Self-Referential Scattering Networks: Connection Matrix Synthesis, J-Unitary Robustness, and Floquet Band-Edge Topology

Sel -Re e en ial Sca e ing Ne wo ks: Connec ion Ma ix
Syn hesis,
J
-Uni a y Robus ness, and Floque Band-Edge
Topology
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Ve sion 1.10
Abs ac
In Sel -Re e en ial Sca e ing Ne wo ks (SSN), his pape p o ides closed-loop me hodol-
ogy om
designimplemen a ion eadou  alsica ion
o
heo em-le el gua an ees
.
Unde
ace-class calib a ion
, es ablish
mod- wo equi alence
be ween global hal -phase
(
√de
co e ing) holonomy, spec al shi , spec al ow h ough
−1
, and disc iminan ans e -
sali y. Compa ed o p e ious d a , his e sion comple es
e iable de ails and checkable
cons an s
a  e key junc u es: (i) Sec ion 3 adds
b idging lemma
o 
log de
egula iza ion
Bi manK en
−1
spec al owmod- wo in e sec ion numbe  wi h hal -page
sel -con ained
p oo
; (ii) Sec ion 4 p o ides
quan i a i e s uc u al lemma
o 
no spu ious c oss-
ing
 a e s a p oduc , speci ying
compa ison p inciple
and
design line
o p incipal block
minimal g adien
gmin
and mu ual coupling uppe bound
max
; (iii) Sec ion 5 o mula es
bi-
na ized p ojec ion + majo i y o ing
as
concen a ion inequali y
, p o iding explici
ela ion be ween misclassica ion a e and sample numbe wi h
co ela ion co ec ion
; (i )
Sec ion 6 cha ac e izes obus domain ia
J
-inne -p oduc no malized
imagina y Rayleigh
quo ien (K en angle)
, cons uc s
pola iza ion homo opy
, p o ides easible uppe bound
o h eshold unc ion
ε0(η, β)
, uni o mly ensu ing Cayley denomina o in e ibili y and dis-
c iminan non-c ossing; ( ) Sec ion 7 o mula es
phase- ype Floque index unca ion
independence
and
gauge independence
as heo ems,
comple ing egula iza ion con-
sis ency and ailu e de ec ion ia
de 2
in Hilbe Schmid scena io
. Enginee ing side
p o ides simula able Schu -closed o m o wo-po couple mic o inggain p o o ype, quan-
i ying squa e- oo scaling and h eshold selec ion basis o g oup delay double-peak me ging.
Keywo ds
: Closed-loop sca e ing; Redhee s a p oduc ; Schu complemen ; He glo z/Ne anlinna;
disc iminan ; spec al shi and spec al ow; mod- wo Le inson;
J
-uni a y; K en angle; Flo-
que phase- ype index.
1 Calib a ion and Basic Objec s
F equency domain and a iables
:
ω
as angula equency (s a ic
ω∈R
; pe iodic sys ems
ω∈[−π/T, π/T]
), uppe hal -plane calib a ion akes
z=ω+ i0+
. Exchange wi h uni ci cle
calib a ion ia Cayley map when necessa y, main aining o ien a ion consis ency.
Sca e ing and s a p oduc
: Po ized node sca e ing
Sj(z)
h ough in e connec ion (in-
cluding eedback) yields closed-loop
S⟲(z)
ia Redhee s a p oduc and Schu complemen .
Cayley duali y
(unied no a ion):
S⟲(z)=(I−iK(z))(I+ iK(z))−1, K(z) = −iI+S⟲(z)−1I−S⟲(z).
1
T ace-class and egula iza ion
: All esul s ela ed o spec al shi
ξ
and Bi manK en
uni o mly assume
S⟲−I∈S1
using
de
. Fo inni e dimensions (e.g., Floque sidebands)  s
unca e ni ely, use
de 2
when necessa y; his pape 's
mod- wo conclusions independen o
de /de 2
choice
(see Appendix F).
