Deriving the Dark Matter Annihilation Channel
from
Metric-Wall Confinement:
A Self-Bonded K=4 Residual at 957 MeV
in the Selection-Stitch Model
Raghu Kulkarni
∗
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
Draft manuscript — May 2026 (revision 2.1: editorial pass)
Abstract
The Selection-Stitch Model identifies dark matter with a K=6 octahedral defect of mass
m
χ
= 1.7195 GeV trapped in the octahedral voids of an FCC vacuum lattice [12]. The
recent Kang et al. [9] detection of a 1.5–1.6 GeV gamma-ray line in three active galactic
nuclei lands about 10% below the 1.7195 GeV energy that a χχ → γγ channel would
predict. We resolve this tension by deriving the annihilation channel from framework
primitives: χ + χ → γ + χ
′
, where χ
′
is a self-bonded K
4
“figure-8” defect with mass
m
χ
′
= (1872/1836) m
p
= 0.9567 GeV. The derivation has three parts. First, a Verification-
Cost Floor that forbids χχ → γγ is shown to be a corollary of the metric-wall confinement
of Kulkarni [11]. Second, FCC crystallography fixes the merger geometry uniquely: direct
enumeration on a 4
3
supercell confirms that all 450 nearest-neighbor oct-void pairs share
exactly one octahedron edge (two bounding vertices joined by one bond of length L). Third,
C
χ
′
= 1872 emerges as the unique lowest-cost final state compatible with energy conserva-
tion, the Verification-Cost Floor, sector identity, and oct-void symmetry, with the figure-8
obtained from the visible proton by removing the c
skew
K = 36 deduction the proton pays for
its color and electric charge. Two-body kinematics then place the photon line at 1.586 GeV,
in 0.17σ agreement with the Kang central value 1.578 ± 0.048 GeV. The line position is a
framework prediction with no fitted parameters. The merger graph is connected through its
single shared edge as a bridge; producing two figure-8 residues in a single event would re-
quire cutting that bridge, so no coincident ∼0.76 GeV doublet is expected, and its detection
would falsify the construction.
Revision note (v2). This revision replaces the static-lemniscate description of χ
′
in §6
with a librating-tetrahedron picture: χ
′
is a regular tetrahedron of edge L that rotates
continuously about a C
2
axis of its host oct void, sweeping through a one-parameter family
of unit tetrahedra. The lemniscate is the projected trajectory of the 4 vertices, not the
instantaneous shape. The instantaneous geometry is identical to that of the visible proton’s
tetrahedron, so the two defects belong literally to the same K
4
-at-edge-L class, differing
only in host void and resulting boundary conditions. The librating picture is forced by the
geometric fact that no 4-clique of oct-void bounding vertices sits at uniform distance L, so
static locking is impossible. This tightens the visible-proton/figure-8 analogy underlying
the subtractive gauging axiom (soft point S4) and closes most of the topology-matches-host-
symmetry gap (soft point S3). The mass prediction m
χ
′
= 0.9567 GeV and the photon-line
prediction E
γ
= 1.586 GeV are unchanged: libration is a Goldstone-like zero mode that
contributes nothing to the rest-mass C-count.
Code availability. A reference implementation of the shared-edge enumeration (§4) is at
https://github.com/raghu91302/ssmtheory/blob/main/verify_shared_edge.py.
∗
raghu@idrive.com
1
1 Introduction
The Selection-Stitch Model (SSM) treats the physical vacuum as a Face-Centered Cubic crys-
tallization of spacetime [10, 11]. Baryonic matter is a K=4 defect trapped at a tetrahedral
interstitial site of the K=12 FCC bulk [11]. Dark matter is a K=6 defect trapped at the oc-
tahedral site [12]. We use K in two senses: as a defect-type label (K=4, K=6 above), and as
the bulk coordination number (K=12) inside the structural-counting formula (K+1)K
2
. Both
usages follow the originating papers; context disambiguates them.
Two parameter-free predictions follow from the bonding-graph counts of these defects:
m
p
m
e
= (K+1)K
2
− c
skew
K = 13 · 144 − 3 · 12 = 1836, (1)
with K=12 here and below, and
m
χ
=
C
DM
C
p
m
p
=
3364
1836
m
p
= 1.7195 GeV, (2)
inheriting only the proton mass as a calibration scale.
A recent Fermi-LAT analysis [9] reports a 1.5–1.6 GeV gamma-ray line detected jointly in
three high-significance active galactic nuclei with combined test statistic TS = 57.77. Per-source
line centroids cluster at 1.578 ± 0.048 GeV. The channel χχ → γγ would predict E
γ
= m
χ
=
1.7195 GeV, sitting ∼9% above the central observed value.
This paper resolves the tension by deriving the annihilation channel from framework primi-
tives. The channel is
χ + χ −→ γ + χ
′
, m
χ
′
=
(K+1)K
2
C
p
m
p
=
1872
1836
m
p
= 0.9567 GeV, (3)
where χ
′
is a stable K=4 residual whose mass comes from the same combinatorial expression as
the visible proton’s, but without the skew-edge deduction. We refer to χ
′
as the figure-8 dark
proton for reasons made precise in §6. Two-body kinematics place the photon line at 1.586 GeV,
within 0.17σ of the Kang centroid.
The derivation proceeds as follows. §2 establishes the Verification-Cost Floor (VCF) as
a corollary of the metric-wall confinement of Kulkarni [11]. The floor forbids χχ → γγ out-
right (§3). FCC crystallography fixes the merger interface uniquely (§4). The mass count
C
χ
′
= 1872 then falls out from four conservation laws plus a structural minimality argument
(§5). The figure-8’s qualitative properties (self-conjugacy, oct-void residence, non-detectability
except gravitationally) follow from the same un-gauging step (§6). Competing channels are
excluded by VCF, sector identity, or merger-graph connectedness (§7). §8 works out the kine-
matic consequences. §9 compares to the Kang detection, including a statistical rejection of the
source-rest-frame interpretation at χ
2
= 118. §10 catalogs the framework’s predictions, five
falsification criteria, and the five places where the derivation rests on plausible-but-unproven
selection principles.
What this paper claims. The line position 1.586 GeV is a framework prediction with no
fitted parameters and no new free inputs. The principal soft point, S4, is the subtractive gauging
step that relates C
χ
′
to C
p
by analogy with the visible proton rather than by independent first-
principles count. We are explicit about what the derivation forces and what it does not, and
we identify in §10 the finite calculations that would close each soft point.
What this paper does not address. The cosmological abundance of either the primary
χ component or the secondary χ
′
component is not derived here. Any abundance prediction
requires both a cross-section for the channel of equation (3) and an SSM analog of the baryon
2
asymmetry parameter η
B
to track matter-antimatter cancellation between formation and the
present epoch. Neither is available in the framework as currently developed. The forward mass
prediction is independent of these.
2 The Verification-Cost Floor
The construction needs one structural principle that the existing SSM literature implies but
does not state in isolation. Define the Verification-Cost Floor (VCF) as the rule that K
4
base structures cannot dissolve back into K=12 bulk coordination at sub-Planckian energies.
We derive VCF as a corollary of the metric-wall confinement of Kulkarni [11], with the original
confinement question recast.
