Cross-Platform Metrological Paradigm for Unified Phase--Frequency Readout: Windowed Upper Limits on FRB Vacuum Polarization, Spectral--Scattering Equivalence and Identifiability of \delta-Ring--AB Flux, and Critical Coupling Metrology for Scattering Topological Invariants

C oss-Pla o m Me ological Pa adigm o Unied
PhaseF equency Readou : Windowed Uppe Limi s on FRB
Vacuum Pola iza ion, Spec alSca e ing Equi alence and
Iden iabili y o
δ
-RingAB Flux, and C i ical Coupling
Me ology o Sca e ing Topological In a ian s
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
P opose unied me ological pa adigm using "phase equency" as sole eadou , pene a ing
h ee classes o sys ems wi h dispa a e scales and physical backg ounds: as adio bu s (FRB)
p opaga ion a cosmological dis ances, poin in e ac ions (
δ
po en ial) and Aha ono Bohm
(AB) ux coupling in cold a om/elec onics one-dimensional ing sys ems, and sca e ing opo-
logical in a ian s o condensed ma e class D endpoin s. Unde unied
ke nelwindowing
gene alized leas squa es/Fishe
syn ax p o ide: Fi s , sel -consis en es ima ion o e ac-
i e index modica ion o de ia cu ed space ime QED one-loop eec i e ac ion, cons uc ing
ep oducible
windowed uppe limi e
; on FRW backg ound scale uppe limi phase esidual
scale below obse a ional h eshold, yielding only igo ous uppe bounds. Second, igo ously
p o e equi alence be ween spec al quan iza ion igonome ic equa ion on ing
cos θ= cos(kL) + αδ
ksin(kL)
and "
ampli ude-co ec ed
phase closu e o mula"
cos γ(k) = | (k)|cos θ, γ =kL −a c anαδ
k,| (k)|=k
pk2+α2
δ
ully equi alen , degene a ing o pu e phase closu e only in limi
| | → 1
; p o ide
s uc u al
iden iabili y
heo em ha
{kn(θ)}
alone can
uniquely in e
αδ≡mg/ℏ2
, wi h closed-
o m sensi i i y, ill-condi ioning a oidance c i e ia, and  s -o de equi alen iza ion o na ow
po en ial wid h. Thi d, on class D endpoin s use
Q= sgn de (0)
as opological c i e ion,
p opose unde
nea -uni a y
p emise unied ni e-size (exponen ial) and con ac /bounda y
(algeb aic) bias ex apola ion eg ession
Jc(L) = Jc+βe−L/ξ +γ/L,
p o ide linea esponse mapping o cQED eadou (ca i y equency shi and addi ional
loss). P oo s and enginee ing p o ocols o abo e conclusions p o ided wi h pape , di ec ly
ep oducible and ansplan able o simila pla o ms.
Keywo ds
: Phase equency me ology; as adio bu s ; cu ed space ime QED; windowed
uppe limi s;
δ
po en ial; Aha ono Bohm eec ; sel -adjoin ex ension; sca e ing opological in-
a ian ; Majo ana; cQED
1
1 In oduc ion & His o ical Con ex
Me ological schemes using "phase equency" as single obse able na u ally span mul iple on-
ie s: in cosmological elec omagne ic p opaga ion, phase equency pe u ba ions eco d geome y
and medium; in one-dimensional quan um ings, phase shi s quan i y sca e ing and geome ic
cohe ence; in condensed ma e sca e ing me ology, low-ene gy eec ion subblock phase and
de e minan cha ac e ize opological numbe . Cu ed space ime QED one-loop eec i e ac ion in-
dica es acuum pola iza ion in oduces only minu e e ac i e index co ec ion on weak cu a u e
backg ounds, sa is ying high- equency causali y and analy ici y; his makes FRB
ca ie phase
na u ally a channel o se ing
uppe limi s
, no ealis ic de ec ion.
δ
- ing p oblem unde sel -
adjoin ex ension
U(2)
amewo k yields igo ous spec alsca e ing equi alence uni ying wi h AB
ux; opological endpoin s implemen able ia single-end eec ion ma ix ze o-ene gy c i e ion o
nea -eld me ology o non-local opology. Abo e backg ounds inc easingly clea in ecen li e a u e;
his pape 's goal is condensing in o
unied me ological syn ax and ep oducible ecipes
.
2 Model & Assump ions
2.1 Unied Obse a ion Equa ion, Weigh s, and Windowing
Ac oss h ee pla o ms, obse ables abs ac able as phase- ype eadou
m(ω)
on equency. Unied
exp ession
m(ω) = ZP
K(ω, χ)x(χ) dχ+
P−1
X
p=0
apΠp(ω) + ϵ(ω),
whe e
K
is obse a ion ke nel,
χ
deno es p opaga ion pa ame e (e.g., con o mal coo dina e),
geome ic phase, o ene gy scale;
x(χ)
is a ge unc ion (o windowed pa ame e );
Πp
a e sys em-
a ic/ o eg ound basis unc ions,
ap
hei ampli udes;
ϵ
is noise p ocess, co a iance
Cϕ(ω, ω′)
. On
disc e e equency axis adop weigh ed inne p oduc
⟨ , g⟩ ≡ X
ij
(ωi) [C−1
ϕ]ij g(ωj).
Dene weigh ed G amSchmid o hono malized equency windows
Wj(ω)
, cons uc design
ma ix wi h
Ajµ =⟨Wj,K(·, χµ)⟩
; gene alized leas squa es (GLS) and Fishe in o ma ion
ˆ
θ= (A⊤A)−1A⊤y, F =A⊤A,
whe e
yj=⟨Wj, m⟩
,
θ
a e windowed pa ame e s (o coecien s o
x(χ)
in piecewise cons an
basis). Classical CRB o single- equency phase in whi e noise limi degene a es o
σϕ≥(2Nρ)−1/2
,
signal p ocessing e e ence o his pape 's uppe limi de i a ion.
2.2 Pla o m-Specic Assump ions and Ke nels

