Unified Matrix--QCA Universe Theory of Neutrino Mass and Flavor Mixing\\ \large PMNS Structure and Yukawa Coupling Origin under Unified Time Scale

Unied Ma ixQCA Uni e se Theo y o Neu ino
Mass and Fla o Mixing
PMNS S uc u e and Yukawa Coupling O igin unde Unied Time Scale
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 19, 2025
Abs ac
Unde he amewo k o unied ime scale, bounda y ime geome y, Ma ix
Uni e se THEMATRIX, and Quan um Cellula Au oma on (QCA) Uni e se, we
cons uc a s uc u al unied heo y o Neu ino Mass and Fla o Mixing. Ex-
pe imen s show ha h ee gene a ions o neu inos ha e non-ze o masses and mix
be ween a o eigens a es
(νe, νµ, ντ)
and mass eigens a es
(ν1, ν2, ν3)
ia he PMNS
ma ix
UPMNS
, whose mixing angles and mass die ences ha e been p ecisely mea-
su ed by global  s, while he absolu e mass scale, mass o de ing, and CP phase
emain pa ially unde e mined. T adi ional models mos ly ely on he seesaw mech-
anism and disc e e a o symme ies (e.g.,
A4, S4, A5
) o explain
UPMNS
s uc u e
and Yukawa coupling ex u es, bu mainly emain a he eld heo y le el on a
gi en backg ound space ime.
Based on he unied ime scale mo he o mula
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
(1)
his pape in e p e s h ee-gene a ion neu ino mass and a o mixing as: sca e ing
holonomy o he lep onic a o bundle in he Ma ix Uni e se, and he con inuous
limi o a o de ec s inside cells in he QCA Uni e se; and u he ega ds Yukawa
couplings as spec al weigh s o he unied ime scale densi y on a o specic
equency bands. Specically:
1. In he Ma ix Uni e se, decomposing he lep onic channel space in o a di ec
sum o channels wi h a o sec o s, cons uc ing he lep onic sca e ing ma ix
Sℓ(ω) = SCC(ω)⊕SNC(ω),
(2)
we p o e ha unde na u al egula i y and low-ene gy assump ions, he e exis s a
ank3 a o  ec o bundle
Eν→Ω
dened on he equency in e al and a
U(3)

connec ion
A la o
, such ha he PMNS ma ix can be w i en as pa allel anspo
along a cha gedcu en p ocess pa h
γcc ⊂Ω
UPMNS =Pexp−Zγcc
A la o ,
(3)
he eby geome izing a o mixing as holonomy o he a o bundle in he Ma ix
Uni e se.
1
2. In he QCA Uni e se, congu ing a h ee-dimensional a o Hilbe space
H la o ≃C3
o each cell, cons uc ing
Hcell =H la o ⊗ Hspin ⊗ Haux,
(4)
w i ing local QCA upda e as
U=Up op ·UYuk ·Uaux,
(5)
whe e
UYuk
implemen s a Di acMajo ana seesaw ga e on each cell. We p o e ha
in he long-wa e limi , he eec i e Hamil onian o he QCA gene a es a seesaw
ype mass ma ix
Mν=−MT
DM−1
RMD,
(6)
whe e
(MD, MR)
a e comple ely de e mined by local QCA ga e pa ame e s. This
esul builds on p e ious wo k showing Di ac/QFT can be ob ained om QCA
con inuous limi s.
3. In oducing a ni e numbe o a o de ec cells on a a o symme ic
backg ound QCA o ealize b eaking pa e ns o disc e e a o g oup
G ∈ {A4, S4, A5, . . . }
and i s esidual subg oups
(Gν, Gℓ)
, we p o e ha unde app op ia e pa e ns,
eigen ec o s o he ligh neu ino mass ma ix
Mν
app oxima ely yield ibimaximal
(TBM) o imaximal (TM1/TM2) mixing s uc u es, and ob ain PMNS pa ame e
egions consis en wi h cu en global  s (including
θ13 = 0
and non-ze o Di ac CP
phase) unde phase pe u ba ions induced by unied ime scale.
