Unified Role of Relative Scattering Determinant in Quantum Gravity: Two-Domain Framework, Fixed-Energy BK ($p\in\{1,2\

Unied Role o Rela i e Sca e ing De e minan in Quan um
G a i y: Two-Domain F amewo k, Fixed-Ene gy BK (
p∈ {1,2}
Unied Ve sion), Closed-Domain
Λ
-Slope, and Black Hole Pole
Spec oscopy
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
Taking  ela i e de e minan  as unied objec , we es ablish igo ous and e iable he-
o y in wo ypes o geome ic-physical scena ios:
(C)
ela i e
ζ
/hea ke nel de e minan o
Euclideanized second a ia ion ope a o amily in compac closed domain and i s olume den-
si y esponse o cosmological cons an e m;
(S)
xed- equency sca e ing ma ix on s a ion-
a y ex e io geome y (Schwa zschildde Si e /Ke de Si e ) wi h ela i e (modied) de e -
minan , spec al shi objec , and quasino mal mode (QNM) spec oscopy. This pape p o-
ides ou main heo ems wi h comple e p oo s: (i) Unde con ol o weigh ed limi ing ab-
so p ion p inciple (LAP) and double ope a o in eg al (DOI), p o e
p∈ {1,2}
unied e -
sion o xed-ene gy Bi manK en equali y: o Lebesgue almos e e ywhe e equency
ω
,
de pSΛ(ω) = exp−2πi Ξ(p)
Λ(ω)
, whe e
p= 1
gi es
Ξ(1) =ξ
as Li shi sK en spec al shi
unc ion,
p= 2
yields
Ξ(2)
as cumula i e an ide i a i e o Koplienko second-o de spec al shi ;
(ii) In closed-domain Mülle ela i e de e minan amewo k, p o e  olume slope heo em:
limµ→0+Vol4(M)−1∂Λℜlog de ζ, el(KΛ+µ2,K0+µ2) = 1
8πG
(pe signa u e con en ion xed
in ex ); (iii) On physical s ip
ℑω > −γ0
, pole se o ela i e sca e ing de e minan
τp(ω) =
de pS(ω)S0(ω)−1
equi alen o QNM (coun ing algeb aic mul iplici y), independen o e e -
ence
S0
choice; (i ) Fo eal equency only phase admi s equali y,
a g de pS=−2πΞ(p)
; while
o
p= 2
Ca leman de e minan
|de 2S|= expPj(1 −cos θj)≥1
, gene ally canno claim
|de 2S|= 1
. Acco dingly in oduce phase-no malized de e minan 
c
de pS:= de pS/|de pS|
as cons ained objec o equency-domain globally me omo phic  ing, p o iding p incipal
angle uppe bound o Fishe in o ma ion. Pape concludes wi h pa ame e s and accep ance
s anda ds o h ee ep oducible expe imen al pipelines: closed-domain el-ze a, ex e io -domain
me omo ph- , channelpseudo-uni a y e ica ion.
1 In oduc ion
Closed-domain ela i e
ζ
/hea ke nel de e minan and ex e io -domain ela i e sca e ing de e mi-
nan sha e essen ial s uc u e ela i e phase. On closed-domain side, his phase eco e s on-shell
ac ion's olume densi y esponse o cosmological cons an
Λ
ia loga i hmic de i a i e; on ex e io -
domain side, i 's con olled by BK/LK (and second-o de Koplienko e sion) spec al shi unc ion
o sca e ing ma ix phase on ene gy be s. This pape unies wo domains in o closed loop o
 e iable hypo hesis
⇒
heo em
⇒
de ailed p oo :
1. Achie e xed-ene gy implemen a ion unde ope a o -Lipschi z and DOI echniques, wi h weigh ed
LAP domina ed con e gence;
1
2. Implemen egula iza ion independence and i em-wise cancella ion o co ne /bounda y/ghos
unde Mülle ela i e de e minan ;
3. Uni y ela i e sca e ing de e minan poles as QNM unde analy ic F edholm amewo k,
p o ing e e ence independence;
4. Sepa a e block-le el modulus conse a ion om global Ca leman modulus non-cons an
iden i y unde pseudo-uni a y (J-uni a y) amewo k, imposing eal-axis modulus cons ain
ia phase-no malized de e minan .
