Boundary, Observer Attention, and Time Axis: Unified Framework of Generalized Entropy Geodesics and Section Universe

Abs ac
Cons uc unied amewo k o ime and obse e wi h bounda y as undamen al
s age. Assume in gauge eld s a e wi hou obse e eadings, bounda y da a o
space ime and elds a e de e mined, bu no p e e ed ime pa ame e exis s; ime
only appea s as geodesic pa ame e  when obse e pe o ms eadings and hei
a en ion selec s bounda y sec ion amily.
Fi s , on sca e ing heo y end, based on scale iden i y
φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω),
dene obse able ime scale as unied eading o bounda y sca e ing phase, ela i e
spec al densi y, Wigne Smi h ime delay ace.
Second, on geome y and g a i y end, based on Eins einHilbe Gibbons
HawkingYo k ac ion a ia ion, iew bounda y as gene a o con e ing no mal
ux in o eadable quan i ies, cha ac e ize gene alized en opy geodesics ia ex-
emali y o gene alized en opy
Sgen
and quan um ocusing inequali y.
In oduce a en ion sec ions and sec ion uni e se: obse e 's a en ion gen-
e a es ime axis calib a ed by scale iden i y and selec ed by gene alized en opy
geodesics h ough selec ing bounda y sec ion amily and accessible algeb a; die en
obse e s' ime axes co espond o die en sec ion amilies on same bounda y geom-
e y, explaining phenomena like double-sli in e e ence whe e obse a ion changes
pa e n geome ically exp essible as en e ing die en sec ions.
Finally p o ide se ies o es able p edic ions and enginee ing p oposals, including
g oup delaygene alized en opy alignmen expe imen s based on elec omagne ic
sca e ing ne wo ks, black hole in o ma ion eadings based on island o mula analog,
double-sli expe imen pa hs con olling classicalquan um ansi ion by adjus ing
obse a ion esolu ion.
Keywo ds:
Bounda y Time Geome y; A en ion Sec ions; Gene alized En opy Geodesics;
Wigne Smi h Time Delay; GibbonsHawkingYo k Bounda y Te m; Quan um Ex emal
Su ace; PageWoo e s Rela ional Time; Double-Sli In e e ence

1 In oduc ion and His o ical Con ex
1.1 Obse e -less Time and Bounda y P io i y
In s anda d o mula ions o gene al ela i i y and quan um eld heo y, ime o en iewed
as p e-gi en ex e nal pa ame e : in gene al ela i i y as pa ame ized geodesic on man-
i old, in quan um heo y as pa ame e gene a ing uni a y e olu ion. Howe e , conside -
ing g a i a ional ac ion wi h bounda y, pu e Eins einHilbe bulk ac ion insucien o
well-dened a ia ion; mus add GibbonsHawkingYo k bounda y e m
SGHY =1
8πZ∂M
ϵ√h K
o elimina e no mal de i a i e con ibu ions, making a ia ion well-dened when bound-
a y geome y xed. This ac shows: in p esence o bounda y, bounda y no passi e
endpoin  bu gene a i e condi ion o bulk geome y and dynamics.
1
In quan um g a i y and holog aphy esea ch, gene alized en opy
Sgen[σ] = A ea(σ)
4Gℏ+Sou [σ]
becomes undamen al quan i y desc ibing bounda y sec ions, whe e
σ
is codimension- wo
sec ion,
Sou
is quan um eld en opy on i s ex e io . Quan um Ex emal Su ace (QES)
gi en by s a iona y condi ion o
Sgen
, co e o cons uc ing g a i a ional sys em en opy
and island o mula.
In pa allel, Quan um Focusing Conjec u e (QFC) p oposes: along any null geodesic
bundle o hogonal o
σ
, quan um expansion o gene alized en opy mono onically non-
inc easing, uni ying Bousso ligh -shee bound wi h classical ocusing heo em, implying
quan um null ene gy condi ion.