2.1 The metric wall in its original form
Section 6 of Kulkarni [11] establishes that a K=4 node trapped at the centroid of a tetrahedral
void cannot escape the void at sub-Planckian energies. The argument is geometric and local.
The trapped node bonds to the 4 bounding vertices of the void. Each bounding vertex is a K=12
node, held in its crystalline equilibrium by 9 external bonds reaching into the surrounding bulk.
Extracting the trapped node requires stretching all 4 internal bonds simultaneously, against a
restoring force from 4 × 9 = 36 bulk bonds. The result is a linear confining potential
V (r) = σ
lat
r, σ
lat
∼ ε/L
2
, (4)
where ε is the unitary entanglement bond energy and L is the lattice spacing. With L ∼ ℓ
P
and ε at the GUT scale, σ
lat
· L ∼ 10
15
GeV. That barrier is many orders of magnitude above
any sub-Planckian scale.
The interstitial type of the void plays no role in this calculation. The 4 bounding vertices
enter as “K=12 bulk nodes to which the trapped node is bonded.” The 9 external bonds per
vertex enter as “the rest of the bounding vertex’s cuboctahedral coordination shell.” Both are
properties of K=12 bulk nodes in any FCC region, not of tet voids specifically. The metric-wall
confinement is therefore general: it applies to any K-class trapped node embedded in a K=12
host, with the number of restoring bonds scaling as K × 9.
2.2 Lift to the dark-matter sector
The K=6 dark matter defect of Kulkarni [12] sits at the centroid of an oct void, bonded to
6 bounding vertices. By the same argument, the restoring force comes from 6 × 9 = 54 bulk
bonds. The confinement barrier is at least as strong as the visible-proton case. Kulkarni [12]
identifies χ as cosmologically stable but does not name the stability mechanism. We name it
here: χ is confined by the same metric wall that confines the trapped quark of Kulkarni [11].
2.3 Reading the wall inward
The metric-wall argument of Kulkarni [11] asks whether the trapped node can escape the void.
The barrier is the energy required to push outward, from the void centroid to a bulk position.
VCF asks a different question: can the trapped node dissolve, that is, relax into bulk K=12
coordination, ceasing to be a defect at all? Both questions trace the same trajectory. To
dissolve, the trapped node must transition from its centroid position (where it has K=4 or
K=6 coordination) to a bulk position (where it would join K=12 coordination). The geometry
between centroid and bulk is the same as between centroid and free-quark exit. The restoring
force is the same. The barrier is the same σ
lat
· L ∼ 10
15
GeV.
The metric wall protects against dissolution by the same physics that protects against escape.
Below the Planck scale, neither door opens.
3
2.4 VCF, derived
Corollary 1 (Verification-Cost Floor). The verification-cost contribution of a K-class trapped
node, namely (K+1)K
2
in un-gauged units (equal to 1872 for K=4 and the un-gauged-baseline
component of 3364 for K=6), is preserved in any sub-Planckian reaction. Gauging structure
attached to the base (skew-edge couplings, anchor selections, color or electromagnetic charge,
antipodal-bond patterns of the K
2,2,2
graph) may be modified by sub-baseline bond rearrangements.
The base itself cannot be eliminated.
Section 6.3 of Kulkarni [11] notes that the metric wall is not literally infinite. If supplied
energy exceeds σ
lat
· L, the lattice un-stitches: a new node nucleates at the fracture point, and
one defect becomes two. VCF accommodates this. The K
4
base count is conserved modulo mul-
tiplication, not modulo dissolution: one defect can become two; one cannot become zero. The
form of VCF used below applies in the sub-Planckian regime, which covers every annihilation
process in the present universe.
3 Why χχ → γγ Is Forbidden
The most obvious annihilation channel for two self-conjugate particles, χχ → γγ, runs into VCF
immediately.
The initial state has two K=6 defects with total verification cost 2 × 3364 = 6728 C-units.
The final state has two photons. Photons are propagating modes of the bulk displacement field;
they carry energy but no localized defect content (C
γ
= 0).
Energy balances. Total rest mass energy 2m
χ
c
2
= 3.439 GeV splits evenly between two
1.7195 GeV photons.
Structure does not. Each input χ carries an un-gauged K
4
base of 1872 C-units as a
subcomponent of its 3364 total. The two inputs together carry 2 × 1872 = 3744 C-units of
un-gauged K
4
structure. The final state, two photons, contains zero defect content of any kind.
For the reaction to proceed, both K
4
bases must dissolve back into bulk K=12 coordination.
This is the K=12 → K=4 inverse phase transition that VCF (Corollary 1) forbids. The metric
wall is the obstruction: the trapped node of each χ would have to traverse the wall from the
oct-void centroid to a bulk position, working against 54 bulk bonds per defect. The barrier is
σ
lat
· L ∼ 10
15
GeV per defect, against an available 1.7195 GeV.
The channel is energetically forbidden in the relevant kinematic regime. The observed
1.586 GeV line [9] must reflect a different channel.
4 The Merger Geometry: A Shared Octahedron Edge
A merger between two K=6 defects requires that they come into causal contact across some
interface. The K=12 bulk between non-adjacent oct voids is impassable below the Planck scale
(the same metric wall again). Causal contact is therefore available only through shared bounding
geometry: vertices and bonds that belong to both defects’ bounding octahedra simultaneously.
This section establishes that nearest-neighbor oct voids in the FCC lattice share exactly one
octahedron edge: two bounding vertices, joined by one bond of length L. The claim is verified
by direct enumeration on a 4
3
supercell.
4.1 FCC oct-void coordinates
The FCC unit cell with cubic lattice constant a contains atoms at the cube corners and face
centers. The nearest-neighbor distance is L = a/
√
2. The cell decomposes into four oct voids:
one at the body center (a/2, a/2, a/2) and three at edge midpoints [1, 3, 12]. Each oct void is
4
bounded by 6 FCC vertices forming a regular octahedron with edge length L and centroid-to-
vertex distance L/
√
2 ≈ 0.707 L.
A 4 ×4 × 4 supercell gives 256 FCC atoms and 256 candidate oct-void centers. Restricting
to oct voids whose 6 bounding vertices are all inside the chunk leaves 108 fully-interior voids.
Enumeration of all pairwise center separations yields the discrete spectrum in Table 1.
Table 1: Oct-void pair separations in a 4
3
FCC supercell. Reproduced by the companion script
(see Code Availability).
Separation d/L Pair count Shared bounding vertices
1 (nearest) 450 2 (one shared octahedron edge)
√
2 216 1 (single shared vertex)
√
3 600 0
2 288 0
Every one of the 450 nearest-neighbor pairs shares exactly 2 bounding vertices. The next
discrete separation, d = L
√
2, gives 1-vertex contact across 216 pairs. Beyond that, contact
drops to zero.
One additional check: are the 2 shared vertices of each nearest-neighbor pair antipodal
within their respective oct voids, or octahedron-edge-adjacent? Antipodal pairs sit at distance
L
√
2 (the octahedron diagonal). Edge-adjacent pairs sit at distance L (the octahedron edge).
Direct computation on all 450 pairs gives distance L for every pair. The shared 2 vertices
are joined by a bond, not separated by an antipodal axis. The shared geometry is exactly an
octahedron edge (Figure 1).
