A geometry of fermions

A geome y o e mions
F ançois Ri e *,1
1Depa men o Geosciences and Na u al Resou ce Managemen , Uni e si y o Copenhagen; Øs e
Voldgade 10, Copenhagen K., 1350, Denma k.
* i e . [email protected]
Con ex & disclaime . This s udy was conduc ed ou side my p ima y ield o expe ise and ou side
wo king hou s. I began as a Sa u day-nigh explo a ion o a e mionic puzzle, which g adually
e ol ed in o a cohe en sys em and, e en ually, a en a i e law whose esul s became oo in iguing o
lea e unpublished. I acknowledge he lack o o mal heo e ical igo , as he app oach is pu ely
geome ical — a ea u e ha can be conside ed bo h a s eng h and a limi a ion. This wo k was
ca ied ou independen ly, in an e o o p ese e a esh pe spec i e and explo e he geome ical
pa e ns as a as possible. I hope he communi y will ecei e his ini ia i e wi h openness and
unde s anding — aking wha is use ul, ejec ing wha is lawed, and, whe e possible, in eg a ing i
in o he igo ous ounda ions o he S anda d Model.
F ançois Ri e
Abs ac
Longs anding ques ions in he S anda d Model’s e mionic sec o — including he unexplained
di e si y and o igin o quan um numbe s, he e mion mass hie a chy, and he mechanism gene a ing
neu ino masses — a e add essed h ough a pu ely geome ical, phenomenological amewo k. The
cons uc ion elies solely on expe imen al inpu s — measu ed e mion masses and quan um
assignmen s — and in oduces ou disc e e numbe s, o seeds (𝑛,𝑘,𝑛′,𝑘′), om which bo h quan um
numbe s and mass ela ions eme ge. Quan um numbe s a e exp essed as second-o de polynomials in
(𝑘,𝑘′), while e mion masses ollow a uni e sal exponen ial law depending only on he seeds and six
cons an s (𝑚𝑒,𝛼,𝛽,𝛼′,𝛽′,𝜆). Wi h he elec on mass 𝑚𝑒 ixed as e e ence, all known e mion masses
a e ep oduced wi hin ins umen al unce ain ies, showing ha a six-pa ame e model su ices o
cap u e he ull hie a chy. The amewo k ex ends na u ally o he case o Di ac- ype neu inos: a
geome ical singula i y in he limi 1/𝑌𝑅→∞ wi h 𝑌𝑅 he igh -handed hype cha ge allows gene a ion
o ex emely small masses consis en wi h cu en obse a ions. Fla o mixing — in he Cabibbo–
Kobayashi–Maskawa (CKM) and Pon eco o–Maki–Nakagawa–Saka a (PMNS) ma ices — eme ges
om ansi ions in he (𝑛,𝑘) seed space, o e ing a uni ied geome ical explana ion o qua k and
lep on la o s uc u es. Th ee main p edic ions a ise. (i) A new cha ged lep on wi h elec on-like
p ope ies and p edic ed mass 8.5938−0.0004
+0.0004 MeV. Rema kably, a 2024 expe imen al epo (Anikina,
Niki in & Rikh i sky, 2024) obse ed a lep on-like s a e a 8.5±2.5 MeV, which hey named he
anomalon (𝑎−), consis en wi h his p edic ion. This obse a ion, i con i med, would cons i u e he
i s expe imen al alida ion o he geome ical mass amewo k. (ii) A hi d e mionic b anch, he
leonids, eme ges o ex ended seed alues, hin ing a hidden o da k sec o s. (iii) A ou h b anch, he
apex, es o es gauge anomaly cancella ion; hese s a es ca y colo 8, o ming a e mionic coun e pa
o he gluons and comple ing he geome ical spec um. In pa allel, he na u al eme gence o he
golden a io
𝜑 and i s conjuga e 𝜑∗ in le - igh symme ic s a es sugges s a deep algeb aic symme y
unde lying e mionic o ganiza ion. Fu u e expe imen al sc u iny, pa icula ly o he anomalon, will
de e mine whe he his geome ical amewo k unco e s he missing s uc u e o ma e .
In oduc ion
The ma e con en o he S anda d Model (SM) consis s o e mions, which include up- ype qua ks
(up 𝑢, cha m 𝑐, op 𝑡), down- ype qua ks (down 𝑑, s ange 𝑠, bo om 𝑏), cha ged lep ons (elec on 𝑒−,
muon 𝜇−, au 𝜏−), and hei associa ed neu inos (𝜈𝑒, 𝜈𝜇, 𝜈𝜏). The SM p o ides an accu a e desc ip ion
o hei in e ac ions and p ope ies, yielding p edic ions o obse ed phenomena anging om be a
decay o high-ene gy collide p ocesses (Glashow, 1961). Despi e i s well-es ablished success o e
mul iple decades, he SM s ill lea es se e al undamen al ques ions abou e mions unanswe ed
(Al mannsho e & G eljo, 2024). In pa icula , ou cen al gaps can be highligh ed:
O igin o quan um numbe s. In he SM, quan um numbe s such as hype cha ge ( o le - and igh -
handed e mions, 𝑌𝐿 and 𝑌𝑅), le -handed weak isospin (𝑇3𝐿), ba yon numbe (𝐵), and lep on numbe
(𝐿) a e no de i ed om i s p inciples; a he , hey a e assigned o ensu e in e nal consis ency. Thei
alues a e chosen o gua an ee gauge anomaly cancella ion, o p ese e empi ical conse a ion laws
(such as 𝐵 and 𝐿 a he pe u ba i e le el, see Hoo , 1976), o de e mine he embedding o e mions
wi hin SU(2) double s ( ia he weak isospin 𝑇3𝐿) and o ep oduce he obse ed elec ic cha ges (𝑄) o
he e mions. Howe e , he SM p o ides no deepe explana ion o how hese pa icula quan um
numbe s eme ge.
Fe mion masses and Yukawa couplings. In he SM, e mion masses o igina e om Yukawa
in e ac ions wi h he Higgs ield a e elec oweak symme y b eaking (Higgs, 1964). Howe e , he
Yukawa couplings hemsel es a e a bi a y ee pa ame e s, ixed empi ically o each pa icle
(Weinbe g, 1967). Thus, while he SM accommoda es he obse ed e mion masses, i does no p edic
hem, lea ing he o igin o he mass hie a chies and he pa e n o h ee gene a ions unexplained.
Neu inos. Neu inos pose a undamen al challenge o he SM, as hei unique p ope ies a e no
na u ally accommoda ed wi hin he amewo k. They a e he only e mions in he SM wi h anishing
elec ic cha ge (𝑄=0). In he minimal SM hey a e s ic ly massless, since igh -handed neu ino
ields a e absen and no Yukawa couplings o he Higgs can be w i en. Expe imen ally, howe e ,
neu ino oscilla ions demons a e ha neu inos ha e nonze o bu ex emely small masses, many
o de s o magni ude below hose o cha ged e mions (Fukuda e al., 1998). This disc epancy indica es
ha he SM is incomple e and sugges s he exis ence o new physics, such as igh -handed neu inos o
al e na i e mass-gene a ion mechanisms (e.g. seesaw models, see Minkowski, 1977). This s uc u al
disc epancy be ween neu inos and he o he e mions challenges he SM’s uni ied desc ip ion o
ma e ields.
Fla o s uc u e. The SM inco po a es qua k mixing h ough he Cabibbo–Kobayashi–Maskawa
(CKM) ma ix, which go e ns la o -changing weak in e ac ions (Cabibbo, 1963). While he
o malism is consis en wi h expe imen , he obse ed hie a chies among he CKM elemen s lack
heo e ical explana ion wi hin he SM (Kobayashi & Maskawa, 1973). The lep onic analogue is he
Pon eco o–Maki–Nakagawa–Saka a (PMNS) ma ix, which desc ibes neu ino oscilla ions (Maki e
al., 1962). The PMNS encodes he misma ch be ween neu ino mass eigens a es and weak in e ac ion
eigens a es. In con as o he CKM case, he PMNS ma ix exhibi s la ge mixing angles and a
ma kedly di e en s uc u e. A p esen , he SM p o ides no mechanism ha ela es he wo ma ices
o accoun s o he s iking dispa i y be ween qua k and lep on la o mixing.
The aim o his s udy is o add ess hese longs anding open ques ions in pa icle physics by adop ing a
pu ely phenomenological and geome ical app oach cons uc ed solely om he obse ed e mion
masses and he quan um numbe s. Th ee main esul s can be summa ized as ollows:
- Common eme gence o quan um numbe s and e mion masses. Bo h quan um numbe s and
e mion masses (cap u ed wi hin ins umen al unce ain ies) a e gene a ed by a no el se o ou
numbe s (𝑛,𝑘,𝑛′,𝑘′), e e ed o as seeds, in oduced in his s udy. Quan um numbe s a e gi en as
second-o de polynomials in he a iables 𝑘 and 𝑘′. Fe mion masses a e modeled by an exponen ial
unc ion depending solely on (𝑛,𝑘,𝑛′,𝑘′) and pa ame ized by six cons an s (𝑚𝑒,𝛼,𝛽,𝛼′,𝛽′ and 𝜆). In
his amewo k, he elec on mass 𝑚𝑒 is ixed as he e e ence cons an (gi en, no p edic ed), om
which all o he e mion masses a e de e mined.
