Unified Time Scale and Boundary Time Geometry:\\ Single Structural Framework of Scattering Phase, Modular Flow, and Gravitational Boundary Terms

Unied Time Scale and Bounda y Time Geome y:
Single S uc u al F amewo k o Sca e ing Phase,
Modula Flow, and G a i a ional Bounda y Te ms
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 cons uc a ime unica ion amewo k wi h bounda y as undamen al s age,
in eg a ing h ee o iginally sepa a e ime s uc u es in o die en p ojec ions o
he same bounda y ime geome y: (1) On sca e ing and spec al heo y end,
based on Bi manK en o mula and Wigne Smi h ime delay, p o e scale iden i y
among o al sca e ing phase de i a i e, ela i e s a e densi y, and g oup delay
ace; (2) On ope a o algeb a and in o ma ion end, based on Tomi aTakesaki
modula heo y and ConnesRo elli he mal ime hypo hesis, cha ac e ize modula
ow pa ame e as in insic ime de e mined by s a ealgeb a pai , in oducing ime
scale equi alence class; (3) On g a i y and geome y end, based on Eins einHilbe 
GibbonsHawkingYo k ac ion and i s bounda y a ia ion, uni y ex insic cu a u e
and ime ansla ion gene a ed by bounda y Hamil onian in o same bounda y ime
geome y.
In unied model, bounda y is desc ibed by iple s uc u e: in insic me ic and
ex insic cu a u e o geome ic bounda y
∂M
, quan um bounda y algeb a
A∂
wi h
s a e
ω
, and sca e ing ma ix
S(ω)
dened in ex e nal egion. Unde well-posed
aceable sca e ing assump ions, cons uc scale iden i y
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
whe e
φ(ω) = 1
2a g de S(ω)
is o al sca e ing phase,
ρ el
is de i a i e o spec al
shi densi y,
Q(ω) = −iS(ω)†∂ωS(ω)
is Wigne Smi h ime delay ma ix. This
scale is s anda dized a modula ime and geome ic ime ends espec i ely h ough
modula Hamil onian ope a o
Kω=−log ∆ω
and Hamil onJacobi unc ional o
GHY bounda y ac ion, hus dening single bounda y ime scale equi alence class
[τ]
.Based on his, we gi e se e al unica ion heo ems: (i) Ca ego ical exis ence
uniqueness o ime scale equi alence class: on common domain o gi en bounda y
algeb a, sca e ing da a, and g a i a ional bounda y geome y, all accep able ime
pa ame e s a e mono onic escalings o one undamen al bounda y ime; (ii) Cosmo-
logical edshi ela ion
1 + z= 1/a( )
can be in e p e ed as global escaling o his
ime equi alence class on la ge scales, hus uni ying local sca e ing ime delay wi h
con o mal ime in FRW backg ound; (iii) In cases wi h ho izons (Rindle wedge and
1
black hole ex e io ), modula ow ime, p ope ime o accele a ed obse e , and
geome ic ou wa d no mal ansla ion ime all in o same equi alence class.
Full ex gi es explici assump ions and heo ems a sca e ingspec al, modula
owin o ma ion, and g a i ybounda y geome y ends espec i ely, wi h de ailed
p oo s o scale iden i y, modula ime equi alence, and bounda y Hamil onian gene -
a ed ime in appendices, nally p oposing enginee ed measu emen schemes based
on wa eguides, mic owa e ca i ies, and Aha ono Bohm ings o c oss-calib a e
h ee ypes o ime scales expe imen ally.
Keywo ds:
Bounda y Time Geome y; Bi manK en Fo mula; Wigne Smi h Time
Delay; Spec al Shi Func ion; Tomi aTakesaki Modula Flow; The mal Time Hypo h-
esis; GibbonsHawkingYo k Bounda y Te m; Time Scale Equi alence Class; Cosmolog-
ical Redshi

1 In oduc ion and His o ical Con ex
Time appea s in physical heo ies in mul iple guises: as coo dina e pa ame izing wo ld-
lines in classical and ela i is ic physics, as con inuous a iable gene a ing uni a y e olu-
ion in quan um mechanics, as imagina y ime pe iod in e sely p opo ional o empe a-
u e in s a is ical physics, as delay o pa icle dwelling in in e ac ion egion in sca e ing
heo y, and in gene al ela i i y eec ing as geome ic s uc u e h ough me ic and
ex insic cu a u e. Die en ime pe spec i es o en based on die en undamen al
objec s and measu emen p ocesses, hus dicul o uni y wi hin single ma hema ical
amewo k o long ime.
