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 identiﬁed

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 ﬁrst-

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

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

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

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 m

= (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 1–2.1σ of each individual source measurement (joint

ﬁt 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, ﬁt 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 identiﬁed 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 veriﬁcation cost E

× 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 L (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 K

2,2,2

and bounding polyhedron (the

regular octahedron with O

symmetry): absence of ﬁrst-order electromagnetic coupling, absence

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

suppressed ﬁrst-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 veriﬁcation cost C

= 1836 extends to the octahedral defect by

inclusion-exclusion on the K

2,2,2

bonding graph, terminating exactly at third order because the

octahedron’s six vertices forbid any 4-matching: C

= 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 [8] 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 1–2.1σ of

each individual source measurement, with the joint ﬁt 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 K

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 ﬁxes 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 ﬁrst-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 m

= 1.719 GeV from the closed inclusion-exclusion expansion

on the K

2,2,2

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

derived from the same structural-counting machinery that yields C

= 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

1–2.1σ of individual source measurements; the ∼ 11% residual oﬀset from the joint ﬁt 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, deﬁnes its bonding

graph K

2,2,2

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

presents the forward derivation of m

= 1.719 GeV via inclusion-exclusion on K

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 falsiﬁability and open calculations. Sec-

tion 7 concludes.

Interactive 3D Visualization. A WebGL visualization of the octahedral defect

geometry — the K

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, ﬁve

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 a contains atoms at the cube corners and face

centers, giving a Bravais lattice with primitive cell volume a

/4 and nearest-neighbor distance

L = a/

√

2. The unit cell decomposes into two distinct interstitial void types [9, 10] (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 L. The centroid-to-vertex distance is L

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 L. The centroid-to-vertex distance is L/

√

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 L. 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 L. 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, reﬂecting the longer centroid-to-vertex distance (L/

√

2 versus L

3/8).

Why K=4 and K=6, and not intermediate values? The selection of the tetrahedral

(K=4) and octahedral (K=6) conﬁgurations 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 L 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 conﬁguration

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 oﬀer

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 conﬁgurations 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 ﬁtting required and

no intermediate-K particle classes admitted.

3.2 Bonding subgraph: K

2,2,2

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

√

2, and each non-antipodal pair at distance L (the nearest-neighbor distance). Bonds at the

SSM unit-bond-length scale L 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 K

(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 K

2,2,2

(solid green); the 3 antipodal pairs at distance L

√

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 (C

) faces is the structural

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

3.3 Suppressed ﬁrst-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 C

) cannot

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

) 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 K

2,2,2

contains 4-cycles only

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

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

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

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

Falsiﬁable consequence. An octahedral-void defect interacts with the electromagnetic ﬁeld

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

intermediaries, or gravitational coupling to bulk photon modes). Direct ﬁrst-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

at ∼ 1 GeV from beam-dump experiments and supernova

cooling [14, 15]) are consistent with the picture predicted here. A complete proof of EM-

neutrality requires deriving the photon coupling vertex from the SSM strain-ﬁeld 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 K

[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 K

2,2,2

(O)

skew





− 6





= 66 − 36 = 30. (5)

This count is veriﬁed by direct enumeration: each of the 12 edges of K

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 conﬁning channels of

the cuboctahedral shell.

Falsiﬁable 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] identiﬁes anti-baryons with spatial inversion of the trapped tetrahedral-void defect:

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 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 O

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.

Falsiﬁable 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 ﬁrst-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 ﬁnd a match in this framework.

3.6 Suppressed ﬁrst-order radiative cooling and clustering asymmetry

A central observational distinction between baryonic and dark matter is the diﬀerence 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

diﬀuse 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 ﬁrst-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 ﬁrst-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 eﬃciently, cannot collapse below its initial velocity-dispersion scale,

and forms diﬀuse halos rather than dense compact objects. This matches the observed phe-

nomenology of dark matter, including the diﬀuse 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 identiﬁcation 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 [11, 12, 13].

4 Forward Derivation of C

via Inclusion-Exclusion

The structural-counting derivation of m

= 1836 for the proton presented in Ref. [2]

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

2,2,2

bonding graph.

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

= 3364,

= 1.719 GeV, with no cosmological input.

