Dark Matter as Incomplete Crystallization: FCC Vacuum Lattice Geometry

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

August 2026

Abstract

In the Selection-Stitch Model (SSM) [1, 2], 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 rst-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

= 25 · 144 − 30 · 10 + 8 · 8 = 3364

. It terminates at third order because the

octahedron's six vertices forbid a 4-matching. Under one stated assumption, the standing-

information postulate of Section 4.1, this corresponds to

= (3364/1836) × m

1.719

GeV, with the proton mass as the sole calibration input and no cosmological tting.

The same geometry xes 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 [3] has weighted centroid

1.578 ±0.048

GeV,

0.3σ

away. No observational input enters

the derivation.

1 Introduction

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

nitude. Candidate frameworks typically introduce one or more new species with adjustable

couplings and masses, t 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 veriable, and is the substance of the paper: the enumerations, the

raghu@idrive.com

uniqueness of the octahedral state, the behavior of the construction on other lattices, 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, 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, not as a standalone physical prediction independent

of it.

The framework.

The Selection-Stitch Model (SSM) [1, 2] proposes that the physical vacuum

is a Face-Centered Cubic (FCC) crystallization of spacetime, with baryonic matter identied

as a single

K=4

node trapped in the tetrahedral interstitial void of the

K=12

FCC bulk. The

framework derives the proton-to-electron mass ratio

= (K + 1)K

− c

skew

K = 13 × 144 −3 × 12 = 1836

(1)

from purely structural counts of the trapped tetrahedral-void defect [1]. An equivalent derivation

via a

[[192, 130, 3]]

CSS code on the FCC lattice [2] reaches the same number through a fault-

tolerant verication cost

× C

= 36 ×51 = 1836

The unifying picture is

incomplete crystallization

. In the SSM the vacuum crystallizes into

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

K=12

. Matter

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

interstitial site, unable to stitch into the surrounding lattice. The companion paper [1] develops

this for the tetrahedral void, where a

K=4

remnant becomes the proton, and shows how its

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

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

void admits a

K=6

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

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

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

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

The second interstitial site.

The FCC unit cell contains two distinct interstitial void types:

8 tetrahedral voids (each bounded by 4 FCC vertices) and 4 octahedral voids (each bounded

by 6 FCC vertices), with all bounding edges at the nearest-neighbor distance

(Section 2).

The framework that traps a defect in the tetrahedral void simultaneously admits an analogous

defect in the octahedral void  the same

K=4

K=12

phase transition, the same kinematic

operators, the same geometric mechanism, applied to the second interstitial site that the FCC

lattice provides. The natural question is what physics this companion defect predicts.

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

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

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

2,2,2

and bound-

ing polyhedron (the regular octahedron with

symmetry): absence of rst-order electro-

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

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

dard requirements for cold dark matter without invoking any free parameters. Second, the

structural-counting framework of Ref. [1] that yields the proton's verication cost

= 1836

extends to the octahedral defect by inclusion-exclusion on the

2,2,2

bonding graph, termi-

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

= 25 ×144 −30 ×10 + 8 ×8 = 3364

. The three terms are the inclusion-exclusion structure

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

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

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

is the number of disrupting nodes (6 bounding

vertices

4 bonds, plus the trapped center),

144 = K

the second-shell footprint at bulk co-

ordination

K=12

the skew-edge pairs of

2,2,2

and

their pairwise rst-shell overlap, and

the two

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

all six by direct enumeration. Under Postulate 1 the count corresponds to a mass:

× m

3364

1836

× 938.272

MeV

= 1.719

GeV

(2)

using only the proton mass [10] as a calibration input. A

∼ 1.5



1.6

GeV gamma-ray line recently

reported in three active galactic nuclei [3] 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.

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. [1], and uses no cosmological observation as input.

What we claim is:

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

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

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

voids.

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

the self-conjugate character and the vanishing rst-order dipole; two are weaker, the sup-

pression of electromagnetic coupling beyond dipole order and the absence of the baryonic

color-generating mechanism.

3. 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. [1]. Under Postulate 1 this

corresponds to

= 1.719

GeV.

4. 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 xes it as

tetrahedral and gives

= 1.591

GeV. The comparison with the reported line [3] is carried

out in Section 5.1 of this paper.

Organization.

Section 2 establishes the crystallographic geometry of the two interstitial void

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

graph

2,2,2

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

presents the forward derivation of

= 3364

via inclusion-exclusion on

2,2,2

, with explicit

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

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

existing dark matter models. Section 6 discusses falsiability and open calculations. Section 7

concludes.

Interactive 3D Visualization.

A WebGL visualization of the octahedral defect

geometry  the

2,2,2

bonding graph among the 6 bounding vertices, the 3 antipodal

non-bonded pairs at second-nearest-neighbor distance

√

2 L

, and the cuboctahedral

coordination cluster around any FCC vertex  accompanies this paper:

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

The visualization makes immediately evident that the octahedral defect cannot be

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

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

outside.

2 Geometry of the Two Interstitial Voids

The FCC unit cell with cubic lattice constant

contains atoms at the cube corners and face

centers, giving a Bravais lattice with primitive cell volume

and nearest-neighbor distance

L = a/

√

. The unit cell decomposes into two distinct interstitial void types [11, 12] (Figure 1).

Tetrahedral voids.

Located at

(a/4)(±1, ±1, ±1)

with all eight sign combinations, giving 8

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

length

. The centroid-to-vertex distance is

3/8 ≈ 0.612 L

Octahedral voids.

Located at the body center

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

and at the 12 edge midpoints.