J
-uni a y calib a ion
:
J=J†=J−1
,
S♯=J−1S†J
,
S†JS =J
. Non-He mi ian obus ness
s a ed unde his calib a ion.
2 Disc iminan , Local Model, and T ans e sali y
On pa ame e mani old
X
dene codimension-one piecewise smoo h submani old amily
D=FbDb
,
whose local models include Jos ze os, h eshold opening/closing, embedded eigen alues, and EP
coalescence. A e emo al
X◦=X D
. Fo closed pa h
γ⊂X◦
, mod- wo in e sec ion numbe
I2(γ, D)
is pa i y o ans e se poin s. Endpoin s/ h esholds excised ia small semici cles me ged
in o
D
bounda y componen . Closed-loop implemen a ion le el can use
D=n(ω, ϑ) : σminI−C(ω, ϑ)Sii(ω, ϑ)= 0o,
equi alen o
de (I−CSii)=0
a gene al posi ion.
3 Mod-Two Hal -PhaseSpec al-Shi Spec al-FlowIn e sec ion:
B idging Lemma and Equi alence Theo em (T ace-Class)
Deni ion 1
(3.1: Global Hal -Phase)
.
ν√de S⟲(γ) = expiIγ
1
2d a g de S⟲∈ {±1}.
Lemma 2
(3.2:
log de
Regula iza ion and Endpoin T ea men )
.
I along closed pa h
γ
almos e -
e ywhe e
S⟲
uni a y wi h
S⟲−I∈S1
, hen
log de S⟲
admi s con inuous con inua ion along
γ
and
is in eg able; a e ea ing endpoin s/ h esholds pe 2 small semici cle excision ule,
Hγ
1
2d a g de S⟲
well-dened (
mod 2π
).
Lemma 3
(3.3: Bi manK en
−1
Spec al Flow B idge: Two-S ep Ve iable De ails)
.
Deno e
ξ(ω)
as spec al shi . Then (i) By BK o mula
de S(ω) = exp{−2πiξ(ω)}
, a bi a y b anch change
makes
ξ7→ ξ+n
(
n∈Z
), con ibu ing
exp{iH1
2d(2πn)}= 1
o hal -phase, hus
mod- wo in a i-
an
; (ii) Le
S(τ) = V(τ)eiΦ(τ)U(τ)
be local Schu o m wi h a
τ=τc
only one simple eigenphase
ϕj
c ossing
π
(
−1
), hen
ξ
jump is
±1
while
S −1= 1
. Mul iple c ossings decompose in o ni e
simple c ossings, hus
exp−iπIγ
dξ= (−1)S −1(S⟲◦γ).
P oposi ion 4
(3.4: Base-Poin and B anch Independen Mod-Two P ope y)
.
Fo a bi a y base
poin and con inuous b anch con inua ion,
ν√de S⟲(γ)
in a ian ; allowed endpoin ea men equi -
alen o adding bounda y homo opy on
D
, mod- wo alue p ese ed.
Theo em 5
(3.5: Fou -Fold Equi alence)
.
I along
γ
almos e e ywhe e
S⟲
uni a y wi h
S⟲−I∈
S1
, hen
ν√de S⟲(γ) = exp−iπIγ
dξ= (−1)S −1S⟲◦γ= (−1)I2(γ,D).
2
4 No Spu ious C ossing A e S a P oduc and
Z2
Combina o ial
Law (Quan i a i e Ve sion)
Block no a ion
:
S(k)= S(k)
ee S(k)
ei
S(k)
ie S(k)
ii !, k = 1,2.
Lemma 6
(4.1: S uc u al Lemma o No Spu ious C ossing: Quan i a i e Condi ions wi h Tubu-
la Sepa a ion)
.
Assume in neighbo hood
U⊂X◦
sa is ying (i) Schu in e ibili y lowe bound:
σminI−S(1)
ii S(2)
ii ≥δ > 0
; (ii) Mu ual coupling smallness:
∥S(1)
ei ∥2,∥S(2)
ie ∥2≤ρ < 1
;
(iii) Tubu-
la sepa a ion
: Fo ze o se s
D(k)
o
k= 1,2
exis s uni o m ubula adius
τ∗>0
, ake
0< τ ≤τ∗
such ha
Nτ(D(1))
,
Nτ(D(2))⊂U
and disjoin .