4.1 The structural-counting formula as inclusion-exclusion

The expression of Ref. [2], C

= (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 K

= 144 vacuum bond states (the second-shell

footprint at the bulk coordination K = 12). Summed over N

= 13 structural nodes, this

gives a base count of 13 × 144 = 1872 bond-state disruptions. Pairs of structural nodes with

mutually disjoint ﬂux channels — the c

skew

= 3 skew-edge pairs of K

— have disruption halos

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

as the bulk ﬁrst-shell intersection |N(e

) ∩ N(e

)| on the FCC lattice, is K

pairwise

= 12 for any

skew pair. Subtracting the double-counted bond states gives 1872 −36 = 1836, the proton’s

veriﬁed veriﬁcation cost. The series terminates: K

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 K

2,2,2

with truncation

The same structure applies to the octahedral defect, with three diﬀerences ﬁxed by the bonded

graph K

2,2,2

1. 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, N

= 25.

2. Skew-edge pair count. Each of the 12 edges of K

2,2,2

has exactly 5 mutually disjoint

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

(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.

3. Pairwise overlap on K

2,2,2

. Computing the bulk ﬁrst-shell intersection directly on the

FCC lattice yields |N(e

) ∩ N(e

)| = 10 uniformly across all 30 skew pairs of K

2,2,2

, in

contrast to K

’s value of 12. The pairwise overlap is a graph-speciﬁc geometric quantity,

not a ﬁxed bulk constant.

Crucially, K

2,2,2

admits the next term in the PIE expansion that K

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 c

(O)

triple

= 8 (veriﬁed 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 ﬂux channels

mutually intersect.

The triple overlap on K

2,2,2

, computed directly as the bulk ﬁrst-shell triple intersection

|N(e

) ∩N(e

)|, evaluates to K

(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 c

(O)

quad

= 0 and all higher orders vanish

identically. The PIE expansion for the octahedral defect is ﬁnite 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 m

= (C

) × m

= (3364/1836) ×

938.27 MeV = 1.719 GeV.

4.3 Explicit enumeration of pairwise and triple overlaps

The values K

(O)

pairwise

= 10 and K

(O)

triple

= 8 appearing in Eq. (6) are not free parameters: they are

determined by direct enumeration on the FCC lattice with no ﬁtting. 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 v 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 =

√

2). For a K

2,2,2

edge

e = {v

, v

} between two such vertices, we deﬁne the ﬁrst-shell neighborhood as the union

N(e) = N

) ∪ N

), (8)

which contains |N(e)| = 12 + 12 −4 = 20 nodes for any edge in K

2,2,2

(the 4 subtracted are the

four common nearest neighbors of v

and v

Pairwise overlap. For each of the 30 skew-edge pairs (e

, e

) in K

2,2,2

, the pairwise intersec-

tion is

|N(e

) ∩ N(e

)| = 10, (9)

uniformly across all 30 pairs. This is veriﬁed 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 K

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) reﬂect the diﬀerent bonding-graph

geometry; neither is a free parameter.

Triple overlap. For each of the 8 perfect matchings (e

, e

) of K

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-speciﬁc 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

speciﬁc orientation of the chosen 3-matching.

The 8 = 8 coincidence. Equation (6) contains two distinct quantities that both equal 8:

• c

(O)

triple

= 8, the number of perfect matchings of the octahedron graph K

2,2,2

. This is a

graph-theoretic invariant, independent of the embedding lattice.

• K

(O)

triple

= 8, the cardinality of the bulk ﬁrst-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 K

n,n,n

at 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

speciﬁc combination of K

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, c

skew

→ 3 specialization of the same PIE structure with

pairwise

= K

bulk

= 12 on K

. The series terminates at second order there because K

admits

no 3-matching. The transition from proton (K

pairwise

= 12) to octahedral defect (K

(O)

pairwise

= 10)

is the only place where a graph-speciﬁc geometric quantity replaces a bulk constant in the

veriﬁcation-cost formula; this replacement is forced by the geometry of K

2,2,2

and is the principal

new combinatorial input of the present paper.

Veriﬁcation artifact. The full enumeration tables for all 30 pairwise intersections and all

8 triple intersections (with explicit 6 + 2 decomposition) are provided in the Appendix, and a

short self-contained Python reference implementation that reproduces every integer in Eq. (6)

from the FCC nearest-neighbor vectors and the six bounding vertex coordinates alone is available

at https://github.com/raghu91302/ssmtheory/blob/main/verify_C_DM.py. The script uses

only the Python standard library and runs in under one second; a reader who wishes to verify

(O)

pairwise

= 10 and K

(O)

triple

= 8 independently can do so either by inspection of the tables or by

execution of the script.

4.4 Embedding Uniformity Lemma and Body-Diagonal Structure

The values K

(O)

pairwise

= 10 and K

(O)

triple

= 8 were established in Section 4.3 by direct enumeration.

