Dark Matter as a Trapped K=6 Remnant in the Octahedral Voids of the FCC Vacuum Lattice

Dark Matter as a Trapped K=6 Remnant

in the Octahedral Voids of the FCC Vacuum Lattice

Raghu Kulkarni

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

Draft manuscript  May 2026

Abstract

Building on the Selection-Stitch Model (SSM) [2, 3], in which baryonic matter is identied

with K=4 remnants trapped in the tetrahedral voids of the K=12 FCC vacuum lattice, we

examine the framework's second interstitial site  the octahedral void  as a candidate dark

matter trap. The bonded subgraph of the octahedral defect is the complete tripartite graph

2,2,2

. Four qualitative predictions follow from its structural symmetry: absence of rst-

order electromagnetic coupling, absence of SU(3) color in the baryonic form, self-conjugate

(Majorana-type) character, and suppressed rst-order radiative cooling. The framework pre-

dicts the dark matter mass directly from a closed inclusion-exclusion expansion on

2,2,2

terminating at third order because the octahedron's six vertices forbid any 4-matching:

= 25 ·144 −30 ·10 + 8 ·8 = 3364

, giving

= (3364/1836) ×m

= 1.719

GeV. This

prediction agrees with the recent Kang et al. 2026 detection of a

1.5



1.6

GeV gamma-ray line

in three active galactic nuclei within



2.1σ

of each individual source measurement (joint

t at

∼ 1.55

GeV,

∼ 11%

below the prediction). Since the line is observed in AGN, which

host supermassive black holes and are predicted to develop dark-matter density spikes, grav-

itational redshift of inner-spike annihilation provides a candidate astrophysical mechanism

mapping the rest-frame prediction to the observed line position.

1 Introduction

The cosmological abundance of dark matter relative to baryonic matter is one of the most

precisely measured ratios in physics:

Ω

= 5.36 ± 0.04

(Planck 2018 [1])

(1)

No mechanism in the Standard Model produces this number; it is an empirical input to current

cosmological models. Candidate dark-matter frameworks typically introduce one or more new

particle species with adjustable couplings and abundances, t to the observed ratio, and test

the result against direct-detection, indirect-detection, and structure-formation constraints.

The Selection-Stitch Model (SSM) [2, 3] proposes that the physical vacuum is a Face-Centered

Cubic (FCC) crystallization of spacetime, with baryonic matter identied as a single K=4 node

trapped in the tetrahedral interstitial void of the K=12 FCC bulk. The framework derives the

proton-to-electron mass ratio

= (K + 1)K

− c

skew

K = 13 ×144 − 3 × 12 = 1836

(2)

raghu@idrive.com

from purely structural counts of the trapped tetrahedral-void defect [2]. An equivalent derivation

via a

[[192, 130, 3]]

CSS code on the FCC lattice [3] reaches the same number through a fault-

tolerant verication cost

× C

= 36 × 51 = 1836

The FCC unit cell contains two distinct interstitial void types: 8 tetrahedral voids (each

bounded by 4 FCC vertices) and 4 octahedral voids (each bounded by 6 FCC vertices), with

all bounding edges at the nearest-neighbor distance

(Section 2). The framework that traps a

defect in the tetrahedral void simultaneously admits an analogous defect in the octahedral void

 the same K=4 to K=12 phase transition, the same kinematic operators, the same geometric

mechanism, applied to the second interstitial site that the FCC lattice provides. The natural

question is what physics this companion defect predicts.

This paper develops the case that the octahedral-void defect is a viable candidate for dark

matter. The case is built in two pieces. First, four qualitative properties of the defect follow rig-

orously from the structural symmetry of its bonding graph

2,2,2

and bounding polyhedron (the

regular octahedron with

symmetry): absence of rst-order electromagnetic coupling, absence

of SU(3) color in the form generated for baryons, self-conjugate (Majorana-type) character, and

suppressed rst-order radiative cooling. These match the standard requirements for cold dark

matter without invoking any free parameters. Second, the structural-counting framework of

Ref. [2] that yields the proton's verication cost

= 1836

extends to the octahedral defect by

inclusion-exclusion on the

2,2,2

bonding graph, terminating exactly at third order because the

octahedron's six vertices forbid any 4-matching:

= 25 ×144 −30 ×10 + 8 ×8 = 3364

. The

framework therefore predicts the dark matter mass directly:

× m

3364

1836

× 938.272

MeV

= 1.719

GeV

(3)

using only the proton mass [5] as a calibration input. The recent detection of a

∼ 1.5



1.6

GeV

gamma-ray line in three active galactic nuclei [4] agrees with this prediction within



2.1σ

each individual source measurement, with the joint t at

∼ 1.55

GeV approximately

11%

below

the predicted rest-frame mass.

What this paper claims and does not claim.

We present a forward derivation of the dark

matter mass via inclusion-exclusion on the

2,2,2

bonding graph (Section 4). The prediction

= 1.719

GeV uses no cosmological observation as input and depends only on the bonded

structure of the octahedral defect, the structural-counting framework of Ref. [2], and the proton

mass.

We do not derive the cosmological abundance ratio

Ω

/Ω

in this paper. The volume-

trapping mechanism that xes the relative formation rates of octahedral and tetrahedral defects,

combined with the early-universe history of matterantimatter annihilation in the tetrahedral

sector and

χχ → γγ

pair-annihilation in the octahedral sector, relates the two abundances.

A rst-principles derivation requires quantitative treatment of both annihilation channels in

the dense post-formation epoch, which is not attempted here. The forward mass prediction is

independent of this calculation.

What we claim is:

1. The framework forces consideration of the octahedral-void defect as a second matter class.

Its existence is not chosen; the crystallography of FCC requires both interstitial sites to be

populated under any defect-generating mechanism that produces baryons from tetrahedral

voids.

2. Four qualitative properties follow strictly from the defect's geometric symmetry and match

what dark matter must do.

3. The framework predicts

= 1.719

GeV from the closed inclusion-exclusion expansion

on the

2,2,2

bonding graph, with no cosmological input. This is a forward prediction

derived from the same structural-counting machinery that yields

= 1836

for the proton

in Ref. [2].