4.2 Three properties of the shared edge
The 2-vertex, 1-bond shared edge is the unique causal interface between two K=6 defects in
adjacent oct voids. Three properties of this interface matter for the merger dynamics.
Connectivity. The compound bonding graph K
2,2,2
∪ K
2,2,2
, formed by identifying the 2
shared vertices and the 1 shared bond, has 10 distinct vertices and 12 + 12 − 1 = 23 distinct
bonds. The graph is connected through the shared edge as a bridge. Removing this single bond
disconnects the compound into the two original K
2,2,2
components.
C
2
symmetry. The shared edge is a 1-dimensional segment with reflection symmetry about
its midpoint. The merger event inherits this C
2
as a global symmetry of the process.
Geometric center. The midpoint of the line joining the two oct-void centers coincides
with the midpoint of the shared edge. This is the unique fixed point of the C
2
axis and, as §5
argues, the natural location of the relaxed composite.
These three properties are direct consequences of the FCC oct-void edge-sharing established
in Table 1. The remainder of the derivation uses them.
5 Deriving C
χ
′
= 1872
The figure-8 mass C
χ
′
= 1872 follows from a small set of framework primitives plus the shared-
edge geometry. The derivation has three parts: identify what is conserved across the reaction,
recognize what is not, and select the figure-8 as the unique minimum final state compatible with
the conservation laws.
5.1 Conserved quantities
Three conservation laws apply to any sub-Planckian reaction in the framework.
5
center A
center B
shared edge = L
d
AB
= L
Figure 1: Shared octahedron edge between two nearest-neighbor oct voids in the FCC lattice.
Yellow stars mark the two oct-void centers (A and B) at separation d
AB
= L. Each oct void is
a regular octahedron with edge length L and 6 bounding vertices (blue for A, red for B). The
two voids share 2 bounding vertices (green, large), joined by a single octahedron edge of length
L (thick green segment). This shared edge is the unique causal interface for a merger event and
is a bridge in the compound graph K
2,2,2
∪ K
2,2,2
: removing it disconnects the compound into
the two original K
2,2,2
components.
Energy. Standard relativistic energy conservation. At low relative velocity (the regime of
galactic halos),
2m
χ
c
2
= m
χ
′
c
2
+ T
χ
′
+ E
γ
, (5)
which evaluates as 3.4383 = 0.9567 + 0.8956 + 1.5860 GeV under the predicted masses, worked
through in §8.
K
4
base content (VCF). §2 establishes that the K
4
base of any defect produced by the
K=4 → K=12 phase transition cannot be dissolved into bulk at sub-Planckian energies. Each
input χ contains one un-gauged K
4
base. The final state must contain at least one.
Sector identity. Defects are confined to whatever interstitial site their bounding geometry
occupies. A residue produced from two oct-void inputs cannot relocate to a tet void without
crossing the K=12 bulk, which the metric wall prevents.
5.2 What is not conserved
The total verification cost C is not a conserved current, and this point deserves a quick aside:
a naive attempt to balance a C-budget across the reaction produces an awkward “doubled K
4
”
framing as a side effect, and the figure-8 mass count emerges cleanly only once the conservation
assumption is dropped.
C is defined as a structural property of a given defect configuration. It counts the bond-state
disruption the configuration imposes on the surrounding lattice. Different configurations have
different C-values. Reactions change the configuration, so they change the total C. What is
conserved is energy, which couples to C through the mass-energy-information correspondence
m = C ·kT ln 2/c
2
[10]: proportional, but not the same. Photons have C
γ
= 0 but carry nonzero
energy; the proportionality decouples whenever a defect-free product appears.
In the present reaction, C
initial
= 2 × 3364 = 6728 and C
final
= 1872 (the figure-8 alone; the
photon contributes zero). The difference ∆C = 4856 is not transported anywhere. It represents
structural cost released into the bulk: the final configuration disrupts the lattice less than the
initial one. The energy that had been sustaining the released C goes into the photon and the
6
figure-8’s kinetic energy, by two-body kinematics. Energy conservation holds. C conservation
does not, and is not required.
With the constraint dropped, the figure-8’s C
χ
′
= 1872 does not need to balance anything.
It just needs to be the C-value of whatever configuration the reaction actually produces. §5.3
identifies that configuration.
5.3 The unique minimum final state
The final state is the lowest-C configuration satisfying all three conservation laws above. Four
constraints select it.
VCF requires at least one K
4
base in the final state. Zero would mean full dissolution into
bulk, which the metric wall forbids. The minimum count is one.
Sector identity requires this K
4
to reside in an oct void.
Oct-void symmetry forces the libration topology, not a tet-void lock. The K
4
at
edge length L realized as a regular tetrahedron of unit edge requires a host region containing
4 vertices mutually at distance L for static locking. The tet void’s 4 bounding vertices satisfy
this and the visible proton locks onto them statically. The oct void’s 6 bounding vertices come
in 3 antipodal pairs at distance L
√
2 and 12 non-antipodal pairs at distance L; no 4-clique at
uniform distance L exists. A regular tetrahedron of edge L therefore cannot lock statically onto
any subset of oct-void vertices. It can only inscribe itself at the void centroid with its orientation
matching one of the void’s C
2
axes, librating freely about that axis (a one-parameter family of
unit tetrahedra related by C
2
rotation). The librating tetrahedron is forced. [S3]
Structural minimality selects one librating tetrahedron over two. One librating K
4
in one
oct void has C = 1872. Two librating K
4
s in two oct voids would have C = 3744. Configurations
with additional defects or modifications have higher C. One librating K
4
is the minimum.
[S1][S2]
The merger of two χ defects produces one librating K
4
in one oct void, plus the photon that
energy conservation requires to carry away the released structural cost. During the merger event
itself, the C
2
symmetry of the shared-edge contact (§4) is preserved: the bond-rewiring cascade
is centered on the shared-edge midpoint and respects the reflection that swaps voids A and B.
The relaxed final state, however, must localize in one void or the other to permit a well-defined
structural-node count from a single host. This means the merger’s C
2
is spontaneously broken
during relaxation, with voids A and B each hosting the composite in roughly 50% of events
across a cosmological population. The two outcomes are physically equivalent: by translational
symmetry of the FCC vacuum, a librating K
4
in any oct void has the same mass and the same
observable signatures. [S5]
5.4 The figure-8 mass
The figure-8 mass C
χ
′
= 1872 is derived from the visible proton’s count by subtracting what the
figure-8 lacks. This is the cleanest framing: the figure-8 and the visible proton are structurally
the same K
4
defect class at unit edge length L, distinguished only by the absence or presence
of gauging structure attached to the K
4
base.
The visible proton’s mass count from equation (1) is
C
p
= (K+1)K
2
− c
skew
K = 1872 − 36 = 1836. (6)
Two pieces enter. The first, (K+1)K
2
= 1872, is the un-gauged base count of a K
4
trapped
node in a K=12 host [11]. The second, −c
skew
K = −36, is the deduction for the three skew-
edge pairs in the proton’s cuboctahedral coordination shell. Those skew pairs generate the down
quark’s ˆr
i
· ˆr
j
= −1/3 anti-alignment [11] and through it the proton’s electric charge and SU(3)
color content. In framework terms, the proton pays a 36-unit C-cost to acquire gauging.