All ma hema ical exp essions in ex p esen ed inline wi h
·
o m, a oiding ambigui y om
display/en i onmen swi ching.
2 Se ing, No a ion, and Ve iable Hypo heses
2.1 Spec um, Ideals, and Modied De e minan s
Take sepa able Hilbe space
H
. Deno e sel -adjoin ope a o pai
(HΛ, H0)
, die ence
V=HΛ−H0
.
Scha en ideal
Sp
s anda d deni ion. Fo
K∈S1
ake F edholm de e minan
de (I+K)
; o
K∈
S2
ake Ca leman de e minan
de 2(I+K) = de (I+K) exp(−K)
. I
U
uni a y wi h
U−I∈S2
,
spec al angles
{θj} ∈ ℓ2
sa is y
|de 2(U)|= expPj(1−cos θj)≥1
,
a g de 2(U) = Pj(θj−sin θj)
.
2.2 Spec al Shi Objec s and DOI
Fi s -o de spec al shi
ξ
and second-o de spec al shi measu e
η
espec i ely sa is y
T ( (HΛ)−
(H0)) = R ′(E)ξ(E)dE
,
T ( (HΛ)− (H0)− ′(H0)V) = R ′′(E)dη(E)
, unc ion class aking
ope a o -Lipschi z/app op ia e Beso in e sec ion. Cumula i e an ide i a i e
Ξ(2)(E) = η((−∞, E))
,
no malized
Ξ(2)(−∞) = 0
. Double ope a o in eg al ep esen a ion
(HΛ)− (H0) = RR Φ (λ, µ)dEΛ(λ)V dE0(µ)
,
whe e
Φ (λ, µ) = ( (λ)− (µ))/(λ−µ)
has Schu /Haage up bound.
2.3 Weigh ed LAP and Ene gy Fibe iza ion
The e exis
s > 1
2
, ene gy window
I
, cons an
CI
such ha
|⟨x⟩−s(H#−λ∓i0)−1⟨x⟩−s| ≤ CI
holds
o
λ∈I
,
#∈ {Λ,0}
. S a iona y ex e io egion (SdS/KdS) s a iona y unde ime Killing eld
wi h equency
ω
, pa ial wa e decomposi ion yields channel ma ix
Sℓm(ω)
.
2.4 Closed-Domain Rela i e
ζ
-De e minan and Volume Slope
Euclideanized second a ia ion ope a o amily
KΛ
wi h e e ence
K0
ma ching p incipal symbol,
bounda y condi ions and Faddee Popo ghos pai ing consis en , ze o modes/ h eshold esonances
emo ed ia dep ojec ion. Die ence hea ke nel
K el( ) = T e− (KΛ+µ2)−e− (K0+µ2)
has sho -
ime expansion, dene
log de ζ, el(KΛ+µ2,K0+µ2) = −R∞
0 −1K el( )d
. Me ic signa u e and
ac ion con en ion xed as
∂ΛSon
-
shell = (8πG)−1Vol4(M)
.
2.5 Ex e io -Domain Re e ence and Pseudo-Uni a y
On s ip
ℑω∈(−γ0,0]
choose e e ence sca e ing ma ix
S0(ω)
, equi e analy ici y on same shee
wi hou ze o/poles. Each channel cons uc s ene gy ux quad a ic o m
η
ia Jos W onskian
no maliza ion making
S†
ℓmηSℓm =η
.
2
2.6 Ve iable Hypo heses (Assump ion Box)
(H
-
AC)
: Wa e ope a o s exis and comple e, AC pa admi s ene gy be iza ion;
(H
-
LAP)
: Weigh ed LAP (pa ame e
s > 1/2
, cons an
CI
);
(H
-
LK/DOI)
: Poisson smoo hing
ε∈OL
,
| ε|OL ≤C/ε
, DOI ke nel has uni o m Schu /Haage up
bound;
(H
-
Sp)
: Fo a.e.
ω∈I
,
χ(−∞,ω](HΛ)−χ(−∞,ω](H0)∈Sp
and
SΛ(ω)S0(ω)−1−I∈Sp
( ypical
p= 2
);
(H
-
elDe )
: P incipal symbol consis en , bounda y/ghos ma ching, no ze o modes o de eso'd,
die ence hea ke nel has sho - ime expansion;
(H
-
Re )
: Re e ence
S0
analy ic on s ip wi hou ze o/poles;
(H
-
Can)
: Channel ene gy ux gauge xed, block-le el pseudo-uni a y holds.
