From the First Triangle to an Error-Correcting Lattice

Proton-mass dark matter from an anchor-free tetrahedral

defect:

an alternative to the octahedral candidate in the

Selection-Stitch Model

Raghu Kulkarni

SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA

raghu@idrive.com

5 November 2026

Abstract

In the Selection-Stitch Model (SSM) a baryon is a node trapped in a tetrahedral void of

the face-centered cubic vacuum lattice. One of its four bonds is the anchor, the single

channel through which charge, momentum and color ux pass to the bulk. We study the

same trapped node with no anchor. Applying the construction that gives the baryon its

charges, colors and mass, we nd that this state is neutral, since every charge in the model

is a projection onto the anchor axis, and closed to color exchange, since color ux leaves the

defect only through the anchor. Its structural count equals the proton's,

= 1836

, because

the mass formula has no anchor term. We take baryon number to count trapped nodes, so

the state carries

B = 1

and shares the cosmic baryon asymmetry. This makes it a candidate

for asymmetric dark matter with a mass of 938.19938.27 MeV and a model uncertainty of

about 0.25 MeV. That places it inside the narrow window

937.900

MeV

< m

+ m

which nuclei, the proton, and the dark matter are all stable. In minimal models of proton-

mass dark matter, the mass is a free parameter that must be set by hand to fall in this

window. Here it follows from the lattice count, provided that a correction the model applies

to the neutron, because it is neutral, does not apply to an anchor-free node. The candidate

predicts no gamma-ray lines, in agreement with our search of 18 years of Fermi-LAT data,

which nds no line at the octahedral candidate's energies. Its signatures are hydrogen

decay,

H → χν

, with a monochromatic neutrino of 0.510.59 MeV, a rare neutron decay

n → χγ

with a photon of 1.291.37 MeV, and heating of old neutron stars. Their rates are

set by a single anchoring amplitude that the model does not yet predict and that existing

data already bound to be very small. The candidate cannot explain the neutron-lifetime

discrepancy. Its main open problems are whether that correction applies, and the ratio of

anchor-free to anchored nodes, which must be about 5.4.

1 Introduction

The Selection-Stitch Model (SSM) treats the vacuum as a face-centered cubic (FCC) lattice

that crystallized from an earlier, less ordered phase. Particles are defects of that lattice, and

their masses are the cost of keeping the defects stable. Section 2 summarizes the parts of the

framework this paper uses, one paper at a time, for readers who have not met it.

The lattice leaves two kinds of gap between its nodes, tetrahedral and octahedral voids.

A baryon is a node trapped in a tetrahedral void and bonded to its four corners; one of the

four bonds, the anchor, couples it to the surrounding lattice [1]. The SSM's rst dark-matter

candidate is a node trapped in an octahedral void [2]. Its main observable, a gamma-ray line at

1.591 GeV, is not seen in 18 years of data from the Large Area Telescope on the Fermi satellite

(Fermi-LAT) (Section 3).

This paper asks whether the tetrahedral void itself can hold dark matter, in the form of a

trapped node with no anchor. We call this state

. It also appears in Ref. [2], as the residue

left when two octahedral nodes merge. We make several modeling choices, state them as such

(Section 5), and work out what follows.

Dark matter with almost exactly the proton's mass has been studied before. Fornal and

Grinstein [3] proposed a dark particle near the proton mass to explain the neutron-lifetime

discrepancy, the disagreement between two methods of measuring the neutron lifetime, and

showed that nuclear stability connes its mass to a window less than 1 MeV wide. Khalaf and

Kuik [4] recently built a minimal model of such dark matter: a new scalar eld with baryon and

lepton number

−1

, produced by freeze-in (slow production from the hot early plasma that never

reaches equilibrium), with hydrogen decay and neutron-star heating as its signatures. In both

minimal models the mass is a free parameter that must be set by hand to fall in the window.

Mirror-matter models [5, 6] reach the proton mass another way. They add a hidden copy of the

Standard Model, and when the mirror symmetry is exact the mirror proton weighs the same as

the proton. The equality then comes from a symmetry, and neutrons may oscillate into mirror

neutrons [7].

