A 1.59 GeV Gamma-Ray Line from Dark Matter
Semi-Annihilation on the FCC Lattice
Raghu Kulkarni
∗
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
Draft manuscript — revised May 2026
Abstract
The Selection-Stitch Model identifies dark matter with a K=6 octahedral defect of mass
m
χ
= 1.719 GeV trapped in the octahedral voids of an FCC vacuum lattice [6]. The Kang
et al. [3] report of a 1.5–1.6 GeV gamma-ray line in three active galactic nuclei lands ∼9%
below the 1.719 GeV that a direct χχ → γγ channel would predict. We resolve this by
deriving the annihilation channel from the lattice geometry. The channel is χ + χ → γ + χ
′
,
a semi-annihilation in which χ
′
is a massive dark residual. Three results fix it, each verified
by direct enumeration on the FCC lattice. (i) Two nearest-neighbor octahedral voids share
exactly one octahedron edge (450/450 pairs on a 4
3
supercell); this shared edge is the unique
causal interface for a merger. (ii) Under the stability rule that a trapped node must be
equidistant from its bounding vertices at a radius above the metric wall L/
√
3, the only closed
cages available are the octahedron and the regular tetrahedron; an octahedral residual is simply
another χ (no line shift), so the residual must be tetrahedral, and the relevant tetrahedral void
is reachable from the shared edge across a triangular face shared with each parent octahedron ,
a path that never crosses K=12 bulk and so never invokes the metric wall. (iii) The tetrahedral
void has full T
d
site symmetry with all four vertices in one orbit, so it admits an anchor-free,
fully S
4
-symmetric occupant: neutral, colorless, and baryon-number-free by symmetry, and
distinct from the (anchor-selected) proton and neutron. This symmetric occupant is a stable
local minimum of the bond-strain energy (positive-definite Hessian; anchor-selecting and face-
escape distortions both raise the energy). The visible baryons are the anchor-selected states of
the same cage, in which one bond is singled out as the bulk-coupling channel, breaking S
4
→ S
3
and generating charge and color through the construction of the companion matter paper [5];
the residual χ
′
is the branch in which no anchor is selected. Its neutrality follows from the
symmetric configuration: the four bond vectors sum to zero, the bulk coupling lies entirely
in the S
4
singlet, and the symmetric superposition is the unique ground state. Its disruption
count is the bare tetrahedral-cage count (K+1)K
2
− c
skew
K = 1872 − 36 = 1836, computed
from the cage directly, giving m
χ
′
= m
p
= 0.938 GeV with no subtraction from the proton.
Two-body kinematics then place the photon line at 1.591 GeV, in 0.27σ agreement with the
Kang centroid 1.578±0.048 GeV. No free parameters are fit. The line prediction depends only
on the two masses and the channel geometry; the visible sector’s fractional charge values, the
program-level value of the lattice scale L, and the compact group su(3) are not derived and
are stated as explicit limitations.
∗
raghu@idrive.com
1
1 Introduction
The Selection-Stitch Model (SSM) treats the physical vacuum as a Face-Centered Cubic crystal-
lization of spacetime [4, 5]. Baryonic matter is a K=4 defect trapped at a tetrahedral interstitial
site of the K=12 FCC bulk [5]; dark matter is a K=6 defect trapped at the octahedral site [6].
Throughout, K denotes both a defect-type label (K=4, K=6) and the bulk coordination number
(K=12) inside the structural-counting formula (K+1)K
2
; context disambiguates.
Two parameter-free results from the companion papers anchor the present work:
m
p
m
e
= (K+1)K
2
−c
skew
K = 13 ·144 −3 ·12 = 1836, m
χ
=
C
DM
C
p
m
p
=
3364
1836
m
p
= 1.719 GeV,
(1)
with K=12. The primary dark-matter mass m
χ
is inherited as an input; it is not re-derived here.
A recent Fermi-LAT analysis [3] reports a 1.5–1.6 GeV line in three high-significance AGN,
joint TS = 57.77, per-source centroids clustering at 1.578 ±0.048 GeV. A direct χχ → γγ channel
predicts E
γ
= m
χ
= 1.719 GeV, ∼2.9σ above the centroid. We resolve the tension by deriving
the annihilation channel from the lattice geometry:
χ + χ −→ γ + χ
′
, (2)
where χ
′
is a stable, neutral, baryon-number-free tetrahedral residual (Figure 1).
