Phase–F equency Uni ied Me ology and
Expe imen al Tes beds
in Compu a ional Uni e se:
Uni ied Time Scale Implemen a ion
om FRB Vacuum Windowing Uppe Limi
o δ-Ring Sca e ing Iden i iabili y
Haobo Ma1Wenlin Zhang2
1Independen Resea che
2Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
In p e ious “compu a ional uni e se” amewo k, uni e se axioma ized as dis-
c e e objec Ucomp = (X, T,C,I), upon which cons uc ed disc e e complexi y ge-
ome y, disc e e in o ma ion geome y, con ol mani old (M, G) induced by uni ied
ime scale, ask in o ma ion mani old (SQ, gQ), and ime–in o ma ion–complexi y
join a ia ional p inciple. Uni ied ime scale gi en by sca e ing mas e scale
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω)
uni ying phase de i a i e, spec al shi densi y, and Wigne –Smi h g oup delay
ace as single scale. Howe e , his amewo k s ill emains mainly a “ heo e ical
geome y” le el, has no sys ema ically gi en how o me ologically measu e and
calib a e uni ied ime scale and compu a ional uni e se s uc u e in ac ual expe i-
men s.
This pape , on basis o compu a ional uni e se–uni ied ime scale–spec al win-
dowing eadou , cons uc s c oss-pla o m me ology pa adigm using “phase– equency”
as sole obse able, implemen ing i on wo ep esen a i e es beds: cosmological-
dis ance Fas Radio Bu s p opaga ion (FRB) and labo a o y-scale δ- ing–Aha ono –
Bohm (AB) lux sca e ing. Co e idea: om compu a ional uni e se pe spec i e,
all obse ables ealized h ough phase– equency eadou unde uni ied ime scale;
FRB and δ- ing sca e ing espec i ely p o ide cosmic-scale and labo a o y-scale
“homologous eadou s”, iewable as implemen a ions o same me ology pa adigm
a di e en scales unde uni ied ime scale and complexi y geome y.
Main esul s o his pape :
1. Unde amewo k o ca ego ical equi alence be ween compu a ional–physical
uni e ses, in oduce “phase– equency eadou unc o ” PhF , sending any
physically ealizable compu a ional uni e se objec o me ology objec con-
aining only phase– equency da a. P o e PhF compa ible wi h uni ied ime
1
scale mas e scale: unde aceable pe u ba ion and wa e ope a o com-
ple eness, PhF ou pu comple ely de e mined by κ(ω) and ini e spec al–
sca e ing in a ian s.
2. Fo FRB, cons uc “ acuum pola iza ion windowing uppe limi ” model:
window FRB equency-domain phase using PSWF/DPSS ype window unc-
ions, p o e unde ixed complexi y budge and cosmological dis ance con-
s ain s, any uni ied ime scale a ia ion δκ(ω) con ibu ion o FRB phase
esidual can be bounded by s ic uppe bound; i obse ed esidual below
his bound, ob ain uni ied ime scale ype uppe limi on acuum pola iza ion
o o he new physics.
3. Fo δ- ing–AB lux sca e ing, es a e equi alence be ween spec al quan iza-
ion equa ion
(k, αδ, θ) = cos(kL)+(αδ/k) sin(kL)−cos θ= 0
and “ampli ude-co ec ed phase closu e”
cos γ(k) = | (k)|cos θ
and p o e unde compu a ional uni e se–con ol mani old pe spec i e: unde
spec al obse a ion {kn(θ)}a ixed (L, θ), δ–coupling s eng h αδand AB
lux θa e iden i iable in non-pa hological domain (Jacobian ull ank), usable
as “labo a o y ule ” o uni ied ime scale–phase me ology.
4. Unde uni ied ime scale–spec al windowing eadou amewo k, embed FRB
and δ- ing sca e ing in same “phase– equency me ology uni e se”, p o e
exis ence o “c oss-pla o m scale uni ica ion condi ion”: when FRB phase
esidual and δ- ing sca e ing spec al shi bo h explained by same κ(ω)
model, hei windowed eadou s belong o same equi alence class on app op i-
a e PSWF/DPSS space, hus can calib a e and consis ency- es uni ied ime
scale h ough join i ing.
5. Embed abo e phase– equency me ology s uc u e in o ime–in o ma ion–
complexi y a ia ional p inciple, o malize “choosing FRB/δ- ing window unc-
ions and con ol pa ame e s” as a ia ional p oblem on join mani old, gi e
a ia ional condi ions o “simul aneously using cosmic-scale and labo a o y-
scale phase– equency eadou s o maximize uni ied ime scale iden i iabili y
unde ini e complexi y budge ”.
