Delay Quantization, Feedback Loops, and \pi-Step Parity Transitions: From Scale Identity to \mathbb Z_2 Topology of Self-Referential Scattering Networks

Delay Quan iza ion, Feedback Loops, and
π
-S ep Pa i y
T ansi ions:
F om Scale Iden i y o
Z2
Topology o Sel -Re e en ial Sca e ing
Ne wo ks
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
Wi hin he unied amewo k o equency-domain sca e ing heo y and eedback ne wo ks,
ne wo ks wi h unable closed-loop delays exhibi highly obus phase-s ep and g oup-delay-pulse
phenomena ac oss a wide ange o physical pla o ms: as he eedback ound- ip ime
τ
a ies
slowly, he o al sca e ing phase and i s equency de i a i e unde go jumps o ampli ude
app oxima ely
π
nea ce ain pa ame e alues, accompanied by e e sals in he di ec ion o
spec al ow. Unde he cons ain o he scale iden i y
κ(ω;τ) = φ′(ω;τ)
π=ρ el(ω;τ) = 1
2π Q(ω;τ)
his pape p o ides a igo ous spec al and opological cha ac e iza ion o he delay quan iza ion
⇒π
-s ep
⇒Z2
pa i y ansi ion phenomenon. He e
S(ω;τ)
is a lossless sca e ing ma ix am-
ily a ying wi h angula equency
ω
and eec i e ound- ip delay
τ
,
φ(ω;τ) = a g de S(ω;τ)
is he o al sca e ing phase, and
Q(ω;τ) = −iS(ω;τ)†∂ωS(ω;τ)
is he Wigne Smi h g oup
delay ma ix.
Unde na u al assump ions o analy ici y, losslessness, and simple ze os/poles, we p o e ha
when
τ
a e ses a amily o delay quan iza ion s eps
τk=τ0+k∆τ, k ∈Z,
he spec al ow o ze os/poles o
de S(ω;τ)
in he complex equency plane unde goes a c ossing
e en ac oss he eal axis; by he a gumen p inciple, his co esponds o a jump o size
±π
in
he o al phase a a xed equency slice; acco dingly, he opological index cons uc ed om
he spec al ow coun
ν(τ)∈ {0,1}, ν(τ+ ∆τ) = ν(τ)⊕1
unde goes a
Z2
pa i y ip a each s ep.
To make esul s compu able and expe imen ally e iable, we  s p o ide an explici ana-
ly ic o m o a one-dimensional single-channel scala model
S o (ω;τ) = 0(ω) + 0(ω)2eiωτ
1− b(ω) eiωτ ,
and igo ously de i e he magni ude and sign o
π
-s eps and uni g oup-delay pulses using he
a gumen p inciple and pole ajec o y analysis. We hen gene alize o mul i-channel ma ix
cases, showing ha upon app op ia e choice o b anch o
de S(ω;τ)
, he main conclusions
depend only on he eigen alue spec al ow o he eec i e eedback block
R(ω)
, hus ha ing
uni e sali y ac oss implemen a ion pla o ms.
1
F om he pe spec i e o he unied ime scale, unable closed-loop delays cons i u e a na u al
opological d i ing pa ame e : each a e sal o a delay quan iza ion s ep co esponds o spec al
ow c ossing he eal axis once in pa ame e space, he eby swi ching be ween wo opological
sec o s in he
Z2
sense. This opological ip mani es s as easily measu able nge p in s o
o al phase and g oup delay
π
-s eps on any linea lossless pla o m, and can be embedded
in o highe -laye s uc u es such as sel - e e en ial sca e ing ne wo ks, spin double co e s, and
NullModula double co e s, p o iding a unied equency-domain opological eadou .