In sca e ing and spec al heo y aspec s, Li shi sK en spec al shi unc ion
ξ(λ)
and i s de i a i e play cen al ole in desc ibing s a e densi y die ence be o e and a e
in e ac ion; Bi manK en o mula gi es p ecise ela ionship be ween sca e ing ma ix
de e minan and spec al shi unc ion
de S(λ) = exp(−2πiξ(λ))
. Unde app op ia e
no maliza ion, de i a i e o spec al shi densi y can be in e p e ed as o al sca e ing
phase de i a i e and equi alen o ace o Wigne Smi h ime delay ma ix, hus iewing
 ime delay as geome ic de i a i e o phasespec al s uc u e.
In ope a o algeb a and quan um s a is ics aspec s, Tomi aTakesaki heo y shows:
gi en on Neumann algeb a
(M, Ω)
wi h cyclic and sepa a ing ec o , can cons uc
modula au omo phism g oup
σΩ
(x)=∆i x∆−i
h ough pola decomposi ion o modula
ope a o
∆
, whose image on ou e au omo phism g oup independen o chosen s a e.
ConnesRo elli he mal ime hypo hesis p oposes: in gene ally co a ian quan um heo y,
physical ime ow de e mined by modula ow o s a ealgeb a pai , modula pa ame e
i sel is ime, hus  ime becomes de i ed objec o s a e s uc u e a he han a p io i
pa ame e .
In g a i y and geome y aspec s, Eins einHilbe ac ion on mani old wi h bounda y
insucien o gi e well-dened a ia ional p inciple, mus add GibbonsHawkingYo k
bounda y e m
SGHY =1
8πG Z∂M
d3y ϵ√h K,
whe e
hab
is induced bounda y me ic,
K
is ace o ex insic cu a u e,
ϵ=±1
depends
on no mal ype. Va ia ion o EH+GHY combined ac ion unde xed bounda y induced
geome y gi es Eins ein equa ions, bounda y e m can be iewed as Hamil onJacobi
2
unc ional, whose unc ional de i a i e wi h espec o bounda y me ic co esponds o
conjuga e momen um and quasilocal ene gy, hus encoding  ime gene a o  o ansla ion
along bounda y no mal.
S uc u al equi alences among sca e ing phase de i a i e ime delay, modula ow
pa ame e  he mal ime, GHY bounda y e mgeome ic ime a e each highly ma u e
ma hema ically and physically meaning ul, bu hei s uc u al equi alence only appea s
sca e ed in exis ing li e a u e. Fo example, sca e ing phase de i a i e can be in e -
p e ed bo h as s a e densi y die ence and as g oup delay h ough equency de i a i e o
S-ma ix; in S-ma ix s a is ical mechanics, sca e ing phase de i a i e en e s s a e den-
si y in eg al, hus connec ing wi h hea and ee ene gy. Modula ow used in AdS/CFT
and en anglemen wedge econs uc ion o dene modula Hamil onian and ene gy o
geome ic egion, connec ing o p opaga ion ime in space ime h ough HKLL/Pe z e-
cons uc ion. GHY bounda y e m iewed as key objec dening g a i a ional ans e
ampli ude and bounda y ime in loop quan um g a i y and quasilocal Hamil onian o -
malism.
Goal o his pape is on igo ously p o able basis o uni y abo e h ee ends in o bound-
a y ime geome y amewo k. This amewo k akes bounda y algeb a, s a e, sca e -
ing ma ix, and bounda y geome y as undamen al objec s, wi h  ime scale equi alence
class as co e, p o ing: unde app op ia e assump ions, sca e ing ime delay, modula
ime, and geome ic bounda y ime belong o same equi alence class, any physical ime
eading can be iewed as mono onic escaling o single bounda y ime pa ame e . Fu -
he mo e, inco po a ing cosmological edshi and scale ac o e olu ion in FRW space-
ime in o same scale s uc u e, connec ing mac oscopic cosmic ime wi h mic oscopic
sca e ing ime delay.