These s uc u es join ly poin o idea: in gauge eld s a e wi hou obse e eadings,
bounda y geome y and gene alized en opy ow a e de e mined, bu no p e e ed ime
pa ame e exis s; e e y hing jus block uni e se geome y and en opy unc ions. Fo
ime o become expe ien ial quan i y, equi es addi ional inpu : obse e and hei
eadings.
1.2 Sca e ing Scale, Phase, Time Delay
In sca e ing heo y, Wigne Smi h ime delay ma ix
Q(ω) = −i S(ω)†∂ωS(ω)
desc ibes g oup delay o inciden s a e a equency
ω
ela i e o ee p opaga ion. I s
ace sa ises scale iden i y wi h o al sca e ing phase
Φ(ω) = a g de S(ω)
de i a i e
and spec al shi unc ion:
φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω),
whe e
φ(ω) = 1
2Φ(ω)
,
ρ el
is K ein spec al shi densi y.
This ype o scale iden i y shows: o gi en sca e ing sys em, obse able ime scale
essen ially de e mined by bounda y sca e ing ma ix
S(ω)
, no by bulk p ese absolu e
ime a iable. Time eading is de i ed quan i y om bounda y spec al da a.
1.3 Rela ional Time and In e nal Re e ence F ame
In quan um ime p oblem, PageWoo e s mechanism p oposes: in o e all s a ic s a e
|Ψ⟩
sa is ying cons ain
ˆ
H o |Ψ⟩= 0
, i in e p e ing ce ain subsys em as clock, condi-
ional s a e ela i e o clock poin e esul s can sa is y eec i e e olu ion equa ion, hus
 ime appea s as pa ame e o sys emclock co ela ion. This mechanism ep esen s in-
e nal e e ence ame pe spec i e: ime no p e-gi en bu gene a ed by obse a ional
co ela ion and condi ioning.
These de elopmen s join ly o m his pape 's his o ical backg ound: bounda y e m
ensu es a ia ional consis ency, gene alized en opy and QES/QFC p o ide dynamical
cons ain s on bounda y sec ions, sca e ing scale iden i y gi es unied p oxy o bound-
a y ime eadings, PageWoo e s ela ional ime schemes show ime de i able om con-
di ional p obabili ies and a en ion selec ion.
2
1.4 Co e Ques ions and App oach
This pape a emp s o answe h ee ela ed ques ions:
1. In gauge eld s a e wi hou obse e eadings, how does bounda y exis wi hou
ime? 2. When obse e pe o ms eadings, how does a en ion selec sec ion amily on
bounda y, hus o ced o mo e along geodesic, how does his geodesic become obse e 's
 ime axis? 3. In his sense, how a e pa e n changes in double-sli in e e ence, coex-
is ence o all possible sec ions in uni e se unied as sec ion uni e se geome y, wi h
obse e me ely selec ing pa h wi h a en ion?
Fo his, in oduce unied amewo k o Bounda y Time Geome y, unde igo -
ous model assump ions gi e p o able heo ems, cha ac e izing ime axis as a en ion
geodesic calib a ed by scale iden i y and selec ed by gene alized en opy ex emali y
and ocusing condi ions.

2 Model and Assump ions
2.1 Unde lying Geome y and Bounda y Da a
Le
(M, gµν)
be Lo en zian mani old wi h bounda y
∂M
, ou wa d uni no mal ec o
nµ
, induced bounda y me ic
hab
. G a i a ional ac ion aken as
Sg a =1
16πG ZM
√−g(R−2Λ) d4x+1
8πG Z∂M
ϵ√h K d3y,
whe e
K
is ex insic cu a u e ace,
ϵ=±1
dis inguishes spacelike/ imelike bounda ies.
Ma e eld bulk ac ion deno ed
Sma e
, o al ac ion
S=Sg a +Sma e
.
Assume bulk eld dynamics gi en by eno malizable o eec i e eld heo y, mak-
ing well-dened pa h in eg al o pa i ion unc ion
Z[∂M]
exis unde gi en bound-
a y condi ions. In semiclassical limi , dominan con ibu ion o gi en bounda y da a
(∂M, hab, Kab)
comes om s a iona y geome ies sa is ying Eins ein equa ions.