The candidate studied here shares the window and several signatures with the minimal

models, but it is a dierent particle, a trapped node with baryon number

whose site symmetry

favors spin 1/2. Its mass is not a free parameter. It follows from the same structural count that

gives the proton its mass. Unlike the mirror case, it needs no second sector: the dark particle

and the proton are two states of one defect in one lattice, and the dark particle is stable only

because its mass falls in the window.

2 The SSM framework

(a)

anchor

(b) (c)

Figure 1: (a) Conventional FCC cell. Each node has 12 nearest neighbors. The orange tetrahe-

dron bounds a tetrahedral void (orange star), and the blue octahedron, whose corners are the

six face centers, bounds an octahedral void (blue star). There are two tetrahedral voids and one

octahedral void per node. (b) A baryon: a node trapped at the center of a tetrahedral void,

bonded to the four corners. One bond is the anchor (red); the other three are the valence bonds

(green) that carry the quark charges. (c) The candidate of this paper: the same trapped node

with no anchor, so all four bonds are equivalent.

2.1 The vacuum lattice

The framework starts from a phase transition in the early universe. Nodes rst arrange with

four neighbors each, in tetrahedral units. Regular tetrahedra cannot ll space, so this phase is

frustrated and gives way to the densest packing of equal spheres, the FCC lattice, in which every

node has

K = 12

neighbors [1]. The bond length

is the diameter of the packed spheres. The

FCC lattice leaves two kinds of gap between its nodes (Fig. 1a). A tetrahedral void is bounded

by four nodes, and an octahedral void by six. Per node there are two tetrahedral voids and one

octahedral void.

2.2 Matter as incomplete crystallization

Ref. [1] identies a baryon with a node that failed to join the ordered phase and stayed trapped

at the center of a tetrahedral void, bonded to the four corners. It is a small pocket of the

old four-neighbor phase inside the twelve-neighbor bulk (Fig. 1b). One of the four bonds, the

anchor, couples the defect to the bulk. The other three are the valence quarks. Because the

angle between tetrahedral bonds has cosine

−1/3

, each valence bond projects

−1/3

onto the

anchor axis. A valence bond can also carry a winding, a topological twist that adds one unit

of charge. This gives the quark charges

−1/3

and

+2/3

and the baryon charges

Q = −1 + W

where

is the number of windings. The six edges of the bounding tetrahedron form exactly

three pairs of edges that share no corner, the skew pairs, which the paper identies with the

three colors. A node cannot be pulled out of its void without passing within the metric wall

√

of three corners at once, which the lattice forbids. The energy needed to pull it farther

grows linearly with displacement, which the paper identies with linear connement. Counting

the disrupted bonds gives

= (K + 1)K

− c

skew

K = 13 · 144 − 3 ·12 = 1836,

(1)

where

skew

= 3

is the number of skew pairs. This is within 0.008% of the measured 1836.15.

2.3 Mass as verication cost

Ref. [8] explains why a count of this kind should be a mass. The vacuum is treated as a quantum

error-correcting code, and a particle is a defect the code's checks can detect but not remove.

Keeping the defect veried costs classical bits, and by Landauer's principle each bit costs

kT ln 2

of energy. With no separate hardware, that cost is the particle's whole rest energy. Mass ratios

then become ratios of integer verication costs,

= C

, and the temperature cancels.

For the proton,

= 36 × 51 = 1836

, where 36 is the number of bonds in the coordination

cluster (a node and its 12 neighbors) and 51 the number of checks that can detect the defect.

The electron is the minimal defect,

= 1

2.4 The vacuum code

Ref. [9] builds the code explicitly. On the FCC lattice the qubits sit on the bonds. One set of

checks sits on the nodes, each acting on its 12 bonds; the other sits on the octahedral voids,

each acting on the 12 bonds of the octahedron. Because one set detects bit ips and the other

phase ips, this is a CSS (CalderbankShorSteane) code. On the smallest periodic block it has

192 qubits, encodes 130, and has distance 3, written [[192,130,3]], a rate of about 68%. It gives

a concrete meaning to a defect the checks detect and to which defects are distinguishable.