Keywo ds
: delay quan iza ion; eedback loops; sca e ing ma ix; Wigne Smi h g oup
delay; scale iden i y; phase s eps;
Z2
pa i y; sel - e e en ial sca e ing ne wo ks; opological
in a ian s
1 In oduc ion & His o ical Con ex
1.1 Delay Feedback Ne wo ks and he
π
Phase Jump Phenomenon
F om be -loop esona o s, in eg a ed mic o- ing esona o s o mic owa e closed-loop ne wo ks and
acous ic ing esona o s, closed-loop eedback s uc u es wi h ni e ound- ip imes epea edly
appea ac oss die en physical pla o ms. Thei common ea u e is ha a wa e packe making a
ound ip in he loop acqui es a o al phase
Φ(ω;τ) = ϕ0(ω) + ωτ,
whe e
ϕ0(ω)
is he addi ional phase in oduced by he co e sca e ing and couple s, and
τ
is he
eec i e ound- ip ime. When
Φ(ω;τ)
sa ises in ege o hal -in ege quan iza ion condi ions, he
ne wo k's esonance, in e e ence, and ansmission ze o s uc u e unde goes signican changes,
igge ing ab up changes in ou pu ampli ude and phase.
In he combina ion s uc u e o op ical ing esonance and MachZehnde in e e ome e s,
π
-
scale phase jumps in he ansmission spec um a ound esonance equencies a e commonly ob-
se ed, along wi h hei co espondence o in e e ence-induced anspa ency eec s. Rela ed ex-
pe imen s and modeling indica e ha hese phase jumps a e closely ela ed o he cohe en in e -
e ence be ween wo o mo e pa hs in he loop, and o he opological s uc u e o esonance modes
in pa ame e space.
Simila
π
-jump phenomena also appea in he ansmission and eec ion phases o sys ems
such as phase-shi ed g a ings and spli - ing esona o s, and a e o en used as indica o s o iden i y
node s uc u e and mode opology. Howe e , abo e hese conc e e s uc u es, he e is s ill a lack
o a unied spec al heo y amewo k ha sys ema ically links  unable delay, phase s eps, and
pa i y opological sec o s.
1.2 Sca e ing Phase, Densi y o S a es, and Time Delay
In quan um sca e ing and wa e sca e ing heo y, he p o ound ela ionship be ween he phase o
he sca e ing ma ix
S(ω)
and ime delay and densi y o s a es has been sys ema ically es ablished
by wo k om mul iple di ec ions. The g oup delay and Wigne Smi h ime delay ma ix in oduced
by Wigne and Smi h
Q(ω) = −iS(ω)†∂ωS(ω)
cha ac e izes he a e age dwell ime o wa e packe s in he sca e ing po en ial eld, ha ing di ec
physical meaning o quan um, acous ic, and elec omagne ic wa e sca e ing.
On he o he hand, F iedel and Le inson- ype heo ems show ha unde app op ia e condi ions,
he e is a linea ela ionship be ween he sca e ing phase de i a i e and he die ence in densi y
2
o s a es wi h and wi hou in e ac ion. Fo one-dimensional o pa ial-wa e sca e ing, one ob ains
he o m
d
dEδl(E)∝ρl(E)−ρ(0)
l(E),
whe e
δl
is he pa ial wa e phase shi , and
ρl
and
ρ(0)
l
a e he densi ies o s a es wi h and wi hou
in e ac ion, espec i ely. Such esul s ha e been e o mula ed in ecen ma hema ical physics wo k
as opological index pai ings be ween spec al shi unc ions and ime delays, in oducing a clea
K- heo y and spec al ow pe spec i e in o sca e ing heo y.
On he expe imen al side, Wigne delay has been di ec ly measu ed in a omic sca e ing, wa eg-
uides, and op ical s uc u es, and linked o esonance li e imes and local densi y o s a es. Thus,
uni ying sca e ing phase, g oup delay, and densi y o s a es unde a single scale iden i y is a na u al
heo e ical de elopmen di ec ion.
1.3 Spec al Flow, Topological In a ian s, and
Z2
S uc u e
Spec al ow cha ac e izes he con inuous e olu ion o ope a o spec a in pa ame e space, and i s
ela ionship wi h opological in a ian s, especially in ege and
Z2
indices, has been sys ema ically
s udied in a ious si ua ions. Fo uni a y sca e ing ma ices, ze o/pole ajec o ies induced by
pa ame e changes can be cha ac e ized h ough he a gumen p inciple and index pai ings, leading
o conclusions simila o  opological Le inson heo ems: he o al change in phase equals he
spec al ow coun .
In many sys ems, each ime spec al ow c osses he eal axis due o pa ame e changes, he
o al phase only unde goes hal a ci cle o winding, co esponding o a jump o
π
a he han
2π
.
