Null--Modular Double Cover and Overlapping Causal Diamond Chains: Total-Order Approximation Bridge for Quadratic Form Localization, Inclusion--Exclusion--Markov Splicing, and Parity Threshold for Distributional Scattering Calibration

NullModula Double Co e and O e lapping Causal Diamond
Chains:
To al-O de App oxima ion B idge o Quad a ic Fo m
Localiza ion, InclusionExclusionMa ko Splicing, and Pa i y
Th eshold o Dis ibu ional Sca e ing Calib a ion
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
We p opose
NullModula double co e
ca ied by ze o-measu e bounda ies o causal di-
amonds, decomposing modula Hamil onians in o local ene gy ux in eg als on wo null shee s in
acuum quad a ic o m sense. Th ough a
o al-o de app oxima ion b idge lemma
, gene al
diamonds a e educed o mono onic hal -space amily limi s on he same ze o-measu e hype -
plane, wi h quad a ic o m closedness and domina ed con e gence ensu ing limi independence o
app oxima ion pa hs. We es ablish
modula Hamil onian inclusionexclusion iden i ies
and
Ma ko splicing
o
o e lapping causal diamond chains
; o non- o ally-o de ed cu s,
we in oduce
Ma ko gap line densi y
quan i a i ely cha ac e izing ailu e wi h compa ison
inequali ies e sus s a ica ion deg ee. On he sca e ing side, unde
dis ibu ional Bi man
K enF iedelLloydWigne Smi h calib a ion
, we in oduce
windowed eadou
, p o-
iding
isible cons an s and h eshold inequali ies
ia Toepli z/Be ezin comp ession,
Eule Maclau in and Poisson disciplines, he eby p o ing
chain
Z2
pa i y h eshold s a-
bili y
wi h obus ness condi ions o
weakly non-uni a y pe u ba ions
. On he geome ic
side,
hal -sided modula inclusion
cons i u es one-pa ame e semig oup o chain ad ance-
men ; in holog aphic limi s,
JLMS equali y
li s bounda y inclusionexclusionMa ko o bulk
en anglemen wedge no mal modula ow, wi h dimensional uppe bounds o subleading
1/N
co ec ions. Finally, we explici ly compu e
GHY join e ms
and
Z2
ledge consis ency
o squa e- oo splicing classes in minimal models in
1+1
and
2+1
dimensions, p o iding
ep oducible expe imen al pa ame e ables
and
e ica ion checklis s
.
1 In oduc ion & His o ical Con ex
Tomi aTakesaki modula heo y endows on Neumann algeb a ec o s a e pai s
(A,Ω)
wi h mod-
ula g oups
∆i
and modula conjuga ion
J
. BisognanoWichmann p ope y geome izes modula
ow as Lo en z boos s on wedge egions. Fo ze o-measu e geome y,
local modula Hamil o-
nians
on hal -spaces and hei smoo h de o ma ions sa is y acuum QNEC sa u a ion, and
ac-
uum Ma ko iani y
on ligh -cones/ligh - on s wi h s ong subaddi i i y sa u a ion o m solid
ounda ions. Algeb aically,
hal -sided modula inclusion
(HSMI) p o ides algeb aic skele on
o inclusionone-pa ame e semig oupBo che s commu a ion ela ions. Holog aphically,
JLMS
equali y
iden ies bounda y and bulk ela i e en opies a leading o de in la ge
N
. On sca e ing
side,
Bi manK en
iden ies de e minan phase wi h spec al shi unc ion,
F iedelLloyd
and
1
Wigne Smi h
uni y densi y-o -s a es die ence wi h g oup delay ace;
Toepli z/Be ezin
com-
p ession wi h
Szeg®/ ace o mulas
p o ide ope a o symbol ools o windowed eadou ;
Eule 
Maclau in
and
Poisson
disciplines yield exponen ial o algeb aic decay e o uppe bounds. This
pape sys ema ically cons uc s in eg a ed heo y o NullModula double co e and o e lapping
diamond chains wi hin his amewo k.
2 Model & Assump ions
2.1 Quad a ic Fo m F amewo k and Na u al Domain
Take Minkowski space ime
R1,d−1
(
d≥2
). Le
D0
be dense domain o ene gy-bounded ec o s in
acuum.
