)/zincblende class.

[11] B. P. Abbott et al. (LIGO/Virgo, Fermi GBM, INTEGRAL), Gravitational waves and gamma-

rays from a binary neutron star merger: GW170817 and GRB 170817A, Astrophys. J. Lett.

848, L13 (2017). doi:10.3847/2041-8213/aa920c.

[12] J. M. Ezquiaga and M. Zumalac´arregui, Dark energy after GW170817: dead ends and the road

ahead, Phys. Rev. Lett. 119, 251304 (2017). doi:10.1103/PhysRevLett.119.251304.

[13] S. Weinberg and E. Witten, Limits on massless particles, Phys. Lett. B 96, 59 (1980).

doi:10.1016/0370-2693(80)90212-9.

[14] G. E. Volovik, The Universe in a Helium Droplet (Oxford University Press, 2003).

doi:10.1093/acprof:oso/9780199564842.001.0001.

[15] A. A. Abdo et al. (Fermi-LAT and Fermi-GBM Collaborations), A limit on the varia-

tion of the speed of light arising from quantum gravity eﬀects, Nature 462, 331 (2009).

doi:10.1038/nature08574.

[16] C. Horsman, A. G. Fowler, S. Devitt, and R. Van Meter, Surface code quantum computing by

lattice surgery, New J. Phys. 14, 123011 (2012). doi:10.1088/1367-2630/14/12/123011.

[17] S. Chadha and H. B. Nielsen, Lorentz invariance as a low energy phenomenon, Nucl. Phys. B

217, 125 (1983). doi:10.1016/0550-3213(83)90081-0.

[18] M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Relativ. Gravit.

42, 2323 (2010). doi:10.1007/s10714-010-1034-0.

[19] B. Swingle, Entanglement renormalization and holography, Phys. Rev. D 86, 065007 (2012).

doi:10.1103/PhysRevD.86.065007.