+ (two
K
= 6 voids)
merger at shared edge
+
0
(single
K
= 4 residual)
+ +
0
: two
K
= 6 octahedral voids sharing an edge merge into one
K
= 4 tetrahedral residual
0
at the interface, plus a 1.591 GeV photon.
Figure 1: The annihilation χ + χ → γ + χ
′
. Two K=6 octahedral voids (the dark-matter particle
χ) share one octahedron edge (green); on merger they form a single K=4 tetrahedral residual χ
′
,
hinged on that edge, plus a photon. An interactive 3D version is available at https://raghu91302.
github.io/ssmtheory/dark_proton_annihilation.html.
The earlier draft of this paper assigned χ
′
a mass of 1872 (0.957 GeV) by adding back the
proton’s skew-edge deduction. That step is withdrawn: as shown below, the residual is the
occupant of a real tetrahedral cage, which carries its skew-edge structure geometrically, so its honest
disruption count is the bare cage count 1872 −36 = 1836 and its mass is m
χ
′
= m
p
= 0.938 GeV.
The number is obtained positively from the cage, with the proton appearing nowhere in the
derivation; the two objects are mass-degenerate because they share the same cage, and differ only
by symmetry.
Plan. Section 2 states the Verification-Cost Floor (VCF) that forbids full annihilation. Section 3
establishes the shared-edge merger interface and the face-adjacency of the tetrahedral residual site.
2
Section 4 proves, by enumeration, that only two stable cages exist, that the residual is therefore
tetrahedral, and that the symmetric (anchor-free) occupant is stable. Section 5 gives the positive
mass count. Section 6 collects the residual’s structural properties. Section 7 works out the
kinematics; Section 9 compares with Kang. Section 10 lists predictions, falsifiers, and the three
stated assumptions.
2 The Verification-Cost Floor
Kulkarni [5, §6] shows that a trapped node cannot escape its void at sub-Planckian energies:
extracting it requires stretching its bonds against the restoring force of the surrounding K=12
shell, giving a linear confining potential V (r) = σ
lat
r with σ
lat
∼ ε/L
2
. With L ∼ ℓ
P
and ε
at the GUT scale, σ
lat
· L ∼ 10
15
GeV. The same barrier obstructs the inverse: a trapped node
cannot dissolve into bulk K=12 coordination, because dissolution traces the same centroid-to-bulk
trajectory as escape.
Corollary 1 (Verification-Cost Floor). A trapped-node base produced by the K=4 → K=12
phase transition cannot be dissolved into bulk at sub-Planckian energies. A reaction may rearrange
a base (one defect may split into two if supplied energy exceeds σ
lat
L), but it cannot reduce the
base count to zero.
The channel χχ → γγ requires both K=6 bases to dissolve entirely into bulk, against a ∼10
15
GeV
barrier per defect with only 1.719 GeV available. It is forbidden. The observed line must reflect
a channel whose final state retains a trapped base. The strength of this conclusion is tied to
L ∼ ℓ
P
; we return to that dependence in Section 10.
3 The Merger Interface and the Residual Site
3.1 Nearest-neighbor octahedral voids share an edge
The FCC unit cell (L = a/
√
2) contains four octahedral voids, each bounded by six FCC vertices
forming a regular octahedron of edge L. A merger between two K=6 defects requires causal
contact, which the bulk forbids except through shared bounding geometry. Direct enumeration
on a 4
3
supercell (108 fully-interior oct voids) gives the separation spectrum in Table 1: every one
of the 450 nearest-neighbor pairs shares exactly two bounding vertices, and those two vertices are
separated by L, an octahedron edge rather than the L
√
2 antipodal diagonal.
Table 1: Oct-void pair separations in a 4
3
FCC supercell (companion script).
Separation d/L Pairs Shared bounding vertices
1 (nearest) 450 2 (one octahedron edge, length L)
√
2 216 1
√
3 600 0
2 288 0
The shared edge (two vertices joined by one bond) is the unique causal interface for a merger.
It is a bridge in the compound graph K
2,2,2
∪K
2,2,2
(removing it disconnects the two defects) and
carries a C
2
reflection about its midpoint.
3
shared edge
oct void A ( )
oct void B ( )
tet void
Two nearest-neighbor octahedral voids share one edge
Figure 2: The shared octahedron edge between two nearest-neighbor oct voids. The two shared
vertices (separation L) form the unique causal interface. The two regular tetrahedral voids that
flank this edge (Section 4) each share a triangular face with both octahedra.