- Neu ino mass gene a ion. The mass law can be na u ally ex ended o accoun o neu ino masses,
he eby co e ing he ull e mion spec um. This is achie ed h ough he geome ical eme gence o a
e m p opo ional o 1/𝑌𝑅, which was ini ially cons uc ed o cha ged lep ons and qua ks only. Fo
neu inos, whe e 𝑌𝑅=0, his e m beha es as a singula i y, which, when p ope ly egula ed, can
p oduce neu ino masses consis en wi h expe imen al obse a ions. Neu inos a e he e o e ea ed as
Di ac- ype pa icles wi hin his amewo k.
- Explana ion o he CKM hie a chy. The obse ed hie a chy in he CKM ma ix is explained
h ough ansi ions in he (𝑛,𝑘) space wi h 𝑛 and 𝑘 he seeds speci ic o he le -handed sec o . The
amewo k can be consis en ly ex ended o he PMNS ma ix. This he e o e p o ides a uni ied
desc ip ion o he qua k and lep on la o s uc u e.
In pa icula , his s udy ad ances h ee p edic ions, one well-suppo ed and he wo o he s mo e
specula i e:
1. A new lep on. I p edic s he exis ence o a new elemen a y cha ged lep on, wi h p ope ies
analogous o hose o he elec on, muon, and au, and a mass o 8.5938−0.0004
+0.0004 MeV.
Independen ly, a 2024 s udy epo ed he obse a ion o a cha ged pa icle ca ying an
elec on-like lep on numbe and a measu ed mass o 8.5 ± 2.5 MeV (Anikina, Niki in &
Rikh i sky, 2024). I iden i y his pa icle wi h he lep on p edic ed he e and adop he name
in oduced in ha s udy, anomalon (symbol 𝑎−). This cha ged lep on necessa ily equi es a
neu ino companion in his amewo k. I con i med, his obse a ion would cons i u e he
mos signi ican alida ion o he model.
2. A hi d e mionic b anch. I u he specula es he exis ence o a hi d e mionic b anch,
complemen ing he qua ks and lep ons. This new b anch, e e ed o as he leonids (wi h hei
ull se o p ope ies de ailed in Table 4 and Fig. 5), a ises na u ally om ex ending he seed 𝑘
beyond i s o iginal ange (0, 1, 2, 3) o encompass (−1, 0, 1, 2, 3, 4).
3. A ou h e mionic b anch. A inal b anch, e med he apex, has been in oduced o es o e
anomaly cancella ion, which is o he wise dis up ed by he inclusion o he anomalon and
leonids. The apex
1
o m a dis inc se o e mions whose ull quan um assignmen s and masses
a e de ailed in Table 5 and Fig. 5.
1
wi h he imposed same o m in singula and plu al: one apex o wo apex.
The esul s o his s udy poin o a deep connec ion be ween he geome ical p ope ies o he
amewo k and he undamen al pa icles and in e ac ions o he SM, o e ing new insigh s in o he
unde lying s uc u e o pa icle physics. Rema kably, he golden a io 𝜑 and i s conjuga e 𝜑∗ eme ge
na u ally wi hin he cons uc ion, associa ed wi h s a es whose le - and igh -handed hype cha ges a e
iden ical. Fu u e expe imen al es s o he p edic ions de i ed in his s udy will ul ima ely de e mine
whe he he sea ch o comple eness should be pu sued u he o abandoned.
The p esen a ion is o ganized as ollows. Fi s , he concep o seeds is in oduced (Fig. 1, Tables 1–2).
Nex , he mass law and i s de ailed de i a ion a e p esen ed (Figs. 2–4, Tables 2–3), ollowed by an
analysis o he CKM and PMNS ma ices wi hin he amewo k o he model. The discussion hen
begins by e iewing he well-suppo ed p edic ions o he s udy, be o e u ning o he mo e specula i e
ex ensions in which he leonids (Fig. 5, Table 4) and apex (Fig. 5, Table 5) a e in oduced o comple e
he geome ical s uc u e o he amewo k. The inal sec ion ou lines he open ques ions ha emain
in ela ion o he SM and p esen s he concluding ema ks.
The seeds
The geome ical cons uc ion o he mass law (de ailed in he ollowing wo sec ions) gi es ise o ou
numbe s 𝑛,𝑘,𝑛′ and 𝑘′, designa ed in his s udy as seeds. Each e mion is associa ed wi h a qua e o
seeds (𝑛,𝑘,𝑛′,𝑘′). The pai (𝑛,𝑘) co esponds o he le -handed sec o , while (𝑛′,𝑘′) co esponds o
he igh -handed sec o . The seeds 𝑘 and 𝑘′ gene a e all known e mionic quan um numbe s (see
below, Table 1 and Fig. 1), while he ull qua e (𝑛,𝑘,𝑛′,𝑘′) gene a es he masses o all e mions
wi hin expe imen al unce ain ies (Table 2). The elec on co esponds o (𝑘,𝑛,𝑘′,𝑛′)=(0,0,0,0) and
is ega ded as he o igin o all e mionic masses, as equi ed by he geome ical cons ain s o he
sys em (Fig. 2).
De ailed explana ion o each seed:
- The gene a ion seed 𝑛={0,1,2,3} is he gene a ion numbe as known in he SM, wi h he excep ion
o he elec on and i s neu ino companion 𝜈𝑒, which a e uniquely assigned 𝑛=0.
- The le -cha ge seed 𝑘={0,1,2,3} gene a es he le -handed hype cha ge (𝑌𝐿), ba yon numbe (𝐵)
and lep on numbe (𝐿). Cha ged lep ons a e associa ed wi h 𝑘=0, neu inos a e associa ed wi h 𝑘=
3. The up- ype and down- ype qua ks a e mixed be ween 𝑘=1 and 𝑘=2: The (𝑢,𝑠,𝑏) qua ks a e
associa ed wi h 𝑘=1, and he (𝑑,𝑐,𝑡) qua ks a e associa ed wi h 𝑘=2. This pe mu a ion 𝑘𝑢↔𝑘𝑑 is
a undamen al cons ain imposed by he geome y (Fig. 2).
- The esidual seed 𝑛′={−1,0,1} has been imposed by geome y o dis ibu e mass esiduals in o
nega i e, neu al, and posi i e linea ela ions ha a e symme ical (Fig. 4). No clea physical
in e p e a ion has eme ged o i ye .
- The igh -cha ge seed 𝑘′={0,1,2} gene a es he igh -handed hype cha ge (𝑌𝑅) and elec ic cha ge
(𝑄). Cha ged lep ons a e associa ed wi h 𝑘′=0, down- ype qua ks wi h 𝑘′=1, and up- ype qua ks
wi h 𝑘′=2. Neu inos a e ma hema ically ex ended o he igh -handed sec o wi h 𝑘𝜈′=−3+√33
2 ( his
will impose 𝑌𝑅=0, see below).
All e mionic quan um numbe s in he SM (𝑌𝐿, 𝐵, 𝐿, 𝑌𝑅, 𝑄, 𝑇3) a e gene a ed om 𝑘 and 𝑘′ ia
second-o de polynomials: 𝑌𝑅(𝑘′)=13𝑘′2+𝑘′−2 ; 𝑄(𝑘′)=16𝑘′2+12𝑘′−1 ; 𝑌𝐿(𝑘)= −23𝑘2+
2𝑘−1 ; 𝐵(𝑘)=−16𝑘2+12𝑘 ; 𝐿(𝑘)=12𝑘2−32𝑘+1 ; and 𝑇3𝐿(𝑘,𝑘′)=12(𝑌𝑅(𝑘′)−𝑌𝐿(𝑘)).
Fig. 1. Polynomial o mula ion o he hype cha ges. The le -handed hype cha ge 𝑌𝐿 and he igh -
handed hype cha ge 𝑌𝑅 can be exp essed as wo polynomials: 𝑌𝐿(𝑘)= −23𝑘2+2𝑘−1 and 𝑌𝑅(𝑘′)=
13𝑘′2+𝑘′−2, wi h 𝑘 and 𝑘′ he le -cha ge and igh -cha ge seeds, espec i ely. Righ -handed
neu inos (𝜐𝑅) a e depic ed as an emp y squa e a 𝑘𝜐′=−3+√33
2, a ising as a ma hema ical ex ension o
he sys em wi h 𝑌𝑅(𝑘𝜐′)=0. The le -handed weak isospin is de ined as 𝑇3𝐿(𝑘,𝑘′)=12(𝑌𝑅(𝑘′)−
𝑌𝐿(𝑘)) o which he golden a io 𝜑 appea s as a oo . The symbol 𝑎 co esponds o a ou h
hypo he ical cha ged lep on called he anomalon.
