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
July 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 [12]. 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, under five structural assumptions stated
explicitly in the text. 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 (another χ) would place the line at 3m
χ
/4 = 1.289 GeV,
disfavored by the data, 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. (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 [11]; 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. 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 [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 octahedral site [12].
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 proton-electron count 1836 is a result of the published matter paper [11]; the
primary dark-matter mass m
χ
is the single input taken from the companion analysis [12], currently
under review, and is not re-derived here. The dependence on that input is deliberately minimal:
m
χ
enters only through the two-body kinematics. As Section 9 shows, the derived channel and
residual mass, combined with the observed line energy, determine m
χ
= 1.707 ± 0.045 GeV from
the data alone—so the counted value 1.719 GeV functions as an independently testable consistency
check rather than a load-bearing foundation.
A recent analysis [3] of Fermi-LAT [1] data 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 (statistical errors; Section 9) 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). A monochro-
matic line is the classic smoking-gun signature of annihilating dark matter [8], and the reaction
above is a semi-annihilation in the established sense of dark-sector reactions whose final state
retains a dark particle [5]; the displacement of the photon line below m
χ
is the characteristic
semi-annihilation signature [6].
A natural alternative would assign χ
′
a disruption count of 1872 (0.957 GeV) by omitting the
skew-edge deduction. That count is incorrect: 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 count
is obtained positively from the cage; the proton enters only as the calibration unit that converts
counts to GeV, as one measured mass must in any framework predicting ratios. The derived
content is the dimensionless equality C
χ
′
= C
p
: the two objects are mass-degenerate because they
occupy 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.
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 five
stated assumptions.
2
+ (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.
2 The Verification-Cost Floor
Kulkarni [11,
§
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 intact
K=12 shell, giving a linear confining potential V (r) = σ
lat
r whose barrier height σ
lat
L is of order
the GUT scale, ∼10
15
GeV, for L ∼ ℓ
P
[11,
§
6]. 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. Both statements are derived for an isolated defect surrounded by
intact bulk.
The reaction studied here involves two trapped nodes in the initial state and one in the final
state, so the bookkeeping of the second node must be stated explicitly. During a merger, the two
defects’ disrupted shells overlap along the shared interface (Section 3); the intact-shell premise of
the barrier derivation does not hold along that interface while it is open. We therefore adopt, as a
stated assumption (A4 of Section 10), that one node can be incorporated into the relaxing interface
region without traversing the metric wall, the released structural cost appearing as the photon
and recoil of Section 7. What the barrier derivation does forbid, without further assumption, is
the dissolution of the last base: once a single trapped node remains, there is no second defect to
maintain an open interface, its surrounding shell heals to intact K=12, and any further dissolution
is the isolated-defect process facing the full barrier. This asymmetry between losing a base and
losing the final base is what the corollary records.
Corollary 1 (Verification-Cost Floor; conditional on A3 and A4). An isolated trapped-node
base surrounded by intact bulk cannot be dissolved at sub-Planckian energies. In a multi-defect
merger, base absorption is possible only while an open interface exists (assumption A4); the last
remaining base is by definition isolated and cannot be dissolved. No sub-Planckian reaction can
reduce the base count to zero.
The channel χχ → γγ ends with zero trapped bases, so it requires the final base to dissolve into
intact bulk against a ∼ 10
15
GeV barrier with only 1.719 GeV available. It is forbidden. The
3
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 shared substructure of the
two cages and hence the causal interface for a merger. It is not, however, a cut element of the
compound graph: the union of the two octahedron bond graphs (ten vertices; twenty-five bonds
of length L, including inter-octahedron bonds through the flanking sites) remains connected when
the shared edge is removed, each shared vertex retaining three bonds into each parent. The edge
carries a C
2
reflection about its midpoint.
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.
(We note that a node crossing a triangular face passes within the wall radius L/
√
3 of the face
vertices—the face centroid sits at exactly L/
√
3—so the transit is the marginal configuration of
the wall rule; whether the face presents a residual barrier is assumption A2 of Section 10.) This
removes the apparent objection 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
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.
