Dark Matter as Incomplete Crystallization

Academic Editor: Mingxing Luo

Received: 19 July 2026

Revised: 24 August 2026

Accepted: 4 September 2026

Published: 5 September 2026

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license.

Article

Dark Matter as Incomplete Crystallization: A Geometric

Construction on the Octahedral Void of the FCC Vacuum Lattice

Raghu Kulkarni

SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA; raghu@idrive.com

Abstract

In the Selection-Stitch Model (SSM), baryonic matter is a

K =

4 remnant trapped in a

tetrahedral void of the

K =

12 FCC vacuum lattice. We examine the second interstitial

site, the octahedral void, as a candidate dark matter trap. Its bonded subgraph is the

complete tripartite graph

2,2,2

. Four structural properties follow from its symmetry. Two

are exact: the defect is self-conjugate, and it has no ﬁrst-order electric dipole. Two are

weaker and are stated as such: the electromagnetic coupling is suppressed at dipole order

rather than at all orders, and the mechanism that generates SU(3) color for the tetrahedral

baryon has no counterpart here. A closed inclusion–exclusion expansion on

2,2,2

gives the

structural count

144

−

3364. It terminates at third order because

the octahedron’s six vertices forbid a four-matching. Under one stated assumption, the

standing-information postulate, this corresponds to

= (

3364

1836

) ×m

1.719 GeV

with the proton mass as the sole calibration input and no cosmological ﬁtting. The same

geometry ﬁxes the annihilation channel. Two octahedral defects can meet only along

a shared octahedron edge, and of the four cages their interface admits, only the two

tetrahedra are products of the merger. The residual therefore has mass

, and the only

channel producing a line gives

1.591 GeV. A recently reported 1.5–1.6 GeV gamma-ray

line has weighted centroid 1.578 ±0.048 GeV, which is 0.3σ away. No observational input

enters the derivation.

Keywords: dark matter; FCC lattice; quantum error correction; topological defects; intersti-

tial voids; Selection-Stitch Model

1. Introduction

The mass of the dark matter particle is unconstrained across more than thirty orders

of magnitude. Candidate frameworks typically introduce one or more new species with

adjustable couplings and masses, ﬁt them to the observed relic density, and test the result

against direct-detection, indirect-detection, and structure-formation constraints. The mass

is an input in almost every such framework, and the freedom to choose it is what allows a

wide range of candidates to survive.

This paper develops a construction in which the mass is not free. The Selection-Stitch

Model assigns a structural count to a lattice defect, and one stated postulate converts that

count into a mass, with the proton as the only calibration input. The construction yields

1.719 GeV, and the value cannot be adjusted without changing the proton value

as well.

We are explicit about what this is and is not. The geometric and combinatorial content

is exact and independently veriﬁable, and is the substance of this paper: the enumerations,

the uniqueness of the octahedral state, the behavior of the construction on other lattices,

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and the annihilation channel all follow from the lattice and are reproduced by scripts. The

step from a structural count to a mass in GeV rests on one assumption, which is isolated

and stated as Postulate 1 in Section 4.1 and revisited in Section 6.1. The mass should

therefore be read as the output of a geometric construction under that postulate rather than

a standalone physical prediction independent of it.

1.1. The Framework

The Selection-Stitch Model (SSM) [

] 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 (1)

from purely structural counts of the trapped tetrahedral-void defect [

]. An equivalent

derivation via a

[[

192, 130, 3

]]

CSS code on the FCC lattice [

] reaches the same number

through a fault-tolerant veriﬁcation cost E

×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 that is unable to stitch into the surrounding lattice. The

companion paper [

] develops this for the tetrahedral void, where a

K =

4 remnant

becomes the proton, and shows how its incomplete bonding generates fractional charge,

color conﬁnement, and the proton mass. This 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.

1.2. The Second Interstitial Site

The FCC unit cell contains two distinct interstitial void types: eight tetrahedral voids

(each bounded by four FCC vertices) and four octahedral voids (each bounded by six

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, which is 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 within the SSM structural-symmetry rules from the bonding graph

2,2,2

and

bounding polyhedron (the regular octahedron with

symmetry): the absence of ﬁrst-

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

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. [

] that yields the proton’s

veriﬁcation cost

1836 extends to the octahedral defect by inclusion–exclusion on the

2,2,2

bonding graph, terminating exactly at the third order because the octahedron’s six

vertices forbid any 4-matching:

144

−

3364. The three terms

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are the inclusion–exclusion structure made explicit: a base veriﬁcation cost, minus the

pairwise overlaps where two ﬂux channels double-count, plus the triple overlaps restored

where three channels coincide. Each coefﬁcient is a ﬁxed structural count rather than a

ﬁtted value: 25 is the number of disrupting nodes (6 bounding vertices

4 bonds, plus

the trapped center), 144

= K

the second-shell footprint at bulk coordination

K =

12, 30

the skew-edge pairs of

2,2,2

and 10 their pairwise ﬁrst-shell overlap, and the two 8s the

perfect matchings of the octahedron and their triple overlap; Section 4 derives all six by

direct enumeration. Under Postulate 1, the count corresponds to a mass:

×m

3364

1836

×938.272 MeV = 1.719 GeV, (2)

using only the proton mass

[3]

as a calibration input. A

∼

1.5–1.6 GeV gamma-ray line

recently reported in three active galactic nuclei [

] sits near this mass. The annihilation

channel that connects the two is derived from the merger geometry in Section 4.8, and the

quantitative comparison is made in Section 5.1.

1.3. What This Paper Claims and Does Not Claim

We present a forward derivation of the structural count

via inclusion–exclusion

on the

2,2,2

bonding graph (Section 4). The count is exact. It depends only on the bonded

structure of the octahedral defect and the structural-counting framework of Ref. [

], and it

uses no cosmological observation as input.

We claim the following:

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.

Four structural properties follow from the defect’s geometric symmetry. Two are exact,

the self-conjugate character and the vanishing ﬁrst-order dipole; two are weaker, the

suppression of electromagnetic coupling beyond dipole order and the absence of the

baryonic color-generating mechanism.

The construction gives

3364 from the closed inclusion–exclusion expansion

on the

2,2,2

bonding graph, with no cosmological input, using the same structural-

counting machinery that yields

1836 for the proton in Ref. [

]. Under Postulate 1,

this corresponds to m

= 1.719 GeV.

The annihilation channel follows from the merger geometry with no observational

input: the residual defect must occupy a cage drawn from both parent voids, which

ﬁxes it as tetrahedral and gives

1.591 GeV. The comparison with the reported

line [4] is carried out in Section 5.1 of this paper.

1.4. 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

2,2,2

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

Section 4 presents the forward derivation of

3364 via inclusion–exclusion on

2,2,2

with explicit 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. Section 7 concludes.

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Interactive 3D Visualization.

A WebGL visualization of the octahedral defect geometry—the

2,2,2

bonding graph

among the six bounding vertices, the three 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 en-

closed 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

contains atoms at the cube corners and

face centers, giving a Bravais lattice with primitive cell volume

4 and nearest-neighbor

distance

L = a/

√

. The unit cell decomposes into two distinct interstitial void types [

]

(Figure 1).

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, reﬂecting the longer

centroid-to-vertex distance (L/

√

2 versus L

√

3/8).

2.1. Tetrahedral Voids

Located at

(a/

)(±

)

with all eight sign combinations, giving eight voids per

cell. Each is bounded by four FCC vertices forming a regular tetrahedron with edge length

L. The centroid-to-vertex distance is L

√

3/8 ≈ 0.612 L.

2.2. Octahedral Voids

Located at the body center

(a/

)

and at the 12 edge midpoints. Each edge

midpoint is shared between four unit cells, contributing 12

3 to the cell count, plus

one body-center, giving four voids per cell. Each is bounded by six FCC vertices forming a

regular octahedron with edge length

. The centroid-to-vertex distance is

√

2 ≈

0.707

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2.3. Notation

The symbol

denotes the coordination number of a node, that is, its degree in the

bond graph, following Ref. [

]. It is used in two ways that should not be confused. A

coordination value is always written

K = n

. Bare,

denotes the bulk value

K =

12, and

the quantity

144 appearing in the counting formulas always refers to this bulk value.

In a phrase such as “a

K =

6 node”, the value labels the coordination of a trapped defect

node, which is a local property of that node alone. Graphs are written with a different

symbol:

and

2,2,2

denote the complete and complete tripartite graphs, and they carry

no coordination meaning.