Each edge midpoint is shared between 4 unit cells, contributing

12/4 = 3

to the cell count, plus

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

octahedron with edge length

. The centroid-to-vertex distance is

√

2 ≈ 0.707 L

Tetrahedral void Octahedral void

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

Trapped node (centroid) Bounding FCC vertex Void interior

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

Left:

A tetrahedral void,

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

. The

trapped

K=4

node (yellow star) sits at the centroid; the yellow translucent sphere illustrates

the void interior.

Right:

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

octahedron with the same edge length

. The trapped

K=6

node sits at the centroid; the yellow

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

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

√

versus

3/8

Notation.

The symbol

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

the bond graph, following Ref. [1]. 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 dierent symbol:

and

2,2,2

denote the complete and complete tripartite graphs, and carry no coordination meaning.

3 The Octahedral-Void Defect and Its Structural Properties

3.1 The defect: trapped

K=6

remnant

During the

K=4 → K=12

phase transition that crystallizes the SSM vacuum [1], the same

kinematics that traps a

K=4

remnant in a tetrahedral void can trap a remnant in an octahedral

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

vertices, creating a local

K=6

pocket inside the

K=12

bulk. This parallels the proton's

K=4

trapped remnant in the tetrahedral case.

Why

K=4

and

K=6

, and not intermediate values?

The selection of the tetrahedral

(

K=4

) and octahedral (

K=6

) congurations as the only viable trapped-remnant classes follows

from two independent constraints that coincide in the FCC lattice.

Crystallographic constraint.

The only positions in the FCC vacuum that admit multiple FCC

nodes at the unit nearest-neighbor distance

are the tetrahedral interstitial site (4 equidistant

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

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

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

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

only

K=4

and

K=6

as candidate trapping geometries.

Strain-balance constraint.

Even hypothetically considering 5 of the 6 vertices of an octahedral

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

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

ˆn

from each

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

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

symmetry  including a square pyramid (5 vertices) 

ˆn

= 0

, and the trapped node's

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

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

dipole, and is not produced by the isotropic kinematics of the

K=4 → K=12

phase transition.

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

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

selects

K=4

and

K=6

as the only stable trapped-remnant classes, with no tting required and

no intermediate-K particle classes admitted.

3.2 Bonding subgraph:

2,2,2

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

√

, and each non-antipodal pair at distance

(the nearest-neighbor distance). Bonds at the

SSM unit-bond-length scale

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

the complete tripartite graph

2,2,2

: 6

vertices

, 12

edges

degree 4 per vertex

, 3

antipodal non-bonded pairs

(3)

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

case where the bonded subgraph is

(4 vertices, 6 edges, degree 3).

3.3 Suppressed rst-order electromagnetic coupling

The face-topology argument.

The argument for the photon coupling channel of baryonic

defects given in Ref. [1] 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

-x

-y

-z

Bond graph K

2; 2; 2

: 12 unit-length edges (green),

3 antipodal non-bonds at L

(red dashed)

All 8 faces are triangular (C

, non-bipartite)

0 square faces ) no C

bipartite EM channel

Face structure: schematic of all 8 triangular faces

Figure 2: Bonded substructure of the octahedral defect.

Left:

The 6 bounding vertices of the

octahedron with the trapped

K=6

node (yellow star) at the centroid. The 12 unit-length bonds

form the complete tripartite graph

2,2,2

(solid green); the 3 antipodal pairs at distance

√

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

Right:

The bounding polyhedron

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

) faces removes the internal

bipartite channel; combined with the full

symmetry of the defect, which forces a vanishing

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

Section 3.3.

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 eld at

rst order, while the square face does. Second, the bulk cuboctahedral coordination shell of each

FCC vertex contains 8 triangular faces and 6 square faces, providing the propagation channel

through which the defect's bipartite oscillation modes couple to bulk photon modes.

The tetrahedral case.

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

tetrahedron with 4 triangular faces and zero square faces. The photon channel of Ref. [1] 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.

The octahedral case.

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

octahedron with 8 triangular faces and

zero

square faces, and the bonded subgraph

2,2,2

con-

tains 4-cycles only between antipodal pairs that are explicitly non-bonded. The defect itself has

no internal

structure on its bonded subgraph.

Why the two cases dier.

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

anchor-selected

: one bond is singled out as the bulk-coupling channel, breaking the site

symmetry

→ S

, and the resulting conguration carries a nonzero dipole moment that the

bulk square faces can carry. The 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 conguration, so its rst-order dipole moment

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

internal

borrowed from the bulkso the suppression is not a property of the bounding faces

alone but of the defect's symmetry: there is no rst-order dipole to couple, whether through

internal faces or through the surrounding cuboctahedral shells. This is the sharp content of

the tetrahedral/octahedral asymmetry: 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.

Scope of the claim.

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

rst-order

dipole

photon coupling. It does not by itself exclude higher-multipole, loop-induced, or baryon-

mediated electromagnetic interactions. The framework's claim is therefore a hierarchy rather

than exact neutrality: the leading dipole term is forbidden by parity, and the surviving higher-

order channels are present but suppressed. EM neutrality below should be read in this sense

neutrality at leading (dipole) order, not at all orders.

Falsiable consequence.

An octahedral-void defect interacts with the electromagnetic eld

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

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

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

parity-even symmetric ground conguration has no

dipole moment. This matches the stan-

dard dark-matter requirement of EM neutrality at the level of leading-order cross-sections.

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

photon mixing parameter

ϵ ≲ 10

−3

∼ 1

GeV from beam-dump experiments and supernova

cooling [16, 17]) are consistent with the picture predicted here. The vanishing of the rst-order

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

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

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

symmetry analog of the Ref. [1] argument applied to the octahedral case, now sharpened to a

parity selection rule for the leading dipole term.