2.5 The length scale of a trapped node

The construction above does not x the lattice spacing, and the mass ratios do not depend on

it. Since baryons are conned within about 1 fm, we take the lattice that carries a trapped

tetrahedral node to have a spacing

of that order. This sets the size of a trapped node, and

so the strength its contact interactions with nuclei would have if nothing suppressed them.

2.6 Migration, spin and antimatter

Ref. [10] asks how a trapped node moves. It works in the four-dimensional

lattice, whose

spatial slice is the FCC lattice. The tetrahedral voids form a simple cubic sublattice, and a defect

migrates by hopping between neighboring voids. Tetrahedral voids come in two orientations,

pointing up or down along a cube diagonal, and a hopping defect alternates between them at each

step. Two results matter here. First, the rotational symmetry of a tetrahedral site is the group

; among half-integer spins only

j = 1/2

stays irreducible under its double cover, so spin

1/2

is the spin the site favors. Second, the matter/antimatter label cannot be the void orientation,

since a moving defect changes orientation at every hop. It is carried by the orientation

of the

defect's worldline. Annihilation requires opposite

at adjacent voids. This corrects the earlier

identication of the two void orientations with matter and antimatter in Ref. [1].

2.7 The octahedral dark-matter candidate

Ref. [2] applies the same idea to the octahedral void. A node trapped there has six bonds and

a verication count

oct

= 3364

, giving

oct

3364

1836

= 1.719

GeV

(2)

We write

oct

for this occupant, to keep

for the candidate of this paper. Its bonded subgraph

has no square plaquettes, the faces that couple to photons in the model, so it is electrically

neutral. Ref. [2] argues that the inversion symmetry of the octahedron makes it its own antipar-

ticle; Section 5 notes a tension with this. Two such defects can merge, leaving a photon and

a tetrahedral residue

′

of proton mass, the state we call

, which gives a line at 1.591 GeV.

Annihilation into two photons, which Ref. [2] forbids, would give a line at 1.719 GeV, and the

transition of an octahedral occupant into a neighboring tetrahedral void, if allowed, a line at

0.603 GeV. Section 3 tests all three energies.

3 Gamma-ray lines at the octahedral energies

Data and method.

We used Fermi-LAT photons in the standard P8R3 SOURCE event class

from August 2008 to October 2026, with a zenith-angle cut at

◦

and energies 0.55 GeV [11].

The test regions surround four sources in which lines near 1.5 GeV have been reported [12, 13]:

4FGL J2329.7

−

2118, J0604.9

−

0000, J0749.6

1324 and J0250.2

−

8224 [14]. For each we took

the photons within

◦

of the source, which we call the patch, and those in a ring 4

◦

around

it. The ring covers 83.8 times the solid angle of the patch. A line from dark matter in the

Milky Way halo is spread smoothly over the sky, so it would put about 84 times more counts in

the ring than in the patch; a line from the source itself would appear only in the patch. Each

spectrum was tted by binned Poisson likelihood with a smooth (log-parabola) background and

a Gaussian line of width

σ/E = 0.09

, the instrument resolution. The line amplitude was free

to be negative. At 0.603 GeV, which lies close to the lower edge of the data, we also used a

more exible background, the exponential of a cubic in

ln E

. We report the signed signicance

s = ±

√

at the three energies, where TS is twice the gain in log-likelihood from adding the

line and the sign is that of the tted line counts. The 1.591 and 0.603 GeV tests were xed in

advance; the 1.719 GeV test was added after earlier scans had passed through that energy.

We also tted four inner elds,

◦

in radius, at Galactic longitude

±30

◦

and latitude

±20

◦

and four outer elds at longitude

150

◦

and

210

◦

and the same latitudes. For an NFW halo [15],

the standard density prole from simulations, the annihilation signal in the inner elds is twelve

times that in the outer ones. Because these elds hold far more photons, we tted them with

the cubic background, chosen before the ts. Photons for the rst three sources came from the

LAT data server; those for J0250.2

−

8224 and the eight elds came from the all-sky weekly les,

binned at 0.9% in energy. Repeating the J2329.7

−

2118 t both ways shows that this binning

shifts

by about 0.2.

