Unified Time Scale and Time Geometry:\\ Equivalence, Domains, and Solvable Models of Spectral--Scattering--Causal--Entropy

Unied Time Scale and Time Geome y:
Equi alence, Domains, and Sol able Models o
Spec alSca e ingCausalEn opy
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 20, 2025
Abs ac
We p opose and igo ously cha ac e ize a unied ime scale amewo k, aligning
phase g adien eadings, ela i e s a e densi y, and ace o Wigne Smi h g oup
delay wi hin s ic sca e ing heo y domain, hus dening
ime scale
as mono onic
epa ame iza ion o a class o spec alsca e ing in a ian s. The iden i y
φ′(ω)
π=ρ el(ω) = 1
2πT Q(ω), Q(ω) = −i S(ω)†∂ωS(ω), φ =1
2a g de S
holds wi hin ene gy windows sa is ying
elas icuni a y sca e ing
and Bi man
K en assump ions; o
abso p i e/non-uni a y
and
long- ange po en ial
cases,
we p opose e iable gene aliza ions: in oducing
complex ime delay
,
dwell
ime
, and
phase eno maliza ion
, using Poissoncon olu ion o gi e exis ence
and ane uniqueness o
windowed clocks
. Pape u he cons uc s model-based
p oo o
eikonal phase de i a i e = geome ic Shapi o delay
in gene al
ela i i y end (Schwa zschild ex e io scala wa e, high- equency/high-angula -
momen um limi ), exp esses edshi as
phase hy hm a io
in cosmological end,
and s a es en opy ex emali y
→
geome ic equa ions as
condi ional p oposi-
ion
in in o ma ionholog aphy end wi h
ela i e en opy mono onici y and
QNEC
as co e assump ions. En i e ex emphasizes
domain o equi alence ela-
ions
and
sol able examples
, gi ing enginee ing- ealizable mul i- equency g oup
delay me ology and lensing delay in e sion schemes.
Keywo ds:
Wigne Smi h g oup delay; spec al shi unc ion; Bi manK en o mula;
eikonal phase; Shapi o delay; BondiSachs ime; TolmanEh en es edshi ; QNEC; gen-
e alized en opy
MSC 2020:
81U40, 47A40, 83C57, 83C45

1 In oduc ion and His o ical Con ex
G oup delay in oduced by Wigne and Smi h in elas ic sca e ing, dened as de i a-
i e o g oup phase wi h espec o equency; ace o i s ma ix o m
Q=−iS†∂ωS
1
equals de i a i e o o al sca e ing phase
Φ = a g de S
, hus xing expe imen al ead-
ing o  ime delay = phase g adien  as in a ian . On o he hand, Bi manK en o mula
connec s sca e ing de e minan wi h spec al shi unc ion
ξ
ia
de S(ω) = e−2πiξ(ω)
,
gi ing
1
2π∂ωΦ = −ξ′=ρ el
. This b idge es ablishes unica ion o phase slope ela i e
s a e densi yg oup delay ace.
In g a i y end, eikonal ampli ude me hod and geome ic op ics show:
eikonal phase
de i a i e wi h espec o ene gy/ equency
gi es
deec ion angle and ime
delay
(Shapi o delay). In cosmology, FRW edshi ela ion
1 + z=a( 0)/a( e)
can be
w i en as phase hy hm a io
(dϕ/d )e/(dϕ/d )0
. Fa -eld null inni y
BondiSachs
amewo k uses e a ded ime
u
o egula ize ou going null su ace, p o iding na u al
bounda y ime o g a i a ional sca e ing and phase eadings.
In in o ma ionholog aphy end,
ela i e en opy mono onici y
and
QNEC
ha e
been p o en in gene al QFT, QFC as conjec u e e ied in wide ange o cases; hese
inequali ies connec second-o de de o ma ion o
gene alized en opy
wi h ene gy con-
di ions, o ming condi ional ou e om en opy ex emali y o geome ic equa ions.
Goal o his pape is: wi hin
s ic domains
o ganize abo e b idges, gi e unied
clock scale co e ing
elas icnon-uni a y, sho - angelong- ange
cases, and con m
phase g adien = geome ic ime delay alignmen wi h
sol able models
.