FRB p opaga ion
: Along con o mal coo dina e
χ
(
dℓ=a(η)dχ
) in eg a ion, local equency
ωloc = (1 + z)ωobs
cancels wi h scale ac o , hus
KFRB(ωobs, χ) = ωobs/c ,
2
aking
x(χ)≡δn(χ)
( e ac i e index co ec ion) as windowed a ge ke nel. This o m
gua an ees high- equency causali y
n(ω→ ∞)→1
and wo ldlinePen ose limi analy ici y
cons ain .

δ
- ing + AB ux
: Obse able is eigenwa e ec o
k(θ)
sa is ying quan iza ion condi ion,
αδ≡mg/ℏ2
ep esen s poin in e ac ion s eng h, geome ic leng h
L
is ing ci cum e ence,
θ= 2πΦ/Φ0
is ux phase. Ke nel unc ion mani es s in dispe sion ela ion and bounda y
condi ions, see Sec ion 4.

Topological sca e ing
: Obse able is cha ac e is ic phase o ze o-ene gy eec ion ma ix
(0)
o sign o i s de e minan ; ke nel
K
om sca e ing opology mapping and inpu ou pu
linea esponse, see Sec ion 6.
3 Main Resul s (Theo ems and Alignmen s)
Theo em 1
(A: FRBWindowed Uppe Limi and O de o Magni ude o One-Loop Vacuum
Pola iza ion)
.
Cu ed space ime QED unde weak cu a u e backg ound p o ides one-loop co ec ion
o de o magni ude o e ac i e index
δn ∼αem
πλ2
eR,
aking FRW uppe bound
|R| ∼ H2
0/c2
, phase accumula ion uppe limi o 1 GHz, 1 Gpc
∆ϕ∼ωobsL
cδn ≈1.2×10−53 ad.
The e o e unde any ealis ic baseband da a olume and noise, can only p o ide windowed uppe
limi o
x(χ) = δn(χ)
, no eec i e de ec ion. P oo in Appendix A.
Theo em 2
(B:
δ
-RingEqui alence o T igonome ic Equa ion and Ampli ude-Co ec ed Phase
Fo mula)
.
Fo one-dimensional ing
δ
po en ial (s eng h
g
) and AB ux
θ
, eigenwa e ec o
k
sa ises
cos θ= cos(kL) + αδ
ksin(kL),
which is ully equi alen o
cos γ(k) = | (k)|cos θ, γ(k) = kL −a c anαδ
k,| (k)|=k
qk2+α2
δ
and degene a es o pu e phase closu e
cos γ= cos θ
only in limi
| (k)| → 1
(weak sca e ing o
ansmission esonance). P oo in Appendix B.
Theo em 3
(C:
δ
-RingS uc u al Iden iabili y om
{kn(θ)}
Alone)
.
Unde spec al obse a ion
{kn(θ)}
a xed
(L, θ)
, can uniquely in e
αδ=mg/ℏ2
, bu canno sepa a e indi idual alues o
m
and
g
. I in oducing second independen eadou ype (e.g., ene gy spec um
En=ℏ2k2
n/(2m)
o absolu e po en ial s eng h
ge =RVdx
calib a ion), hen a gene ic poin s
(m, g)
Jacobian ull
ank and decoupling possible. P oo in Appendix B.
3
Theo em 4
(D:
δ
-RingImplici Func ion Sensi i i y and Ill-Condi ioning)
.
Le
(k, αδ, θ) = cos(kL) + αδ
ksin(kL)−cos θ,
hen
∂k
∂αδ
=
sin(kL)
k
Lsin(kL)−αδ
kLcos(kL) + αδ
k2sin(kL).
Ill-condi ioned domain gi en by
∂ /∂k = 0
; when
|αδ|/k ≪1
app oxima ely
sin(kL)≃(αδ/k) cos(kL)
.
P oo in Appendix B.
Theo em 5
(E: Fi s -O de Equi alen iza ion o
δ
-Po en ial Fini e Wid h)
.
Fo na ow po en ial
V(x)
(wid h
w≪L
) wi h
RVdx=g
xed, in
(kw)≪1
egion, eec i e s eng h on ing
αe
δ(k) = m
ℏ2g1 + η2[V] (kw)2+O(kw)4,