4
3.2 The residual site is face-adjacent, not bulk-separated
The residual forms on this interface. The question of where it relaxes is settled in Section 4: the
only stable cage compatible with a line shift is a regular tetrahedron, and the relevant tetrahe-
dral voids are the two that flank the shared edge, centered (in the supercell coordinates of the
companion script) at (
1
2
,
3
2
,
1
2
) and (
1
2
,
3
2
,
3
2
). Direct enumeration of the triangular faces of the two
parent octahedra and of these tetrahedra shows that each flanking tetrahedral void shares a full
triangular face with each parent octahedron, and every such shared face contains both vertices of
the merger edge. The residual therefore reaches its tetrahedral site by sliding across a shared
triangular face, a two-dimensional interface, and never traverses three-dimensional K=12 bulk.
The metric wall, which obstructs bulk traversal, is not invoked. This removes the apparent ob-
jection that a residual “born between two octahedra” has no business in a tetrahedral void: the
tetrahedral cage is the completion of the merger edge by a single additional vertex across a face
common to both reactants.
4 Only Two Stable Cages; the Residual Is a Stable Anchor-Free
Tetrahedron
4.1 Enumeration of stable cages
A trapped node is stable only if it sits at a position equidistant from its bounding vertices (a unique
strain-balanced circumcenter exists), at a common radius above the metric wall L/
√
3 ≈ 0.577 L.
Enumerating all connected vertex subsets at the two-octahedron interface and retaining those
that (a) admit a unique equidistant center, (b) have radius ≥ L/
√
3, (c) are strain-balanced (
P
unit bond vectors = 0), and (d) are non-coplanar (a coplanar set leaves the node free to slide
normal to the plane, so it is not a trap) yields exactly two cage types:
Result. The only stable equidistant cages at the interface are the regular tetrahedron (4 vertices,
radius 0.612 L) and the regular octahedron (6 vertices, radius 0.707 L). No third cage survives
the constraints.
An octahedral residual is itself a K=6 defect (another χ) and carries no mass change, predicting
the line at m
χ
= 1.719 GeV, which the data exclude. The residual must therefore be tetrahedral.
Combined with Section 3.2, the tetrahedral residual is reachable from the merger edge without
crossing bulk.
4.2 The tetrahedral void admits an anchor-free occupant
In Kulkarni [5], every baryon (∆
−
, n, p, ∆
++
) is built by selecting an anchor: one of the four
centroid-to-vertex bonds becomes the bulk-coupling junction, breaking the tetrahedron’s S
4
sym-
metry to S
3
and generating the 1+3 valence split that carries electric charge, color, and baryon
number. We ask whether the lattice forces an anchor. It does not. Enumerating the cubic point-
group operations that fix the tetrahedral void center and preserve both the tetrahedron and the
surrounding FCC lattice returns 24 operations realizing the full S
4
on the four vertices, with all
four vertices in a single orbit. The tetrahedral interstitial site has full T
d
site symmetry: no vertex
is geometrically distinguished, so a fully S
4
-symmetric, anchor-free occupant is admissible. By
the same symmetry the three skew-edge (color) pairs lie in one orbit, so a symmetric occupant
assigns them equivalently, color-neutral by symmetry rather than by tuning.
This anchor-free occupant is the residual χ
′
. Because it selects no anchor, it has no valence
split (no fractional charge; net charge zero by symmetry), no color triplet (net color zero by
5
symmetry), and no anchor junction to carry baryon number. It is distinct from the neutron,
which is the anchor-selected, baryon-number-carrying W =1 state.
4.3 Stability of the symmetric occupant
A highly symmetric defect can be unstable to a symmetry-lowering (Jahn–Teller) distortion, here
the spontaneous anchor selection that would turn χ
′
into a baryon. We test this in the framework’s
native energetics: model each of the four bonds as a harmonic spring of natural length equal to
the equilibrium radius r
0
, so the trapped-node energy is E(x) =
1
2
P
i
(|x −V
i
|−r
0
)
2
. The Hessian
at the symmetric center is isotropic and positive definite, with all three eigenvalues equal to +
4
3
;
there is no soft mode. Displacing the node toward one vertex (the anchor-selecting distortion) or
toward a triangular face (the escape distortion) raises the energy monotonically. The symmetric
occupant is a stable local minimum.
Result. In the SSM bond-strain model, the anchor-free tetrahedral occupant is a stable local
minimum (Hessian eigenvalues +
4
3
; anchor and face distortions both raise E). It does not spon-
taneously relax to a baryon.
