Boundary Time Geometry: Unified Theory of Time Scale, Resolution Hierarchy, and Interaction

Bounda y Time Geome y:
Unied Theo y o Time Scale, Resolu ion Hie a chy,
and In e ac ion
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
Cons uc unied heo e ical sys em wi h bounda y as on ology and ime as
geome ic scale. Basic assump ion: physical eali y  s mani es s as bounda y ob-
se able algeb a and i s spec al da a; bulk dynamics a e ex ensions de e mined by
bounda y da a. All obse able ime scalessca e ing ime, modula ime, geome -
ic imebelong o same equi alence class. Obse e 's ni e esolu ion geome i-
cally mani es s as esolu ion be bundle wi h connec ion and cu a u e.
Ma hema ically, in oduce noncommu a i e geome ic s uc u e o spec al iple
wi h bounda y; uni y B ownYo k bounda y s ess enso wi h AdS/CFT bounda y
s ess enso , Wigne Smi h ime delay ma ix wi h Bi manK ein spec al shi
unc ion, Tomi aTakesaki modula ow wi h he mal ime hypo hesis wi hin single
Bounda y Time Geome y (BTG) amewo k.
P o e unde app op ia e ma ching condi ions, exis s unique (up o ane escal-
ing) bounda y ime gene a o making sca e ing ime, modula ime, geome ic ime
dene same ime scale equi alence class. All classical  o ces mani es as p ojec-
ions o unied bounda y connec ion cu a u e in die en be di ec ions, no longe
undamen al objec s bu eme gen p ope ies o bounda y geome y and esolu ion
s uc u e.
Fu he es ablish phenomenal hie a chy eme gence heo em on esolu ion be
bundle, cla i ying how high- esolu ion quan um sca e ing and modula ime s uc-
u es degene a e in o mac oscopic g a i y and classical mechanics ia comple ely
posi i e coa se-g aining maps.
Finally p o ide BTG e o mula ions o black hole he modynamics, cosmological
edshi , mesoscopic anspo ; p opose expe imen al e ica ion p o ocols imple-
men able in mic owa e ne wo ks, a omic clock ne wo ks, mesoscopic conduc o s.
Keywo ds:
Bounda y Time Geome y; Noncommu a i e Geome y; Spec al T iple;
Wigne Smi h Time Delay; Bi manK ein Spec al Shi ; B ownYo k S ess Tenso ;
The mal Time Hypo hesis; Resolu ion Fibe Bundle; Holog aphic Bounda yBulk Co -
espondence; Reno maliza ion G oup

1
1 In oduc ion and His o ical Con ex
In gene al ela i i y, GibbonsHawkingYo k bounda y e m and B ownYo k quasilocal
s essene gy enso show ha well-dened a ia ion o g a i a ional ac ion and deni-
ion o quasilocal ene gymomen um undamen ally depend on bounda y geome y and
conjuga e a iables. Va ia ion o bounda y h ee-me ic de i es su ace s ess enso
Tab
BY
eco e ing ADM ene gy in app op ia e limi s, p o iding quasilocal ene gy meaning o
black hole he modynamics.
In AdS/CFT holog aphic amewo k, Balasub amanianK aus s ess enso iews
eno malized bounda y s essene gy as ene gymomen um enso o dual con o mal eld
heo y, u he deepening bounda y-domina ed pe spec i e.
On sca e ing heo y side, Wigne Smi h ime delay ma ix
Q(ω) = −iS(ω)†∂ωS(ω)
cha ac e izes a e age esidence ime o wa e packe s in sca e ing egion; i s ace igh ly
connec ed o de i a i e o spec al shi unc ion ia Bi manK ein o mula: o al sca -
e ing phase de i a i e, Wigne Smi h g oup delay ace, and ela i e s a e densi y a e
die en mani es a ions o same objec .
In algeb aic quan um eld heo y and quan um s a is ics, Tomi aTakesaki modula
heo y e eals: gi en obse able algeb a and s a e, na u ally exis s one-pa ame e au o-
mo phism g oup
σω
whose pa ame e
in e p e able as modula ime. ConnesRo elli
he mal ime hypo hesis u he p oposes physical ime unde s andable as modula ow
pa ame e de e mined by s a ealgeb a pai ; adi ional ime becomes de i ed concep .
