Dark Matter Quantum Numbers from Interstitial Site Symmetry

Dark Matter Quantum Numbers from Interstitial Site

Symmetry:

Electromagnetic Neutrality and Self-Conjugacy from the

Centrosymmetry of the Octahedral Void

Raghu Kulkarni

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

June 2026

Abstract

In the Selection-Stitch Model (SSM), baryonic matter and dark matter are trapped nodes

in the two interstitial voids of the FCC vacuum lattice: the proton in the tetrahedral void,

the dark matter candidate in the octahedral void. The companion papers derive the masses

and the color structure of both defects from the combinatorics of their bonded graphs.

Here we identify a second, distinct structural origin for the remaining quantum numbers.

We show that the electromagnetic and matter/antimatter properties of a trapped defect

are xed not by its bonded graph but by the point-group symmetry of its interstitial

site

and that a single discriminatorwhether the site is an inversion centerseparates the two

sectors. The octahedral void is centrosymmetric (

); the tetrahedral void is not (

We prove that centrosymmetry forces the rst-order electric dipole moment to vanish as a

protected selection rule, valid for

any

charge arrangement consistent with the site symmetry,

not merely the symmetric one. Under a stated identication of charge conjugation with

the site's parity structure, the same centrosymmetry forces the defect to be self-conjugate

(Majorana-type) [4]. The non-centrosymmetric tetrahedral site provides neither protection:

its dipole is generic and a distinct antiparticle exists, matching the proton. Extending the

analysis to the full multipole tower, we nd that the magnetic dipole is also forbiddennot

by centrosymmetry, to which an axial vector is blind, but by the cubic rotational symmetry

and, more robustly, by self-conjugacyand that the unique surviving electromagnetic form

factor is a momentum-suppressed anapole moment, the one coupling a Majorana particle can

carry. We are explicit about the epistemic status of each resultthe dipole theorem is pure

geometry; the self-conjugacy theorem rests on the stated charge-conjugation modeland

we show that the color property does

not

reduce to site symmetry but to the bonded-graph

combinatorics of the companion work. The electromagnetic and conjugation properties of

dark matter thus follow from one geometric fact: the dark defect occupies an inversion center

of the vacuum lattice.

1 Introduction

The Selection-Stitch Model (SSM) [1, 2] treats the physical vacuum as a face-centered cubic

(FCC) crystallization of spacetime, and matter as the places where that crystallization is in-

complete: a node trapped below bulk coordination at an interstitial site, unable to stitch into

the surrounding lattice. The FCC lattice provides exactly two interstitial void types, and the

raghu@idrive.com

program identies a particle with each. The proton is a

K = 4

remnant in the tetrahedral

void [1]; the dark matter candidate is a

K = 6

remnant in the octahedral void [3].

The masses of both defects, and the color structure of the proton, are derived in the com-

panion papers from the combinatorics of the defect's bonded graph: the proton-to-electron mass

ratio from the tetrahedral graph

, the dark matter mass from the octahedral graph

2,2,2

and the three QCD colors from the skew-pair count of

. These are

internal

propertiesthey

depend on how the defect's own bonds are arranged, and they are computed by counting on the

graph.

This paper isolates a second and structurally distinct origin for the

remaining

quantum

numbers. We show that the electromagnetic behavior and the matter/antimatter (conjugation)

character of a trapped defect are xed not by its bonded graph but by the point-group symmetry

of the interstitial

site

it occupiesa property of the defect's

placement

in the lattice rather than

of its internal bonding. The central observation is that a single feature of the sitewhether

it is an inversion centerseparates the two sectors and accounts, by one mechanism, for two

properties that the companion work states but does not derive in this way.

The octahedral void is centrosymmetric: it possesses a center of inversion, and its point

group is

. The tetrahedral void is not: it has no inversion center, and its point group is

We prove that this dierence forces the dark defect's rst-order electric dipole moment to vanish

identically, as a selection rule protected by the site symmetry, while the proton's site provides no

such protection. Under a stated identication of charge conjugation with the parity structure

of the site, the same centrosymmetry forces the dark defect to be its own antiparticle, while

the proton's non-centrosymmetric site admits a distinct antiparticle. We are careful throughout

to separate what follows from pure geometry from what follows under a physical model, and

we show explicitly that the color property belongs to the graph-combinatorial category, not the

site-symmetry one.