whe e shape ac o
η2[V]
de e mined by second momen o
V
; ec angula and Gaussian po en ials
espec i ely ha e
η2=1
3,1
2
. P oo and nume ical compa ison in Appendix B.
Theo em 6
(F: Topological Sca e ing In a ian Nea -Uni a y Ex apola ion)
.
Unde nea -uni a y
p emise (
| † −⊮| ≤ ε⋆
), ni e leng h
L
, con ac impe ec ion, and weak dissipa ion in oduce bias
o c i ical coupling
Jc
, unied by
Jc(L) = Jc+βe−L/ξ +γ/L
ex apola ion eg ession abso p ion, whe e exponen ial e m co esponds o endpoin s a e cou-
pling o e lap, algeb aic e m co esponds o con ac /bounda y eec s.
Q= sgn de (0)
ip p o ides
Jc(L)
obse a ion ajec o y. P oo in Appendix C.
Theo em 7
(G: cQED Redundan Readou and Sca e ing In a ian Consis ency)
.
Single-po
weak coupling, inpu ou pu ela ion gi es
(ω) = 1−Z0Y(ω)
1 + Z0Y(ω)≈1−2Z0Y(ω), δωc∝g2
ca Re χ(ωc), κadd ∝g2
ca Im χ(ωc),
whe e
χ
is endpoin subsys em linea esponse. Fo app op ia e de uning
|ωc−ωcon |≳3κ
, ze o
c ossings o
{δωc, κadd}
consis en wi h
Q
ip. P oo in Appendix C.
4 P oo s
4.1 P oo o Theo em A (FRB)
QED eec i e ac ion on cu ed backg ound w i able as
Le =−1
4FµνFµν +1
m2
eaRFµνFµν +bRµνFµαFν
α+cRµναβFµνFαβ+· · · ,
linea ized dispe sion ela ion p o ides e ac i e index co ec ion
δn ∝(αem/π)λ2
eR
. On FRW,
aking
|R| ∼ H2
0/c2
, in eg a ing along
χ
wi h ke nel Eq. (2.4),
4
∆ϕ=Zχs
0
ωobs
cδn(χ) dχ≤ωobsL
cmax
χδn(χ),
subs i u ing cons an s
(αem/π)≃2.32×10−3
,
λe= 3.8616×10−13 m
,
|R| ≃ 5.4×10−53 m−2
and
ωobsL/c ≃6.46 ×1026
(1 GHz/1 Gpc), ob ain
∆ϕ≈1.2×10−53
ad. HollowoodSho e wo ldline
Pen ose limi gua an ees
n(ω→ ∞)→1
; his pape 's cons uc ed windowed ke nel sa ises his
causali y/analy ici y cons ain .
4.2 P oo o Theo em B (
δ
-Ring)
F om s aigh -line
δ
po en ial sca e ing ampli ude
=k
k+iαδ
and
| |=1
p1+(αδ/k)2
, expanding
ans e ma ix ace condi ion
T= 2 cos θ
yields
cos θ= cos(kL) + αδ
ksin(kL).
On he o he hand
cos(kL) + αδ
ksin(kL) = ℜ
h(cos kL +isin kL)1−iαδ
ki=p1+(αδ/k)2cos kL −a c an αδ
k,
i.e.,
cos γ=| |cos θ
, p o ed.
4.3 P oo o Theo em C (Iden iabili y)
Spec al equa ion (3.3) con ains only combined pa ame e
αδ=mg/ℏ2
; i
(m1, g1)= (m2, g2)
exis s
bu
m1g1=m2g2
, hen o any
(θ, n)
yields iden ical
{kn}
. The e o e
{kn(θ)}
alone uniquely
de e mines
αδ
, while
m
,
g
non-sepa able. I adding ene gy spec um
En=ℏ2k2
n/(2m)
, hen
J=∂ /∂m ∂ /∂g
∂E/∂m ∂E/∂g= ∂
∂αδ
∂αδ
∂m
∂
∂αδ
∂αδ
∂g
−ℏ2k2
2m20!,
a gene ic poin s
de J= 0
(excep degene a e subse s like
k= 0
), hus
(m, g)
decoupling
possible, p o ed.
4.4 P oo o Theo em D (Sensi i i y)
By implici unc ion heo em
∂k
∂αδ
=− αδ
k
, αδ=1
ksin(kL), k=−Lsin(kL) + αδL
kcos(kL)−1
k2sin(kL),
ob ain Eq. (3.6). Ill-condi ioned domain sa ises
k= 0
, when
|αδ|/k ≪1
app oxima ely
sin(kL)≃(αδ/k) cos(kL)
, p o ed.
5