Noncommu a i e geome y p o ides language dening geome y ia spec al da a:
spec al iple
(A,H, D)
consis s o algeb a, Hilbe space, Di ac- ype ope a o ; o
compac Riemannian mani olds, me ic s uc u e uniquely econs uc ible om Di ac
spec um; his amewo k p o ides na u al pla o m uni ying bounda y geome y wi h
bounda y obse able algeb a.
This pape 's basic s ance: glue abo e h ee h eadsbounda y g a i y, sca e ing
ime, modula imein unied Bounda y Time Geome y amewo k, aking bounda y
as on ology, ime as scale, esolu ion as be , cons uc ing unied heo e ical sys em
accommoda ing exis ing heo ies while yielding new p edic ions.

2 Model and Assump ions
2.1 Axioms: Bounda y P io i y, Time Equi alence, Resolu ion
Hie a chy
Axiom 1
(Bounda y P io i y)
.
Gi en space ime egion
(M, g)
wi h good causal s uc u e,
con aining opologically well-beha ed bounda y
∂M
(including imelike, spacelike, o null
bounda ies), undamen al desc ip ion o physical obse ables gi en by bounda y obse -
able algeb a
A∂
and s a e se
S∂
; bulk obse ables and dynamics iewable as ex ensions
de e mined by
(A∂,S∂)
in app op ia e sense.
Axiom 2
(Time Scale Equi alence)
.
Exis s ime scale equi alence class
[τ]
whose ele-
men s a e ime pa ame e s unde die en cons uc ions: sca e ing ime
τsca
, modula
2
ime
τmod
, geome ic ime
τgeom
. Any wo ime scales equi alen ia ane ans o ma ion
τ(2) =aτ(1) +b
(
a > 0
) on common domain.
Axiom 3
(Resolu ion Hie a chy)
.
Fo each conc e e expe imen al a angemen o ob-
se e , exis s esolu ion pa ame e
Λ
(unde s andable as UV cu o, coa se-g aining s age,
o RG scale) such ha a die en
Λ
, same bounda y geome ic da a p ojec s ia com-
ple ely posi i e map o die en coa se-g ained eec i e algeb as
AΛ⊆ A∂
.
2.2 Bounda y Spec al Da a
Deni ion 2.1
(Bounda y Spec al T iple)
.
Bounda y spec al iple is uple
(A∂,H∂, D∂)
whe e:
1.
A∂
is dense
∗
-algeb a dened on bounda y ( ypically
C∞(∂M)
o noncommu a i e
gene aliza ion);
2.
H∂
is
Z2
-g aded Hilbe space ca ying
∗
- ep esen a ion o
A∂
;
3.
D∂
is sel -adjoin ,  s -o de ellip ic ope a o (Di ac- ype) wi h compac esol en ,
sa is ying commu a o
[D∂, a]
bounded o any
a∈ A∂
.
This is bounda y e sion o Connes (e en) spec al iple.
Theo em 2.2
(Spec al Recons uc ion o Bounda y Me ic)
.
I
∂M
is compac spin
Riemannian mani old, iple
(A∂,H∂, D∂) = (C∞(∂M), L2(S∂), D∂)
de e mines unique Riemannian me ic
hab
such ha Connes dis ance
d(x, y) = sup{|a(x)−a(y)|:a∈C∞(∂M),|[D∂, a]| ≤ 1}
equals geodesic dis ance on
(∂M, hab)
.
Thus in BTG, bounda y me ic need no be gi en a p io i bu dened by spec al
s uc u e o
D∂
; his p o ides na u al channel embedding ime scale in o Di ac spec um.
2.3 Bounda y S ess Tenso and Quasilocal Hamil onian
In ou -dimensional gene al ela i i y, a e in oducing GHY bounda y e m, a ia ion
o ac ion wi h espec o bounda y h ee-me ic
hab
denes B ownYo k su ace s ess
enso
Tab
BY := 2
√−h
δSg a
δhab
.
I s ze o componen 's app op ia e p ojec ion gi es quasilocal ene gy densi y; in eg a ed
quasilocal ene gy equals Hamil onian gene a ing uni p ope ime ansla ion on bound-
a y.In AdS scena io, holog aphic eno maliza ion p ocess de i es eno malized bounda y
s ess enso
Tab
en
, in e p e able as dual CFT expec a ion alue
⟨Tab⟩
. These esul s show:
bounda y s ess enso na u ally ca ies Hamil onian gene a ing bounda y  ime ow.