2.2 Bounda y Obse able Algeb a and S a es
Le
A∂
be obse able algeb a a ached o bounda y
∂M
, aken as well-dened local
ope a o algeb a's
C∗
o on Neumann algeb a; s a es ep esen ed by posi i e no malized
linea unc ionals
ω:A∂→C
. Gi en pa i ion
∂M=R∪¯
R
, can dene local subalgeb a
AR⊂ A∂
and i s es ic ed s a e
ωR
.
In case wi h g a i a ional dual, assume o each well-dened bounda y egion
R
ex-
is s codimension- wo sec ion
σR
(like HRT su ace o QES) as co esponding en opy
bounda y, whose gene alized en opy
Sgen[σR]
equals on Neumann en opy o
ωR
plus
geome ic a ea e m.
2.3 Gene alized En opy Geodesics and Quan um Focusing
Gi en smoo h codimension- wo sec ion amily
{σ(λ)}
de o med along null geodesic bundle
N
pa ame ized by
λ
. Dene gene alized en opy
Sgen(λ) = A ea[σ(λ)]
4Gℏ+Sou [σ(λ)].
3
Quan um Focusing Conjec u e p oposes gene alized en opy's quan um expansion
Θ(λ) = 4Gℏ
A ea[σ(λ)]
dSgen
dλ
mono onically non-inc easing along
N
:
dΘ/dλ ≤0
, implying quan um null ene gy con-
di ion and Bousso ligh -shee bound in app op ia e limi s.
Call cu e amilies sa is ying ollowing condi ions gene alized en opic geodesics:
•
Each sec ion
σ(λ)
makes
Sgen
s a iona y unde small ans e se a ia ions:
δSgen[σ(λ)] =
0
;
•
Quan um expansion mono onically non-inc easing along amily di ec ion:
dΘ/dλ ≤
0
.In semiclassical limi , abo e condi ions degene a e o a ea ex emali y and classical
ocusing heo em, aligning wi h usual ex emal su ace and geodesic concep s.
2.4 Sca e ing Scale Mo he Rule and Time Reading
Conside sca e ing p ocess on bounda y wi h S-ma ix spec al decomposi ion
S(ω)
.
Dene scale mo he ule
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
whe e
Q(ω) = −iS(ω)†∂ωS(ω)
. Scale iden i y p o able unde app op ia e ace-class
pe u ba ion condi ions ia ela ion be ween Bi manK ein spec al shi unc ion and
Wigne Smi h ime delay ma ix.
In enginee ing con ex , can implemen ime scale calib a ion by measu ing g oup
delay spec um
(2π)−1 Q(ω)
o sca e ing phase de i a i e
φ′(ω)/π
. He e iew
κ(ω)
as bounda y ime scale mo he ule scale s uc u e al eady exis ing in obse e -less
backg ound bu no ye selec ed as specic ime axis.
2.5 Obse e , A en ion Sec ions, Time Axis
Obse e
O
cha ac e ized by h ee elemen s:
1. Timelike cu e (candida e wo ldline)
γ:τ7→ xµ(τ)
, pa ame e
τ
no necessa ily
iden ied wi h p ope ime;
2. A en ion sec ion amily
{Στ}
, each
Στ
codimension-one sec ion embedded in
∂M
, ans e se o
γ(τ)
;
3. Accessible algeb a
AΛ
Στ⊂ A∂
and co esponding condi ional s a e
ωΛ
Στ
a ached o
Στ
, whe e
Λ
ep esen s obse a ion esolu ion o coa se-g aining scale.
A en ion o malized as amily o comple ely posi i e ace-p ese ing maps
Eτ,Λ:
A∂→ AΛ
Στ
plus Bayes- ype upda e om global s a e
ω
o condi ional s a e
ωΛ
Στ
.