The picture that organizes the paper is the contrast between the two defects' placements.

The tetrahedral remnant sits within a single coordination cluster of the lattice; the octahedral

remnant sits symmetrically about an inversion center, its six bounding nodes falling into three

antipodal pairs (Figure 1). It is this symmetric placementvisualizable as a defect shared evenly

across the interface between adjacent cellsthat carries the electromagnetic and conjugation

content. The remainder of the paper makes that statement precise.

What this paper claims, and does not.

We claim that (i) the octahedral interstitial site is

centrosymmetric and the tetrahedral is not; (ii) centrosymmetry forces the vanishing of the rst-

order electric dipole as a protected selection rule, which we prove as pure geometry; (iii) under

a stated charge-conjugation model, centrosymmetry forces self-conjugacy; (iv) the permanent

magnetic dipole and the low multipoles are likewise forbidden, with the anapole the unique

surviving electromagnetic moment; and (v) the color property does not reduce to site symmetry

but to the bonded-graph combinatorics of Refs. [1, 3]. We do not claim that all quantum

numbers reduce to a single fact; on the contrary, we identify two distinct structural origins and

assign each property to its own. We do not re-derive the masses, which are established in the

companion work and used here only as context.

2 The two interstitial sites and their point groups

The FCC lattice with cubic constant

has atoms at the cube corners and face centers, with

nearest-neighbor distance

L = a/

√

. It admits two interstitial void types.

The

tetrahedral void

is bounded by four lattice nodes forming a regular tetrahedron of edge

. Taking the representative void at

(a/4)(1, 1, 1)

, its four bounding nodes are

{(0, 0, 0), (

, 0), (

, 0,

), (0,

)},

(1)

and the site symmetry is the tetrahedral point group

The

octahedral void

is bounded by six lattice nodes forming a regular octahedron of edge

Taking the body-center void at

(

)

, its six bounding nodes are

{(

, 0), (

, a), (

, 0,

), (

, a,

), (0,

), (a,

)},

(2)

and the site symmetry is the full octahedral point group

The single structural fact this paper turns on is the following.

Proposition 1

(Site centrosymmetry)

The octahedral void is a center of inversion of the

bounding-node set: under

r 7→ 2c − r

about the void center

, the six bounding nodes map

onto themselves, falling into three antipodal pairs. The tetrahedral void is not: no inversion

about its center maps its four bounding nodes onto themselves.

Proof.

Direct computation. For the octahedral void, inversion about the center sends each

bounding node to the diametrically opposite one (

(

, 0) ↔ (

, a)

, and the two analogous

pairs), so the set is invariant and partitions into three antipodal pairs. For the tetrahedral void,

inversion about the centroid maps the four vertices to four points that are not lattice nodes of

the void (a regular tetrahedron has no central inversion symmetry, its point group

lacking

the element

), so the set is not invariant.

The contrast in Proposition 1 is the origin of everything that follows.

= T

× {e, i}

contains the inversion

;

does not. We now show that this single group-theoretic dierence

xes the electromagnetic dipole and the conjugation character.

Tetrahedral void (proton site)

no inversion center

Octahedral void (dark matter site)

inversion center

Figure 1: The two interstitial sites and their symmetry.

Left:

the tetrahedral void (proton site),

four bounding nodes, point group

, no inversion center.

Right:

the octahedral void (dark

matter site), six bounding nodes in three antipodal pairs about a center of inversion, point group

. The trapped defect (center) sits at the inversion center of the octahedral site; the bounding

nodes' position vectors sum to zero by the antipodal pairing.

3 Theorem 1: the vanishing dipole is a protected selection rule

The electric dipole moment of a localized charge distribution on the bounding nodes is

d =

− c),

(3)

Tetrahedral void location in one FCC cell

(proton site)

Octahedral void location across two FCC cells

(dark matter site)

Figure 2: Location of the two void sites within the FCC lattice (supplemental to Figure 1).