3. The Octahedral-Void Defect and Its Structural Properties

3.1. The Defect: Trapped K = 6 Remnant

During the

K =

→ K =

12 phase transition that crystallizes the SSM vacuum [

], 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 Rather Than 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

are the tetrahedral interstitial site

(four equidistant nearest neighbors) and the octahedral interstitial site (six 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 ﬁve (or seven, eight, 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 ﬁve of the six 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

∑

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 (ﬁve vertices) —

∑

= 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 six 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 four non-antipodal

partners, generating the complete tripartite graph

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2,2,2

: 6 vertices, 12 edges, degree 4 per vertex, 3 antipodal non-bonded pairs. (3)

This is the bonded subgraph of the octahedral-void defect (Figure 2), which is 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

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 ﬁrst-order

dipole moment, this forbids a ﬁrst-order dipole photon coupling, as discussed in Section 3.3.

3.3. Suppressed First-Order Electromagnetic Coupling

3.3.1. The Face-Topology Argument

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

]

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 as

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 ﬁeld at ﬁrst order, while the square face does. Second, the bulk cuboctahedral

coordination shell of each FCC vertex contains eight triangular faces and six square faces,

providing the propagation channel through which the defect’s bipartite oscillation modes

couple to bulk photon modes.

3.3.2. The Tetrahedral Case

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

with four triangular faces and zero square faces. The photon channel of Ref. [

] is not

strictly forbidden because the dipolar oscillation can be carried on the square faces of the

surrounding bulk cuboctahedral shells, which the defect’s external bonds traverse.

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3.3.3. The Octahedral Case

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

eight triangular faces and zero square faces, and the bonded subgraph

2,2,2

contains four

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

internal C

structure on its bonded subgraph.

3.3.4. Why the Two Cases Differ

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

natural 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 conﬁguration carries a nonzero dipole moment that the bulk square faces can

carry. The octahedral 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 conﬁguration, so its ﬁrst-order dipole moment

vanishes identically by symmetry. A vanishing dipole has nothing for any channel to

carry—internal or 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 ﬁrst-order dipole to couple,

whether through internal faces or through the surrounding cuboctahedral shells. This is

the sharp content of the tetrahedral/octahedral asymmetry: anchor 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.

3.3.5. Scope of the Claim

To be precise about scope: this establishes the absence of a ﬁrst-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 but not at all orders.

3.3.6. 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 (dipole)

coupling is forbidden by the centrosymmetry of the defect’s bounding octahedron: a parity-

even symmetric ground conﬁguration has no

dipole moment. 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 ≲

−3

∼

1 GeV from beam-dump experiments and

supernova cooling [

]) are consistent with the picture predicted here. The vanishing of

the ﬁrst-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-ﬁeld dynamics, which we do not attempt here. The

argument given is the structural-symmetry analog of the Ref. [

] argument applied to

the octahedral case, which is now sharpened to a parity selection rule for the leading

dipole term.

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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

[

]. The combinatorial identity

4, 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. (4)

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

2,2,2

has exactly

ﬁve skew partners (the edges connecting the four remaining vertices, minus the one

antipodal non-bond), giving 12

30 skew pairs. It is also obtained by counting the

two-matchings of the octahedron graph.

The skew-pair count 30 does not reproduce the speciﬁc color-generating mechanism

that the SSM uses for baryons. In Ref. [

], the three QCD colors arise because the

skew-

pair count is exactly 3, in bijection with the three conﬁning 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 speciﬁc three-color conﬁning 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 conﬁning

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 [

] identiﬁes anti-baryons with spatial inversion of the trapped tetrahedral-

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

(a/

)

and

(

4, 3

)

), related by inversion

r → −

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 octahedral void (body-center or edge-midpoint) is therefore its own image under

inversion. The framework 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

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

universe history of pair-annihilation rather than not by an asymmetry between particle

and antiparticle.

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•

The pair annihilation of two octahedral defects is in principle allowed (

χχ

, not

)

with the 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

≈

−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. Uniqueness of the Octahedral State

The tetrahedral cage does not support a single particle. It supports a multiplet. Refer-

ence [

] obtains four charge states from one trapped-node geometry, and spatial inversion

doubles them to eight conﬁgurations in all. It is therefore fair to ask what the octahedral

cage supports and why one state rather than a family is identiﬁed with dark matter. The

answer is that the two multiplicities have different origins, and neither survives the change

of cage.

3.6.1. Where the Baryon Multiplet Comes From

The four baryon states of Ref. [

] require two ingredients. First, one of the four bonds

is selected as the anchor that couples the defect to the bulk. This breaks the site symmetry

→ S

and leaves three valence bonds. Second, those three valence bonds are mutually

equivalent under the residual symmetry and share the projection

−

3 onto the anchor

direction, so each may carry an independent winding number

∈ {

}

. The total

winding

W =

∑

∈ {

0, 1, 2, 3

}

then generates the charge ladder

Q = −

+ W

and the

four states

∆

−

∆

. The multiplet is a consequence of three equivalent bonds rather

than the trapped node itself.

3.6.2. The Octahedral Cage Has No Equivalent Valence Bonds

On the octahedron, the second ingredient fails for a reason that can be stated exactly.

The point group of the regular tetrahedron has order 24, and the stabilizer of one bond

direction, of order 6, acts transitively on the remaining three: a single orbit of size 3. The

point group of the regular octahedron has order 48, and the stabilizer of one bond direction,

of order 8, splits the remaining ﬁve bonds into two orbits, of sizes 4 and 1. The antipodal

bond and the four equatorial bonds are not equivalent to one another. Their projections

onto the selected bond are

−

1 for the antipode and 0 for each of the four equatorial bonds

rather than the common value −1/3 of the tetrahedral case.

There is accordingly no set of mutually equivalent valence bonds to carry independent

winding numbers, and no charge ladder can be built. The projections are also integers, so

even a formal ladder would not produce fractional charges. Charge conservation still holds,

since 1

+ (−

) +

0, but it holds trivially rather than through a balance among

equivalent bonds.

3.6.3. No Anchor Is Selected in the First Place

The argument above is a backstop. As established in Section 3.3, the octahedral defect

is anchor-free and retains its full

site symmetry, which is what forbids its ﬁrst-order

dipole. With no anchor selected, the winding construction has no starting point at all.

The two arguments are independent and point the same way: the octahedral defect is

not anchor-selected, and were it anchor-selected, the residual symmetry would still not

generate a multiplet.

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3.6.4. No Inversion Partner Either

The second source of tetrahedral multiplicity also disappears. The regular tetrahedron

is not centrosymmetric, so the two void orientations in the FCC unit cell are distinct and

support matter and antimatter separately. The regular octahedron contains the inversion

operator in its point group, so it maps to itself. This is the same centrosymmetry that gives

the defect its self-conjugate character in Section 3.5. There is no second orientation and

therefore no partner state.

3.6.5. One Crystallographic Site

Finally, the FCC unit cell contains four octahedral voids, at the body center and the

twelve edge midpoints, and all four belong to a single Wyckoff orbit. They are related by

lattice translation and are physically identical. The eight tetrahedral voids, by contrast, fall

into two inversion-related orientations. Table 1 collects the comparison.

Table 1. Origin of the tetrahedral multiplet and its absence on the octahedral cage. The baryon

multiplet requires three mutually equivalent valence bonds and two distinct void orientations. The

octahedral cage provides neither.

Tetrahedral Cage Octahedral Cage

Point group order 24 48

Stabilizer of one bond 6 8

Orbits on remaining bonds {3} {4, 1}

Projections onto that bond −1/3, −1/3, −1/3 −1, 0, 0, 0, 0

Winding ladder yes, W ∈ {0, 1, 2, 3} none

Inversion in point group no yes

Distinct orientations 2 (matter, antimatter) 1 (self-conjugate)

Geometric states 4 ×2 = 8 1

3.6.6. Scope

The statement proved here is that the octahedral cage admits one geometric conﬁgu-

ration, so the dark matter candidate is not one member of a family that observation must

select among. It is not a claim that the octahedral void can host no other physics. Vibrational

and higher-shell excitations, and defects extending beyond the ﬁrst coordination shell, lie

outside the ﬁrst-shell enumeration used throughout this paper and in Ref. [

], Section 9.1.

Those would be radial excitations of the same state rather than additional ﬁrst-shell states.

3.6.7. Veriﬁcation: Symmetry Orbits and Bond Projections

The point group orders, stabilizer orbits, bond projections, and the presence or absence

of inversion are computed directly by

verify_octahedral_states.py

, which enumerates

every orthogonal map permuting each vertex set. It requires numpy only, runs in about

two seconds, and is available at https://github.com/raghu91302/ssmtheory/blob/main/

verify_octahedral_states.py (accessed on 3 September 2026).