This sugges s he exis ence o a na u al double-co e s uc u e: each c ossing e en in he base
pa ame e space co esponds o wo sec o s in he li ed space, dis inguished by
Z2
pa i y. This
s uc u e is o mally isomo phic o spin double co e s, page-change phenomena in Fe mi s a is ics,
and double-co e sec o s in NullModula geome y.
1.4 Goals and S uc u e o This Pape
This pape ocuses on sca e ing ne wo ks wi h unable closed-loop delays, o malizing hem as a
pa ame e amily
S(ω;τ)∈U(N),
whe e
ω∈R
is he equency and
τ∈R
is he con ollable eec i e ound- ip ime. Unde he
cons ain o he scale iden i y
κ(ω;τ) = 1
π∂ωφ(ω;τ) = ρ el(ω;τ) = 1
2π Q(ω;τ)
his pape es ablishes he ollowing h ee main conclusions:
1. Unde na u al assump ions o analy ici y and non-degene acy, he ze o/pole spec al ow
a ying wi h
τ
o ms a se ies o isola ed c ossing e en s in he complex equency plane,
each co esponding o a pole o ze o c ossing he eal axis once.
2. Each c ossing e en induces a jump o size
±π
in he o al phase
φ(ω;τ)
a a xed equency
slice; he co esponding jump in he equency in eg al o scale densi y o g oup delay is one
uni .
3
3. The
Z2
index
ν(τ)
dened by
N(τ) mod 2
, whe e
N(τ)
is he opological coun cons uc ed
om spec al ow, ips once a each delay quan iza ion s ep, o ming he unied s uc u e
delay quan iza ion
⇒π
-s ep
⇒Z2
pa i y ansi ion.
Theo e ically, his pape p o ides spec al and opological p oo s o he abo e s uc u e and
illus a es i s uni e sali y h ough one-dimensional scala and mul i-channel ma ix models; in ap-
plica ions, his pape p oposes a se ies o expe imen al schemes based on op ical, mic owa e, and
acous ic pla o ms o measu e
π
-s eps and econs uc
Z2
indices, p o iding equency-domain ead-
ou s o sel - e e en ial sca e ing ne wo ks and double-co e s uc u es.
2 Model & Assump ions
2.1 F equency-Domain Sca e ing Ma ix, To al Phase, and G oup Delay
Conside a linea lossless ne wo k wi h
N
ex e nal channels, whose equency-domain sca e ing
ma ix is deno ed
S(ω;τ)∈CN×N, ω ∈R, τ ∈I⊂R,
whe e
I
is a pa ame e in e al. Losslessness means ha o each eal equency
ω
and
τ∈I
, we
ha e
S(ω;τ)†S(ω;τ) = IN.
Fo xed
τ
, assume
S(·;τ)
admi s analy ic con inua ion in o he uppe hal -plane, wi h poles co -
esponding o esonances o quasi-bound s a es; o xed
ω
, assume
S(ω;·)
is analy ic on
I
. Dene
he o al sca e ing phase
φ(ω;τ) = a g de S(ω;τ)∈R/2πZ,
and x a con inuous b anch in a neighbo hood o a chosen e e ence poin
(ω∗, τ∗)
such ha
φ(ω∗, τ∗)=0
.
The Wigne Smi h g oup delay ma ix is dened as
Q(ω;τ) = −iS(ω;τ)†∂ωS(ω;τ).
Fo uni a y ma ix amilies, we ob ain
∂ωφ(ω;τ) = ℑ∂ωlog de S(ω;τ) = 1
2 Q(ω;τ),
yielding he scale densi y
κ(ω;τ) := 1
π∂ωφ(ω;τ) = 1
2π Q(ω;τ).
In he s anda d sca e ing se ing,
κ(ω;τ)
can be iden ied wi h he ela i e densi y o s a es
ρ el(ω;τ)
, he die ence in densi y o s a es wi h and wi hou he sca e ing po en ial. This iden i-
ca ion makes he scale iden i y
κ(ω;τ) = φ′(ω;τ)
π=ρ el(ω;τ) = 1
2π Q(ω;τ)
he uni ying mo he o mula connec ing phase, ime, and densi y o s a es.