This pape hus comple es expe imen al implemen a ion design o “phase– equency
uni ied me ology” wi hin compu a ional uni e se amewo k: FRB and δ- ing sca -
e ing become wo-end es beds o uni ied ime scale and complexi y geome y,
PSWF/DPSS window unc ions become na u al ools o e o con ol, bo h join ly
cons uc ing c oss-scale, c oss-pla o m, ye comple ely uni ied phase– equency
me ology sys em unde compu a ional uni e se pe spec i e.
Keywo ds: Compu a ional uni e se; Phase- equency me ology; FRB; δ- ing sca e -
ing; Uni ied ime scale; PSWF/DPSS; Expe imen al es bed
1 In oduc ion
In p e ious se ies wo ks, we ha e comple ed cons uc ions a ollowing le els:
2
1. A disc e e le el, abs ac uni e se as axioma ic compu a ional uni e se objec
Ucomp = (X, T,C,I), upon which cons uc complexi y g aph Gcomp = (X, E, C),
complexi y dis ance dcomp, complexi y dimension and disc e e Ricci cu a u e.
2. A uni ied ime scale–sca e ing heo y le el, in oduce
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω),
as “ ini y mas e scale”, uni ying sca e ing phase de i a i e, spec al shi densi y,
and g oup delay ace.
3. Th ough uni ied ime scale and complexi y geome y cons uc con ol mani old
(M, G), p o e disc e e complexi y dis ance con e ges o geodesic dis ance dGin
e inemen limi .
4. Th ough obse e amily and ela i e en opy second-o de s uc u e cons uc
ask in o ma ion mani old (SQ, gQ), and gi e join a ia ional p inciple o ime–
in o ma ion–complexi y on join mani old EQ=M×SQ.
5. In spec al windowing e o con ol wo k, in oduce PSWF/DPSS window unc-
ions, show in uni ied ime scale– equency domain, hey a e op imal eadou win-
dows unde ini e ime–bandwid h–complexi y budge .
These esul s lay ounda ion o cons uc ing “uni ied ime scale–compu a ional uni-
e se” heo e ical sys em, bu do no di ec ly answe key ques ion: how o me ologically
**measu e and calib a e** his uni ied ime scale mas e scale h ough conc e e physical
expe imen s? Ma hema ical exis ence o uni ied ime scale insu icien o demons a e
i s physical measu abili y; we need o connec sca e ing mas e scale wi h ac ually ob-
se able phase– equency da a, and design c oss-pla o m me ology s a egy so phase–
equency eadou s om cosmic scale and labo a o y scale can be join ly used o es and
calib a e uni ied ime scale.
In his con ex , Fas Radio Bu s s (FRB) and δ- ing–AB lux sca e ing become wo
e y na u al es beds:
FRB a e sho -du a ion b oadband adio signals a e sing cosmological dis ances,
whose p opaga ion phase, g oup delay and dispe sion s uc u e con ain in eg a ed
in o ma ion abou cosmological medium and acuum p ope ies; unde uni ied ime
scale–sca e ing pe spec i e, FRB essen ially “cosmic-le el sca e ing expe imen ”.
δ- ing–AB lux sca e ing is p ecise measu emen o spec al–sca e ing s uc u e o
one-dimensional ing geome y, poin po en ial and AB lux a labo a o y scale; i s
spec al quan iza ion equa ion and phase closu e p o ide highly con ollable phase–
equency es bed, usable o “ e e se calib a ion” o uni ied ime scale model unde
known geome ic pa ame e s and coupling cons an s.
Goal o his pape : uni y embedding o FRB and δ- ing sca e ing in o compu a-
ional uni e se–uni ied ime scale amewo k, es ablish me ology pa adigm using phase–
equency as sole eadou , so wo ypes o expe imen s can mu ually calib a e and consis ency-
es on same uni ied ime scale mas e scale.
3
2 Phase–F equency Readou Func o in Compu a-
ional Uni e se
This sec ion, unde backg ound o compu a ional–physical uni e se ca ego ical equi a-
lence, in oduces “phase– equency eadou unc o ” PhF , whose ou pu con ains only
phase– equency da a, di ec ly connec ed wi h uni ied ime scale mas e scale.
2.1 Re iew o Physical–Compu a ional Uni e se Equi alence
In p e ious ca ego ical equi alence wo k, we cons uc ed physical uni e se ca ego y PhysUni QCA
and compu a ional uni e se ca ego y CompUni phys, gi ing mu ually in e se unc o s
F:PhysUni QCA →CompUni phys,
G:CompUni phys →PhysUni QCA.