A single node bonded to four springs has a non-degenerate, s-like ground state, so the Jahn–
Teller theorem does not force a distortion; geometrically, moving toward any one vertex stretches
the other three, and the three losers outvote the one winner. We note the converse implication:
in this same bond-strain model the symmetric occupant lies below the anchor-selected (baryon)
configuration, so the persistence of baryons as tetrahedral occupants is not an equilibrium effect of
the isolated cage but is set by the bulk-coupling/anchor-selection construction of the companion
papers (Section 4.4).
4.4 The residual is the symmetric (anchor-free) occupant, derived
The residual χ
′
is the fully S
4
-symmetric occupant of the tetrahedral cage: the state in which
no bond is singled out. We can now say more than “neutral by symmetry.” The four centroid-
to-vertex bonds carry the permutation representation of S
4
, which decomposes canonically and
without any choice as
4 = 1 (trivial) ⊕ 3 (standard). (3)
Three facts about the symmetric state follow by direct computation and together establish that
χ
′
is neutral, isotropic, and stable without invoking any anchor.
(i) No spatial direction. The four unit bond vectors sum exactly to zero, ˆr
A
+ ˆr
B
+ ˆr
C
+ ˆr
D
= 0,
so the symmetric occupant has no preferred axis; the trivial sub-representation is isotropic. (This
is also why no spatial pinning or “anchor” direction can be singled out: there is none to single
out.)
(ii) The bulk coupling lives entirely in the singlet. Embedding the void in the FCC bulk,
each bounding vertex has exactly nine bulk neighbors (its twelve FCC neighbors minus the three
other bounding vertices), so the coupling-strength vector over the four bonds is (9, 9, 9, 9). De-
composed into S
4
irreducibles this is pure trivial-representation, with identically zero standard-
representation component: the bulk coupling is carried by the singlet alone. The “anchor versus
valence” division of labor that the matter sector assigns by hand is, for the symmetric state, the
representation-theoretic split of Eq. (3), not a chosen bond.
(iii) The symmetric state is the ground state. A four-bond hopping model with equal inter-
bond amplitude has a unique ground state, the uniform superposition (the singlet), separated
from the threefold-degenerate excited combinations; any localized, single-bond (“anchor”) state
6
is not an energy eigenstate. The symmetric occupant is therefore the physically stable object,
consistent with the bond-strain stability of Section 4.3.
Hence χ
′
is neutral (net charge zero as the singlet weight plus traceless part, 1 + 3(−
1
3
) = 0),
colorless, and baryon-number-free, with these properties derived from the symmetric configuration
rather than assumed. The visible baryons of Kulkarni [5], by contrast, are the charged states,
which require the S
4
→ S
3
breaking that produces a distinguished bond. That breaking, and the
values of the resulting fractional charges, are not derived in the companion papers or here; they
are the open problem of the matter sector (Section 10.1). The present paper requires only the
symmetric branch, whose neutrality and stability are established above and which is all that the
annihilation residual needs to be.
5 The Residual Mass, Counted from the Cage
5.1 What is conserved, and what is not
Three quantities constrain any sub-Planckian reaction in the framework. Energy is conserved in
the usual relativistic sense. The K=4 base content is conserved in the VCF sense of Corollary 1:
a base cannot dissolve to zero, so the final state must retain at least one trapped base. Sector
identity is preserved: a residue formed from oct-void inputs cannot relocate to a tet void by
crossing bulk. As Section 3.2 showed, the face-adjacent tet site is reachable without any bulk
crossing, so sector identity is respected here.
The total verification cost C is not a conserved current, and saying so explicitly avoids a
common misreading. C is a structural property of a given configuration (the bond-state disruption
it imposes on the surrounding lattice), so different configurations carry different C, and a reaction
that changes the configuration changes the total C. What couples C to a conserved quantity is
energy, through m = C kT ln 2/c
2
[4]: proportional, but not identical, and the proportionality
decouples whenever a defect-free product (a photon, C
γ
= 0) carries energy away. In the present
reaction C
initial
= 2 × 3364 = 6728 while C
final
= 1836 (the residual alone). The difference is
not transported anywhere; it is structural cost released into the bulk as the final configuration
disrupts the lattice less than the initial one, and the corresponding energy appears as the photon
and the residual’s recoil. A naive attempt to “balance a C-budget” across the reaction is therefore
a category error; only energy balances. With that understood, the residual’s mass is fixed not
by a C-balance but by the C-value of the configuration the reaction actually produces, which
Section 4 identified as the anchor-free tetrahedral occupant.
