Dark Matter as Incomplete Crystallization

Dark Matter as Incomplete Crystallization:

Neutrality, Abundance, and a 1.719 GeV Mass from the

Octahedral Void

Raghu Kulkarni

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

June 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 properties follow from its structural symmetry: absence

of rst-order electromagnetic coupling, absence of the baryonic SU(3) color mechanism,

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

framework's central testable output is the dark matter mass, predicted directly from a

closed inclusion-exclusion expansion on

2,2,2

 a base verication cost, minus pairwise ux-

channel overlaps, plus triple overlaps  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, with the proton mass as the sole calibration input and no

cosmological tting. A geometric plausibility estimate of the cosmic abundance ratio follows

from the same lattice: weighting the nucleation volumes of the octahedral and tetrahedral

interstitial sites by their multiplicities and by the defect mass ratio gives

Ω

/Ω

≈ 5.7

(Appendix A), within

of the observed

5.36

and accounting for the order of magnitude of

the ratio from void geometry alone, though it depends on the choice of nucleation volume

and is not a derivation of the precise value. A recently reported

1.5



1.6

GeV gamma-ray

line [4] is consistent with annihilation of a defect of this mass; the annihilation channel

and the quantitative comparison are deferred to a companion paper [5], and the predictive

content of the present work is the

1.719

GeV mass itself, not the gamma-ray signal.

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.

raghu@idrive.com

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)

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 unifying picture is

incomplete crystallization

. In the SSM the vacuum crystallizes into

the FCC lattice, and every node in the perfect bulk reaches full coordination

K = 12

. Matter

is where this crystallization fails to complete: a node trapped below bulk coordination at an

interstitial site, unable to stitch into the surrounding lattice. The companion paper [2] develops

this for the tetrahedral void, where a

K = 4

remnant becomes the proton, and shows how its

incomplete bonding generates fractional charge, color connement, and the proton mass. The

present paper applies the same picture to the lattice's other interstitial site: the octahedral void

admits a

K = 6

remnant, a second form of incomplete crystallization at a more symmetric

site, which we identify as a dark matter candidate. The two particles are then not independent

constructions but the two ways the FCC crystal can fail to close around an interstitial node 

the tetrahedral remnant giving visible matter, the octahedral remnant giving dark matter.

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 fol-

low within the SSM structural-symmetry rules from the bonding graph

2,2,2

and bound-

ing polyhedron (the regular octahedron with

symmetry): absence of rst-order electro-

magnetic coupling, absence of the baryonic SU(3) color-generating mechanism, self-conjugate

(Majorana-type) character, and suppressed rst-order radiative cooling. These match the stan-

dard 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, termi-

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

= 25 ×144 −30 ×10 + 8 ×8 = 3364

. The three terms are the inclusion-exclusion structure

made explicit: a base verication cost, minus the pairwise overlaps where two ux channels

double-count, plus the triple overlaps restored where three channels coincide. Each coecient

is a xed structural count, not a tted value:

is the number of disrupting nodes (6 bounding

vertices

4 bonds, plus the trapped center),

144 = K

the second-shell footprint at bulk coor-

dination

K = 12

the skew-edge pairs of

2,2,2

and

their pairwise rst-shell overlap, and

the two

s the perfect matchings of the octahedron and their triple overlap; Section 4 derives

all six by direct enumeration. The framework therefore predicts the dark matter mass directly:

× m

3364

1836

× 938.272

MeV

= 1.719

GeV

(3)

using only the proton mass [9] as a calibration input. A

∼ 1.5



1.6

GeV gamma-ray line re-

cently reported in three active galactic nuclei [4] sits near this mass; the annihilation channel

that connects the two, and the quantitative comparison, are treated in a companion paper [5]

(Section 5.1).

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 give a geometric plausibility estimate of the cosmological abundance ratio

Ω

/Ω

Appendix A: the nucleation volumes of the octahedral and tetrahedral voids, weighted by the

defect mass ratio, yield

Ω

/Ω

≈ 5.7

, within

of the observed value. This xes the order

of magnitude from lattice geometry alone. A rst-principles derivation from the early-universe

formation and annihilation history is not attempted here, and the forward mass prediction is

independent of it.