Dene p incipal disc iminan
Φk(ϑ) = de I−CS(k)
ii 
and p incipal block minimal g adien
gmin = in
ϑ∈Umin|∇ϑΦ1(ϑ)|,|∇ϑΦ2(ϑ)|.
Then mu ual coupling esidual uppe bound exis s
max ≤∥S(1)
ei ∥2∥S(2)
ie ∥2
δ2,
and when
max <(τgmin)2
ne wo k disc iminan is ans e se disjoin union o subne disc iminan s:
Dne =D(1) ⊔D(2)
,
wi h
I2(γ, Dne ) = I2(γ, D(1)) + I2(γ, D(2)) mod 2.
P oo essen ials
: By ubula sepa a ion and mean alue heo em, ou side ubes ha e
|Φ1(ϑ)| ∧
|Φ2(ϑ)| ≥ τgmin
, hus
|Φ1(ϑ)Φ2(ϑ)| ≥ (τgmin)2
; when
max <(τgmin)2
, mu ual coupling emainde
insucien o in oduce new ze os ou side ubes. In each single ube use implici unc ion heo em
ob aining ze o se as no mal small de o ma ion o o iginal ze o se , ob aining ans e se disjoin
union since ubes non-in e sec ing.
Enginee ing design line ( e iable)
Take
δ= in Uσmin(I−S(1)
ii S(2)
ii )
. I
∥S(1)
ei ∥2∥S(2)
ie ∥2≤1
2δ2(τgmin)2,
hen
max ≤1
2(τgmin)2<(τgmin)2
, sa is ying lemma sucien condi ion.
Bounda y and ailu e mode 4.2
: When
δ→0+
o
ρ→1−
, nea - angen c ossing and
esonance-induced spu ious c ossing may occu . Should eal- ime moni o
σmin(I−CSii)
ma gin
and
gmin
nume ical es ima e, sh ink
U
o econgu e po s when necessa y.
Theo em 7
(4.3:
Z2
Combina o ial Law)
.
Unde lemma condi ions,
νne =ν(1) ⊙ν(2)
(componen -
wise mul iplica ion, mod- wo addi ion).
3
5 Bina ized P ojec ion, Concen a ion Inequali y, and Dealiasing
Phase inc emen and bina iza ion
:
∆ϕab =1
2ha g de Sγa;θb+δ−a g de Sγa;θb−δicon ,Π(∆ϕab) = 1{|∆ϕab|≥π/2}.
Deni ion 8
(Eec i e Phase Window)
.
Assume expe imen /simula ion uses measu emen g id
G={(a, b)}
, dene
∆ϕab
and bina iza ion ule
Π(·)
pe abo e o mula. Dene
∆ϕe := ess in
(a,b)∈G |∆ϕab|.
When bounded addi i e noise
ε
exis s wi h
|ε| ≤ ϵ
, adop h eshold condi ion
∆ϕe >π
2+ 2ϵ
;
acco dingly ob ain Theo em 5.1's misclassica ion a e and sample complexi y es ima e.
Hypo hesis (independence and sub-Gaussian)
Measu emen samples
{c
∆ϕ(n)
ab }N
n=1
inde-
penden ,
Ec
∆ϕ(n)
ab = ∆ϕab
, sub-Gaussian wi h p oxy a iance
σ2
.
Theo em 9
(5.1: E o Bound and Sample Complexi y o Majo i y Vo ing)
.
Assume measu emen
samples
{c
∆ϕ(n)
ab }N
n=1
mu ually independen ,
Ec
∆ϕ(n)
ab = ∆ϕab
, sub-Gaussian wi h p oxy a iance
σ2
;
wi h bounded addi i e bias
|ε| ≤ ϵ
. Deno e
m:= ∆ϕe −π
2−2ϵ > 0, q := P|c
∆ϕ(n)
ab |<π
2.