3.7. 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, clus-

ters, 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 com-

pact objects, while dark matter remains diffuse on scales below the cluster halo with no

equivalent of stellar or planetary collapse.

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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 kinetic 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 emission. The same suppression that yields EM neutrality also sup-

presses 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

phenomenology of dark matter, including the diffuse extension of galactic and cluster ha-

los 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, which is consistent with current observational constraints on dark-matter

self-interaction [9–11].

4. Forward Derivation of C

via Inclusion–Exclusion

The structural-counting derivation of

1836 for the proton presented in

Ref. [

] (Equation

(1)

) extends to the octahedral defect by inclusion–exclusion on the

2,2,2

bonding graph. We present this extension here and obtain

3364, corresponding to

= 1.719 GeV, with no cosmological input.

4.1. From a Count to a Mass

The quantity

is not a generic “structural complexity” but a count of the bond-state

disruptions a trapped defect imposes on the surrounding lattice. Converting that count

into a mass is the one step in this paper that is not geometry. The correspondence used for

that step was introduced in Ref. [

]; we restate it here in a form that separates what follows

from the code from what is assumed.

4.1.1. Step 1: The Vacuum Codeword Carries No Syndrome

The empty lattice satisﬁes every check. Its syndrome is identically zero, and no

classical record is required to describe it.

4.1.2. Step 2: A Defect Obliges the Code to Carry Standing Syndrome Information

A trapped node is a pattern the checks ﬂag but cannot repair. It is not a transient error

that is detected and removed; it persists, and so does the syndrome it produces. The code

must therefore carry

bits of classical information that the codeword does not require

for as long as the defect exists. This information is standing rather than transient: it is

regenerated identically by every round of checks, and it never resolves, because there is no

repair operation that removes the defect.

4.1.3. Step 3: That Information Cannot Be Exported

For a physical memory, standing information is held by hardware, and any thermo-

dynamic cost is paid into the environment surrounding it. The vacuum is the case with

neither. If spacetime is itself the code state, there is no substrate to hold the record and no

environment to export it to. The information can be neither erased nor shed.

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4.1.4. Step 4: Unexportable Standing Information Is Localized Energy

This step is not derived. It is the single assumption on which every mass in this

framework rests, and we set it out as such.

Postulate 1 (Standing information). In a code with no independent substrate, standing syndrome

information that cannot be exported is localized energy at the site that requires it at a ﬁxed rate

per bit:

= C

. (5)

By mass–energy equivalence [12], the corresponding rest mass is

= C

. (6)

Every mass quoted in this paper follows from Postulate 1 and from nothing else that is

not geometry.

Why This Is Not Landauer’s Principle

The relation resembles the Landauer bound

E ≥ kT ln

2 per bit [

], and it is

worth saying precisely why it is not that bound. Landauer’s principle governs the energy

dissipated when information is erased: erasure is logically irreversible, and the cost is incurred

by discarding a record into an environment. Neither condition holds here. Nothing is

erased, because the defect is never repaired and its syndrome is constant; and there is no

environment to dissipate into, as seen in Step 3. A bound on the cost of erasure therefore

does not apply, and Equation

(6)

does not follow from it. What Landauer’s principle does

supply is the precedent that information and energy are convertible at all. The conversion

asserted here is of a different kind: not erasure-as-dissipation but rather unexportable

information as standing energy.

What Is Established and What Is Assumed

Steps 1 to 3 are properties of the code and of the substrate-free hypothesis, and Step 5

below is algebra. Step 4 is Postulate 1, and it is the assumption on which every mass in

this framework rests. We state it as such rather than presenting Equation

(6)

as a theorem.

It is not derived from quantum ﬁeld theory, general relativity, or thermodynamics, and

we make no claim that it is. Its justiﬁcation is that a substrate-free code has no other place

to put the information a defect obliges it to carry, and its content is tested by whether the

resulting integer ratios match observation.

4.1.5. Step 5: Only Ratios Are Claimed, and ϵ

Is Not Needed

Equation (6) is linear in C

, so for any two defects

, (7)

and

cancels exactly. Its value is not predicted by the framework and is never used. The

cancellation requires only that the same

applies to both defects, which holds because

both are excitations of the same lattice. No lattice temperature is assumed, and none

is needed.

The linearity is the substantive content, and it has a reason. Bits are additive: a defect

that obliges the code to carry twice as much standing information costs twice as much. This

is why a structural count determines a mass ratio rather than merely correlating with one.

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Two consequences follow. The dimensionless ratio 3364

1836 can be read directly as a

mass ratio with the proton entering only as the single dimensionful calibration. And the

correspondence contains no free constant that could be tuned:

cancels identically, so the

proton and the dark matter candidate cannot be calibrated independently of one another.

4.2. The Structural-Counting Formula as Inclusion–Exclusion

The expression of Ref. [

= (K +

− c

skew

K =

144

−

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

structural nodes, this gives a base count of 13

144

1872 bond-state disruptions. Pairs of

structural nodes with mutually disjoint ﬂux 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 ﬁrst-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

−

1836, which is the proton’s veriﬁed veriﬁcation cost. The

series terminates:

has only four vertices, so it admits no triple of mutually disjoint edges

(a three-matching requires six vertices), and the third-order PIE term is identically zero.

4.3. Extension to K

2,2,2

with Truncation

4.3.1. Carrying the Structure Across

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

bonded graph K

2,2,2

Structural node count. Each of the six bounding vertices contributes four bonds

within

2,2,2

, giving 6

24 boundary structural nodes; with the trapped center,

= 25.

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

2,2,2

has exactly ﬁve mutually disjoint

partners (12 total edges minus one self minus six edges sharing a vertex), giving

(O)

skew

= (

30 skew pairs. The same count is obtained by direct enumeration

of two-matchings of the octahedron graph and by the closed form

(

)

−

(

)

66 −36 = 30.

Pairwise overlap on

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

2,2,2

, which is in contrast to

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

geometric quantity; it is not a ﬁxed bulk constant.

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

does not. With

six vertices, the octahedron graph hosts triples of mutually disjoint edges (three-matchings,

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

(O)

triple

(veriﬁed by direct enumeration; see Section 4.4). 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

2,2,2

, computed directly as the bulk ﬁrst-shell triple intersec-

tion

|N(e

) ∩ N(e

, evaluates to

(O)

triple

8 uniformly across all eight perfect

matchings of the octahedron.

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4.3.2. Why the Series Terminates

The series terminates exactly at third order. A four-matching requires eight mutually

distinct vertices; the octahedron graph has only six. Therefore,

(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. (8)

Under Equation

(6)

, this count corresponds to

= (C

) ×m

= (

3364

1836

) ×

938.27 MeV = 1.719 GeV.

4.4. Explicit Enumeration of Pairwise and Triple Overlaps

The values

(O)

pairwise

10 and

(O)

triple

8 appearing in Equation

(8)

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.

4.4.1. 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)} (9)

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

L =

√

). For a

2,2,2

edge

e = {v

}

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

N(e) = N

) ∪N

), (10)

which contains

|N(e)| =

−

20 nodes for any edge in

2,2,2

(the four subtracted

are the four common nearest neighbors of v

and v

4.4.2. Pairwise Overlap, and the Single Rule That Fixes It

For each of the 30 skew-edge pairs (e

, e

) in K

2,2,2

, the pairwise intersection is

|N(e

) ∩N(e

)| = 10, (11)

uniformly across all 30 pairs. The same quantity is deﬁned on the tetrahedral void, and we

compute it there by the identical procedure. On K

, it returns

|N(e

) ∩N(e

)| = 12, (12)

uniformly across the three skew pairs with the same edge neighborhood size |N(e)| = 20

in both cages.

This settles a question a reader may raise about Equation

(8)

. The proton formula

of Ref. [

] carries the factor 12 in its correction term, and 12 is also the bulk coordination

number, so the two readings coincide numerically on

. They are not the same quantity.

The number that belongs in the correction term is the ﬁrst-shell overlap, and on the

tetrahedral cage, that overlap happens to equal 12. Equation

(12)

establishes this by direct

enumeration rather than by inspection of the published formula.

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One rule therefore covers both cages: the correction term uses the computed ﬁrst-shell

overlap of the cage in question. It returns 12 on

and 10 on

2,2,2

. No choice is made

between a bulk value and a geometric value, because only the geometric value is ever used.

The apparent switch from 12 to 10 is the same rule evaluated on two different graphs.