7
The figure-8 has the same base structure (K
4
at edge L, instantaneously a regular tetrahe-
dron), but no gauging. Its host is the oct void rather than a tet void, and as established in §5.3
the tetrahedron librates rather than locking statically. The bonds of the librating K
4
do not
reach into a fixed cuboctahedral coordination shell to inherit skew-edge handedness, because
the libration averages over the surrounding shell’s orientation-dependent structure. There is no
skew-edge structure attached, so there is no 36-unit deduction. The figure-8’s mass count is the
un-gauged base count alone:
C
χ
′
= (K+1)K
2
= 1872 = C
p
+ 36. [S4] (7)
This is the framework-internal meaning of the figure-8: it is the visible proton’s K
4
defect class
without the cuboctahedral gauging that produces the proton’s charge and color. The 36-unit
increment over C
p
is exactly the C-cost the visible proton spent on gauging that the figure-8
forfeits.
With this caveat [S4], the figure-8 mass follows via the mass-energy-information correspon-
dence:
m
χ
′
=
C
χ
′
C
p
m
p
=
1872
1836
· 938.272 MeV = 0.9567 GeV. (8)
The ratio m
χ
′
/m
p
= 1872/1836 = 1.0196 is the framework-internal cost of forfeiting the pro-
ton’s gauging. The figure-8 is heavier than the proton by exactly c
skew
K · m
p
/C
p
= 36 ×
938.272/1836 ≈ 18.4 MeV. The proton mass used here is the CODATA 2022 recommended
value [13].
6 Structural Properties of the Figure-8 Residual
The mass count derived above is the most consequential property of χ
′
. The same un-gauging
step that produces it generates four further properties: its topology and dynamics, three cor-
related structural traits (heavier rest mass, no color, self-conjugacy), its preferred residence in
oct voids, its stability, and its complete non-detectability except through gravitation.
6.1 Topology and naming: the librating tetrahedron
The figure-8 dark proton χ
′
is a K=4 defect of the same structural class as the visible proton:
4 nodes pairwise bonded at the FCC nearest-neighbor distance L, realizing the complete graph
K
4
(6 edges, each vertex of degree 3) with all edges at unit length. The two defects differ only
in their host void and the boundary conditions that follow from it.
The visible proton: a static tetrahedron in a tet void. The visible proton sits at the
centroid of a tetrahedral void. Its 4 nodes coincide with the 4 bounding vertices of that void,
forming a rigid regular tetrahedron of edge L. The host void’s C
3
symmetry axes are compatible
with the tetrahedron’s own C
3
axes, so the configuration is static: the K
4
is locked into the tet
void’s bounding geometry, and the surrounding cuboctahedral coordination shell carries the 3
skew-edge pairs that produce color charge and the c
skew
K = 36 deduction in the mass count.
The figure-8: a librating tetrahedron in an oct void. The figure-8 χ
′
sits at the centroid
of an octahedral void. Its 4 nodes form a regular tetrahedron of edge L as well (at any instant,
all 6 pairwise distances equal L), but this tetrahedron does not coincide with any rigid subset
of the 6 oct-void bounding vertices. The octahedral void has no 4-vertex subset whose pairwise
distances are all equal to L: its bounding vertices come in 3 antipodal pairs at distance L
√
2
and 12 non-antipodal pairs at distance L, and no 4-clique at uniform distance L exists. The K
4
tetrahedron therefore cannot lock onto the oct void’s vertex set the way the visible proton’s K
4
locks onto its tet void.
8
What the K
4
can do is inscribe itself inside the oct void at the centroid, with the tetrahe-
dron’s orientation chosen so that its symmetry is compatible with one of the oct void’s three C
2
axes (the axes through antipodal vertex pairs). Compatibility along one C
2
axis does not fix the
tetrahedron rigidly: the configuration is free to rotate continuously about that axis, sweeping
through a one-parameter family of unit tetrahedra all sharing the void centroid and all related
by the host C
2
. This continuous rotation is the libration of the figure-8.
The figure-8 is a projected trajectory, not an instantaneous shape. The instantaneous
shape during libration is always a regular tetrahedron of edge L: the bonding graph K
4
is
preserved at every instant, and all 6 edge lengths remain equal to L throughout the motion.
What changes over time is the orientation of the tetrahedron about its libration axis. Projected
onto the plane perpendicular to the libration axis (the oct void’s equatorial plane), the 4 vertices
trace out a lemniscate (a figure-8) as the tetrahedron rotates. This projected trajectory is the
geometric origin of the name “figure-8 dark proton.” We use the terms “librating K
4
” and
“figure-8” interchangeably; the former describes the instantaneous geometry and dynamics, the
latter the time-averaged trace.
The structural contrast between the visible and dark K=4 defects is shown in Figure 2.
Three consequences. The librating-tetrahedron picture has three consequences worth stat-
ing explicitly before the structural-properties discussion that follows.
The K=4 label is literal. Both the visible proton and χ
′
instantiate the K
4
bonding graph at
unit edge length L. “K=4 defect class” means the K
4
-at-edge-L structure, realized in different
host voids with different boundary conditions. The subtractive gauging axiom (S4 of §10)
asserts that two K
4
-at-edge-L defects differing only in skew-edge gauging differ in C by exactly
c
skew
K = 36; it applies to a pair of configurations with identical internal bonding graph and
identical instantaneous geometry. The visible proton and χ
′
are such a pair.
The tet-void / oct-void distinction is forced, not chosen. A regular tetrahedron of edge
L can sit statically in a tet void (the 4 bounding vertices are mutually at distance L, so the
tetrahedron locks onto them). A regular tetrahedron of edge L cannot sit statically in an oct
void (no 4-clique of bounding vertices at uniform distance L exists). The defect must therefore
librate if it resides in an oct void. The libration is not an added dynamical assumption; it is
the only way a K
4
-at-edge-L defect can occupy an oct void.
The C
2
matching of §5.3 is now dynamical rather than static. The earlier discussion in §5.3
motivated the figure-8 residence in an oct void by invoking a “topology matches host symmetry”
selection principle (soft point S3 of §10). In the librating-tetrahedron picture, the libration axis
is the oct void’s C
2
axis. The configuration matches host symmetry by sweeping through it,
not by sitting statically aligned to it. This closes most of the S3 gap: the matching is now a
forced geometric consequence of K
4
-at-edge-L residing in an oct void, not a separate selection
principle.
6.2 Three correlated structural properties
The structural distinction between the visible and dark K=4 defects generates three correlated
properties of the figure-8.
(i) Heavier rest mass. m
χ
′
= 1872/1836 · m
p
= 0.9567 GeV. The absence of skew-edge
deduction in the un-gauged mass count yields m
χ
′
= m
p
+ 18.4 MeV. The increment is
exactly the c
skew
K · m
p
/C
p
contribution that the visible proton subtracts via the down
quark’s fractional-charge structure.