5
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 of the ten interface vertices (sizes 3–10, connectivity in
the interface bond graph) 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): by the two-body kinematics of Section 7
with residual mass m
χ
, it predicts a line at E
γ
= 3m
χ
/4 = 1.289 GeV, which the data disfavor at
6.0σ (statistical; 2.7σ with the energy-scale systematic of Section 9). 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 [11], 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
symmetry 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
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.
6
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
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, at the level of geometry, the symmetric occupant has no preferred direction, a bulk
coupling carried entirely by the S
4
singlet, and no anchor junction. The translation of these geo-
metric facts into quantum numbers—that χ
′
is electrically neutral, colorless, and baryon-number-
free—holds under the matter-sector identification of Kulkarni [11], in which charge and color are
properties of anchor-selected bond structure and baryon number is carried by the anchor junction:
the identification locates the quantum numbers on structure the symmetric state provably lacks,
and so assigns it none. These statements are consequences of a published identification applied
to a derived geometry, not independent derivations of gauge quantum numbers from the lattice;
they inherit the status of that identification, whose fractional charge values remain underived
(Section 10.1). (Illustratively, with the matter sector’s charge identification, the singlet weight
plus traceless part gives 1 + 3(−
1
3
) = 0; we do not rely on this, since the fractional charge values
7
themselves are not derived, cf. Section 10.1.) The visible baryons of Kulkarni [11], 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 compan-
ion 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 trapped-base count is protected in the VCF sense of Corollary 1:
absorption of a base is possible only while a multi-defect interface is open (assumption A4), and
the last base cannot dissolve, so the final state must retain at least one trapped base. Spatial
sector identity is preserved: a residue formed at the oct-void interface 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
[10]: 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 [11,
§
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
[10],
m
χ
′
=
C
χ
′
C
p
m
p
=
1836
1836
m
p
= m
p
= 0.938 GeV. (5)
8
The proton’s role in Eq. (5) is that of the calibration unit: the derived statement is the dimen-
sionless count equality C
χ
′
= C
p
, and one measured mass converts it to physical units.
Why the count is 1836 and not 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. Charge conjugation flips the handedness of the skew structure; 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. These phenomenological statements, like the quantum
numbers themselves, hold under the matter-sector identification discussed in Section 4.4.
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
′
.
9
Stability of the residual population. Two residuals in bulk-separated tetrahedral voids—the
generic case for residuals produced in distinct, uncorrelated merger events—cannot contact for
a χ
′
χ
′
reaction without traversing intact bulk, which the metric wall forbids. The accumulated
residual population is therefore non-depleting on cosmological timescales. One configuration is
exempt from this argument and we flag it: the pair produced in a single two-residue event (Sec-
tion 8) is born in the two edge-sharing flanking voids, 0.707 L apart and sharing two bounding
vertices—an open interface by the same criterion that defined the χχ merger. Whether the re-
coiling pair separates before a secondary reaction can act, or promptly reacts (χ
′
χ
′
→ γ + χ
′
,
allowed under A4, with a two-body line at 3m
χ
′
/4 = 0.704 GeV), is not computed here; any
such secondary feature is contingent on the underived two-residue branching and on the prompt-
reaction probability. χ
′
χ
′
→ γγ remains forbidden in either case: it ends with zero bases, the
same last-base protection that forbids χχ → γγ (cf. falsifier F4).
Stability of χ itself. The same intact-bulk/open-interface distinction that underlies assump-
tion A4 supplies a derived stability argument for the dark-matter particle. Every isolated χ is face-
adjacent to eight tetrahedral voids (the octahedron’s eight faces), and a transit into any one would
realize the decay χ → χ
′
+ γ with a monochromatic photon at (m
2
χ
−m
2
χ
′
)/(2m
χ
) = 0.604 GeV—
an unobserved line. The decay does not occur for the same reason isolated dissolution does not:
an isolated χ has no open interface, its bounding faces sit in intact bulk, and the face crossing
is the isolated-defect process facing the full barrier of Section 2. Face transit is unlocked only by
a second defect (assumption A2 operates on the open merger interface). This converts a latent
vulnerability into a falsifiable corollary: no 0.604 GeV decay line is predicted; the detection of one
would falsify the intact-bulk protection, with the observed rate measuring the finite face barrier
(falsifier F5).