In his amewo k, he le weak isospin connec s he le -handed and he igh -handed sec o h ough
𝑇3𝐿(𝑘,𝑘′)=12(𝑌𝑅(𝑘′)−𝑌𝐿(𝑘)). In e es ingly, o 𝑘′=𝑘= 𝜑= 1+√5
2 ( he golden a io), one inds
𝑇3𝐿(𝜑,𝜑)=0. A his special alue, he e is no dis inc ion be ween le - and igh -handed
hype cha ges due o 𝑌𝑅(𝜑)=𝑌𝐿(𝜑)= −1+2√5
3≈0.4907 (Fig. 1).

Righ -handed neu inos can be ma hema ically ex ended wi h 𝑘𝜈′=−3+√33
2, which sa is ies 𝑌𝑅(𝑘𝜈′)=
0 and 𝑄(𝑘𝜈′)=0. This is necessa y o assign 𝑇3=𝑇3𝐿(𝑘=3,𝑘′=𝑘𝜈′)= +12 o he neu inos, bu
also o gene a e hei mass wi h a ia ions a ound 𝑘𝜈′ (see he sec ion ex ension o he neu inos).
Colo , being a ec o p ope y in SU(3) space, is no a scala quan um numbe and is no gene a ed
wi h he seed numbe s. Finally, ans o ming ma e in o an ima e simply e e ses he signs o all
coe icien s in he polynomials de ining he quan um numbe s, lea ing he seeds unchanged.
Seeds
Quan um numbe s
𝑘
𝑘′
𝑌𝐿
𝑌𝑅
𝑄
𝑇3𝐿
𝐿
𝐵
𝑒−
0
0
→
→
→
→
-1
-2
-1
-1/2
1
0
𝑎−
0
0
-1
-2
-1
-1/2
1
0
𝜇−
0
0
-1
-2
-1
-1/2
1
0
𝜏−
0
0
-1
-2
-1
-1/2
1
0
𝑑
2
1
→
→
→
+1/3
-2/3
-1/3
-1/2
0
+1/3
𝑠
1
1
+1/3
-2/3
-1/3
-1/2
0
+1/3
𝑏
1
1
+1/3
-2/3
-1/3
-1/2
0
+1/3
𝑢
1
2
→
→
→
+1/3
+4/3
+2/3
+1/2
0
+1/3
𝑐
2
2
+1/3
+4/3
+2/3
+1/2
0
+1/3
𝑡
2
2
+1/3
+4/3
+2/3
+1/2
0
+1/3
𝜐𝑒
3
𝑘𝜈′
→
→
→
→
-1
0
0
+1/2
1
0
𝜐𝑎
3
𝑘𝜈′
-1
0
0
+1/2
1
0
𝜐𝜇
3
𝑘𝜈′
-1
0
0
+1/2
1
0
𝜐𝜏
3
𝑘𝜈′
-1
0
0
+1/2
1
0
Table 1. Gene a ion o he quan um numbe s om he seeds 𝒌 and 𝒌′. The le -handed
hype cha ge 𝑌𝐿(𝑘), he igh -handed hype cha ge 𝑌𝑅(𝑘′), he elec ic cha ge 𝑄(𝑘′), he le -handed
weak isospin 𝑇3𝐿(𝑘,𝑘′)=12(𝑌𝑅(𝑘′)−𝑌𝐿(𝑘)) , he lep on numbe 𝐿(𝑘) and he ba yon numbe 𝐵(𝑘)
can be exp essed as second-o de polynomials aking in o a gumen s he le -cha ge seed 𝑘 (associa ed
wi h he le -handed sec o ) and he igh -cha ge seed 𝑘′ (associa ed wi h he igh -handed sec o ). The
alue 𝑘𝜈′=(−3+√33)/2 ep esen s a ma hema ical ex ension o neu inos in o he igh -handed
sec o . The symbol 𝑎− co esponds o a ou h hypo he ical cha ged lep on called he anomalon, wi h
𝜐𝑎 i s neu ino companion.
The mass law
P esen a ion o he law
The ollowing law ep oduces he masses o he cha ged lep ons and qua ks wi hin hei expe imen al
unce ain ies (Table 2). I s ex ension o neu inos a ises na u ally h ough he singula limi 1
𝑌𝑅→ ∞,
which enables he gene a ion o ex emely small masses consis en wi h measu emen s (see sec ion
ex ension o he neu inos).
𝑚p ed(𝑛,𝑘,𝑛′,𝑘′)=𝑚𝑒𝑒(𝑘+𝑛−3)𝑒𝛼𝑘+𝛽−3(𝑘−𝑒𝛽)𝜆𝑘′+𝑛′(𝛼′𝑘′+3
𝑌𝑅(𝑘′)+𝛽′)
wi h:
- 𝑚p ed he p edic ed mass o a e mion.
- (𝑛,𝑘,𝑛′,𝑘′) he ou seeds.
- 𝑚𝑒 he obse ed mass o he elec on. This alue is gi en, no p edic ed.
- 𝑌𝑅(𝑘′)=13𝑘′2+𝑘′−2 he igh -handed hype cha ge
- 𝛼,𝛽,𝛼′,𝛽′ and 𝜆 i e empi ical cons an s ( alues shown in Table 3).
The anomalon: a ou h cha ged elemen a y lep on
An emp y slo o a new pa icle appea s a (𝑛,𝑘,𝑛′,𝑘′)=(1,0,1,0), co esponding o a p edic ed
mass o 8.5938−0.0004
+0.0004 MeV and cha ge 𝑄(𝑘′=0)=−1. A sea ch in he li e a u e e ealed a s iking
candida e: Anikina, Niki in, and Rikh i sky (2024) epo ed he disco e y o a cha ged pa icle o
mass 8.5±2.5 MeV sha ing he lep onic numbe o an elec on. They named his pa icle he
anomalon, and i s na u e emained elusi e. I adop his e minology he e, deno e i by he symbol 𝑎−
associa ed wi h an obse ed mass o 𝑚𝑎=8.5±2.5 MeV (Fig. 2,3,4). The anomalon has been ully
in eg a ed in he p esen amewo k, and is in e p e ed as a new elemen a y cha ged lep on be ween
he elec on and he muon. The anomalon necessa ily equi es a neu ino companion 𝜈𝑎 o ill he slo
(𝑛,𝑘)=(1,3) (see sec ion neu inos).
Seeds
P edic ed mass
Obse ed mass
𝑛
𝑘
𝑛′
𝑘′
𝑚p ed (MeV)
𝑚obs (MeV)
𝑒−
0
0
0
0
→
gi en, no p edic ed
0.51099895000−0.00000000015
+0.00000000015
𝑎−
1
0
1
0
→
8.5938−0.0004
+0.0004
8.5−2.5
+2.5
𝜇−
2
0
-1
0
→
105.6583755−0.0000022
+0.0000022
105.6583755−0.0000023
+0.0000023
𝜏−
3
0
0
0
→
1776.93−0.09
+0.09
1776.93−0.09
+0.09
𝑑
1
2
1
1
→
4.69−0.01
+0.01
4.70−0.07
+0.07
𝑠
2
1
0
1
→
93.4−0.2
+0.2
93.5−0.8
+0.8
𝑏
3
1
-1
1
→
4183−7
+7
4183−7
+7
𝑢
1
1
0
2
→
2.18−0.01
+0.01
2.16−0.07
+0.07
𝑐
2
2
1
2
→
1273.2−3.1
+3.1
1273.0−4.6
+4.6
𝑡
3
2
-1
2
→
172570−280
+280
172570−290
+290
Ex ension o he neu inos (hypo hesis: No mal O de , mυe=0 MeV, and la o masses
p esen ed he e as mass eigens a es o simplici y.)
𝜐𝑒
0
3
0
𝑘𝜈′
→
0
consis en wi h 0 MeV
𝜐𝑎
1
3
±1
𝑘𝜈′
±𝜖𝑎
→
≳45600
unobse ed ye
𝜐𝜇
2
3
±1
𝑘𝜈′
∓𝜖𝜇
→
(0.00865−0.00011
+0.00011)×10−6
(0.00865−0.00011
+0.00011)×10−6
𝜐𝜏
3
3
±1
𝑘𝜈′
∓𝜖𝜏
→
(0.05013−0.00020
+0.00020)×10−6
(0.05013−0.00021
+0.00021)×10−6
Table 2. Gene a ion o he e mionic masses. All p edic ed masses (in MeV) a e gene a ed om he
mass law 𝑚p ed(𝑛,𝑘,𝑛′,𝑘′) pa ame ized by six cons an s (𝑚𝑒,𝛼,𝛽,𝛼′,𝛽′ and 𝜆). P edic ed alues
co espond o median and 2.5 h and 97.5 h pe cen iles o ~105 se s ha indi idually ep oduce he
obse ed masses (PDG 2024) wi hin expe imen al unce ain ies (de ails in Table 3). The symbol 𝑎−
co esponds o a ou h hypo he ical cha ged lep on called he anomalon (Anikina, Niki in &
Rikh i sky, 2024), wi h 𝜐𝑎 i s neu ino companion. Ex ension o he neu inos (NuFi 6.0, see Es eban
e al., 2024) is pe o med wi h h ee addi ional cons an s (𝜖𝑎,𝜖𝜇 and 𝜖𝜏) ha egula e he singula i y
1
𝑌𝑅(𝑘𝜈′)→ ∞ wi h 𝑌𝑅(𝑘′)=13𝑘′2+𝑘′−2 he igh -handed hype cha ge and 𝑘𝜈′=(−3+√33)/2. Fo
he neu inos 𝜐𝑎, 𝜐𝜇 and 𝜐𝜏, he alue o 𝑛′ is ei he 1 o −1 (unde e mined ye ).