A reader is entitled to ask whether the observed uniformity across the 30 skew pairs and the

8 perfect matchings is itself a structural fact or a coincidence of independent computations.

The following lemma settles this question and identiﬁes a clean combinatorial labeling of the 8

matchings by the 4 body diagonals of the cube circumscribing the octahedral void.

Lemma (Embedding Uniformity). Embed the octahedral void at the origin of the FCC

lattice as in Section 4.3, and let O

denote its point stabilizer (the symmetry group of the regular

octahedron, of order 48). Then:

(i) The 8 perfect matchings of K

2,2,2

form a single O

-orbit. The triple ﬁrst-shell intersection

size |N(e

) ∩ N(e

)| is therefore constant across all matchings, and equals 8 by

direct evaluation on any representative matching. Hence K

(O)

triple

= 8 is O

-forced.

(ii) The 30 skew pairs of K

2,2,2

split into two O

-orbits, of sizes 24 and 6. The pairwise

ﬁrst-shell intersection size is constant within each orbit by symmetry; direct evaluation

on a representative of each orbit gives the value 10 in both cases. Hence K

(O)

pairwise

= 10

uniformly.

Status of (i) versus (ii). The triple statement (i) is fully O

-forced: a single orbit means

the symmetry alone determines uniformity. The pairwise statement (ii) is stronger than O

symmetry alone can force. With two orbits, O

-invariance guarantees uniformity within each

orbit but allows the two orbits to give diﬀerent values; direct enumeration is required to conﬁrm

that they happen to give the same value 10. We record this as an empirical fact about the

particular embedding of K

2,2,2

in the FCC lattice rather than as a consequence of pure symmetry.

The orbit decomposition and both intersection-size evaluations are veriﬁed by the standalone

script https://github.com/raghu91302/ssmtheory/blob/main/verify_uniformity.py.

Body-diagonal labeling of the matchings. The 8 perfect matchings admit a natural geo-

metric labeling by the 4 body diagonals of the cube circumscribing the octahedral void — the

cube whose 8 corners sit at displacement (±1, ±1, ±1) from the void center, and whose 6 face

centers are the bounding vertices A, B, C, D, E, F of the octahedron. For each matching M :

(a) The 2 “matching-speciﬁc” bulk nodes appearing in the 6 + 2 decomposition of the triple

intersection (Table 4) are exactly the two antipodal corners of one body diagonal of this

cube. Each matching is therefore labeled by a unique body diagonal.

(b) Each body diagonal labels exactly two matchings; the 8 matchings group into 4 pairs in

bijection with the 4 body diagonals.

r 7→ −r (in coordinates relative to the void center). This is the identical centrosymme-

try −I ∈ O

that underlies the self-conjugate (Majorana-type) character of the defect

established in Section 3.5.

(d) The three edge midpoints of any matching M lie in the plane through the void center

perpendicular to M’s labeling body diagonal.

These four properties are veriﬁed by the same standalone script.

Implication for the “8 = 8” coincidence. The factorization c

(O)

triple

· K

(O)

triple

= 8 × 8 = 64

in Eq. (6) multiplies two structurally independent quantities (Section 4.3, “8 = 8 coincidence”):

the graph-theoretic 8 (number of perfect matchings of K

2,2,2

, ﬁxed by the symmetry K

n,n,n

at n = 2) and the geometric 8 (size of the bulk ﬁrst-shell triple intersection, ﬁxed by the FCC

embedding). The body-diagonal labeling gives each of these 8’s an independent structural origin

— the matchings are in bijection with the 4 cube diagonals via the inversion pairing, and the

triple intersection decomposes as 6 + 2 where the 2 are precisely the diagonal’s endpoints. The

numerical equality of the two 8’s in this speciﬁc embedding remains a non-trivial fact about

2,2,2

in FCC, but it is no longer an unstructured coincidence: each side has its own geometric

reading.

4.5 QEC dual: an open question

The structural counting result C

= 1836 of Ref. [2] has an independent realization as a fault-

tolerant veriﬁcation cost E

×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 ﬁts inside this single cluster: its 4 bounding vertices

are mutually at the nearest-neighbor distance L.

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 K

2,2,2

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

than an asymptotic series.