Obse e 's ime axis selec ed by join condi ion o
τ
and
γ
, making
τ
simul ane-
ously sa is y:
•
Main ain linea i y ela i e o scale mo he ule
κ(ω)
eading:
τ∼
Rκ(ω)w(ω)dω
, whe e
w(ω)
weigh unc ion ela ed o expe imen al appa a us;
•
Co e-
sponding sec ion amily
{σ(τ)}
(de e mined by
Στ
and bulk geome y) sa ises gene al-
ized en opy geodesic condi ions.
In his sense, ime axis is gene alized en opy geodesic selec ed by a en ion and
calib a ed by sca e ing scale mo he ule pa ame e .

4
3 Main Resul s (Theo ems and Alignmen s)
This sec ion gi es h ee main esul s. P oo s expanded in Sec ion 4 and appendices.
Theo em 3.1
(Obse e -less Time and Bounda y Block S uc u e)
.
Le
(M, gµν)
sa is y
Eins ein equa ions, gi en g a i a ionalma e bounda y da a on
∂M
making gene al-
ized en opy
Sgen(λ)
well-dened o any codimension- wo sec ion amily
{σ(λ)}
. I no
selec ing any a en ion sec ion amily
{Στ}
and accessible algeb a amily
{AΛ
Στ}
, hen:
1. Can dene global scale mo he ule
κ(ω)
, bu no selec ed single ime pa ame e
τ
exis s, only scale eld wi hou p e e ed di ec ion;
2. Any desc ip ion o e olu ion es a able as au omo phism on
(∂M,A∂, ω)
, equi -
alen o coo dina e escaling on block uni e se.
Thus in obse e -less sense, bounda y has de e mined geome icen opyscale s uc-
u e bu no p e e ed ime axis.
Theo em 3.2
(A en ion Geodesic Theo em)
.
Unde assump ions 2.12.5, assume exis s
imelike cu e candida e amily
γ
and a en ion map
Eτ,Λ
sa is ying:
1. Fo each
τ
, codimension- wo sec ion
σ(τ)
de e mined by
γ(τ)
and bounda y da a
makes
Sgen[σ(τ)]
s a iona y o small ans e se a ia ions;
2. Along null geodesic bundle de e mined by
σ(τ)
, gene alized en opy's quan um
expansion s ic ly mono onically non-inc easing;
3. Time eading
τ
gi en by scale mo he ule , i.e., exis s non-nega i e in eg able
weigh unc ion
w(ω)
making
dτ
dλ =Zκ(ω)wλ(ω)dω,
whe e
wλ
only depends on
Eλ,Λ
and bounda y sca e ing s uc u e.
Then exis s one- o-one co espondence (modulo mono onic epa ame iza ion) be ween:
•
A en ion ime axis amilies
{γ, τ, Eτ,Λ}
sa is ying abo e 13 condi ions;
•
Timelike
geodesic amilies ex emizing ac ion unc ional
J[γ] = RLe (x, ˙x)dτ
, whe e
Le
is eec-
i e Lag angian including gene alized en opy and scale e ms.
In o he wo ds, a en ion ime axes sa is ying scale iden i y and gene alized en opy
geodesic condi ions equi alen o geodesics on ce ain eec i e geome y.
Theo em 3.3
(Sec ion Uni e se and Obse a ion B anches)
.
Unde Theo em 3.2 se -
ing, conside amily o die en obse a ion esolu ions
{Λi}
and co esponding a en ion
sec ion amilies
{Σ(i)
τ}
, dening a en ion ime axis amilies
{γ(i), τ(i)}
. I o each pai
(i, j)
exis s comple ely posi i e map amily be ween sec ions
Φij,τ :AΛi
Σ(i)
τ→ AΛj
Σ(j)
˜τ
sa is ying ela i e en opy mono onici y
S(ω(i)
Σ(i)
τ∥ω(i)
Σ(i)
τ◦Φij,τ )≥0,
hen can:
1. Embed all a en ion ime axes in o sec ion uni e se space
S
, whose poin s a e
equi alence classes
[Σ,AΛ
Σ, ωΛ
Σ]
;
2. Unde s and each obse e 's expe ien ial his o y as pa h on
S
; die ences be ween
obse e s co espond o selec ing die en geodesic amilies on
S
.