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

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

tetrahedron

[1]. The combinatorial identity

C(4, 2)/2 = 3

generates the three-color SU(3)

representation on baryon-like defects.

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

2,2,2

(O)

skew





− 6





= 66 −36 = 30.

(4)

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

2,2,2

has exactly 5 skew

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

giving

12×5/2 = 30

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

graph.

The skew-pair count 30 does not reproduce the specic color-generating mechanism that

the SSM uses for baryons. In Ref. [1] the three QCD colors arise because the

skew-pair

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

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

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

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

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

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

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

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

Consequence.

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

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

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

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

mechanism the framework supplies.

3.5 Self-conjugate (Majorana-type) character

Reference [1] identies anti-baryons with spatial inversion of the trapped tetrahedral-void defect:

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

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

and

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

), related by inversion

ˆr → −ˆr

that exchanges the sign of the anchor

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

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

over the other.

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

invariant under inversion through its centroid: the symmetry group

contains the inversion

operator

−I

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

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

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

Majorana-type dark-matter particle.

Falsiable consequences.



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

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

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



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

χχ

, not

χ¯χ

), with rate

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

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

⟨σv⟩

γγ

≈

2 ×10

−28

/s required by the Kang et al. 2026 AGN line [3].



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

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

3.6 Uniqueness of the octahedral state

The tetrahedral cage does not support a single particle. It supports a multiplet. Reference [1]

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

to eight congurations in all. It is therefore fair to ask what the octahedral cage supports, and

why one state rather than a family is identied with dark matter. The answer is that the two

multiplicities have dierent origins, and neither survives the change of cage.

Where the baryon multiplet comes from.

The four baryon states of Ref. [1] 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

−1/3

onto the anchor direction, so each may carry an independent winding number

∈ {0, +1}

. The total winding

W =

∈ {0, 1, 2, 3}

then generates the charge ladder

Q = −1 + W

and the four states

∆

−

∆

. The multiplet is a consequence of three

equivalent bonds, not of the trapped node itself.

The octahedral cage has no equivalent valence bonds.

On the octahedron the second

ingredient fails, and it 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 ve 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

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 wind-

ing 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 + (−1) + 4 × 0 = 0

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

bonds.

No anchor is selected in the rst 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 rst-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.

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.

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 Wycko 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.

Scope.

The statement proved here is that the octahedral cage admits one geometric congu-

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 rst coordination shell, lie outside the

rst-shell enumeration used throughout this work and in Ref. [2], Section 9.1. Those would be

radial excitations of the same state rather than additional rst-shell states.

Verication artifact.

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

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

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.

3.7 Suppressed rst-order radiative cooling and clustering asymmetry

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

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

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

potential wells. The asymmetry is in how each species behaves

after

gravitational infall: baryons

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

observational constraints on dark-matter self-interaction [13, 14, 15].

4 Forward Derivation of

via Inclusion-Exclusion

The structural-counting derivation of

= 1836

for the proton presented in Ref. [1] (Eq. (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. [2]; we restate it here, in a form that separates what follows from the code from what is

assumed.

Step 1: the vacuum codeword carries no syndrome.

The empty lattice satises every

check. Its syndrome is identically zero, and no classical record is required to describe it.

Step 2: a defect obliges the code to carry standing syndrome information.

A trapped

node is a pattern the checks ag 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.

Step 3: that information cannot be exported.

For a physical memory, standing informa-

tion is held by hardware, and any thermodynamic 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.

Step 4: unexportable standing information is localized energy.

This step is not de-

rived. 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 syn-

drome information that cannot be exported is localized energy at the site that requires it, at a

xed rate

per bit:

= C

(5)

By massenergy equivalence [8] 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 [7, 9], and it is worth saying precisely why it is not that bound. Landauer's prin-

ciple 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, by Step 3. A bound on the cost of erasure therefore

does not apply, and Eq. (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 dierent kind: not erasure-as-dissipation, but 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 Eq. (6) as a theorem. It is not derived from quantum eld theory, general relativity,

or thermodynamics, and we make no claim that it is. Its justication 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.

Step 5: only ratios are claimed, and

is not needed.

Equation (6) is linear in

, 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.

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 correspon-

dence 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. [1],

= (K + 1)K

− c

skew

K = 13 × 144 − 3 × 12 = 1836

, admits

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

structural node of the defect carries a disruption halo of

= 144

vacuum bond states (the

second-shell footprint at the bulk coordination

K=12

). Summed over

= 13

structural nodes,

this gives a base count of

13 ×144 = 1872

bond-state disruptions. Pairs of structural nodes with

mutually disjoint ux channels  the

skew

= 3

skew-edge pairs of

 have disruption halos

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

as the bulk rst-shell intersection

|N(e

) ∩ N(e

on the FCC lattice, is

pairwise

= 12

for any

skew pair. Subtracting the double-counted bond states gives

1872 −36 = 1836

, the proton's

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

has only 4 vertices, so it admits no triple

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

identically zero.

4.3 Extension to

2,2,2

with truncation

Carrying the structure across.

The same structure applies to the octahedral defect, with

three dierences xed by the bonded graph

2,2,2

Structural node count.

Each of the 6 bounding vertices contributes 4 bonds within

2,2,2

, giving

6 ×4 = 24

boundary structural nodes; with the trapped center,

= 25

Skew-edge pair count.