Atoms and their nearest-neighbor bonds are drawn uniformly in gray; the diagonal bond pattern

is the signature of the face-centered structure. In each panel the trapped defect (green) sits in

the void and is connected by dotted lines to its bounding nodes (highlighted), the lattice atoms

that dene the void.

Left:

the tetrahedral void within a single FCC cell, four bounding nodes

(blue), proton site.

Right:

the octahedral void across two FCC cells stacked along a shared

face (faint plane), six bounding nodes (amber), the site shared symmetrically across the interface

(dark matter site). This gure locates the sites; their bounding-node geometry and symmetry

are detailed in Figure 1.

where

is the eective charge on bounding node

and

− c

its position relative to the void

center. We make no assumption about the values

beyond their being consistent with the site

symmetry, i.e. invariant under the operations of the site point group. This is the key point: the

result below is not that a particular (e.g. uniform) charge arrangement has no dipole, but that

site-symmetric arrangement can.

Theorem 1

(Dipole selection rule)

At a centrosymmetric site, the rst-order electric dipole mo-

ment of any charge distribution invariant under the site point group vanishes identically. At the

octahedral void this forbids a rst-order dipole for the dark defect. At the non-centrosymmetric

tetrahedral void no such constraint holds, and a site-symmetric charge distribution generically

carries a nonzero dipole.

Proof.

At a centrosymmetric site, inversion

about the center is a symmetry operation, and

it pairs each bounding node

with its antipode

′

− c = −(r

− c)

. Site-symmetry

invariance of the charge distribution requires

′

= q

(the inversion maps the conguration to

itself). Then the contributions of each pair cancel:

− c) + q

′

− c) = q

− c) + q



− (r

− c)



= 0,

(4)

and summing over the three antipodal pairs of the octahedron gives

d = 0

. The cancellation

uses only the pairing and the equality

′

= q

forced by inversion symmetry; it is independent

of the actual values

. In representation-theoretic terms [8, 9], the dipole operator transforms

as the parity-odd vector representation (

), which has zero overlap with the parity-even

symmetric ground conguration (

); the matrix element vanishes by parity.

At the tetrahedral site there is no inversion to pair the nodes, so no equality among the

forced by parity, and the four position vectors do not cancel pairwise. A generic site-symmetric

charge distribution therefore has a nonzero dipole.

We veried Theorem 1 numerically as a guard against algebraic error: over

5000

random

charge assignments to the octahedral nodes, each made inversion-symmetric by equating an-

tipodal pairs, the dipole magnitude was zero to machine precision (

< 10

−15

); over

5000

random

charge assignments to the tetrahedral nodes, the dipole magnitude ranged from

0.009

0.456

in units of

, i.e. generically nonzero.

The physical reading is that the dark defect's electromagnetic neutrality at leading (dipole)

order is not a property that must be arranged but one that cannot be avoided: it is protected by

the centrosymmetry of the site. The proton's site oers no such protection, consistent with the

proton's nonzero charge structure derived in Ref. [1]. We emphasize the scope: this establishes

the absence of a

rst-order dipole

coupling. It does not exclude higher-multipole, loop-induced,

or mediated electromagnetic interactions, which remain open and are expected to be present

but suppressed; the claim is neutrality at leading order, protected by parity.

4 Theorem 2: self-conjugacy from the site's parity structure

A particle is self-conjugate (Majorana-type) when it carries no conserved label that charge

conjugation

would reverse to produce a distinct antiparticle. In the lattice-defect picture, the

labels that distinguish a baryon from its antibaryonfractional electric charge and colorare

orientational (signed) assignments on the defect's bonds, and the antiparticle is the sign-reversed

conguration. The defect is self-conjugate precisely when no such signed assignment survives

the site symmetry.

To make this precise we must state what

is at the level of a lattice defect. We adopt

the identication, consistent with the parity-sector analysis of the electromagnetic coupling in

Ref. [1] and with the particle/antiparticle picture of the SSM, that charge conjugation acts on

the signed bond labels as the site's inversion (parity) operation acts on the bounding nodes:

reverses the orientation that inversion would exchange. This is a physical model of

, not a

theorem, and we ag it as such; the result below holds under this identication.