3 Main Theo ems and Conclusions
Theo em 1
(3.1: Fixed-Ene gy BK:
p∈ {1,2}
Unied Ve sion)
.
Unde
(H
-
AC)
,
(H
-
LAP)
,
(H
-
LK/DOI)
,
(H
-
Sp)
, o Lebesgue almos e e ywhe e
ω∈I
: when
p= 1
,
de SΛ(ω) = exp −
2πi ξΛ(ω)
; when
p= 2
,
de 2SΛ(ω) = exp −2πi Ξ(2)
Λ(ω)
. Thus
a g de pSΛ(ω) = −2πΞ(p)
Λ(ω)
.
Theo em 2
(3.2: Closed-Domain Volume Slope)
.
Unde
(H
-
elDe )
and de esonance p ojec ion,
limµ→0+Vol4(M)−1∂Λℜlog de ζ, el(KΛ+µ2,K0+µ2) = 1
8πG
(pe ex signa u e con en ion).
Theo em 3
(3.3:
τp
Poles = QNM, Re e ence Independen )
.
Le
τp(ω) = de pS(ω)S0(ω)−1
.
Unde
(H
-
Re )
, on s ip
ℑω∈(−γ0,0]
, pole se o
τp
coincides wi h
S
poles (QNM) coun ing
algeb aic mul iplici y. I changing e e ence o
e
S0
s ill sa is ying
(H
-
Re )
, hen
τp/eτp
is analy ic
ou e unc ion wi hou ze os/poles on s ip, lea ing pole se unchanged.
Theo em 4
(3.4: Real-F equency Phase and Modulus; Phase-No malized De e minan )
.
Block-
le el: I
S†
ℓm(ω)ηSℓm(ω) = η
, hen
|de Sℓm(ω)|= 1
. Global: gene ally only
a g de pS(ω) =
−2πΞ(p)(ω)
holds. When
S(ω)
uni a y wi h
S(ω)−I∈S2
,
|de 2S(ω)|= exp Pj(1−cos θj(ω))≥
1
. Dene
c
de pS(ω) = de pS(ω)/|de pS(ω)|
,
bτp(ω) = τp(ω)/|τp(ω)|
as eal-axis modulus equals 1
cons ained objec s.
4 P oo o Theo em 3.1 (DOILAP Domina ed Con e gence o
Fixed Ene gy)
P oo s a egy o e iew
: App oxima e s ep unc ion ia Poisson smoo hing
ε
, apply DOI ex-
p ession wi h weigh ed LAP es ablishing uni o m
Sp
domina ion inequali y, hen exchange limi
ε↓0
a Lebesgue poin s o spec al shi objec , nally iden i y sca e ing phase ia AC be iza-
ion and exponen ia e o de e minan equali y. Die ence be ween
p= 1
and
p= 2
ca ied by
 s /second-o de ace o mulas.
S ep 1 (Poisson smoo hing and DOI ke nel bound)
: Take
ε(λ) = 1
2+1
πa c an((ω−λ)/ε)
.
Then
ε∈OL
wi h
| ε|OL ≤C/ε
. DOI exp ession
ε(HΛ)− ε(H0) = RR Φ ε(λ, µ)dEΛ(λ)V dE0(µ)
,
whe e
|Φ ε|Schu ≤C/ε
.
S ep 2 (Weigh ed LAP and Scha en domina ion)
: W i e weigh ed p ojec ion bounda y
alue esol en o m ia S one o mula, apply
(H
-
LAP)
yielding
|⟨x⟩−sR#(ω±i0)⟨x⟩−s| ≤ CI
. By
Bi manSolomyak ype es ima e ob ain
| ε(HΛ)− ε(H0)|Sp≤CI(C/ε)Mp(I)
, whe e
Mp(I) =
supλ∈I|⟨x⟩−s(RΛ(λ±i0) −R0(λ±i0))⟨x⟩−s|Sp
bounded.