2 Model and Assump ions
This sec ion gi es ma hema ical and physical model suppo ing unied amewo k, ex-
plici ly speci ying assump ion domain.
2.1 Sca e ing and Spec al End
Conside sel -adjoin ope a o pai
(H, H0)
ac ing on sepa able Hilbe space
H
, sa is ying
ollowing s anda d sca e ing assump ions:
(1)
H0
possesses absolu ely con inuous spec al subspace
Hac(H0)
, in which ene gy
ep esen a ion can be es ablished, making
H0
mul iplica ion ope a o
E7→ E
in
ha ep esen a ion;
(2) Pe u ba ion
V=H−H0
such ha o some
p≤1
,
(H+i)−p−(H0+i)−p∈S1
,
hus sa is ying basic condi ion o ace-class sca e ing heo y;
(3) Wa e ope a o s
W±= s
-
lim →±∞ ei He−i H0Pac(H0)
exis and a e comple e, hus
sca e ing ope a o
S=W†
+W−
well-dened on
Hac(H0)
.
In ene gy ep esen a ion,
S
be s in o amily o uni a y ma ices
S(ω)
, whe e
ω
deno es ene gy o equency a iable. Assume o almos e e ywhe e
ω
,
S(ω)−⊮∈S1
,
3
hus de e minan
de S(ω)
and Wigne Smi h ime delay ope a o
Q(ω) = −iS(ω)†∂ωS(ω)
well-dened and aceable.
Dene spec al shi unc ion
ξ(ω;H, H0)
as unc ion sa is ying Li shi sK en ace
o mula, whose de i a i e
ξ′(ω)
gi es ela i e s a e densi y die ence:
ρ el(ω) := −ξ′(ω)
.
2.2 Modula Flow and The mal Time End
Le
M⊂B(H)
be on Neumann algeb a,
Ω∈ H
be cyclic and sepa a ing ec o ,
|Ω|= 1
.
Deno e ec o s a e
ω(x) = (xΩ,Ω)
. Tomi aTakesaki heo y gi es pola decomposi ion
S=J∆1/2
o closed ope a o
S
, whe e
J
is modula conjuga ion,
∆
is modula ope a o .
Modula au omo phism g oup dened as
σω
(x) = ∆i x∆−i , ∈R.
σω
is one-pa ame e au omo phism g oup o
M
, and
ω
is i s KMS s a e.
Connes p o ed: o any wo ai h ul s a es
ω, ω′
, hei modula ows' images in ou e
au omo phism g oup
Ou (M)
a e consis en , i.e., he e exis s 1cocycle
u ∈M
such ha
σω′
= Ad(u )◦σω
, hus ob aining s a e-independen geome ic ime on
Ou (M)
.
The mal ime hypo hesis p oposes: physical ime ow de e mined by modula ow,
i.e., modula pa ame e
i sel is ime scale, a he han gi en by a p io i backg ound.
2.3 G a i y and Bounda y Geome y End
Conside ou -dimensional space ime mani old
(M, gµν)
wi h bounda y
∂M
. G a i a-
ional ac ion chooses Eins einHilbe plus GibbonsHawkingYo k sum
Sg a =1
16πG ZM
d4x√−g R +1
8πG Z∂M
d3y ϵ√h K.
Assume bounda y is smoo h h ee-dimensional mani old o non-ze o measu e, dis inguish-
ing spacelike and imelike bounda ies. When a ying while keeping bounda y induced
me ic
hab
xed, olume e m gi es Eins ein equa ions, bounda y e m a ia ion de e -
mines bounda y conjuga e momen um and quasilocal ene gy.
In some cases (such as acuum egion, pa ial LQG cons uc ion), olume e m an-
ishes on shell, Hamil onJacobi ac ion comple ely gi en by GHY bounda y e m, hus
bounda y ac ion i sel becomes objec gene a ing no mal ime e olu ion.
2.4 Unied Bounda y Sys em and Time Scale Equi alence
We call iple da a
B= (A∂, ω∂, S(ω); hab, Kab)
a bounda y sys em, whe e
A∂
is bounda y algeb a gene a ed by sca e ing channels and
nea -bounda y elds,
ω∂
is ai h ul s a e on i (can be gi en by sca e ing incoming s a e
o bounda y CFT s a e),
S(ω)
is sca e ing ma ix on ene gy shell,
hab, Kab
a e in insic
and ex insic da a o geome ic bounda y.