4. This mass prediction agrees with the recently reported 1.51.6 GeV gamma-ray line within



2.1σ

of individual source measurements; the

∼ 11%

residual oset from the joint t is

consistent with gravitational redshift of inner-spike annihilation in the host AGN environ-

ment (Section 5).

Organization.

Section 2 establishes the crystallographic geometry of the two interstitial void

types in the FCC unit cell. Section 3 introduces the octahedral-void defect, denes its bonding

graph

2,2,2

, and derives the four qualitative properties from the geometric symmetry. Section 4

presents the forward derivation of

= 1.719

GeV via inclusion-exclusion on

2,2,2

, with ex-

plicit enumeration of all combinatorial inputs. Section 5 compares the prediction with current

observational constraints, including the Fermi-LAT 1.5 GeV line, and contrasts the framework

with existing dark matter models. Section 6 discusses falsiability and open calculations. Sec-

tion 7 concludes.

Interactive 3D Visualization.

A WebGL visualization of the octahedral defect

geometry  the

2,2,2

bonding graph among the 6 bounding vertices, the 3 antipodal

non-bonded pairs at second-nearest-neighbor distance

√

2 L

, and the cuboctahedral

coordination cluster around any FCC vertex  accompanies this paper:

https://raghu91302.github.io/ssmtheory/oct_void_3D.html

The visualization makes immediately evident that the octahedral defect cannot be

enclosed by any single 13-node coordination cluster: of the six bounding vertices, ve

lie within the cluster of any chosen anchor, but the sixth (the anchor's antipode) lies

outside.

2 Geometry of the Two Interstitial Voids

The FCC unit cell with cubic lattice constant

contains atoms at the cube corners and face

centers, giving a Bravais lattice with primitive cell volume

and nearest-neighbor distance

L = a/

√

. The unit cell decomposes into two distinct interstitial void types [6, 7] (Figure 1).

Tetrahedral voids.

Located at

(a/4)(±1, ±1, ±1)

with all eight sign combinations, giving 8

voids per cell. Each is bounded by 4 FCC vertices forming a regular tetrahedron with edge

length

. The centroid-to-vertex distance is

3/8 ≈ 0.612 L

Octahedral voids.

Located at the body center

(a/2, a/2, a/2)

and at the 12 edge midpoints.

Each edge midpoint is shared between 4 unit cells, contributing

12/4 = 3

to the cell count, plus

1 body-center, giving 4 voids per cell. Each is bounded by 6 FCC vertices forming a regular

octahedron with edge length

. The centroid-to-vertex distance is

√

2 ≈ 0.707 L

3 The Octahedral-Void Defect and Its Structural Properties

3.1 The defect: trapped K=6 remnant

During the K=4

→

K=12 phase transition that crystallizes the SSM vacuum [2], the same

kinematics that traps a K=4 remnant in a tetrahedral void can trap a remnant in an octahedral

void. The trapped node sits at the centroid of the octahedral void and bonds to the six bounding

vertices, creating a local K=6 pocket inside the K=12 bulk. This parallels the proton's K=4

trapped remnant in the tetrahedral case.

Tetrahedral void Octahedral void

Both voids drawn at the same physical scale (edge length L)

Trapped node (centroid) Bounding FCC vertex Void interior

Figure 1: The two interstitial void types of the FCC unit cell.

Left:

A tetrahedral void,

bounded by 4 FCC vertices (dark blue) forming a regular tetrahedron with edge length

. The

trapped K=4 node (yellow star) sits at the centroid; the yellow translucent sphere illustrates

the void interior.

Right:

An octahedral void, bounded by 6 FCC vertices forming a regular

octahedron with the same edge length

. The trapped K=6 node sits at the centroid; the yellow

translucent sphere illustrates the void interior. The octahedral void is geometrically larger than

the tetrahedral void, reecting the longer centroid-to-vertex distance (

√

versus

3/8

Why K=4 and K=6, and not intermediate values?

The selection of the tetrahedral

(K=4) and octahedral (K=6) congurations as the only viable trapped-remnant classes follows

from two independent constraints that coincide in the FCC lattice.

Crystallographic constraint.

The only positions in the FCC vacuum that admit multiple FCC

nodes at the unit nearest-neighbor distance

are the tetrahedral interstitial site (4 equidistant

nearest neighbors) and the octahedral interstitial site (6 equidistant nearest neighbors). Generic

positions in FCC have a unique closest FCC node and cannot host a trapped conguration

with multiple equal-length bonds. There is no position in FCC with exactly 5 (or 7, 8, etc.)

equidistant nearest neighbors. The interstitial sites of the perfect FCC lattice therefore oer

only K=4 and K=6 as candidate trapping geometries.

Strain-balance constraint.

Even hypothetically considering 5 of the 6 vertices of an octahedral

void as the bounding set of a square-pyramidal defect, the trapped node would not sit at the

geometric centroid in mechanical equilibrium. The inward unit-vector sum

ˆn

from each

bounding vertex toward the centroid vanishes identically for the regular tetrahedron and the

regular octahedron, by their inversion or rotational symmetries. For any subset that breaks this

symmetry  including a square pyramid (5 vertices) 

ˆn

= 0

, and the trapped node's

strain-energy minimum is displaced along the symmetry-breaking axis (toward or away from the

removed vertex). Such a defect would be intrinsically asymmetric, carrying a built-in structural

dipole, and is not produced by the isotropic kinematics of the K=4

→

K=12 phase transition.

The two constraints reinforce each other: the symmetric strain-balanced congurations are

exactly the symmetric crystallographic interstitial voids. The SSM framework therefore naturally

selects K=4 and K=6 as the only stable trapped-remnant classes, with no tting required and

no intermediate-K particle classes admitted.