3
S ep 3 (
p= 1
: spec al shi and BK)
: Fi s -o de ace o mula yields
T ( ε(HΛ)−
ε(H0)) = R ′
ε(E)ξ(E)dE
. Taking
ε↓0
wi h domina ed con e gence yields
T (χ(−∞,ω](HΛ)−
χ(−∞,ω](H0)) = ξ(ω)
a Lebesgue poin s o
ω
. AC be iza ion wi h s a iona y sca e ing shows
de S(ω) = exp(−2πi ξ(ω))
.
S ep 4 (
p= 2
: Koplienko phase)
: Second-o de ace o mula yields
T  ε(HΛ)− ε(H0)−
′
ε(H0)V=R ′′
ε(E)dη(E)
. In eg a e igh side wice by pa s, aking
ε↓0
yields
Ξ(2)(ω) =
η((−∞, ω))
. Fixed-ene gy implemen a ion same as abo e, hus
de 2S(ω) = exp(−2πi Ξ(2)(ω))
.
QED.
5 P oo o Theo em 3.2 (Rela i e Hea Ke nel I em-Wise Cancel-
la ion, Taube ian Exchange, and Signa u e Con en ion)
S ep 1 (Loga i hmic de i a i e hea ke nel ep esen a ion)
:
∂Λlog de ζ, el =−R∞
0 −1∂ΛK el( )d
,
whe e
K el( ) = T (e− (KΛ+µ2)−e− (K0+µ2))
.
S ep 2 (Sho - ime expansion and i em-wise cancella ion)
: Unde p incipal symbol con-
sis ency, bounda y/ghos pai ing consis ency, mul iplica i e anomaly anishing,
K el( )∼Pk≥0a el
k (k−d)/2
(
d= 4
), local coecien s (including GHY, co ne s, ghos pai ing) cancel i em-wise excep olume
e m
a el
0
, i.e.,
a el
k>0= 0
.
S ep 3 (Taube ian exchange and olume slope)
: In oduce small mass
µ > 0
con olling
la ge
pa , spli
R∞
0=R 0
0+R∞
0
. Fo me domina ed by
a el
0
, la e unde de esonance p ojec ion
has uni o m bound. Exchanging
µ↓0
wi h olume densi y limi yields
Vol−1
4∂Λℜlog de ζ, el =∂Λc0
.
By ex con en ion
∂ΛSon
-
shell = (8πG)−1Vol4
, alignmen yields
∂Λc0=1
8πG
. QED.
6 P oo o Theo em 3.3 (Analy ic F edholm and Re e ence Inde-
pendence)
S ep 1 (Analy ic F edholm)
: On s ip
ℑω∈(−γ0,0]
, w i e
S(ω) = I+K(ω)
whe e
K(ω)
is
Sp
- alued me omo phic amily. De e minan
Dp(ω) = de p(I+K(ω))
me omo phic, i s ze o o de
equals ke nel dimension (algeb aic mul iplici y) o
I+K(ω)
.
S ep 2 (Rela i iza ion and pole coun ing)
: Dene
τp(ω) = de p(S(ω)S0(ω)−1)
. I
S0
analy ic nonze o on s ip,
τp
sha es poles and o de s wi h
S
, poles being QNM.
S ep 3 (Re e ence independence)
: I choosing ano he
e
S0
also sa is ying condi ion, hen
τp/eτp= de p(S0e
S−1
0)
is analy ic ou e unc ion wi hou ze os/poles, lea ing pole se unchanged.
QED.
7 P oo o Theo em 3.4 (Block-Le el Pseudo-Uni a y and Global
Ca leman Modulus)
Block le el
: By
S†
ℓmηSℓm =η
and
de (η−1S†
ℓmηSℓm)=1
ob ain
|de Sℓm|= 1
.
Global phase
: By Theo em 3.1 ob ain
a g de pS(ω) = −2πΞ(p)(ω)
.
Global modulus (
p= 2
)
: I
S(ω)
uni a y wi h
S(ω)−I∈S2
, spec al angles
{θj(ω)} ∈ ℓ2
yield
|de 2S(ω)|= exp Pj(1 −cos θj(ω))≥1
. In gene al J-uni a y case modulus non-cons an ,
hus phase-no maliza ion
c
de p
na u al objec o eal-axis modulus cons ain . QED.