(ii) No SU(3) color charge. The visible proton’s three-color structure derives from the three
skew-edge pairs in its cuboctahedral coordination shell, each carrying one color syn-
9
Visible proton
static tetrahedron, tet void
C
p
= 1872 − 36 = 1836
m
p
= 938.3 MeV
charge, color, distinct anti
→
un-gauge
∆C = +36
C
2
libration
Figure-8 dark proton
librating tetrahedron, oct void
projected trace (schematic)
C
χ
′
= 1872
m
χ
′
= 956.7 MeV
neutral, color singlet, self-conjugate
Figure 2: Structural contrast between the visible proton (left) and the figure-8 dark proton
(right). Both are K=4 defects with internal bonding graph K
4
at unit edge length L; at any
instant, a regular tetrahedron with all 6 pairwise distances equal to L. The visible proton’s
tetrahedron is static, locked onto the 4 bounding vertices of a tetrahedral void; the surrounding
cuboctahedral coordination shell carries 3 skew-edge pairs that produce the c
skew
K = 36 syn-
drome deduction in the mass count. The figure-8 dark proton’s tetrahedron librates (rotates
continuously) about a C
2
axis of an octahedral void, shown as the orange dashed line through
the void centroid. No 4-clique of oct-void bounding vertices at uniform distance L exists, so
static locking is geometrically impossible; the tetrahedron must rotate. The solid purple tetra-
hedron is the current libration phase; the faded purple tetrahedron is an earlier phase. The
inset at lower left is a schematic of the projected vertex trajectory in a plane oblique to the
C
2
axis (the lemniscate shape that gives the defect its name); it is the time-averaged trace, not
the instantaneous shape. The figure-8’s bounding shell carries no skew-edge inheritance, so no
c
skew
K deduction applies. The mass-count difference ∆C = 36 corresponds to the absence of
skew-edge structure and yields the mass relation m
χ
′
/m
p
= 1872/1836 = 1.0196.
drome [11]. The librating K
4
averages over the surrounding shell’s orientation, break-
ing the rigid skew-edge handedness that the tet-void-locked proton inherits. No triplet
structure remains: no color charge, no strong coupling, no quark substructure.
(iii) Self-conjugacy (Majorana-type). The skew-edge handedness of the visible proton’s cuboc-
tahedral shell distinguishes proton from antiproton: charge conjugation C flips the hand-
edness of all three skew pairs, mapping one to the other. The figure-8 has no skew-edge
handedness, and is therefore mapped to itself under C by this structural argument alone.
The librating-tetrahedron picture gives a second, complementary derivation. Inversion
through the oct void’s centroid (the symmetry −I ∈ O
h
that the oct void inherits from its
O
h
point group) maps the librating tetrahedron to itself: −I rotates the tetrahedron by
π about any axis through the centroid, and on the libration’s C
2
axis this is exactly a π
rotation that lies inside the libration orbit. The configuration is therefore invariant under
−I, and C (which acts as inversion on the defect’s spatial structure in the framework
of Kulkarni [12]) maps the libration to itself. The figure-8 is its own antiparticle by the
dynamics of the libration, independently of (and in addition to) the absence of skew-edge
handedness. It has no distinct anti-state.
These three properties are not independent. All three follow from the single structural fact
10
that the figure-8 K
4
is un-gauged from the FCC’s skew-edge structure. The visible proton pays
for its color charge and matter–antimatter doubling with a 36/1872 = 1.92% mass reduction;
the figure-8 forfeits both and recovers the full (K+1)K
2
mass count.
6.3 Geometric residence in octahedral voids
The librating-tetrahedron picture makes the question “why oct void rather than tet void”
sharper than it appeared in earlier framings.
A regular tetrahedron of edge L can sit statically in a tet void: the 4 bounding vertices
are mutually at distance L, so the tetrahedron locks onto them and inherits the surrounding
cuboctahedral coordination shell with its skew-edge handedness. This is exactly the visible
proton. A K
4
-at-edge-L defect in a tet void is a visible proton, with charge, color, and a
distinct anti-state.
The defining feature of the figure-8 is the absence of skew-edge gauging: the un-gauged
structure that yields C
χ
′
= 1872. Such an un-gauged K
4
cannot be a tet-void resident, because
the tet-void boundary conditions automatically generate the gauging. The figure-8 must reside
in a host whose boundary conditions do not produce skew-edge handedness. The oct void is
exactly such a host: the K
4
tetrahedron cannot lock onto its 6 bounding vertices (no 4-clique
at uniform distance L), so it librates, and the libration averages over the surrounding shell’s
orientation, breaking the rigid handedness inheritance.
The tet-void and oct-void residences therefore select different gauge structures by their host
geometries:
• Tet void ⇒ static lock ⇒ rigid skew-edge handedness ⇒ visible proton (charged, colored).
• Oct void ⇒ libration ⇒ averaged-out handedness ⇒ figure-8 (neutral, color-singlet).
The defect-class distinction is not a choice; it is a consequence of which interstitial site the K
4
occupies, which in turn is set by the kinematics of the original K=4 → K=12 phase transition
and by the merger geometry of §4.
The geometric size of the two voids is also consistent with this picture. The tetrahedral
void’s inscribed sphere radius is r
T
= 0.225 R ≈ 0.112 L, where R is the FCC sphere radius;
the octahedral void’s inscribed sphere is r
O
= 0.414 R ≈ 0.207 L, roughly twice the linear size
and 6.26× the volume. The oct void provides room for the libration’s full range of orientations,
while the tet void’s narrower interior matches the static lock there.
6.4 Stability: a three-pronged argument
A natural concern about the figure-8 population is whether the residuals themselves annihilate,
χ
′
+ χ
′
→ γγ. The reaction is kinematically allowed (the χ
′
is self-conjugate), and would slowly
deplete the dark proton population if available.
The framework forbids the reaction by three complementary mechanisms.
Geometric isolation. The figure-8 sits inside an oct void. The void is bounded by 6 FCC
vertices and the surrounding cuboctahedral structure of the K=12 bulk. Two figure-8 dark
protons in two different oct voids are separated by the lattice walls. They cannot come into
contact for a χ
′
+χ
′
reaction without traversing the intervening K=12 bulk, which is structurally
prohibitive at the relevant energy scale.
Topological saturation. The K
4
graph of the figure-8 is internally complete: every vertex
bonds to every other vertex (degree 3 for each of 4 vertices, 6 edges total). There are no unused
bonding sites for an external defect to attach to. Compare with the K
2,2,2
graph of the K=6
defect: each vertex bonds to 4 of the other 5 (missing only its antipodal partner), leaving 6
11
unused antipodal-axis bonds available for external extension. The K=6’s “unused capacity”
along antipodal axes is what enables the χ + χ annihilation reaction by allowing the two defects’
bonding graphs to merge across the void boundary. The figure-8 has no such capacity. Even
if two figure-8s were forced into contact (counterfactually, ignoring geometric isolation), the
fully-bonded K
4
has no available stubs for merger.
Libration zero-mode and topological saturation. A potential concern about the librating-
tetrahedron picture is whether the libration itself costs energy. If libration carried a finite fre-
quency, it would contribute kinetic energy to the rest-frame configuration and shift C
χ
′
above
1872.
It does not. The libration is a continuous symmetry direction of K
4
-at-edge-L inside the
oct void: every configuration in the libration orbit is structurally identical, with the same K
4
bonding graph, the same 6 unit edges, the same internal node count, and therefore the same C-
count. There is no restoring force along the libration direction; only perpendicular to it, where
deviations from the unit-edge-length condition or from C
2
-axis alignment would cost energy.
The libration is therefore a Goldstone-like zero mode with vanishing frequency, contributing
zero to the rest-mass C-count.