3.2 Bonding subgraph:

2,2,2

The 6 bounding vertices of the octahedron sit at three antipodal pairs: each pair at distance

√

, and each non-antipodal pair at distance

(the nearest-neighbor distance). Bonds at the

SSM unit-bond-length scale

connect each vertex to its 4 non-antipodal partners, generating

the complete tripartite graph

2,2,2

: 6

vertices

, 12

edges

degree 4 per vertex

, 3

antipodal non-bonded pairs

(4)

This is the bonded subgraph of the octahedral-void defect (Figure 2), in contrast to the proton

case where the bonded subgraph is

(4 vertices, 6 edges, degree 3).

-x

-y

-z

Bond graph K

2; 2; 2

: 12 unit-length edges (green),

3 antipodal non-bonds at L

(red dashed)

All 8 faces are triangular (C

, non-bipartite)

0 square faces ) no C

bipartite EM channel

Face structure: schematic of all 8 triangular faces

Figure 2: Bonded substructure of the octahedral defect.

Left:

The 6 bounding vertices of the

octahedron with the trapped K=6 node (yellow star) at the centroid. The 12 unit-length bonds

form the complete tripartite graph

2,2,2

(solid green); the 3 antipodal pairs at distance

√

are non-bonded under the SSM unit-bond rule (red dashed).

Right:

The bounding polyhedron

has 8 triangular faces and 0 square faces. The absence of square (

) faces is the structural

origin of the suppressed rst-order electromagnetic coupling discussed in Section 3.3.

3.3 Suppressed rst-order electromagnetic coupling

The argument for the photon coupling channel of baryonic defects given in Ref. [2] proceeds

in two steps. First, the defect's coupling to the bulk lattice is mediated by oscillation modes

hosted on the bounding faces of its internal bonded structure: triangular faces (cycle

) cannot

carry the bipartite alternation that supports a dipolar mode, while square faces (cycle

) can.

Second, the bulk cuboctahedral coordination shell of each FCC vertex contains 8 triangular

faces and 6 square faces, providing the propagation channel through which the defect's bipartite

oscillation modes couple to bulk photon modes.

For the tetrahedral-void defect, the bounding polyhedron is a regular tetrahedron with 4

triangular faces and zero square faces. The photon channel of Ref. [2] is not strictly forbid-

den because the dipolar oscillation can be carried on the square faces of the surrounding bulk

cuboctahedral shells, which the defect's external bonds traverse.

For the octahedral-void defect, the bounding polyhedron is a regular octahedron with 8

triangular faces and

zero

square faces, and the bonded subgraph

2,2,2

contains 4-cycles only

between antipodal pairs that are explicitly non-bonded. The defect itself has no internal

structure on its bonded subgraph and is bounded by 8 triangular faces. The defect-to-bulk

coupling that would mediate rst-order photon emission has no preferred bipartite mode at the

defect side: the rst-order EM coupling channel is suppressed at the defect-internal level.

Falsiable consequence.

An octahedral-void defect interacts with the electromagnetic eld

only through higher-order channels (multi-photon processes, mediation through virtual baryonic

intermediaries, or gravitational coupling to bulk photon modes). Direct rst-order single-photon

coupling is suppressed by the absence of bipartite face structure on the defect's bounding poly-

hedron. This matches the standard dark-matter requirement of EM neutrality at the level of

leading-order cross-sections.

Direct-detection bounds on dark-matterphoton couplings (currently constraining the dark

photon mixing parameter

ϵ ≲ 10

−3

∼ 1

GeV from beam-dump experiments and supernova

cooling [11, 12]) are consistent with the picture predicted here. A complete proof of EM-

neutrality requires deriving the photon coupling vertex from the SSM strain-eld dynamics; the

argument given here is the structural-symmetry analog of the Ref. [2] argument applied to the

octahedral case.

3.4 Absence of SU(3) color in the form generated for baryons

In the tetrahedral case, three color charges arise from the three skew-edge pairs of the bounding

tetrahedron

[2]. The combinatorial identity

C(4, 2)/2 = 3

generates the three-color SU(3)

representation on baryon-like defects.

For the octahedral case, the skew-edge pair count of

2,2,2

(O)

skew





− 6





= 66 − 36 = 30.

(5)

This count is veried by direct enumeration: each of the 12 edges of

2,2,2

has exactly 5 skew

partners (the edges connecting the 4 remaining vertices, minus the one antipodal non-bond),

giving

12×5/2 = 30

skew pairs. It is also obtained by counting the 2-matchings of the octahedron

graph.

The skew-pair count 30 does not factor as a small representation analogous to the three-

color case. The natural color structure on an octahedral defect is therefore not SU(3) of the

standard QCD form. The defect does not couple through the three-color conning channels of

the cuboctahedral shell.

Falsiable consequence.

An octahedral-void defect does not participate in standard strong-

force interactions. It cannot bind to baryons through gluon exchange, cannot form bound states

with quarks, and cannot hadronize. This matches the standard dark-matter requirement of

strong-force neutrality.

3.5 Self-conjugate (Majorana-type) character

Reference [2] identies anti-baryons with the spatial inversion of the trapped tetrahedral-void de-

fect: the FCC unit cell contains two distinct tetrahedral-void orientations (centered at

(a/4, a/4, a/4)

and

(3a/4, 3a/4, 3a/4)

), related by the inversion

ˆr → −ˆr

that exchanges the sign of the anchor

projection. The two orientations support matter and anti-matter respectively; the cosmological

matterantimatter asymmetry corresponds to the local cosmological excess of one orientation

over the other.

The octahedral void has no analogous orientation degeneracy. The regular octahedron is

invariant under inversion through its centroid: the symmetry group

contains the inversion

operator

−I

, which maps the octahedron to itself rather than to a distinct partner. Each octahe-

dral void (body-center or edge-midpoint) is therefore its own image under inversion. The frame-

work predicts that the octahedral-void defect is its own anti-particle: a stable, self-conjugate

Majorana-type dark-matter particle.

Falsiable consequences.



No matterantimatter asymmetry is required (or possible) in the dark sector. The cos-

mological abundance of the dark species is set by formation rates and the early-universe

history of pair-annihilation, not by an asymmetry between particle and antiparticle.



