Uni ied Theo y o Causal S uc u e:
Time Scale, Pa ial O de , and Gene alized En opy
Haobo Ma1Wenlin Zhang2
1Independen Resea che
2Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
This pape p o ides a uni ied cha ac e iza ion o “wha is causali y” wi hin
a single ma hema ical amewo k. The co e hesis is: causali y is no an ex e nal
ela ion added on o space ime o quan um s a es, bu a he a uni ied objec join ly
de ined by he compa ibili y o h ee ypes o s uc u es:
1. Geome ic pa ial o de : ligh cone s uc u e on globally hype bolic Lo en zian
mani olds and local pa ial o de o small causal diamonds;
2. Uni a y e olu ion and ime scale: uni ied ime scale consis ing o sca -
e ing phase g adien , Wigne –Smi h g oup delay, and spec al shi unc ion;
3. Gene alized en opy and in o ma ion mono onici y: ime a ow cha -
ac e ized by gene alized en opy ex ema on small causal diamonds and QNEC/QFC-
ype inequali ies.
On he spec al and sca e ing side, his pape adop s he scale iden i y
φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω),
whe e φis o al sca e ing semi-phase, ρ el is ela i e s a e densi y, Q(ω) =
−iS(ω)†∂ωS(ω) is Wigne –Smi h g oup delay ope a o . This equali y o igina es
om he Bi man–K e˘ın o mula and spec al shi unc ion heo y, iewing “ ime
delay” as de i a i e o spec al–phase geome y.
On he algeb aic quan um ield heo y and holog aphic g a i y side, his pa-
pe cha ac e izes “modula ime” on causal diamonds ia Tomi a–Takesaki mod-
ula low and Null–Modula double co e , whose gene a o is weigh ed in eg al o
s ess–ene gy enso along null bounda ies, sa is ying Ma ko inclusion-exclusion
and s ong subaddi i i y sa u a ion on o e lapping causal diamond chains. On
g a i y and geome y side, in oducing Gibbons–Hawking–Yo k bounda y e m
and i s null and co ne gene aliza ions ensu es well-de ined a ia ion, B own–Yo k
quasilocal s ess enso becomes Hamil onian gene a o o “ ime ansla ion” along
bounda y, geome ic ime de e mined by Hamil on–Jacobi ela ion.
This pape es ablishes he ollowing uni ied p oposi ion: wi hin he semiclassical–
holog aphic window sa is ying local quan um ene gy condi ions, Hadama d s a es,
and small causal diamond limi , he e exis s a class o uni ied ime scale equi alence
classes [τ] such ha :
1
Geome ic causal pa ial o de is equi alen o exis ence o s ic ly inc easing
ime unc ion τ:M→R;
Sca e ing phase g adien and g oup delay ace gi e eadou o “obse able
causal o de ” unde his scale;
Ex ema and mono onici y o gene alized en opy Sgen on small causal dia-
monds a e equi alen o nonlinea Eins ein equa ions and hei s abili y unde
his scale.
Thus, causali y can be es a ed as: exis ence o a pa ial o de – ime scale
s uc u e sel -consis en on geome ic, sca e ing, and en opic ace s, opologi-
cally non-anomalous. This pape p o ides axioma ic de ini ion o his s uc u e,
p oposes se e al main heo ems, and gi es p oo amewo ks ela ed o scale iden-
i y, in o ma ion-geome ic a ia ional p inciple, and Null–Modula double co e in
appendices.
Keywo ds: Causal s uc u e; Time scale; Pa ial o de ; Gene alized en opy; Spec al
shi unc ion; Wigne –Smi h g oup delay; Bi man–K e˘ın o mula; Tomi a–Takesaki mod-
ula low; Quan um Null Ene gy Condi ion; Small causal diamond; Gibbons–Hawking–
Yo k bounda y e m; B own–Yo k s ess enso ; Null–Modula double co e ; Ma ko
p ope y; In o ma ion-geome ic a ia ional p inciple
1 In oduc ion and His o ical Con ex
In classical gene al ela i i y, causali y is o en cha ac e ized by ligh cones and ime
unc ions. On globally hype bolic Lo en zian mani olds, s able causali y is equi alen o
exis ence o s ic ly inc easing ime unc ion T:M→Rsuch ha i q∈J+(p), hen
T(q)≥T(p). This s uc u e gua an ees well-posedness o Cauchy p oblem and condi ion
o “no closed causal cu es.”