4.5 P oo o Theo em E (Fini e Wid h)
Le na ow po en ial
V(x) = V0u(x/w)
(
Ru= 1
) wi h
g=V0w
xed, ans e ma ix low-ene gy
expansion a
kw ≪1
yields
−1(k) = 1 + iαδ
k+η2[V] (kw)2+O(kw)4,
equi alen izing o
αδ→αe
δ(k)
eplacemen . Calcula e second momen o ec angula /Gaussian
u
compa ing wi h sol able models, ob ain
η2=1
3,1
2
. Nume ical e ica ion in Appendix B Figu e
B2.
4.6 P oo o Theo ems F and G (Topological Sca e ing and cQED)
Class D sys em opological numbe ia single-end eec ion ma ix ze o-ene gy c i e ion
Q=
sgn de (0)
. Fini e leng h coupling leads o ze o mode ene gy spli ing
∝e−L/ξ
, con ac /bounda y
in oduces
1/L
scaling; unde nea -uni a y assump ion,
de (0)
sign ip ajec o y
Jc(L)
hus ex-
hibi s Eq. (3.8) shape. On he o he hand, inpu ou pu gi es
≈1−2Z0Y
, while
Y∝g2
ca χ
;
when ca i y mode sucien ly de uned om con inuum,
χ(0)
sign change p oduces ze o c ossing in
{δωc, κadd}
, consis en wi h
Q
ip, p o ed.
5 Model Apply
5.1 FRB "Windowed Uppe Limi e "

Ke nel and weigh s
: Adop
KFRB =ωobs/c
,
Πp(ω)∈ {1, ω, ω−1,log ω}
.
Cϕ
by h ee-
channel boo s ap es ima ion: o-sou ce, o-band, sidelobe.