5

When applying his s uc u e o double-sli in e e ence expe imen , cases wi h/wi hou
pa h de ec o co espond o die en a en ion pa hs in sec ion uni e se: appea ance/disappea ance
o in e e ence inges is geome ic exp ession o sec ion selec ion no single wa e unc-
ion collapse.
P oposi ion 3.4
(Mac oscopic G a i yMic oscopic Time Delay B idge)
.
In weak eld,
slowly a ying g a i a ional backg ound, Shapi o delay p oduced by mac oscopic g a i a-
ional po en ial
Φg a
along obse e wo ldline w i able a  s -o de app oxima ion as
eec i e con ibu ion o bounda y sca e ing phase, hus uni ying mac oscopic g a i a-
ional ime delay wi h mic oscopic Wigne Smi h ime delay on scale mo he ule . Can
in e p e g a i a ional po en ial as geome ic eec o scale mo he ule in eg a ion,
no exis ence o addi ional o ce.

4 P oo s
This sec ion p o ides p oo skele ons o Theo ems 3.13.3 and P oposi ion 3.4; de ailed
calcula ions pos poned o appendices.
4.1 Theo em 3.1
Key o Theo em 3.1 is sepa a ing exis ing s uc u e om selec ed pa ame e .
1. F om gi en bounda y da a and bulk eld equa ions, can dene pa h in eg al
Z[∂M]
o co esponding bounda y s a e
|Ψ∂⟩
con aining all allowed bulk his o ies.
2. Fo any codimension- wo sec ion amily
{σ(λ)}
, can calcula e gene alized en opy
Sgen(λ)
ia semiclassical me hods, ob aining scala unc ion amily.
3. Scale mo he ule
κ(ω)
comple ely de e mined by bounda y sca e ing ma ix
S(ω)
, which de e mined by
(M, gµν,
bounda y condi ions
)
, independen o obse e exis-
ence.
Howe e , combining
κ(ω)
and
Sgen(λ)
in o one-dimensional ime pa ame e 
τ
e-
qui es selec ing:
•
Specic cu e amily
{γ}
;
•
Specic sec ion amily
{Στ}
and accessible
algeb a amily;
•
Specic weigh unc ion amily
wλ(ω)
.
Wi hou hese selec ions,
κ(ω)
and
Sgen(λ)
jus elds and unc ions on block uni e se,
no p e e ed pa ame iza ion. Any e olu ion iewable as au omo phism o bounda y
s a e
ω
, i.e., coo dina e escaling, simila o o e all s a ic s a e
|Ψ⟩
si ua ion in Page
Woo e s o malism. Thus Theo em 3.1 di ec ly holds.
[P oo s o Theo ems 3.2, 3.3, and P oposi ion 3.4 condensed due o leng h...]

5 Model Applica ions
This sec ion demons a es amewo k applica ions in h ee ypical scena ios: black hole
in o ma ion, cosmological edshi , double-sli in e e ence.
6
5.1 Black Hole In o ma ion and Island Fo mula
In black hole e apo a ion phenomenon, la e- ime Hawking adia ion on Neumann en-
opy g ows wi h ime un il Page ime icini y eaching peak, hen declines; o e all cu e
gi en by island o mula
S(R) = min
QES A ea(∂I)
4Gℏ+Sbulk(I∪R),
whe e
R
is adia ion egion,
I
is island egion.
In his amewo k:
•
Dis an obse e 's a en ion sec ion amily
{Σou
τ}
mainly sup-
po ed nea inni y bounda y, co esponding QES pene a es black hole in e io ;
•
F ee-
all obse e 's sec ion amily
{Σin
τ}
local o nea -ho izon egion, co esponding QES di -
e en om ex e nal sec ions.
Two obse e ypes' a en ion ime axes co espond o die en pa hs in sec ion uni-
e se
S
. Fo ex e nal obse e , island egion in o ma ion encoded in adia ion; o in-
e nal obse e , same in o ma ion exis s locally inside black hole. In o ma ion p ese ed
bu dis ibu ed ac oss die en sec ion amilies, elie ing exp essi e ension o whe e is
in o ma ion.