4.4.3. Why the Skew Condition and Not Some Other Selection

The restriction of the correction terms to mutually disjoint edge sets is also ﬁxed rather

than chosen. On

, the three skew pairs give overlap 12, while all twelve vertex-sharing

pairs give 15. On

2,2,2

, the 30 skew pairs give 10, while the 36 vertex-sharing pairs give 14

or 15. In both graphs, the skew pairs are exactly the pairs of minimal ﬁrst-shell overlap. The

same holds at third order: among all triples of edges of

2,2,2

, the eight perfect matchings

are the triples of minimal overlap, and they are the only class on which the overlap is

constant. The selection rule can be stated without reference to either graph: correct on the

mutually disjoint sub-conﬁgurations, which are the conﬁgurations of minimal overlap.

4.4.4. Triple Overlap

For each of the eight perfect matchings (e

, e

) of K

2,2,2

, the triple intersection is

|N(e

) ∩N(e

)| = 8, (13)

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

|{z}

bounding vertices

+ 2

|{z}

matching-speciﬁc bulk nodes

= 8. (14)

The six 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 two nodes vary between matchings and lie at the FCC nodes most strongly

correlated with the speciﬁc orientation of the chosen three-matching.

4.4.5. The 8 = 8 Coincidence

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

• c

(O)

triple

8, the number of perfect matchings of the octahedron graph

2,2,2

. This is a

graph-theoretic invariant that is 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 that is 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 speciﬁc 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 Equation

(8)

multiplies these two as independent

inputs and does not double-count a single combinatorial quantity.

4.4.6. Robustness Check

A useful sanity check is that the proton derivation of Ref. [

] arises as the

K = K

bulk

N → N

13,

skew

→

3 specialization of the same PIE structure with

pairwise

= K

bulk

12 on

. The series terminates at second order there because

admits no three-matching.

The transition from proton (

pairwise

12) to octahedral defect (

(O)

pairwise

10) is the only

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

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cost formula; this replacement is forced by the geometry of

2,2,2

and is the principal new

combinatorial input of this paper.

4.4.7. Veriﬁcation: Cage Enumerations and Sensitivity Scan

The full enumeration tables for all 30 pairwise intersections and all eight triple inter-

sections (with explicit 6

2 decomposition) are provided in the Appendix, and a short

self-contained Python (3.8 or later, requiring only

numpy

) reference implementation that

reproduces every integer in Equation

(8)

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 (accessed on 3 September 2026). 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.

4.5. Embedding Uniformity Lemma and Body-Diagonal Structure

The values

(O)

pairwise

10 and

(O)

triple

8 were established in Section 4.4 by direct

enumeration. A reader is entitled to ask whether the observed uniformity across the 30

skew pairs and the eight 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 eight matchings by the four body diagonals of the cube

circumscribing the octahedral void.

Lemma 1 (Embedding Uniformity). Embed the octahedral void at the origin of the FCC lattice as

in Section 4.4, and let

denote its point stabilizer (the symmetry group of the regular octahedron,

of order 48). Then,

(i)

The eight perfect matchings of

2,2,2

form a single

-orbit. The triple ﬁrst-shell intersection

size

|N(e

) ∩N(e

is therefore constant across all matchings, and it equals eight

by direct evaluation on any representative matching. Hence, K

(O)

triple

= 8 is O

-forced.

(ii)

The 30 skew pairs of

2,2,2

split into two

-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,

(O)

pairwise

10 uniformly.

4.5.1. 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 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

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 (accessed on 3 September 2026).

4.5.2. Body-Diagonal Labeling of the Matchings

The eight perfect matchings admit a natural geometric labeling by the four body diag-

onals of the cube circumscribing the octahedral void—the cube whose eight corners sit at

displacement

(±

)

from the void center and whose six face centers are the bound-

ing vertices A, B, C, D, E, F of the octahedron. For each matching M, the following apply:

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(a)

The two “matching-speciﬁc” bulk nodes appearing in the 6

2 decomposition of

the triple intersection (Table A3) 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 eight matchings group into four

pairs in bijection with the four body diagonals.

(c)

Within each such pair, the two matchings are related by inversion through the void

center,

r 7→ −r

(in coordinates relative to the void center). This is the identical

centrosymmetry

−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 M’s labeling body diagonal.

These four properties are veriﬁed by the same standalone script.

4.5.3. Implication for the “8 = 8” Coincidence

The factorization

(O)

triple

·K

(O)

triple

64 in Equation

(8)

multiplies two structurally

independent quantities (Section 4.4, “8 = 8 coincidence”): the graph-theoretic 8 (number

of perfect matchings of

2,2,2

, ﬁxed by the symmetry

n,n,n

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 four cube diagonals via the inversion pairing, and the triple

intersection decomposes as 6

2 where the two 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.

4.6. The Expansion Coefﬁcients Are Not Adjustable

The six integers in Equation

(8)

are each ﬁxed by enumeration, as set out above. It is a

separate question of how much the prediction would move if they were treated as free. We

answer it directly, because the answer bears on whether the derivation could have been

steered toward a target value.

4.6.1. The Two Inputs a Reader Might Question

Two quantities are natural candidates for doubt. The ﬁrst is the pairwise overlap

(O)

pairwise

10, which differs from the value 12 that appears in the proton formula. The

second is the third-order term

(O)

triple

(O)

triple

8, which has no counterpart in the proton

case. Table 2 gives the predicted mass when each is altered.

Table 2. Sensitivity of the predicted mass to the two inputs that a reader might treat as adjustable.

Either input, altered on its own, shifts the result by less than 2 percent; the fourth row alters both at

once. The last row is the degenerate case in which the expansion is truncated at zeroth order and is

included only for scale.

Variant C

[GeV] Shift

As derived 3364 1.7191 —

Triple term dropped 3300 1.6864 −1.9%

pairwise

set to 12 3304 1.6885 −1.8%

Both changes together 3240 1.6558 −3.7%

All corrections dropped 3600 1.8397 +7.0%

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4.6.2. The Variants in the Measured Channel

Carried through the kinematics of Section 5.1, the four variants give photon energies

of 1.591, 1.556, 1.558 and 1.523 GeV, respectively. All four lie within 1.2

of the measured

centroid of Equation

(21)

, and the three single changes within 0.5

. The agreement is not

sensitive to the counting scheme.

4.6.3. The Levers Are Small and the Derivation Never Consults the Data

Neither disputed input moves the prediction by as much as 2%, and all four non-

degenerate variants fall inside a band of about 4%. More to the point, no observational

quantity enters Equation

(8)

at any stage. The six integers come from the bonding graph

and from ﬁrst-shell enumeration on the lattice. A value that is never compared with a

measurement during its own derivation cannot have been tuned to that measurement.

4.6.4. The Third-Order Term Is Not Doing the Work

The 8

64 contribution is 1.9% of

. Removing it entirely gives 1.686 GeV

instead of 1.719 GeV. Both values sit in the same region, and neither is singled out by any

observation used in this paper. The term is included because the principle of inclusion and

exclusion requires it once the graph admits three-matchings rather than because it improves

an agreement. The numerical coincidence between the number of perfect matchings and

the size of the triple intersection, discussed in Section 4.4, therefore affects the result at the

2% level and cannot account for the prediction.

4.6.5. What Would Count as a Fitted Parameter

A ﬁtted parameter is one whose value is selected by comparing outcomes with data.

No quantity in Equation

(8)

has that character. Each of the six integers is recovered by

a script that reads only the FCC nearest-neighbor vectors and the six bounding-vertex

coordinates, and the same script computes the

values of Equation

(12)

by the identical

procedure. The sensitivity ﬁgures in Table 2 are reproduced by the same script.

4.7. Robustness to the Choice of Lattice

The construction so far has been carried out on the FCC lattice. A reader may reason-

ably ask whether the proton count

1836 and the dark matter count

3364 are

properties of FCC or artifacts of an arbitrary choice of lattice. We settle this by running the

same construction on the two standard alternatives, BCC and HCP.

4.7.1. Method

Each lattice is generated at nearest-neighbor distance

L =

1. HCP is taken at the

ideal axial ratio

c/a =

√

8/3

. In each lattice, we search for the two cage types used

above: a tetrahedral cage, which is deﬁned as four mutually nearest-neighbor nodes

clique in the bond graph), and an octahedral cage, which is deﬁned as six nodes

carrying twelve bonds with every node of degree 4 (a

2,2,2

cage). For every fully interior

cage, we then compute the ﬁrst-shell overlaps

pairwise

= |N(e

) ∩ N(e

and

triple

|N(e

) ∩ N(e

exactly as in Section 4.4. Nothing in the procedure is adjusted

between lattices.

4.7.2. BCC Admits No Cage at All

BCC has interior coordination

K =

8. Its bond graph contains no

clique anywhere:

two nearest neighbors of a BCC node are separated by

, while the nearest-neighbor

distance is

√

≈

0.866

, so they are not bonded to each other. No four mutually

nearest-neighbor nodes exist, and therefore neither cage type exists. The construction

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cannot be written down on BCC. This is a stronger statement than a disagreement in the

ﬁnal number.