Table 1: Signed signicance of a line at the three octahedral energies. Ring values test Milky

Way dark matter; patch values test the sources.

1.591 GeV 1.719 GeV 0.603 GeV (ring)

source patch ring ring smooth bkg exible bkg

J2329.7

−

2118

+1.74 −1.40 −0.88 +0.93 −0.79

J0604.9

−

0000

+0.71 −1.78 −2.44 +1.61 −0.78

J0749.6

1324

+1.05 −1.93 −1.68 +1.13 +0.11

J0250.2

−

8224

+0.52 +0.46 −0.81 +3.76 +1.65

Results.

No line appears at 1.591 or 1.719 GeV in any ring (Table 1). The four rings combine

−2.3

at 1.591 GeV and

−2.9

at 1.719 GeV. These negative values reect a residual wave of

35% in the diuse background near these energies, seen in every eld we examined. In the inner

elds, where a halo line would be brightest, the combined signicance at 1.591 GeV is

−2.2

; in

the outer elds it is

−0.4

Limits.

For the merger channel

oct

→ γχ

′

, with one photon per merger, and an NFW

halo with scale radius 20 kpc, solar distance 8.2 kpc and local density 0.4 GeV cm

−3

, the inner

elds give

⟨σv⟩ < 2 × 10

−29

−1

from statistics alone, and

1 × 10

−28

−1

after adding

5% of the background to allow for the residual wave. Both are uncertain by about a factor of

two because the exposure was estimated rather than computed.

The 0.603 GeV line.

At 0.603 GeV the four rings combine to

+3.7

with the smooth back-

ground, driven mostly by J0250.2

−

8224, but to

+0.1

with the exible one. The line sits just

above the 0.5 GeV edge of the data, where a line and a bend in the background cannot be told

apart, so this is not evidence.

The excess near the sources.

The patches show a weak common excess near 1.57 GeV

(Fig. 2a). It cannot be Milky Way dark matter. If it were, the rings would show 84 times as

many line counts, and they exclude that by factors of 5 to 30 (Fig. 2b).

What this means for the octahedral candidate.

Its rates are not predicted, so these

results constrain it without ruling it out. But its main observable is not seen, the reported

blazar lines cannot be cited in its support, and a thermal relic (one whose abundance was set by

annihilation in the early universe) annihilating from rest at this mass is excluded by the cosmic

microwave background (CMB) [16, 17], and also by these limits if the line channel dominates.

Under the baryon-number rule adopted in Section 5, the merger channel is forbidden outright.

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

photon energy (GeV)

−

excess over background (%)

(

)

∘

around the four sources

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

photon energy (GeV)

−

(

)

−

∘

rings around them

Figure 2: Excess over a smooth background, summed over the four sources. (a) Within

◦

the sources. (b) In the 4

◦

rings. The dotted line marks 1.591 GeV. The dashed curve in (b)

is what the rings would show if the excess in (a), tted at each source's reported line energy

(1.531.62 GeV), came from dark matter in the Milky Way halo.

4 The anchor and what it does

In Ref. [1] the four bonds of a trapped tetrahedral node split into one anchor and three valence

bonds. The anchor is described as the gluon junction through which charge, momentum, and

color ux are exchanged with the bulk. Choosing it breaks the

symmetry of the tetrahedron

. The three valence bonds carry the quark charges. Each projects onto the anchor axis

with weight

ˆr

· ˆr

= −1/3

, and a winding

w ∈ {0, 1}

adds one unit, so the baryon charge is

Q = −1 + W

with

W =

An anchor-free occupant is the same trapped node with all four bonds equivalent (Fig. 1c).

Three properties follow from the published construction; the third needs one assumption.

Charge.

Every charge in the model is the projection of a valence bond onto the anchor axis,

shifted by its winding. With no anchor there is no axis to project onto and no bond singled out

as a valence bond, so no charge can be assigned. The four bond directions also sum to zero,

i=A

ˆr

= 0,

(3)

so the occupant has no preferred direction from which a charge could arise. It is neutral.

Color.

The three colors come from the three skew-edge pairs of the bounding tetrahedron.