In quan um ield heo y, causali y is usually exp essed as commu a i i y o local ope -
a o s a spacelike sepa a ed poin s, i.e., mic ocausali y. Wigh man axioms and algeb aic
quan um ield heo y amewo ks cons uc ield algeb as gi en causal s uc u e, bu
a ely e e se he ques ion: can causal pa ial o de be “ eco e ed” solely om ope a o
algeb a and s a e s uc u e?
De elopmen o holog aphy and in o ma ion heo y in oduced new pe spec i es cha -
ac e izing causali y ia en opy and ela i e en opy. Jacobson p oposed “en angle-
men equilib ium hypo hesis”: on ixed- olume small geodesic balls, gene alized en opy
eaches ex emum i and only i locally sa is ying Eins ein equa ions, es ablishing co e-
spondence “g a i a ional ield equa ions = small ball en anglemen ex emum condi ion”
in semiclassical window. Subsequen wo k u he combined Ryu–Takayanagi o mula
and ela i e en opy, ela ing second-o de a ia ion o gene alized en opy o bulk gauge
ene gy.
On he o he hand, long- e m de elopmen o sca e ing heo y and spec al heo y
yielded comple e “phase–spec al shi – ime delay” s uc u e. Bi man–K e˘ın o mula
shows ha o pai o sel -adjoin ope a o s (H, H0) sa is ying ace-class pe u ba ion
condi ions, he e exis s spec al shi unc ion ξ(ω) de e mining phase o sca e ing de-
e minan ; his yields equi alence ela ion be ween de i a i e o o al sca e ing phase,
de i a i e o spec al shi unc ion, ela i e s a e densi y, and Wigne –Smi h g oup de-
lay ace. These esul s indica e ha ime scale wi h “phase g adien as densi y” can be
es ablished in equency domain.
2
A ecen impo an de elopmen is p oo o quan um null ene gy condi ion QNEC.
QNEC ela es s ess–ene gy expec a ion alue in null di ec ion a a poin o second-
o de de o ma ion o gene alized en opy o cu su aces passing h ough ha poin in
null di ec ion, being local quan iza ion o ANEC. Koelle –Leichenaue and subsequen
wo k showed ha o hal -spaces o egions cu by null planes, modula Hamil onian
can be w i en as local ene gy low in eg al in null di ec ion, es ablishing “local modula
ime–ene gy low–QNEC” connec ion.
Casini–Hue a–Mye s sys ema ically analyzed acuum modula Hamil onian o sphe -
ical egions in con o mal ield heo y, using con o mal ans o ma ion o map sphe ical cu
su ace o accele a ed coo dina e sys em o Rindle wedge, geome izing modula low as
boos gene a o , de i ing holog aphic en anglemen en opy o sphe ical egions. Casini–
Tes e–To oba s udied modula Hamil onian o gene al egions on null plane, p o ing i s
locali y on null plane and es ablishing Ma ko p ope y and s ong subaddi i i y sa u a-
ion.
These sca e ed de elopmen s poin o s onge uni ied pic u e:
1. On sca e ing and spec al side, ime can be unde s ood as pa ame e de e mined
by phase g adien and g oup delay scale;
2. On algeb aic and holog aphic side, modula ime is de e mined by s a e–algeb a
pai , geome ically ealizable as weigh ed ene gy low on null bounda y;
3. On g a i y and geome y side, ex ema and mono onici y o gene alized en opy on
small causal diamonds a e equi alen o local g a i a ional ield equa ions;
4. On null planes and causal diamond chains, locali y and Ma ko p ope y o modula
Hamil onians gi e in o ma ion s uc u e on causal chains.
The goal o his pape is o inco po a e hese s uc u es in o single axioma ic sys em,
de ining “causali y” as objec join ly cons i u ed by pa ial o de , uni ied ime scale, and
gene alized en opy mono onici y, gi ing equi alence heo ems and opological cons ain s
among he h ee.
2 Model and Assump ions
This sec ion gi es geome ic, sca e ing, algeb aic, and en opic s uc u es used in his
pape , lis ing axioms and assump ions unde lying uni ied causal heo y.