5.2 The mass count
The residual occupies a regular tetrahedral cage. Its structural-disruption count is therefore the
tetrahedral-cage count computed directly in Kulkarni [5, §7]:
C
χ
′
= (K+1)K
2
− c
skew
K = 13 · 144 −3 · 12 = 1872 −36 = 1836. (4)
Both terms are properties of the cage. The base (K+1)K
2
= 1872 counts the 13 structural nodes
of the trapped-tetrahedron defect, each disrupting K
2
= 144 second-neighbor bond states. The
deduction −c
skew
K = −36 is the double-count along the three skew-edge pairs; the tetrahedron
possesses these three pairs as a matter of geometry (the complete graph K
4
has exactly C(4, 2)/2 =
3 skew pairs), present whether or not an anchor is selected. Equation (4) is thus a direct count
of the cage, not a modification of the proton. The visible proton equals 1836 for the same reason:
it is the same cage. The two objects are therefore mass-degenerate; they are distinguished by
symmetry (anchor vs. no anchor), not by mass. Via the mass-energy-information correspondence
7
[4],
m
χ
′
=
C
χ
′
C
p
m
p
=
1836
1836
m
p
= m
p
= 0.938 GeV. (5)
Remark on the withdrawn 1872. A residual mass of 1872 (0.957 GeV) would require a
tetrahedral base without the skew-edge deduction, an “un-gauged” K
4
that does not couple to a
surrounding shell. The only such object is a self-bonded cluster with no surrounding lattice to
disrupt; but in the SSM, mass is surrounding-lattice disruption, so such a cluster carries a count
of order its own internal bonds (∼ 10), not 1872. A nucleon-scale mass and the absence of the
skew deduction are mutually exclusive. The honest tetrahedral residual carries the skew structure
and counts 1836.
6 Structural Properties of the Residual
All of the residual’s qualitative properties follow from its being the anchor-free S
4
-symmetric
occupant of a tetrahedral cage.
(i) Mass. m
χ
′
= m
p
= 0.938 GeV, mass-degenerate with the nucleon by shared cage (Eq. 5).
(ii) No electric charge. No anchor selection means no valence split and no fractional charge; net
charge is zero by S
4
symmetry.
(iii) No SU(3) color. The three skew-edge pairs are present but, lying in one T
d
orbit, are assigned
equivalently; net color is zero by symmetry. The residual does not hadronize.
(iv) Self-conjugacy (Majorana-type). Charge conjugation flips the handedness of the skew struc-
ture; with no anchor to break the symmetry, the residual maps to itself. It is its own
antiparticle.
(v) No external coupling. With no anchor junction and no net gauge charge, the residual has
no first-order coupling to ordinary matter: no electromagnetic, color, or weak vertex. It is
detectable only gravitationally.
Baryon (proton / neutron)
S
4
S
3
: anchor selected, charge·color·B = 1
Dark residual
0
S
4
preserved: neutral, colorless, B = 0
Same regular-tetrahedral cage; skew-edge (color) pairs present in both. Only the four center bonds differ: one promoted to an anchor (left) vs all equal (right).
Figure 3: The visible proton and the dark residual are the same regular-tetrahedral cage distin-
guished only by anchor selection. Selecting an anchor (left) breaks S
4
to S
3
and produces a baryon;
the anchor-free occupant (right) is the stable, neutral residual χ
′
.
8
Stability of the residual population. Two residuals in distinct tetrahedral voids are sepa-
rated by the metric wall and cannot contact for a χ
′
χ
′
reaction without traversing bulk. The
residual population is therefore non-depleting on cosmological timescales: it accumulates as χχ
annihilation continues.
7 Kinematics
The reaction is a two-body annihilation of identical Majorana-type particles into a massive residual
and a massless photon. In the center-of-momentum frame at low relative velocity (⟨v⟩ ∼ 10
−3
c in
galactic haloes), with
√
s = 2m
χ
,
E
γ
= m
χ
−
m
2
χ
′
4m
χ
, E
χ
′
= m
χ
+
m
2
χ
′
4m
χ
. (6)
With m
χ
= 1.719 GeV and m
χ
′
= 0.938 GeV,
E
γ
= 1.719 −
(0.938)
2
4 ·1.719
= 1.719 −0.128 = 1.591 GeV, (7)
E
χ
′
= 1.847 GeV, T
χ
′
= 0.909 GeV, v
χ
′
= 0.861 c. (8)
Energy balances: 2m
χ
= 3.438 = m
χ
′
+ T
χ
′
+ E
γ
.