4.7.3. HCP Admits Both Cages but Gives the Wrong Proton

HCP is the demanding comparison. Like FCC, it saturates the Kepler bound, it has

interior coordination

K =

12, and it contains both cage types. The local geometry is

identical in every respect we can measure: the tetrahedral and octahedral cages have the

same circumradii, 0.6124

and 0.7071

, and every cage edge has the same ﬁrst-shell size

|N(e)| =

20. The lattices differ only in their second shells, which is exactly where the

disruption count K

= 144 lives.

That difference is enough. On HCP, the tetrahedral ﬁrst-shell overlap is

pairwise

11,

uniformly across all three skew pairs of every cage, rather than the value 12 found on FCC.

The proton count becomes

(HCP)

= 13 ×144 −3 ×11 = 1839, (15)

which misses the observed ratio 1836.153 by 0.155%. The FCC value 1836 misses it by

0.008%. The proton mass ratio therefore distinguishes the two close packings by a factor of

about twenty in accuracy. Table 3 collects the comparison.

Table 3. The same construction applied to three candidate lattices, all at nearest-neighbor distance

L =

1 and HCP at the ideal axial ratio. BCC contains no cage of either type. HCP contains both,

but its tetrahedral count misses the observed proton-to-electron ratio, and its octahedral overlaps

are not uniform, so the expansion coefﬁcients are not well deﬁned there. Only FCC yields uniform

coefﬁcients and the observed 1836.

Tetrahedral Cage Octahedral Cage

Lattice K K

pairwise

triple

BCC 8 no cage undeﬁned no cage no cage

HCP 12 11 (uniform) 1839 11 or 10 8 or 6

FCC 12 12 (uniform) 1836 10 (uniform) 8 (uniform)

4.7.4. On HCP, the Octahedral Coefﬁcients Are Not Well Deﬁned

The sharper failure appears in the octahedral cage. Within a single HCP octahe-

dral cage, the thirty skew pairs split into twelve with

pairwise

11 and eighteen with

pairwise

= 10

, and the eight perfect matchings split into six with

triple

8 and two with

triple

6. The overlaps are not constant. Equation

(8)

therefore cannot be written on

HCP without ﬁrst choosing some averaging prescription, and any such choice would be an

adjustable input. On FCC, no choice arises, because all thirty pairs give 10 and all eight

matchings give 8. This is the content of the Embedding Uniformity Lemma of Section 4.5,

and the comparison with HCP shows that the lemma is a property of FCC rather than a

general feature of close packing.

4.7.5. Why FCC and Not HCP

The reason is site symmetry. In FCC, the octahedral interstitial site has the full point

symmetry

, of order 48, which acts on the thirty skew pairs with two orbits and on the

eight matchings with one. In HCP, the same site has only

of order 12. The smaller

group breaks the skew pairs and the matchings into more orbits, and the ﬁrst-shell overlaps

then take different values on different orbits. The uniformity that makes the expansion well

deﬁned is supplied by the larger symmetry group, which only the cubic close packing has.

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4.7.6. Scope of This Test

Only three lattices have been tested. There are inﬁnitely many close-packed stackings

beyond ABCABC and ABAB, and we have not surveyed them. We therefore do not claim

that FCC is unique. What the test establishes is narrower and is what the question requires:

the count 1836 is not obtained on either standard alternative, it is not obtainable at all

on BCC, and on HCP, the octahedral expansion has no well-deﬁned coefﬁcients. The

agreement with the proton mass is a property of the FCC lattice speciﬁcally. We note in

passing that the HCP tetrahedral count 1839 coincides numerically with the neutron count

of Ref. [

], which is reached there by an unrelated route (36

3); the two have no

structural connection.

4.7.7. Veriﬁcation: Lattice Comparison

All values in Table 3 are reproduced by a self-contained script, verify_lattices.py,

which generates each lattice, ﬁnds every interior cage, and computes the overlaps di-

rectly. It requires numpy only and runs in under a second. The script is available at

https://github.com/raghu91302/ssmtheory/blob/main/verify_lattices.py (accessed on 3

September 2026).

4.8. The Merger Interface and the Residual Defect

The construction so far concerns a single defect. Two defects that come into contact

are also ﬁxed by the lattice, and the geometry determines what a merger can leave behind.

Nothing in this subsection uses an observation; the consequences for the gamma-ray

spectrum are taken up in Section 5.1.

4.8.1. Two Octahedral Defects Can Only Meet Along a Shared Edge

A merger requires contact. Direct enumeration of the octahedral voids of the FCC

lattice shows that every nearest-neighbor pair, at separation

, shares exactly two bounding

vertices, and that those two vertices are themselves separated by

. The shared object is an

octahedron edge, not the

√

antipodal diagonal. The union of the two bounding sets is

a ten-vertex interface, and any residual defect must be trapped in a cage built from those

ten vertices.

4.8.2. The Interface Admits Four Cages but Only Two Are Products

A trapped node is stable only if it sits at a position equidistant from its bounding

vertices at a radius above the metric wall. We therefore enumerate every connected subset

of the ten interface vertices, of every size from 3 to 10, and retain those that (a) admit a

unique equidistant center, (b) have a radius of at least

√

, (c) are strain-balanced, so that

the unit bond vectors from the center sum to zero, and (d) are non-coplanar, since a coplanar

set leaves the node free to slide normal to its plane and is not a trap. Of the 968 subsets,

794 are connected and exactly four survive: two regular tetrahedra of circumradius 0.612

and two regular octahedra of circumradius 0.707

. The enumeration is exhaustive rather

than a search for expected cage types, and it returns the same counts on every interface in

the supercell, so no third cage type exists. Both radii clear the metric wall

√

, so both

are geometrically admissible traps, and at this level the residual appears ambiguous. It is

not. Write each cage as

( n

shared

, n

), (16)

counting its vertices that are shared, exclusive to the ﬁrst parent void, and exclusive to

the second. Enumeration over every nearest-neighbor pair of octahedral voids gives,

without exception, the result shown in Table 4.

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Table 4. Composition of the four cages admitted by the merger interface, counted over all 240

nearest-neighbour pairs of octahedral voids in the supercell. Only the tetrahedra draw vertices from

both parents.

Cage Composition Draws from Both Parents Count

Tetrahedron 1 (2, 1, 1) yes 240

Tetrahedron 2 (2, 1, 1) yes 240

Octahedron 1 (2, 4, 0) no 240

Octahedron 2 (2, 0, 4) no 240

The two octahedral cages contain no vertex exclusive to the other parent. They are the

parent voids themselves, which are returned by an enumeration that searched the interface

and found the objects already sitting in it. A cage that is one of the reactants is not a product

of the reaction.

The tetrahedra are different in kind. Each takes the shared edge together with one

vertex exclusive to each parent, so each exists only because the two defects are in contact.

Their centers straddle the midpoint of the two void centers, and every one of them is a

genuine FCC tetrahedral void rather than incidental geometry. No octahedral cage other

than the two parents occurs in any interface.

4.8.3. The Residual Is Tetrahedral

The residual must occupy a cage formed by the merger, that is, one containing vertices

from both parent voids. Only the tetrahedra qualify. The residual is therefore a trapped

node in an FCC tetrahedral void, whose structural count is C

= 1836, so

′

1836

= m

= 0.938 GeV. (17)

The residual is mass-degenerate with the proton because it occupies the same cage. It is not

a proton, which is for the reason given next.

4.8.4. The Residual Site Is Reached Across a Face, Not Through Bulk

Each of the two tetrahedral cages shares exactly one full triangular face with each

parent octahedron, and every one of those shared faces contains both vertices of the merger

edge. The residual therefore reaches its site by sliding across a two-dimensional interface

and never traverses three-dimensional

K =

12 bulk. This matters because a residue formed

at the octahedral interface could not relocate to a distant tetrahedral void: the metric wall

that traps the defect in the ﬁrst place would have to be crossed. The face-adjacent site is

reachable without paying that cost.

4.8.5. The Residual Is Anchor-Free and Stable

In Ref. [

], every baryon is built by selecting an anchor: one of the four center-to-

vertex bonds becomes the bulk-coupling junction, breaking the site symmetry

→ S

and

generating the 1

3 valence split that carries electric charge, color and baryon number.

The lattice does not force that choice. Enumerating the orthogonal maps that ﬁx the

tetrahedral void center and preserve both the tetrahedron and the surrounding FCC lattice

returns 24 operations, realizing the full

on the four vertices with all four in a single

orbit. No vertex is geometrically distinguished, so a fully symmetric, anchor-free occupant

is admissible; by the same symmetry, the three skew-edge pairs lie in one orbit and are

assigned equivalently, so the occupant is color-neutral by symmetry rather than by tuning.