Each of the 12 edges of

2,2,2

has exactly 5 mutually disjoint

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

(O)

skew

(12 × 5)/2 = 30

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

matchings of the octahedron graph and by the closed form





− 6





= 66 −36 = 30

Pairwise overlap on

2,2,2

Computing the bulk rst-shell intersection directly on the

FCC lattice yields

|N(e

) ∩ N(e

)| = 10

uniformly across all 30 skew pairs of

2,2,2

, in

contrast to

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

not a xed bulk constant.

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

does not. With 6

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

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

(O)

triple

= 8

(veried by direct

enumeration; see Section 4.4). By PIE, the triple-overlap contribution must be added back

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

mutually intersect.

The triple overlap on

2,2,2

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

|N(e

) ∩N(e

, evaluates to

(O)

triple

= 8

uniformly across all 8 perfect matchings of the

octahedron.

Why the series terminates.

The series terminates exactly at third order. A 4-matching

requires 8 mutually distinct vertices; the octahedron graph has only 6. Therefore

(O)

quad

= 0

and

all higher orders vanish identically. The PIE expansion for the octahedral defect is nite and

closed:

= N

· K

bulk

− c

(O)

skew

· K

(O)

pairwise

+ c

(O)

triple

· K

(O)

triple

= 25 ×144 − 30 × 10 + 8 × 8

= 3600 −300 + 64 = 3364.

(8)

Under Eq. (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 Eq. (8) are not free parameters: they are

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

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

First shell of an FCC vertex.

Each FCC node

has exactly 12 nearest neighbors (its

cuboctahedral coordination shell), at the 12 displacements

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

(9)

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

L =

√

). For a

2,2,2

edge

e = {v

, v

}

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

N(e) = N

) ∪N

(10)

which contains

|N(e)| = 12 + 12 −4 = 20

nodes for any edge in

2,2,2

(the 4 subtracted are the

four common nearest neighbors of

and

Pairwise overlap, and the single rule that xes it.

For each of the 30 skew-edge pairs

, e

)

2,2,2

, the pairwise intersection is

|N(e

) ∩N(e

)| = 10,

(11)

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

compute it there by the identical procedure. On

it returns

|N(e

) ∩N(e

)| = 12,

(12)

uniformly across the 3 skew pairs, with the same edge neighborhood size

|N(e)| = 20

in both

cages.

This settles a question a reader may raise about Eq. (8). The proton formula of Ref. [1]

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 rst-shell overlap, and on the tetrahedral cage that overlap

happens to equal 12. Eq. (12) establishes this by direct enumeration rather than by inspection

of the published formula.

One rule therefore covers both cages: the correction term uses the computed rst-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 dierent graphs.

Why the skew condition, and not some other selection.

The restriction of the correction

terms to mutually disjoint edge sets is also xed rather than chosen. On

the three skew pairs

give overlap 12, while all twelve vertex-sharing pairs give 15. On

2,2,2

the thirty skew pairs

give 10, while the thirty-six vertex-sharing pairs give 14 or 15. In both graphs the skew pairs

are exactly the pairs of minimal rst-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-congurations, which are the

congurations of minimal overlap.

Triple overlap.

For each of the 8 perfect matchings

, e

)

2,2,2

, the triple intersection

|N(e

) ∩N(e

)| = 8,

(13)

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

|{z}

bounding vertices

+ 2

|{z}

matching-specic bulk nodes

= 8.

(14)

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

every

matching

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

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

specic orientation of the chosen 3-matching.

The 8 = 8 coincidence.

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



(O)

triple

= 8

, the number of perfect matchings of the octahedron graph

2,2,2

. This is a

graph-theoretic invariant, independent of the embedding lattice.



(O)

triple

= 8

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

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

bonding graph and the embedding lattice.

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

symmetry

n,n,n

n = 2

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

structure and decomposes as

6 + 2

as above. The numerical equality is a coincidence of the

specic combination of

2,2,2

bonding graph and FCC lattice; either factor would change under

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

that Eq. (8) multiplies these two as independent inputs and does not double-count a single

combinatorial quantity.

Robustness check.

A useful sanity check is that the proton derivation of Ref. [1] arises as

the

K = K

bulk

N → N

= 13

skew

→ 3

specialization of the same PIE structure with

pairwise

= K

bulk

= 12

. The series terminates at second order there because

admits

no 3-matching. The transition from proton (

pairwise

= 12

) to octahedral defect (

(O)

pairwise

= 10

)

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

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

2,2,2

and is the principal

new combinatorial input of the present paper.

Verication artifact.

The full enumeration tables for all 30 pairwise intersections and all

8 triple intersections (with explicit

6 + 2

decomposition) are provided in the Appendix, and a

short self-contained Python reference implementation that reproduces every integer in Eq. (8)

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

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

. The script uses

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

(O)

pairwise

= 10

and

(O)

triple

= 8

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

execution of the script.

4.5 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

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

The following lemma settles this question and identies a clean combinatorial labeling of the 8

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

Lemma (Embedding Uniformity).

Embed the octahedral void at the origin of the FCC

lattice as in Section 4.4, and let

denote its point stabilizer (the symmetry group of the regular

octahedron, of order 48). Then:

(i)

The 8 perfect matchings of

2,2,2

form a single

-orbit. The triple rst-shell intersection

size

|N(e

) ∩ N(e

is therefore constant across all matchings, and equals 8 by

direct evaluation on any representative matching. Hence

(O)

triple

= 8

-forced.

(ii)

The 30 skew pairs of

2,2,2

split into two

-orbits, of sizes 24 and 6. The pairwise

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

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

(O)

pairwise

= 10

uniformly.

Status of (i) versus (ii).

The triple statement (i) is fully

-forced: a single orbit means

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

symmetry alone can force. With two orbits,

-invariance guarantees uniformity

within

each

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

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

particular embedding of

2,2,2

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

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

script

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

Body-diagonal labeling of the matchings.