7 Kinematics
The reaction is a two-body annihilation of identical self-conjugate defects into a massive residual
and a massless photon. (We deliberately do not assign point-particle quantum numbers: for a
spin-
1
2
Majorana pair the s-wave state is
1
S
0
, and J=0 → γ + spin-0 has no allowed amplitude, so
the vertex would require either a nonzero defect spin or a p-wave rate suppressed by v
2
∼ 10
−6
; the
defect spin spectrum is not derived, and this is stated as assumption A5 of Section 10. The energies
below are pure two-body kinematics and independent of the spin assignment.) 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
10
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.
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 Channel Selection and Line Selectivity
The channel χχ → γ + χ
′
is not merely compatible with the framework; it is the unique two-
body monochromatic channel among the enumerated stable residual classes of two adjacent K=6
defects. The alternatives fall to the conservation laws of Section 5.1, are disfavored by the observed
line energy, or are spectrally distinguished.
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. Like the derived channel, this outcome
requires one interface absorption and is therefore allowed under assumption A4; it is selected
against by the data, not by the VCF. With residual mass m
χ
the two-body kinematics give
E
γ
= 3m
χ
/4 = 1.289 GeV, disfavored at 6.0σ (statistical; 2.7σ with the energy-scale systematic),
versus 1.591 GeV at 0.27σ for the tetrahedral residual.
χχ → γ + (K=2). The framework has no stable K=2 defect class: Kulkarni [11,
§
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.
11
χχ → γ + (K
4
in a tet void as a baryon). A gauged (anchor-selected) K
4
residue would be a
baryon. This outcome is excluded by the energetics and symmetry of Section 4.4: the symmetric
(anchor-free) occupant is the ground state of the cage, anchor selection has no derived trigger (the
void is exactly symmetric, including its FCC embedding), and a relaxing merger settles into the
ground state. Producing a baryon would additionally require creating baryon number from a state
with none, via a symmetry-breaking mechanism the framework does not supply. The residual is
therefore the anchor-free occupant: dark in, dark out.
The two-residue channel χχ → 2γ + 2χ
′
. This channel is not excluded, and we say so plainly.
It conserves the trapped-node count (2 → 2) and therefore requires no interface absorption at
all—under Corollary 1 it is the least obstructed outcome—and both flanking tetrahedral voids are
face-adjacent and contain both shared-edge vertices (Section 3.2), so forming one residual on each
side severs nothing. Nor does graph topology forbid it: the shared edge is not a cut element of
the compound graph (Section 3). What distinguishes the two channels is their spectral signature.
The two-residue final state is four-body; its photon spectrum is a continuum, not a line, with
endpoint
E
max
γ
=
s − 4m
2
χ
′
2
√
s
=
m
2
χ
− m
2
χ
′
m
χ
= 1.207 GeV, (9)
below the 1.591 GeV line. The separation is emphatic: the endpoint sits 0.384 GeV (24%) below
the line—several times the per-source centroid uncertainties (0.07–0.10 GeV) and the combined
energy-scale systematic (∼0.11 GeV)—and the endpoint is the continuum’s maximum; the bulk of
its photons are softer still. The two-residue channel therefore cannot broaden, shift, or masquerade
as the monochromatic feature. The relative branching between the two channels is not derived
here; what the framework fixes is that the only monochromatic feature produced by any allowed
channel is the γ + χ
′
line at 1.591 GeV.
Line selectivity. With full annihilation excluded by the VCF, the K=6 residue disfavored by
its 1.289 GeV line energy, the K=2 residue excluded by the cage enumeration, and the tet-void
baryon excluded by the ground-state result of Section 4.4, two channels survive: γ + χ
′
and the
node-conserving 2γ + 2χ
′
. Only the former is monochromatic. χ + χ → γ + χ
′
is therefore the
unique two-body monochromatic channel among the enumerated stable residual classes, and the
observed line energy tests it regardless of the underived branching between the two. (Excited
occupant states of the surviving cages and higher radiation multiplicities are not enumerated
here; the former would shift a line, the latter produce continua.)