A highly symmetric defect may be unstable to a symmetry-lowering distortion, which

is here the spontaneous anchor selection that would turn

′

into a baryon. Modeling each

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of the four bonds as a harmonic spring of natural length equal to the equilibrium radius,

the Hessian of the trapped-node energy at the symmetric center is isotropic and positive

deﬁnite with all three eigenvalues equal to

. There is no soft mode, and displacement

either toward a vertex (anchor selection) or toward a triangular face (escape) raises the

energy. The anchor-free occupant is a stable local minimum.

Because it selects no anchor,

′

carries no fractional valence split, no color triplet, and

no anchor junction to carry baryon number. It is distinct from the neutron, which is the

anchor-selected W = 1 state of the same cage.

4.8.6. What Is Conserved, and What Is Not

Three quantities constrain any reaction of this kind. Energy is conserved in the usual

relativistic sense. The trapped-node count is protected in the sense of the barrier below:

the absorption of a node is possible only while a multi-defect interface is open, and the

last remaining node cannot dissolve, so the ﬁnal state must retain at least one trapped

defect. Spatial sector identity is preserved in that a residue formed at the octahedral

interface cannot relocate by crossing bulk; the face-adjacency result above is what allows

the tetrahedral site to be reached without violating this.

The total veriﬁcation cost

is not a conserved quantity, and saying so avoids a natural

misreading.

is a structural property of a conﬁguration, namely the bond-state disruption

it imposes on the surrounding lattice, so different conﬁgurations carry different

, and a

reaction that changes the conﬁguration changes it. Here,

initial

3364

6728, while

ﬁnal

1836. The difference is not transported anywhere: it is structural cost released as

the ﬁnal conﬁguration disrupts the lattice less than the initial one, and it leaves as photon

energy. What couples

to a conserved quantity is energy, through Equation

(6)

, and that

proportionality decouples whenever a defect-free product carries energy away.

4.8.7. Complete Annihilation Is Forbidden

The channel

χχ → γγ

would leave no trapped node at all. Extracting a node from

its void requires stretching its bonds against the restoring force of the surrounding intact

K =

12 shell, giving the linear conﬁning potential

V(r) = σ

lat

of Ref. [

]. A lattice string

tension is an energy per unit length, so

lat

= ε/L

with

the entanglement bond energy,

and the barrier at one lattice spacing is

V(L) = σ

lat

L = ε ∼ 10

GeV, (18)

taking

at the GUT scale as in Ref. [

]. The barrier is one bond energy and is independent

, so it does not depend on which length scale is assigned to the lattice spacing. Against

GeV, there is 2

3.4 GeV available. Dissolution of the last remaining node is

forbidden, and the ﬁnal state must retain one trapped defect.

4.8.8. The Other Candidate Channels

Four alternatives are worth checking against the constraints above. A residue with

K = 6

is one of the parent octahedra, which is excluded by the composition argument as

an unmerged reactant rather than a product. A residue with

K =

2 has nowhere to reside,

since no such cage survives the stability constraints. A residue that is an anchor-selected

K =

4 occupant would be a baryon; this is excluded because the anchor-free occupant

is the ground state of the cage, the void is exactly symmetric so nothing triggers anchor

selection, and producing a baryon would require creating a baryon number from a state

carrying none.

The fourth alternative is not excluded. The two-residue channel

χχ →

γ +

′

conserves the trapped-node count, so it requires no interface absorption at all, and both

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ﬂanking tetrahedral voids are face-adjacent, so forming one residual on each side severs

nothing. What distinguishes it is spectral rather than structural. Its ﬁnal state is four-body,

so its photons form a continuum rather than a line, with endpoint

max

s −4m

′

√

−m

= 1.207 GeV, (19)

which lies 0.384 GeV, or 24%, below the line derived next. That separation is several times

the per-source centroid uncertainties and the energy-scale systematic, and the endpoint

is the continuum’s maximum, so the bulk of its photons are softer still. The two-residue

channel cannot broaden, shift, or imitate a monochromatic feature. Its branching fraction

relative to the one-residue channel is not derived here.

4.8.9. Veriﬁcation: Merger Interface and Residual Site

The shared-edge result, the four-cage enumeration and the composition table are

reproduced by

verify_residual_cages.py

, which runs over every nearest-neighbor pair

of octahedral voids in the supercell rather than a single representative pair, and which also

veriﬁes the site symmetry and Hessian of the residual site. It requires numpy only and runs

in about twenty seconds. It is available at https://github.com/raghu91302/ssmtheory/

blob/main/verify_residual_cages.py (accessed on 3 September 2026).

4.8.10. Assumptions

Two are stated rather than derived. The ﬁrst is that a reaction product is a cage formed

by the merger, which is what excludes the parent octahedra; this is a premise about what

counts as a product, not a geometric result, though the geometric asymmetry it acts on is

exact. The second is that one node may be absorbed while the merger interface is open,

the barrier of Equation

(18)

applying to the last remaining node, whose shell has healed to

intact K = 12.

4.9. QEC Dual: An Open Question

The structural counting result

1836 of Ref. [

] has an independent realization

as a fault-tolerant veriﬁcation cost

× C

1836 in a CSS code on the FCC

lattice [

]. 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 four bounding vertices are mutually at the nearest-neighbor distance L.

The octahedral defect’s six bounding vertices include three antipodal pairs at second-

nearest-neighbor distance

√

2 L

, which prevents the defect from being captured by any

single coordination cluster and requires a multi-cluster QEC footprint. Constructing the

QEC dual of Equation

(8)

is left to future work. We note that the truncation of the PIE series

at third order, by the six-vertex constraint of

2,2,2

, suggests that the QEC dual should

likewise be a closed expression rather than an asymptotic series.

4.10. 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, the absence of the baryonic SU(3) color

mechanism, the 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 m

= 1.719 GeV, which is derived from the

closed PIE expansion (Equation (8)) and the proton mass alone.

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5. Comparison with Observations

5.1. The Reported 1.5–1.6 GeV Gamma-Ray Line

Kang et al. [

] report a narrow gamma-ray feature near 1.5–1.6 GeV in Fermi-LAT

observations of three active galactic nuclei. The predicted mass

1.719 GeV lies

above that range, so any comparison depends on which annihilation channel produces the

photon. The channel is ﬁxed by the lattice rather than by the measurement. Section 4.8

shows that two octahedral defects can meet only along a shared octahedron edge, that

the resulting interface admits exactly four stable cages, and that only the two tetrahedra

are products of the merger. The residual is therefore a tetrahedral-void defect of mass

That derivation uses no observational input. Here, we take its kinematic consequences and

compare them with the reported line.

5.1.1. The Monochromatic Channel and Its Photon Energy

Complete annihilation is excluded, a

K =

2 or baryonic residue is excluded,

and the two-residue channel produces a continuum (Section 4.8). The only channel yielding

a line is therefore

χχ → γ + χ

′

, E

= m

−

′

= m

−

= 1.591 GeV, (20)

from two-body kinematics at threshold with

√

s =

. Every quantity is already ﬁxed:

1.719 GeV from Section 4 and

′

= m

from Equation

(17)

. The residual recoils with

kinetic energy 0.909 GeV at

v =

0.861

. No parameter is adjustable and no observational

input enters. The framework therefore ﬁxes the position of the only monochromatic feature

any allowed channel can produce.

5.1.2. Comparison with the Measurement

The three reported line energies are 1.55

0.10, 1.53

0.09 and 1.62

0.07 GeV. Their

inverse-variance weighted mean is

obs

= 1.578 ±0.048 GeV (statistical), χ

= 0.72 for 2 d.o.f. (p = 0.70), (21)

so the three measurements are consistent with a single energy. The derived value of

Equation

(20)

lies 0.3

from this centroid. Inverting Equation

(20)

through the measured

centroid gives

1.707

0.045 GeV against the counted 1.719 GeV, which is a difference

of 0.28σ.

For orientation, the excluded channels would have given

= m

1.719 GeV for

complete annihilation, which sits 2.9

from the centroid, and

1.289 GeV

for an octahedral residual, which sits 6.0

away. Both are disfavored by the data as well as

excluded by the geometry. The two-residue continuum endpoint of Equation

(19)

lies at

1.207 GeV, which is further still. These are consistency checks on the derivation rather than

the reason for it.