Pair-annihilation of two octahedral defects is in principle allowed (

χχ

, not

χ¯χ

), with rate

set by the available coupling channels. With suppressed rst-order EM, annihilation rates

are loop-suppressed. This is structurally consistent with the loop-suppressed

⟨σv⟩

γγ

≈

2 × 10

−28

/s required by the Kang et al. 2026 AGN line [4].



Searches for a Dirac-type dark matter particle with distinct anti-particle (e.g., dark-

asymmetric models) would not nd a match in this framework.

3.6 Suppressed rst-order radiative cooling and clustering asymmetry

A central observational distinction between baryonic and dark matter is the dierence in their

large-scale clustering behavior. Both species clump under gravity  galaxies, clusters, and the

cosmic web are dark-matter-dominated gravitational structures, with baryons collected in their

potential wells. The asymmetry is in how each species behaves

after

gravitational infall: baryons

collapse to high densities through radiative cooling (atomic line emission, bremsstrahlung, free-

free emission), forming stars, planets, and dense compact objects, while dark matter remains

diuse on scales below the cluster halo, with no equivalent of stellar or planetary collapse.

The framework predicts this asymmetry from the same structural feature already invoked

for EM-suppression. Tetrahedral-void defects (baryons) couple to photons through the bulk

cuboctahedral coordination shell, providing a rst-order radiative channel through which ki-

netic energy can be shed. Octahedral-void defects, by contrast, lack the defect-internal bipartite

structure required to source the dipolar oscillation modes that mediate rst-order photon emis-

sion. The same suppression that yields EM-neutrality also suppresses radiative cooling.

A non-radiatively-cooling gravitating species is collisionless on astrophysical timescales: it

cannot shed kinetic energy eciently, cannot collapse below its initial velocity-dispersion scale,

and forms diuse halos rather than dense compact objects. This matches the observed phe-

nomenology of dark matter, including the diuse extension of galactic and cluster halos beyond

the visible baryonic component, and the absence of dark stellar or planetary analogues at any

scale. The recent identication of almost dark galaxies dominated by dark matter halos with

negligible stellar content provides further phenomenological support; we discuss this class of

observation in Section 5.4. The prediction is therefore that octahedral defects are collisionless

on cosmological timescales relative to baryonic radiative cooling rates, consistent with current

observational constraints on dark-matter self-interaction [8, 9, 10].

4 Forward Derivation of

via Inclusion-Exclusion

The structural-counting derivation of

= 1836

for the proton presented in Ref. [2]

(Eq. (2)) extends to the octahedral defect by inclusion-exclusion on the

2,2,2

bonding graph.

We present this extension here and obtain the framework's forward prediction

= 3364

= 1.719

GeV, with no cosmological input.

4.1 The structural-counting formula as inclusion-exclusion

The expression of Ref. [2],

= (K +1)K

−c

skew

K = 13×144−3×12 = 1836

, admits a natural

interpretation in the language of the Principle of Inclusion-Exclusion (PIE). Each structural

node of the defect carries a disruption halo of

= 144

vacuum bond states (the second-shell

footprint at the bulk coordination

K = 12

). Summed over

= 13

structural nodes, this

gives a base count of

13 × 144 = 1872

bond-state disruptions. Pairs of structural nodes with

mutually disjoint ux channels  the

skew

= 3

skew-edge pairs of

 have disruption halos

that overlap, double-counting bond states. The size of each pairwise overlap, computed directly

as the bulk rst-shell intersection

|N(e

) ∩ N(e

on the FCC lattice, is

pairwise

= 12

for any

skew pair. Subtracting the double-counted bond states gives

1872 −36 = 1836

, the proton's

veried verication cost. The series terminates:

has only 4 vertices, so it admits no triple

of mutually disjoint edges (a 3-matching requires 6 vertices), and the third-order PIE term is

identically zero.

4.2 Extension to

2,2,2

with truncation

The same structure applies to the octahedral defect, with three dierences xed by the bonded

graph

2,2,2

Structural node count.

Each of the 6 bounding vertices contributes 4 bonds within

2,2,2

, giving

6 × 4 = 24

boundary structural nodes; with the trapped center,

= 25

Skew-edge pair count.

Each of the 12 edges of

2,2,2

has exactly 5 mutually disjoint

partners (12 total edges minus 1 self minus 6 edges sharing a vertex), giving

(O)

skew

(12 × 5)/2 = 30

skew pairs. The same count is obtained by direct enumeration of 2-

matchings of the octahedron graph and by the closed form





− 6





= 66 − 36 = 30

Pairwise overlap on

2,2,2

Computing the bulk rst-shell intersection directly on the

FCC lattice yields

|N(e

) ∩ N(e

)| = 10

uniformly across all 30 skew pairs of

2,2,2

, in

contrast to

's value of 12. The pairwise overlap is a graph-specic geometric quantity,

not a xed bulk constant.

Crucially,

2,2,2

admits the next term in the PIE expansion that

does not. With 6 ver-

tices, the octahedron graph hosts triples of mutually disjoint edges (3-matchings, equivalently

perfect matchings of the octahedron), of which there are exactly

(O)

triple

= 8

(veried by direct

enumeration; see Section 4.3). By PIE, the triple-overlap contribution must be added back

to compensate for the pairwise subtraction's overcounting in regions where three ux channels

mutually intersect.

The triple overlap on

2,2,2

, computed directly as the bulk rst-shell triple intersection

|N(e

) ∩N(e

, evaluates to

(O)

triple

= 8

uniformly across all 8 perfect matchings of the

octahedron.

The series terminates exactly at third order. A 4-matching requires 8 mutually distinct

vertices; the octahedron graph has only 6. Therefore

(O)

quad

= 0

and all higher orders vanish

identically. The PIE expansion for the octahedral defect is nite and closed:

= N

· K

bulk

− c

(O)

skew

· K

(O)

pairwise

+ c

(O)

triple

· K

(O)

triple

= 25 × 144 − 30 × 10 + 8 × 8

= 3600 − 300 + 64 = 3364.