4.6 Status of the forward derivation

The qualitative predictions of Section 3 stand independently of the veriﬁcation-cost calculation:

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

pressed radiative cooling all follow from the structural symmetry of the K

2,2,2

bonding graph and

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

= 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 signiﬁcance > 5σ (joint test statistic

T S = 57.77). The individual source measurements and the SSM forward mass prediction m

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

Source (4FGL designation) E

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 ﬁt ∼ 1.55 +0.169 GeV ∼ 11%

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

prediction m

= 1.719 GeV (assuming χχ → γγ annihilation, in which case E

line

= m

The ﬁrst three sources are reported as primary detections by [4] with line-detection signiﬁcances

2.8–4.1σ; J0357.0−4955 is a marginal candidate at 1.6σ detection signiﬁcance 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 σ

(not the

detection signiﬁcance). The forward prediction agrees with each individual source measurement

within 1–2.1 such units. The joint ﬁt 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

oﬀset is discussed in Section 5.

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]. Conﬁrmation requires additional observations

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

beyond the mass agreement.

Cross-section structure. The Kang et al. 2026 ﬁt 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 ﬁrst-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 ﬁt 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 [26]. Because annihilation ﬂux scales as ρ

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

/R, where R

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 ﬁt 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 eﬀective emission radii in the

range R ≈ 4–9 R

, plausibly reﬂecting source-to-source variation in spike proﬁle, 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 suﬃciently 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 1–2.1σ level per source, with a ∼ 11%

residual on the joint ﬁt, does not by itself establish that the line is dark matter of the SSM

type. Two cautions deserve explicit statement. First, the most direct historical precedent for a

Fermi-LAT gamma-ray line claim at comparable per-source signiﬁcance is the ∼ 130 GeV feature

identiﬁed by Bringmann et al. and Weniger in 2012 [5, 6], which generated extensive theoretical

follow-up before being substantially attenuated by additional data and reprocessed instrument

response functions and ultimately attributed in signiﬁcant part to instrumental systematics [7].

The Kang et al. 2026 detection is at a diﬀerent energy and in a diﬀerent source class (extragalactic

AGN rather than Galactic Center), so the instrumental story is not the same; but the base

rate for line claims at ∼ 3–4σ per source surviving further data is poor, and conﬁrmation by

an independent analysis or instrument is required before the present mass agreement carries

decisive weight. Second, the gravitational-redshift mechanism that closes the ∼ 11% residual

is itself a free degree of freedom: as noted above, an emission radius R ≈ 5 R

produces the

observed shift, but the same mechanism (which is strictly one-sided, observed ≤ rest-frame)

would accommodate any rest-frame mass between the observed line and ∼ 1.2× the observed

line by an appropriate choice of R, covering the entire range plausibly populated by spike-

dominated emission. The mass-prediction falsiﬁability is therefore O(20%) once the redshift

is left unconstrained; a sharper test requires either an independent spike-proﬁle measurement

at one of the AGN or a line-shape comparison of the kind noted at the end of the preceding

paragraph. What Table 1 does establish is that the framework’s forward prediction (derived from

the structural-counting expansion on K

2,2,2

, with no cosmological input) is consistent with the

principal observational anchor available, and that conﬁrmation 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 ﬁrst-order EM and no strong coupling has recoil energies

in the few-keV range when scattering oﬀ 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 [21, 22] 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

ﬁelds 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 cm

/g for L in the QCD-scale range 0.5–

2 fm. This places the framework in the “essentially collisionless” regime, consistent with the

Bullet Cluster bound σ/m ≲ 0.47 cm

/g [13] 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 ≈ 0.1–2 cm

/g [20, 19]

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 deﬁning the strain ﬁeld of the octahedral defect quantitatively, which

is left to future work.

5.4 Diﬀuse dark-matter halos and almost-dark galaxies

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

at ﬁrst order, and therefore forms diﬀuse 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 identiﬁcation of Candidate Dark Galaxy-2 (CDG-2) in the Perseus cluster [23]

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

globular-cluster overdensity rather than diﬀuse 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 signiﬁcant diﬀuse 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 L

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 [24] and [25], 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 identiﬁed.

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

tor: the dark-matter halo is extended and diﬀuse, the baryonic component is minimal and

conﬁned 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

ﬁrst-order dissipation channel.