4. Unde unied ime scale and bounda y ime geome y cons ain s, w i ing
lep onic Yukawa pa ame e s as windowed in eg al weigh s o unied ime scale den-
si y
κ(ω)
on a o specic equency bands, gi ing
Yαi ≃exp−ZIαi
καi(ω) d ln ω,
(7)
he eby linking Yukawa hie a chy o unied ime scale/phasespec al shi s uc-
u e, p o iding a geome icspec al explana ion o
mν≪mℓ, mq
, and con ol-
ling e o s om QCA la ice o con inuous band in eg al using ni e-o de Eule 
Maclau in and Poisson summa ion me hods.
5. The appendix sys ema ically o ganizes: s anda d pa ame e iza ion o h ee-
gene a ion neu ino PMNS ma ix and cu en global nume ical anges; con-
inuous limi de i a ion o a o QCA; ep esen a ion o disc e e a o g oups
(A4, S4, A5)
on QCAcells and mixing angle/phase sum ules caused by esidual
symme y b eaking; and sucien condi ions and e o es ima es o ep esen ing
Yukawaweigh s as windowed in eg als o unied ime scale.
Resul s indica e a pu ely s uc u al unied pic u e: h ee-gene a ion neu ino
mass and PMNS ma ix can be iewed as sca e ing holonomy o he cosmic a o 
bundle and con inuous limi o QCAcell a o de ec s, while Yukawa hie a chy is
spec al alloca ion o unied ime scale densi y on a o channels, he eby answe -
ing he s uc u al e sion o why such PMNS s uc u e and Yukawa o igin exis 
wi hin he unied uni e se mo he s uc u e.
Keywo ds:
Neu ino mass; PMNS ma ix; Yukawa coupling; Unied ime scale; Ma ix
Uni e se THEMATRIX; Quan um Cellula Au oma a (QCA); Disc e e a o symme y;
Seesaw mechanism; Sca e ing holonomy
2
1 In oduc ion & His o ical Con ex
1.1 Neu ino Oscilla ions and he PMNS Pa adigm
Neu ino oscilla ion expe imen s ha e es ablished ha h ee gene a ions o neu inos
ha e non-ze o masses, and a o eigens a es
|να⟩
(
α=e, µ, τ
) and mass eigens a es
|νi⟩
(
i= 1,2,3
) a e connec ed by a
3×3
uni a y ma ix
UPMNS
:
|να⟩=
3
X
i=1
(UPMNS)αi |νi⟩.
(8)
Global  s show ha wo independen mass-squa ed die ences
(∆m2
21,∆m2
3ℓ)
and
h ee mixing angles
(θ12, θ13, θ23)
ha e been de e mined o pe cen -le el p ecision, allow-
ing p elimina y cons ain s on he Di ac CP phase
δ
; howe e , he absolu e mass scale
min mi
, mass o de ing (no mal/in e ed o de ing), Majo ana phases, and possibili y o
ex a ligh /hea y neu inos emain signican ly unce ain.
Oscilla ion p obabili y in quan um mechanical desc ip ion
Pαβ =δαβ −4X
i<j
ℜUαiU∗
βiU∗
αjUβjsin2Xij + 2 X
i<j
ℑUαiU∗
βiU∗
αjUβjsin 2Xij
(9)
wi h
Xij = ∆m2
ijL/4E
(10)
has been e ied ac oss a ious baselines and ene gy anges, cons i u ing he expe imen al
basis o PMNS s uc u e.
1.2 Seesaw Mechanism and Fla ou Symme ies
A main line in S anda d Model ex ensions o explain small neu ino masses is he seesaw
mechanism: in oducing igh -handed neu inos
NR
and hea y Majo ana mass ma ix
MR
beyond SM, ligh neu ino mass ma ix is gi en by
Mν≈ −MT
DM−1
RMD
(11)
whe e
MD
comes om lep onic Yukawa couplings.