Then
q≤2 exp−m2
2σ2,P(
majo i y o ing misjudgmen
)≤exp−2N1
2−q2.
Gi en a ge e o
δ∈(0,1)
, sucien condi ion is
N≥log(1/δ)
21
2−2e−m2/(2σ2)2,
equi ing
m>σ√2 log 4
ensu ing
q < 1
2
o majo i y o ing con e gence. I o e sampling
co ela ion exis s, eplace
N
in abo e o mula wi h eec i e sample numbe
Ne
.
P oposi ion 10
(5.2: Dealiasing and Seconda y E idence Fusion)
.
Fo mul iple c ossings o nea -
h eshold masking: (i) Adop mul i-window sliding and ancho con inui y s a egy; (ii) Fuse g oup
delay double-peak me ging, con ming c ossing only when bo h nge p in s synch onize; (iii) When
c oss alk occu s, add edundan columns nea ly o hogonal o exis ing sensi i i y, ecalcula e G am
c i e ion un il ull ank.
6
J
-Uni a y Robus ness: K en Angle, Pola iza ion Homo opy, and
Th eshold Func ion
K en angle and angle gap
:
Assume
⟨ψj(τ), Jψj(τ)⟩ = 0
(non-neu al eigens a e), o he -
wise
κj
undened, and ha pa ame e poin iewed as obus domain bounda y and
excluded.
Dene
4
κj(τ) = Im ⟨ψj(τ), J S−1(∂τS)ψj(τ)⟩
⟨ψj(τ), J ψj(τ)⟩,
consis en wi h
∂τϕj
in uni a y limi
J=I
, used as
phase slope
.
Angle gap
dened as
η:= min
jin
τdis ϕj(τ), π + 2πZ∈(0, π].
Nea
J
-uni a y he e exis
c±(ε)→1
making
c−|∂τϕj| ≤ |κj| ≤ c+|∂τϕj|
.
No e:
κj
only as
slope con ol e m, no pa icipa ing in
η
deni ion
.
Lemma 11
(6.1: Es ima e om
(S†JS −J)
o
(K−K♯)
)
.
Assume
S(τ)
poin wise in e ible along
conside ed pa ame e domain, wi h
sup
τ|S(τ)|<∞,sup
τ|S(τ)−1|<∞,|S†JS −J| ≤ ε, β = in
τσmin(I+ iK(τ)) >0.
Then cons an
C=Cβ, sup |S|,sup |S−1|
exis s making
|K−K♯| ≤ C ε.
P oo essen ials
: Use F éche die en ial o Cayley in e se map
K=−i(I+S)−1(I−S)
wi h
J
-conjuga ion and bounded mul iplie inequali y.
Cons uc ion 6.2 (pola iza ion homo opy)
Le
K = (1 − )K+ 1
2(K+K♯), S = (I−iK )(I+ iK )−1, ∈[0,1].
By Lemma 6.1 and Neumann lemma ob ain
σminI+ iK (τ)≥β−
2C ε,
hus when
ε < 2β/C
,
(I+iK )
in e ible h oughou
∈[0,1]
,
S
well-dened. In nea
J
-uni a y
calib a ion, cons an
α > 0
exis s (depending on
|S|
,
|S−1|
,
β
) making
ε0(η, β) := min 2β
C, α sin2η
2
easible uppe bound
: when
∥S†JS −J∥ ≤ ε<ε0(η, β)
wi h
η > 0
, homo opy
{S }
doesn'
in e sec
D
, wi h Cayley denomina o in e ible h oughou .
Theo em 12
(6.3: Homo opy Robus ness)
.
I homo opy
{S }
sa is ying abo e o mula exis s wi h-
ou in e sec ing
D
h oughou , hen
ν√de S⟲
equals uni a y limi alue.