On bounda y sys em, we allow h ee ypes o one-pa ame e e olu ion:
4
(1) Phasedelay scale on sca e ing ene gy pa ame e
ω
, de e mined by
S(ω)
and
Q(ω)
;
(2) Modula pa ame e
mod
, gene a ed by modula ow o
(A∂, ω∂)
;
(3) Geome ic pa ame e
geom
, gene a ed by ansla ion along bounda y no mal (o
e olu ion d i en by ex insic cu a u e).
Co e assump ion o unied amewo k is: unde app op ia e physical si ua ions (such
as asymp o ically a o AdS space ime inni e bounda y, Rindle wedge nea black hole
ho izon, con o mal bounda y o FRW uni e se), abo e h ee ypes o pa ame e s can all
be dened and sa is y common equi alence ela ion, o ming ime scale equi alence class
[τ]
.
3 Main Resul s (Theo ems and Alignmen s)
This sec ion s a es main heo ems and co espondences o unied amewo k, s ic ly
dis inguishing known esul s om newly in oduced s uc u es.
Theo em 3.1
(Sca e ing PhaseSpec al Shi G oup Delay Scale Iden i y (Theo em
1))
.
Unde a o emen ioned sca e ing assump ions, le spec al shi unc ion be
ξ(ω;H, H0)
,
dene ela i e s a e densi y
ρ el(ω) := −ξ′(ω).
Le o al sca e ing phase
Φ(ω) := a g de S(ω), φ(ω) := 1
2Φ(ω),
Wigne Smi h delay ope a o
Q(ω) = −iS(ω)†∂ωS(ω).
Then o almos e e ywhe e
ω
, scale iden i y holds
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω).
B ie : Fi s equali y om Bi manK en o mula
de S(ω) = exp(−2πiξ(ω))
and e-
la ionship be ween spec al shi unc ion and s a e densi y; second equali y om con-
nec ion be ween de i a i e o
ln de S(ω)
wi h espec o equency and Wigne Smi h
ope a o ace.
Theo em 3.2
(Time Scale Equi alence Class o Modula Flow (Theo em 2))
.
Le
(M, Ω)
be on Neumann algeb a wi h cyclic sepa a ing ec o as abo e,
ω, ω′
be wo ai h ul s a es,
dening modula ows
σω
, σω′
espec i ely.
(1) The e exis s amily o uni a y ope a o s
u ∈M
sa is ying 1cocycle condi ion
u +s=u σω
(us)
, such ha
σω′
= Ad(u )◦σω
.
5

(2) In ou e au omo phism g oup
Ou (M)
,
[σω
] = [σω′
]
, ime pa ame e
scale only
unde goes linea escaling when s a e changes.
(3) I he e exis s geome ic clock (such as ine ial o uni o mly accele a ed obse e )
whose p ope ime
τphys
ela es o modula pa ame e
mod
h ough KMS empe a u e
β
(such as Un uh empe a u e
T=a/2π
o he mal equilib ium s a e), hen
τphys =
α mod
, whe e
α
de e mined by
β
. The e o e modula pa ame e denes ime scale
equi alence class
[τmod]
.
Theo em 3.3
(GHY Bounda y Ac ion and Geome ic Time (Theo em 3))
.
Unde Eins ein
Hilbe GHY ac ion, o gi en bounda y
∂M
egion
Σ
, dene Hamil onJacobi unc ional
SHJ[hab] = 1
8πG ZΣ
d3y ϵ√h K,
in case whe e acuum Eins ein equa ion holds and olume e m anishes on shell, a i-
a ion o his unc ional wi h espec o bounda y induced me ic
hab
gi es conjuga e mo-
men um
πab =δSHJ
δhab
=ϵ
16πG√h(Kab −Khab),
dening quasilocal ene gy densi y and gene a o o ansla ion along bounda y no mal.