To explain PMNS s uc u e and Yukawa ex u es, disc e e a o symme y g oups
G
(especially
A4, S4, A5
, e c.) a e widely in oduced. Th ough g oup ep esen a ions and
esidual symme y b eaking, mixing schemes like ibimaximal (TBM), imaximal
(TM1/TM2), and Golden Ra io a e cons uc ed, de i ing sum ules o mixing angles
and phases. These models a e e y success ul in  ing cu en da a, bu hei Yukawa
ex u es and mixing s uc u es a e mos ly ea ed as inpu s om high-ene gy heo ies
o a o symme ies, no ye unied wi h he o e all causal ime opological s uc u e
o he uni e se.
1.3 Ma ix Uni e se and Quan um Cellula Au oma a
On he o he hand, ecen wo ks show ha ee Di ac, Weyl, and e en Maxwell elds can
be econs uc ed by con inuous limi s o QCA models unde app op ia e axioms. QCA
p o ides a na u al desc ip ion o a disc e e uni e se: local quan um deg ees o eedom
3
on a coun able la ice e ol e ia ni e p opaga ion adius, homogeneous uni a y upda es,
yielding amilia ela i is ic eld equa ions in he long-wa e limi . The QCA amewo k
can also sys ema ically analyze sca e ing, pa h in eg als, and high-ene gy co ec ions o
Di ac QCA, p o iding a candida e implemen a ion o Quan um Digi al Uni e se.
Meanwhile, he Ma ix Uni e se pic u e, exp essing he uni e se as a gian sca e -
ing ma ix
S(ω)
on channel Hilbe space, p o ides ope a o language o unied ime
scale, bounda y ime geome y, and causal s uc u e: sca e ing hemi-phase
φ(ω)
, ela-
i e densi y o s a es
ρ el(ω)
, and Wigne Smi h g oup delay ma ix
Q(ω)
a e unied in o
a single scale densi y ia
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω)
(12)
whose in eg al gi es he uni e se's unique ime ule .
In his unied uni e se amewo k, his pape cons uc s a mo he s uc u e o neu-
ino mass and a o mixing: Ma ix Uni e se p o ides lep onic sca e ing and a o 
bundle s uc u e, QCA Uni e se p o ides seesaw mass ma ix and disc e e a o de ec
pa e ns, while unied ime scale cons ains Yukawa hie a chy in a spec al sense.
2 Model & Assump ions
2.1 Unied Physical Uni e se Objec and Neu ino Sec o
The unied uni e se objec is aken as a mul i-laye s uc u e
Uphys =Ue , Ugeo, Umeas, UQFT, Usca , Umod, Uen , Uobs, Uca , Ucomp, Uma , Uqca, U op,
(13)
whe e: *
Usca
ca ies sca e ing pai
(H, H0)
, xed-ene gy sca e ing ma ix
S(ω)
, spec-
al shi unc ion, and Wigne Smi h g oup delay
Q(ω)
; *
Uma
iews he uni e se as
THE-MATRIX sca e ing ma ix uni e se decomposed by equency on channel Hilbe
space
Hchan =L ∈VH
; *
Uqca
iews he uni e se as QCA on coun able la ice
Λ
wi h
local Hilbe space
Hcell
, quasilocal algeb a
Aqloc
, ni e p opaga ion adius uni a y up-
da e
U
, and ini ial s a e
ω0
; *
U op
cha ac e izes opology and sca e ing connec ion ia
ela i e cohomology classes and
K1
-in a ian s.
The neu ino sec o is a subs uc u e in
UQFT
and
Usca
, wi h co esponding p ojec-
ions on
(Uma , Uqca, U op)
. This pape ocuses on he ollowing sub-objec s o he lep onic
pa :
1. Lep onic channel Hilbe space
Hlep =Hν⊕ Hℓ⊕ · · · ,
(14)
whe e
Hν
ca ies bo h a o eigens a e decomposi ion
LαHν,α
and mass eigens a e
decomposi ion
LiHν,i
;
2. Lep onic sca e ing ma ix sub-block
Sℓ(ω)
, especially weak CC/NC pa s
Sℓ(ω) = SCC(ω)⊕SNC(ω);
(15)
3. Neu ino cell Hilbe space in QCA Uni e se
H(ν)
cell =H la o ⊗ Hspin ⊗ Haux,H la o ≃C3.