Physical uni e se objec abs ac able as
Uphys = (M, g, F, κ, S),
compu a ional uni e se objec as
Ucomp = (X, T,C,I).
Func o s F, G p ese e s uc u e o uni ied ime scale densi y κ(ω) and sca e ing da a
S(ω): om physical side o compu a ional side, uni ied ime scale disc e ized as single-
s ep cos ; om compu a ional side o physical side, complexi y geome y con inuized as
con ol–sca e ing mani old.
2.2 Phase–F equency Da a Objec s
De ine “phase– equency da a objec ” as
UPhF = (Ω,Θ(ω), κ(ω)),
whe e Ω ⊂Ris e ec i e equency band, Θ(ω) is o al sca e ing phase (o i s no -
maliza ion), κ(ω) is uni ied ime scale densi y. Acco ding o uni ied ime scale mas e
scale, unde aceable pe u ba ion condi ion
κ(ω) = Θ′(ω)/π
holds up o addi i e cons an .
2.3 Phase–F equency Readou Func o
De ini ion 2.1 (Phase–F equency Readou Func o ).De ine unc o
PhF :PhysUni QCA →PhF Uni ,
whe e PhF Uni objec s a e UPhF , mo phisms a e equency-domain ans o ma ions
p ese ing ωand Θ(ω), κ(ω) s uc u e.
4
Fo physical uni e se objec Uphys,PhF (Uphys) = (Ω,Θ(ω), κ(ω)) de e mined by i s
sca e ing da a and uni ied ime scale mas e scale.
Th ough ca ego ical equi alence, gi en compu a ional uni e se objec Ucomp, i s use
G(Ucomp) = Uphys o ob ain physical uni e se, hen apply PhF o ob ain phase– equency
da a objec . Thus ob ain composi e unc o
PhF ◦G:CompUni phys →PhF Uni .
P oposi ion 2.2 (Consis ency o Uni ied Time Scale and Phase–F equency Readou ).
Unde aceable pe u ba ion and wa e ope a o comple eness condi ions, phase– equency
eadou objec UPhF = (Ω,Θ(ω), κ(ω)) comple ely de e mined by uni ied ime scale densi y
κ(ω)and cons an phase shi , i.e.,
Θ(ω) = πZω
κ(ω′) dω′+ Θ0.
In pa icula , any wo pai s (Θ, κ),(Θ′, κ′)wi h κ≡κ′and Θ−Θ′cons an co espond
o same uni ied ime scale s uc u e.
P oo omi ed.
3 FRB Vacuum Pola iza ion Windowing Uppe Limi :
Compu a ional Uni e se Pe spec i e
This sec ion iews FRB p opaga ion as cosmic-scale sca e ing–p opaga ion p ocess, con-
s uc s “ acuum pola iza ion windowing uppe limi ” unde uni ied ime scale–spec al
windowing.
3.1 Sca e ing–P opaga ion Model o FRB P opaga ion
Fo simplici y, conside FRB signal complex ampli ude A(ω) in equency domain, whose
phase pa w i able as
A(ω) = |A(ω)|exp(iΦFRB(ω)).
I p opaga ion includes only known dispe sion and eionized medium con ibu ions,
hen
ΦFRB(ω) = Φknown(ω)+Φnew(ω),
whe e Φknown om con en ional dispe sion measu e and medium model, Φnew ep-
esen s possible con ibu ions om acuum pola iza ion, new pa icles, o uni ied ime
scale pe u ba ions.
Unde uni ied ime scale–sca e ing pe spec i e, ΦFRB(ω) unde s andable as e ec i e
sca e ing phase ΘFRB(ω), whose de i a i e gi es e ec i e ime scale densi y pe u ba ion
δκFRB(ω) = 1
π∂ωΦnew(ω).
5
3.2 Windowed FRB Phase and E o Uppe Bound
Obse a ionally, we can only measu e phase– equency da a in ini e equency band
ΩFRB = [ωmin, ωmax] wi h ini e esolu ion. In oduce window unc ion WFRB(ω) (e.g.,
gene a ed by PSWF/DPSS spec um) and de ine windowed esidual
RFRB =ZΩFRB
WFRB(ω)ΦFRB(ω)−Φknown(ω)dω.
I uni ied ime scale pe u ba ion δκFRB(ω) o FRB signal has cons ain |δκFRB| ≤
Λ unde some spec al no m, hen h ough in eg a ion and Cauchy–Schwa z inequali y
ob ain
|RFRB| ≤ Λ|WFRB|L2(ΩFRB )CFRB,
whe e CFRB de e mined by cosmological p opaga ion ke nel and geome ic ac o s.