No a ion and measu e con en ion
: Ze o-measu e bounda y decomposes in o wo shee s
e
E=E+⊔E−
; no a ion
REσ(···)dλ dd−2x⊥
e e s o s anda d measu e in eg a ion on his shee by
ane pa ame e
λ
and ans e se coo dina e
x⊥
.
Assume o any egion
R
he e exis s lowe bounded closed quad a ic o m
kR[ψ] := X
σ=±ZEσ
g(R)
σ(λ, x⊥)⟨ψ, Tσσ(λ, x⊥)ψ⟩dλ dd−2x⊥, ψ ∈ D0,
hus he e exis s sel -adjoin ope a o
KR
sa is ying
⟨ψ, KRψ⟩=kR[ψ]
. CFT's sphe ical e-
gions/wedges and hei con o mal images yield exac geome ic equali ies.
Le
kR
ha e lowe bound
aR∈R
, i.e.,
kR[ψ]≥aR|ψ|2
. Take any
cR>−aR
, dene
shi ed
g aph no m
|ψ|2
kR,cR:= |ψ|2+kR[ψ] + cR|ψ|2,
hen
(D(kR),|·|kR,cR)
is comple e, compa ible wi h ep esen a ion heo em o sel -adjoin op-
e a o
KR
.
2.2 Ze o-Measu e Localiza ion and QNEC
In ze o-measu e hal -space
RV={u= 0, ≥V(x⊥)}
(
V∈C2
),
KV= 2πZdd−2x⊥Z∞
V(x⊥)
( −V)T ( , x⊥)d
holds as quad a ic o m iden i y; i s second-o de a ia ional ke nel is
2π T
, consis en wi h
acuum QNEC sa u a ion.
2.3 Double Co e and Splicing, Squa e-Roo Co e and Ledge
Ze o-measu e bounda y decomposes in o wo shee s
e
E=E+⊔E−
. Modula conjuga ion
J
ex-
changes wo shee s and e e ses o ien a ion, modula g oup gene a es in eg able ow along ane
pa ame e
λ
in geome izable cases. Seam splicing accoun ed by
ϵi∈ {±1}
. On sca e ing side
in oduce squa e- oo co e 
P√S={(E, σ) : σ2= de S(E)}
as
Z2
p incipal bundle s uc u e;
splicing class o closed chain loops sha es same
Z2
ledge wi h join e m o ien a ion signs.
2
2.4 Sca e ingIn o ma ion Calib a ion and Windowing
Uni a y sca e ing ma ix
S(E)
piecewise
C2m
wi h
S(E)−I
ace-class wi hin ene gy band; dene
Q(E) := −iS†∂ES, φ(E) := 1
2a g de S(E), ρ el(E) := 1
2π Q(E).
Employ window unc ion
h∈ S(R)
(e.g., Gaussian),
o
h∈C2m+1
c(R)
wi h endpoin je s up o
2m
o de anishing (
m≥1
). In his case
b
h(ω) = O(|ω|−(2m+1))
. I
h
only piecewise
C2m
wi h com-
pac suppo (endpoin s allow co ne s, including Kaise Bessel), adop
co ne ail bound
(a leas
O(|ω|−2)
), whe eby Theo em G's Poisson aliasing se ies con e ges. Co esponding Toepli z/Be ezin
comp ession and ace o mulas ollow e o decomposi ion in 3.5, whe e
endpoin emainde
REM
: o
C∞
c
windows ake
O(ℓ−(m−1))
; o piecewise
C2m
compac suppo windows (including
Kaise Bessel) adop
co ne e sion
es ima e (o de gene ally d ops o
O(ℓ−1)
), inco po a ed in o
o al e o budge
Eh(γ)
.
Addi ional assump ion (Toepli z commu a o in eg abili y)
: On any examined ene gy
band
I
,
∂ES(E)∈S2
and
ZI|∂ES(E)|2dE < ∞
. Thus
RT≤CTℓ−1/2RI|∂ES|2dE
is bounded.
Global con en ion (window and ail e m)
: Se
RRh= 1
and
h≥0
, scale
hℓ(E) =
ℓ−1h(E/ℓ)
. Dene
R ail(ℓ, I, E0) := ZR I(γ)|hℓ(E−E0)|dE ∈[0,1].