Self-consistency in C-units. Converting the balance to C-units via C/C
p
= m/m
p
provides
a check. The input is 2 ×3364 = 6728; the output is the residual rest mass 1836 plus the photon
energy and residual kinetic energy, which convert to 4892, summing to 6728. The framework’s
mass-counting principle is conservative under the derived reaction even though C is not a con-
served current (Section 5.1): only the energy-weighted combination balances.
9
0.938
(27%)
0.909
(26%)
1.591
(46%)
= 2
m
= 3.438 GeV
Energy budget of + +
0
(GeV)
residual rest mass
m
0
residual kinetic
T
0
photon
E
Figure 4: Energy budget of χ + χ → γ + χ
′
. Input rest-mass energy 2m
χ
= 3.438 GeV partitions
into the residual rest mass (27%), residual kinetic energy (26%), and the photon (46%). All values
follow from two-body kinematics once m
χ
and m
χ
′
are fixed.
8 Uniqueness of the Channel
The channel χχ → γ + χ
′
is not merely compatible with the framework; it is the unique sub-
Planckian outcome of two adjacent K=6 defects. The alternatives fall to the conservation laws
of Section 5.1 or to merger-graph topology.
χχ → γγ. Forbidden by VCF (Section 2): both K=6 bases would have to dissolve into bulk
against the ∼10
15
GeV metric-wall barrier.
χχ → γ + (K=6). A single K=6 residue is itself a χ: it requires destroying one input’s entire
base, which VCF forbids, and in any case produces no line shift (E
γ
= m
χ
= 1.719 GeV), excluded
by the data.
χχ → γ + (K=2). The framework has no stable K=2 defect class: Kulkarni [5, §3] establishes
that only the symmetric interstitial sites (tet and oct voids) host stable defects, and the enumer-
ation of Section 4 confirms no K=2 equidistant cage survives the stability constraints. A K=2
residue has nowhere to reside.
χχ → γ + (K
4
in a tet void as a baryon). A gauged (anchor-selected) K
4
residue would be a
baryon. The initial state carries zero baryonic content, so producing one violates sector identity.
The anchor-free residual of Section 4 carries no baryon number and preserves the sector: dark in,
dark out.
10
The two-residue channel χχ → 2γ + 2χ
′
. This is the one alternative that satisfies VCF
(two bases), sector identity (both in oct-adjacent tet voids), and energy conservation (two softer
photons at ∼ 0.76 GeV). It is excluded instead by the topology of the merger interface. The
compound bonding graph K
2,2,2
∪ K
2,2,2
is connected through the shared edge as a bridge (its
removal disconnects the two components). Partitioning the compound into two separate K
4
residues, one per original void, requires cutting that bridge, the bond that brought the two
defects into causal contact. A single merger event cannot both be initiated by the shared edge
and terminate by destroying it; equivalently, the C
2
symmetry of the merger maps the two halves
onto each other, and a two-residue final state breaks that symmetry. Two residues in two voids
would require two distinct encounters , a sequence of two single-residue reactions rather than one
branching of a single event. The single-event two-residue channel therefore does not occur, and
no coincident ∼0.76 GeV companion line is predicted.
Uniqueness. With full annihilation, the K=6 and K=2 residues, and the tet-void baryon
excluded by VCF or sector identity, and the two-residue channel excluded by merger-graph con-
nectedness, χ + χ → γ + χ
′
is the unique sub-Planckian outcome.
9 Comparison with the Kang et al. Detection
Kang et al. [3] report a line near 1.5–1.6 GeV in three high-confidence AGN (Table 2). If the line
originated in the source rest frame, the rest-frame energies E
obs
(1 + z) would scatter to 1.58, 2.84,
and 3.32 GeV; a common rest-frame fit is rejected at χ
2
= 118 (dof = 2, p ≪ 10
−6
). The line must
therefore sit at fixed observed energy in our local frame, consistent with annihilation in foreground
(Milky Way halo or intervening) cold dark matter, with the AGN serving as bright backlights.
Under this interpretation the three energies are mutually consistent at
¯
E
obs
= 1.578 ±0.048 GeV
(χ
2
= 0.72, p = 0.70).
Table 2: The three Kang et al. [3] AGN sources. E
obs
is the observed line centroid; σ
E
its
uncertainty; z the source redshift.