Fig. 2. O igin o he mass law. Th ee linea ela ionships (panel a) appea in loga i hmic space wi h
espec o na u al in ege s. They a e de i ed om speci ic a ios o obse ed e mionic masses gi en
in se s 𝐴𝑘 and 𝐵. The anomalon 𝑎 (emp y ci cle) is included wi h i s obse ed mass o 𝑚𝑎=8.5 MeV
o illus a e ha i ills he o he wise emp y slo a (𝑛,𝑘)=(1,0). The slopes o hese h ee
ela ionships a e in e connec ed and ake he o m 𝑒𝛼𝑘+𝛽 wi h 𝑘={0,1,2}. The inal ela ionship
(panel b) links he h ee lines, allowing he en i e sys em o be exp essed as 𝑚(𝑘,𝑛)=
𝑚𝑒𝑒(𝑘+𝑛−3)𝑒𝛼+𝑘𝛽−3(𝑘−𝑒𝛽). Residuals om his simpli ied model emain la ge and equi e u he
co ec ion (Fig. 3 hen Fig. 4).
Demons a ion o he mass law: i s pa
Fou linea ela ionships eme ge when ou well-chosen se s o masses a e placed in a loga i hmic
space e sus na u al in ege s be ween 0 and 3 (Fig. 1a). Le 𝑁1={0,2,3} and 𝑁2={1,2,3}, hen he
ollowing ela ionships a e linea (wi h ze o in e cep by cons uc ion):
wi h 𝐴0={1,𝑚𝜇
𝑚𝑒,𝑚𝜏
𝑚𝑒}, log (𝐴0) e sus 𝑁1 is associa ed wi h a slope o 𝑒𝛼×0+𝛽
wi h 𝐴1={1,𝑚𝑠
𝑚𝑢,𝑚𝑏
𝑚𝑢}, log (𝐴1) e sus (𝑁2-1) is associa ed wi h a slope o 𝑒𝛼×1+𝛽
wi h 𝐴2={1,𝑚𝑐
𝑚𝑑,𝑚𝑡
𝑚𝑑}, log (𝐴2) e sus (𝑁2-1) is associa ed wi h a slope o 𝑒𝛼×2+𝛽
wi h 𝐵={1, 𝑚𝑠
𝑚𝑑,𝑚𝜏
𝑚𝑑}, log (𝐵) e sus (𝑁2-1) is associa ed wi h a slope o 3
These linea ela ionships show ha masses in each se 𝐴𝑘 a e ela ed wi h an exponen ial ac ion
𝑒𝛼𝑘+𝛽 wi h 𝑘={0,1,2} and 𝛼, 𝛽 wo cons an s (Table 3). The in ege s 𝑁1 and 𝑁2 co espond o he
gene a ion numbe 𝑛, a he excep ion o he elec on which is se o 𝑛=0. The se s 𝐴𝑘 con ain all
𝑛(up− ype)−𝑛(down− ype)=[0 −1 −2
1 0 −1
2 1 0]
This achie es o p o e he connec ion be ween he CKM and ansi ions in he (𝑛,𝑘) space.
PMNS ma ix
The PMNS ma ix is he lep onic coun e pa o he CKM ma ix, encoding he ampli udes o
ansi ions om cha ged lep ons o neu ino mass eigens a es h ough he cha ged weak cu en
media ed by he bosons 𝑊±. Explici ly, i akes he ollowing o m (indices 1, 2, 3 a e eplaced by
𝜈𝑒,𝜈𝜇 and 𝜈𝜏 as each label deno es a single neu ino s a e, analogous o qua k no a ion):
𝑉𝑃𝑀𝑁𝑆=[𝑉𝑒𝜈𝑒𝑉𝑒𝜈𝜇𝑉𝑒𝜈𝜏
𝑉𝜇𝜈𝑒𝑉𝜇𝜈𝜇𝑉𝜇𝜈𝜏
𝑉𝜏𝜈𝑒𝑉𝜏𝜈𝜇𝑉𝜏𝜈𝜏]
wi h ampli ude (simpli ied om NuFi 6.0 wi h a mosphe ic da a, see Es eban e al., 2024):
|𝑉𝑃𝑀𝑁𝑆|=[0.822−0.020
+0.020 0.550−0.031
+0.031 0.149−0.007
+0.007
0.377−0.125
+0.125 0.588−0.092
+0.092 0.704−0.052
+0.052
0.397−0.121
+0.121 0.579−0.094
+0.094 0.690−0.053
+0.053]
As in he qua k sec o , uni a i y ensu es p obabili y conse a ion, bu in con as o he CKM ma ix,
he PMNS en ies display no s ong hie a chical pa e n: all elemen s a e o compa able size, wi h
la ge mixing be ween la o s. This ea u e unde lies he phenomenon o neu ino oscilla ions and
poin s o a quali a i ely di e en la o s uc u e in he lep on sec o , o which he SM o e s no
explana ion.
In his s udy, he exponen ial pa ame iza ion o he CKM ma ix has been mimicked o he PMNS
wi h success by simply eplacing he (𝑛,𝑘) indices associa ed wi h he down- ype ( esp. up- ype)
qua ks wi h he neu inos ( esp. cha ged lep ons). The ou ma ices 𝑛(cha ged−lep ons), 𝑛(neu inos),
𝑘(cha ged−lep ons) and 𝑘(neu inos) a e de ined as:
{
𝑤(cha ged−lep ons)= [𝑤𝑖𝑗(cha ged−lep ons)]=[𝑤𝑒𝑤𝑒𝑤𝑒
𝑤𝜇𝑤𝜇𝑤𝜇
𝑤𝜏𝑤𝜏𝑤𝜏]
𝑤(neu inos)= [𝑤𝑖𝑗(neu inos)]=[𝑤𝜐𝑒𝑤𝜐𝜇𝑤𝜐𝜏
𝑤𝜐𝑒𝑤𝜐𝜇𝑤𝜐𝜏
𝑤𝜐𝑒𝑤𝜐𝜇𝑤𝜐𝜏]
wi h 𝑤=𝑛 o 𝑤=𝑘 ( alues in Table 2). The s uc u e di ec ly pa allels ha o he CKM mixing
ma ix:
{𝑎𝑖𝑗
𝑃𝑀𝑁𝑆=−𝑏𝜐(𝑛𝑖𝑗
(neu inos)−𝑛𝑖𝑗
(cha ged−lep ons))𝜆𝜐𝑑𝑖𝑗𝑧𝜐
𝛿3𝑗𝛿𝑖1𝑧𝜐𝛿1𝑗𝛿3𝑖
𝑑𝑖𝑗=2|𝑛𝑖𝑗
(neu inos)−𝑛𝑖𝑗
(cha ged−lep ons)|+|𝑘𝑖𝑗
(neu inos)−𝑘𝑖𝑗
(cha ged−lep ons)|−1

which co espond o:
𝐴𝑃𝑀𝑁𝑆= [𝑎𝑖𝑗
𝑃𝑀𝑁𝑆]= 𝑏𝜐[0 −2𝜆𝜐6−3𝜆𝜐8(𝑥𝜐−𝑖𝑦𝜐)
2𝜆𝜐60 −𝜆𝜐4
3𝜆𝜐8(𝑥𝜐+𝑖𝑦𝜐) 𝜆𝜐40]
The PMNS is ep oduced wi hin expe imen al unce ain ies wi h he cons an 𝑏𝜐=𝑏𝑞=1.231, 𝑦𝜐=
𝑦𝑞=0.2215, 𝑥𝜐=2.2643 and 𝜆𝜐= 0.8602. Only 𝑥𝜐 and 𝜆𝜐 ha e changed om he CKM case,
which was necessa y o mi iga e he hie a chical pa e n (𝜆𝜐 is close o 1). The compu ed modules
co espond o:
|𝑉𝑃𝑀𝑁𝑆|=[0.827 0.544 0.142
0.300 0.640 0.707
0.475 0.543 0.693]
The eade should no ice ha he e y la ge unce ain ies o he PMNS ma ix allow o a ious o ms
o 𝐴𝑃𝑀𝑁𝑆 o ma ch he a ge ma ix, and his appa en success migh be an a i icial cons uc . The key
inding he e is ha he CKM s uc u e de ined in he (𝑛,𝑘) space emains compa ible when applied o
he PMNS ma ix.