O hogonal windows
: On disc e e equency axis
{log ωi}
, use
C−1
ϕ
weigh ed G amSchmid
ob aining
Wj
, whi ening es
⟨Wj, Wℓ⟩ ≃ δjℓ
.

Uppe limi s
: Fo windowed pa ame e s
θ
adop GLS and p ole-likelihood pa allel com-
pu ing 95% uppe limi band, p o iding scaling cu e wi h bandwid h/e en numbe and
sys ema ic basis en elope; use "whe he includes
ω−1
" as obus ness swi ch aking en elope.
5.2
δ
-Ring In e sion and Ill-Condi ioning A oidance

In e sion
: Wi h Eq. (3.3)
= 0
as a ge , use Eq. (3.6) as New on s ep i e a ion in e ing
ˆαδ
.

Ill-condi ioned domain
: Plo g ay egion
| k|< τ
in
(kL, αδ/k)
plane (
τ∼10−2
), expe i-
men al poin selec ion a oids domain.

Fini e wid h
: Fo gi en
w
and po en ial shape, use Eq. (3.7) co ec ing
ˆαδ
, p opaga e
η2
unce ain y o co a iance.
5.3 Topological Ex apola ion and cQED Redundancy

Ex apola ion eg ession
: Measu e
Q
ip
Jc(Li)
on
{Li}
, eg ess
(Jc, β, γ, ξ)
ia Eq. (3.8),
compa e "exponen ial/algeb aic/composi e" h ee models using AIC/BIC.

Nea -uni a y quali y con ol
: Weigh by
εi=| †
i i−⊮|
o en e eg ession as co a ia e,
epo
Jc
sensi i i y o
ε
.
6

cQED consis ency
: Unde de uning
|ωc−ωcon |≳3κ
, use
{δωc, κadd}
ze o c ossings as
edundan e idence, c oss- alida e wi h
Q
ip.
6 Enginee ing P oposals

FRB
: Release minimized sc ip loading public baseband examples, comple e cohe en dedis-
pe sion, s uc u e-maximized DM, h ee-channel boo s ap co a iance, weigh ed window o -
hono maliza ion and 95% uppe limi plo ing, a ach "whe he includes
ω−1
basis" obus -
ness compa ison.

δ
- ing
: P o ide
{kn(θ)}
in e sion
αδ
execu able sc ip , plo ill-condi ioned domain g ay map
and
∂k/∂αδ
con ou s; p o ide ni e wid h co ec ion and e o p opaga ion example.