I choosing imelike di ec ion and uni no mal on bounda y, decomposing ex insic
cu a u e as
K=K +Kspa ial
, hen he e exis s geome ic ime pa ame e
geom
such
ha o small ime ansla ion
δ geom
, ac ion change sa ises
δSHJ =Eq.l.δ geom,
whe e
Eq.l.
is quasilocal ene gy, hus
geom
uniquely de e mined by bounda y geome y and
g a i a ional ac ion, o ming geome ic ime scale.
Deni ion 3.4
(Bounda y Time Scale Equi alence Rela ion (Deni ion 4))
.
On bound-
a y sys em
B
, conside h ee ypes o ime pa ame e s: sca e ing ime pa ame e
sca
(e.g., in eg al o Wigne Smi h delay), modula pa ame e
mod
, and geome ic ime
geom
.
Dene equi alence ela ion
∼
: i he e exis
C1
s ic ly mono onic unc ion
such
ha
sca = sm( mod), mod = mg( geom),
and de i a i e a
0
ni e and non-ze o, hen h ee belong o same ime scale equi alence
class, w i en
[τ]
.
Theo em 3.5
(Exis ence and (Local) Uniqueness o Time Scale Equi alence Class (The-
o em 4))
.
Le bounda y sys em
B
sa is y:
(1) Sca e ing side sa ises Theo em 1 assump ions, dening g oup delay ace scale
dτsca (ω) = 1
2π Q(ω) dω;
(2) Ope a o algeb a side sa ises Theo em 2 assump ions, he e exis s modula ow
σω
wi h co esponding modula Hamil onian ope a o
Kω=−log ∆ω
;
(3) G a i y side sa ises Theo em 3 assump ions, he e exis s bounda y Hamil on
Jacobi unc ional
SHJ
, whose pa ame e o imelike ansla ion is
geom
.
6
And assume in some ene gy window and geome ic egion he e exis s AdS/CFT o
sca e inggeome y co espondence, such ha s uc u e-p ese ing isomo phism exis s
be ween sca e ing channels and bounda y algeb ageome y (such as co espondence be-
ween Rindle wedge o sphe ical egion and i s CFT modula Hamil onian).
Then on his common domain he e exis s ime scale equi alence class
[τ]
sa is ying:
(1) Fo any obse a ion p ocess, i s ime eading
obs
is
C1
mono onic unc ion o
τ
;
(2) I in oducing ano he ime pa ame e
˜
and equi ing i s uni in e al equi alen o
uni changes o sca e ing phase de i a i e, modula Hamil onian expec a ion alue,
and GHY bounda y ac ion, hen
˜
mus locally be linea escaling o
τ
, he e o e
[τ]
locally unique.
Theo em 3.6
(Cosmological Redshi as Global Rescaling o Time Scale (Theo em 5))
.
In spa ially iso opic, homogeneous FRW uni e se, me ic can be w i en as
ds2=−d 2+a( )2γijdxidxj,
whe e
a( )
is scale ac o . In oduce con o mal ime
η
sa is ying
d =a(η)dη
.
Le he e be adia ion geodesic be ween sou ce and obse e a gi en edshi
z
, e-
quency edshi ela ion
1 + z=1
a( em),
whe e
em
is emission ime, a obse a ion
a( 0)=1
.
I iewing sou ceobse e sys em sca e ing as eec i e  a - egion sca e ing on cos-
mological backg ound, hen he e exis s bounda y ime scale
τ
such ha local obse e 's
con o mal ime inc emen
dη
and Wigne Smi h delay scale
dτsca
belong o same equi -
alence class, while cosmological edshi mani es s as global escaling o
τ
dτcosmo = (1 + z) dτlocal,
hus mac oscopic cosmic ime e olu ion can be iewed as scale ac o e olu ion o unied
ime scale equi alence class.

4 P oo s
This sec ion gi es p oo skele ons o main heo ems, comple e echnical de ails placed in
appendices.
4.1 P oo o Theo em 1
Bi manK en o mula gi es
de S(ω) = exp(−2πiξ(ω)).
Le
Φ(ω) := a g de S(ω)
, hen he e exis s con inuous b anch choice such ha
Φ(ω) = −2πξ(ω).
7
Dene hal -phase
φ(ω) = 1
2Φ(ω)
, hen
φ′(ω) = −πξ′(ω) = πρ el(ω).
Thus  s equali y holds.