5.2 Cosmological Redshi and Time Scale Rescaling
In s anda d FRW cosmology, como ing obse e 's edshi  ime ela ion con olled by
scale ac o
a( )
, equency sa ises
ω∝a−1
. In his amewo k, sca e ing ma ix on
cosmological ho izon o la ge-scale bounda y e ol es wi h ime, i s phase de i a i e gi es
scale mo he ule
κ(ω)
. Cosmological edshi es a able as:
•
Backg ound geome y de e mines scale mo he ule escaling wi h equency;
•
Obse e s a die en cosmic epochs selec die en weigh unc ions
wλ(ω)
ia espec-
i e a en ion sec ion amilies and accessible equency bands;
•
Time axis compa ison
becomes scale in eg al compa ison be ween die en obse e s, no ow o some absolu e
ime.
This es a emen p o ides unied bounda ysca e ingscale in e p e a ion o cos-
mic ime, acili a ing analysis o ela ions among g a i yda k ene gyin o ma ion ow.
5.3 Sec ion Selec ion in Double-Sli In e e ence
Conside s anda d double-sli expe imen . Wi hou in oducing pa h de ec o , inciden
pa icle s a e
|ψin⟩
e ol es h ough wo sli s o de ec ion sc een bounda y s a e
|ψou ⟩
,
in ensi y dis ibu ion con olled by in e e ence e ms. Adding pa h de ec o achie able
ia en anglemen wi h en i onmen , leading o eec i e s a e on sc een app oxima ely
mixed s a e wi hou in e e ence e ms.
In his pape 's amewo k:
•
Wi hou de ec ion, a en ion sec ion amily
{Σ ee
τ}
co esponds o accessible alge-
b a main aining wo-sli cohe ence, i s gene alized en opy s uc u e allowing c oss-sli
cohe ence;
•
Wi h de ec ion, a en ion map
Epa h
τ,Λ
comp esses accessible algeb a o pa h-dis inguishable
subalgeb a, i s ela i e en opy s uc u e and en i onmen al in o ma ion ow de e mining
new sec ion amily
{Σdecoh
τ}
.
Two sec ion amilies co espond o die en pa hs in sec ion uni e se
S
. Expe imen e
by choosing whe he o enable de ec o ac ually selec s hei a en ion ime axis in sec ion
7
uni e se, no di ec ly changing uni e se s a e. Uni e se s uc u ally accommoda es all
sec ion pa hs; obse a ion me ely selec s one.

6 Enginee ing P oposals
6.1 ScaleEn opy Alignmen Expe imen Based on Elec omag-
ne ic Sca e ing Ne wo ks
Cons uc mul i-po elec omagne ic sca e ing ne wo k, use ec o ne wo k analyze
o measu e equency-dependen S-ma ix
S(ω)
, ob ain Wigne Smi h ma ix
Q(ω)
and
i s ace ia nume ical die en ia ion. Simul aneously, pa i ion ne wo k in e io in o
accessible egion and en i onmen egion, con ol gene alized en opy
Sou
dis ibu ion
by a ying coupling s eng h and lossy elemen congu a ion.
Measu emen scheme includes: 1. Es ima e on Neumann o Rényi en opy app oxi-
ma ion o ou pu s a e ia po powe spec um; 2. Ve i y co ela ion be ween g oup delay
spec um changes and en opy changes, es scale iden i y s abili y in con olled en i on-
men ; 3. Implemen double-sli -like scena ios ia unable swi ches and pa h-iden iable
de ec ion po s, compa e ela ions among g oup delayen opypa e n unde die en
a en ion sec ions in same ne wo k.
6.2 PageWoo e s Scheme Based on Cold A oms and Quan um
Op ics
Implemen PageWoo e s ela ional ime expe imen s on cold a om o apped-ion pla -
o ms, use pa o deg ees o eedom as clock, ano he pa as sys em, p epa e o e all
app oxima ely s a ic cons ained s a e. Recons uc sys em's eec i e e olu ion ia con-
di ional measu emen o clock esul s.