No e
: In his case
R ail = 1 −RI(γ)hℓ(E−E0)dE
.
No a ion con en ion (Poisson s ep size)
: Deno e
∆>0
as ene gy band segmen a ion/ equency
sampling s ep size (g id spacing); in Poisson esumma ion es ima e ake
ZI|RP|dE ≤ChX
|q|≥1b
h(2πq ℓ/∆) ,
consis en ly using his
∆
as in 3.5's iden ically named e m.
2.5 Chain and O e lap, Algeb aic Assump ions
Chain
{Dj}
adjacen o e laps on same su ace; o each ans e se poin
x⊥
o al-o de cu is de aul
assump ion. Algeb aically adop s anda d assump ions o spli p ope y and s ong addi i i y;
HSMI as algeb aic ealiza ion o chain ad ancemen .
3 Main Resul s (Each Resul Labeled wi h Signicance/Domain)
3.1 Double-Shee Geome ic Decomposi ion and To al-O de App oxima ion
B idge
Theo em 1
(A: Double-Shee Geome ic Decomposi ion)
.
KD= 2πX
σ=±ZEσ
gσ(λ, x⊥)Tσσ(λ, x⊥)dλ dd−2x⊥,
whe e
T++ =T
,
T−− =Tuu
. In CFT sphe ical diamonds
gσ(λ) = λ(1 −λ)
.
[
Quad a ic o m; domain: acuum, CFT exac equali y
]
3
Assump ion 2
(A
′
: Null Ene gy Flux Uni o m In eg abili y)
.
Fo any
ψ∈ D0
and geome ically
bounded mono onic app oxima ion amily
{R±
Vα}
, he e exis s
Hσ∈L1
loc(Eσ×Rd−2)
such ha
g(α)
σ(λ, x⊥)⟨ψ, Tσσ(λ, x⊥)ψ⟩≤Hσ(λ, x⊥)
holds almos e e ywhe e, and
supαRKHσ<∞
o any compac se
K ⊂ Eσ×Rd−2
.
Lemma 3
(A: O de ed Cu App oxima ion)
.
The e exis s mono onic hal -space amily
{R±
Vα}
along
E±
such ha
⟨ψ, KDψ⟩= lim
α→∞ X
σ=±
2πZEσ
g(α)
σ⟨ψ, Tσσψ⟩, g(α)
σ→gσ
in
L1
loc,
and limi is independen o chosen o de ed app oxima ion.
[
Quad a ic o m con e gence; domain: acuum, acuum QNEC sa u a ion
]
Exclusion ema k
: Wi hou BW/HSMI o bounda y oughness b eaking acuum QNEC sa -
u a ion, abo e decomposi ion may no hold.
Assump ion 4
(A
′′
: Quad a ic Fo m Lowe Bound and Closedness Th eshold)
.
Assume all pa -
icipa ing egions
R
ha e quad a ic o ms
kR
wi h uni o m lowe bound
a∈R
, i.e.,
kR[ψ]≥a|ψ|2
.
Take any
c > −a
dening shi ed g aph no m
|ψ|2
kR,c =|ψ|2+ (kR[ψ] + c|ψ|2)
, hen
kR
closed and
D(kR)
comple e unde
|·|kR,c
.
P oposi ion 5
(A.1: Necessa y and Sucien Condi ion o Limi Pa h Independence)
.
Unde
Assump ions A
′
and A
′′
, i along any wo mono onic app oxima ion amilies
{RVα}
,
{Re
Vβ}
we ha e
g(α)→g
,
eg(β)→g
in
L1
loc
, hen o each
ψ∈ D0
,
lim
α→∞X
σZg(α)
σ⟨ψ, Tσσψ⟩= lim
β→∞X
σZeg(β)
σ⟨ψ, Tσσψ⟩.
Reason: Domina ed con e gence iden ies each app oxima ion's limi wi h
g
; closedness and
lowe bound yield quad a ic o m con inui y, hus independen o app oxima ion pa h.
3.2 InclusionExclusion and Closedness
Theo em 6
(B: InclusionExclusion Iden i y)
.