C
χ
′
= 1872 is the rest-frame value, with libration treated as a zero mode. The figure-8 mass
m
χ
′
= 0.9567 GeV is unaffected by the libration.
This is consistent with the topological-saturation stability mechanism just stated: a con-
figuration whose bonding graph is internally complete (K
4
has no available stubs) and which
possesses a continuous symmetry direction librates freely along that direction without energy
cost. The libration is the geometric realization of topological saturation: saturation says no
further bonds can attach; libration says the saturated configuration explores its symmetry orbit
freely.
These three mechanisms together imply absolute stability of the figure-8 population on
cosmological timescales: the residual count is monotonically non-decreasing over future cosmic
time as K=6 annihilation continues to produce new figure-8s.
6.5 Non-detectability except gravitationally
The figure-8 has no first-order coupling channel to ordinary matter:
• No electromagnetic charge (no skew-edge structure to produce fractional charge);
• No SU(3) color (no triplet structure in the bonding shell);
• No weak charge (no chirality-carrying bond-handedness);
• No external bond stubs available for any other lattice-mediated coupling.
In direct-detection experiments such as SENSEI [14], DAMIC [5], CRESST-III [4], and
SuperCDMS [15] that target sub-GeV dark matter via nuclear or electron scattering, the figure-
8 produces zero signal at all orders. In indirect-detection searches for dark matter annihilation
gamma rays, the figure-8 cannot annihilate (§6.4) and produces no signal. At colliders, the figure-
8 cannot be pair-produced from Standard Model initial states because there is no Standard
Model coupling to it; LHC and future-collider searches in the sub-GeV mass window will find
nothing.
The figure-8 is therefore detection-immune except through its gravitational contribution to
the total dark matter density. This is not a parametric statement (small cross-section that future
improvements might reach) but a structural one: the framework predicts no non-gravitational
signature at any order.
12
7 Excluding Alternative Channels
The previous sections show that χχ → γ + χ
′
is compatible with the framework. We now show
it is unique among the kinematically allowed alternatives.
7.1 Channels excluded by VCF or sector inventory
Three classes of alternatives fall to the conservation laws of §5 alone.
χχ → γ + (K=6) residue. A single K=6 defect as residue carries C = 3364, with 3364
units released as photon energy. But this would require destroying one χ entirely, including its
full K
4
base. VCF forbids this.
χχ → γ + (K=2) residue. The framework has no stable K=2 defect class. Kulkarni [11]
establishes that only the symmetric interstitial sites of FCC, tet and oct voids, host stable defects.
Intermediate-K configurations are asymmetric and not produced by the isotropic kinematics of
the K=4 → K=12 phase transition. A K=2 residue has nowhere to reside.
χχ → γ +(K
4
in tet void). A K
4
residue in a tet void would be structurally a visible proton
(a baryon). The reaction’s initial state contains zero baryonic content. Creating a baryon from
two dark-matter defects violates sector identity. The figure-8’s residence in an oct void preserves
sector identity: dark in, dark out.
7.2 The two-figure-8 channel
One channel survives the constraints above: χχ → 2γ + 2χ
′
, with two figure-8 residues and two
softer photons (∼ 0.76 GeV each). This channel satisfies VCF (two K
4
bases, more than the
minimum of one), satisfies sector identity (both residues in oct voids), is energetically allowed
(2m
χ
− 2m
χ
′
= 1.526 GeV for the photon pair), and has higher final-state C (3744 vs. 1872).
Structural minimality disfavors it but does not strictly forbid it. A stronger argument is needed.
The argument comes from the topology of the merger graph. The compound K
2,2,2
∪K
2,2,2
has 10 vertices and 23 bonds, connected through the shared edge as a unique bridge (§4).
Partitioning this graph into two disjoint K
4
-bonded sub-clusters, one in each original oct void,
requires cutting the shared edge. The shared edge is the bond that brought the two χ defects
into causal contact. The merger cannot simultaneously be initiated by the shared edge and
terminate by destroying it. Equivalently, the C
2
symmetry that the merger geometry imposes
maps the two halves of the compound onto each other; a final state with two separate composites
breaks this symmetry, contradicting the unbroken C
2
of the merger event.
Two figure-8 residues in two separate oct voids would require two distinct merger events
with two separate pair encounters. That is a different reaction at a different rate, namely a
sequence of two χχ → γ + χ
′
reactions, not a single χχ → 2γ + 2χ
′
event with a different
branching. The single-event two-residue channel does not occur.
7.3 Uniqueness
With all alternatives excluded (full annihilation by VCF, K=6 and K=2 and tet-void residues
by VCF or sector identity, and the two-figure-8 channel by merger-graph connectedness), the
channel
χ + χ −→ γ + χ
′
(9)
is the unique sub-Planckian outcome of two K=6 defects in adjacent oct voids.
8 Kinematic Consequences
The reaction is a two-body annihilation between identical Majorana-type particles, producing
a massive residual and a massless photon. In the center-of-momentum frame, the photon and
13
residual emerge back-to-back with equal momentum magnitudes. Standard relativistic two-body
kinematics give
E
γ
=
s − m
2
χ
′
2
√
s
, E
χ
′
=
s + m
2
χ
′
2
√
s
,
√
s = 2m
χ
, (10)
where
√
s is the invariant mass of the two-χ system at low relative velocity (the relevant regime
in galactic halos, where ⟨v⟩ ∼ 10
−3
c). Substituting,
E
γ
= m
χ
−
m
2
χ
′
4m
χ
, E
χ
′
= m
χ
+
m
2
χ
′
4m
χ
. (11)
With m
χ
= 1.7195 GeV and m
χ
′
= 0.9567 GeV,
E
γ
= 1.7195 −
(0.9567)
2
4 · 1.7195
= 1.7195 − 0.1331 = 1.586 GeV, (12)
E
χ
′
= 1.7195 + 0.1331 = 1.853 GeV, (13)
T
χ
′
= E
χ
′
− m
χ
′
= 0.896 GeV, (14)
|p
χ
′
| = |p
γ
| = E
γ
= 1.586 GeV, v
χ
′
= |p
χ
′
|/E
χ
′
= 0.856 c. (15)
Energy-budget self-consistency in C-units. Converting the rest-mass values to C-units
via C/C
p
= m/m
p
provides a consistency check. Input: 2 × 3364 = 6728. Output: 1872
(figure-8 rest mass) plus 4856 (photon energy plus figure-8 kinetic energy, converted at C
p
/m
p
).
The total balances exactly. The framework’s mass-counting principle is conservative under the
derived reaction, even though C itself is not a conserved current (§5).
9 Comparison with the Kang et al. 2026 Detection
9.1 Per-source measurements
Kang et al. [9] report a gamma-ray line near 1.5–1.6 GeV in three high-confidence AGN sources
from a 17-year Fermi-LAT [2] blind scan of bright high-latitude AGN. The per-source data are
reproduced in Table 2.
Table 2: Three Kang et al. [9] AGN sources reporting a 1.5–1.6 GeV spectral line. E
obs
is the
observed Gaussian line centroid; σ
E
is the centroid uncertainty; TS is the test statistic; σ
1
is
the local single-degree-of-freedom significance; z is the source redshift.