4
2.2 Tunable-Delay Feedback Loop Model
Gi en a delay- ee co e ne wo k
S0(ω)
, in oduce a closed-loop b anch be ween some o i s po s
wi h ound- ip delay
τ
, whose equency-domain desc ip ion is
D(ω;τ)=eiωτ IM, M ≤N.
Using he Redhee s a p oduc o Schu complemen , he co e ne wo k and delay block can be
combined in o an eec i e sca e ing ma ix
S(ω;τ) = S0(ω) + S1(ω)IM−R(ω)eiωτ −1S2(ω),
whe e
R(ω)
is an eec i e eedback block, and
S1, S2
a e coupling ma ices. When he co e is lossless
and he delay block is pu e phase,
S(ω;τ)
emains a uni a y ma ix o each eal equency
ω
. Poles
and some ze os a e con olled by
de IM−R(ω)eiωτ = 0.
Le
λj(ω)
be he eigen alues o
R(ω)
; he co esponding poles sa is y
1−λj(ω)eiωτ = 0 ⇐⇒ eiωτ =λj(ω)−1.
In a one-dimensional scala minimal model, he co e ne wo k is desc ibed by complex eec ion
coecien
0(ω)
and ansmission coecien
0(ω)
, wi h eedback b anch eec ion coecien
b(ω)
.
The o al sca e ing coecien is
S o (ω;τ) = 0(ω) + 0(ω)2eiωτ
1− b(ω) eiωτ ,
whose denomina o
1− b(ω)eiωτ
ze o poin s gi e pole ajec o ies.
2.3 Analy ici y and Non-Degene acy Assump ions
P oo s o subsequen heo ems ely on he ollowing assump ion.
Assump ion 1
(Assump ion A (Analy ici y and Non-Degene acy))
.
1. Fo each
τ∈I
,
S(·;τ)
admi s analy ic con inua ion in o he uppe hal -plane, wi h all ze os/poles o ni e o de and
only ni ely many in compac egions.
2. Fo each
ω∈R
,
S(ω;·)
is analy ic on
I
.
3. The e exis s a sequence
{τk} ⊂ I
wi h co esponding equencies
{ωk} ⊂ R
such ha in a
neighbo hood o each
(ωk, τk)
,
de S(ω;τ)
has exac ly one ze o o pole
zk(τ)
c ossing he eal
axis, sa is ying
zk(τk) = ωk, ∂τℑzk(τk)= 0,
and no o he ze os/poles simul aneously c oss he eal axis in he same neighbo hood.
When Assump ion A is sa ised,
(ωk, τk)
is called a c ossing e en , and
{τk}
is called a amily
o delay quan iza ion s eps. We will see ha when he eigen alues o
R(ω)
mo e along he uni
ci cle wi h app oxima ely equal spacing,
τk
app oxima ely o ms an a i hme ic sequence
τk≃τ0+k∆τ, k ∈Z,
whe e
∆τ
is gi en by he a e age ound- ip phase quan iza ion condi ion.
5

3 Main Resul s (Theo ems and Alignmen s)
This sec ion p esen s he main heo ems on delay-d i en spec al ow,
π
-s eps, and
Z2
indices unde
Assump ion A, and aligns hem wi h he scale iden i y.
3.1 Delay-D i en Spec al Flow and he A gumen P inciple
Fo xed
τ∈I
, suppose
de S(·;τ)
has ze os
{zj(τ)}
and poles
{pk(τ)}
(coun ed wi h mul iplic-
i y) in he uppe hal -plane, sa is ying app op ia e g ow h condi ions. Taking a closed con ou
Γ
su ounding he eal axis in e al
[ω1, ω2]
, he a gumen p inciple gi es
1
2π∆Γa g de S(·;τ) = Nze o(τ)−Npole(τ),
whe e
Nze o(τ), Npole(τ)
a e he numbe s o ze os and poles inside
Γ
, espec i ely. Choosing he
s anda d keyhole pa h, we ob ain he eal-axis in eg al o m
1
πφ(ω2;τ)−φ(ω1;τ)=Nze o(τ)−Npole(τ).