Fo
{RVi}N
i=1
on same ze o-measu e su ace,
K∪iRVi=
N
X
k=1
(−1)k−1X
1≤i1<···<ik≤N
KRVi1∩···∩RVik
.
De i ed om poin wise iden i y
( −miniVi)+=Pk≥1(−1)k−1P|I|=k( −maxi∈IVi)+
.
[
Quad a ic o m; domain: acuum,
Vi
piecewise smoo h
]
P oposi ion 7
(B: Closedness)
.
Deno e
k:= k∪iRVi
as con aine domain closed quad a ic o m,
wi h lowe bound
a∈R
. Take any
c > −a
. I
ψn, ψ ∈ D(k)∩
I=∅D(kRVI), ψn→ψ
unde shi ed g aph no m
|·|k,c,
hen inclusionexclusion iden i y's bo h sides o quad a ic o m alues on
ψn
con e ge simul a-
neously o alues on
ψ
; hus iden i y closes on abo e o m domain. Whe e
4
|ψ|2
k,c := |ψ|2+k[ψ] + c|ψ|2.
[
Quad a ic o m closedness
]
Ope a ional domain ema k
: Abo e closedness holds on common o m domain
D∗:= D(k)∩
TI=∅D(kRVI)
; o chain applica ions, aking
Vi
piecewise
C1
wi h uni o m Lipschi z cons an ensu es
D∗
non-emp y and dense.
3.3 Ma ko Splicing, Pe z Reco e y, and Non-To al-O de Gap
Theo em 8
(C: Ma ko Splicing)
.
Unde same-su ace o al o de , acuum sa ises
I(Dj−1:Dj+1 |Dj)=0, KDj−1∪Dj+KDj∪Dj+1 −KDj−KDj−1∪Dj∪Dj+1 = 0.
[
In o ma ion equi alence; domain: acuum, spli /s ong addi i i y
]
Theo em 9
(C
′
: Ma ko Gap o Non-To al-O de )
.
Deni ion (s a ica ion deg ee)
: Le
V±
i(x⊥)
be h esholds on
E±
espec i ely, dene
κ(x⊥) := #{(a, b) : a < b, (V+
a−V+
b)(V−
a−V−
b)<0}.
No e
: Unde o al-o de cu
κ≡0
. Thus
ι
mono onically non-dec easing in
κ
yields compa ison
inequali y.
To bound
ι( , x⊥)
's
domain, deno e
−(x⊥) := min
iV+
i(x⊥), +(x⊥) := max
iV+
i(x⊥),
i.e., eec i e suppo in e al endpoin s co e ed by chain on
E+
shee ; below s a emen s abou
unde s ood wi hin
[ −(x⊥), +(x⊥)]
.
Ma ko gap line densi y
ι( , x⊥)≥0
dened by ela i e en opy densi y ke nel sa ises
I(Dj−1:Dj+1 |Dj) = ZZ ι( , x⊥)d dd−2x⊥, ι
mono onically non-dec easing in
κ.
Pa icula ly, unde o al o de
κ≡0
and
I(Dj−1:Dj+1 |Dj) = 0
(Ma ko sa u a ion).
[
Inequali y; domain: acuum
]
Lemma 10
(C.1: S a ica ion Deg eeGap Compa ison)
.
Assume
V±
i
piecewise
C1
wi h only
ni ely many c ossings a each
x⊥
. Then he e exis s cons an
c∗>0
(depending on
sup |∂V ±
i|
and
c ossing numbe uppe bound) such ha in dis ibu ional sense
ι( , x⊥)≥c∗κ(x⊥)1{ ∈[ −(x⊥), +(x⊥)]}.
Combined wi h FawziRenne lowe bound, yields quan i a i e gap lowe bound unde non- o al-
o de .
Fideli y con en ion
: This pape uni o mly akes Uhlmann deli y (no squa ed)
F(ρ, σ) := √ρ√σ1∈[0,1].
Acco dingly, FawziRenne inequali y w i es
I(A:C|B)≥ −2 ln F,
equi alen ly
F≥e−I(A:C|B)/2.
5

Theo em 11
(D: Pe z Reco e y and S abili y  Sel -Consis en Ve sion)
.