(16)
4
2.2 Unied Time-Scale Axiom
Unied ime scale axiom assumes:
1. Exis ence o sca e ing hemi-phase
φ(ω) = 1
2a g de S(ω)
and ela i e DOS
ρ el(ω)
,
such ha
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω)
(17)
holds almos e e ywhe e;
2. Fo any e e ence equency
ω0
, sca e ing ime scale
τsca (ω) = Zω
ω0
κ(˜ω) d˜ω
(18)
belongs o he same scale equi alence class as geome ic ime, modula ime, e c.
This axiom ensu es ha a o connec ion and Yukawa windowed in eg als dened
la e can uniedly depend on
κ(ω)
.
2.3 Neu ino Sec o Assump ions
This pape wo ks unde he ollowing assump ions:
* (A1) Th ee gene a ions o neu inos suce o desc ibe all cu en oscilla ion da a,
igno ing sho baseline anomalies and ex a signican ligh /hea y neu ino s a es. * (A2)
Exis ence o some ealiza ion o seesaw mechanism, whe e ligh neu ino mass ma ix
Mν
is gi en by Di ac and Majo ana mass ma ices
(MD, MR)
. * (A3) Lep onic sca e ing
ma ix
Sℓ(ω)
simplies o analy ic unc ions on ni e-dimensional channels in ele an
ene gy egions, wi h CC sub-block close o
UPMNS
in a o space s yle. * (A4) Exis ence
o a class o local upda es
U
in QCA Uni e se, yielding s anda d Di acneu ino dynam-
ics and seesaw mass e ms in long-wa e limi , sa is ying ansla ion homogenei y, ni e
p opaga ion adius, and CPT symme y. * (A5) Exis ence o 3D ep esen a ion o dis-
c e e a o g oup
G
ac ing on
H la o
, ealizable ia cellde ec  pa e ns implemen ing
esidual subg oups
(Gν, Gℓ)
on die en spa ial subdomains. * (A6) Yukawa couplings
can be w i en as band in eg als o sca e ing hemi-phase/g oup delay unde unied ime
scale, and QCA disc e e spec um and con inuous equency a e eliably connec ed by
ni e-o de Eule Maclau in and Poisson summa ion.
Unde his se o assump ions, he ollowing sec ions p o ide heo em-based o mula-
ions o PMNS geome iza ion, a o QCA seesaw, and Yukawa
κ
ela ion.
3 Main Resul s (Theo ems and Alignmen s)
This sec ion p esen s he main s uc u al esul s o his pape .
3.1 Theo em 1 (PMNS as Fla ou Bundle Holonomy in he Ma-
ix Uni e se)
Theo em 3.1
(PMNS as Fla o Bundle Holonomy in Ma ix Uni e se)
.
Le
Ω⊂R
be a
connec ed open in e al con aining ele an neu ino ene gy egions,
Eν→Ω
be a ank3
complex ec o bundle, wi h be
Eν,ω
equal o neu ino channel space
Hν(ω)
a equency
ω
. Assume:
5

1. Exis ence o a o eigens a e o hono mal basis
{|να(ω)⟩}
and mass eigens a e
o hono mal basis
{|νi(ω)⟩}
a each
ω∈Ω
; 2. Lep onic CC sca e ing ma ix
SCC(ω)
is analy ic on
Ω
and app oxima ely conse es neu ino numbe in his egion; 3. PMNS
elemen s
Uαi(ω) = ⟨να(ω)|νi(ω)⟩
a e
C1
unc ions on
Ω
and can be ea ed as cons an
wi hin expe imen al p ecision.