The 8 perfect matchings admit a natural geo-

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

cube whose 8 corners sit at displacement

(±1, ±1, ±1)

from the void center, and whose 6 face

centers are the bounding vertices

A, B, C, D, E, F

of the octahedron. For each matching

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

6 + 2

decomposition of the triple

intersection (Table 6) are exactly the two antipodal corners of one body diagonal of this

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

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

bijection with the 4 body diagonals.

r 7→ −r

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

try

−I ∈ O

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

established in Section 3.5.

(d) The three edge midpoints of any matching

lie in the plane through the void center

perpendicular to

's labeling body diagonal.

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

Implication for the 8 = 8 coincidence.

The factorization

(O)

triple

· K

(O)

triple

= 8 × 8 = 64

in Eq. (8) multiplies two structurally independent quantities (Section 4.4, 8 = 8 coincidence):

the graph-theoretic 8 (number of perfect matchings of

2,2,2

, xed by the symmetry

n,n,n

n = 2

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

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

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

triple intersection decomposes as

6 + 2

where the 2 are precisely the diagonal's endpoints. The

numerical equality of the two 8's in this specic embedding remains a non-trivial fact about

2,2,2

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

reading.

4.6 The expansion coecients are not adjustable

The six integers in Eq. (8) are each xed by enumeration, as set out above. It is a separate

question 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.

The two inputs a reader might question.

Two quantities are natural candidates for doubt.

The rst is the pairwise overlap

(O)

pairwise

= 10

, which diers from the value 12 that appears

in the proton formula. The second is the third-order term

(O)

triple

(O)

triple

= 8 × 8

, which has no

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

Variant

[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%

Table 2: Sensitivity of the predicted mass to the two inputs that a reader might treat as ad-

justable. 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.

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

1.2σ

of the measured centroid of Eq. (21), and the three single changes within

0.5σ

. The

agreement is not sensitive to the counting scheme.

The levers are small and the derivation never consults the data.

Neither disputed

input moves the prediction by as much as 2 percent, and all four non-degenerate variants fall

inside a band of about 4 percent. More to the point, no observational quantity enters Eq. (8)

at any stage. The six integers come from the bonding graph and from rst-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.

The third-order term is not doing the work.

The

8 ×8 = 64

contribution is 1.9 percent

. 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 3-matchings,

not 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 aects the

result at the 2 percent level and cannot account for the prediction.

What would count as a tted parameter.

A tted parameter is one whose value is selected

by comparing outcomes with data. No quantity in Eq. (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 Eq. (12) by the

identical procedure. The sensitivity gures 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 reasonably 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.

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, dened as four mutually nearest-neighbor nodes (a

clique in the bond

graph), and an octahedral cage, dened 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 rst-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.

BCC admits no cage at all.

BCC has interior coordination

K=8

. Its bond graph contains

clique anywhere: two nearest neighbors of a BCC node are separated by

, while the

nearest-neighbor distance is

√

3/2 ≈ 0.866 a

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

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

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

number.

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 L

and

0.7071 L

, and every cage edge has the

same rst-shell size

|N(e)| = 20

. The lattices dier only in their second shells, which is exactly

where the disruption count

= 144

lives.

That dierence is enough. On HCP the tetrahedral rst-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

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.

Tetrahedral cage Octahedral cage

Lattice

K K

pairwise

triple

BCC 8 no cage undened no cage no cage

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

FCC 12 12 (uniform)

1836

10 (uniform) 8 (uniform)

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 coecients are not well dened there.

Only FCC yields uniform coecients and the observed

1836

On HCP the octahedral coecients are not well dened.

The sharper failure appears

in the octahedral cage. Within a single HCP octahedral 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 rst 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.

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 rst-shell

overlaps then take dierent values on dierent orbits. The uniformity that makes the expansion

well dened is supplied by the larger symmetry group, which only the cubic close packing has.

Scope of this test.

Three lattices have been tested, not all lattices. There are innitely

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-dened

coecients. The agreement with the proton mass is a property of the FCC lattice specically.

We note in passing that the HCP tetrahedral count

1839

coincides numerically with the neutron

count of Ref. [2], which is reached there by an unrelated route (

36 × 51 + 3

); the two have no

structural connection.

Verication artifact.

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

verify_lattices.py

which generates each lattice, nds every interior cage, and computes the overlaps directly. 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

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

xed 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.

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.

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 radius 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 L

and two regular octahedra of circumradius

0.707 L

. 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 rst parent void, and exclusive to the sec-

ond. Enumeration over every nearest-neighbor pair of octahedral voids gives, without exception:

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, 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 dierent 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.

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

= 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, for the reason given next.

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 rst place would have to be crossed.

The face-adjacent site is reachable without paying that cost.

The residual is anchor-free and stable.

In Ref. [1] 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 x 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, here the

spontaneous anchor selection that would turn

′

into a baryon. Modeling each 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 denite, 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.

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: absorption of a node is possible only while a multi-defect interface

is open, and the last remaining node cannot dissolve, so the nal 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 verication cost

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

misreading.

is a structural property of a conguration, namely the bond-state disruption it

imposes on the surrounding lattice, so dierent congurations carry dierent

and a reaction

that changes the conguration changes it. Here

initial

= 2 × 3364 = 6728

while

ﬁnal

= 1836

The dierence is not transported anywhere: it is structural cost released as the nal conguration

disrupts the lattice less than the initial one, and it leaves as photon energy. What couples

a conserved quantity is energy, through Eq. (6), and that proportionality decouples whenever a

defect-free product carries energy away.

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 conning potential

V (r) = σ

lat

of Ref. [1].