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. The predicted mass sits near the reported 1.51.6 GeV gamma-ray line [4]; the annihi-

lation channel and observational comparison are carried out in the companion paper [5]

(Section 5.1), and a geometric plausibility estimate of

Ω

/Ω

is given in Appendix A.

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 [10, 11] (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

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

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.

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).

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.

The distinction is the standard graph-theoretic one between odd and even cycles. A dipolar

oscillation requires the charge displacement to alternate in sign around the faceone sublattice

swinging positive while the other swings negativewhich is a two-coloring of the cycle's vertices

into

and

−

. An even cycle such as

admits this two-coloring (

+, −, +, −

); an odd cycle such

does not, because closing the loop after three vertices forces two like-signed vertices to be

adjacent, so no consistent alternating (bipartite) mode exists. The triangular face therefore has

no dipolar oscillation to couple to the eld at rst order, while the square face does. 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.

-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 removes the internal

bipartite channel; combined with the full

symmetry of the defect, which forces a vanishing

rst-order dipole moment, this forbids a rst-order dipole photon coupling, as discussed in

Section 3.3.

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.

This is where the octahedral and tetrahedral cases part, and the distinction answers the nat-

ural objectionwhy the tetrahedral defect may borrow the bulk cuboctahedral square channels

while the octahedral defect does not. The tetrahedral baryon is

anchor-selected

: one bond is

singled out as the bulk-coupling channel, breaking the site symmetry

→ S

, and the resulting

conguration carries a nonzero dipole moment that the bulk square faces can carry. The octa-

hedral defect is anchor-free and retains its full

site symmetry. Under

the position vectors

of the six bounding nodes sum to zero and span no symmetry-singlet dipole; the electric-dipole

operator transforms as the

representation, which does not appear in the defect's symmetric

ground conguration, so its rst-order dipole moment vanishes identically by symmetry. A van-

ishing dipole has nothing for any channel to carryinternal

borrowed from the bulkso the

suppression is not a property of the bounding faces alone but of the defect's symmetry: there

is no rst-order dipole to couple, whether through internal faces or through the surrounding

cuboctahedral shells. This is the sharp content of the tetrahedral/octahedral asymmetry: an-

chor selection breaks the symmetry and produces a dipole for the baryon; the absence of anchor

selection preserves it and forbids one for the dark defect.

To be precise about scope: this establishes the absence of a

rst-order dipole

photon coupling.

It does not by itself exclude higher-multipole, loop-induced, or baryon-mediated electromagnetic

interactions. The framework's claim is therefore a hierarchy rather than exact neutrality: the

leading dipole term is forbidden by parity, and the surviving higher-order channels are present

but suppressed. EM neutrality below should be read in this senseneutrality at leading

(dipole) order, not at all orders.

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

(dipole) coupling is forbidden by the centrosymmetry of the defect's bounding octahedron: a

parity-even symmetric ground conguration has no

dipole moment. This matches the stan-

dard 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 [15, 16]) are consistent with the picture predicted here. The vanishing of the rst-order

dipole is a symmetry (parity) result and is robust; a complete treatment of the residual higher-

multipole and higher-order couplings requires deriving the photon coupling vertex from the SSM

strain-eld dynamics, which we do not attempt here. The argument given is the structural-

symmetry analog of the Ref. [2] argument applied to the octahedral case, now sharpened to a

parity selection rule for the leading dipole term.

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 reproduce the specic color-generating mechanism that

the SSM uses for baryons. In Ref. [2] the three QCD colors arise because the

skew-pair

count is exactly 3, in bijection with the three conning channels of the cuboctahedral shell;

the octahedral defect's count of 30 has no such bijection with a three-channel structure, so the

baryonic color-generating construction does not carry over. We state this as the more limited

claim it is: the octahedral defect does not acquire color through the SSM mechanism that

produces it for baryons. We do not claim to exclude every possible SU(3) representation or

composite color assignment by group theory alone; that would require a separate argument.