Theorem 2

(Self-conjugacy at a centrosymmetric site)

Under the stated identication of charge

conjugation with the site's parity structure, a defect at a centrosymmetric site carries no signed

(particle versus antiparticle) charge: its symmetric conguration is invariant under

, so the

defect equals its own conjugate and is self-conjugate. A defect at a non-centrosymmetric site is

not so constrained and admits a distinct antiparticle.

Proof.

Decompose any bond labeling

on the bounding nodes into parts even and odd under

the site inversion,

q = q

(g)

+ q

(u)

, with

(g)

(q + P q)

and

(u)

(q − P q)

, where

is the

inversion (antipodal exchange). The signed,

-odd contentthe part that distinguishes particle

from antiparticle under the stated identicationis

(u)

. At a centrosymmetric site, the physical

defect occupies the symmetric ground conguration, which is inversion-invariant:

P q

(g)

= q

(g)

hence its odd content is

(g)

− P q

(g)

)/2 = 0

. There is no signed charge to reverse;

acts

trivially and the defect is self-conjugate. At a non-centrosymmetric site there is no inversion

pairing, the decomposition into

and

sectors under a site inversion does not exist, and a

signed labeling is unconstrained: a distinct

-conjugate conguration exists, and the defect is

not self-conjugate.

We veried the computation underlying Theorem 2: projecting

5000

random symmetric con-

gurations of the octahedral nodes onto their inversion-odd part returned zero to machine preci-

sion, conrming that no signed charge survives the centrosymmetric site. The result matches the

phenomenology the companion work assigns to the two sectors: the proton (tetrahedral, non-

centrosymmetric) has a distinct antiproton, while the dark defect (octahedral, centrosymmetric)

is its own antiparticlea Majorana-type dark matter candidate requiring no particle/antiparticle

asymmetry in the dark sector.

We restate the epistemic status plainly. Theorem 1 is pure geometry: it uses only the antipo-

dal pairing and parity, with no physical model beyond the identication of the dipole operator

with the position-weighted charge sum. Theorem 2 additionally requires the identication of

charge conjugation with the site's parity structure, which is a physical model. A reader who

rejects that identication retains Theorem 1 in full; the self-conjugacy result stands or falls with

the model of

, and we present it as such.

5 The multipole tower: which electromagnetic moments survive

Theorem 1 forbids the rst-order electric dipole. It is natural to ask how far the prohibition

extendswhether a magnetic dipole, or a higher multipole, could carry a faint electromagnetic

signature accessible to direct-detection experiments. The answer requires care, because the

electric and magnetic dipoles transform dierently under inversion, and the centrosymmetry

argument of Theorem 1 does

not

transfer directly to the magnetic case. We work through the

leading moments and identify exactly what is forbidden, by which symmetry, and what survives.

Electric versus magnetic dipole: dierent transformations.

The electric dipole

is a

polar vector: under inversion

it changes sign,

d 7→ −d

, which is precisely why a centrosym-

metric site forbids it (Theorem 1). The magnetic dipole

is an axial (pseudo)vector: under

inversion it is

invariant

µ 7→ +µ

. Inversion alone therefore places no constraint on a magnetic

dipole, and the argument of Theorem 1 cannot be carried over to it. Centrosymmetry is blind

to axial vectors.

The magnetic dipole is nonetheless forbidden, by two independent routes.

First, by

the full point group: in

a polar vector transforms as

and an axial vector as

, and

neither contains the trivial representation

. A permanent moment requires its operator to

contain

, so the static

site supports neither a permanent electric nor a permanent magnetic

dipole. The electric prohibition is carried by the inversion element; the magnetic prohibition is

carried by the cubic rotational axes. The conclusion is the same but the mechanism is dierent,

and we are explicit about which symmetry does the work in each case.

Second, and more robustly, by self-conjugacy: a magnetic dipole moment reverses sign under

charge conjugation,

µ 7→ −µ

. If the dark defect is the self-conjugate (Majorana-type) state of

Theorem 2, a permanent magnetic dipole is forbidden by

-invariance: the moment would have

to equal its own negative. The same fact that makes the defect its own antiparticle forbids its

magnetic dipole. This route does not depend on the spatial point group at all, and it shows

that Theorem 2, granted its premise, does more than establish self-conjugacy: it closes the

magnetic-dipole channel as well.