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. [1]. The barrier is one bond energy and is independent of

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

GeV

there is

= 3.4

GeV available. Dissolution of the last remaining node is forbidden, and the

nal state must retain one trapped defect.

The other candidate channels.

Four alternatives are worth checking against the constraints

above. A residue with

K=6

is one of the parent octahedra, excluded by the composition ar-

gument 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 baryon number from a state carrying none.

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

χχ → 2γ +2χ

′

conserves the

trapped-node count, so it requires no interface absorption at all, and both anking tetrahedral

voids are face-adjacent, so forming one residual on each side severs nothing. What distinguishes

it is spectral rather than structural. Its nal 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.

Verication artifact.

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

veries 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

Assumptions.

Two are stated rather than derived. The rst 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 Eq. (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. [1] has an independent realization as a fault-

tolerant verication cost

×C

= 36 × 51 = 1836

in a CSS code on the FCC lattice [2]. That

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

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

are mutually at the nearest-neighbor distance

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

neighbor distance

√

2 L

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

tion cluster and requires a multi-cluster QEC footprint. Constructing the QEC dual of Eq. (8) is

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

constraint of

2,2,2

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

than an asymptotic series.

4.10 Status of the forward derivation

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

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

Majorana character, and suppressed radiative cooling all follow from the structural symmetry

of the

2,2,2

bonding graph and the bounding octahedron. The quantitative mass prediction

in this paper is

= 1.719

GeV, derived from the closed PIE expansion (Eq. (8)) and the

proton mass alone.

5 Comparison with Observations

5.1 The reported 1.51.6 GeV gamma-ray line

Kang et al. [3] 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

xed by the lattice, not 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.

The monochromatic channel and its photon energy.

Complete annihilation is excluded,

K=6

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 = 2m

. Every quantity is already xed:

1.719

GeV from Section 4 and

′

= m

from Eq. (17). The residual recoils with kinetic energy

0.909

GeV at

v = 0.861c

. No parameter is adjustable and no observational input enters. The

framework therefore xes the position of the only monochromatic feature any allowed channel

can produce.

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 Eq. (20)

lies

0.3σ

from this centroid. Inverting Eq. (20) through the measured centroid gives

1.707 ±0.045

GeV against the counted

1.719

GeV, a dierence of

0.28σ

For orientation, the excluded channels would have given

= m

= 1.719

GeV for complete

annihilation, which sits

2.9σ

from the centroid, and

= 3m

/4 = 1.289

GeV for an octahedral

residual, which sits

6.0σ

away. Both are disfavored by the data as well as excluded by the

geometry. The two-residue continuum endpoint of Eq. (19) lies at

1.207

GeV, further still.

These are consistency checks on the derivation, not the reason for it.

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 ≈ 2 × 10

−26

. Taken at face value the measurements therefore require a foreground origin at

z ≃ 0

, which is what Eq. (21) assumes. This is a sharp and falsiable statement: any further

source at a substantially dierent redshift must report the same observed energy, not the same

emitted energy.

Caveats.

Three should be recorded. First, Ref. [3] is a preprint and has not been refereed, and

the history of tentative gamma-ray lines counsels patience: the reported 130 GeV feature [5, 4]

did not survive further data [6]. Second, the reported signicances are local. The strongest

single source has

T S = 23.03

, corresponding to

4.8σ

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 diuse, present across the sky

and brightest toward the Galactic center [28], 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 o a xenon nucleus at

v ∼ 10

−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 de-

tectors (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 [23, 24] 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 qualication before

the numbers rather than after them. It depends on identifying the interaction length scale with

L ∼ 0.5



fm, which this paper does not derive, and it will change once the strain prole of

the octahedral defect is available. It is oered to show that the framework does not obviously

conict with cluster-scale bounds, not to demonstrate agreement with them.

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

strain elds in the FCC bulk. Taking the geometric cross-section

∼ π(2L)

= 4πL

and scaling

by mass gives

σ/m ∼ 0.01



0.1

/g over that range of

. For orientation, the Bullet Cluster

bound is

σ/m ≲ 0.47

/g [15]. The estimate sits below it, and also below the window

σ/m ≈

0.1



/g [22, 21] in which self-interacting dark matter models address small-scale structure.

Both comparisons are order-of-magnitude only. The framework as constituted predicts cold dark

matter that is largely collisionless on observational scales. A more precise calculation requires

dening the strain eld of the octahedral defect quantitatively, which is left to future work.

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

Scope of this comparison.

We state at the outset that CDG-2 does not conrm 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 [25]

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

not evidence.

The SSM prediction in Section 3.7 that octahedral-void dark matter cannot dissipate energy

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

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

low-surface-brightness morphologies and minimal stellar content.

The recent identication of Candidate Dark Galaxy-2 (CDG-2) in the Perseus cluster [25]

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

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

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

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

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

genuine galaxy. The system contains four globular clusters within a

∼ 1.2

kpc diameter and a

total galaxy luminosity

V,gal

≈ 6.2 × 10

⊙

, with the GC light fraction estimated at

∼ 16.6%

(rising to

∼ 33%

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

mass scaling relations of [26] and [27], these data imply a dark-matter halo mass fraction of

99.94



99.99%

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

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

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

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

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

2 is consistent with a dark sector lacking a rst-order dissipation channel, as are many other

models.

5.5 Comparison with existing dark matter models

Composite dark matter.