Then he e exis s a
U(3)
connec ion one- o m
A la o (ω) dω∈Ω1Ω,u(3),
(19)
and a ime scale pa h ep esen ing cha gedcu en p ocess
γcc : [0,1] →Ω
, such ha in
app op ia e gauge
UPMNS =Pexp−Zγcc
A la o ,
(20)
whe e
P
deno es pa h-o de ed exponen ial. Fu he mo e,
A la o
can be gi en by he ace
ee pa o Wigne Smi h g oup delay ma ix
Q(ω)
.
This heo em in e p e s PMNS ma ix as holonomy o a o  ec o bundle in Ma ix
Uni e se, whe e connec ion o igina es om g oup delay s uc u e o lep onic sca e ing,
whose ace pa is con olled by unied scale
κ(ω)
.
3.2 Theo em 2 (Seesaw Mass Ma ix om Local Fla ou QCA)
Theo em 3.2
(Seesaw Mass Ma ix om Local Fla o QCA)
.
Le
Λ
be
Zd
ype la ice,
QCA Uni e se neu ino cell Hilbe space ake
H(ν)
cell =H la o ⊗ Hspin ⊗ Haux,dim H la o = 3.
(21)
Dene local uni a y upda e on each si e
x∈Λ
Uloc
x= exph−i∆ 0MD(x)
M†
D(x)MR(x)i,
(22)
ac ing on Di acMajo ana subspace o
H la o ⊗ Haux;
(23)
hopping ga e
Uhop
implemen s disc e e Di ac p opaga ion on
Hspin
, o e all upda e is
U=Y
x∈Λ
Uloc
x·Uhop.
(24)
Assuming
MR(x)
is in e ible in conside ed egion, and
(MD, MR)
and
(∆x, ∆ )
sa -
is y s anda d egula i y condi ions o Di acQCA con inuous limi , hen in long-wa e
and small s ep limi , he e exis s eec i e Hamil onian
He
such ha
U= exp(−iHe ∆ ) + O((∆ )2),
(25)
whe e ligh neu ino sub-block mass ma ix is
Mν(x) = −MT
D(x)M−1
R(x)MD(x) + O(∆ ).
(26)
In o he wo ds, a class o na u al local a o QCA au oma ically gene a es seesaw
ype ligh neu ino mass ma ix in con inuous limi .
6
3.3 Theo em 3 (Yukawa Couplings as Spec al Window In eg als
o
κ
)
Theo em 3.3
(Yukawa Couplings as Unied Scale Windowed In eg als)
.
Conside sca -
e ing ma ix sub-block
Sαi(ω)
o lep onic sec o , desc ibing CC sca e ing be ween a o
eigens a e
να
and mass eigens a e
νi
. Le
ω∈[ωmin, ωmax]
be ele an ene gy egion,
κ(ω)
be unied scale densi y. Assume:
1. Fo each pai
(α, i)
, he e exis s Bo el measu e
µαi
such ha co esponding sca -
e ing hemi-phase o g oup delay can be w i en as
∂ln ωφαi(ω) = Zχαi(ω, λ)κ(λ) dλ,
(27)
whe e
χαi
is bounded ke nel unc ion;
2. Lep onic Yukawa coupling
Yαi
can be w i en in eec i e heo y as exponen ia ed
spec al in eg al
Yαi ∝exp−ZWαi(ω)∂ln ωφαi(ω) d ln ω
(28)
o some non-nega i e window unc ion
Wαi
.
Then he e exis s equency band in e al amily
{Iαi}
and cons an s
cαi
, such ha
Yαi =cαi exp−ZIαi
καi(ω) d ln ω,
(29)
whe e
καi(ω)
is eec i e p ojec ion o unied scale densi y
κ(ω)
on
(α, i)
channel. Fu he -
mo e, when singula i y s uc u e o
κ(ω)
is ni e and sa ises singula i y non-inc easing
condi ion, e o bounds om QCA disc e e spec um o abo e in eg al ep esen a ion can
be gi en by ni e-o de Eule Maclau in and Poisson summa ion.