Deno e
A=Dj−1
,
B=Dj
,
C=Dj+1
. Take o ge ing channel
ΦBC→B(XBC ) = T C[XBC],Φ∗(YB) = YB⊗IC.
Wi h
σBC =ρBC
as e e ence s a e ( hus
σB=ρB
), Pe z eco e y map
RB→BC
dened as
RB→BC(XB) = σ1/2
BCσ−1/2
BXBσ−1/2
B⊗ICσ1/2
BC ,
whe e in e se akes pseudo-in e se on
supp(σB)
.
I and only i
I(A:C|B) = 0
pe ec
eco e y exis s
(idA⊗RB→BC)(ρAB) = ρABC.
Gene ally he e exis s
o a ionally a e aged Pe z eco e y
R o
B→BC
such ha
I(A:C|B)≥ −2 ln FρABC,(idA⊗R o
B→BC)(ρAB),
equi alen ly
F≥e−I(A:C|B)/2.
Abo e inequali y gene ally no gua an eed o un o a ed
RB→BC
; his pape uni o mly adop s
R o
B→BC
o s abili y p oposi ions.
[
Pe ec eco e y/s abili y; domain: Ma ko sa u a ion
]
3.4 Hal -Sided Modula Inclusion and Chain Ad ancemen
Theo em 12
(E: HSMI Ad ancemen )
.
I
(A(Dj)⊂ A(Dj+1),Ω)
is igh HSMI, hen he e exis s
posi i e-ene gy one-pa ame e semig oup co a ian wi h
∆i
A(Dj+1)
, in insically ad ancing
A(Dj)
o
A(Dj+1)
.
[
Algeb aic s uc u e; domain: HSMI
]
3.5 Dis ibu ional KFLWS Calib a ion and Windowed Pa i y Th eshold
Non-smoo h window ansi ion and e o inco po a ion
: I window
h∈C0
c
piecewise
C2m
wi hin suppo (endpoin s allow co ne s), ake s anda d smoo hing ke nel
ρδ
and dene
hℓ,δ :=
hℓ∗ρδ
. Then o each xed
ℓ > 0
,
hℓ,δ −hℓL1(R)=O(δ),
and Theo em F, Toepli z/Be ezin comp ession and ace o mula  s apply o
hℓ,δ
; by iangle
inequali y
Rsmoo h(δ) := ZI(γ)|hℓ,δ −hℓ|dE
inco po a es in o o al e o budge
Eh(γ)
. Unde Theo em G h eshold condi ions, choose
δ=δ(ℓ, m)
making
Rsmoo h(δ)≤1
2δ∗(γ)
, p ese ing same pa i y h eshold conclusion as
hℓ
.
Theo em 13
(F: Dis ibu ional Calib a ion Iden i y)
.
Fo
h∈C∞
c(R)
(o
h∈ S(R)
),
Z∂Ea g de S(E)h(E)dE =Z Q(E)h(E)dE =−2πZξ′(E)h(E)dE,
6
whe e
ξ
is spec al shi unc ion. (Con en ion: Bi manK en akes
de S(E) = e−2πiξ(E)
.)
Ene gy band h esholds and embedded eigens a es a oided by choosing
supp h
; long- ange po en ials
equi e co esponding gene alized KFL.
[
Dis ibu ional equali y; domain:
S−I∈S1
, piecewise smoo h
]
P oposi ion 14
(F
′
: Rela i e/Modied Calib a ion)
.
I
S0(E)
is e e ence sca e ing co-analy ically
segmen -wise wi hin ene gy band, wi hou ze os/poles, and
U(E) := S(E)S0(E)−1, U(E)−I∈S2, ∂EU∈S2,ZI|∂EU|2<∞,
hen Ca leman de e minan sa ises
Z∂Ea g de
2U(E)h(E)dE =Z Q(E)−Q0(E)h(E)dE,
whe e
Q=−iS†∂ES
,
Q0=−iS†
0∂ES0
. I
S
uni a y and
S0=I
, abo e educes o Theo em
F. This p oposi ion yields phaseg oup delayspec al shi consis ency unde non- ace-class bu
ela i ely second-o de aceable window.