Squa e oo asymp o ics (enginee ing nge p in )
: Dominan b anching
2×2
eec i e
subspace yields
a g de S( ) = a g de S( c)±a c anκ1/2| − c|1/2+O| − c|3/2,
g oup delay exhibi s symme ic double-peak me ging, peak sepa a ion
∆ω=Cp| − c|+O| −
c|3/2
. Highe -o de oo s (EP o de
>2
) mod- wo equi alen o squa e oo .
5

7 Floque -SSN: Phase-Type Band-Edge Index, T unca ion and Gauge
Independence
Deni ion 13
(7.1: Phase-Type Index)
.
T unca e sideband o
|n| ≤ N
ob aining ni e-dimensional
S(N)
F(ω)
, dene
ν(N)
F= expi
2Zπ/T
−π/T
∂ωa g de S(N)
F(ω) dω∈ {±1}.
Equi alen ly,
ν(N)
F= expi
2Zπ/T
−π/T
Im ∂ωlog de S(N)
F(ω) dω.
Theo em 14
(7.2: T unca ion Independence: No m/HS Ve sion)
.
I
S(N)
F→SF
in ope a o no m
o Hilbe Schmid opology, wi h endpoin s
ω=±π/T
ha ing no
N
-mig a ing b anching, hen
N∗
exis s making
ν(N)
F
s able o
N≥N∗
; dene
νF=ν(N∗)
F
. In
Hilbe Schmid scena io
, eplace
de
die en ial wi h Koplienko spec al shi and
de 2
de i a i e ecipe, combined wi h Appendix F
mod- wo consis ency, ob aining same
νF
.
S ong con e gence alone insucien o ensu e
de 2
and second-o de ace o mula well-denedness, hus no included in heo em
p emise.
Lemma 15
(7.3: Gauge Independence)
.
I
SF(ω)7→ UL(ω)SF(ω)UR(ω)
whe e
UL,R
con inuous
wi h
|de UL,R|= 1
, sa is ying band-edge gluing condi ion
[a g de UL+ a g de UR]π/T
−π/T ∈4πZ,
hen
Zπ/T
−π/T
∂ωa g de SF(ω) dω
mod- wo alue in a ian .
Theo em 16
(7.4: Band-Edge Equi alence)
.
I endpoin s sa is y squa e oo local model wi h The-
o em 7.2 holding, hen
νF= (−1)I2[−π/T,π/T ],DF.
Failu e mode 7.5 (de ec ion and aul ole ance h eshold)
When unca ion induces
endpoin pseudo-b anching, moni o
min
ω∈{±π/T}σmin
I−CS(N)
ii (ω)≥δF,
equi ing
ν(N)
F=ν(N+1)
F=ν(N+2)
F
con inuous h ee-o de consis ency o judge s able.
6
8 P o o ype and SOP
8.1 Two equi alen pa hs o single c ossing
: (i) Complex pa ame e small loop:
λ
ci cles
Jos ze o once in complex plane; (ii) Real pa ame e a e se + equency domain eadou : Real
pa ame e a e ses
D
, selec single b anch in equency domain using ancho con inui y. Bo h
homo opy equi alen . Along any
closed loop
om complex pa ame e small loop o  eal pa ame e
a e se + e u n closu e, ha e
Id a g de S=±2π, expi
2Id a g de S=−1,
ully consis en wi h 3 hal -phasespec al-ow pa i y.
8.2 Two-po couple mic o inggain p o o ype (simula able)
: Couple
C(κ) = √1−κ2iκ
iκ√1−κ2,C(ω, ) = ρeiϕ(ω, ).
Eec i e sca e ing (Schu -closu e)
S⟲=See +SeiI−CSii−1CSie.
In c i ical neighbo hood o
CSii →I
alls back o 8.1 squa e oo model, di ec ly ep oducing
∆φ=π
and g oup delay double-peak me ging.
8.3 SOP and c i e ia
: Po de-embedding

sweep eedback c ossing
D

use
∆ϕe >
π/2+2ϵ
and double-peak me ging as passing condi ion; iple nge p in asynch ony e oes causal
chain, e u ning o column con ollabili y p ocedu e o edundan column addi ion o equency
window eselec ion.