On o he hand, using
ln de S(ω) = ln S(ω)
and chain ule,
∂ωln de S(ω) = (S(ω)−1∂ωS(ω)).
By uni a i y o
S(ω)
,
S(ω)−1=S(ω)†
. W i ing Wigne Smi h ope a o as
Q(ω) = −iS(ω)†∂ωS(ω),
hen
∂ωln de S(ω) = i Q(ω).
On o he hand, by Bi manK en o mula
∂ωln de S(ω) = −2πiξ′(ω) = 2πiρ el(ω),
compa ing wo exp essions yields
i Q(ω)=2πiρ el(ω)⇒ρ el(ω) = 1
2π Q(ω).
Combining wi h a o emen ioned
φ′/π =ρ el
gi es scale iden i y. Technical equi emen s
o abo e de i a ion (such as ace-class condi ions, loga i hm b anch choice) igo ously
e ied in Appendix A.
4.2 P oo o Theo em 2
(1) and (2) a e s anda d conclusions o Tomi aTakesaki heo y and Connes modula
cobounda y heo y: h ough closu e and pola decomposi ion o
S0:mΩ7→ m∗Ω
con-
s uc modula ope a o
∆
, hen using
∆i
ealize modula ow, can p o e o any wo
ai h ul s a es, modula ows' images in
Ou (M)
a e consis en .
(3) When modula ow desc ibes ime e olu ion o KMS s a e, KMS condi ion con-
nec s modula pa ame e
mod
wi h empe a u e
β−1
. In Un uh eec , obse e wi h
accele a ion
a
expe iences empe a u e
T=a/2π
, co esponding o pe iod
β= 2π/a
.
Modula ow pe iod in imagina y ime di ec ion is
β
, hus xing p opo ion cons an
be ween modula pa ame e and obse e p ope ime
τphys
, ob aining
τphys =α mod
,
α∝β
.
4.3 P oo o Theo em 3
Va ying EH+GHY o al ac ion, using Pala ini iden i y and
δ√−g=−1
2√−ggµνδgµν
,
olume e m a ia ion gi es Eins ein enso
Gµν
, bounda y e m a ia ion unde xed
hab
condi ion o ganizes in o combina ion o ex insic cu a u e and
δhab
. S anda d de i a ion
shows
δSGHY =1
16πG Z∂M
d3y ϵ√h(Kab −Khab)δhab.
Viewing
δSGHY
as a ia ion o Hamil onJacobi unc ional, ob ain conjuga e momen um
πab
as s a ed in heo em.
8
Choosing imelike angen ec o eld and uni no mal on bounda y, decomposing
hab
in o ime and space pa s, in ADM decomposi ion
K
oge he wi h lapse unc ion
N
de e mine no mal ansla ion. Res ic ing
δhab
o pu e ime escaling (keeping spa ial
sec ion shape unchanged), can w i e
δSHJ
as quasilocal ene gy imes ime inc emen ,
ob aining
δSHJ =Eq.l.δ geom
.
4.4 P oo o Theo em 4
On common domain, by assump ion he e exis s sca e inggeome ymodula ow co -
espondence:
(1) Sca e ing side gi es amily o equency scales
dτsca (ω)
, co esponding o ex e nal
geome y h ough eikonal app oxima ion and lensing/Shapi o delay;
(2) Modula ow side h ough JLMS- ype ela ion o AdSRindle co espondence
maps modula Hamil onian o ene gy ope a o o geome ic egion, hus modula
ime linea ly ela ed o geome ic ime locally;
(3) Geome y side h ough Eins ein equa ions and bounda y condi ions connec s GHY
ac ion wi h ex insic cu a u e, quasilocal ene gy, and ime ansla ion.
Since sca e ing ime scale de e mined by
Q
and
∂ωφ
, while modula ime and
geome ic ime bo h scaled by bounda y ene gy (modula Hamil onian expec a ion alue,
quasilocal ene gy), and all h ee ac on same bounda y algeb ageome ic s uc u e, hei
scaling can only die by posi i e ni e ac o , hus he e exis s undamen al scale
τ
.