(6)

The forward prediction from this derivation is

= (C

) × m

= (3364/1836) ×

938.27

MeV

= 1.719

GeV.

4.3 Explicit enumeration of pairwise and triple overlaps

The values

(O)

pairwise

= 10

and

(O)

triple

= 8

appearing in Eq. (6) are not free parameters: they are

determined by direct enumeration on the FCC lattice with no tting. We make the enumeration

explicit here so a reader can verify the computation without recourse to external code.

First shell of an FCC vertex.

Each FCC node

has exactly 12 nearest neighbors (its

cuboctahedral coordination shell), at the 12 displacements

(v) − v ∈ {(±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1)}

(7)

in lattice units (parity-even FCC convention with NN distance

L =

√

). For a

2,2,2

edge

e = {v

, v

}

between two such vertices, we dene the rst-shell neighborhood as the union

N(e) = N

) ∪ N

(8)

which contains

|N(e)| = 12 + 12 −4 = 20

nodes for any edge in

2,2,2

(the 4 subtracted are the

four common nearest neighbors of

and

Pairwise overlap.

For each of the 30 skew-edge pairs

, e

)

2,2,2

, the pairwise intersec-

tion is

|N(e

) ∩ N(e

)| = 10,

(9)

uniformly across all 30 pairs. This is veried by direct enumeration on the FCC lattice; the

value 10 emerges from the geometry of the octahedral void (with the void center at

(1, 0, 0)

and

bounding vertices at

{(2, 0, 0), (0, 0, 0), (1, ±1, 0), (1, 0, ±1)}

) and is invariant under permutation

of the three coordinate axes. For comparison, on the

tetrahedral void the analogous quantity

yields

|N(e

) ∩ N(e

)| = 12

uniformly across the 3 skew pairs  the value used implicitly in

the proton derivation of Ref. [2]. The two values (10 vs. 12) reect the dierent bonding-graph

geometry; neither is a free parameter.

Triple overlap.

For each of the 8 perfect matchings

, e

)

2,2,2

, the triple intersection

|N(e

) ∩ N(e

)| = 8,

(10)

uniformly across all 8 matchings. The triple intersection set decomposes structurally as

|{z}

bounding vertices

+ 2

|{z}

matching-specic bulk nodes

= 8.

(11)

The 6 bounding vertices of the octahedral void are common to the intersection of

every

matching

 they are the lattice nodes adjacent to all 12 octahedron edges by construction. The remaining

2 nodes vary between matchings and lie at the FCC nodes most strongly correlated with the

specic orientation of the chosen 3-matching.

The 8 = 8 coincidence.

Equation (6) contains two distinct quantities that both equal 8:



(O)

triple

= 8

, the number of perfect matchings of the octahedron graph

2,2,2

. This is a

graph-theoretic invariant, independent of the embedding lattice.



(O)

triple

= 8

, the cardinality of the bulk rst-shell triple intersection, computed on the FCC

lattice for any one such matching. This is a geometric quantity, dependent on both the

bonding graph and the embedding lattice.

These quantities are independent. The graph-theoretic 8 (matching count) arises from the

symmetry

n,n,n

n = 2

. The geometric 8 (overlap size) arises from the FCC neighborhood

structure and decomposes as

6 + 2

as above. The numerical equality is a coincidence of the

specic combination of

2,2,2

bonding graph and FCC lattice; either factor would change under

deformation of the lattice neighborhood, while the matching count would not. We emphasize

that Eq. (6) multiplies these two as independent inputs and does not double-count a single

combinatorial quantity.

Robustness check.

A useful sanity check is that the proton derivation of Ref. [2] arises as

the

K = K

bulk

N → N

= 13

skew

→ 3

specialization of the same PIE structure with

pairwise

= K

bulk

= 12

. The series terminates at second order there because

admits

no 3-matching. The transition from proton (

pairwise

= 12

) to octahedral defect (

(O)

pairwise

= 10

)

is the only place where a graph-specic geometric quantity replaces a bulk constant in the

verication-cost formula; this replacement is forced by the geometry of

2,2,2

and is the principal

new combinatorial input of the present paper.

4.4 QEC dual: an open question

The structural counting result

= 1836

of Ref. [2] has an independent realization as a fault-

tolerant verication cost

×C

= 36 ×51 = 1836

in a CSS code on the FCC lattice [3]. That

construction is built on the 13-node cuboctahedral coordination cluster around a single FCC

node, and the proton's tetrahedral defect ts inside this single cluster: its 4 bounding vertices

are mutually at the nearest-neighbor distance

The octahedral defect's 6 bounding vertices include 3 antipodal pairs at second-nearest-

neighbor distance

√

2 L

, which prevents the defect from being captured by any single coordina-

tion cluster and requires a multi-cluster QEC footprint. Constructing the QEC dual of Eq. (6) is

left to future work. We note that the truncation of the PIE series at third order, by the 6-vertex

constraint of

2,2,2

, suggests that the QEC dual should likewise be a closed expression rather

than an asymptotic series.

4.5 Status of the forward derivation

The qualitative predictions of Section 3 stand independently of the verication-cost calculation:

the absence of rst-order EM coupling, absence of SU(3) color, self-Majorana character, and sup-

pressed radiative cooling all follow from the structural symmetry of the

2,2,2

bonding graph and

the bounding octahedron. The quantitative mass prediction in this paper is

= 1.719

GeV,

derived from the closed PIE expansion (Eq. (6)) and the proton mass alone.

5 Comparison with Observations

5.1 The Fermi-LAT 1.51.6 GeV gamma-ray line

A recent Fermi-LAT analysis [4] reports a gamma-ray spectral line at approximately

1.5



1.6

GeV

in three active galactic nuclei (AGN), with combined signicance

> 5σ

(joint test statistic

T S = 57.77

). The individual source measurements and the SSM forward mass prediction

1.719

GeV (Eq. (6)) are compared in Table 1.