The higher tower.

The same

criterion disposes of the next moments. The electric

quadrupole transforms as

+ T

and the magnetic quadrupole as

+ T

under

; neither

contains

, so both permanent quadrupoles are forbidden. The electric monopole (charge)

transforms as

and is therefore symmetry-allowed, but it is set to zero independently by the

neutrality established in the companion work [3]. The permanent-multipole tower is thus empty

through the orders that direct detection would probe coherently.

What survives: the anapole.

One electromagnetic form factor remains compatible with all

of the above, and it is the unique one a Majorana particle can carry [5, 6, 7]: the anapole (toroidal

dipole) moment. The anapole is parity-odd and charge-conjugation-odd, the same quantum

numbers as the source current

rather than the charge density, and it is the single moment that

survives self-conjugacy. Physically it corresponds to a toroidal current congurationa solenoid

bent into a ringand it couples not to the electromagnetic eld directly but to its sources,

i.e. to the nuclear current in a scattering event. Its hallmark is momentum suppression: the

anapole coupling vanishes as the momentum transfer

q → 0

, scaling as

relative to a standard

charge coupling, which gives a recoil-energy dependence distinct from ordinary spin-independent

scattering.

The anapole is therefore the one genuine, if faint, electromagnetic handle on this candidate.

We state its status precisely. Its existence is contingent on the defect being spin-

: a scalar

(spin-

) defect has no magnetic moment and no anapole, so this channel presumes the fermionic

interpretation that underlies Theorem 2. And while symmetry

permits

the anapole, it does not

x its

magnitude

: the coupling strength depends on how the defect couples to lattice currents,

which is not derived here and which we identify, consistently with the companion work, as

awaiting the defect's strain prole. The result of this section is thus a symmetry statementthe

permanent electric and magnetic dipoles and the low multipoles are forbidden, the anapole alone

is allowedand not yet a prediction of an observable rate. A defect that is electromagnetically

silent at every permanent-multipole order, and whose only allowed coupling is a momentum-

suppressed anapole, is by construction extraordinarily weakly interacting, consistent with its

having escaped detection.

6 Why color does not reduce to site symmetry

It is natural to ask whether the remaining quantum numbercoloralso reduces to the site

symmetry, completing a single-fact account of the property list. It does not, and the reason

is instructive: it locates color rmly in the graph-combinatorial category rather than the site-

symmetry one.

In the SSM the three QCD colors arise because the tetrahedral graph

has exactly three

skew (disjoint) edge pairs, in bijection with three conning channels [1]. The octahedral graph

2,2,2

has thirty skew pairs, with no three-fold bijection, so the baryonic color-generating mech-

anism does not carry over [3]. These countsthree and thirtyare properties of the bonded

graphs, established by direct enumeration.

One might hope to recover them from the site point groups, but the attempt fails cleanly.

Both site groups possess three-dimensional irreducible representations:

has two (

and

has four (

, T

). The existence of a three-dimensional irrep therefore does

not distinguish the two sites and cannot, by itself, explain why the tetrahedral defect carries

three colors and the octahedral does not. The decisive quantity is the skew-pair count, which is

combinatorial, not the parity structure that xes the dipole and the conjugation character.

The site symmetry does

illuminate

the color structure without

determining

it. Under

the thirty octahedral skew pairs split into two orbits, of sizes six and twenty-four, and neither

is a three-element orbit; this is consistent with the absence of a three-color bijection. Under

the three tetrahedral skew pairs form a single orbit of size three, consistent with their bijection

to three colors. So the orbit structure under the site group is in harmony with the color count,

but the count itself is set by the graph combinatorics of the companion work, not by whether

the site is an inversion center.

This is the cleanest statement of the paper's architecture. A trapped defect's quantum

numbers have two structural origins. The mass and the color follow from the combinatorics

of the bonded graphthey are internal properties, xed by how the defect's own bonds are

arranged. The electromagnetic dipole and the conjugation character follow from the point-

group symmetry of the sitethey are placement properties, xed by where the defect sits in

the lattice. The inversion center is the discriminator for the second category and the second

category only.