What follows for the dark-matter phenomenology is that the defect does not couple through the

specic three-color conning channels by which baryons hadronize.

Consequence.

An octahedral-void defect does not participate in strong-force interactions

through the baryonic color mechanism: it does not acquire one of the three cuboctahedral

conning charges, so it cannot bind to baryons through that channel or hadronize as quarks do.

This matches the standard dark-matter requirement of strong-force neutrality at the level of the

mechanism the framework supplies.

3.5 Self-conjugate (Majorana-type) character

Reference [2] identies 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

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 [12, 13, 14].

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.

Interpretation of the verication cost.

The quantity

is not a generic structural com-

plexity but a count of the bond-state disruptions a trapped defect imposes on the surrounding

lattice. In the SSM, mass is identied with exactly this disruption through the massenergy

information correspondence of Ref. [3]: each disrupted bond state carries a xed quantum

of energy

kT ln 2

(the Landauer cost of one resolved bit), so the rest energy of a defect is

= C kT ln 2

, linear in the disruption count

. The lattice temperature scale

and the con-

version are common to every defect, so they cancel in any ratio of masses:

= C

exactly, independent of the (program-level) value of

. This is why the dimensionless count

ratio

3364/1836

can be read directly as a mass ratio, and why the proton mass enters only as

the single dimensionful calibration. The linear scaling is the content of the correspondence, not

an assumption introduced here; the present paper inherits it and applies it to the octahedral

defect.

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.

The octahedral graph admits a further term in the PIE expansion that

does not. With 6

vertices, 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.

Verication 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

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

(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

(O)

pairwise

= 10

and

(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 identies 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

denote its point stabilizer (the symmetry group of the regular

octahedron, of order 48). Then:

(i)

The 8 perfect matchings of

2,2,2

form a single

-orbit. The triple rst-shell intersection

size

|N(e

) ∩ N(e

is therefore constant across all matchings, and equals 8 by

direct evaluation on any representative matching. Hence

(O)

triple

= 8

-forced.

(ii)

The 30 skew pairs of

2,2,2

split into two

-orbits, of sizes 24 and 6. The pairwise

rst-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

(O)

pairwise

= 10

uniformly.

Status of (i) versus (ii).

The triple statement (i) is fully

-forced: a single orbit means

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

symmetry alone can force. With two orbits,

-invariance guarantees uniformity

within

each

orbit but allows the two orbits to give dierent values; direct enumeration is required to conrm

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

particular embedding of

2,2,2

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

The orbit decomposition and both intersection-size evaluations are veried 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

(a) The 2 matching-specic bulk nodes appearing in the

6 + 2

decomposition of the triple

intersection (Table 3) 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

lie in the plane through the void center

perpendicular to

's labeling body diagonal.

These four properties are veried by the same standalone script.

Implication for the 8 = 8 coincidence.

The factorization

(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

2,2,2

, xed by the symmetry

n,n,n

n = 2

) and the geometric 8 (size of the bulk rst-shell triple intersection, xed 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 specic 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

= 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.

4.6 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 the baryonic SU(3) color mechanism, self-

Majorana character, and suppressed 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 reported 1.51.6 GeV gamma-ray line

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

1.5



1.6

GeV in three

active galactic nuclei (joint test statistic

T S ≃ 57.77

> 5σ

combined; individual signicances

2.8



4.1σ

). The line sits below the

1.719

GeV that a direct

χχ → γγ

annihilation of the present

candidate would produce. The annihilation channel of the octahedral defect is

not

direct two-

photon annihilation: it is xed by the lattice geometry to be a semi-annihilation

χχ → γ χ

′

with a massive dark residual, which places the photon line below

. That derivation, the

resulting line energy, and its quantitative comparison with the Kang et al. [4] measurement

are the subject of the companion paper [5]; we do not reproduce or defend the observational

comparison here. The present paper establishes only the candidate, its mass

= 1.719

GeV,

and its abundance (Appendix A); the observational test is deferred to the companion work.