Source z E
obs
[GeV] σ
E
[GeV] TS Type
4FGL J0250.2−8224 0.830 1.55 0.10 23.03 BCU blazar
4FGL J2329.7−2118 0.031 1.53 0.09 21.72 Radio galaxy
4FGL J0749.6+1324 1.050 1.62 0.07 12.92 Blazar
Joint — — — 57.77 —
The framework prediction is E
γ
= 1.591 GeV, deviating from the centroid by
1.591 −1.578
0.048
= 0.27σ, (9)
well within tolerance, while the naive γγ channel sits at 2.9σ. No parameters are fit.
Caveats. The result should be read with two cautions. First, look-elsewhere effects: Kang et
al. [3] scanned ∼2000 AGN over a broad energy range, so the ∼4.9σ maximum local significance
corresponds to ∼ 1.9σ globally; individual sources are not on their own robust detections, and
the strength rests on the three-source coincidence at one observed energy. Second, the historical
base rate for Fermi-LAT line claims at ∼3–4σ per source surviving further data is poor (cf. the
11
0.0 0.2 0.4 0.6 0.8 1.0
source redshift
z
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
line energy (GeV)
J0250.2-8224
J2329.7-2118
J0749.6+1324
Local-frame vs source-rest-frame interpretation of the Kang line
framework prediction
E
= 1.591 GeV
source-rest-frame
E
obs
(1 +
z
) (
2
= 118, rejected)
observed
E
obs
(consistent,
2
= 0.72)
Figure 5: Source-rest-frame vs. local-frame interpretation. Source-rest-frame energies diverge
with z (χ
2
= 118, rejected); observed energies are mutually consistent at 1.578 ± 0.048 GeV. The
framework prediction E
γ
= 1.591 GeV is the dashed line.
∼ 130 GeV feature of 2012, later attributed largely to instrumental systematics [2]). Indepen-
dent confirmation is required before the agreement carries decisive weight. What the analysis
establishes is that the framework’s prediction, derived from lattice geometry with no cosmological
input and no fitted mass, is consistent with the principal observational anchor available.
10 Predictions, Falsifiers, and Stated Assumptions
Predictions. P1: a single line at 1.591 GeV from χχ → γ +χ
′
, not a γγ pair at 1.719 GeV. P2:
a stable, self-conjugate, gravitationally-only residual χ
′
at 0.938 GeV. P3: no coincident softer
line. The single-event two-residue channel is excluded (Section 8) because partitioning the merger
graph into two tetrahedra requires cutting the shared-edge bridge that initiated contact.
Falsifiers. F1: a confirmed line at any energy other than 1.591 GeV (within the Fermi-LAT
∼ 5–7% systematic at 1.5 GeV) falsifies the derivation; no parameters were fit. F2: a (1 + z)
shift of the line across sources (the source-rest-frame interpretation, already rejected at χ
2
= 118)
falsifies the local-frame requirement. F3: direct (non-gravitational) detection of a 0.94 GeV dark
particle falsifies the un-gauged structure. F4: detection of a dark γγ line at 0.94 GeV would
falsify the residual’s stability against χ
′
χ
′
annihilation.
Stated assumptions. The derivation rests on three assumptions, each narrow and each closable
by a finite calculation.
Assumption (Harmonic-model stability, A1). The anchor-free occupant is a stable minimum of
the SSM bond-strain energy (Section 4.3). This is an internal-consistency result, not a quantum
proof; stability against tunneling and beyond-harmonic terms is assumed. The same model implies
the symmetric branch lies below the baryon branch, so baryon persistence requires the gauging
mechanism of Kulkarni [5] to stabilize the anchor-selected state.
12
Assumption (Face transit, A2). The residual physically migrates from the oct interface to the
tetrahedral void across the shared triangular face (Section 3.2). The sites are face-connected
(verified), so no bulk is crossed; that the node transits the face, rather than the face presenting a
residual barrier, is assumed and not computed here.
Assumption (Lattice scale, A3). The confinement barrier σ
lat
L ∼ 10
15
GeV that forbids γγ
(Section 2) assumes L ∼ ℓ
P
. The SSM uses L at different scales in different contexts across the
companion papers; fixing L to a single value and recomputing all L-dependent quantities is an
outstanding program-level task. The mass prediction m
χ
′
= m
p
is a dimensionless ratio and is
independent of the value of L; only the γγ-forbidding barrier depends on it.