Fo a xed equency window
[ω1, ω2]
, as
τ
a ies con inuously, he ze o/pole ajec o ies
{zj(τ), pk(τ)}
e ol e con inuously in he complex equency plane. Whene e a ze o o pole c osses he eal axis,
he coun on he igh changes by
±1
, inducing a s ep in he o al phase wi hin ha equency
window.
3.2 Delay Quan iza ion S eps and C ossing E en s
In ne wo ks wi h delay b anches, he ze o/pole equa ion can o en be w i en as
de IM−R(ω)eiωτ = 0.
Le
λj(ω)
be eigen alues o
R(ω)
; he pole condi ion is
1−λj(ω)eiωτ = 0.
I
λj(ω) = |λj(ω)|eiϕj(ω)
, aking loga i hms gi es he app oxima e pole loca ion
ωj,n(τ) = 1
τϕj(ωj,n)+2πn −i ln |λj(ωj,n)|−1.
When
|λj(ω)|≲1
and
τ
a ies on mac oscopic scales, he eal pa is app oxima ely
ℜωj,n(τ)≃ϕj+ 2πn
τ.
Imagining
n
xed and
τ
inc easing, poles mo e along ajec o ies con ac ing om he a end
owa d he o igin, app oaching he eal axis unde app op ia e condi ions. Th ough small loss o
coupling adjus men s, one can cons uc si ua ions whe e poles c oss he eal axis, ealizing c ossing
e en s in Assump ion A.
Since
ωτ
is dimensionless, c ossing e en s ypically co espond o he condi ion
ωkτk+ϕj(ωk)≃(2mk+ 1)π, mk∈Z,
i.e., ound- ip phase sa ises hal -in ege quan iza ion, na u ally dening a amily o app oxima ely
equally-spaced delay s eps
{τk}
.
6
3.3 Main Theo em:
π
-S eps and
Z2
Pa i y T ansi ions
Nea a c ossing e en ,
de S(ω;τ)
can be w i en in local ac o iza ion
de S(ω;τ)=(ω−zk(τ))mkgk(ω;τ),
whe e
mk= +1
co esponds o a ze o,
mk=−1
o a pole, and
gk
is analy ic and nonze o in a
neighbo hood. Dene he local phase jump a
τk
∆φk= lim
ϵ→0+φ(ωk;τk+ϵ)−φ(ωk;τk−ϵ),
and he no malized jump numbe
∆nk=1
π∆φk.
Theo em 2
(Theo em 3.1:
π
-S ep and Uni Jump)
.
Unde Assump ion A, o each c ossing e en
(ωk, τk)
, he local phase change sa ises
∆φk=±π, ∆nk=±1.
See Sec ion 4 and Appendix A o he p oo . The co e is ha when
zk(τ)
c osses he eal axis,
ωk−zk(τ)
w aps a ound he o igin by hal a ci cle in he complex plane, so
a g(ωk−zk(τ))
jumps
by
±π
, while he analy ic ac o
gk
con ibu es con inuous phase no aec ing he jump coun .
By accumula ing all c ossing e en s wi h delay less han
τ
, dene he global spec al ow coun
N(τ) = X
τk<τ
∆nk∈Z, ν(τ) = N(τ) mod 2 ∈ {0,1}.
Theo em 3
(Theo em 3.2:
Z2
Pa i y T ansi ion)
.
Unde Assump ion A, he opological index
ν(τ) = N(τ) mod 2
unde goes a pa i y ip a each delay s ep
τk
, i.e.,
ν(τk+ 0) = ν(τk−0) ⊕1.
In pa icula , i
τk
o ms an app oxima ely a i hme ic sequence
τk≃τ0+k∆τ
, hen as
τ
inc eases
mono onically along
I
,
ν(τ)
execu es an app oxima ely ideal
Z2
squa e wa e.
The abo e esul s make p ecise he ela ionship be ween delay quan iza ion and
π
-s eps,
Z2
opological sec o s: each pole o ze o c ossing he eal axis co esponds o a uni spec al ow
e en , d i ing a ip in he opological index.