No e (
π/2
bue o igin)
: In pa i y de e mina ion,
(−1)⌊Θ/π⌋
only ips when
Θ
c osses odd
mul iples o
π
. Con e ging pe u ba ion o al o
< π/2
ensu es no c ossing nea es in ege mul iple
o
π
, hus consis en wi h unpe u bed pa i y; aking
δ∗(γ) = min{π/2, δgap(γ)} − ε
is explici
o mula ion o his bue .
B anch con en ion (a g egula iza ion)
: Take con inuous b anch o
a g de S
dened wi hin
ene gy band excep coun able disc e e se ; i s dis ibu ional de i a i e
∂Ea g de S
independen o
b anch's
2π
jump choice, as
h∈C∞
c
annihila es jumps and ma ches
Q
ia DOI/Hele Sjös and
ep esen a ion.
Theo em 15
(G: Windowed Pa i y Th eshold; Wi h-Gap Th eshold)
.
Le
Θh(γ) := 1
2ZI(γ)
Q(E)hℓ(E−E0)dE, νchain(γ) := (−1)⌊Θh(γ)/π⌋.
Dene unwindowed limi
Θgeom(γ) := 1
2ZI(γ)
Q(E)dE =ZI(γ)
φ′(E)dE =φ(E2)−φ(E1),
whe e
I(γ)=[E1, E2]
,
φ(E) = 1
2a g de S(E)
. Dene gap
δgap(γ) := dis Θgeom(γ), πZ.
Unde
RRh= 1
, and se ing
Eh(γ) := ZI|REM|dE
| {z }
EM endpoin
+ZI|RP|dE
| {z }
Poisson aliasing
+CTℓ−1/2ZI|∂ES|2dE
|{z }
Toepli z commu a o
+R ail(ℓ, I, E0)
| {z }
ou -o -in e al ail
≤δ∗(γ),
whe e
R ail(ℓ, I, E0) := ZR I(γ)|hℓ(E−E0)|dE ∈[0,1].
7
No e
: I
h≥0
and
RRh= 1
, hen
R ail = 1−RI(γ)hℓ(E−E0)dE
. Se
δ∗(γ) := min π
2, δgap(γ)−
ε
. I he e exis
ℓ > 0
,
∆>0
,
m∈N
and
ε∈(0, δgap(γ))
making abo e inequali y hold, hen o
any window cen e
E0
sa is ying abo e window quali y condi ions,
νchain(γ) = (−1)⌊Θh(γ)/π⌋= (−1)⌊Θgeom(γ)/π⌋.
He e
•REM
is Eule Maclau in endpoin emainde , sa is ying
R|REM| ≤ Cmℓ−(m−1)
;
•RP
is
Poisson aliasing, sa is ying
ZI|RP|dE ≤ChX
|q|≥1b
h2πq ℓ
∆,
whe e
∆>0
is
ene gy sampling s ep size (ene gy band la ice spacing)
used in Poisson
summa ion;
•RT
is Toepli z commu a o e m, unde assump ion
∂ES∈S2
and
RI|∂ES|2dE < ∞
sa is ying
RT≤CTℓ−1/2RI|∂ES|2dE
;
•
Ou -o -in e al ail e m
:
R ail(ℓ, I, E0) := ZR I(γ)|hℓ(E−E0)|dE ∈[0,1].
No e
: I
h≥0
and
RRh= 1
, hen
R ail = 1 −RI(γ)hℓ(E−E0)dE
.
No e
: Fo piecewise smoo h compac suppo windows (e.g., Kaise ), abo e
REM
's
Cmℓ−(m−1)
should be eplaced wi h co ne es ima e (e.g.,
O(ℓ−1)
), o he h ee e ms
RP, RT, R ail
unchanged. By
abo e decay o de s,
RP≤ChX
|q|≥1b
h(2πq ℓ/∆) 
ni e
, main aining same o de as co ne es ima e.
[
Windowed dis ibu ional equali y + explici h eshold; domain: uni a y sca e ing,
h∈C∞
c
o
h∈ S
]
Lemma 16
(T: Toepli z/Be ezin Comp ession E o )
.