The octahedral-void defect predicted here shares qualitative fea-

tures with several composite-dark-matter scenarios in the literature. GeV-scale strongly-coupled

hidden-sector models [18, 19] produce dark baryons whose mass arises from connement dynam-

ics and whose direct-detection signatures are suppressed relative to weakly-interacting massive

particles. Strongly-interacting massive particle (SIMP) models [20] predict sub-GeV to GeV-scale

dark particles with non-trivial self-interaction. The SSM prediction lies in a similar parameter

region (1.7 GeV mass, suppressed electromagnetic coupling) but diers in origin: the existence

and structural properties of the dark matter candidate are derived from FCC vacuum geome-

try rather than from cosmological freeze-out, asymmetric mechanisms, or hidden-sector gauge

dynamics.

The sexaquark.

The closest phenomenological analog in the existing literature is the

sex-

aquark

[29, 30]: a hypothesized neutral, avor-singlet, scalar bound state of

uuddss

quarks with

baryon number

B = 2

and strangeness

S = −2

, proposed by Farrar as a dark matter candidate.

The sexaquark and the SSM

K=6

octahedral defect agree on several non-trivial phenomeno-

logical features. Both are GeV-scale particles with mass

∼ 2m

(sexaquark stability requires

≲ 2054

MeV, with cosmological relic-abundance ts favoring 1.51.8 GeV [30]; 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 neu-

tral with suppressed hadronic couplings (sexaquark: avor-singlet decoupling from pions; SSM:

absence of square plaquettes in the bonded subgraph

2,2,2

). Both are consistent with the dif-

fuse, non-collisional halo phenomenology required by direct-detection nulls and self-interaction

constraints.

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 = −2

S = +2

); the SSM defect, by contrast, is its own antiparticle by the inversion sym-

metry of the regular octahedron (Section 3.5). The Majorana-vs-Dirac character is empirically

distinguishable in indirect-detection signatures (annihilation

χχ

vs.

χ¯χ

) and in cosmological

asymmetry channels. Second, the sexaquark mass is a free phenomenological parameter that

lattice QCD cannot yet predict to the precision required for stability; the SSM mass follows from

a closed combinatorial expansion on

2,2,2

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

strongly-interacting QCD bound state (a avor-singlet hadron), whereas the SSM defect carries

no SU(3) color charge (Section 3.4) and does not participate in strong interactions. The two

frameworks are therefore not in direct competition: a denitive sexaquark detection would not

falsify the SSM defect, and vice versa, though discriminating direct-detection, indirect-detection,

and accelerator signatures can be designed.

6 Discussion

6.1 What this framework is and is not

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

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

enters and how much weight it carries.

The mass-information correspondence is an input.

The step from a count

to a mass,

Eq. (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 eld 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 en-

vironment to dissipate into. Landauer's principle supplies the precedent that information and

energy are convertible; it does not supply Eq. (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 tted independently.

No equation of motion is oered.

The framework has no Lagrangian, no equation of

motion, and no microscopic interaction between lattice nodes. Its outputs are counts on a xed

geometry, 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 predict anything about processes rather than about spectra, and constructing

one is beyond the present work.

Cross-sections are not derived, which limits what can be tested.

The framework

xes a mass and a set of symmetry properties. It does not x the strength of any interaction.

The rst-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.

The framework has not been independently reproduced.

The Selection-Stitch Model

is the work of a single author, and no independent group has veried 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 standard libraries only 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 Falsiable predictions

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

be falsied by observation. Falsifying them would bear on the construction together with that

postulate, not on the geometry alone.

Mass scale.

The framework predicts a single dark matter species with mass

1.719

GeV, derived from the closed inclusion-exclusion expansion on

2,2,2

(Eq. (8)) and

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

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

framework.

Gamma-ray line energy.

The annihilation channel is xed 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 falsiable. A conrmed line from this mass scale at

itself

(

1.719

GeV) or at

(

1.289

GeV) would contradict the derivation. The two-residue

channel contributes a continuum with endpoint

1.207

GeV (Eq. (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 dierent redshift must report the

same observed energy, not the same emitted energy.

Suppressed rst-order EM coupling.

The octahedral defect has suppressed single-

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

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

framework.

Absence of the baryonic SU(3) color mechanism.

The octahedral defect does not

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

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

falsify this identication.

Single dominant dark-matter species.

The 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). Denitive

observation of multiple dark-matter species at distinctly dierent mass scales 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 an-

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

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

Collisionless halos and absence of dark stellar collapse.

The framework predicts

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

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

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

would falsify the framework. The recent identication of CDG-2 [25] (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 are, in order of priority:

The abundance ratio

Ω

/Ω

Deriving this requires the dynamics of the

K=4 → 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 verication cost on a CSS code on the FCC

lattice [2]. Constructing the analogous multi-cluster QEC dual of

= 3364

for the

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

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

coordination clusters (Section 4.9).

Self-interaction cross-section.

A quantitative SSM calculation of the strain-eld overlap

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

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

of dark-matter halos.

Halo formation phenomenology.

The free-streaming length of a

∼ 1.7

GeV particle decou-

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

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

treatment.

7 Conclusions

The Selection-Stitch Model identies baryonic matter with

K=4

remnants trapped in the tetra-

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

stitial site  the octahedral void  that admits an analogous

K=6

trapped remnant under

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

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

tromagnetic coupling (the bonding graph

2,2,2

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

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

generated for baryons (the skew-edge pair count

(O)

skew

= 30

does not factor as a three-color rep-

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

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

baryonicdark-matter clustering asymmetry.

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

2,2,2

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

= 25 ·144 − 30 · 10 + 8 · 8 = 3364, m

× m

= 1.719

GeV

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

Ref. [1], 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, not 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 veriable. The reported

1.5



1.6

GeV gamma-ray line [3]

sits near this mass. Section 4.8 derives the annihilation channel from the merger interface, and

Section 5.1 obtains

= 1.591

GeV,

0.3σ

from the measured centroid, with no observational

input. The Fermi-LAT signal itself awaits conrmation.