The match should be interpreted carefully. The Kang et al. 2026 line, while detected at

> 5σ

in joint analysis, remains at

∼ 2.8



4.1σ

in individual sources, and the Galactic Center analysis

did not converge in the discoverers' work [4]. Conrmation requires additional observations

and analysis. We note three features that connect the Fermi-LAT signal to the SSM prediction

beyond the mass agreement.

Source (4FGL designation)

line

(GeV) Deviation from 1.719 GeV

|∆|/σ

J0749.6+1324 (blazar)

1.62 ± 0.07 +0.099

GeV

1.4

J0250.2

−

8224 (blazar)

1.55 ± 0.10 +0.169

GeV

1.7

J2329.7

−

2118 (radio galaxy)

1.53 ± 0.09 +0.189

GeV

2.1

J0357.0

−

4955 (marginal candidate)

1.50 ± 0.21 +0.219

GeV

1.0

Joint t

∼ 1.55 +0.169

GeV

∼ 11%

Table 1: Fermi-LAT line position measurements [4] compared with the SSM forward mass

prediction

= 1.719

GeV (assuming

χχ → γγ

annihilation, in which case

line

= m

The rst three sources are reported as primary detections by [4] with line-detection signicances

2.8



4.1σ

; J0357.0

−

4955 is a marginal candidate at

1.6σ

detection signicance and is included

here for completeness. The fourth column shows the deviation of the measured line position

from the SSM prediction, expressed in units of the line measurement uncertainty

(

not

the

detection signicance). The forward prediction agrees with each individual source measurement

within



2.1

such units. The joint t at

∼ 1.55

GeV is approximately

11%

below the prediction,

slightly outside Fermi-LAT's

∼ 10%

instrumental energy resolution at 1.5 GeV; the residual

oset is discussed in Section 5.

Cross-section structure.

The Kang et al. 2026 t requires

⟨σv⟩

γγ

≈ 2 × 10

−28

/s, ap-

proximately

150×

smaller than the canonical thermal-relic WIMP cross-section. The discover-

ers attribute this to one-loop EM suppression of the

χχ → γγ

channel. The SSM prediction

of suppressed rst-order EM coupling (Section 3.3) is structurally consistent with this loop-

suppression: a particle whose tree-level photon coupling is absent or strongly suppressed will

produce loop-suppressed annihilation rates without requiring the EM-suppression to be put in

by hand.

Self-conjugate annihilation.

The Kang et al. 2026 analysis treats the line as

χχ

annihilation

(not

χ¯χ

), consistent with the SSM prediction that the octahedral defect is its own antiparticle

(Section 3.5). Models with distinct particle/antiparticle would require additional channels or

assumptions to t the same data.

Source environment and gravitational redshift.

The Fermi-LAT line is observed in active

galactic nuclei, which host supermassive black holes (SMBHs) and are predicted to develop dark-

matter density spikes through adiabatic contraction [23]. Because annihilation ux scales as

emission is dominated by material at small radii in the spike. Photons emerging from this

inner region experience gravitational redshift by a factor

1 − R

, where

is the SMBH's

Schwarzschild radius. For emission at

R ≈ 5 R

, the redshift factor is

≈ 0.90

, mapping a

rest-frame line at

1.719

GeV to

∼ 1.55

GeV observed  consistent with the joint t of [4].

The individual sources tabulated (three primary detections plus the marginal candidate), with

observed energies spanning

1.50



1.62

GeV, would correspond to eective emission radii in the

range

R ≈ 4



9 R

, plausibly reecting source-to-source variation in spike prole, SMBH spin,

or line-of-sight geometry. We emphasize this is a candidate astrophysical interpretation, not a

derivation: the same redshift mechanism would map any suciently heavier rest-frame mass

to the observed line for a suitably chosen emission radius. A more discriminating test would

compare predicted line shapes (broadened by the spike's radial extent and by the orbital velocity

dispersion of the annihilating particles) with observed source spectra; we identify this as an

additional observational direction for future work.

Scope of agreement.

The mass agreement at the



2.1σ

level per source, with a

∼ 11%

residual on the joint t, does not by itself establish that the line is dark matter of the SSM type.

It does establish that the framework's forward prediction (derived from the structural-counting

expansion on

2,2,2

, with no cosmological input) is consistent with the principal observational

anchor available, and that conrmation or refutation of the Fermi-LAT signal in future work

will directly bear on the framework.

5.2 Direct detection

A 1.7 GeV particle with suppressed rst-order EM and no strong coupling has recoil energies

in the few-keV range when scattering o nuclei via available higher-order channels. Standard

WIMP detectors (xenon-based TPCs, germanium detectors) have thresholds in the few-keV

range and are optimized for higher-mass candidates. Direct detection by current experiments

at WIMP scales [18, 19] is not expected for this mass with this coupling structure. Future

light-dark-matter detectors (DAMIC, SENSEI, low-threshold cryogenic detectors) operating in

the sub-keV range may achieve sensitivity if any non-gravitational coupling persists through

higher-order channels.

5.3 Self-interaction cross-section

Two octahedral-void defects passing close to each other interact through their respective strain

elds in the FCC bulk. A leading-order geometric estimate (cross-section

∼ π(2L)

= 4πL

scaled by mass to give

σ/m

) yields

σ/m ∼ 0.01



0.1

/g for

in the QCD-scale range

0.5



fm. This places the framework in the essentially collisionless regime, consistent with the

Bullet Cluster bound

σ/m ≲ 0.47

/g [10] and with the absence of strong dark-sector self-

interaction signatures in cluster collisions.

The leading-order estimate sits below the parameter window

σ/m ≈ 0.1



/g [17, 16]

where self-interacting dark matter (SIDM) models address small-scale-structure observations.

The framework as constituted predicts cold dark matter that is largely collisionless on observa-

tional scales, with a small residual self-interaction below current detection sensitivity. A more

precise calculation requires dening the strain eld of the octahedral defect quantitatively, which

is left to future work.

5.4 Diuse dark-matter halos and almost-dark galaxies

The SSM prediction in Section 3.6 that octahedral-void dark matter cannot dissipate energy

at rst order, and therefore forms diuse halos rather than collapsed compact structures, is

observationally testable through the existence of dark-matter-dominated galaxies with extended,

low-surface-brightness morphologies and minimal stellar content.