Combining his amewo k:
•
App oxima e expe imen al appa a us op ical bound-
a y as
∂M
;
•
Recons uc s a e changes on bounda y obse able algeb a ia quan um
omog aphy, e i y consis ency o ime pa ame iza ions co esponding o die en a -
en ion sec ion amilies;
•
Measu e condi ional s a e en opy changes, es app oxima e
condi ions o gene alized en opy geodesics.
6.3 G a i a ional Lensing and FRB Time Delay
Use mul i-image ime delays and equency dispe sion cha ac e is ics o Fas Radio Bu s s
(FRBs) o FRB-like pulse signals o analyze ela ions be ween mac oscopic g a i a ional
po en ial and mic oscopic equency scale. In his amewo k, obse e 's a en ion sec-
ion amilies a die en image su aces co espond o die en bounda y sca e ing pa hs,
hei g oup delay die ences on scale mo he ule di ec ly gi ing mac oscopic g a i a-
ional eec s. Combining s a is ics o nume ous e en s, po en ially gi e as onomical-le el
es s o P oposi ion 3.4 b idge ela ion.

8
7 Discussion (Risks, Bounda ies, Pas Wo k)
7.1 Risks and Applicabili y Domain
Main isks o his amewo k:
•
Dependence on QES and island o mula makes s ic applicabili y limi ed o s a es
wi h semiclassical g a i a ional duals; o gene al QFT backg ounds, cons uc ion o gen-
e alized en opy geodesics and sec ion uni e se may be insucien .
•
Assump ion o QFC cu en ly s ill conjec u al s a us; hough pa ial p oo s and
s ong suppo examples exis , lacks comple ely gene al ma hema ical p oo .
•
Scale iden i y s ic ness in sca e ing heo y depends on specic ace-class pe u -
ba ion condi ions and spec al assump ions; equi es ca e ul ex apola ion o complex
open sys ems.
Thus his amewo k should be unde s ood as unied pic u e unde iple condi ions
o semiclassical g a i y + well-beha ed sca e ing + con ollable obse a ion; ex apo-
la ion o s ong quan um g a i y, non-local in e ac ions, highly non-equilib ium sys ems
equi es addi ional a gumen s.
7.2 Rela ion o Exis ing Wo k on Obse e TimeMeasu emen
This pape closely connec ed o PageWoo e s, in e nal e e ence ames and ela ional
ime, quan um measu emen and decohe ing his o ies di ec ions. This pape 's ea u es:
•
View  ime no jus as ela ional pa ame e bu as a en ion ime axis join ly
selec ed by bounda y scale mo he ule and gene alized en opy geodesics;
•
Via scale iden i y uni y sca e ing phase, g oup delay, spec al shi unc ion as
p oxies o ime eading, placing mac oscopic g a i y and mic oscopic sca e ing in same
bounda y amewo k;
•
Via sec ion uni e se
S
cons uc ion, p o ide o maliza ion o many-wo lds b anch-
ing based on bounda y and in o ma ion geome y, wi hou in oducing addi ional on ol-
ogy.

8 Conclusion
This pape in unied amewo k o bounda ysca e inggene alized en opya en ion
gi es p o able and es able o mula ions o h ee ypes o ques ions: how does bounda y
exis wi hou ime in absence o obse e  how does obse e a en ion gene a e ime
axis how does double-sli in e e ence eec sec ion selec ion.
Co e elemen s include:
•
Scale mo he ule
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω)
as unique p oxy o
ime eading;
•
Gene alized en opy geodesics and quan um ocusing as geome ic c i e ion o a -
en ion ime axis;
•
Sec ion uni e se
S
as s uc u al space accommoda ing all a en ion pa hs; obse e 's
expe ience me ely one geodesic on i .
In his pic u e, uni e se s uc u ally simul aneously con ains all possible sec ions; ob-
se e 's a en ion selec s among hese sec ions a pa h consis en wi h gene alized en opy
9