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 [22, 23] 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 [14] and with the absence of strong dark-sector self-

interaction signatures in cluster collisions. This estimate should be read only as an order-of-

magnitude dimensional check until the SSM strain prole of the octahedral defect is derived: it

depends on identifying the interaction length scale with

L ∼ 0.5



fm, which the present paper

does not derive.

The leading-order estimate sits below the parameter window

σ/m ≈ 0.1



/g [21, 20]

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 [24]

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 [25] and [26], 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. The

99.94



99.99%

halo-mass fraction is inferred

from GC-count scaling relations rather than direct kinematic measurement; the authors of [24]

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 [17, 18]

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 [19] 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

[28, 29]: 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 [29]; 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 the baryonic SU(3) color mechanism.

The octahedral defect does not

acquire SU(3) color through the SSM baryonic color-generating mechanism. Observation

that dark matter hadronizes or binds through ordinary QCD-like color channels would

falsify this identication.

Single dominant dark-matter species.

The minimal framework predicts a single dom-

inant 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 frame-

work beyond its minimal form (for example, additional stable defect classes, excitations,

or bound states).

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 [24] (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:

First-principles derivation of

Ω

/Ω

Appendix A gives a geometric plausibility estimate

(

≈ 5.7

, within

of the observed

5.36 ± 0.04

[1]) from the void nucleation volumes and

the defect mass ratio. A full derivation of the precise value requires modeling the K=4

→

K=12 phase transition dynamics, the relative formation rates of the two defect types, and

the early-universe annihilation history in both sectors (including the origin of the baryon

asymmetry). 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 4.5).

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 reported

1.5



1.6

GeV gamma-ray line [4] sits near this mass; the geometry-xed

semi-annihilation channel that connects them, and the quantitative comparison, are carried out

in the companion paper [5] (Section 5.1), and the Fermi-LAT signal itself awaits conrmation.

A geometric plausibility estimate of the abundance ratio,

Ω

/Ω

≈ 5.7

(within

of the

observed

5.36

), is given in Appendix A; a rst-principles derivation of the precise value, together

with the origin of the baryon asymmetry, remains the principal open problem.

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

sence of the baryonic SU(3) color-generating mechanism, 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 [24] 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 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

= 3364

is available at

https://github.com/raghu91302/ssmtheory/blob/main/verify_C_DM.py

, and

a companion script that veries 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 nancial interests or personal relationships that could

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

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A A Geometric Estimate of the Abundance Ratio

The two interstitial sites that host the baryonic and dark defects are the tetrahedral and octahe-

dral voids of the FCC lattice. Their geometry alone xes the order of magnitude of the cosmic

abundance ratio, without any abundance tting. We make the estimate explicit and state its

assumptions.

Treat each void as a nucleation region for a new node, and assume nodes form at a constant

rate per unit nucleation volume, so that the expected number of defects of a given type is

proportional to the total nucleation volume of that void class: the number of voids times the

nucleation volume per void. The lattice nodes are dimensionless points; the relevant volume is

therefore not a hard-sphere interstitial gap but the region in which a new node may be created.

A new node cannot nucleate arbitrarily close to an existing bounding node: the minimum

separation is the Voronoi half-spacing

L/2

, where

is the nearest-neighbor lattice distance.

The nucleation region of a void is thus the sphere, centered on the void, of radius

r = d −

(12)

where

is the distance from the void center to its bounding nodes. In units of the nearest-

neighbor distance

, the centroid-to-vertex distances are

oct

√

L, d

tet

(13)

so subtracting the half-spacing

L/2

gives the nucleation radii in closed form,

oct



√

−



L = (

√

2 − 1)

≈ 0.414

(14)

tet



−



L =



− 1



≈ 0.225

(15)

The decimals

0.414

and

0.225

are therefore the exact surds

√

2 − 1

and

3/2 − 1

, not tted

quantities; both are xed entirely by the lattice spacing, with no atomic size invoked. The two

voids and their size dierencethe longer octahedral centroid-to-node distance, and hence the

larger nucleation sphereare shown in Figure 1.