5.1.3. The Line Is Not at the Source Redshift

The three sources lie at

z =

0.830, 0.031 and 1.050. Were the feature emitted in each

active galaxy, the rest-frame energies would be 2.836, 1.577 and 3.321 GeV, which are not

consistent with a common value:

118 for 2 degrees of freedom,

p ≈

−26

. Taken

at face value, the measurements therefore require a foreground origin at

z ≃

0, which is

what Equation

(21)

assumes. This is a sharp and falsiﬁable statement: any further source

at a substantially different redshift must report the same observed energy rather than the

same emitted energy.

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5.1.4. Caveats

Three should be recorded. First, Ref. [

] is a preprint and has not been refereed, and the

history of tentative gamma-ray lines counsels patience: the reported 130 GeV feature [

]

did not survive further data [

]. Second, the reported signiﬁcances are local. The strongest

single source has

TS =

23.03, corresponding to 4.8

before any trials correction, and the

trials factor over a scan of roughly two thousand sources and a range of line energies is

substantial. Third, and most seriously, a foreground origin sits awkwardly with the way

the signal was found. A line produced by annihilation in the Milky Way halo should be

diffuse, present across the sky and brightest toward the Galactic center [

], rather than

concentrated in three of some two thousand active-galaxy sightlines. We do not resolve this

tension. Its resolution, in blank-sky and Galactic-center data, is the decisive test of whether

the feature is dark matter at all, and it is independent of the derivation above.

5.2. Direct Detection

A 1.7 GeV particle scattering elastically off a xenon nucleus at

v ∼

−3

deposits

at most a few keV by kinematics alone. This sets the energy scale of any signal but says

nothing about its rate. The rate requires the higher-order coupling that survives the dipole

suppression of Section 3.3, and we do not derive it. No cross-section is claimed here, and

the statements below concern detectability thresholds rather than expected event rates.

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 [

] 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

The following is a dimensional estimate, not a calculation, and we place that qualiﬁca-

tion before the numbers rather than after them. It depends on identifying the interaction

length scale with

L ∼

0.5–2 fm, which this paper does not derive, and it will change

once the strain proﬁle of the octahedral defect is available. It is offered to show that the

framework does not obviously conﬂict with cluster-scale bounds rather than demonstrate

agreement with them.

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

tive strain ﬁelds in the FCC bulk. Taking the geometric cross-section

∼ π(

πL

and scaling by mass gives

σ/m ∼

0.01–0.1 cm

/g over that range of

. For orientation,

the Bullet Cluster bound is

σ/m ≲

0.47 cm

/g [

]. The estimate sits below it and also

below the window

σ/m ≈

0.1–2 cm

/g [

] in which self-interacting dark matter models

address small-scale structure. Both comparisons are order-of-magnitude only. The frame-

work as constituted predicts cold dark matter that is largely collisionless on observational

scales. 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

Scope of This Comparison

We state at the outset that CDG-2 does not conﬁrm the SSM and cannot. Any dark-

matter model with a non-dissipative dark sector, including standard collisionless cold

dark matter and many wave-like or fuzzy alternatives, is consistent with the existence of

almost-dark galaxies. The 99.94–99.99% halo-mass fraction is inferred from globular-cluster

scaling relations rather than from direct kinematic measurement, and the authors of [

]

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note that kinematic and spectroscopic follow-up is required. What follows is a consistency

check—it is not evidence.

The SSM prediction in Section 3.7 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 clus-

ter [

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

detected through globular-cluster overdensity rather than diffuse stellar emission—the

standard search methodology that fails for galaxies whose stellar component is too faint

to detect directly. The 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 glob-

ular clusters within a

∼

1.2 kpc diameter and a total galaxy luminosity

V,gal

≈

6.2

⊙

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

[24,25],

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

sector: 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. CDG-2 is consistent with a dark sector lacking a ﬁrst-order dissipation channel,

as are many other models.

5.5. Comparison with Existing Dark Matter Models

5.5.1. Composite Dark Matter

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 [

] 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 [

] 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.

5.5.2. The Sexaquark

The closest phenomenological analog in the existing literature is the sexaquark [

a hypothesized neutral, ﬂavor-singlet, scalar bound state of

uuddss

quarks with baryon

number

B =

2 and strangeness

S = −

2, which was 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

∼

(sexaquark

stability requires

≲

2054 MeV with cosmological relic-abundance ﬁts favoring 1.5–

1.8 GeV [

]; 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: ﬂavor-

singlet decoupling from pions; SSM: absence of square plaquettes in the bonded subgraph

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2,2,2

). Both are consistent with the diffuse, non-collisional halo phenomenology required

by direct-detection nulls and self-interaction constraints.

5.5.3. What Distinguishes the Two

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 = −

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.

) 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 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. What This Framework Is and Is Not

The results above are combinatorial. Their physical interpretation rests on a correspon-

dence that this paper assumes rather than establishes. This subsection sets out where that

assumption enters and how much weight it carries.

6.1.1. The Mass-Information Correspondence Is an Input

The step from a count

to a mass, Equation

(6)

, is set out in full in Section 4.1 rather

than referenced. Three of its steps follow from the code and the substrate-free hypothesis:

the vacuum codeword carries no syndrome, a defect that cannot be repaired obliges the

code to carry standing syndrome information for as long as it exists, and a vacuum with no

substrate has neither hardware to hold that record nor an environment to export it to. The

fourth step is Postulate 1: that unexportable standing information is localized energy. We

do not claim it follows from quantum ﬁeld theory, general relativity, or thermodynamics. It

does not. Every mass in this framework rests on it.

We are explicit in Section 4.1 that this is not Landauer’s principle. That bound governs

the energy dissipated when information is erased, and neither of its conditions holds here:

nothing is erased, since the defect is never repaired and its syndrome is constant, and

there is no environment to dissipate into. Landauer’s principle supplies the precedent that

information and energy are convertible; it does not supply Equation (6).

Two things limit the damage. The correspondence is linear, and the linearity has a

reason: bits are additive, so a defect obliging the code to carry twice as much standing

information costs twice as much. All dependence on the per-bit energy cancels in the

ratio

= C

, and only ratios are ever claimed. And the correspondence is not

adjustable: it contains no free constant that could be tuned to a target, which is why the

proton and the dark matter candidate cannot be ﬁtted independently.

6.1.2. No Equation of Motion Is Offered

The framework has no Lagrangian, no equation of motion, and no microscopic interac-

tion between lattice nodes. Its outputs are counts on a ﬁxed geometry that are converted to

masses by the correspondence above. This is a genuine limitation and not a presentational

one. A dynamical formulation would be required before the framework could be said to

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predict anything about processes rather than about spectra, and constructing one is beyond

the scope of this paper.

6.1.3. Cross-Sections Are Not Derived, Which Limits What Can Be Tested

The framework ﬁxes a mass and a set of symmetry properties. It does not ﬁx the

strength of any interaction. The ﬁrst-order dipole vanishes by symmetry, but the surviving

higher-order couplings are not computed, so the framework cannot presently predict a

direct-detection rate, an annihilation cross-section, or a self-interaction cross-section. What

can be tested now is therefore restricted to the mass, to the symmetry-derived properties,

and to consistency with existing bounds. A detection at 1.719 GeV would be strong

evidence; a null result at any sensitivity cannot falsify the framework until the couplings

are calculated. Section 6.2 lists the tests that do bite.

6.1.4. The Framework Has Not Been Independently Reproduced

The Selection-Stitch Model is the work of a single author, and no independent group

has veriﬁed or reproduced its results. This is a real limitation for a claim of this scope. Every

numerical claim in this paper is therefore reproducible from short scripts that require only

Python and

numpy

and run in seconds, and the text states which quantities are computed,

which are assumed, and which are taken from earlier work in the series.

6.2. Falsiﬁable Predictions

Under Postulate 1, the construction has the following consequences, which could

in principle be falsiﬁed by observation. Falsifying them would bear on the construction

together with that postulate rather than on the geometry alone.

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

= 1.719 GeV

, which is derived from the closed inclusion–exclusion expansion

2,2,2

(

Equation (8)

) and the proton mass alone with no cosmological input. The

deﬁnitive identiﬁcation of dark matter at a different mass scale, with the structural

properties predicted here, would falsify the framework.

Gamma-ray line energy. The annihilation channel is ﬁxed by the merger geometry

(Section 4.8), so the framework predicts a line at

1.591 GeV and at no other

energy. Three consequences are falsiﬁable. A conﬁrmed line from this mass scale at

itself (1.719 GeV) or at 3

4 (1.289 GeV) would contradict the derivation. The two-

residue channel contributes a continuum with endpoint 1.207 GeV (Equation

(19)

), so

accompanying photons above that endpoint but below the line would also contradict

it. And because the emission is local, any further source at a substantially different

redshift must report the same observed energy rather than the same emitted energy.