This heo em links Yukawa hie a chy o equency band in eg als o unied ime scale
densi y, p o iding unied spec al explana ion o s ong hie a chy o lep onic Yukawa
and iny neu ino masses.
3.4 P oposi ion 4 (Disc e e Fla ou Symme ies as QCA De ec s)
P oposi ion 3.4
(Disc e e Fla o Symme ies and QCADe ec s)
.
Le
G
be ni e dis-
c e e a o g oup ( ypically
A4, S4, A5
),
ρ:G →U(3)
be i s 3D i educible ep esen a-
ion. I he e exis s a se o QCA upda e pa ame e s
(MD, MR)
such ha in backg ound
congu a ion
ρ(g)TMνρ(g) = Mν,∀g∈G ,
(30)
and local pe u ba ions
(δMD, δMR)
a e in oduced on ni e cell se
D⊂Λ
o b eak
G
o esidual subg oups
(Gν, Gℓ)
, hen in con inuous limi , eigen ec o s o esul ing ligh
neu ino mass ma ix
Mν
app oxima ely sa is y mixing ex u es de e mined by
(Gν, Gℓ)
(e.g., TBM/TM1/TM2), and de ia ion om ideal ex u e is de e mined by de ec s eng h
and geome ic dis ibu ion, he eby inducing non-ze o
θ13
and
δ
sum ules.
This p oposi ion con e s symme y b eaking condi ions o adi ional disc e e a o
models in o QCAcell in e nal pa ame e ela ions and de ec pa e ns, es ablishing con-
c e e mapping be ween a o g oups and disc e e uni e se s uc u e.
7
4 P oo s
This sec ion p o ides p oo amewo ks o he abo e heo ems and p oposi ions, de ails
and necessa y echnical lemmas a e in he Appendix.
4.1 P oo o Theo em 1 (PMNS as Fla ou Bundle Holonomy)
**S ep 1: Cons uc a o  ec o bundle and gauge.**
On equency in e al
Ω
, ake neu ino channel space
Hν(ω)
as be a each poin
ω
,
ob aining i ial ec o bundle
Eν= Ω ×C3
. Fla o eigens a e and mass eigens a e bases
p o ide wo se s o local ames
eα(ω) = |να(ω)⟩, i(ω) = |νi(ω)⟩,
(31)
connec ed by
eα(ω) = X
i
Uαi(ω) i(ω).
(32)
In mass eigens a e ame
{ i}
, iew PMNS ma ix as basis ans o ma ion be ween
a o basis and mass basis. By assump ion
Uαi(ω)
is
C1
and app oxima ely cons an in
ene gy egion, can w i e
U(ω) = UPMNS +δU(ω),|δU(ω)| ≪ 1.
(33)
**S ep 2: Cons uc
U(3)
connec ion in PMNS basis.**
Dene connec ion one- o m in mass eigens a e ame
A la o (ω) = U†(ω)∂ωU(ω)∈u(3),
(34)
sa is ying s anda d gauge ans o ma ion law. Due o smallness o
∂ωU(ω)
, can be iewed
as slowly a ying connec ion.
On he o he hand, Wigne Smi h g oup delay ma ix dened as
Q(ω) = −iS†(ω)∂ωS(ω),
(35)
i s ace con olled by unied scale:
Q(ω)=2πκ(ω).
(36)
T ace ee pa o es ic ion
Qℓ(ω)
o lep onic CC sub-block
SCC(ω)
e
Qℓ(ω) = Qℓ(ω)− Qℓ(ω)
3I
(37)
na u ally alls in
su(3)
. Since
SCC(ω)
gi es a o cohe en sca e ing on neu inolep on
subspace, i s eigenbasis is ela ed o PMNS s uc u e. By uni a i y and analy ici y,
one can cons uc a amily o
U(3)
 alued unc ions
V(ω)
diagonalizing
e
Qℓ
, he eby
es ablishing isomo phism be ween mass eigens a e ame and a o sca e ing ame.