These pairs exist whether or not an anchor is chosen, so the anchor-free occupant has the same

internal color ux as a baryon and is a color singlet in the same way. What it lacks is the channel

through which that ux is exchanged with the bulk. It is therefore closed to color exchange with

other hadrons. Direct-detection limits require dark matter to have this property.

Mass.

Equation (1) is the product of 13 structural nodes and the

second neighbors each one

disrupts, less three skew-pair overlaps. None of these terms refers to the anchor, so the anchor-

free occupant has the same base count, 1836. One correction must be considered, which we call

the neutral-state correction. Ref. [8] adds

D = 3

to the neutron's count,

= 1836 + 3 = 1839

because the boundary of a neutral baryon looks like empty vacuum and the code must probe

the three color-ux arms of its Y-junction to tell it apart. We identify that Y-junction with the

anchor, which Ref. [1] calls the gluon junction. An anchor-free node then has no junction for the

probe to act on, so we take

= 1836

. The candidate depends on this choice. With

= 1839

the mass would be close to

, above

+ m

, and

would decay like a neutron.

Taking the count literally gives

= 1836 m

= 938.19

MeV, and applying the proton's

own 0.008% residual gives

= m

= 938.27

MeV. The model's neutronproton splitting,

= 1.53

MeV against the measured 1.29 MeV, shows that splittings between nearly equal

counts are reliable to about 0.25 MeV. We therefore quote

= 938.19



938.27

MeV with a

model uncertainty of about 0.25 MeV.

5 Modeling choices

The published papers disagree on, or do not settle, several points that matter here. Together

with the mass count above, we resolve them as follows.

Stability.

Ref. [1] states that one bond must couple to the surrounding

K = 12

bulk to remain

stable, while Ref. [2] calls the anchor-free state the ground state of the cage. We read the rst

statement as stability

as a baryon

: the anchor holds the charge conguration. The node itself

cannot leave the void, since the metric wall blocks extraction [1], and a single anchor-free node

has no channel through which to emit bulk excitations such as photons or mesons. We therefore

take the anchor-free occupant to be stable. Kinematics supports this: the decays

χ → p e

−

¯ν

and

p → χ e

require

> m

+ m

and

< m

− m

, and both are closed for

in the

predicted range.

Baryon number.

We take baryon number to count trapped nodes, anchored or not, so

= 1

We argue that this is the quantity the lattice conserves: the metric wall forbids creating or

removing a single trapped node, and choosing an anchor changes which bond plays which role

without adding or removing a node. It is also consistent with the result of Ref. [10] that species

is preserved under migration. This choice revises Ref. [2] in two places. First, the anchor-free

residue carries

B = 1

, not zero. Second, the octahedral occupant is also a trapped node with

B = 1

, so its merger into

γχ

′

changes

by one unit and is forbidden if

is conserved. The

octahedral-to-tetrahedral transition at 0.603 GeV conserves

Antiparticle.

Ref. [10] places the matter and antimatter label in the worldline orientation

rather than in the void orientation. An anchor-free occupant carries

like any other trapped

node, so it has a distinct antiparticle

¯χ

with

B = −1

. A

χ¯χ

pair at adjacent voids annihilates by

the cancellation described in Ref. [10]: both nodes leave their voids, the bonds they held return

to full coordination, and the energy goes into lattice modes. This is a two-node process and

does not need an anchor. It is the channel that removes the symmetric part of the population

(Section 7). The same rule would give the octahedral node a distinct antiparticle, in tension

with the self-conjugacy argued in Ref. [2]. We leave this open.

Spin.

The rotational site symmetry of a tetrahedral void is

∼

. Among half-integer spins

only

j = 1/2

stays irreducible under its double cover [10]. We take

to be a spin-1/2 fermion.

This is favored by the site symmetry, not derived.

6 The stability window

A neutral particle with

B = 1

and mass near the proton's is stable, and leaves ordinary matter

stable, only in a narrow window. Three bounds apply [3, 4]; the nuclear bound supersedes the

proton bound:



Proton stability,

p → χe

> m

− m

= 937.761

MeV.