Discussion and ex ension
Be o e add essing po en ial con lic s wi h he SM, i is impo an o clea ly s a e he main claims o
his wo k. The claims a e p esen ed in wo g oups: i s he s ong, well-suppo ed claims o he
known e mions (C), ollowed by mo e specula i e claims (SC) ha p edic wo new e mionic
b anches ( he leonids and he apex) by ex ending his amewo k. The discussion hen p oceeds o lis
he open ques ions (O), ollowed by a conclusion.
Suppo ed claims
C1. Fou seeds pe e mion. Each e mion is associa ed wi h ou seeds (Table 2), which gene a e
bo h i s quan um numbe s and i s mass (excep o he elec on). The seeds (𝑛,𝑘) co espond o he
le -handed sec o , while he seeds (𝑛′,𝑘′) co espond o he igh -handed sec o .
C2. Eme gence o quan um numbe s. Quan um numbe s eme ge om second-o de polynomials
aking in o a gumen he le -cha ge seed 𝑘 and/o igh -cha ge seed 𝑘′: 𝑌𝑅(𝑘′)=13𝑘′2+𝑘′−2 ;
𝑄(𝑘′)=16𝑘′2+12𝑘′−1 ; 𝑌𝐿(𝑘)= −23𝑘2+2𝑘−1 ; 𝐵(𝑘)=−16𝑘2+12𝑘 ; 𝐿(𝑘)=12𝑘2−32𝑘+1
and 𝑇3𝐿(𝑘,𝑘′)=12(𝑌𝑅(𝑘′)−𝑌𝐿(𝑘)).
In his amewo k, quan um numbe s a e no longe ea ed as undamen al; ins ead, hey eme ge om
a deepe unde lying s uc u e. An ima e co esponds o e e sing he sign o each coe icien .
C3. Mass law. All e mionic masses eme ge om he ollowing law:
𝑚p ed(𝑛,𝑘,𝑛′,𝑘′)=𝑚𝑒𝑒(𝑘+𝑛−3)𝑒𝛼𝑘+𝛽−3(𝑘−𝑒𝛽)𝜆𝑘′+𝑛′(𝛼′𝑘′+3
𝑌𝑅(𝑘′)+𝛽′)
wi h 𝑚𝑒 he obse ed mass o he elec on and 𝛼,𝛽,𝛼′,𝛽′ and 𝜆 i e empi ical cons an s (Table 3).
C4. The elec on as e e ence. The elec on is no longe conside ed pa o he i s gene a ion.
Ins ead, i se es as he undamen al e e ence poin o he en i e e mionic ealm, cha ac e ized by
anishing seeds. I s mass is no eme gen bu is ea ed as a undamen al cons an . This aligns wi h he
ac ha he elec on is he only e mion ha is s able and does no unde go decay. All o he e mions
a e uns able and decay in o ligh e ones, a p ocess which co esponds o ansi ions be ween seeds.
Wi hin his amewo k, all e mions can be in e p e ed as exci a ions o he elec on in he unde lying
seed space, wi h hei masses and p ope ies co esponding o speci ic seed con igu a ions.
C5. The anomalon. A new elemen a y lep on, he anomalon, sha es he same quan um numbe s as
o he cha ged lep ons (lep on numbe , ba yon numbe , elec ic cha ge, igh - and le -handed
hype cha ges). I s mass is p edic ed o be 8.5938−0.0004
+0.0004 MeV and was mos likely obse ed in 2024
(Anikina, Niki in & Rikh i sky, 2024), p io o he p edic ion. An associa ed neu ino companion is
necessa y o he wise he slo (𝑛,𝑘)=(1,3) would emain open. The na u e o his neu ino (ei he
s e ile o ac i e) emains unknown.
C6. Neu ino masses. The ex emely small masses o he neu inos o igina e om a singula i y
ela ed o he igh -handed hype cha ge in 1/𝑌𝑅 (o possibly he elec ic cha ge 𝑄 as hey a e ela ed
by 𝑄=𝑌𝑅/2). Neu inos ollow he same gene a ion as hei cha ged lep on companions.
C7. CKM hie a chy. The weigh ed Manha an dis ance 𝑑𝑖𝑗=2|𝑛𝑖−𝑛𝑗|+|𝑘𝑖−𝑘𝑗|−1 p o ides a
quan i a i e measu e o he ela i e supp ession o la o -changing ansi ions be ween qua ks 𝑖 and 𝑗
in he (𝑛,𝑘) seed space, e ec i ely encoding he obse ed hie a chy o he CKM ma ix elemen s.
Specula i e claims
SC1. A hi d e mionic b anch: he leonids. This new e mionic b anch (Fig. 5, Table 4) would exis
in 𝑘=(−1,4) alongside he qua ks (mixed be ween he up- ype and down- ype qua ks a 𝑘=1 and
𝑘=2) and he lep ons (cha ged lep ons a 𝑘=0 and neu inos a 𝑘=3). Two nume ical coincidences
mo i a e he explo a ion o 𝑘=(−1,4). Fi s , 𝑇3𝐿(−1,−1)=+12, which is highly non i ial since i
a ises om he di e ence be ween wo polynomials ha p e iously sa is y 𝑇3𝐿(0,0)=−12, 𝑇3𝐿(1,1)=
−12 and 𝑇3𝐿(2,2)=+12 (see Fig. 5). The second coincidence is ha 𝑌𝐿(−1) and 𝑌𝐿(4) a e bo h equal o
−11
3. This specula ion is summa ized in Table 4 and p edic s six new pa icles ( h ee wi h imagina y
masses) ha a e nei he lep ons no qua ks. This new b anch is named he leonids (singula : one
leonid), and i mimics he qua ks and lep ons in i s s uc u e (le -handed double s, igh -handed
single s, and 𝑇3𝐿=±12 ). The leonids a e composed o h ee lions – galion (Λ𝑔), melion (Λ𝑚) and
solion (Λ𝑠) - associa ed wi h 𝑇3𝐿=+12 ( h ee e mions wi h eal masses). Thei companions a e he
h ee chime as — g i in (χ𝑔), man ico e (χ𝑚), and sphinx (χ𝑠) — associa ed wi h 𝑇3𝐿=−12 ( h ee
e mions wi h complex masses). These names come om he ac ha he h ee chime as do no exis
in he eal wo ld (imagina y mass) bu a e all composed o body pa s o lions. Chime as a e na u al
candida es o he da k sec o , as hey possess complex masses due o hei igh -cha ge seed 𝑘𝜒′=
−3+𝑖√23
2 ha e i ies 𝑌𝑅(𝑘𝜒′)=𝑌𝐿(4)+2𝑇3𝐿= −11
3−1=−14
3 . This ex ension is simila o he
igh -handed neu inos ex ension o he lep ons, whe e 𝑘𝜈′=−3+√33
2 e i ied 𝑌𝑅(𝑘𝜈′)=𝑌𝐿(3)+
2𝑇3𝐿= −1+1=0. The leonids a e necessa ily colo less o he gauge anomaly cancella ion
pe o med in SC2.
names
seeds
p edic ed mass (MeV)
quan um numbe s
leonids
𝑛
𝑘
𝑘′
i 𝑛′=−1
i 𝑛′=0
i 𝑛′=1
𝑌𝐿
𝑌𝑅
𝑄
𝑇3𝐿
𝐿
𝐵
lions
galion
Λ𝑔
1
-1
-1
89.30
±0.27
100.73
±0.32
113.62
±0.38
−11
3
−83
−43
+12
3
−23
melion
Λ𝑚
2
-1
-1
620.6
±1.6
700.1
±1.9
789.7
±2.3
−11
3
−83
−43
+12
3
−23
solion
Λ𝑠
3
-1
-1
4313.4
±9.4
4865
±11
5488
±13
−11
3
−83
−43
+12
3
−23
chime as
g i in
χ𝑔
1
4
𝑘𝜒′
(1.16
± 0.03)107
+𝑖 (1.63
± 0.02)106
(1.32
± 0.03)107
+𝑖 (1.71
± 0.03)106
(1.50
± 0.04)107
+𝑖 (1.79
± 0.04)106
−11
3
−14
3
−73
−12
3
−23
man ico e
χ𝑚
2
4
𝑘𝜒′
(4.19
± 0.14)1011
+𝑖 (5.93
± 0.08)1010
(4.78
± 0.16)1011
+𝑖 (6.22
± 0.09)1010
(5.45
± 0.18)1011
+𝑖 (6.49
± 0.10)1010
−11
3
−14
3
−73
−12
3
−23
sphinx
χ𝑠
3
4
𝑘𝜒′
(1.52
± 0.07)1016
+𝑖 (2.15
± 0.04)1015
(1.74
± 0.07)1016
+𝑖 (2.26
± 0.04)1015
(1.98
± 0.09)1016
+𝑖 (2.36
± 0.04)1015
−11
3
−14
3
−73
−12
3
−23
Table 4. Cha ac e is ics o he leonids. The names, symbols, seeds, p edic ed mass and quan um
numbe s o he new e mionic b anch called he leonids (pa i ioned in o he lions and he chime as)
a e summa ized he e. Because 𝑛′ is unknown (dis ibu ed be ween −1,0 and 1), he e a e h ee
possibili ies o he mass o each leonid. P edic ed masses co espond o median and 2.5 h and 97.5 h
pe cen iles o p edic ions om ~105 se s o (𝛼,𝛽,𝛼′,𝛽′ and 𝜆) ha indi idually ep oduce he
obse ed masses o known e mions (PDG 2024) wi hin expe imen al unce ain ies (de ails in Table
3). The alue 𝑘𝜒′=(−3+𝑖√23)/2 is se o e i y 𝑌𝑅(𝑘𝜒′)=−14/3 and espec he s uc u e o 𝑇3𝐿=
±1/2. The complex seed 𝑘𝜒′ leads o complex masses h ough he mass law, which make chime as
na u al candida es o he da k sec o . The leonids a e necessa ily colo less o cancel he gauge
anomalies (see SC2).