9 Readou and Wigne Smi h Unied Calib a ion
∂ωlog de S= S−1∂ωS, ∂ωa g de S= Im S−1∂ωS.
In uni a y case le
Q(ω) = −iS†∂ωS
, hen
∂ωa g de S= Q(ω)
. In
J
-uni a y case can also
w i e
∂ωa g de S= Im (S♯∂ωS)
.
10 Conclusion
This pape igo izes hal -phasespec al-shi spec al-owin e sec ion ou - old equi alence ia
b idging lemma
; ensu es s a -p oduc in e connec ion disc iminan ans e se disjoin union and
Z2
combina o ial law ia
quan i a i e s uc u al lemma
; gua an ees bina ized p ojec ion ep o-
ducibili y ia
concen a ion inequali y
; cha ac e izes nea
J
-uni a y obus domain and h esh-
old ia
K en angle and pola iza ion homo opy
; closes band-edge opological eadou chain
ia
phase- ype Floque index
and
unca ion/gauge independence heo ems
. Combined
wi h minimal p o o ype and SOP, p o ides p og ammable implemen a ion and coun e ac ual e -
ica ion o closed-loop sel -consis ency
⇒
squa e- oo c i icali y
⇒
double Riemann shee
⇒Z2
hal -phase.
7
Re e ences (Selec ed)
[1] R. Redhee , On a Ce ain Linea F ac ional T ans o ma ion,
Pacic J. Ma h.
,
9
(1959) 871
893.
[2] J. Gough, M. R. James, The Se ies P oduc and I s Applica ion o Quan um Feed o wa d
and Feedback Ne wo ks,
IEEE TAC
,
54
(2009) 25302544.
[3] B. Simon,
T ace Ideals and Thei Applica ions
, 2nd ed., AMS (2005).
[4] M. Sh. Bi man, M. G. K en, On he Theo y o Wa e and Sca e ing Ope a o s,
So . Ma h.
Dokl.
,
3
(1962) 740744.
[5] L. Koplienko, T ace Fo mula o Pe u ba ions o Class
S2
,
Sb. Ma h.
,
122
(1983) 457486.
[6] E. P. Wigne , Lowe Limi o he Ene gy De i a i e o he Sca e ing Phase Shi ,
Phys.
Re .
,
98
(1955) 145147; F. T. Smi h, Li e ime Ma ix in Collision Theo y,
Phys. Re .
,
118
(1960)
349356.
[7] T. Ya. Azizo , I. S. Iokh ido ,
Linea Ope a o s in Spaces wi h an Indeni e Me ic
, Wiley
(1989).
[8] D. Z. A o , H. Dym,
J
-Con ac i e Ma ix-Valued Func ions and Rela ed Topics
, CUP (2008).
[9] I. C. Fulga, F. Hassle , A. R. Akhme o , Sca e ing Fo mula o he Topological Quan um
Numbe ,
Phys. Re . B
,
85
(2012) 165409.
[10] M. S. Rudne , N. H. Lindne , E. Be g, M. Le in, Anomalous Edge S a es...,
Phys. Re .
X
,
3
(2013) 031005.
A No a ion and Regula iza ion
No a ion
:
J
(K en me ic),
S♯=J−1S†J
;
S1/S2
( ace-class/HS);
de ⋆
(ni e-dimensional ake
de
, HS ake
de 2
);
D, DF
(disc iminan /band-edge disc iminan );
Λ
,
Πτ
,
η
,
Q=−iS†∂ωS
,
ν√de S⟲
,
νF
.
Calib a ion
: Main ex adop s
S1
. HS case s abilizes wi h
de 2
; ou - old equi alence and
νF
mod- wo alue independen o
de /de 2
choice (see Appendix F).
B Co e ingLi ing and
Z2
Reduc ion
Squa e co e ing
p:z7→ z2
co esponds o mul iplica ion by wo in cohomology:
s:X◦→U(1)
exis s making
s2= de S⟲
i and only i
[de S⟲]∈2H1(X◦;Z)
. I s
Z2
educ ion is his pape 's
global hal -phase in a ian .