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).
The framework prediction is E
γ
= 1.591 GeV, deviating from the centroid by
1.591 − 1.578
0.048
= 0.27σ, (10)
12
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 —
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.
13
well within tolerance, while the naive γγ channel sits at 2.9σ. These figures use the statistical
uncertainty on the centroid alone. Folding in the Fermi-LAT energy-scale systematic at these
energies (∼ 5–7%, i.e. ∼ 0.08–0.11 GeV; cf. F1 of Section 10) gives a combined uncertainty of
∼0.09–0.12 GeV, softening the comparison to ≈0.1σ for the derived channel versus ≈1.2–1.5σ
for γγ: the data are fully consistent with the derived line and disfavor, but do not yet sharply
discriminate against, the direct channel. No parameters are fit.
Inverting the dependency. The comparison can be run in reverse, and doing so isolates
what this paper takes from the in-review companion [12]. Everything else in the derivation—the
channel, the cage enumeration, and the residual mass m
χ
′
= m
p
via the 1836 count—rests on the
published matter and mass papers [10, 11] and on enumerations verified in this paper. Holding
those fixed and treating m
χ
as unknown, the kinematic relation E
γ
= m
χ
−m
2
p
/(4m
χ
) inverts to
m
χ
=
1
2
E
γ
+
q
E
2
γ
+ m
2
p
; the observed centroid 1.578 ± 0.048 GeV then gives
m
(obs)
χ
= 1.707 ± 0.045 GeV (statistical), (11)
independently of any octahedral count. The companion’s counted value 1.719 GeV lies 0.28σ
inside this determination. Equations (11) and the 0.27σ agreement above are of course the same
single number read in two directions; the point of the inversion is logical, not statistical: the
paper’s empirical content—the channel, the residual at m
p
, and the relation between the line
energy and the dark-matter mass—stands on published foundations, with the companion’s count
entering as a corroborated input rather than a prerequisite.
A fourth source. A subsequent single-source analysis by the same group [4] reports, in 17.75
years of Fermi-LAT data toward the blazar 4FGL J0604.9−0000, a tentative excess at 1.59 ±
0.07 GeV (local significance 2.6σ, width fixed to the instrumental resolution), together with
a second excess at 11.15 ± 0.61 GeV. The lower centroid coincides with the prediction E
γ
=
1.591 GeV within 0.01σ. We record this as supportive but tentative: the 1.59 GeV feature is
individually weak, and the 11 GeV excess—whatever its origin—is not addressed by the present
framework, which produces no line at that energy.
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. The trials correction applies per
source: the strongest single-source local significance (∼4.9σ, T S = 23.03) corresponds to ∼1.9σ
globally, so no individual source is on its own a robust detection. The joint statistic (T S = 57.77,
nominally ∼7.6σ local) is stronger, but a global figure for a three-source coincidence at a single
observed energy requires its own trials analysis, and we conservatively do not claim one here; the
evidential weight rests on the coincidence structure rather than on any quoted global significance.
Second, the historical base rate for Fermi-LAT line claims at ∼3–4σ per source surviving further
data is poor (cf. the ∼ 130 GeV feature of 2012 [7], later attributed largely to instrumental
systematics [2]). Third, the local-frame interpretation implies annihilation in foreground cold dark
matter, which must also be consistent with sightlines of far larger J-factor: the same mechanism
predicts line emission toward the Galactic center and dwarf spheroidals, where dedicated line
searches place strong constraints [2, 9], and the absence (or presence) of a 1.59 GeV feature
there is a direct consistency test that we flag but do not perform; a quantitative flux comparison
requires the halo profile and the underived annihilation cross-section. Independent confirmation
is required before the agreement carries decisive weight. What the analysis establishes is that the
14
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 second monochro-
matic feature from primary annihilation. The two-residue channel 2γ + 2χ
′
is allowed (Section 8)
but yields a photon continuum with endpoint 1.207 GeV, spectrally distinct from and below the
1.591 GeV line. A contingent secondary line at 3m
χ
′
/4 = 0.704 GeV from prompt adjacent-pair
reactions (Section 6) is possible but depends on two underived quantities; its absence does not
falsify, while its presence alongside the 1.591 GeV line would support the two-residue branch. A
confirmed second line at any other energy would falsify the channel analysis.