Nuclear stability,

Be →

Be + χ

and similar:

> 937.900

MeV.



Dark-matter stability,

χ → p e

−

¯ν

< m

+ m

= 938.783

MeV.

The allowed range is 937.900938.783 MeV, a width of 0.88 MeV (Fig. 3).

937.6 937.8 938.0 938.2 938.4 938.6 938.8 939.0

dark-matter mass (MeV)

allowed window: stable nuclei, stable proton, stable

Be stable

937.900

938.783

−

937.761

= 938.272

SSM:

= 1836

Figure 3: The window in which nuclei, the proton and a neutral

B = 1

dark particle are all stable.

The SSM value for

= 1836

(orange bar: the range between the literal count, 938.19 MeV,

and

) lies inside it, 0.290.37 MeV above the lower edge and 0.510.59 MeV below the upper

edge. The light orange band shows the model uncertainty of about 0.25 MeV.

Khalaf and Kuik [4] study dark matter in this window. In their model, as in that of Fornal

and Grinstein [3], the mass is a free parameter that must be set by hand to fall in the window.

They call this a ne-tuning of about

2 × 10

−3

and suggest it may be anthropic. Mirror-matter

models avoid the tuning with a symmetry, at the cost of a complete second sector [5, 6]. In the

SSM the mass follows from structure. The anchored and anchor-free occupants are the same

trapped node, and the mass count does not see the anchor. The predicted range sits inside the

window. The upper margin is about twice the model uncertainty, but the lower margin only

slightly exceeds it, so a sharper calculation of the splitting is needed.

7 Abundance

With

= 1

and a distinct antiparticle,

shares the baryon asymmetry. Its present abundance

is the asymmetry carried by anchor-free nodes, as for baryons. This is the asymmetric dark-

matter mechanism [18, 19], in which dark matter, like ordinary matter, survives as a small excess

of particles over antiparticles. This sidesteps a problem of the octahedral candidate. Being its

own antiparticle, that candidate must stop annihilating at just the right abundance, and the

usual thermal way of doing so is excluded (Section 3). Both

and the baryons are what remains

after particleantiparticle annihilation, and neither needs a separate account of why it survived.

The observed densities of cold dark matter and baryons [16] give

Ω

0.1200

0.02237

= 5.36.

(4)

Because

≈ m

, this translates directly into a ratio of number densities, with

vis

the density

of ordinary (visible) baryons,

vis

≃ 5.4.

(5)

Equivalently,

5.36/6.36 ≈ 84%

of trapped tetrahedral nodes must be anchor-free. We do not

derive this. Any account must explain why anchoring was the less common outcome at crystal-

lization.

8 Signatures

The anchor-free and anchored states dier by a single bond role. A transition between them is

a change of anchoring. We describe its strength by one amplitude, written as a mixing angle

between

and

, which the model does not yet predict. The laboratory and stellar signatures

below scale as

No gamma-ray lines.

Asymmetric dark matter has almost no antiparticles left to annihilate

with. The candidate predicts no annihilation line. The null results of Section 3 are what it

expects.

Hydrogen decay.

Within the window, atomic hydrogen can decay by electron capture into

the dark particle [20],

H → χ ν

, E

= m

+ m

− m

≈ 0.51



0.59

MeV

(6)

conserving charge, baryon number and lepton number. The radiative mode

H → χνγ

is limited

by a recast of data from the Borexino solar-neutrino detector. McKeen and Pospelov nd

that the limit restricts the neutrondark-particle mixing to

θ ≲ 10

−11

over a wide range of

parameters [20].

Neutron dark decay.

A free neutron can decay as

n → χγ, E

− m

≈ 1.29



1.37

MeV

(7)

The hydrogen limit above caps the branching fraction near

−4

, well below the roughly 1%

needed to explain the dierence between beam and bottle measurements of the neutron life-

time [20, 21]. That dierence is about

4.5σ

: the most precise beam measurement gives

887.7 ±

2.2

s [22] and the most precise bottle measurement

877.75 ±0.34

s [23]. The candidate does not

explain it. A search for a monochromatic photon near 1.3 MeV from ultracold neutrons, at the

−5



−4

branching level, would test the part of parameter space the hydrogen bound allows,

if the neutron-star bound below turns out to be weaker.