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

photon coupling at ﬁrst order. The observation of ﬁrst-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. The

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

channels would falsify this identiﬁcation.

Single dominant dark-matter species. The framework predicts a single dominant

dark-matter species at the mass scale

∼

1.7 GeV; the octahedral cage admits exactly one

geometric state with no charge multiplet and no inversion partner (Section 3.6). The

deﬁnitive observation of multiple dark-matter species at distinctly different mass scales

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would require extension of the framework 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 antipar-

ticle. The 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 ﬁrst order and therefore forms diffuse non-

collisional halos. The 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 [

]

(see Section 5.4) is consistent with this prediction, though it does not discriminate

between non-dissipative models.

6.3. Open Calculations

The most important open calculations follow in order of priority:

The abundance ratio

Ω

/Ω

. Deriving this requires the dynamics of the

K =

→

K =

12 phase transition, the relative formation rates of tetrahedral and octahedral

defects, and the annihilation history of both sectors, including the origin of the baryon

asymmetry. It is the principal open calculation of the framework. The mass prediction

= 1.719 GeV is independent of it.

QEC dual of the PIE expansion. The structural counting

1836 for the proton has

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

FCC lattice [

]. 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 non-trivial because the octahedral defect’s bounding vertices span

multiple coordination clusters (Section 4.9).

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.

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 tetrahedral interstitial voids of the FCC vacuum lattice. The same lattice contains a

second interstitial site—the octahedral void—that admits an analogous

K =

6 trapped rem-

nant 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 electromagnetic coupling (the bonding graph

2,2,2

is bipartite-deﬁcient at the

relevant four-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 representation), 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.

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The structural count follows from the closed inclusion–exclusion expansion on the

2,2,2

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

the octahedron:

= 25 ·144 −30 ·10 + 8 · 8 = 3364, m

×m

= 1.719 GeV.

The count uses only the bonded structure of the defect, the structural-counting framework

of Ref. [

], and the proton mass; no cosmological observation is consumed. The step from

the count

to a mass in GeV is not part of that geometry. It rests on Postulate 1, which

nothing here establishes. The mass values quoted throughout should be read accordingly:

as what this geometric construction yields under a stated assumption rather than as a

prediction independent of it. What does stand on its own is the geometric and combinatorial

content—the enumerations and overlap counts, the uniqueness of the octahedral state,

the behavior of the construction on BCC and HCP, and the derivation of the annihilation

channel from the merger interface—all of which is exact and independently veriﬁable.

The reported

1.5–1.6 GeV

gamma-ray line [

] sits near this mass. Section 4.8 derives the

annihilation channel from the merger interface, and Section 5.1 obtains

1.591 GeV,

0.3

from the measured centroid with no observational input. The Fermi-LAT signal itself

awaits conﬁrmation.

The qualitative predictions of the framework—the absence of ﬁrst-order EM coupling,

the absence of the baryonic SU(3) color-generating mechanism, the 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 they are independent of the

veriﬁcation-cost calculation. The recent identiﬁcation of the almost-dark galaxy CDG-2 [

]

is consistent with the collisionless diffuse-halo prediction, as it is with other non-dissipative

dark-matter models.

Funding: This research received no external funding. The APC was funded by the author.

Data Availability Statement: The original contributions presented in this study are included in

the article. Further inquiries can be directed to the corresponding author. 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 veriﬁed by three independent methods) are standard crystallographic

and combinatorial data. The Fermi-LAT line measurements are from [

]. The full enumeration of the

30 pairwise and 8 triple overlaps used in Equation

(8)

is given in the Appendix; a ﬁve self-contained

scripts reproduce every computed quantity in this paper. All require only Python 3.8 or later and

numpy

, and each runs in seconds.

verify_C_DM.py

—the cage enumerations on both the tetrahedral

and octahedral voids, the assembled counts

1836 and

3364, and the sensitivity scan

of Table 2 (Sections 4.4 and 4.6): https://github.com/raghu91302/ssmtheory/blob/main/verify_

C_DM.py (accessed on 3 September 2026);

verify_uniformity.py

—the orbit decomposition, body-

diagonal labeling, and inversion pairing of the Embedding Uniformity Lemma (Section 4.5): https:

//github.com/raghu91302/ssmtheory/blob/main/verify_uniformity.py (accessed on 3 September

2026) HCP and FCC, reproducing Table 3 (Section 4.7): https://github.com/raghu91302/ssmtheory/

blob/main/verify_lattices.py (accessed on 3 September 2026) orbits and bond projections of Table 1

(Section 3.6): https://github.com/raghu91302/ssmtheory/blob/main/verify_octahedral_states.py

(accessed on 3 September 2026) four-cage enumeration and the composition result that ﬁxes the

residual as tetrahedral (Section 4.8): https://github.com/raghu91302/ssmtheory/blob/main/verify_

residual_cages.py (accessed on 3 September 2026) the enumerations, the overlap counts, the symmetry

orbits, and the cage stability conditions, all of which are combinatorial facts about the lattice. They

cannot test Postulate 1, which is where the physical content of the framework sits, nor any other

claim that is not geometry. The interactive 3D visualization referenced in the introduction is hosted at

https://raghu91302.github.io/ssmtheory/oct_void_3D.html (accessed on 3 September 2026).

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Conﬂicts of Interest: Raghu Kulkarni is an employee of SSMTheory Group, IDrive Inc. The views

expressed in this paper are those of the author and do not necessarily represent the views of IDrive

Inc. The author declares no known competing ﬁnancial interests or personal relationships that could

have appeared to inﬂuence the work reported in this paper.

Appendix A. Enumeration Tables for Pairwise and Triple Overlaps

This appendix provides the explicit enumeration of the 30 pairwise and eight triple

ﬁrst-shell intersections that determine

(O)

pairwise

10 and

(O)

triple

8 in Equation

(8)

. The

setup follows Section 4.4 exactly: the octahedral void is centered at

(

1, 0, 0

)

, the 12 FCC

nearest-neighbor displacements are

{(±

1, 0

)

(±

1, 0,

)

(

)}

, and the ﬁrst-

shell neighborhood of an edge is N(e) = N

) ∪N

) with |N(e)| = 20.

Appendix A.1. Vertex Labels

We label the six bounding vertices of the octahedral void as in Table A1.

Table A1. 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).

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)

The 12 edges of

2,2,2

are then

{AC

DF}

(every vertex-pair from distinct antipodal classes). The three antipodal non-bonds

EF are at distance L

√

2 and do not contribute to K

2,2,2

Appendix A.2. All 30 Pairwise Intersections

Table A2 lists every skew pair of

2,2,2

together with its ﬁrst-shell intersection size.

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

Table A2. All 30 skew-edge pairs of

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 Equation (8) is the common value of

this list; it is not an average over a non-uniform distribution.

Pair: |N(e

) ∩N(e

)| Pair: |N(e

) ∩N(e

)| Pair: |N(e

) ∩N(e

(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

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Appendix A.3. All Eight Perfect Matchings and Their Triple Intersections

Table A3 lists every perfect matching (three-matching) of

2,2,2

together with the

explicit eight elements of its triple ﬁrst-shell intersection. The decomposition into six

bounding-vertex nodes plus two matching-speciﬁc bulk nodes is shown directly.

Table A3. All eight perfect matchings of

2,2,2

with their ﬁrst-shell triple intersections decomposed

as 6

2. The “Bounding” column counts the six bounding vertices

(common to every

matching by construction), and the “Other” column counts the two matching-speciﬁc bulk nodes

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

(O)

triple

= 8 uniformly.

# 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)

Appendix A.4. Structure of the Matching-Speciﬁc Nodes

The two matching-speciﬁc bulk nodes for each matching sit at displacement

(±

)

from the void center

(

1, 0, 0

)

on the four body diagonals of the unit cube

whose face centers are the bounding octahedral vertices. The eight matchings group into

four pairs

↔ M

)

↔ M

)

↔ M

)

↔ M

)

, with the two matchings

in each pair sharing the same two-node companion set. The four distinct companion sets

are in bijection with the four body diagonals of that cube. The two nodes within each

companion set are inversion-related through the void center, reﬂecting the centrosymmetry

∋ −I that also underlies the self-conjugate character of the defect (Section 3.5).

Appendix A.5. Assembly

Combining Tables A2 and A3 with

25 and the per-node disruption count

= 144 reproduces Equation (8) by direct substitution:

= 25 ×144 − 30 ×10 + 8 ×8 = 3600 −300 + 64 = 3364, (A1)

and

= (

3364

1836

) × m

1.7191 GeV using the CODATA proton mass

938.272 MeV [3].

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