Le
Tℓ
be windowed comp ession ope a o on
ene gy axis (ke nel is con olu ion wi h
hℓ(E−E′)
), le
Q(E) = −iS(E)†∂ES(E)
, wi h
∂ES∈S2
sa is ying
RI|∂ES|2dE < ∞
. Then he e exis s cons an
CT>0
such ha
 Q∗hℓ−ZQ(E)hℓ(E−E0)dE≤CTℓ−1/2ZI|∂ES|2dE.
P oo essen ials: W i e comp ession e o as
[Tℓ,·]
commu a o , do one mean alue es ima e on
ene gy de i a i e; use Hilbe Schmid  ace Hölde wi h window expansion scale
R(E−E0)2hℓ∼
ℓ−1
o ob ain
ℓ−1/2
decay.
Lemma 17
(P: Poisson/EM Window Condi ions)
.
I
h∈C2m+1
c
wi h endpoin
≤2m
o de je s
anishing, hen
b
h(ω) = O(|ω|−(2m+1))
, hus
X
|q|≥1b
h2πq ℓ
∆<∞,Z|REM| ≤ Cmℓ−(m−1).
Fo piecewise
C2m
compac suppo windows wi h endpoin co ne s, use co ne es ima e o e-
place
REM
o de , main aining Poisson se ies con e gence.
Lemma 18
(G: Windowed Phase Pe u ba ion)
.
I wo sca e ing g oups
S
,
˜
S
sa is y on ene gy
egion
IZI|S−˜
S|2|∂ES|2+|∂ES−∂E˜
S|1dE ≤η,
8
hen
Θh[S]−Θh[˜
S]≤Chη, Ch= sup
EZ|hℓ(E−E′)|dE′.
Co olla y 19
(G: Weakly Non-Uni a y S abili y)
.
Dene
∆nonU(E) = |S†S−I|1
. Le
δgap(γ) :=
dis Θgeom(γ), πZ
. I
ZI(γ)
∆nonU(E)dE ≤ε, Eh(γ)≤δ∗(γ) := min π
2, δgap(γ)−ε,
whe e
ε∈(0, δgap(γ))
,
Eh(γ) := RI(γ)|REM|dE +RI(γ)|RP|dE +CTℓ−1/2RI(γ)|∂ES|2dE +
R ail(ℓ, I, E0)
, hen
νchain(γ)=(−1)⌊Θh(γ)/π⌋
in a ian , consis en wi h unwindowed limi
(−1)⌊Θgeom(γ)/π⌋
.
(Th eshold ully aligned wi h Theo em G.)
[
S abili y; domain: weak dissipa ion
]
Lemma 20
(N: Weakly Non-Uni a y Phase Die ence Bound)
.
W i e pola decomposi ion
S=
U(I−A)
,
U
uni a y,
A≥0
. I
RI|S†S−I|1dE ≤ε
, hen he e exis s cons an
CN
such ha
ZI
Q(S)hℓ−ZI
Q(U)hℓ≤CNε.
P oo essen ials:
Q(S) = Im (S−1∂ES)
, o nea -uni a y
S
ha e
∥S−1∥ ≤ (1 − ∥A∥)−1
; use
∥∂ES∥1≤ ∥∂EU∥1+∥∂EA∥1
wi h
∥A∥1≲∥S†S−I∥1
o con ol die ence and in eg a e.
3.6 Join Te ms and
Z2
Ledge
Theo em 21
(H: Ledge Consis ency and Gauge T ans o ma ion)
.
A nullnull and nullspacelike
co ne s,
Ijoin =εJ
8πG Z√γΞdd−2x,
whe e
Ξ = ln |k1·k2|
2
(nullnull) o
Ξ = ln |n·k|
(nullspacelike).
Unde independen escaling
ki→αiki
,
n→βn
,
Ξ7→ Ξ + ln |α1α2|
(nullnull),
Ξ7→ Ξ + ln |α|+ ln |β|
(nullspacelike).
Only when no mal ips
k→ −k
(o
n→ −n
),
εJ
changes sign while
Ξ
unchanged. Thus single
co ne 's
Ijoin
no pu ely sign in a ian ; bu a e closing along chain wi h squa e- oo splicing class
ϵi
accoun ing, ne eec only depends on
Qiϵi
pa i y, consis en wi h
⌊Θh/π⌋
pa i y.
[
Gauge ans o ma ion; domain: anely pa ame ized null bounda ies
]
9