Not addressed. The cosmological abundances of χ and χ
′
are not derived: an SSM analogue
of the baryon-asymmetry parameter η
B
is not available, and no abundance prediction follows
without it. The mass and line-position predictions are independent of this.
10.1 Stated limitations of the broader program
The result of this paper, the line at 1.591 GeV, depends only on the two masses and the channel
geometry, and is independent of the items below. We nonetheless state them explicitly, because
χ
′
is identified within a larger framework whose color and charge sector has known boundaries; a
companion scope assessment collects the supporting computations.
Electric charge values are not derived. The neutrality of χ
′
is derived (Section 4.4). The
nonzero fractional charges of the visible quarks are not: the value −
1
3
is the tetrahedral bond-angle
cosine cos 109.47
◦
, a geometric projection identified with an electric charge, and the symmetry
breaking that would produce a charged baryon (S
4
→ S
3
) has no derived trigger, since the void
is exactly symmetric including its FCC embedding. Charge in the visible sector is therefore an
identification awaiting a mechanism. This does not affect χ
′
, which is the neutral symmetric
branch.
The lattice scale L is not fixed across the program. As noted in assumption A3, L is used at
the Planck scale in some companion papers and the fermi scale in others. Every dimensionful
quantity that invokes the confinement barrier inherits this ambiguity. The present line prediction
is a dimensionless mass ratio and is unaffected, but the program-level resolution of L remains
outstanding and is the prerequisite for any absolute energy or cross-section claim.
The non-abelian color group is not hosted by the program’s code. The geometry yields the
color count (three, from the K
4
skew pairs) and, with complex sheet structure and a stacking-
closure (tracelessness) condition, the complexified algebra sl(3, C) with the correct A
2
root system.
Selecting the compact real form su(3) requires unitarity, whose natural home is the program’s
stabilizer code; but that code is a qubit code, and su(3)’s fundamental representation requires
a qutrit (a three-level system C
3
), whereas a localized logical space on the void is computed to
be three logical qubits (C
8
), not a qutrit. The dimension three matches; the algebraic type does
not. The defensible color claim is therefore that the representation dimension and sl(3, C) are
geometric, while the compact group su(3) and its qutrit host are not derived from the present
code. Again this is a matter- sector limitation and does not bear on the neutral residual χ
′
of
this paper.
11 Conclusion
The dark-matter annihilation channel χχ → γ + χ
′
follows from lattice geometry. Two K=6
octahedral defects share an octahedron edge (450/450 pairs); the only stable equidistant cages
13
are the octahedron and the regular tetrahedron; an octahedral residual produces no line shift, so
the residual is tetrahedral; and the relevant tetrahedral void is reachable from the merger edge
across a shared triangular face, never crossing bulk. The tetrahedral void’s T
d
site symmetry
admits an anchor-free occupant (neutral, colorless, baryon-number-free by symmetry, distinct
from the proton and neutron), which is a stable minimum of the bond-strain energy. Its mass is
the bare tetrahedral-cage count 1872 − 36 = 1836, giving m
χ
′
= m
p
= 0.938 GeV positively from
the cage, with the proton appearing nowhere in the derivation. Two-body kinematics place the
line at 1.591 GeV, 0.27σ from the Kang centroid, with no fitted parameters. Three assumptions
remain (A1–A3) and are stated explicitly; closing them tightens the derivation. The earlier
“figure-8” residual at 1872 and its subtractive-gauging axiom are withdrawn: the residual is a real
tetrahedral occupant and carries its skew structure, so its honest count is 1836.
Code and Data Availability
The computations supporting Sections 3 and 4 are reproduced by four self-contained Python
scripts, each depending only on numpy:
• Shared-edge enumeration (Table 1, Figure 2): https://github.com/raghu91302/ssmtheory/
blob/main/dm_verify_shared_edge.py
• Equidistant-cage enumeration (Section 4, Result on the two cages): https://github.com/
raghu91302/ssmtheory/blob/main/dm_verify_cages.py
• Face-adjacency check (Section 3.2): https://github.com/raghu91302/ssmtheory/blob/
main/dm_verify_face_adjacency.py
• Bond-strain stability / Hessian test (Section 4.3): https://github.com/raghu91302/ssmtheory/
blob/main/dm_verify_stability.py
No external data are used beyond the CODATA 2022 proton mass (m
p
= 938.272 MeV) and the
Kang et al. [3] line measurements as tabulated in Table 2.
Declaration of Competing Interest
The author declares no competing interests.
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