3.4 Unied Time Readou Unde he Scale Iden i y
By he scale iden i y
κ(ω;τ) = 1
π∂ωφ(ω;τ) = 1
2π Q(ω;τ),
aking he equency window
[ωk−δω, ωk+δω]
, dene
I(τ) = Zωk+δω
ωk−δω
κ(ω;τ) dω=1
πφ(ωk+δω;τ)−φ(ωk−δω;τ).
7
P oposi ion 4
(P oposi ion 3.3: Uni Jump in Scale Densi y In eg al)
.
Unde Assump ion A, o
each c ossing e en
(ωk, τk)
, he e exis s sucien ly small
δω > 0
such ha
I(τk+ 0) −I(τk−0) = ∆nk=±1.
Tha is, he in eg al o he g oup delay ace
Q(ω;τ)
in a small equency window jumps by
one uni a each delay quan iza ion s ep.
Thus, he opological index
ν(τ)
can be dened no only h ough he jumps in o al phase in
pa ame e space, bu also equi alen ly desc ibed h ough jumps in he equency in eg al o scale
densi y o ela i e densi y o s a es.
4 P oo s
This sec ion p o ides p oo ou lines o he main heo ems, de e ing echnical de ails o Appendices
A and B.
4.1 Local A gumen Analysis and P oo o Theo em 3.1
In a neighbo hood o he c ossing e en
(ωk, τk)
, w i e
de S(ω;τ)
as
de S(ω;τ)=(ω−z(τ))mg(ω;τ),
whe e
z(τk) = ωk
,
∂τℑz(τk)= 0
,
m=±1
, and
g
is analy ic wi h
g(ωk;τk)= 0
.
Fo xed
ω=ωk
, conside he unc ion
h(τ) = ωk−z(τ) = a(τ)−ib(τ),
whe e
a(τ) = ωk− ℜz(τ)
,
b(τ) = ℑz(τ)
. Nea
τk
,
a(τk)= 0
,
b(τk)=0
, and
∂τb(τk)= 0
, so when
τ
a e ses
τk
, he ec o
h(τ)
c osses he eal axis in he complex plane.
S anda d complex analysis geome y shows:
∆ a g h:= lim
ϵ→0+a g h(τk+ϵ)−a g h(τk−ϵ)=±π,
wi h sign de e mined by he signs o
a(τk)
and
∂τb(τk)
. Since
φ(ωk;τ) = ma g h(τ) + a g g(ωk;τ),
and
g
is nonze o in a neighbo hood,
a g g(ωk;τ)
can be chosen as a con inuous b anch, so in he
local limi
∆φk=m∆ a g h=±π.
Thus
∆nk= ∆φk/π =±1
, p o ing Theo em 3.1. See Appendix A o de ailed p oo .
4.2 Scale Densi y In eg al and P oo o P oposi ion 3.3
By
I(τ) = 1
πφ(ωk+δω;τ)−φ(ωk−δω;τ),
we can w i e
I(τk+ 0) −I(τk−0) = 1
π∆φ(ωk+δω)−∆φ(ωk−δω).
8
Choose
δω
sucien ly small such ha in he ec angula egion
[ωk−δω, ωk+δω]×[τk−δτ, τk+δτ]
he e is only one c ossing ze o o pole, and i s ajec o y c osses he midline o he equency
window. Using he analysis o he local ac o
(ω−z(τ))m
in Appendix A, we know ha in his
egion
∆φ(ω;·)
is a piecewise cons an unc ion o
ω
, wi h he die ence in alues on ei he side o
ωk
being
±π
. Thus
I(τk+ 0) −I(τk−0) = ±1.
This p o es P oposi ion 3.3. By he deni ion o
N(τ)
and in ege addi ion s uc u e, clea ly
ν(τ) = N(τ) mod 2
ips once a each s ep, hence Theo em 3.2 holds.
4.3 Fini e-O de Eule Maclau in and Nume ical E o Con ol
In ac ual nume ical and expe imen al da a p ocessing, scale densi y in eg als a e o en app oxima ed
by ni e sampling as disc e e sums. Le
Ih(τ) = h
N
X
n=0
κ(ωn;τ), ωn=ωk−δω +nh,
whe e
h= (2δω)/N
. The Eule Maclau in o mula gi es
Ih(τ) = Zωk+δω
ωk−δω
κ(ω;τ) dω+O(h2),
p o ided
κ(ω;τ)
has bounded second de i a i es in he equency window. As long as he sampling
s ep
h
is sucien ly small, he phase s ep heigh o
±1
is only smoo hed, no e ased o ipped.