C Squa e Roo Puiseux Asymp o ics and E o
Fo
2×2
eec i e subblock
M( )
wi h
∆( )≈∆′( c)( − c)
, Cayley mapping o
S( )=(I−iM)(I+
iM)−1
has leading e m
±a c anκ1/2| − c|1/2+O| − c|3/2,
g oup delay exhibi s symme ic double-peak me ging, peak sepa a ion
∆ω=Cp| − c|+O| −
c|3/2
. Subs i u ing in o 5's
∆ϕe
yields sample complexi y es ima e.
8
D Ta ge - o-De ice Execu ion Checklis and
ℓ
Open-loop calib a ion and de-embedding

choose
W
and s ep size

compu e
Sθb(ω) = ∂θba g de S(ω)
in eg a ing in o
M

ob ain
Π(M)
ia
π/2±τ
checking ank

sol e
Π(M)x=ℓ
selec ing columns

igge column-by-column wi h
∆φ=π
, double-peak me ging as s op

coun e ac ual e ica-
ion and a chi e.
ℓ= (1 −ν⋆)/2∈ {0,1}m
(
ν⋆= +1 7→ 0
,
ν⋆=−17→ 1
).
E K en Angle and Homo opy Th eshold unde
J
-Uni a y
Assume
S†JS =J
, deno e
X=S−1˙
S
. By die en ia ion
X♯=−X
(
J
-skew-He mi ian). Fo
(λj= eiθj, j)
:
˙
θj=Im ⟨ j, JX j⟩
†
jJ j
,κj=Im ⟨ j, JX j⟩
†
jJ j
.
F om abo e wo o mulas poin wise ob ain
κj=˙
θj
, hus
|˙
θ1−˙
θ2|=|κ1−κ2|.
By Lemma 6.1 ob ain
∥K−K♯∥ ≤ C ε
. Take
K = (1 − )K+ (K+K♯)/2
,
S = (I−iK )(I+
iK )−1
. By Neumann lemma ob ain
σmin(I+iK )≥β−
2C ε
, hus when
ε < 2β/C
,
S
well-dened.
I
∥S†JS −J∥ ≤ ε
wi h
η > 0
, cons an
α > 0
exis s making
ε0(η, β) = min{2β/C, α sin2(η/2)}
easible uppe bound, uni o mly ensu ing Cayley denomina o in e ibili y and disc iminan non-
c ossing.
F Regula iza ion Independence (Unied P oposi ion)
P oposi ion 17
(F.1:
de /de 2
Mod-Two Consis ency)
.
Assume along closed pa h
γ
almos e -
e ywhe e
S⟲
uni a y. I
S⟲−I∈S1
hen
expiIγ
1
2∂log de S⟲= (−1)S −1= (−1)I2.
I only
S⟲−I∈S2
, ake
S1
app oxima ion amily
S⟲
ϵ→S⟲
in HS opology, dene hal -phase
ia
de 2
, hen
lim
ϵ→0expiIγ
1
2∂log de S⟲
ϵ= expiIγ
1
2∂log de
2S⟲,
wi h bo h sides' mod- wo alue consis en wi h
S −1
,
I2
.
G Floque T unca ion, Con e gence, and Failu e Mode
T unca e
SF
sideband o
|m| ≤ M
ob aining
S(M)
F
. I
P|m|>M ∥Km∥ → 0
(
ope a o no m
o HS con e gence; his condi ion implies ope a o no m con e gence
), wi h endpoin s
ha ing no
M
-mig a ing b anching, hen
supω∥S(M+1)
F−S(M)
F∥ → 0
,
M0
exis s making
ν(M)
F
s able
o
M≥M0
. De ec ion quan i ies: endpoin singula alue h eshold
δF
and pla eau s abili y
c i e ion (
ν(M)
F=ν(M+1)
F=ν(M+2)
F
). When abno mal d i occu s inc ease
M
o educe coupling
bandwid h o a oid pseudo-b anching.
9