The conventional cubic cell contains

oct

= 4

octahedral and

tet

= 8

tetrahedral voids.

The octahedral sites host the dark defect and the tetrahedral sites the baryon, so the expected

number

ratio of dark to baryonic defects is

oct

tet

4 (

√

2 − 1)

8 (

3/2 − 1)

4 (0.414)

8 (0.225)

≈ 3.13,

(16)

Weighting by the defect mass ratio

= 3364/1836 = 1.833

gives the energy-density

ratio

Ω

≈ 3.13 ×1.833 ≈ 5.7.

(17)

The observed value is

Ω

/Ω

= 5.36 ± 0.04

(Planck 2018 [1]). The geometric estimate lands

∼ 7%

high. This accounts for the order of magnitude of the ratiowhy it is of order

rather

than order

0.5

from the void geometry and the defect mass ratio alone, with no free

parameter and no abundance tting. We do not represent it as a derivation of the precise value:

the residual

is unexplained, and closing it would require either an occupancy model with a

tunable rate or a dierent nucleation-radius prescription, both of which forfeit the parameter-free

character that makes the estimate meaningful.

We checked one framework-internal alternative for the nucleation radius: the metric-wall

radius

√

that bounds the trapped-node stability region (cf. [5]). Using the cage radii relative

to that wall in place of the

d−L/2

nucleation sphere gives a markedly worse result (o by factors

of several, depending on the precise reading), so the

L/2

minimum-separation measure is not a

tuned choice but the one natural scale that lands near the observed ratio. The estimate xes only

the abundance ratio; the origin of the baryon asymmetrywhy baryonic defects exceed anti-

defectsis a separate mechanism not addressed here, and the dark sector, being self-conjugate

(Majorana), has no analogous asymmetry to explain.

B Enumeration Tables for Pairwise and Triple Overlaps

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

sections that determine

(O)

pairwise

= 10

and

(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 rst-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 1.

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 1: 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

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

are

at distance

√

and do not contribute to

2,2,2

All 30 pairwise intersections.

Table 2 lists every skew pair of

2,2,2

together with its rst-

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

veried by direct enumeration:

splits the 30 skew pairs into two orbits (of sizes 6 and 24)

and guarantees a common value within each orbit, and direct evaluation shows both orbits give

the same value, 10.

(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 2: All 30 skew-edge pairs of

2,2,2

with their rst-shell intersection sizes

|N(e

) ∩N(e

Every entry is 10. The summary statistic

(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 3 lists every perfect match-

ing (3-matching) of

2,2,2

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

The decomposition into 6 bounding-vertex nodes plus 2 matching-specic bulk nodes is shown

directly.

# Matching Bounding Other Other-node coordinates

{AC, BE, DF }

6 2

(0, +1, −1)

(2, −1, +1)

{AC, BF, DE}

6 2

(0, +1, +1)

(2, −1, −1)

{AD, BE, CF }

6 2

(0, −1, −1)

(2, +1, +1)

{AD, BF, CE}

6 2

(0, −1, +1)

(2, +1, −1)

{AE, BC, DF }

6 2

(0, −1, +1)

(2, +1, −1)

{AE, BD, CF }

6 2

(0, +1, +1)

(2, −1, −1)

{AF, BC, DE}

6 2

(0, −1, −1)

(2, +1, +1)

{AF, BD, CE}

6 2

(0, +1, −1)

(2, −1, +1)

Table 3: All 8 perfect matchings of

2,2,2

with their rst-shell triple intersections decomposed

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-specic bulk

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

(O)

triple

= 8

uniformly.

Structure of the matching-specic nodes.

The 2 matching-specic 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)

(M4 ↔ 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, reecting the centrosymmetry

∋ −I

that also

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

Assembly.

Combining Tables 2 and 3 with

= 25

and the per-node disruption count

= 144

reproduces Eq. (6) by direct substitution:

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

(18)

and

= (3364/1836)×m

= 1.7191

GeV using the CODATA proton mass

= 938.272

MeV [9].