2 Model and Assump ions
2.1 Sca e ing Pai and Spec al Shi F amewo k
Le
(H, H0)
be pai o sel -adjoin ope a o s sa is ying
ace-class/quasi- ace-class
pe u ba ion assump ion (e.g.,
H−H0∈S1
o
(H−i)−1−(H0−i)−1∈S1
). Then he e
exis s
spec al shi unc ion
ξ(λ)
such ha o sucien ly smoo h
T  (H)− (H0)=ZR
′(λ)ξ(λ) dλ.
I absolu ely con inuous spec al ene gy window
I⊂R
has wa e ope a o s exis ing
and sca e ing ma ix
S(ω)
die en iable and
uni a y
, hen Bi manK en o mula
de S(ω) = e−2πi ξ(ω)
holds and con inuous b anch o
Φ(ω) = a g de S(ω)
can be chosen.
Deni ion 2.1
(Rela i e S a e Densi y (Deni ion 2.1))
.
Deno e
ρ el(ω) := −ξ′(ω)
. A
Lebesgue-a.e. poin s in
I
ha e
1
2π∂ωΦ(ω) = ρ el(ω).
Domain ema k:
Abo e equali y may hold only in dis ibu ional o bounded a i-
a ion (BV) sense a
h esholds, bound s a es, and esonance poin s
; b anch o
Φ
xed join ly by analy ic con inua ion o
S(ω)
and a -eld no maliza ion (Appendix A).
2
2.2 Wigne Smi h G oup Delay
Fo uni a y
S(ω)
dene
Q(ω) = −i S(ω)†∂ωS(ω),
hen
Q
sel -adjoin , and
ace iden i y
∂ωΦ(ω) = T Q(ω)
holds in
I
, hus
φ′(ω)
π=ρ el(ω) = 1
2πT Q(ω), φ =1
2Φ.
This is scale iden i y in
elas icuni a y
domain.
Coun e example and lowe bound:
G oup delay can ake
nega i e
alues nea
an i- esonances (anomalous delay); bu Wigne causali y gi es
lowe bound
on ene gy
de i a i e and o e all sum cons ain . This pape ob ains weak mono onici y and ane
uniqueness unde
windowed clocks
(4.2, Appendix B).
2.3 Non-Uni a y/Abso p i e and Gene alized Time Delay
When ex e nal isible channels incomple e o abso p ion exis s (black hole ho izon, lossy
media, open ca i ies),
S
non-uni a y. Take
Qgen(ω) := −i S(ω)−1∂ωS(ω),
whose ace gene ally complex; can dene
eal pa
as gene alized Wigne delay,
imag-
ina y pa
ela ed o abso p ion/gain; can also in oduce dwell ime and ansmission
eec ion decomposi ion. This pape in 4.3 gi es ela ionship wi h
∂ωa g de S
and
me ological meaning.
2.4 Long-Range Po en ial and Phase Reno maliza ion
Fo Coulomb/g a i y
1/
long- ange po en ials, need use
modied wa e ope a o s
and
phase eno maliza ion
(Dolla d/IsozakiKi ada ype), emo ing loga i hmic e ms in
asymp o ic phase. This pape o Schwa zschild ex e io scala wa e unde
o oise
coo dina es
and ReggeWheele equa ion cons uc s
eno malized phase
Φ en(ω)
,
p o ing
∂ωΦ en(ω)=∆TShapi o(ω) + o(1)
holds in high- equency/high-angula -momen um limi (5, Appendix D).
2.5 Geome y and Bounda y Time
Local clock a e/ edshi in s a ic space ime con olled by
g
o TolmanEh en es law;
lapse
N
in ADM decomposi ion gi es a io o coo dina e ime o p ope ime; emo e
bounda y
BondiSachs e a ded ime
u
p o ides na u al sca e ing ime a null
inni y.
3
2.6 In o ma ionHolog aphy Assump ion Domain
Rela i e en opy mono onici y and
QNEC
hold in gene al QFT; QFC as conjec u e
p o ides s onge s uc u e. This pape s a es en opy ex emali y
→
eld equa ions
as
condi ional p oposi ion
, asse ing only unde small causal diamonds, Hadama d
s a es, weak cu a u e, and app op ia e de o ma ion classes (6, Appendix F).