The recent identication of Candidate Dark Galaxy-2 (CDG-2) in the Perseus cluster [20]

provides an observational example of this phenomenology. CDG-2 was rst detected through

globular-cluster overdensity rather than diuse stellar emission  the standard search method-

ology that fails for galaxies whose stellar component is too faint to detect directly. Subsequent

stacking of HST imaging and analysis of Euclid Early Release Observations revealed extremely

faint but signicant diuse emission consistent across both datasets, validating CDG-2 as a

genuine galaxy. The system contains four globular clusters within a

∼ 1.2

kpc diameter and a

total galaxy luminosity

V,gal

≈ 6.2 × 10

⊙

, with the GC light fraction estimated at

∼ 16.6%

(rising to

∼ 33%

if a canonical GC luminosity function is assumed). Applied to the GC-to-halo-

mass scaling relations of [21] and [22], these data imply a dark-matter halo mass fraction of

99.94



99.99%

, making CDG-2 plausibly the most dark-matter-dominated galaxy yet identied.

This is exactly the morphological signature predicted by a non-radiatively-cooling dark sec-

tor: the dark-matter halo is extended and diuse, the baryonic component is minimal and

conned to the residual stellar systems (globular clusters and faint inter-cluster light), and there

is no indication of collapsed dark-matter structure analogous to baryonic stellar formation. The

discovery of CDG-2 strengthens the broader phenomenological case that the dark sector lacks a

rst-order dissipation channel.

Scope of this comparison.

We do not claim that CDG-2 uniquely conrms the SSM. Any

dark-matter model with a non-dissipative dark sector  including the standard collisionless

cold-dark-matter paradigm and many wave-like or fuzzy-dark-matter alternatives  is consistent

with the existence of almost-dark galaxies. Furthermore, the

99.94



99.99%

halo-mass fraction is

inferred from GC-count scaling relations rather than direct kinematic measurement; the authors

of [20] note that high-precision kinematic and spectroscopic follow-up is needed to conrm the

dark-matter content directly. CDG-2 is therefore one observational data point consistent with

the SSM phenomenology, not a discriminating test against alternative dark-matter models. The

discovery of further almost-dark galaxies, particularly with direct kinematic conrmation of

their halo masses, would extend this class of observation and tighten the constraint on the

morphological predictions of any non-dissipative dark-matter framework.

5.5 Comparison with existing dark matter models

The octahedral-void defect predicted here shares qualitative features with several composite-

dark-matter scenarios in the literature. GeV-scale strongly-coupled hidden-sector models [13, 14]

produce dark baryons whose mass arises from connement dynamics and whose direct-detection

signatures are suppressed relative to weakly-interacting massive particles. Strongly-interacting

massive particle (SIMP) models [15] predict sub-GeV to GeV-scale dark particles with non-trivial

self-interaction. The SSM prediction lies in a similar parameter region (1.7 GeV mass, suppressed

electromagnetic coupling) but diers in origin: the existence and structural properties of the

dark matter candidate are derived from FCC vacuum geometry rather than from cosmological

freeze-out, asymmetric mechanisms, or hidden-sector gauge dynamics.

The closest phenomenological analog in the existing literature is the

sexaquark

[24, 25]: a

hypothesized neutral, avor-singlet, scalar bound state of

uuddss

quarks with baryon number

B = 2

and strangeness

S = −2

, proposed by Farrar as a dark matter candidate. The sexaquark

and the SSM K=6 octahedral defect agree on several non-trivial phenomenological features. Both

are GeV-scale particles with mass

∼ 2m

(sexaquark stability requires

≲ 2054

MeV, with cos-

mological relic-abundance ts favoring 1.51.8 GeV [25]; the SSM prediction

= 1.719

GeV

lies inside this preferred range). Both involve a six-fold structure (sexaquark: six quarks; SSM

defect: six bounding vertices). Both are electromagnetically neutral with suppressed hadronic

couplings (sexaquark: avor-singlet decoupling from pions; SSM: absence of square plaquettes

in the bonded subgraph

2,2,2

). Both are consistent with the diuse, non-collisional halo phe-

nomenology required by direct-detection nulls and self-interaction constraints.

Three theoretical features distinguish the two models. First, the sexaquark carries

B = 2

and

S = −2

and is therefore strictly distinct from its antiparticle (

, with

B = −2

S = +2

);

the SSM defect, by contrast, is its own antiparticle by the inversion symmetry of the regular

octahedron (Section 3.5). The Majorana-vs-Dirac character is empirically distinguishable in

indirect-detection signatures (annihilation

χχ

vs.

χ¯χ

) and in cosmological asymmetry channels.

Second, the sexaquark mass is a free phenomenological parameter that lattice QCD cannot yet

predict to the precision required for stability; the SSM mass follows from a closed combinatorial

expansion on

2,2,2

with no tted parameters. Third, the sexaquark is a strongly-interacting

QCD bound state (a avor-singlet hadron), whereas the SSM defect carries no SU(3) color charge

(Section 3.4) and does not participate in strong interactions. The two frameworks are therefore

not in direct competition: a denitive sexaquark detection would not falsify the SSM defect, and

vice versa, though discriminating direct-detection, indirect-detection, and accelerator signatures

can be designed.

6 Discussion

6.1 Falsiable predictions

The framework makes the following predictions that could in principle be falsied by observation:

Mass scale.

The framework predicts a single dark matter species with mass

1.719

GeV, derived from the closed inclusion-exclusion expansion on

2,2,2

(Eq. (6)) and

the proton mass alone, with no cosmological input. Denitive identication of dark matter

at a dierent mass scale, with the structural properties predicted here, would falsify the

framework.

Suppressed rst-order EM coupling.

The octahedral defect has suppressed single-

photon coupling at rst order. Observation of rst-order DMphoton coupling at the

level expected for charged or magnetic-moment-bearing dark matter would falsify the

framework.