7 Discussion

The result of this paper is a partition. The qualitative properties that distinguish the dark octa-

hedral defect from the baryonic tetrahedral one fall into two groups with two distinct structural

origins. The mass and the absence of the baryonic color mechanism are graph-combinatorial:

they are computed by counting on the bonded graph, and they are derived in the companion

papers [1, 3]. The electromagnetic and conjugation propertiesthe vanishing electric dipole,

the self-conjugate character, the forbidden magnetic dipole and low multipoles, and the anapole

as the lone survivorare site-symmetric: they follow from one fact, that the octahedral void is

a center of inversion while the tetrahedral void is not. The electromagnetic content reduces to

a single statement: the defect is silent at every permanent-multipole order, and its only allowed

coupling is a momentum-suppressed anapole, contingent on the fermionic interpretation.

The asymmetry in rigor between the two site-symmetric results is real and worth restating.

The dipole selection rule (Theorem 1) is pure geometry, as rm as the parity argument that

forbids a permanent dipole in any centrosymmetric system. The self-conjugacy result (Theo-

rem 2) is a theorem under a stated identication of charge conjugation with the site's parity

structure; it is as secure as that identication, which we regard as physically natural but do

not derive from a more primitive principle. A sharpened account of charge conjugation in the

SSMideally one that derives the action of

on bond labels from the lattice dynamics rather

than positing itwould settle the status of the self-conjugacy result, and we identify it as the

natural next step.

One implication of Theorem 1 deserves to be drawn out, because it marks a genuine departure

from how neutrality is understood in the Standard Model. There, a particle's electric neutrality

follows from the Gell-MannNishijima relation

Q = T

+ Y /2

: it requires a specic arrangement

of weak isospin and hypercharge assignments, the hypercharge tied to the electroweak and

Higgs sector, and those assignments are further constrained by anomaly cancellation across a

fermion generation. Neutrality is, in that account, an electroweak and representation-theoretic

statement. The mechanism here is entirely dierent. The dark defect's neutrality follows from

the inversion symmetry of its interstitial sitepure three-dimensional spatial geometrywith

no reference to weak isospin, hypercharge, a Higgs coupling, or anomaly cancellation. The

bounding-node contributions sum to zero because the site is a center of inversion, and that

is the whole of it. The dark sector's electromagnetic quietness is therefore decoupled from

the electroweak sector: it need not participate in anomaly cancellation and requires no tuned

Higgs-sector charge assignment to be neutral. We state the scope precisely. This is a geometric

protection of neutrality at the level of the dipole and the permanent multipole tower (Section 5),

not a derivation of a quantized total charge from electroweak principles; the anapole survives

and loop-induced couplings remain open, so the defect is electromagnetically quiet rather than

sterile. What is geometric is the

mechanism

of neutrality, and that mechanism owes nothing to

the electroweak structure that supplies it in the Standard Model.

The broader signicance is for the coherence of the program. The companion papers show

that one mechanisma node trapped in an interstitial voidyields both the proton and the

dark matter candidate, with their masses xed by the respective graph counts. This paper

adds that the same geometry xes their electromagnetic and conjugation properties through a

single, orthogonal discriminator: the symmetry of the site. The matter sector occupies a non-

centrosymmetric site and is charged, colored, and has a distinct antiparticle; the dark sector

occupies a centrosymmetric site and is dipole-free and self-conjugate. The contrast between

visible and dark matter, in this picture, is at bottom the contrast between two interstitial sites

of one latticeone with a center of inversion and one without.

Data and code availability

The single self-contained Python script

verify_quantum_numbers.py

(NumPy only) reproduces

every computational claim in this paper: Proposition 1 (site centrosymmetry), Theorem 1 (the

dipole selection rule and the tetrahedral contrast), Theorem 2 (self-conjugacy under the stated

charge-conjugation identication), the skew-pair counts and orbit split of Section 6, and the mul-

tipole/anapole bookkeeping of Section 5. The script is available in the SSMTheory repository:

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

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