3 Main Resul s (Theo ems and Alignmen s)
3.1 Domain Theo em o Scale Iden i y
Theo em 3.1
(Elas icUni a y Domain (Theo em 3.1))
.
Le
(H, H0)
be sel -adjoin sca -
e ing pai sa is ying 2.1 ace-class assump ion. Le
I⊂R
be absolu ely con inuous
spec al ene gy window,
S(ω)∈C1(I;U(N(ω)))
wi h isola ed se
Σ⊂I
o h esholds and
esonances absen . Then in
I Σ
ha e
φ′(ω)
π=ρ el(ω) = 1
2πT Q(ω)
(Lebesgue-a.e.)
.
On
Σ
his equali y holds in BV/dis ibu ional sense, jumps o
Φ
wi h bound s a e
esonance con ibu ions gi en by Le inson/F iedel in eg al (Appendix A).
P oo :
See Appendix A (Bi manK en + ace iden i y + die en iabili y and b anch
choice).
No e (long- ange eno maliza ion):
I po en ial long- ange, hen he e exis s
eno malized phase
Φ en
such ha iden i y holds a e eno maliza ion; p oo in Appendix
D.1 (Dolla d/IsozakiKi ada amewo k).
3.2 Exis ence and Ane Uniqueness o Windowed Clocks
Deni ion 3.1
(PoissonWindowed Clock (Deni ion 3.2))
.
Take Poisson ke nel o wid h
∆>0
P∆(x) = 1
π
∆
x2+ ∆2,ZR
P∆(x) dx= 1.
Dene
windowed scale densi y
Θ∆(ω) := ρ el ∗P∆(ω) = 1
2πT Q∗P∆(ω)
and
clock
∆(ω)− ∆(ω0) = Zω
ω0
Θ∆(˜ω) d˜ω.
Theo em 3.2
(Weak Mono onici y and Ane Uniqueness (Theo em 3.3))
.
I
S
analy ic
in uppe hal -plane wi h no uppe hal -plane poles, and
∆
o cons an o de la ge han
minimum esonance wid h/spacing wi hin gi en ene gy window, hen
Θ∆(ω)>0
holds in
measu e sense, hus
∆
s ic ly inc easing; i
˜
∆
is clock gi en by ano he window amily
sa is ying same window condi ion, hen he e exis
a > 0, b ∈R
such ha
˜
∆=a ∆+b.
4
P oo key poin s:
log de S
is Ne anlinnaHe glo z ype unc ion, whose bounda y
imagina y pa is dis ibu ion
−2πξ′
; Poisson smoo hing gi es ha monic con inua ion
and supp esses oscilla ion e ms o local nega i e delay; window wid h condi ion ensu es
posi i e ma gin co e s an i- esonance nega i e lobes (Appendix B; coun e examples and
nume ics in 5.3).
Commen :
This heo em esponds o ac ha g oup delay can be locally nega i e:
clock d i en by windowed s a e densi y
, sa is ying weak mono onici y and ane
uniqueness, no poin wise mono onici y.
3.3 Gene alized Iden i y o Non-Uni a y/Abso p i e
P oposi ion 3.3
(Gene alized Time Delay and Phase (P oposi ion 3.4))
.
Fo non-
uni a y
S
dene
Qgen =−iS−1∂ωS
. Then
∂ωlog de S(ω) = iT Qgen(ω), ∂ωa g de S=ℜT Qgen,
can dene
eal delay
τRe := (1/2π)ℜT Qgen
and
abso p ion a e
α:= (1/2π)ℑT Qgen
.
In small abso p ion limi
|S†S−1| ≪ 1
ha e
τRe = (2π)−1T Q+O(|S†S−1|)
.
3.4 Eikonal Phase and Geome ic Shapi o Delay
Theo em 3.4
(High-F equency/High-
l
Limi (Theo em 3.5))
.
Reno malized phase
Φ en(ω)
o Schwa zschild ex e io scala wa e ( equency
ω
) sa ises in eikonal limi
∂ωΦ en(ω) = ∆TShapi o(ω) + O(ω−1),
whe e
∆TShapi o
is Shapi o delay o geome ic ay pa h.
P oo :
See 5 (WKB phase die ence = ac ion die ence, using o oise coo dina es
and high- equency decomposi ion o ReggeWheele po en ial; phase b anch no malized
wi h eld- ee e e ence).
3.5 Redshi = Phase Rhy hm Ra io and Bounda y Time
Unde FRW me ic, ime de i a i e o pho on phase
ϕ
p opo ional o obse ed equency,
ob aining
1 + z=νe
ν0
=(dϕ/d )e
(dϕ/d )0
=a( 0)
a( e),
his o mula unies cosmological edshi as
bounda y phase hy hm a io
.
3.6 En opy Ex emali y
→
Geome ic Equa ions: Condi ional
P oposi ion
P oposi ion 3.5
(Condi ional (P oposi ion 3.6))
.
Unde small causal diamond limi ,
Hadama d s a e, weak cu a u e, and app op ia e de o ma ion class, i assuming ela i e
en opy mono onici y and
QNEC
, hen second-o de de o ma ion o gene alized en opy
combined wi h Raychaudhu i equa ion yields
Rµν −1
2Rgµν + Λgµν = 8πG ⟨Tµν⟩.
5