4
8 Globally Me omo phic Fi ing and Fishe P ojec ion Geome y
(Fo Da a-Side Implemen a ion)
On s ip
ℑω∈[−γ0,0]
pa ame ize
log bτp(ω) = PJ
j=1 log ω−ωj
ω−ωj+iQ(ω)
, whe e
ωj
a e lowe hal -plane
poles,
Q
low-o de en i e unc ion aking pu ely imagina y alues on eal axis. En o ce conjuga e
pai ing and phase-no malized modulus equals 1, supp ess alse poles ia s ip c oss- alida ion.
P oposi ion 5
(8.1: Fishe P incipal Angle Uppe Bound)
.
Fo whi ened obse a ion
yk=ℑlog bτp(ωk)+
ϵk
, Jacobian
J
wi h cons ain submani old angen space p ojec ion
PM
yields es ic ed Fishe
FM= (PMJ)⊤(PMJ)
. I
ϑ
is maximum p incipal angle be ween
ange(J)
and
ange(PM)
, hen
a iance educ ion ac o
R ≤ 1/|sin ϑ|
. P oo in Appendix F.
9 Rep oducible Expe imen al P o ocols (P1P3)
P1
|
el-ze a (closed domain)
: G id s ep
h
( h ee le els), hea ke nel window
∈[ min, max]
(
min ∼c h2
), ex apola ion o de
N∈ {2,3}
, small mass
µ
( h ee o  e loga i hmic poin s).
Ta ge quan i y
Vol−1
4∂Λℜlog de ζ, el
. Accep ance: slope e o
<1%
; d i unde die en co ne
iangula ions/gauges
<0.5%
.
P2
|
me omo ph- (ex e io domain)
: Fi
log bτp
eco e ing
{ωj}
. P io s: pai wise sym-
me ic, s ip analy ic, eal-axis modulus cons ain (on
bτp
), and
ℜlog de 2(ω)≥0
(i using
p= 2
).
Accep ance: CRLB imp o emen o e mode-by-mode
≥1.3×
; alse ala m a e
≤5%
; c oss-s ip
consis en .
P3
|
bh-channels (pseudo-uni a y and BK phase)
: Jos W onskian no maliza ion con-
s uc s
η
, compu e
|S†
ℓmηSℓm −η|
and phase closu e
a g c
de pS+2πΞ(p)
. Accep ance: pseudo-uni a y
esidual
<10−12
, phase closu e
<10−3
adians; con e ges as
a+b/ℓmax
wi h
ℓmax
.
10 Discussion and Ou look
Unde explici ly e iable analy ic hypo heses, his pape comple es ou main conclusions:
p∈
{1,2}
unied e sion o xed-ene gy BK, closed-domain olume slope, e e ence independence o
ela i e sca e ing de e minan poles = QNM, and eal- equency phasemodulus decomposi ion,
p o iding ep oducible expe imen al pipelines. Limi a ions: LAP cons an may de e io a e unde
s ong apping o ex eme spin; non-local bounda ies and singula geome y equi e sepa a e e -
ica ion o mul iplica i e anomaly; s a is ical side needs obus egula iza ion agains model bias.
Fu u e wo k includes: ex ending modulusphase o mula o
de p
unde K en spaces; seamlessly
inco po a ing BK e sion o die en ial o ms/elec omagne ic elds; es ing s abili y o  e e ence
independen  poles using mul i-s a ion s ip da a.
A Comple e De i a ion o DOILAP Domina ed Con e gence
A.1 Ke nel Bound and Weigh Inse ion
Take
ε(λ) = 1
2+1
πa c anω−λ
ε
.
ε∈OL
,
| ε|OL ≤C/ε
. DOI exp ession yields
ε(HΛ)− ε(H0) =
RR Φ ε(λ, µ)dEΛ(λ)V dE0(µ)
, whe e
Φ ε
sa ises
supλR|Φ ε(λ, µ)|dµ ≤C/ε
,
supµR|Φ ε(λ, µ)|dλ ≤
C/ε
.
Inse
⟨x⟩±s
ob aining
ε(HΛ)− ε(H0) = RR(⟨x⟩−sdEΛ(λ)) (⟨x⟩sV⟨x⟩s) (dE0(µ)⟨x⟩−s) Φ ε(λ, µ)
.