Appendix B p o ides s anda d es ima es o he Eule Maclau in emainde , showing ha he
singula i y nea poles is only smoo hed a ni e esolu ion, wi hou changing he spec al ow
coun . Thus he opological index
ν(τ)
is obus o ni e esolu ion and noise.
5 Model Applica ions
This sec ion e u ns o conc e e models, demons a ing he implemen a ion o he main heo ems
in single-channel scala and mul i-channel ma ix models, and discussing local linea iza ion in sel -
e e en ial sca e ing ne wo ks.
5.1 Single-Channel Reec ion-Type Feedback Model
Conside he single-channel model
S o (ω;τ) = 0(ω) + 0(ω)2eiωτ
1− b(ω) eiωτ .
In a small equency window, adop he slow- a ia ion app oxima ion, ea ing
0, 0, b
as con-
s an s
0, 0, b ∈C
, sa is ying
| 0|2+| 0|2= 1,| b| ≤ 1.
9
C Rela ionship wi h Sel -Re e en ial Sca e ing Ne wo ks and Double-
Co e Geome y
This appendix discusses he posi ion o his pape 's esul s in he b oade con ex o sel - e e en ial
sca e ing ne wo ks and double-co e s uc u es.
C.1 Sel -Re e en ial Sca e ing Ne wo ks and Nonlinea Feedback
In sel - e e en ial sca e ing ne wo ks, he esponse in eedback loops can depend on he ne wo k's
own ou pu s a es o his o y, making he sca e ing ma ix a nonlinea o ime- a ying objec .
Typical examples include eedback s uc u es wi h gain sa u a ion, nonlinea phase modula ion, o
adap i e con ol.
Nea a wo king poin , he nonlinea ne wo k can be linea ized o ob ain an eec i e sca e ing
ma ix
Se (ω;τ)
and co esponding eedback block
Re (ω)
. As long as he linea ized sys em sa ises
Assump ion A, all conclusions abou spec al ow,
π
-s eps, and
Z2
indices in his pape emain alid.
This shows ha in he pa ame e space o sel - e e en ial ne wo ks, a amily o local egions can be
iden ied whe e he ne wo k's opological beha io is assembled om se e al
π
-s ep uni s.
C.2
Z2
Double Co e and Hal -Phase Winding
The
π
-s eps disco e ed in his pape a e essen ially hal -ci cle windings in phase space. Conside ing
he phase map o
de S(ω;τ)
in he complex plane, i s na u al alue space is
R/2πZ
. I li ed o
he double-co e space
R/πZ
, hen each
π
jump co esponds o one ull winding in he double-co e
space, and he
Z2
index
ν(τ)
cha ac e izes he numbe o page- u ns o he li ed pa h be ween wo
pages.
This s uc u e has o mal pa allelism wi h he spin double co e
Spin(n)→SO(n)
and he
e mion s a is ics phenomenon o wo-winding iden i y: in sca e ing phase space, he e a e only
wo ypes o sec o s, dis inguished by odd o e en numbe s o
π
-jumps.
C.3 Role in Unied Time Scale and Bounda y Geome y
In he amewo k o unied ime scale and bounda y ime geome y, he scale iden i y unies
sca e ing phase de i a i e, ela i e densi y o s a es, and Wigne Smi h g oup delay ace in o a
single ime scale densi y. This pape shows ha when unable delay eedback exis s in ne wo ks,
ime scale densi y exhibi s a amily o disc e e opological ecalib a ion poin s in pa ame e space,
like inse ing a se ies o  empo al hal -la ice poin s on he pa ame e axis.
These ecalib a ion poin s o m
π
-s eps in he equency domain and co espond o ips o
Z2
sec o s opologically, which can be iewed as opological ma ke s o he unied ime scale in
pa ame e space. Fo highe -laye sel - e e en ial uni e se models and NullModula double-co e
s uc u es, his pape 's delay eedback ne wo ks cons i u e compu able, expe imen ally e iable
undamen al opological modules.
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