Scope of this comparison. We do not claim that CDG-2 uniquely conﬁrms 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 [23] note that high-precision kinematic and spectroscopic follow-up is needed to conﬁrm 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 conﬁrmation 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 [16, 17]

produce dark baryons whose mass arises from conﬁnement dynamics and whose direct-detection

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

massive particle (SIMP) models [18] 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 diﬀers 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 [27, 28]: a

hypothesized neutral, ﬂavor-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 m

≲ 2054 MeV, with cos-

mological relic-abundance ﬁts favoring 1.5–1.8 GeV [28]; the SSM prediction m

= 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: ﬂavor-singlet decoupling from pions; SSM: absence of square plaquettes

in the bonded subgraph K

2,2,2

). Both are consistent with the diﬀuse, 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 (

S, 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 K

2,2,2

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

QCD bound state (a ﬂavor-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 deﬁnitive 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 Falsiﬁable predictions

The framework makes the following predictions that could in principle be falsiﬁed by observation:

1. Mass scale. The framework predicts a single dark matter species with mass m

1.719 GeV, derived from the closed inclusion-exclusion expansion on K

2,2,2

(Eq. (6)) and

the proton mass alone, with no cosmological input. Deﬁnitive identiﬁcation of dark matter

at a diﬀerent mass scale, with the structural properties predicted here, would falsify the

framework.

2. Suppressed ﬁrst-order EM coupling. The octahedral defect has suppressed single-

photon coupling at ﬁrst order. Observation of ﬁrst-order DM–photon coupling at the

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

framework.

3. 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.

4. Single dark-matter species. The framework predicts a single dark-matter species at the

mass scale ∼ 1.7 GeV. Deﬁnitive observation of multiple dark-matter species at distinctly

diﬀerent mass scales would require extension of the framework.

5. 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.

6. Collisionless halos and absence of dark stellar collapse. The framework predicts

that dark matter cannot radiatively cool at ﬁrst order and therefore forms diﬀuse 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 identiﬁcation of CDG-2 [23] (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:

1. 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 ﬁrst-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 m

= 1.719 GeV is independent of this calculation.

2. QEC dual of the PIE expansion. The structural counting C

= 1836 for the proton has

an independent realization as a fault-tolerant veriﬁcation cost on a CSS code on the FCC

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

= 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 4.5).

3. Self-interaction cross-section. A quantitative SSM calculation of the strain-ﬁeld 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.

4. 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 identiﬁes 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, identiﬁed with dark

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

tromagnetic coupling (the bonding graph K

2,2,2

is bipartite-deﬁcient 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 c

(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 ﬁrst-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 K

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 1–2.1σ of each individual source measurement (joint ﬁt 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 ﬁrst-order EM coupling. We caution that the Fermi-LAT signal remains at ∼ 2.8–

4.1σ in individual sources and requires conﬁrmation.

The cosmological abundance ratio Ω

/Ω

is not derived in this paper. A ﬁrst-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 ﬁrst-order EM coupling, absence

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

from the structural symmetry of the K

2,2,2

bonding graph and the bounding octahedron, and are

independent of the veriﬁcation-cost calculation. The recent identiﬁcation of the dark-matter-

dominated almost-dark galaxy CDG-2 [23] provides observational support for the collisionless

diﬀuse-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 c

(O)

skew

= 30 veriﬁed 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 full enumeration of the

30 pairwise and 8 triple overlaps used in Eq. (6) is given in the Appendix; a self-contained

reference implementation that reproduces every integer in the forward prediction C

= 3364

is available at https://github.com/raghu91302/ssmtheory/blob/main/verify_C_DM.py, and

a companion script that veriﬁes the orbit decomposition, body-diagonal labeling, and inversion

pairing of the Embedding Uniformity Lemma (Section 4.4) is available at https://github.com/

raghu91302/ssmtheory/blob/main/verify_uniformity.py. 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 ﬁnancial interests or personal relationships that could

have appeared to inﬂuence the work reported in this paper.

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100423

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Appendix: Enumeration Tables for Pairwise and Triple Overlaps

This appendix provides the explicit enumeration of the 30 pairwise and 8 triple ﬁrst-shell inter-

sections that determine K

(O)

pairwise

= 10 and K

(O)

triple

= 8 in Eq. (6). The setup follows Section 4.3

exactly: the octahedral void is centered at (1, 0, 0), the 12 FCC nearest-neighbor displace-

ments are {(±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1)}, and the ﬁrst-shell neighborhood of an edge is

N(e) = N

) ∪ N

) with |N(e)| = 20.

Vertex labels. We label the six bounding vertices of the octahedral void as in Table 2.

Label Coordinates

A (2, 0, 0)

B (0, 0, 0)

C (1, +1, 0)

D (1, −1, 0)

E (1, 0, +1)

F (1, 0, −1)

Table 2: The six bounding vertices of the octahedral void at (1, 0, 0). The three antipodal non-

bonded pairs are (A, B), (C, D), (E, F ).