3
3 Main Resul s (Theo ems and Alignmen s)
3.1 Unied Time Scales on Bounda y
Dene ime scales om sca e ing, modula ow, geome ic pe spec i es espec i ely:
1. Sca e ing ime
τsca
Conside ni e-channel sca e ing ma ix
S(ω)
a xed ene gy; dene Wigne Smi h
ime delay ma ix
Q(ω) = −iS(ω)†∂ωS(ω).
I s ace
τW(ω) := Q(ω)
gi es o al g oup delay. In Bi manK ein amewo k,
spec al shi unc ion
ξ(ω)
sa ises
de S(ω) = exp(−2πiξ(ω)),
hus
ξ′(ω) = 1
2π Q(ω).
Gi en e e ence ene gy
ω0
and window
I⊂R
, dene sca e ing ime scale
τsca (ω) := Zω
ω0
ξ′(˜ω)d˜ω=ξ(ω)−ξ(ω0).
2. Modula ime
τmod
Fo bounda y obse able algeb a
A∂
and s a e
ω
, assuming sepa a ingcyclic ec o
exis s making Tomi aTakesaki modula da a
(J, ∆ω)
well-dened; modula g oup
σω
(A) := ∆i
ωA∆−i
ω
denes one-pa ame e au omo phism g oup. The mal ime hypo hesis sugges s app o-
p ia e physical ime pa ame e
τmod
die s om modula pa ame e
only by cons an
ac o ;
σω
plays  ime e olu ion ole in equilib ium s a es.
3. Geome ic ime
τgeom
In gene al ela i i y wi h bounda y, choose uni imelike ec o eld
ua
on bounda y
wi h co esponding Killing o app oxima e Killing gene a o
ξa
; B ownYo k Hamil onian
w i able as
H∂[ξ] = ZΣ∩∂M
√σ uaTab
BYξbdd−2x,
whe e
σ
is induced me ic on c oss-sec ion. Canonical e olu ion pa ame e gene a ed by
H∂
denes geome ic ime scale
τgeom
.
In BTG amewo k, we don' p esuppose h ee ime scales mu ually independen , bu
uni y ia ollowing heo em:
Theo em 3.1
(Bounda y Time Scale Equi alence Theo em)
.
Le
∂M
be benign bound-
a y sa is ying:
1. Exis s bounda y spec al iple
(A∂,H∂, D∂)
and B ownYo k bounda y s ess en-
so
Tab
BY
;
2. Bounda y admi s sca e ing p ocess wi h sca e ing ma ix
S(ω)
con inuously di -
e en iable in ene gy
ω
, sa is ying Hilbe Schmid locali y and BK condi ions on ene gy
window
I
;
3. Fo same bounda y egion exis s on Neumann algeb a
A′′
∂
and KMS s a e
ω
whose
modula g oup
σω
physically ep esen s he mal equilib ium ime e olu ion;
4
4. B ownYo k Hamil onian
H∂[ξ]
gene a ed bounda y ime ansla ion induces au o-
mo phism g oup
ατ
on obse able algeb a compa able o sca e ing e olu ion and mod-
ula ow in same ene gy equency window, i.e., exis s common in a ian subalgeb a
Acom ⊂ A∂
.
Then exis s unique ime scale equi alence class
[τ]
, plus h ee posi i e cons an s
asca , amod, ageom >
0
and h ee ansla ion cons an s
bsca , bmod, bgeom
, such ha on common domain:
τsca =asca τ+bsca , τmod =amodτ+bmod, τgeom =ageomτ+bgeom.
In o he wo ds, sca e ing ime, modula ime, geome ic ime in BTG only ep esen
die en no maliza ions and ze o-poin choices o same ime scale.
Rigo ous p oo gi en in P oo s sec ion and appendices, co e being:
•
Use BKWigne 
Smi h iden i y o exp ess sca e ing ime as in eg al o ela i e spec al densi y;
•
Via
he mal ime hypo hesis and bounda y KMS s a e, align modula pa ame e wi h ela i e
spec al densi y;
•
Via B ownYo k Hamil onian and bounda y s ess enso 's spec al
ep esen a ion, associa e geome ic ime ow gene a o wi h same spec al measu e.
3.2 No Fundamen al Fo ces: Cu a u e o Unied Bounda y
Connec ion
Deni ion 3.2
(Bounda y To al Bundle and Unied Connec ion)
.