Source z E
obs
[GeV] σ
E
[GeV] TS σ
1
Type
4FGL J0250.2−8224 0.830 1.55 0.10 23.03 4.86 BCU blazar
4FGL J2329.7−2118 0.031 1.53 0.09 21.72 4.66 Radio galaxy
4FGL J0749.6+1324 1.050 1.62 0.07 12.92 3.59 Blazar
Joint — — — 57.77 7.05 —
9.2 Rejection of the AGN-rest-frame interpretation
If the line originates in the source rest frame (for example, from dark matter annihilation in a
Gondolo–Silk spike [8] around each AGN’s central black hole), the rest-frame line energies are
E
rest
= E
obs
(1 + z), giving 1.58, 2.84, and 3.32 GeV for the three sources. A weighted χ
2
test
against a single common rest-frame energy yields
χ
2
source-frame
= 118.13, dof = 2, p ≪ 10
−6
. (16)
14
The source-rest-frame interpretation is rejected at extreme significance.
The signal must therefore originate at fixed observed energy in our local frame: in the Milky
Way halo along the line of sight to each AGN, or in intervening cold dark matter structure not
subject to host-frame cosmological redshift. The AGN serve as bright continuum backlights
against which the foreground line becomes detectable.
9.3 Comparison with the framework prediction
Under the local-frame interpretation, the three observed energies are mutually consistent with
a single value:
¯
E
obs
= 1.578 ± 0.048 GeV, χ
2
= 0.72, p = 0.70. (17)
The framework’s kinematic prediction is E
γ
= 1.586 GeV from equation (12). The deviation
from the central observed value is
E
SSM
γ
−
¯
E
obs
σ
¯
E
=
1.586 − 1.578
0.048
= 0.17σ, (18)
well within statistical tolerance. By contrast, the naive χχ → γγ channel would predict E
γ
=
m
χ
= 1.7195 GeV, sitting at (1.7195 − 1.578)/0.048 = 2.95σ from the centroid. The γ + χ
′
channel agrees with the measured value without introducing free parameters.
9.4 Caveats
The match should be interpreted carefully on two counts.
Look-elsewhere effects. Kang et al. [9] scanned 2004 AGN over a broad energy range (∼ 25
effective line-width bins per source for a ∼ 10% Gaussian width on a log-energy axis spanning
100 MeV–1 TeV), giving an effective trials count ∼ 5 × 10
4
. The maximum local significance
(4.86σ in J0250.2−8224, p
local
= 5.9 ×10
−7
) corresponds to a global p-value of ∼0.03, or ∼1.9σ
globally. Individual sources are not, on their own, robust detections. The strength of the result
rests on the spatial coincidence: three sources with detections at the same observed energy,
with joint TS = 57.77 that cannot be replicated by trials. This is consistent with one common
physical mechanism in the local frame, and inconsistent with three independent statistical flukes
at the same energy.
Precedent. The most direct historical precedent for a Fermi-LAT gamma-ray line claim at
comparable per-source significance is the ∼ 130 GeV feature identified in 2012, which generated
extensive theoretical follow-up before being substantially attenuated by additional data and
reprocessed instrument response functions [7]. The Kang detection is at a different energy and
in a different source class (extragalactic AGN rather than Galactic Center), so the instrumental
story is not the same. But the base rate for line claims at ∼3–4σ per source surviving further
data is poor, and confirmation by an independent analysis or instrument is required before the
present agreement carries decisive weight. What the analysis above does establish is that the
framework’s forward prediction (derived from the structural-counting expansion and the metric-
wall confinement, with no cosmological input) is consistent with the principal observational
anchor available.
10 Predictions, Falsifiers, and Soft Points
10.1 Predictions
P1: Single line, not doublet. The framework predicts a single line at 1.586 GeV. The two-
figure-8 channel would have produced two softer photons at ∼ 0.76 GeV each; §7 excludes this
15
channel by merger-graph connectedness. No coincident ∼0.76 GeV companion line is expected.
P2: Figure-8 stability and undetectability. The figure-8 χ
′
at 0.9567 GeV is self-
conjugate, geometrically isolated in oct voids, and topologically saturated against further anni-
hilation. It contributes to the dark matter density but couples to ordinary matter only gravita-
tionally.
P3: The channel is forced. Under VCF, sector identity, and energy conservation, no
other sub-Planckian annihilation channel is available to two K=6 defects in adjacent oct voids.
Any future detection of dark matter annihilation gamma rays in this mass range, in the SSM
framework, must reflect this channel.
10.2 Falsifiers
F1: Coincident line at ∼ 0.76 GeV. A future joint analysis of Fermi-LAT, DAMPE [6], or
HESS data finding a second line at ∼0.76 GeV in the same AGN sample (or in stacked galactic-
halo searches), co-occurring with the 1.586 GeV line, falsifies the merger-graph connectedness
argument of §7.
F2: Off-prediction line position. A confirmed line in the same AGN sample at any
energy other than 1.586 GeV (within the Fermi-LAT ∼5–7% systematic at 1.5 GeV) falsifies
the derivation. No free parameters were fit, so the prediction is sharp.
F3: Line in source rest frame. The line must be observed at fixed observed energy
in the local frame, with no (1 + z) shift across sources. §9 rejected the source-rest-frame
interpretation at χ
2
= 118, p ≪ 10
−6
across the three Kang AGN. Future detections must
satisfy this constraint.
F4: Direct detection of χ
′
. The figure-8 at 0.9567 GeV has no first-order coupling to
ordinary matter (no EM, no SU(3) color, no chirality). Any direct detection of a dark matter
particle near this mass with non-gravitational coupling falsifies the framework, because the
figure-8 K
4
provides no external bond stubs for any mediated interaction.
F5: Figure-8 self-annihilation. The framework predicts that two figure-8s cannot anni-
hilate with each other on cosmological timescales: geometric isolation in separate oct voids plus
topological saturation (no external bond stubs). A detection of a γγ line at the self-conjugate
energy 0.9567 GeV in any astrophysical source would falsify the stability mechanism of §6.4.
10.3 Soft points
Five places in the derivation rest on framework principles that are plausible and consistent with
the rest of the SSM but not formally established. They are flagged inline at the point each
is invoked; we consolidate them here so each can be tightened in future work. The librating-
tetrahedron picture of §6.1 tightens S3 and S4 noticeably; the wording below reflects that.
S1: Structural minimality as a selection rule. §5.3 invokes minimization of final-state
C as the selection rule between competing allowed outcomes. The framework has not formally
established that reactions select lowest-C final states. In standard physics, branching fractions
are determined by phase space and matrix elements; exothermic reactions do not always reach
the ground state on every event. The argument here rests on the absence of an intermediate
metastable state between the figure-8 and the next-heavier final state (S2 below), but a complete
derivation requires a cross-section calculation that the framework has not yet developed.
S2: Discrete spectrum of allowed configurations. §5.3 implicitly assumes no K
4
-in-oct-
void configuration exists with C between 1872 and the next-heavier candidate. The 25-candidate
enumeration of Kulkarni [10] covers the first coordination shell only. Second-shell extensions
could in principle produce intermediate defects. None are known; the absence is not formally
16
established. Extending the enumeration to second-shell K
4
-in-oct-void configurations is a finite
calculation that should be performed.