5

A.2 Scha en Domina ion Inequali y
By Haage up/Schu bound wi h Hölde inequali y (on
Sp
),
| ε(HΛ)− ε(H0)|Sp≤ |Φ ε|Schu ·
supλ∈I|⟨x⟩−sE′
Λ(λ)⟨x⟩−s| · |⟨x⟩sV⟨x⟩s|Sp·supµ∈I|⟨x⟩−sE′
0(µ)⟨x⟩−s|
.
Use S one o mula
E′
#(λ) = π−1ℑR#(λ+i0)
wi h
(H
-
LAP)
ob aining
| ε(HΛ)− ε(H0)|Sp≤
CIε−1|⟨x⟩sV⟨x⟩s|Sp
.
In sca e ing se ing be e use die ence esol en o m
|⟨x⟩−s(RΛ(λ±i0)−R0(λ±i0))⟨x⟩−s|Sp
con olling
|⟨x⟩sV⟨x⟩s|Sp
, hus ob aining unied domina ion
| ε(HΛ)− ε(H0)|Sp≤CIε−1Mp(I)
.
A.3 Lebesgue Poin s and Limi Exchange
Le
ω
be Lebesgue poin o spec al shi objec . By abo e ob ain amily o
ε
-uni o mly in eg able
domina ing unc ion
M(ω) = supε<ε0| ε(HΛ)− ε(H0)|Sp∈L1
loc(I)
. Thus limi o DOI- ace as
ε↓0
commu es wi h local in eg a ion o e
ω
, ob aining xed-ene gy e sion o  s /second-o de
ace o mula, comple ing Theo em 3.1 p oo .
B Rela i e Hea Ke nel I em-Wise Cancella ion and
Λ
-Slope Re-
nemen
B.1 Sho -Time Expansion and Local Coecien s
Fo Laplace- ype ope a o
K#
(including gauge-ghos pai ing) ha e
T (e− K#)∼Pj≥0aj(K#) (j−d)/2
.
Unde p incipal symbol consis ency wi h bounda y/ghos ma ching, ela i e die ence
a el
j>0=
aj(KΛ)−aj(K0)=0
; co ne and bounda y e m coecien s also cancel in  ela i e die ence
(Wodzicki esidue ze o ensu es mul iplica i e anomaly absen ).
B.2 Loga i hmic De i a i e and Volume Te m
Rela i e
ζ
w i en as
ζ el(s;µ) = 1
Γ(s)R∞
0 s−1e− µ2K el( )d
. Die en ia ing wi h espec o
Λ
, only
a el
0= Vol4(M)c0
con ibu es, yielding
∂Λζ′
el(0; µ) = −∂Λa el
0R∞
0 −1e− µ2d +
ni e e ms. Volume
densi y wi h
µ↓0
exchange, by ex con en ion
∂Λc0=1
8πG
, ob aining Theo em 3.2.
B.3 Taube ian Exchange E o Es ima e
Take
0=µ−2α
(
α∈(0,1)
),
R∞
0 −1e− µ2K el( )d
con olled by spec al gap and de esonance p o-
jec ion as
O(µ2(1−α))
, while
(0, 0)
segmen e o a e highe -o de coecien cancella ion becomes
O( 1/2
0) = O(µ−α)
coecien nullica ion e m, o e all can ake
α
making o al e o
o(1)
.
C Re e ence Independence o Rela i e Sca e ing De e minan and
Pole Coun ing
C.1 Analy ic F edholm Tools
W i e
S(ω) = I+K(ω)
,
K(ω)
being
Sp
- alued me omo phic amily. Then
Dp(ω) = de p(I+K(ω))
me omo phic wi h ze o o de equal o
dim ke (I+K(ω))
.
6
C.2 Rela i iza ion and Pole T ans e
Assume
S0
analy ic wi hou ze os/poles on s ip, dene
τp(ω) = de p(S(ω)S0(ω)−1)
=
de p(I+
K(ω)) ·de p(S0(ω)−1)
. La e ac o analy ic nonze o, hus
τp
poles synch onize wi h
Dp
, o de
being QNM algeb aic mul iplici y.
C.3 Re e ence Independen Ou e Func ion Fac o
I changing e e ence o
e
S0
, hen
τp/eτp= de p(S0e
S−1
0)
is analy ic wi hou ze os/poles (ou e unc-
ion), lea ing pole se and mul iplici y unchanged.