1. On geome ic
gauge esolu ion h ee-laye deg ees o eedom, dene bounda y o al bundle
π:B →
∂M
wi h be
F=Fin ×F es,
ca ying in e nal gauge deg ees o eedom and esolu ion scale deg ees o eedom e-
spec i ely.
2. S uc u e g oup aken as
G o = SO(1,3)↑×GYM ×G es,
whe e
G es
is scale g oup equi alen o eno maliza ion g oup o coa se-g aining ans o -
ma ions.
3. Unied bounda y connec ion dened as
Ω∂=ωLC ⊕AYM ⊕Γ es,
co esponding o Le iCi i a spin connec ion, YangMills connec ion, esolu ion connec-
ion; co esponding cu a u e
R∂=R∂⊕F∂⊕R es.
Theo em 3.3
(No Fundamen al Fo ces Theo em)
.
Unde BTG amewo k, conside
cha gedcolo ed es pa icle o eec i e mass ajec o y li
γ(τ)⊂ B
on bounda y; i s
p ojec ion o
∂M
is
xµ(τ)
, in insic deg ees o eedom ia ep esen a ion
ρ:GYM →
Au (Fin )
and
G es
one-dimensional ep esen a ion. Then ollowing holds:
1. Fo ce- ee mo ion o ajec o y
γ(τ)
is pa allel anspo o unied connec ion
Ω∂
:
Dτ˙γ= 0
.
2. I s base ajec o y
xµ(τ)
sa ises equa ion w i able as
mD2xµ
Dτ2=qFµν˙xν+ µ
es,
5

whe e
Fµν
is YangMills cu a u e p ojec ion unde ep esen a ion
ρ
,
µ
es
is esolu ion
cu a u e
R es
p ojec ion in app op ia e eec i e ac ion.
3. Classical g a i a ional  o ce co esponds o geodesic de ia ion eec o
R∂
; hus
all  o ces unde s andable as die en p ojec ions and ep esen a ions o unied bounda y
connec ion cu a u e, no longe undamen al objec s.
The e o e in BTG heo y,  o ces no independen axioma ic en i ies bu eme gen
mani es a ions o bounda y ime geome y; all in e ac ionsincluding g a i y, gauge in-
e ac ions, esolu ion-d i en en opic o cesjoin ly a ise om unied connec ion cu -
a u e.
3.3 Resolu ion Hie a chy and Eme gen Phenomena
Deni ion 3.4
(Resolu ion Fibe Bundle and Coa se-G aining Maps)
.
1. On bounda y
o al bundle dene  esolu ion be bundle
P es = (B, ∂M, G es, π es)
wi h be coo di-
na e unde s andable as esolu ion o eno maliza ion scale
Λ
.
2. Each
Λ
induces comple ely posi i e, uni -p ese ing map
ΦΛ:A∂→ AΛ⊆ A∂,
iewable as coa se-g aining om high- esolu ion bounda y algeb a o low- esolu ion e -
ec i e algeb a.
Theo em 3.5
(Phenomenal Hie a chy Eme gence Theo em)
.
When sa is ying:
1.
{ΦΛ}Λ
o ms no mal
∗
-homomo phism amily o semig oup (o g oup) sa is ying
ΦΛ1◦ΦΛ2= ΦΛ1◦Λ2
;
2. Fo any local obse able
A∈ A∂
, i s image
ΦΛ(A)
in
Λ→0
limi (coa ses )
con e ges o classical unc ion o ope a o
Acl
;
3. Unied connec ion
Ω∂
's connec ion o m
Γ es
in esolu ion di ec ion sa ises Callan
Symanzik-like equa ion: pa allel anspo along
Λ
ow equi alen o eno maliza ion
g oup ow.
Then:
1. In high- esolu ion limi , desc ip ion o
A∂
is ull quan um sca e ing and modula
ime s uc u e;
2. A medium esolu ion, cu a u e expec a ion alues in coa se-g ained algeb a
AΛ
mani es as gauge o ces, en opic o ces, opological eec s;
3. In
Λ→0
mac oscopic limi , geome ic cu a u e and B ownYo k enso domi-
na e, dynamics degene a ing o classical g a i y and he modynamics; all  o ces eec-
i ely iewable as geome ic eec s o me ic and eec i e po en ials.