Neutron stars.

Inside a neutron star, neutrons can convert to

. Fast conversion would soften

the equation of state so much that two-solar-mass stars could not exist [24, 25]; slow conversion

heats old, cold stars. Ref. [4] nds that heating of the coldest known pulsar gives the strongest

present bound in its model. We have not translated that bound into

for this candidate. It is

likely to be the most restrictive.

Direct detection and self-interaction.

An anchor-free node has no channel for exchanging

momentum or color ux with ordinary hadrons at leading order. Its scattering on nuclei and on

itself is expected to be suppressed, and its halo collisionless. The size of the residual coupling is

not computed here.

9 Comparison

Table 2 sets the candidate beside the octahedral one and the minimal proton-mass model of

Khalaf and Kuik.

Table 2: The two SSM candidates and the minimal proton-mass model of Ref. [4].

octahedral anchor-free tetrahedral KhalafKuik (

)

mass 1.719 GeV 938.19938.27 MeV set by hand in window

origin of mass count 3364 count 1836 free parameter

charge 0 0 0

baryon number 1 here; 0 in [2] 1

−1

(and

L = −1

)

spin not xed 1/2 (favored) 0

own antiparticle yes in [2]; in tension no no

abundance open (thermal excluded) asymmetric; ratio 5.4 open freeze-in

gamma-ray line 1.591 GeV, not seen none none

lab signature none identied

H → χν

n → χγ H → ϕ

∗

The two SSM candidates are not exclusive. If both kinds of trapped node exist, the octahedral

one could be a minor or unstable component. In Ref. [2] the merger residue

′

is the anchor-free

occupant studied here; under the baryon-number rule of Section 5 that merger is forbidden, so

the two candidates are not linked by it.

10 Open problems

The neutral-state correction.

The candidate requires that the

probe Ref. [8] applies to

the neutron does not apply to an anchor-free node. If it does,

is unstable. This is the

most important point to settle.

The anchoring amplitude.

The mixing

sets every laboratory signature. The model must

give a reason it is below

−11

, and neutron-star heating probably requires smaller.

The abundance ratio.

About 84% of trapped tetrahedral nodes must be anchor-free. This

needs a dynamical account of anchoring at crystallization.

Stability and baryon number.

Both are modeling choices made here to resolve a disagree-

ment between Refs. [1] and [2]. They should be derived from the lattice dynamics.

Spin.

The site symmetry favors

j = 1/2

without determining it [10].

Mass splitting.

The literal count and the measured proton mass dier by 0.08 MeV, and

the model's splittings are reliable to about 0.25 MeV. A sharper calculation of

− m

would x the neutrino and photon energies and show how securely

sits above the nuclear

bound.

11 Conclusions

The SSM's baryon construction, applied to a tetrahedral node with no anchor, gives a neutral

particle closed to color exchange with the proton's structural count. With baryon number

counting trapped nodes, it is asymmetric dark matter with a mass close to

, inside the

narrow window where it, the proton and nuclei are all stable. In minimal models of proton-mass

dark matter the mass is a free parameter that must be set by hand to fall in that window,

and mirror-matter models obtain it by duplicating the Standard Model; here it follows from the

count, provided the neutral-state correction applied to the neutron does not apply to an anchor-

free node. The candidate predicts no gamma-ray lines, consistent with present data, and can be

tested by hydrogen decay, rare neutron decay and neutron-star heating. Its open problems are,

in order of importance, the neutral-state correction, the anchoring amplitude, and the fraction

of nodes left without an anchor.

Code and data availability

The analysis scripts, the in-browser lter used to build the full-mission spectra, the spectra

themselves (

hists.csv

), and the scripts for Figs. 13 are available as a single archive at

https:

//github.com/raghu91302/ssmtheory/blob/main/anchorfree_dm_scripts.zip

. The

Fermi-LAT photon data are public from the Fermi Science Support Center and the HEASARC

archive.

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Proton-mass dark matter from an anchor-free tetrahedral defect - good file name