Dene
τ
as ime pa ame e making h ee uni changes consis en in e e ence case (such
as low-ene gy limi o xed e e ence obse e ), i.e.,
δτ =φ′(ω)
πδω =1
2π Q(ω)δω =1
E∗
δSHJ,
whe e
E∗
is e e ence ene gy unde uni scale. Fo any o he ime pa ame e
obs
, i
equi ing i s uni in e al consis en wi h abo e h ee eadings, mus ha e
d obs =αdτ
,
hus
obs
linea ly equi alen o
τ
.
Local uniqueness s ems om: i he e exis s ano he pa ame e
˜
simul aneously sa -
is ying consis ency o sca e ing scale and ene gy scale, hen
d˜
/dτ
locally non-ze o and
cons an , hus
˜
locally only ane escaling o
τ
.
4.5 P oo o Theo em 5
In FRW me ic con o mal ime sa ises
d =a(η)dη
, he e o e con o mal ime in e al
dη
ep esen s ligh a el ime pulled back o a me ic. Fo high- equency elec o-
magne ic wa es, eikonal app oxima ion shows phase a ies linea ly wi h con o mal ime.
Viewing cosmic signal (such as FRB o dis an quasa ) as sca e ing p ocess om emis-
sion bounda y o obse a ion bounda y, equency edshi
1 + z= 1/a( em)
means in
e ms o sou ce's p ope ime scale, obse ed equency scaled by
1/(1 + z)
compa ed o
local equency.
I unied ime scale
τ
dened as local obse e 's sca e ing scale, hen be ween emis-
sion and obse a ion ends, ime scale needs global escaling by scale ac o
a( )
, hus
dτcosmo =dφ
π·1
ωobs
=dφ
π·1
ωem/(1 + z)= (1 + z) dτlocal,
9
A.3 Wigne Smi h Ope a o T ace and Spec al Shi Densi y
In ene gy ep esen a ion,
S(λ)
is amily o uni a y ope a o s on
Hλ
, whose loga i hmic
de i a i e sa ises
∂λln de S(λ) = (S(λ)−1∂λS(λ)).
Since
de S(λ) = exp(−2πiξ(λ))
,
∂λln de S(λ) = −2πiξ′(λ) = 2πiρ el(λ).
On o he hand, dening Wigne Smi h ope a o
Q(λ) = −iS(λ)†∂λS(λ),
hen
(S(λ)−1∂λS(λ)) = (S(λ)†∂λS(λ)) = i Q(λ).
Compa ing yields
ρ el(λ) = (2π)−1 Q(λ)
.
A.4 One-Dimensional Case and Le inson Theo em
Unde one-dimensional sho - ange po en ial, eigens a es in ni e box app oxima ion
sa is y
knR+δ(kn) = nπ
. Changing po en ial o box leng h, eigen alue densi y die ence
can be exp essed as phase shi de i a i e, hus
ρ el(k) = 1
πδ′(k).
Le inson heo em gi es
δ(0) −δ(∞) = πNbound
and o he bounda y condi ions, hus
uni ying sca e ing phase and bound s a e coun ing. This cons uc ion compa ible wi h
spec al shi unc ion deni ion, p o iding conc e e implemen a ion o scale iden i y in
one-dimensional models.

B Modula Flow, KMS Condi ions, and The mal Time
B.1 Tomi aTakesaki Cons uc ion
S a ing om
M
and
Ω
, dene densely dened an ilinea ope a o
S0:mΩ7→ m∗Ω, m ∈M.
Closu e
S
admi s pola decomposi ion
S=J∆1/2
, whe e
J
is an ilinea isome ic modula
conjuga ion,
∆
is posi i e, sel -adjoin modula ope a o . Modula ow dened as
σω
(m)=∆i m∆−i .
Tomi aTakesaki heo em asse s:
σω
is one-pa ame e au omo phism g oup o
M
, and
ω
sa ises KMS condi ion o i , i.e., he e exis s analy ic unc ion in s ip egion such
ha
F( ) = ω(aσω
(b)), F( +i) = ω(σω
(b)a).
16

B.2 Connes 1Cocycle and S a e-Independen Time
Gi en wo ai h ul s a es
ω, ω′
, can cons uc Connes 1cocycle
u
such ha
σω′
(m) = u σω
(m)u−1
,
wi h
u +s=u σω
(us)
. This means modula ow's image in ou e au omo phism g oup
Ou (M) = Au (M)/Inn(M)
independen o s a e, only dening geome ic ime di ec-
ion.