D Pseudo-Uni a y: Channel Cons uc ion and Global Ca leman
Modulus
D.1 Ene gy Flux Quad a ic Fo m and J-Uni a y
Take Jos solu ions
uin/ou
a bo h ends o adial equa ion wi h W onskian no maliza ion, making
ene gy ux
F=ℑ(u ∂ u)
consis en a bo h ends. Acco dingly dene channel quad a ic o m
η= diag(1,−1)
making
S†
ℓmηSℓm =η
.
D.2 Block-Le el De e minan Uni Modulus and Global Phase
Fini e-dimensional block di ec ly yields
|de Sℓm|= 1
. Global di ec sum unde
S2
se ing only
p ese es phase equali y; i globally uni a y wi h
S−I∈S2
, spec al angle expansion yields
|de 2S|= exp(P(1 −cos θj)) ≥1
.
E Koplienko Phase Fixed-Ene gy Cons uc ion (
p= 2
)
E.1 Second-O de T ace Fo mula and DOI
Fo
ε
ha e
T  ε(HΛ)− ε(H0)− ′
ε(H0)V=R ′′
ε(E)dη(E)
. In eg a e igh side wice by pa s
yielding
−R ′
ε(E)dΞ(2)(E)
.
E.2 Domina ed Con e gence and Lebesgue Poin s
By Appendix A's
S2
domina ion wi h
| ′
ε|L1≤C
ob ain in eg able domina ion, as
ε↓0
,
′
ε
con e ges
o
δω
(weak sense) eco e ing
Ξ(2)(ω)
a Lebesgue poin s.
E.3 Sca e ing Phase and De e minan
Fixed-ene gy implemen a ion wi h AC be iza ion iden ies
Ξ(2)(ω)
as sca e ing phase second-
o de spec al shi an ide i a i e, exponen ia ing yields
de 2S(ω) = exp(−2πi Ξ(2)(ω))
.
F P oo o Fishe P ojec ion Geome y Uppe Bound
F.1 Model and P ojec ion
Whi ened obse a ion
y=Jθ +ϵ
, ha d cons ain
C(θ) = 0
dening die en iable submani old
M
wi h angen space p ojec ion
PM
sa is ying
P2
M=PM
.
7
F.2 P incipal Angle and Spec al Bound
Le
ϑ
be maximum p incipal angle be ween
ange(J)
and
ange(PM)
, ha e
|PM |≥|sin ϑ| | |
o
all
∈ ange(J)
. Thus
⊤FM =|PMJ |2≥sin2ϑ|J |2= ⊤(sin2ϑ F)
, yielding
FM⪰sin2ϑ F
.
Taking maximum eigen alue yields
T (F−1
M)≤T (F−1)/sin2ϑ
, i.e., a iance educ ion ac o
R ≤
1/|sin ϑ|
. QED.
G Rep oducible Expe imen Pa ame e s and E o Budge (B ie
Table)
G.1 P1 (closed domain)
:
h∈ {h0, h0/2, h0/4}
;
min ∼c h2
,
max
sa is ying semiclassical window;
µ
aking
{µ0, µ0/3, µ0/9, µ0/27}
. Ex apola ion using bilinea ( o
(µ, h)
) and Richa dson ( o
-
window) hyb id. Tole ance: slope
<1%
; d i unde die en co ne iangula ions/gauges
<0.5%
.
G.2 P2 (ex e io domain)
: S ip
ℑω∈[−γ0,0]
uni o m sampling; pole numbe
J
join ly
de e mined by AIC/BIC and s ip c oss- alida ion; penal y e m cons ains
Q(ω)
deg ee and eal-
axis pu ely imagina y condi ion; p io
ℜlog de 2≥0
only as so egula iza ion. Tole ance: CRLB
imp o emen
≥1.3
, alse ala m
≤5%
.
G.3 P3 (channels)
: Ex apola ion adius, ma ching adius, in eg a ion s ep calib a ed ia
g id sea ch; W onskian no maliza ion die ence
<10−12
;
|S†
ℓmηSℓm −η|∞<10−12
; phase closu e
<10−3
adians; con e ges as
a+b/ℓmax
wi h
ℓmax
.
End o Main Tex and Appendices
8