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

of the baryonic SU(3) color-generating mechanism, self-conjugate Majorana character, and sup-

pressed radiative cooling  follow from the structural symmetry of the

2,2,2

bonding graph

and the bounding octahedron, and are independent of the verication-cost calculation. The

recent identication of the almost-dark galaxy CDG-2 [25] is consistent with the collisionless

diuse-halo prediction, as it is with other non-dissipative dark-matter models.

Data Availability

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

2,2,2

for the octahedron, skew-pair count

(O)

skew

= 30

veried by three independent methods)

are standard crystallographic and combinatorial data. The Fermi-LAT line measurements are

from [3]. The full enumeration of the 30 pairwise and 8 triple overlaps used in Eq. (8) is given

in the Appendix; a ve self-contained scripts reproduce every computed quantity in this paper.

All require standard libraries only 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 (Sec-

tions 4.4 and 4.6):

https://github.com/raghu91302/ssmtheory/blob/main/verify_C_

DM.py



verify_uniformity.py

 the orbit decomposition, body-diagonal labeling, and inver-

sion pairing of the Embedding Uniformity Lemma (Section 4.5):

https://github.com/

raghu91302/ssmtheory/blob/main/verify_uniformity.py



verify_lattices.py

 the same construction applied to BCC, ideal HCP and FCC,

reproducing Table 3 (Section 4.7):

https://github.com/raghu91302/ssmtheory/blob/

main/verify_lattices.py



verify_octahedral_states.py

 the point groups, stabilizer orbits and bond projec-

tions of Table 1 (Section 3.6):

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

verify_octahedral_states.py



verify_residual_cages.py

 the shared-edge interface, the four-cage enumeration and

the composition result that xes the residual as tetrahedral (Section 4.8):

https://

github.com/raghu91302/ssmtheory/blob/main/verify_residual_cages.py

What the scripts do and do not establish should be stated plainly. They verify the enu-

merations, 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 geome-

try. The interactive 3D visualization referenced in the introduction is hosted at

https:

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

Declaration of Competing Interest

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

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

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A Enumeration Tables for Pairwise and Triple Overlaps

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

sections that determine

(O)

pairwise

= 10

and

(O)

triple

= 8

in Eq. (8). The setup follows Section 4.4

exactly: the octahedral void is centered at

(1, 0, 0)

, the 12 FCC nearest-neighbor displace-

ments are

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

, and the rst-shell neighborhood of an edge is

N(e) = N

) ∪N

)

with

|N(e)| = 20

Vertex labels.

We label the six bounding vertices of the octahedral void as in Table 4.

The 12 edges of

2,2,2

are then

{AC, AD, AE, AF, BC, BD, BE, BF, CE, CF, DE, DF }

(ev-

ery vertex-pair from distinct antipodal classes). The 3 antipodal non-bonds

are

at distance

√

and do not contribute to

2,2,2

Label Coordinates

A (2, 0, 0)

B (0, 0, 0)

C (1, +1, 0)

D (1, −1, 0)

E (1, 0, +1)

F (1, 0, −1)

Table 4: 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 )

All 30 pairwise intersections.

Table 5 lists every skew pair of

2,2,2

together with its rst-

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

veried by direct enumeration:

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

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

the same value, 10.

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

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

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

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

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

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

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

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

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

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

Table 5: All 30 skew-edge pairs of

2,2,2

with their rst-shell intersection sizes

|N(e

) ∩N(e

Every entry is 10. The summary statistic

(O)

pairwise

= 10

used in Eq. (8) is the common value of

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

All 8 perfect matchings and their triple intersections.

Table 6 lists every perfect match-

ing (3-matching) of

2,2,2

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

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

directly.

Structure of the matching-specic nodes.

The 2 matching-specic bulk nodes for each

matching sit at displacement

(±1, ±1, ±1)

from the void center

(1, 0, 0)

, on the four body diago-

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

group into 4 pairs

(M1 ↔ M8)

(M2 ↔ M6)

(M3 ↔ M7)

(M4 ↔ M5)

, with the two match-

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

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

are inversion-related through the void center, reecting the centrosymmetry

∋ −I

that also

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

Assembly.

Combining Tables 5 and 6 with

= 25

and the per-node disruption count

= 144

reproduces Eq. (8) by direct substitution:

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

(22)

and

= (3364/1836)×m

= 1.7191

GeV using the CODATA proton mass

= 938.272

MeV [10].

# Matching Bounding Other Other-node coordinates

{AC, BE, DF }

6 2

(0, +1, −1)

(2, −1, +1)

{AC, BF, DE}

6 2

(0, +1, +1)

(2, −1, −1)

{AD, BE, CF }

6 2

(0, −1, −1)

(2, +1, +1)

{AD, BF, CE}

6 2

(0, −1, +1)

(2, +1, −1)

{AE, BC, DF }

6 2

(0, −1, +1)

(2, +1, −1)

{AE, BD, CF }

6 2

(0, +1, +1)

(2, −1, −1)

{AF, BC, DE}

6 2

(0, −1, −1)

(2, +1, +1)

{AF, BD, CE}

6 2

(0, +1, −1)

(2, −1, +1)

Table 6: All 8 perfect matchings of

2,2,2

with their rst-shell triple intersections decomposed

6 + 2

. The Bounding column counts the 6 bounding vertices

A, B, C, D, E, F

(common to

every matching by construction), and the Other column counts the 2 matching-specic bulk

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

(O)

triple

= 8

uniformly.