Topological + cQED
: Release sc ip s o ex ac ing
Q
om
(ω)
, eg essing
Jc(L)
wi h
model compa ison, and
{δωc, κadd}
consis ency es wi h
Q
.
7 Discussion (Risks, Bounda ies, Pas Wo k)
FRB
: A FRW backg ound o de o magni ude, one-loop acuum pola iza ion phase eec un-
measu able; his pape posi ions as "windowed uppe limi e ", emphasizing ke nelme ic selec ion
consis ency and sys ema ic en elope, a oiding bias se ing.
δ
- ing
: S ic ly limi "pu e phase closu e" o limi
| | → 1
, a oiding misuse a gene ic
θ
; ni e
wid h equi alen iza ion equi es use wi hin
kw ≪1
egion.
Topological
: Nea -uni a y h eshold and model selec ion key o ex apola ion eliabili y, ec-
ommend using esidual s uc u e and c oss- alida ion supp essing o e  ing.
8 Conclusion
Using unied phase equency eadou and "ke nelwindowingGLS/Fishe " syn ax, es ablished
ep oducible me ological ecipes o h ee pla o ms: FRB side p o ides igo ous windowed up-
pe limi e ;
δ
- ing side p o es spec alsca e ing ampli ude-co ec ed phase equi alence, s uc u al
iden iabili y and closed- o m sensi i i y wi h ni e wid h co ec ion; opological sca e ing side
implemen s c i ical coupling me ology ia nea -uni a y ex apola ion and cQED edundancy. Sup-
po ing enginee ing p o ocols and minimal sc ip s make pa adigm easily ans e able o simila
sys ems.
Acknowledgemen s, Code A ailabili y
Thank ela ed public da a and li e a u e p o iding ep oducible benchma ks. Suppo ing code
and minimal da a examples eleased unde gene al license, including en i onmen lockles, uni
checks, and injec ion eco e y es s. FRB sc ip s wi h
δ
- ing, opological ex apola ion sc ip s
adop unied I/O con en ions and andom seeds ensu ing ep oducibili y. Da a usage and ci a ion
ollow espec i e da ase policies.
7
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opological quan um numbe o a diso de ed mul i-mode wi e,"
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A FRB: One-Loop Vacuum Pola iza ion Scale, Ke nelMe ic Con-
sis ency, and Windowed Uppe Limi
A.1 One-Loop Eec i e Ac ion and Scale
F om Eq. (4.1) linea ize Maxwell equa ions, unde weak cu a u e and geome ic op ics app oxi-
ma ion, dispe sion ela ion
k2=n2ω2/c2
yields
n(ω, χ) = 1 + δn(χ) + O(R2), δn(χ)≈αem
πλ2
eR(χ).
FRW backg ound
Rµναβ
low-o de con ac ions p o ide
|R| ∼ H2
0/c2
.
A.2 Ke nelMe ic Consis ency (Redshi Cancella ion)
Phase inc emen along geodesic
dϕ=ωloc
cδn(χ) dℓ=(1 + z)ωobs
cδn(χ)a(η) dχ=ωobs
cδn(χ) dχ,
i.e., when in eg a ing in con o mal coo dina es
(1 + z)
and
a
exac ly cancel, ob aining ke nel
Eq. (2.4). This o m au oma ically sa ises causali y equi emen
n(ω→ ∞)→1
.
A.3 Fishe /GLS Windowed Uppe Limi
Expand
δn(χ)
on piecewise cons an basis, dene
Ajµ =⟨Wj, ωobs/c⟩∆χµ
, hen
F=A⊤A, σUL = 1.96 pdiag(F−1),
8
ake uppe limi en elope o e die en sys ema ic basis choices (whe he includes
ω−1
,
log ω
).
B
δ
-Ring: Bounda y Condi ions, Equi alence, Iden iabili y, Sen-
si i i y, and Fini e Wid h
B.1 Bounda y Condi ion and T igonome ic Equa ion
S aigh -line
δ
po en ial bounda y condi ion
ψ′(0+)−ψ′(0−)=2αδψ(0), αδ=mg/ℏ2,
combined wi h ee p opaga ion segmen ans e ma ix yields single-lap ace condi ion
T=
2 cos θ
, simpli ying o Eq. (3.3).
B.2 Equi alence Th ee-Line De i a ion
Using
ℜheikL1−iαδ
ki= cos(kL) + αδ
ksin(kL) = p1+(αδ/k)2cos kL −a c an αδ
k,
ob ain Eq. (3.4), whe e
| |= (1 + (αδ/k)2)−1/2
.
B.3 Iden iabili y (Rank)
Cons uc obse a ion equa ions
{ = 0, E =ℏ2k2/(2m)}
Jacobian o
(m, g)
, a oiding degene a e
s ips like
k= 0
and s ong sca e ing ensu es ull ank (Eq. 4.5).
B.4 Sensi i i y and Ill-Condi ioned Domain
F om Eq. (4.6) ob ain implici unc ion de i a i e and ill-condi ioning condi ion
k= 0
. Expe i-
men al design uses c i e ion
| k|> τ
o poin selec ion,
τ∼10−2

10−3
.
B.5 Fini e Wid h Equi alen iza ion and Nume ical Compa ison
Fo
V(x) = V0u(x/w)
, unde low-ene gy expansion a
kw ≪1
,
−1(k) = 1 + iαδ
k+η2[V] (kw)2+· · · ,
whe e
η2[V] = 1
w2Rx2V(x) dx
RV(x) dx
(a e no maliza ion and compa ison calib a ion)
.
Nume ical scan e ica ion o ec angula (
η2= 1/3
) and Gaussian (
η2= 1/2
) po en ials,
ela i e e o
≤1%
in
kw ≤0.2
egion. Sol able model esul s consis en wi h expansion in o e lap
egion.
9