Absence of standard SU(3) color.

The octahedral defect does not participate in QCD

interactions. Observation of dark-matter hadronization or quark-binding signatures would

falsify the framework.

Single dark-matter species.

The framework predicts a single dark-matter species at the

mass scale

∼ 1.7

GeV. Denitive observation of multiple dark-matter species at distinctly

dierent mass scales would require extension of the framework.

Self-conjugate (Majorana-type) dark matter.

The octahedral defect is its own an-

tiparticle. Observation of a Dirac-type dark sector with distinct particle/antiparticle, or

asymmetric dark matter with a baryogenesis-like origin, would falsify the framework.

Collisionless halos and absence of dark stellar collapse.

The framework predicts

that dark matter cannot radiatively cool at rst order and therefore forms diuse non-

collisional halos. Observation of dense compact dark-matter objects analogous to stars or

planets, or evidence of strong dark-sector radiative cooling producing baryon-like collapse,

would falsify the framework. The recent identication of CDG-2 [20] (see Section 5.4) is a

positive observation consistent with this prediction.

6.2 Open calculations

The most important open calculations are, in order of priority:

Forward derivation of

Ω

/Ω

The cosmological abundance ratio observed by Planck

(

5.36 ± 0.04

[1]) is not derived in the present paper. A rst-principles SSM computation

requires modeling the K=4

→

K=12 phase transition dynamics, the relative formation

rates of octahedral and tetrahedral defects, the early-universe history of matterantimatter

annihilation in the tetrahedral sector, and

χχ → γγ

pair-annihilation in the octahedral

sector. We identify this as the principal remaining open problem. The forward mass

prediction

= 1.719

GeV is independent of this calculation.

QEC dual of the PIE expansion.

The structural counting

= 1836

for the proton has

an independent realization as a fault-tolerant verication cost on a CSS code on the FCC

lattice [3]. Constructing the analogous multi-cluster QEC dual of

= 3364

for the

octahedral defect would provide an independent check on the forward derivation. This

is technically nontrivial because the octahedral defect's bounding vertices span multiple

coordination clusters (Section 4.4).

Self-interaction cross-section.

A quantitative SSM calculation of the strain-eld overlap

between two octahedral defects would produce a self-interaction cross-section that can be

compared to the Bullet Cluster bound and to the inferred small-scale-structure properties

of dark-matter halos.

Halo formation phenomenology.

The free-streaming length of a

∼ 1.7

GeV particle decou-

pling at the FCC crystallization epoch determines whether the framework is consistent

with cold-dark-matter structure formation or requires a warm/intermediate dark-matter

treatment.

7 Conclusions

The Selection-Stitch Model identies baryonic matter with K=4 remnants trapped in the tetra-

hedral interstitial voids of the FCC vacuum lattice. The same lattice contains a second inter-

stitial site  the octahedral void  that admits an analogous K=6 trapped remnant under

the same kinematic rules. We have shown that this companion defect, identied with dark

matter, has the structural properties expected of cold dark matter: suppressed rst-order elec-

tromagnetic coupling (the bonding graph

2,2,2

is bipartite-decient at the relevant 4-cycle

level and the bounding polyhedron has zero square faces), no SU(3) color charge in the form

generated for baryons (the skew-edge pair count

(O)

skew

= 30

does not factor as a three-color rep-

resentation), a self-conjugate (Majorana-type) character due to the inversion symmetry of the

regular octahedron, and suppressed rst-order radiative cooling, which accounts for the observed

baryonicdark-matter clustering asymmetry.

The dark matter mass is predicted directly via the closed inclusion-exclusion expansion on

the

2,2,2

bonding graph, terminating exactly at third order by the 6-vertex constraint of the

octahedron:

= 25 · 144 − 30 · 10 + 8 · 8 = 3364, m

× m

= 1.719

GeV

This is a forward prediction using only the bonded structure of the defect, the structural-counting

framework of Ref. [2], and the proton mass. No cosmological observation is consumed in the

derivation. The recently reported

1.5



1.6

GeV gamma-ray line in three AGN [4] agrees with

this prediction within



2.1σ

of each individual source measurement (joint t at

∼ 1.55

GeV,

∼ 11%

below the predicted rest-frame mass; the residual is consistent with gravitational redshift

of inner-spike annihilation in the host SMBH environment). The cross-section structure required

by the line (

⟨σv⟩

γγ

≈ 2 × 10

−28

/s, loop-suppressed) is consistent with the SSM prediction

of suppressed rst-order EM coupling. We caution that the Fermi-LAT signal remains at

∼ 2.8



4.1σ

in individual sources and requires conrmation.

The cosmological abundance ratio

Ω

/Ω

is not derived in this paper. A rst-principles

SSM derivation requires modeling the K=4

→

K=12 phase transition dynamics, the relative

formation rates of octahedral and tetrahedral defects, and the early-universe history of pair-

annihilation in both the tetrahedral sector (matterantimatter annihilation) and the octahedral

sector (

χχ → γγ

). We identify this as the principal remaining open problem.

The qualitative predictions of the framework  absence of rst-order EM coupling, absence

of SU(3) color, self-conjugate Majorana character, and suppressed radiative cooling  follow

from the structural symmetry of the

2,2,2

bonding graph and the bounding octahedron, and are

independent of the verication-cost calculation. The recent identication of the dark-matter-

dominated almost-dark galaxy CDG-2 [20] provides observational support for the collisionless

diuse-halo prediction.

Data Availability

The geometric quantities used in this paper (FCC void radii, void counts per unit cell, edge graph

2,2,2

for the octahedron, skew-pair count

(O)

skew

= 30

veried by three independent methods)

are standard crystallographic and combinatorial data. The Planck cosmological parameters are

from [1]. The Fermi-LAT line measurements are from [4]. The interactive 3D visualization

referenced in the introduction is hosted at

https://raghu91302.github.io/ssmtheory/oct_

void_3D.html

Declaration of Competing Interest

The author declares no known competing nancial interests or personal relationships that could

have appeared to inuence the work reported in this paper.

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