S3: Topology matches host symmetry. The original derivation invoked a “topology
matches host symmetry” selection rule to argue that the figure-8 K
4
must reside in a host
region with C
2
symmetry. The librating-tetrahedron picture of §6.1 closes most of this gap: a
K
4
-at-edge-L defect cannot sit statically inside an oct void (no 4-clique of bounding vertices
at uniform distance L exists), so it must librate, and the libration axis necessarily coincides
with one of the oct void’s three C
2
axes. The symmetry matching is now a forced geometric
consequence of K
4
-at-edge-L residing in an oct void rather than a separate selection principle.
The residual gap is narrower than the original S3 but not zero. The oct void offers three
perpendicular C
2
axes through its antipodal vertex pairs, and the libration must align with one
of them. The framework has not formally established which C
2
axis is selected in any given
merger event, nor whether all three axes are populated equally across a cosmological population
of figure-8s. By the FCC translational and rotational symmetry of the vacuum, the three choices
are physically equivalent (a figure-8 librating about any one of the three axes has the same mass
and the same observable signatures), so the question is one of microscopic dynamics rather than
of any observational consequence. We flag it here for completeness.
S4: Subtractive gauging axiom. §5.4 derives C
χ
′
= 1872 from C
p
= 1836 by subtracting
the proton’s c
skew
K = 36 deduction. This treats the visible proton and the figure-8 as two
instances of the same K
4
-at-edge-L defect class with and without cuboctahedral gauging, and
asserts that the C-difference between gauged and ungauged variants is exactly c
skew
K.
The librating-tetrahedron picture of §6.1 brings the two configurations into sharper compar-
ison. Both are regular tetrahedra of edge L with bonding graph K
4
. At any instant during the
figure-8’s libration, the instantaneous geometry is identical to a snapshot of the visible proton’s
tetrahedron: same 4 nodes, same 6 unit edges, same K
4
structure. The two configurations differ
only in (i) their host void and (ii) the resulting boundary conditions: the visible proton’s tet-
void residence locks the tetrahedron onto 4 bounding vertices, embedding it in a cuboctahedral
shell with skew-edge structure; the figure-8’s oct-void residence requires libration and provides
no skew-edge inheritance.
Under the librating-tetrahedron picture, S4 reduces to the claim that the C-cost of the
cuboctahedral skew-edge gauging the visible proton inherits is exactly c
skew
K = 36 units, and
that the absence of this gauging accounts for the entire C-difference between the two configura-
tions. Every defect count in the framework is compatible with this rule, including the electron
(C
e
= 1, un-gauged single-node defect), the muon (C
µ
= 207, un-gauged 3-sheet defect with
kinematic shedding), the pion (C
π
= 273, gauged 2-sheet defect with boundary closure), the pro-
ton (gauged K
4
, C
p
= 1836), the neutron (C
n
= 1839, gauged K
4
with neutral-baryon probe),
and the dark matter K=6 defect (C
DM
= 3364). It has not, however, been derived from exist-
ing framework material. An independent first-principles count of the figure-8’s structural-node
geometry, analogous to the 13-node derivation for the visible proton in Kulkarni [11], would
close this gap.
Until that independent count is performed, C
χ
′
= 1872 is rigorous relative to C
p
= 1836
(the two are constrained to differ by exactly c
skew
K under the librating-tetrahedron picture),
but not in an absolute sense. S4 remains the most consequential of the five soft points: it is
the place where the figure-8 mass is currently established by analogy with the visible proton
rather than by independent count. The librating-tetrahedron picture tightens the analogy but
does not eliminate the need for the independent count.
S5: Internal excited states and oct-void localization. The C-count of 1872 corresponds
to the figure-8’s ground-state structural configuration with libration treated as a zero mode.
17
The framework has not enumerated possible internal excited states of the figure-8 (oscillations
perpendicular to the libration axis, breathing modes of the unit-edge-length constraint, or higher-
order coordination shells) that could carry slightly higher effective masses, by analogy with
excited states of atomic systems. If such states exist, they would decay back to the ground
state by emitting bulk modes, producing additional spectral features below the 1.586 GeV line.
Relatedly, the localization of the relaxed composite in one of the two host oct voids (§5.3) is
by spontaneous symmetry breaking of the merger’s C
2
. The choice between voids A and B
is physically equivalent by FCC translational symmetry, but the framework has not formally
established the relaxation dynamics that select one void over the other in any given event.
Summary. None of these soft points is independently dispositive. An honest reading of the
present derivation is that it forces χχ → γ + χ
′
as the unique channel modulo five plausible
selection principles. The librating-tetrahedron picture of §6.1 tightens S3 and S4 noticeably but
does not eliminate either. Closing any of them tightens the derivation. Closing all five makes
it tight.
11 Conclusion
The dark matter annihilation channel χχ → γ + χ
′
is forced by framework primitives. The
derivation can be summarized as follows.
The Verification-Cost Floor, on which the entire uniqueness argument rests, is a corollary of
the metric-wall confinement of Kulkarni [11], applied to K-class trapped nodes and read inward
against dissolution.
Full annihilation χχ → γγ violates VCF: both K
4
bases would have to dissolve into the bulk
against a ∼ 10
15
GeV barrier with only 1.7195 GeV available per defect.
The merger geometry is fixed by FCC crystallography. Nearest-neighbor oct voids share an
octahedron edge, verified at 450/450 pairs on a 4
3
supercell. The shared edge is the unique
causal interface and carries a natural C
2
symmetry axis.
The figure-8 mass C
χ
′
= 1872 emerges as the unique minimum-C final state compatible with
energy conservation, VCF, sector identity, and oct-void symmetry. The factor (K+1)K
2
= 1872
comes from the un-gauged K
4
base count, established here via the librating-tetrahedron picture
(§6.1): χ
′
and the visible proton are both regular tetrahedra of edge L with bonding graph K
4
,
differing only in host void (oct vs. tet) and the resulting boundary conditions. The tet void’s
static lock generates skew-edge gauging and a 36-unit deduction; the oct void’s forced libration
averages over the surrounding shell and recovers the full un-gauged count.
Alternative residues are excluded by VCF (full annihilation, K=6 and K=2 residues) or by
sector identity (K
4
in tet void). The two-figure-8 channel is excluded by the connectedness of
the merger graph through its single shared edge.
Two-body kinematics then place the photon line at 1.586 GeV. The Kang detection at
1.578 ± 0.048 GeV agrees at 0.17σ. No free parameters were fit.
Five soft points remain (S1–S5); the librating-tetrahedron picture tightens S3 (oct-void C
2
matching is now a forced consequence of the no-4-clique fact) and S4 (the visible-proton/figure-
8 pair has identical instantaneous geometry, sharpening the subtractive gauging axiom), but
neither is closed entirely. S4 remains the most consequential: an independent first-principles
count of the figure-8’s structural-node geometry is the calculation that would close it.
The cosmological abundance of either the primary χ component or the secondary χ
′
compo-
nent is not addressed: the SSM analog of the baryon asymmetry parameter η
B
is not derived
in the framework, and no rigorous abundance prediction is available without it. The forward
mass prediction is independent of this calculation, and the principal predictions, falsifiers, and
soft points stand on the derivation above.
18
Declaration of Competing Interest
The author declares no known competing financial interests or personal relationships that could
have appeared to influence the work reported in this paper.
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