Explana ion:
QFC no uni e sal heo em, his pape does no use i as sucien
condi ion; p oposi ion only holds unde abo e assump ions and local window, echnically
suppo ed by Jacobson equa ion o s a e and subsequen JLMS/de o ma ion modula
Hamil onian (Appendix F).

4 P oo s (Summa y; De ails in Appendices)
4.1 Theo em 3.1
Bi manK en gi es
de S=e−2πiξ
; die en ia ing wi h espec o
ω
gi es
Φ′=−2πξ′=
2πρ el
. On o he hand
T Q=∂ωT log S=∂ωΦ
. Combining gi es iden i y; unde s ood
as BV/dis ibu ion a h esholds and esonances (Appendix A).
4.2 Theo em 3.3
log de S
is Ne anlinnaHe glo z unc ion; i s bounda y imagina y pa is dis ibu ion
−2πξ′
. Poisson smoo hing equals bounda y alue o ha monic con inua ion o uppe
hal -plane; choosing
∆
la ge han minimum esonance wid h, local uc ua ions o neg-
a i e delay co e ed by posi i e en elope, hus
Θ∆>0
a.e.; ane uniqueness om uni
no maliza ion and addi i e cons an eedom (Appendix B). Coun e examples (nega i e
delay) and window h eshold quan i a i ely shown in one-dimensional sol able po en ials
(5.3).
4.3 P oposi ion 3.4
Fo in e ible
S
use Jacobi iden i y
∂ωlog de S= T (S−1∂ωS) = iT Qgen
. Taking eal
and imagina y pa s gi es s a emen ; small abso p ion expansion in Appendix C.
4.4 Theo em 3.5
In Schwa zschild ex e io , exp ess ansmission/ eec ion phase using WKB solu ion o
ReggeWheele equa ion; a high equency/high
l
phase die ence equals geome ic ac-
ion die ence,
∂ω
gi es Shapi o delay; long- ange phase ea ed wi h o oise coo dina es
and e e ence phase eno maliza ion (Appendix D).
4.5 P oposi ion 3.6
Rela i e en opy mono onici y gi es linea ela ionship be ween modula Hamil onian
and ene gy-momen um enso ; QNEC ela es lowe bound o second-o de de o ma ion
o gene alized en opy wi h
Tkk
, combined wi h Raychaudhu i equa ion and ex emali y
condi ion yields enso o m in each null di ec ion;
Λ
as in eg a ion cons an (Appendix
F).
6
5 Model Applica ions
5.1 Schwa zschild Ex e io :
∂ωΦ
and Shapi o Delay
S a ing om ReggeWheele equa ion, cons uc eikonal solu ion and phase eno maliza-
ion
Φ en
, nume ical/asymp o ic compa ison shows
∂ωΦ en(ω)
consis en wi h geome ic
∆TShapi o
(de ia ion
O(ω−1)
). P o ides end- o-end chain om
wa e equa ion
→
S-
ma ix
→
phase de i a i e
→
geome ic ime delay
.
5.2 Lensing:
∂ω(Φi−Φj) = ∆ ij
De i a i e o phase o Ki chho in eg al amplica ion ac o
F(ω)
wi h espec o
ω
gi es
Fe ma a i al ime delay; in hin lens limi wi h poin mass/SIS model ob ains unied
equency-domain ime-domain  ing o mul i-image ime delays.
5.3 One-Dimensional Sol able Po en ial and Nega i e Delay
Choose sol able po en ial con aining an i- esonance, showing local nega i e alues and
sum ule o
T Q(ω)
; e i y weak mono onici y c i ical wid h o
windowed clock
wi h
∆
as a iable. Re e ence Win ul's e iew on Ha man/anomalous delay and elec omag-
ne ic/acous ic ex ensions.