B.3 The mal Time Hypo hesis and Tempe a u eTime Rela ion
In adi ional quan um s a is ics, gi en Hamil onian
H
and in e se empe a u e
β
, KMS
s a e unde ime e olu ion
α (a) = ei Hae−i H
sa ises
ω(aα (b)) = ω(α +iβ(b)a).
The mal ime hypo hesis e e ses his logic:  s gi en s a e
ω
and algeb a
M
, hen
in e p e modula ow
σω
pa ame e as ime. I ex e nal obse e 's physical ime
τphys
exis s, can ob ain
τphys =α mod
by compa ing modula ow wi h physical Hamil onian
gene a ed e olu ion, whe e
α
gi en by empe a u e o accele a ion. Un uh eec 's
T=
a/2π
p o ides conc e e example o his p opo ion.

C GHY Bounda y Te m, Hamil onJacobi Func ional,
and Quasilocal Time
C.1 Va ia ion o EH+GHY Ac ion
S a ing om
Sg a =1
16πG ZM
√−g R d4x+1
8πG Z∂M
ϵ√h K d3y,
a y wi h espec o
gµν
. EH e m a ia ion can be w i en as sum o olume in eg al and
bounda y in eg al, la e exac ly canceled by GHY e m a ia ion, hus unde
δhab = 0
condi ion, o al a ia ion only con ains olume in eg al, gi ing Eins ein equa ions.
C.2 Hamil onJacobi Func ional and Conjuga e Momen um
Viewing
SHJ[hab]
as unc ion o ac ion ob ained by sol ing Eins ein equa ions unde gi en
bounda y geome y, a ia ion wi h espec o
hab
gi es conjuga e momen um
πab =δSHJ
δhab
=ϵ
16πG√h(Kab −Khab).
In ADM decomposi ion, me ic w i en as
ds2=−N2d 2+hij(dxi+Nid )(dxj+Njd ),
ex insic cu a u e de e mined h ough lapse
N
and shi
Ni
. Choosing app op ia e
gauge (such as
Ni= 0
), no mal ime ansla ion co esponds o change o
, he e o e
δSHJ/δ
gi es quasilocal ene gy.
17
C.3 Rindle and Black Hole Cases
In Rindle wedge o Schwa zschild black hole ex e io , nea -ho izon egion ex insic cu -
a u e p opo ional o su ace g a i y, GHY bounda y e m in Euclidean pa h in eg al
gi es black hole ee ene gy and empe a u e ela ion. By pai ing Euclidean ime pe iod
β
wi h modula ow pe iod, can show geome ic ime, modula ime, and he mal ime
belong o same scale equi alence class.

D Ca ego ical S uc u e o Time Scale Equi alence Class
D.1 Objec s and Mo phisms
Cons uc ca ego y
BTG
:
(1) Objec s a e bounda y sys ems
B= (A∂, ω∂, S;hab, Kab)
;
(2) Mo phisms a e mappings
Φ : B1→ B2
p ese ing physical s uc u e, sa is ying:
•Φ
gi es
∗
isomo phism
ϕ:A∂,1→ A∂,2
on algeb a;
•Φ
maps s a e and S-ma ix o objec s p ese ing BK and scale iden i y s uc u e;
•Φ
maps bounda y geome y o embedding p ese ing me ic and ex insic cu a-
u e (o hei equi alence class).
In his ca ego y, ime scale equi alence class
[τ]
can be iewed as unc o om objec s
o
R
, assigning each bounda y sys em se o ime pa ame e s, mo phisms co esponding
o mono onic escaling o ime scales.
D.2 Ca ego ical S a emen o Local Uniqueness
Theo em 4 can be es a ed as: in some local subca ego y o gi en objec
B
, i equi ing
unc o
T:BTG →Time
simul aneously p ese e uni in e als o sca e ing scale,
modula ime, and geome ic ime, hen unique in sense o na u al isomo phism.
This p o ides basis o u u e connec ion o ime scale equi alence class wi h highe -
le el ca ego ical s uc u es (such as b a ion, na u al ans o ma ion), bu hese ex en-
sions beyond scope o his pape .
18