4 P oo s
This sec ion p o ides p oo skele ons o main heo ems; de ails and echnical lemmas in
appendices.
6
4.1 P elimina ies: Sca e ing, Time Delay, Spec al Shi
Le
H0
and
H=H0+V
be sel -adjoin ope a o s on some Hilbe space sa is ying wa e
ope a o exis ence condi ions o gene al sca e ing heo y. Bi manK ein heo y p o ides
spec al shi unc ion
ξ(ω)
gi ing ace o mula o smoo h unc ions
o
H, H0
:
T ( (H)− (H0)) = Z ′(ω)ξ(ω)dω.
Unde sui able condi ions, sca e ing ma ix
S(ω)
sa ises
de S(ω) = exp(−2πiξ(ω)).
Fo Theo em 2, only need local BK o mula on ene gy window
I
and die en iabili y
o Wigne Smi h ma ix. Le
Q(ω) = −iS(ω)†∂ωS(ω),
hen
ξ′(ω) = (2π)−1 Q(ω)
, hus sca e ing ime scale
τsca (ω) = ξ(ω)−ξ(ω0)
well-dened on
I
.
4.2 P oo Ske ch o Theo em 2 (Time Scale Equi alence)
S ep 1: Unied Spec al Measu e
On ene gy window
I
, dene spec al measu e ia BK:
µsca (dω) := 1
2π Q(ω)dω.
On o he hand, KMS s a e
ω
on bounda y on Neumann algeb a
A′′
∂
induces modula
ope a o
∆ω
whose spec al measu e
µmod
de e mines modula g oup
σω
gene a o
Kω:=
−log ∆ω
. The mal ime hypo hesis equi es cons an
cmod >0
exis s making physical
Hamil onian
Hmod =cmodKω
.
On geome ic end, B ownYo k Hamil onian w i able as unc ional on Di ac o Laplace
ope a o spec um: unde app op ia e bounda y condi ions, i s expec a ion alue on en-
e gy eigens a e
|E⟩
gi es spec al unc ion
hgeom(E)
, in oducing measu e
µgeom(dE) =
hgeom(E)dE
. In AdS/CFT con ex , his measu e equi alen o bounda y CFT ene gy
momen um enso spec al measu e.
S ep 2: Ma ching Condi ions and Measu e Equi alence
Assume sca e ing p ocess, modula ow, geome ic ime ansla ion ac on common
decomposable subalgeb a
Acom
; wi hin ene gy window
I
, h ee dynamics' spec al de-
composi ions ep esen able in same Hilbe space; his is compa abili y condi ion in
heo em s a emen .
Unde his condi ion, can p o e:
1. Exis s amily o mono onic die en iable ene gy escaling unc ions making
µsca
,
µmod
,
µgeom
mu ually absolu ely con inuous on
I
wi h cons an RadonNikodym de i a-
i es;
2. This ensu es h ee ime gene a o s equi alen on
L2(I, µ)
, die ing only by cons an
ac o s and addi i e cons an s.
7
S ep 3: Uniqueness
I ano he ime scale
˜τ
anely equi alen o all h ee abo e, hen
˜τ
also anely
equi alen o
τ
; hus equi alence class
[τ]
unique.
Comple e p oo in ol es ne con ol o spec al decomposi ion, KMS condi ions,
B ownYo k Hamil onian spec al ep esen a ion; see Appendix A.
4.3 P oo Ske ch o Theo em 3 (No Fundamen al Fo ces)
Unde unied connec ion
Ω∂
, conside cu e
γ(τ)
on o al bundle
B
. I s co a ian de i a-
i e
Dτ=d
dτ + Ω∂(˙γ).
Dene  ee mo ion as
Dτ˙γ= 0
.
Expanding his condi ion in die en be di ec ion componen s yields:
1. In
SO(1,3)
pa : s anda d geodesic equa ion; 2. In
GYM
pa : Wong-equa ion-like
gauge o ce e m: pa icle pa allel anspo in in e nal space induces
Fµν ˙xν
e m on
base ajec o y; 3. In
G es
pa : esolu ion connec ion
Γ es
cu a u e ia eec i e ac ion's
scale dependence gi es en opic o ce o in o ma ion o ce, specic o m depending on
chosen eec i e ee ene gy unc ional.