SC2. A ou h e mionic b anch: he apex. This ou h e mionic b anch (Fig. 5, Table 5) has been
in oduced o cancel he gauge anomalies (see he sc ip in he supplemen a y). In he SM, he
cancella ion o gauge anomalies is essen ial o ensu e he in e nal consis ency and eno malizabili y o
he heo y. In pa icula , ou ypes o anomalies mus anish: he cubic 𝑈(1)𝑌 anomaly (∑𝑌3 o e all
e mions), he mixed 𝑆𝑈(2)𝐿2−𝑈(1)𝑌 anomaly (∑𝑌 o e le -handed double s only), he 𝑆𝑈(3)𝐶2−
𝑈(1)𝑌 anomaly (∑𝑌 o e all colo ed e mions), and he g a i a ional–𝑈(1)𝑌 anomaly (∑𝑌 o e all
e mions). The adi ional SM anomaly budge has been dis u bed by he in oduc ion o he anomalon
and i s neu ino companion (leading o ou gene a ions o lep ons ins ead o h ee) and he leonids,
which is a new colo less e mionic b anch con aining h ee gene a ions o e mions (see SC1). A leas
one new e mionic b anch is equi ed o cancel hese dis up ions. Geome ically, a ema kable poin is
loca ed a 𝑘=𝑘A=3/2 (see Fig. 5) and co esponds o he maximum le -handed hype cha ge (𝑌𝐿=
12). The alue o 𝑌𝐿=12 can be used o cons uc a double , wi h an uppe membe associa ed wi h 𝑌𝑅=
𝑌𝐿+1=32 and lowe membe associa ed wi h 𝑌𝑅=𝑌𝐿−1=−12. Le me call his specula i e
e mionic b anch he apex (using he e apex o bo h singula and plu al: one apex, wo apex), wi h i s
uppe membe s e e ed o as he high- ype apex (symbol Δ, 𝑇3𝐿=+12) and he lowe membe s as he
low- ype apex (symbol ∇, 𝑇3𝐿=−12) o mimic he qua ks. When assigning h ee gene a ions o he
apex (see de ails in Table 5) and gi ing hem a colo ep esen a ion o 8 (analogous o he gluons), all
gauge anomalies a e cancelled (see supplemen a y). This gauge anomaly cancella ion is ema kable
because i occu s ac oss h ee gene a ions o leonids, qua ks, and apex, bu ou gene a ions o lep ons.
Fig. 5. Two new e mionic b anches: he leonids and he apex. The le -handed and igh -handed
hype cha ge polynomials p esen ed in Fig. 1 ha e been ex ended o he alues 𝑘=−1 and 𝑘=4 and
lead o 𝑌𝐿(−1)=𝑌𝐿(4)= −11/3. Coinciden ly, 𝑇3𝐿(−1,−1)=+12 which sugges s ha his
ex ension could be a ibu ed o a new e mionic b anch named he leonids. The leonids a e pa i ioned
be ween he lions (𝑘=−1, 𝑇3𝐿(−1,−1)=+12) and he chime as (𝑘=4, 𝑇3𝐿(4,𝑘𝜒′)=−12) wi h 𝑘𝜒′=
(−3+𝑖√23)/2. The golden a io 𝜑 and i s conjuga e 𝜑∗ appea a he in e sec ion be ween 𝑌𝐿 and 𝑌𝑅.
A las e mionic b anch ( he apex) loca ed a (𝑘,𝑌𝐿)=(32,12) has been in oduced o en i ely cancel he
gauge anomalies (see sec ion SC2).
names
seeds
p edic ed mass (MeV)
quan um numbe s
apex
𝑛
𝑘
𝑘′
i 𝑛′=−1
i 𝑛′=0
i 𝑛′=1
𝑌𝐿
𝑌𝑅
𝑄
𝑇3𝐿
𝐿
𝐵
high- ype
high
Δℎ
1
32
𝑘Δ′
1.878
±0.011
2.313
±0.011
2.850
±0.012
+12
+32
+34
+12
−18
+38
ise
Δ𝑟
2
32
𝑘Δ′
171.03
±0.72
210.69
±0.68
259.55
±0.78
+12
+32
+34
+12
−18
+38
zeni h
Δ𝑧
3
32
𝑘Δ′
15578
±44
19191
±40
23642
±56
+12
+32
+34
+12
−18
+38
low- ype
low
∇𝑙
1
32
𝑘∇′
2.2808
±0.0059
2.1947
±0.0062
2.1119
±0.0097
+12
−12
−14
−12
−18
+38
all
∇𝑓
2
32
𝑘∇′
207.75
±0.45
199.91
±0.28
192.36
±0.61
+12
−12
−14
−12
−18
+38
nadi
∇𝑛
3
32
𝑘∇′
18923
±59
18209
±20
17522
±36
+12
−12
−14
−12
−18
+38
Table 5. Cha ac e is ics o he apex. The names, symbols, seeds, p edic ed mass and quan um
numbe s o he new e mionic b anch called he apex (pa i ioned in o he high- ype apex and he low-
ype apex) a e summa ized he e. Because 𝑛′ is unknown (dis ibu ed be ween −1,0 and 1), he e a e
h ee possibili ies o he mass o each apex. P edic ed masses co espond o median and 2.5 h and
97.5 h pe cen iles o p edic ions om ~105 se s o (𝛼,𝛽,𝛼′,𝛽′ and 𝜆) ha indi idually ep oduce he
obse ed masses o known e mions (PDG 2024) wi hin expe imen al unce ain ies (de ails in Table
3). The alue 𝑘Δ′=(−3+√51)/2 and 𝑘∇′=(−3+√27)/2 a e se o espec he s uc u e o 𝑇3𝐿=
±1/2. The apex ha e necessa ily a colo ep esen a ion o 8 o cancel he gauge anomalies (see SC2).
SC3. A p ima y cha ge seed 𝜿. This SC3 assumes SC1 o be alid, and he claim is ha 𝜅=
{−1,0,1} (deno ed as he “cha ge seed”) is su icien o p oduce all le -cha ge and igh -cha ge seeds
o he leonids, lep ons and qua ks. The apex and he golden a ios 𝜑 and 𝜑∗ seem o eme ge om a
di e en mechanism (see sec ion O2). Cu en ly, he le -cha ge and igh -cha ge seed alues span a
wide ange: 𝑘= −1 o 4 (including he leonids), 𝑘′=−1,0,1,2 and suddenly 𝑘𝜈′= −3+√33
2 and 𝑘𝜒′=
−3+𝑖√23
2. This likely poin s o a deepe mechanism gene a ing such di e si y in seed alues. F om his
hypo hesis, wo unc ions na u ally a ise: 𝑓𝐿(𝑘)=3−𝑘 and 𝑓𝑅(𝑘′)=−3+√−20𝑘′2+36𝑘′+33
2 . The le -
cha ge and igh -cha ge seeds o each double ’s membe s a e connec ed o 𝜅 h ough he 𝑓𝐿 and 𝑓𝑅
ans o ma ions:
- leonids: he lion double s a e associa ed wi h (𝑘,𝑘′)=(−1,−1), while he chime a double s a e
associa ed wi h (𝑘,𝑘′)=(4,𝑘𝜒′)=(𝑓𝐿(−1),𝑓𝑅(−1)). Chime as co espond o he lions unde
(𝑓𝐿,𝑓𝑅) applied o −1.
- lep ons: he cha ged lep on double s a e associa ed wi h (𝑘,𝑘′)=(0,0), while he neu ino double s
a e associa ed wi h (𝑘,𝑘′)=(3,𝑘𝜈′)=(𝑓𝐿(0),𝑓𝑅(0)). Neu inos co espond o he cha ged-lep ons
unde (𝑓𝐿,𝑓𝑅) applied o 0.