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 last-base protection of Corollary 1 itself, since χ
′
χ
′
→ γγ ends with zero trapped bases.
F5: detection of a 0.604 GeV decay line (χ → χ
′
+ γ; Section 6) would falsify the intact-bulk face
barrier that stabilizes isolated χ: the line energy is fixed by the two masses, and the observed
rate—equivalently the implied lifetime, which decay searches currently constrain to far beyond
the Hubble time—would measure the barrier’s scale.
Stated assumptions. The derivation rests on five 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 [11] to stabilize the anchor-selected state.
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.
Assumption (Interface absorption, A4). The reaction takes two trapped nodes to one; the sec-
ond node is incorporated into the relaxing merger interface without traversing the metric wall
(Section 2). The barrier derivation applies to an isolated defect in intact bulk and is assumed not
15
to apply along the open two-defect interface; computing the interface pathway’s actual barrier
is a finite lattice-relaxation calculation not performed here. The asymmetry that protects the
γγ exclusion is independent of this assumption’s quantitative details: the last base has no open
interface and always faces the intact-bulk barrier.
Assumption (Defect spin and vertex structure, A5). The angular-momentum content of the
trapped defects is not derived. If χ were a point spin-
1
2
Majorana fermion and χ
′
spin-0, the s-
wave two-body channel χχ → γ + χ
′
would be forbidden (
1
S
0
→ γ + scalar admits no amplitude),
and the p-wave rate would carry a v
2
∼ 10
−6
suppression fatal to the flux. The channel as derived
therefore assumes the extended lattice defects are not subject to this point-particle selection rule—
equivalently, that the defect spin assignments differ from the Majorana/scalar pairing. Computing
the defect spin spectrum from the lattice construction is an open task; the two-body kinematics
and the line energy are independent of 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 relative to the matter-
sector identification of charge with anchor-selected structure (Section 4.4): the geometric inputs—
no preferred direction, singlet-only bulk coupling, no anchor junction—are computed in this paper,
while the identification that converts them into quantum-number statements is inherited from
Kulkarni [11]. 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
16
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 as the unique two-
body monochromatic channel among the enumerated stable residual classes; the node-conserving
two-residue channel is allowed but yields only a continuum below 1.207 GeV. Two K=6 octahe-
dral defects share an octahedron edge (450/450 pairs); the only stable equidistant cages are the
octahedron and the regular tetrahedron; an octahedral residual would place the line at 1.289 GeV,
disfavored by the data, 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, and baryon-number-free
under the matter-sector identification, 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,
equal to the proton’s count for the same cage; the dimensionless equality C
χ
′
= C
p
is the derived
content, with the proton entering only as the calibration unit, giving m
χ
′
= m
p
= 0.938 GeV.
Two-body kinematics place the line at 1.591 GeV, 0.27σ from the Kang centroid (statistical;
≈0.1σ with the energy-scale systematic folded in), with no fitted parameters. Five assumptions
remain (A1–A5) and are stated explicitly; closing them tightens the derivation. The geometry
and kinematics fix the line position; the vertex structure and the annihilation rate remain to be
derived (assumption A5), so the present result is a line-energy and channel-geometry prediction,
not yet a complete annihilation model.
Code and Data Availability
The computations supporting Sections 3 and 4 are reproduced by four self-contained Python
scripts, each depending only on numpy:
dm verify shared edge.py — shared-edge enumeration (Table 1, Figure 2);
dm verify cages.py — full connected-subset equidistant-cage enumeration (Section 4, Re-
sult on the two cages);
dm verify face adjacency.py — face-adjacency check (Section 3.2);
dm verify stability.py — bond-strain stability / Hessian test (Section 4.3).
All four scripts are archived together at https://github.com/raghu91302/ssmtheory/blob/
main/dark_proton_scripts_github.zip. 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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