Thus any seemingly  o ced mo ion iewable as pa allel anspo unde some unied
connec ion, we simply igno e ce ain be di ec ions in p ojec ion. Theo em 3 con en
me ely o malizes his geome ic ac .

5 Model Applica ions
5.1 Black Hole The modynamics in BTG
Fo space ime wi h e en ho izon, ea ho izon as special null bounda y; in oduce null
B ownYo k s ess enso and co esponding quasilocal ene gy, ew i ing black hole he -
modynamics ou laws in pu e bounda y language.
P oposi ion 5.1
(Bounda y Res a emen o Black Hole The modynamics)
.
1. Hawking
empe a u e
TH=κ/(2π)
comes om modula ow pe iod
2π/κ
on ho izon, whe e
κ
is
su ace g a i y;
2. Bekens einHawking en opy
SBH =A/(4G)
in e p e able as on Neumann en opy
o ho izon bounda y algeb a o en opy densi y on ype ac o ;
3. Hawking adia ion pu i y p oblem o mulable in BTG as Ma ko p ope y s abili y
p oblem be ween ho izon and inni y bounda y algeb as: i bounda y ela i e en opy
slice-independen sa is ying app op ia e quan um ocusing condi ions, o e all e olu ion
can p ese e pu e s a es.
In BTG language, black hole he modynamics no longe mix u e o bulk singula i y
and ho izon s uc u e, bu comple ely desc ibed by bounda y ime geome y and modula
ime scale.
8
5.2 Cosmological Redshi as Bounda y Time Rescaling
In FRW uni e se, s anda d edshi o mula
1 + z=a( 0)/a( e)
ew i able in BTG as:
P oposi ion 5.2
(Bounda y In e p e a ion o Cosmological Redshi )
.
Choose cosmo-
logical bounda y as con o mal inni y o como ing obse e amily wo ld ube bounda y,
wi h ime scale dened by bounda y ime scale
τ∂
; exis s ane ans o ma ion making
1 + z=τ∂( 0)/τ∂( e).
This shows edshi iewable as o e all escaling o bounda y ime scale, no bulk
p ope ime die ence; BTG di ec ly connec s edshi o bounda y spec al da a e o-
lu ion.
5.3 Mesoscopic T anspo and F iedelWigne Consis ency
In mesoscopic conduc o s o AB ings, Wigne Smi h ime delay ma ix and F iedel sum
ule p o ide connec ions be ween local densi y o s a es, phase shi , anspo p ope -
ies. In BTG, hese esul s in e p e able as bounda y spec al iple p ojec ions a ni e
esolu ion:
•
Phase shi de i a i e
∂ωϕ(ω)
a io o local densi y o s a es di ec ly gi es sca e -
ing ime scale;
•
Via Theo em 2, his scale equi alen o modula and geome ic ime,
making mesoscopic anspo expe imen s di ec e ica ion pla o m o BTG ime scale
equi alence.

6 Enginee ing P oposals
6.1 Mic owa e Sca e ing Ne wo ks as Disc e e Bounda y Mod-
els
Cons uc mul i-po mic owa e ne wo k iewing as disc e ized bounda y
∂M
model:
1. Measu e mul i-po sca e ing ma ix
S(ω)
ia ec o ne wo k analyze ; nume i-
cally cons uc Wigne Smi h ma ix
Q(ω)
and spec al shi unc ion
ξ(ω)
, dene sca -
e ing ime scale
τsca
.
2. In oduce unable geome ic pa ame e s a ne wo k nodes (e.g., elec ical leng h,
lossy elemen s); econs uc
τgeom
ia ne wo k Lag angian o eec i e RLC model in e -
sion.
3. Place ne wo k in con olled noise en i onmen ; dene s a is ical s eady s a e and
cons uc equi alen modula ow; measu e
τmod
p oxy quan i ies (e.g., co ela ion unc-
ion decay pa ame e s).
BTG p edic ion: Wi hin ene gy window and esolu ion condi ions sa is ying Theo em
2 assump ions, a ios o h ee ime scales should be cons an ; de ia ions a ibu able o
esolu ion connec ion
Γ es
cu a u e and expe imen al non-ideali ies.
6.2 A omic Clock Ne wo ks and G a i a ional Redshi
Deploy a omic clock ne wo k a die en g a i a ional po en ials; use wo-way ime ans-
e p o ocol o measu e equency a io
ν2/ν1
. In BTG language, his equency a io
9