- qua ks: he down- ype double s a e associa ed wi h (𝑘,𝑘′)=(1,1), while he up- ype double s a e
associa ed wi h (𝑘,𝑘′)=(2,2)=(𝑓𝐿(1),𝑓𝑅(1)). Up- ype qua ks co espond o he down- ype qua ks
unde (𝑓𝐿,𝑓𝑅) applied o 1.

The down qua k and he up qua k a e an excep ion: (𝑘,𝑘′)=(2,1) and (𝑘,𝑘′)=(1,2), espec i ely.
Howe e , his excep ion is handled by he ac ha 𝑓𝐿(2)=1 and 𝑓𝑅(2)=1, which is ema kable!
E en mo e ema kably, 𝑓𝑅 e i ies ha 𝑓𝑅(𝜑)=𝜑 and 𝑓𝑅(𝜑∗)=𝜑∗ and 𝑓𝐿 e i ies ha 𝑓𝐿(𝑘A)=𝑘A
wi h 𝑘A=32 he unique le -cha ge seed o he apex. The e o e, he h ee seeds ha ha e been le ou
− 𝑘A, 𝜑 and 𝜑∗ − a e in a ian ei he unde 𝑓𝐿 o 𝑓𝑅. These coincidences a e epo ed in he sec ion
O2.
I has jus been p o en ha 𝜅={−1,0,1} is su icien o p oduce all cha ge seeds o he leonids,
lep ons and qua ks. This sugges s an eme gence o he chime as, neu inos and up- ype qua ks om
hei o he hal (see SC4).
SC4. A e symme y b eaking in a p e-chi al space, chime as eme ged om lions, neu inos
om cha ged lep ons, and up- ype qua ks om down- ype qua ks.
This SC4 assumes SC1 and SC3 o be alid. SC4 is mo i a ed by ou obse a ions:
(1) in he SM, e mion masses and elec ic cha ges a e co ela ed — excep o he up and down
qua ks. This anomaly is no explained in he SM bu is in e p e ed he e as he signa u e o a b oken
symme y ( ep esen ed by he pe mu a ion 𝑘𝑢↔𝑘𝑑 in Fig. 2).
(2) he gene a ion 𝑛=0 associa ed wi h (𝑒−,𝜐𝑒) canno be ex ended o qua ks due o expe imen al
cons ain s — no ou h gene a ion o qua ks has been obse ed. This sugges s ha 𝜐𝑒 may ha e
inhe i ed 𝑛=0 om he elec on alone. Ex ending his easoning o all double s implies ha each
membe o a double may ha e “eme ged” om hei o he hal .
(3) he cha ge seed 𝜅={−1,0,1} alone p oduces all le -cha ge and igh -cha ge seeds o he leonids,
qua ks and lep ons unde he (𝑓𝐿,𝑓𝑅) ans o ma ion (see SC3).
(4) he e is a ema kable symme y 𝑘=𝑘′ o he lions, cha ged-lep ons and down- ype qua ks. The
excep ion is, again, he down qua k and up qua k, who b eak he symme y be ween 𝑘 and 𝑘′.
These obse a ions can be uni ied h ough he ollowing ch onological na a i e:
1) lions (𝜅=−1), cha ged-lep ons (𝜅=0) and down- ype qua ks (𝜅=1) ini ially li ed in a p e-
chi al space cha ac e ized by he cha ge seed 𝜅={−1,0,1}. A his s age, hey we e single s, massless,
wi hou le -handed coun e pa s, and he chime as, neu inos, and up- ype qua ks had no ye
eme ged.
2) A symme y b eaking occu ed (associa ed wi h he pe mu a ion 𝑘𝑢↔𝑘𝑑), du ing which 𝜅 spli
in o he le -cha ge seed 𝑘 and igh -cha ge seed 𝑘′. This b eaking ma ked he onse o chi ali y,
leading o he o ma ion o le -handed double s. Each missing pa ne wi hin a double subsequen ly
eme ged acco ding o he ans o ma ions (𝑓𝐿(𝑘),𝑓𝑅(𝑘′)) c ea ing he new ollowing seed alues: 𝑘=
2,3,4 and 𝑘′=2,𝑘𝜈′, 𝑘𝜒′ (see SC3). A e symme y b eaking, he le -hype cha ge polynomial 𝑌𝐿(𝑥)
became ela ed o he igh -hype cha ge polynomial 𝑌𝑅(𝑥) h ough he ela ion: 𝜙(𝑥) =𝑌𝑅(𝑥)−
𝑌𝐿(𝑥) whe e 𝜙(𝑥)=𝑥2−𝑥−1 is he golden polynomial, sa is ying 𝜙(𝜑)=𝜙(𝜑∗)=0.
The beha io o 𝑛′ and 𝑛 in his na a i e emains unde e mined a his s age, as does he ole o he
apex and he golden a io 𝜑 and 𝜑∗ (see sec ion O2). I is in iguing ha 𝑛′={−1,0,1} sha es he
same ange o alues as 𝜅={−1,0,1}, bu I do no ye unde s and how he gene a ion seed 𝑛 could
spli om 𝑛′ in he same manne as he le - and igh -cha ge seeds did om 𝜅. I he lions exis , hei
𝑛′ will be de e mined wi h hei mass (Table 4), and he ela ionship be ween 𝑛 and 𝑛′ will be easie o
unde s and.
SC5. Chi al eo ganiza ion o 𝑸 and 𝑻𝟑𝑳. The elec ic cha ge 𝑄 and he le -handed weak isospin 𝑇3𝐿
canno be exp essed in e ms o he le -cha ge seed 𝑘. Ins ead, 𝑄 is a pu ely polynomial unc ion o
he igh -cha ge seed 𝑘′: 𝑄(𝑘′)=16𝑘′2+12𝑘′−1, while 𝑇3𝐿 depends on bo h 𝑘 and 𝑘′: 𝑇3𝐿(𝑘,𝑘′)=
12(𝑌𝑅(𝑘′)−𝑌𝐿(𝑘)). This s uc u e sugges s he ollowing bold claim: he elec ic cha ge 𝑄 is an
in insic p ope y o he igh -handed sec o alone, whe eas he weak isospin 𝑇3 ac s as a b idge
linking he le - and igh -handed sec o s (i could be enamed 𝑇3𝐿𝑅) a he han being in insic o he
le -handed sec o . I con i med, his would compel a e ision o he SM’s gauge s uc u e, eshaping
ou unde s anding o cha ge assignmen s and weak in e ac ions.
Open ques ions
This sec ion ou lines a se ies o un esol ed aspec s and specula i e di ec ions eme ging om he
p esen amewo k. While he co e s uc u e cap u es he obse ed e mionic pa e ns, se e al
elemen s emain only pa ially cons ained o in i e deepe in e p e a ion beyond he es ablished
model.
O1. The physical sense o 𝒏′. The esidual seed 𝑛′ has been in oduced o co ec ly accoun o he
esiduals obse ed in Fig. 4. The alues a e o 𝑛′ a e elegan ly dis ibu ed be ween −1,0 and 1 o
e mions sha ing same igh -handed hype cha ge ( ixed 𝑘′). Values o he seed 𝑛′ echoes hose o he
cha ge seed 𝜅={−1,0,1} (see SC4), and 𝑛′ o e s a igh -handed coun e pa o he le -handed
gene a ion seed 𝑛. Howe e , 𝑛′ is no associa ed wi h any known quan um numbe s such as he
hype cha ges, no wi h any es ablished expe imen al obse a ions, such as he h ee e mionic
gene a ions. The absence o clea p edic i e pa e ns o his seed leads o un esol ed alues o he
neu inos, leonids, and apex (Tables 2, 4, and 5). Ne e heless, i is assumed ha 𝑛′ can only ake
alues in he ange −1,0 o 1, consis en wi h all known e mions.
O2. The golden a ios 𝝋/𝝋∗ and he apex. The golden a io 𝜑 and i s conjuga e 𝜑∗appea wice in
his s udy: (i) A he in e sec ion be ween 𝑌𝐿(𝑥) and 𝑌𝑅(𝑥) in Fig. 5, and (ii) as in a ian s unde he 𝑓𝑅
ans o ma ion (see SC3). They a e associa ed wi h a anishing weak isospin 𝑇3𝐿𝑅, howe e , i emains
unclea why i a ional numbe s such as 𝜑 and 𝜑∗ a ise in a amewo k o he wise cen e ed on na u al
seeds 𝜅={−1,0,1} (see SC4). I is possible ha hese alues ac as seeds gene a ing exo ic s a es.