The 12 edges of K

2,2,2

are then {AC, AD, AE, AF, BC, BD, BE, BF, CE, CF, DE, DF } (ev-

ery vertex-pair from distinct antipodal classes). The 3 antipodal non-bonds AB, CD, EF are

at distance L

√

2 and do not contribute to K

2,2,2

All 30 pairwise intersections. Table 3 lists every skew pair of K

2,2,2

together with its ﬁrst-

shell intersection size. All 30 entries equal 10; no pair gives any other value. The uniformity is

a consequence of the O

symmetry of the octahedral void acting transitively on skew pairs.

(AC, BD) : 10 (AD, BE) : 10 (AE, CF ) : 10

(AC, BE) : 10 (AD, BF ) : 10 (AE, DF ) : 10

(AC, BF ) : 10 (AD, CE) : 10 (AF, BC) : 10

(AC, DE) : 10 (AD, CF ) : 10 (AF, BD) : 10

(AC, DF ) : 10 (AE, BC) : 10 (AF, BE) : 10

(AD, BC) : 10 (AE, BD) : 10 (AF, CE) : 10

(AE, BF ) : 10 (BC, DE) : 10 (AF, DE) : 10

(BC, DF) : 10 (BE, CF ) : 10 (BD, CE) : 10

(BE, DF ) : 10 (BF, CE) : 10 (BD, CF ) : 10

(BF, DE) : 10 (CE, DF ) : 10 (CF, DE) : 10

Table 3: All 30 skew-edge pairs of K

2,2,2

with their ﬁrst-shell intersection sizes |N(e

) ∩N(e

)|.

Every entry is 10. The summary statistic K

(O)

pairwise

= 10 used in Eq. (6) is the common value of

this list, not an average over a non-uniform distribution.

All 8 perfect matchings and their triple intersections. Table 4 lists every perfect match-

ing (3-matching) of K

2,2,2

together with the explicit 8 elements of its triple ﬁrst-shell intersection.

The decomposition into 6 bounding-vertex nodes plus 2 matching-speciﬁc bulk nodes is shown

directly.

# Matching Bounding Other Other-node coordinates

M1 {AC, BE, DF } 6 2 (0, +1, −1), (2, −1, +1)

M2 {AC, BF, DE} 6 2 (0, +1, +1), (2, −1, −1)

M3 {AD, BE, CF } 6 2 (0, −1, −1), (2, +1, +1)

M4 {AD, BF, CE} 6 2 (0, −1, +1), (2, +1, −1)

M5 {AE, BC, DF } 6 2 (0, −1, +1), (2, +1, −1)

M6 {AE, BD, CF } 6 2 (0, +1, +1), (2, −1, −1)

M7 {AF, BC, DE} 6 2 (0, −1, −1), (2, +1, +1)

M8 {AF, BD, CE} 6 2 (0, +1, −1), (2, −1, +1)

Table 4: All 8 perfect matchings of K

2,2,2

with their ﬁrst-shell triple intersections decomposed

as 6 + 2. The “Bounding” column counts the 6 bounding vertices A, B, C, D, E, F (common to

every matching by construction), and the “Other” column counts the 2 matching-speciﬁc bulk

nodes (listed explicitly in the last column). Every triple intersection has size 8, so K

(O)

triple

= 8

uniformly.

Structure of the matching-speciﬁc nodes. The 2 matching-speciﬁc bulk nodes for each

matching sit at displacement (±1, ±1, ±1) from the void center (1, 0, 0), on the four body diago-

nals of the unit cube whose face centers are the bounding octahedral vertices. The 8 matchings

group into 4 pairs (M1 ↔ M8), (M2 ↔ M6), (M3 ↔ M7), (M 4 ↔ M5), with the two match-

ings in each pair sharing the same 2-node companion set. The 4 distinct companion sets are

in bijection with the 4 body diagonals of that cube. The two nodes within each companion set

are inversion-related through the void center, reﬂecting the centrosymmetry O

∋ −I that also

underlies the self-conjugate character of the defect (Section 3.5).

Assembly. Combining Tables 3 and 4 with N

= 25 and the per-node disruption count

= 144 reproduces Eq. (6) by direct substitution:

= 25 ×144 − 30 × 10 + 8 × 8 = 3600 − 300 + 64 = 3364, (12)

and m

= (3364/1836)×m

= 1.7191 GeV using the CODATA proton mass m

= 938.272 MeV [8].