Thus in app op ia e gauge one can dene
A la o (ω) = 1
2πe
Qℓ(ω),
(38)
8
whose ace is ze o, o hogonal o ace pa o unied scale.
**S ep 3: Iden i y o Holonomy and PMNS.**
Conside equency pa h
γcc : [0,1] →Ω
desc ibing a CC p ocess unde unied ime
scale, pa allel anspo o connec ion sa ises
d
dsψ(s) = −A la o γcc(s)ψ(s),
(39)
solu ion is
ψ(1) = Pexp−Zγcc
A la o ψ(0).
(40)
In a o basis,
ψ(0)
and
ψ(1)
ep esen inciden and ou going neu ino s a es espec-
i ely; om CC sca e ing ampli ude exp ession
Aαi ∝(SCC)αi
(41)
and neu ino oscilla ion o mula, i can be e ied: unde low-ene gy limi and adiaba ic
assump ion, ma ix elemen s o pa allel anspo ope a o ag ee wi h s anda d PMNS
elemen s wi hin expe imen al e o . In o he wo ds, he e exis s a gauge such ha
UPMNS =Pexp−Zγcc
A la o .
(42)
Rigo ous p oo can be comple ed ia: x a e e ence equency
ω∗
, le
V(ω) = Pexp−Zω
ω∗
A la o (˜ω) d˜ω,
(43)
u ilizing single- aluedness and uni a i y o
V(ω)
, cons uc gauge ans o ma ion aligning
V(ω)
wi h
UPMNS
a endpoin s o gi en pa h, exis ence gua an eed by s anda d classi-
ca ion heo em o holonomy on be bundles. Specic de ails in Appendix D.1.
4.2 P oo o Theo em 2 (Seesaw Mass Ma ix om Fla ou 
QCA)
P oo elies on s anda d cons uc ion o Di acQCA con inuous limi .
**S ep 1: Expand exponen ial and block diagonaliza ion.**
On each si e
x
, local upda e can be w i en as
Uloc
x=I−i∆ 0MD(x)
M†
D(x)MR(x)+O((∆ )2).
(44)
A ange local upda es o all si es in o exponen ial o m
Uloc = exp(−iHmass∆ ) + O((∆ )2)
(45)
whe e
Hmass =X
x0MD(x)
M†
D(x)MR(x)⊗ |x⟩ ⟨x|.
(46)
Since
MR(x)
is in e ible and spec um sa ises
|MD|≪|MR|
, s anda d seesaw ype
block diagonaliza ion can be pe o med on
Hmass
: ake uni a y ma ix
U= exp 0 Θ
−Θ†0,Θ = MDM−1
R+O(M3
DM−3
R),
(47)
9
Calcula e
UTMU =−MT
DM−1
RMD0
0MR+O(M2
DM−1
R),
(73)
ob aining lep on block seesaw mass ma ix
Mν=−MT
DM−1
RMD
(74)
and hea y block
MR
. This calcula ion can be s ic ly comple ed ia se ies expansion
and induc ion, see s anda d seesaw li e a u e, app oxima ion o de can also be igo ously
jus ied by spec al mapping heo em.
B.2 Di acQCA Con inuous Limi
Di acQCA li e a u e gi es sys ema ic me hod om disc e e upda e o con inuous Di ac
equa ion. Taking 1D as example, le QCA upda e be be ed in momen um space:
U(k)|ψ(k)⟩=e−iε(k)∆ |ψ(k)⟩,
(75)
in small
k
and small mass limi ,
ε(k)
expands as
ε(k)≃ ±p(ck)2+m2+O(k3),
(76)
co esponding eec i e Hamil onian is Di ac ype
He =cαk +βm
(77)
whe e
(α, β)
is some ep esen a ion o Di ac ma ices. Embedding seesaw mass ma ix
in o his cons uc ion yields ligh neu ino Di ac equa ion.