6 Enginee ing P oposals
1.
Mul i- equency Shapi og oup delay pa allel in e sion:
Measu e phase
Φ(ω)
in plane a y occul a ion geome y, compu e
∂ωΦ
pa allel decon olu ion wi h
co onal plasma dispe sion, combined wi h hyd ogen clock and s able link gi es
absolu e phase e e ence
.
2.
On-chip Wigne Smi h omog aphy me ology:
Cons uc
Q=−iS†∂ωS
in
mul i-po S-pa ame e me ology, use ace in a iance o de ice ole ance in e -
sion and g oup delay imaging.
3.
Wa e lensing b oadband ime delay spec um:
Fi mul i-image a i al ime
delays and dispe sion using
∂ωΦ
, educing ime delay cosmology sys ema ic e o s.

7 Discussion (Risks, Bounda ies, Pas Wo k)

Domain and egula i y:
Scale iden i y clea es unde
elas icuni a y
and
sho - ange
classes; needs BV/dis ibu ional unde s anding a h esholds/ esonances;
long- ange po en ials need eno maliza ion.

Nega i e delay and windowing:
G oup delay can be locally nega i e;
Poisson
windowing
p o ides weakly mono onic clock. Sucien condi ions and minimum
window wid h o his cons uc ion depend on esonance spec um.
7

Non-uni a y gene aliza ion:
In abso p i e/open sys ems,
ℜT Qgen
gi es mea-
su able  eal delay,
ℑT Qgen
measu es abso p ion; quan i a i e ela ionship wi h
dwell ime/ene gy s o age exis s.

Geome y end:
Eikonalgeome ic op ics connec ion mos di ec in s a ic/weak
elds; s ong elds and o a ion need mo e ened cohe en anspo and nume ical
ay acing.

In o ma ionholog aphy:
This pape a oids ea ing QFC as heo em, only gi -
ing condi ional p oposi ion unde QNEC and ela i e en opy mono onici y.

8 Conclusion
Wi hin s ic sca e ing domain, his pape denes
ime scale
as mono onic epa ame iza-
ion o spec alsca e ing in a ian , co e objec being
φ′(ω)
π≡ρ el(ω)≡1
2πT Q(ω).
We speci y i s
domain
(elas icuni a y, sho - ange, ene gy windows away om
h esholds/ esonances) and
gene aliza ions
(non-uni a y/abso p i e, phase eno mal-
iza ion o long- ange po en ials), p opose
Poissonwindowed clock
p o ing weak
mono onici y and ane uniqueness, gi e end- o-end model-based p oo o
eikonal phase
Shapi o delay
in Schwa zschild ex e io , and w i e cosmological edshi as
phase
hy hm a io
. In in o ma ionholog aphy end, s a e condi ional p oposi ion o en opy
ex emali y
→
geome ic equa ions based on QNEC/ ela i e en opy mono onici y. Thus
o ming
unied ime geome y
om spec alsca e ing o causalen opy.

Acknowledgemen s, Code A ailabili y
Thanks o public ex books and pape s; phase eno maliza ion and Schwa zschild eikonal
nume ical sc ip s, windowed clock demons a ion, and g oup delay cu e  ing code o
one-dimensional po en ials a ailable upon eques .

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A Rigo ous Domain o Scale Iden i y (Elas icUni a y,
Sho -Range)
A.1 SSF and Bi manK en
Unde
H−H0∈S1
o esol en die ence ace-class, spec al shi unc ion
ξ
exis s
sa is ying ace o mula and
de S(ω) = e−2πi ξ(ω).
Choosing con inuous b anch sa is ying
a g de S(ω)→0
(
|ℑω| → ∞
), ob ain
Φ(ω) =
−2πξ(ω) mod 2π
. A.e. de i a i e wi h espec o
ω
gi es
1
2πΦ′(ω) = ρ el(ω), ρ el =−ξ′.
Unde s ood as BV/dis ibu ion a h eshold/ esonance poin s
Σ
; Le inson/F iedel in e-
g al con ols
Rρ el
and bound s a e coun ing.
9