Con e sely, i obse a ionally esidual |RFRB| ≤ εobs, ob ain uppe bound on uni ied
ime scale pe u ba ion
Λ≥Λmin ≥|RFRB|
|WFRB|CFRB
.
W i ing op imal window unc ion choice p oblem as cons ained minimiza ion o
|WFRB|, classical esul s show PSWF/DPSS ype windows minimize e o uppe bound
unde gi en ime– equency–complexi y budge , hus gi ing “FRB acuum pola iza ion
windowing limi e ”.
4δ-Ring-AB Flux Sca e ing Spec al-Phase-Sca e ing
Equi alence and Iden i iabili y
This sec ion, om compu a ional uni e se–con ol mani old pe spec i e, es a es spec al–
phase–sca e ing s uc u e o δ- ing–AB lux sca e ing, gi es iden i iabili y heo em.
4.1 δ-Ring-AB Flux Model
Conside one-dimensional ing, ci cum e ence L, coo dina e x∈[0, L) wi h pe iodici y
x∼x+L. In oduce poin δ–po en ial on ing, s eng h αδ, and AB lux θ∈[0,2π).
Co esponding Hamil onian (uni mass, igno ing cons an s) w i able as
H=−∂2
x+αδδ(x),
bounda y condi ion includes AB phase:
ψ(L−) = eiθψ(0+), ψ′(L−) = eiθψ′(0+).
Sol ing eigenequa ion Hψ =k2ψyields spec al quan iza ion condi ion
(k, αδ, θ) = cos(kL)+(αδ/k) sin(kL)−cos θ= 0.
Co esponding sca e ing ampli ude (k) and phase γ(k) sa is y ce ain phase closu e,
ypical o m
6
cos γ(k) = | (k)|cos θ,
whe e unc ional ela ionship be ween γ(k) and kL,αδgi en by sca e ing heo y.
4.2 δ-Ring Con ol Mani old in Compu a ional Uni e se
View δ- ing sca e ing as compu a ional uni e se on low-dimensional con ol mani old:
con ol pa ame e space
Mδ- ing ={(L, αδ, θ)},
equipped wi h me ic G, e.g.,
G=gLLdL2+gααdα2
δ+gθθdθ2.
Spec al obse a ion {kn(θ)}co esponds o da a poin s on in o ma ion mani old,
while uni ied ime scale densi y gi en by sca e ing phase de i a i e and g oup delay. δ-
ing hus becomes highly con ollable “compu a ional sub-uni e se” on h ee-dimensional
con ol mani old, usable o uni ied ime scale and phase– equency me ology.
4.3 Spec al–Phase–Sca e ing Equi alence Theo em
Theo em 4.1 (Spec al Quan iza ion and Phase Closu e Equi alence).In δ- ing–AB
lux model, spec al quan iza ion condi ion
(k, αδ, θ)=0
and phase–ampli ude closu e
cos γ(k) = | (k)|cos θ
equi alen unde usual sca e ing egula i y condi ions; in pa icula , when | (k)| → 1
(weak sca e ing o ansmission esonance), educes o pu e phase closu e cos γ(k) =
cos θ.
P oo ske ch. F om bounda y condi ions and δ–po en ial jump condi ions de i e ans e
ma ix, equi e wa e unc ion o ma ch i sel a e one loop a ound ing, ob ain spec-
al quan iza ion equa ion; on o he hand, compu e sca e ing ma ix elemen s (k), (k)
and phase shi γ(k), ew i e spec al condi ion as phase–ampli ude closu e. Algeb aic
equi alence be ween hem e i ied h ough di ec subs i u ion and simpli ica ion. See
Appendix B.1 o de ails.
4.4 Pa ame e Iden i iabili y Theo em
Theo em 4.2 (Local Iden i iabili y o δ-Ring).Gi en Land se e al AB lux alues θj,
i obse e su icien ly many eigenwa enumbe s {kn(θj)}, and Jacobian ma ix a hese
poin s
J= (∂ /∂k, ∂ /∂αδ, ∂ /∂θ)
has ull ank a (k, αδ, θ), hen (αδ, θ)a e locally iden i iable pa ame e s o spec al da a
nea his poin , i.e., local in e se unc ion exis s w i ing (αδ, θ)as unc ion o {kn(θj)}.