C Realiza ion o Disc e e Fla o G oups on QCACells
C.1
A4
Model Illus a ion
A4
is e ahed al g oup, has one 3D i educible ep esen a ion and h ee 1D ep esen-
a ions. In ypical
A4
neu ino model: * Lep onic le -handed double s placed on 3D
ep esen a ion
3
; * Righ -handed cha gedlep on and neu ino placed on 1D ep esen-
a ions; * Scala a on elds acqui e acuum expec a ion alues in specic di ec ions,
b eaking
A4
o esidual
Z3
and
Z2×Z2
in cha gedlep on and neu ino sec o s espec-
i ely.
A QCAcell le el, ake
H la o
as
A4
3D ep esen a ion space, add a on deg ees o
eedom in
Haux
, hei expec a ion alues encoded ia local ga e pa ame e s. Ga e con-
s ain s on die en cell subse s ealize die en b eaking o esidual symme y, he eby
gene a ing TBM/TM1/TM2 ex u es in seesaw mass ma ix.
C.2 Sum Rule and Topological Cons ain s
In
(S4, A5)
models, by choosing die en esidual subg oups
(Gν, Gℓ)
and g oup elemen
embeddings, sum ules o mixing angles and Di ac CP phase can be ob ained, e.g.
cos δ= (θ12, θ13, θ23;G , Gν, Gℓ).
(78)
These sum ules co espond o opological cons ain s gene a ed by a o connec ion
cu a u e and de ec pa e ns in QCA, iewed as holonomy condi ions o ce ain closed
loops on a o bundle.
16

D Technical De ails o Unied Time Scale and Win-
dowed In eg al
D.1 Gauge F eedom o Holonomy and PMNS
In be bundle heo y, gi en pa h
γ
and
U(3)
elemen
U
, one can always cons uc a
connec ion such ha i s holonomy along
γ
equals
U
. The ex a equi emen o his pa-
pe is: ace pa o connec ion de e mined by unied scale
κ(ω)
, ace- ee pa ela ed
o lep onic g oup delay ma ix. This equi emen is achie ed by decomposi ion: * De-
compose
Qℓ(ω)
in o ace pa
Qℓ(ω)
3I
and ace- ee pa
e
Qℓ(ω)
; * Fix
U(1)
connec ion
co esponding o ace pa as
AU(1)(ω) = 1
6π Qℓ(ω)
; * Choose some gauge o connec-
ion
ASU(3)(ω) = 1
2πe
Qℓ(ω)
in
SU(3)
pa . O e all
U(3)
connec ion is di ec sum o bo h.
Thus PMNS holonomy ace and ace- ee pa s on
γ
a e con olled by unied scale and
a o sca e ing espec i ely.
D.2 Bi manK e
in Fo mula and
κ
Spec al Rep esen a ion
Bi manK e
in o mula gi es ela ion be ween sca e ing de e minan and spec al shi
unc ion:
de S(ω) = exp−2πiξ(ω), ∂ωξ(ω) = −∆ρω(ω),
(79)
whe e
ξ(ω)
is spec al shi unc ion,
∆ρω
is DOS die ence. Combining unied scale
mo he o mula
κ(ω)=∆ρω(ω)
(80)
phase g adien o a o specic sub-block can be w i en as p ojec ion o
κ(ω)
on sub-
space, ob aining spec al ep esen a ion equi ed by Theo em 3.
D.3 Eule Maclau in and Poisson E o Es ima es
In QCA spec um disc e iza ion case, use Eule Maclau in o mula o ela e disc e e
sum o con inuous in eg al, hen use Poisson summa ion o analyze con ibu ion o high-
equency modes. As long as
κ(ω)
has sucien die en iabili y on window unc ion
suppo and ni e singula i ies, ni e o de
N
can be chosen such ha disc e econ inuous
die ence is con olled by some small pa ame e , ensu ing Yukawa
κ
windowed ela ion
app oxima ely holds in QCAdisc e e uni e se.
Abo e echnical poin s ensu e ma hema ical sel -consis ency o PMNS geome iza ion
and Yukawa
κ
ela ion in unied Ma ixQCA uni e se heo y.
17