Rema kably, pos ula ing new ec o -like pa icles eme ging om 𝜑 and/o 𝜑∗ would no a ec he
gauge anomaly calcula ion because 𝑌𝐿(𝜑)=𝑌𝑅(𝜑) and 𝑌𝐿(𝜑∗)=𝑌𝑅(𝜑∗) and hei con ibu ions
would cancel each o he . This ou e is explo ed in he supplemen a y ma e ial wi h new hypo he ical
golden pa icles, he au ions. In pa allel, apex a e a cons uc ion geome ically consis en wi h he le -
handed hype cha ge 𝑌𝐿 o o he e mions (Fig. 5) and nume ically consis en wi h a ull cancella ion o
he gauge anomalies. Thei colo cha ge o 8 is analogous he adjoin ep esen a ion o gluons,
sugges ing a possible connec ion be ween he apex and gauge bosons — an idea eminiscen o
supe symme ic co espondences be ween e mionic and bosonic deg ees o eedom (Haag,
Łopuszański, & Sohnius 1975). While hei le -cha ge seed 𝑘𝐴=32 is in a ian unde he 𝑓𝐿
ans o ma ion, he igh -cha ge seeds associa ed wi h he low- ype apex (𝑘∇′=−3+√27
2) and high- ype
apex (𝑘Δ′=−3+√51
2) a e no consis en wi h he 𝑓𝑅 ans o ma ion: 𝑘Δ′≠𝑓𝑅(𝑘∇′), unlike all o he
e mionic b anches. Howe e , 𝑘∇′ and 𝑘Δ′ a e exp essed as −3+√𝑋
2 which in iguingly mimics he
s uc u e o 𝑓𝑅. All hese e idences sugges ha apex and he golden a ios eme ged om a di e en
mechanism han he h ee o he e mionic b anches ha a e all cons uc ed on he same cha ge seed
𝜅={−1,0,1} (see SC3). The in a iance 𝑓𝐿(𝑘A)=𝑘A, 𝑓𝑅(𝜑)=𝜑 and 𝑓𝑅(𝜑∗)=𝜑∗ may pe haps
indica e — hough no conclusi ely — ha hey a e a e ac s o he p e-chi al space pos ula ed in SC4.
O3. The anomalon and i s a ached neu ino. The anomalon is a cha ged lep on wi h a mass o ~8.6
MeV and iden ical quan um numbe s o hose o he known cha ged lep ons. I s exis ence equi es an
associa ed neu ino o occupy he (𝑛,𝑘)=(1,3) slo in he mass law. Expe imen al e idence is
limi ed, wi h a single indica ion (Anikina, Niki in & Rikh i sky, 2024) om a p opane bubble
chambe , whose supe hea ed liquid medium is uniquely sensi i e o localized ene gy deposi ion. The
pa icle’s appa en absence in mode n de ec o s may be explained by se e al specula i e mechanisms:
i may decay in o neu al o e y so inal s a es, a el slowly p oducing sho acks below de ec ion
h esholds, o m neu al o weakly in e ac ing composi es, o be p oduced only a ely wi h a small
c oss sec ion. These explana ions a e no mu ually exclusi e and emain un es ed. The na u e o he
associa ed neu ino — whe he ac i e o s e ile — is en i ely unknown.
O4. The leonids. The exis ence o he leonids aises p essing heo e ical and expe imen al ques ions.
The chime as ca y complex masses, implying in insic ins abili y, and he double s mix eal and
complex componen s — a s uc u e wi hou p eceden in he SM. The p edic ed complex masses o
he chime as each absu dly high alues (See Table 4), a exceeding any known SM e mions,
po en ially app oaching scales eminiscen o he p imo dial ene gies o he ea ly uni e se. Such
ex eme mass scales may hin a a connec ion o physics nea he Big Bang o o hidden sec o s ha
we e ac i e only a ul a-high ene gies. The exo ic cha ges, ba yon/lep on assignmen s, and pa ially
complex masses aise ques ions abou he compa ibili y o he leonids wi h cosmological,
as ophysical, and collide cons ain s. Decay channels, li e imes, and couplings emain uncons ained,
lea ing expe imen s wi h minimal guidance. Ye he 2024 hin o he anomalon, consis en wi h his
amewo k, sugges s ha p e iously unseen lep onic s a es may exis , mo i a ing explo a ion o he
leonids in hidden o pa ially decoupled sec o s.
O5. Higgs mechanism. In he SM, e mion masses a ise h ough Yukawa couplings o he Higgs ield.
Fo a e mion ( ), he mass is gi en by 𝑚𝑓=𝑦𝑓𝑣
√2 whe e 𝑦𝑓 is he Yukawa coupling ( ee pa ame e )
and 𝑣 ~ 246 GeV is he Higgs acuum expec a ion alue (VeV). The le - and igh -handed
componen s o he e mion a e connec ed ia he Higgs ield, gi ing ise o he obse ed mass a e
spon aneous symme y b eaking. Wi hin he p esen amewo k, he elec on plays he ole o a
undamen al e e ence: i s mass 𝑚𝑒 is conside ed non-eme gen and ea ed as a undamen al cons an ,
he eby eplacing he 𝑣
√2. All o he e mions a ise as exci a ions o his unde lying elec on ield,
cha ac e ized by speci ic seed con igu a ions (𝑛,𝑘,𝑛′,𝑘′). The mass law o all e mions na u ally
ac o izes in o a le - e m 𝑒(𝑘+𝑛−3)𝑒𝛼+𝑘𝛽−3(𝑘−𝑒𝛽) mul iplied by a igh - e m 𝜆𝑘′+𝑛′(𝛼′𝑘′+3
𝑌𝑅(𝑘′)+𝛽′), which
esona es wi h he SM s uc u e whe e le - and igh -handed componen s a e coupled. Howe e , a
mechanism analogous o he Higgs is equi ed o connec he le -handed double s wi h he igh -
handed single s and gene a e mass. A possible escape ou e is he exis ence o a second Higgs boson
wi h a VeV equal o 𝑚𝑒√2 ha would be speci ic o he e mionic sec o .
O6. Neu inos. In his amewo k, neu inos a e na u ally o Di ac ype, as hei igh -handed
componen s a e explici ly de ined h ough 𝑘𝜈′=−3+√33
2. This choice ensu es ha neu inos acqui e
mass h ough he same exponen ial law as he cha ged e mions, bu wi h a dis inc i e supp ession
ac o a ising om he 1/𝑌𝑅 dependence in he exponen . A his alue o 𝑘𝜈′, he denomina o 𝑌𝑅
anishes, in oducing a s ong enhancemen in he mass exponen ha d i es he neu ino masses o
ex emely small alues. This mechanism p o ides a pu ely algeb aic explana ion o he obse ed
neu ino mass hie a chy, wi hou in oking Majo ana e ms o seesaw dynamics. Howe e , wo
heo e ical issues emain open in his app oach: (i) wha mechanism ensu es 𝑌𝑅(𝑘𝜈′)≠0, p e en ing a
singula i y o a leas wo neu inos, and (ii) how he esidual seed 𝑛′ is dis ibu ed among neu ino
la o s.
Poin (i). The simples esolu ion p oposed in his s udy in oduces a small pe u ba ion 𝜖𝜈≪1 such
ha 𝑘𝜈,e
′=𝑘𝜈′±𝜖𝜈, wi h 𝜖𝜈 depending on he la o (Table 3). O he o mula ions, howe e , a e
possible. No ably, since 𝑓𝑅(𝜅)=𝑘𝜈′ o 𝜅=0 (as shown in SC3), his issue can equi alen ly be
exp essed as a pe u ba ion on 𝜅, namely 𝜅𝜈,e
′=±𝛿𝜈 wi h 𝛿𝜈≪1. In ui i ely, he pa ame e s 𝛿𝜈
could be assumed o ollow a symme ic dis ibu ion a ound a non-ze o mean 𝛿e (mo i a ed by 𝜖𝜇≠
𝜖𝜏 in Table 3). In his o mula ion, one may w i e 𝜅𝜈,e
′=𝛿e ±𝑛′×𝛿0 whe e 𝛿e and 𝛿0 a e ixed
cons an s independen o la o , and 𝑛′ de e mines he sign. This cons uc ion emains specula i e. As
long as he mechanism esponsible o egula ing he singula beha io o 1/𝑌𝑅 is no iden i ied, he
p oblem emains open.
Poin (ii). The second open ques ion hen conce ns he p ecise assignmen o he esidual seed 𝑛′.
While he cha ged lep ons include wo s a es wi h 𝑛′=0 ( he elec on and he au), he neu inos
appea o mi o his pa e n only pa ially. The simples esolu ion p oposed in his s udy is o p oduce
only one s a e 𝑛′=0 assigned o 𝜐𝑒, while o he neu inos ollow 𝑛′=±1. This b eaks he symme y
in 𝑛′ wi h he cha ged-lep ons, bu i be e ma ches he expe imen al cons ain s. Al e na i ely, one
can assign 𝑛′=0 o 𝜐𝑒 and 𝜐𝑎, and le e age he inde e minacy 0 × ∞ (in he e m ~𝜆𝑘′+ 0 × ∞) by
assigning 0 × ∞ → −∞ o 𝜐𝑒 (which leads o a ze o mass) and assigning 0 × ∞ → 0 o 𝜐𝑎, which
would lead o a mass o 𝑚𝜈𝑎=422.5±1.4 MeV. This seems o con adic he expe imen al
cons ain s on he mass o a hypo he ical ou h neu ino, bu his would es o e he symme y in 𝑛′.