7
P oo ske ch. Apply implici unc ion heo em: i ∂ /∂k = 0 and pa ial de i a i e sub-
ma ix wi h espec o (αδ, θ) ull ank, can sol e k=k(αδ, θ) locally, hen cons uc
composi e map o mul iple θj, ob ain local in e ibili y. Fo mul iple eigen alues, s ack
componen s; i combined Jacobian ull ank, o e all iden i iabili y holds. See Appendix
B.2 o de ails.
4.5 Pa hological Domains and Condi ion Numbe s
Th ough explici compu a ion
∂k
∂αδ
=− αδ
k
,
whe e αδ= (sin(kL))/k, k=−Lsin(kL)+αδ(. . . ), can de ine pa hological condi ion
numbe egion k≈0, co esponding o spec al quan iza ion cu e highly sensi i e o
pa ame e s o non-in e ible. Unde compu a ional uni e se–complexi y geome y pe -
spec i e, hese pa hological egions co espond o egions on con ol mani old wi h la ge
cu a u e, spec al–phase in o ma ion’s “geodesic sensi i i y” o pa ame e s d ama ically
ampli ied, equi ing window unc ions and expe imen al design o a oid o specially han-
dle wi hin complexi y budge .
5 FRB and δ-Ring Uni ied Phase-F equency Me ol-
ogy
This sec ion embeds FRB and δ- ing sca e ing in same phase– equency me ology ame-
wo k, gi es geome ic condi ions o “c oss-pla o m scale uni ica ion”.
5.1 Jux aposi ion o Phase–F equency Readou Objec s
Fo FRB and δ- ing, we espec i ely ob ain phase– equency da a objec s
UFRB
PhF = (ΩFRB,ΘFRB(ω), κFRB(ω)),
Uδ
PhF = (Ωδ,Θδ(ω), κδ(ω)).
Unde uni ied ime scale hypo hesis, exis s “mas e scale densi y” κuni (ω) such ha
e ec i e ime scales co esponding o FRB and δ- ing espec i ely
κFRB(ω) = gFRB(ω)κuni (ω),
κδ(ω) = gδ(ω)κuni (ω),
whe e gFRB, gδa e weigh unc ions de e mined by geome y and p opaga ion ke nels.
8
5.2 C oss-Pla o m Scale Uni ica ion Condi ion
De ini ion 5.1 (C oss-Pla o m Scale Uni ica ion).FRB and δ- ing sca e ing called
scale-uni ied on uni ied ime scale i he e exis mas e scale densi y κuni and weigh s
gFRB, gδsuch ha windowed phase esiduals sa is y consis en in e p e a ion:
RFRB(WFRB)≈ZWFRB(ω)δgFRB(ω)κuni (ω) dω,
Rδ(Wδ)≈ZWδ(ω)δgδ(ω)κuni (ω) dω,
o window unc ion amily WFRB, Wδ.
Theo em 5.2 (C oss-Pla o m Consis ency Tes o Uni ied Time Scale).I he e exis
mas e scale densi y κuni and weigh s gFRB, gδsuch ha o PSWF/DPSS ype window
unc ion amily {Wj}, windowed esiduals o FRB and δ- ing sa is y
RFRB(Wj) = λjRδ(Wj) + O(εj),
whe e λja e a ios p ecompu able om geome ic ac o s, when εjaccep able wi hin
expe imen al e o , hen phase– equency da a o FRB and δ- ing sca e ing consis en
wi h uni ied ime scale model.
Con e sely, i o some window unc ions Wjsys ema ic de ia ion exis s exceeding
e o ole ance, can de e mine uni ied ime scale model has inconsis ency in his equency
band and scale, need o co ec κuni o weigh model.
P oo ske ch. Using comple eness o PSWF/DPSS, expand κuni in window unc ion
space, w i e FRB and δ- ing esiduals as coe icien ec o s o same basis, es whe he
hey sa is y p especi ied linea ela ionship. See Appendix B.3 o de ails.
6 Join Va ia ion o Window Func ions and Con ol
Pa ame e s: Op imal C oss-Pla o m Me ology
S a egy
This sec ion inco po a es FRB and δ- ing window unc ion and con ol pa ame e choices
in o ime–in o ma ion–complexi y join a ia ional p inciple, gi es a ia ional o m o
“op imal c oss-pla o m me ology unde ini e complexi y budge ”.
6.1 Ex ended Join Mani old
P e ious join mani old EQ=M × SQ. Now in oduce wo ypes o addi ional deg ees o
eedom:
1. FRB window unc ion pa ame e space WFRB, e.g., spanned by linea coe icien s
o se e al PSWF modes;
2. δ- ing con ol pa ame e space Mδ={(L, αδ, θ)}and co esponding window unc-
ion space Wδ.
9