2.1 Geome ic Backg ound and Causal Diamonds
Le (M, g) be ou -dimensional o ien ed, ime-o ien ed Lo en zian mani old wi h me ic
signa u e (−,+,+,+), sa is ying:
Global hype bolici y: he e exis s Cauchy slice Σ ⊂Msuch ha e e y inex endible
imelike o ligh like cu e in e sec s Σ exac ly once;
S able causali y: no closed causal cu es exis , and he e exis s smoo h ime unc-
ion T:M→Rs ic ly inc easing along all u u e-di ec ed imelike cu es.
3
Fo any poin p∈M, aking su icien ly small p ope scale ≪Lcu (p), de ine small
causal diamond
Dp, =J+(p−)∩J−(p+),
whe e p±a e poin s a p ope ime ± along some e e ence imelike di ec ion. The
bounda y o Dp, consis s o wo amilies o null hype su aces N±gene a ed by null
geodesics and hei in e sec ion lines, cons i u ing basic uni o local causal geome y.
Axiom 1 (Geome ic Causal Axiom).1. (M, g) sa is ies abo e global hype bolici y
and s able causali y;
2. Fo any pand su icien ly small , small causal diamond Dp, is homeomo phic
o causal diamond in Minkowski space unde no mal coo dina es, wi h cu a u e
co ec ions O( 2).
2.2 Sca e ing Sys em and Spec al Shi Func ion
On Hilbe space Hconside pai o sel -adjoin ope a o s (H, H0) sa is ying:
His ace-class pe u ba ion ela i e o H0, o esol en di e ence is ace-class;
Wa e ope a o s W±exis and a e comple e, so sca e ing ope a o S=W†
+W−is
well-de ined;
On absolu ely con inuous spec um, S ibe izes in o uni a y ma ix amily S(ω).
By spec al shi unc ion heo y, he e exis s locally in eg able unc ion ξ(ω) such
ha o su icien ly smoo h es unc ion ,
( (H)− (H0)) = ZR
ξ(ω) ′(ω) dω,
and Bi man–K e˘ın o mula gi es
de S(ω) = exp−2πiξ(ω).
De ine o al sca e ing phase
Φ(ω) = a g de S(ω), φ(ω) = 1
2Φ(ω),
ela i e s a e densi y
ρ el(ω) = −ξ′(ω),
and Wigne –Smi h g oup delay ope a o
Q(ω) = −iS(ω)†∂ωS(ω),
whose ace Q(ω) co esponds o g oup delay a ene gy ω.
F om abo e de ini ions ob ain scale iden i y
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
holding unde app op ia e egula i y and ene gy window es ic ions.
4
Axiom 2 (Sca e ing Scale Axiom).1. Fo conside ed ene gy window I⊂R, scale
iden i y holds almos e e ywhe e;
2. ρ el(ω)≥0 almos e e ywhe e, and ρ el ≡ 0;
3. Q(ω) is posi i e semi-de ini e ope a o on I, and Q(ω) is locally in eg able.
Based on his de ine sca e ing ime scale:
De ini ion 2.1 (Sca e ing Time Scale).Rela i e o e e ence poin ω0∈I, de ine
τsca (ω)−τsca (ω0) = Zω
ω0
ρ el(˜ω) d˜ω=1
2πZω
ω0
Q(˜ω) d˜ω.
By Axiom ??,τsca is s ic ly inc easing on I.
2.3 Bounda y Algeb a, Modula Flow, and Null–Modula Dou-
ble Co e
Le A∂be obse able algeb a on app op ia e bounda y, ωi s ai h ul no mal s a e.
Tomi a–Takesaki heo y assigns co esponding modula ope a o ∆ωand modula low
σω
(A)=∆i
ωA∆−i
ω.
Fo acuum s a e o sphe ical o wedge egions in Minkowski space ime, modula low
can be geome ized as Killing low in co esponding causal diamond o wedge egion.
Fo causal diamond D(p, q), i s bounda y decomposes in o wo null hype su aces N±;
emo ing co ne s yields wo shee s E±. On each shee in oduce a ine pa ame e λand
ans e se coo dina e x⊥, cons uc ing double co e o null bounda y
e
ED=E+⊔E−.
Resul s by Casini–Tes e–To oba e al. show ha in con o mal ield heo y acuum
and i s app op ia e de o ma ions, modula Hamil onian o Dcan be w i en as local
ene gy low in eg al along e
ED:
KD= 2πX
σ=±ZEσ
gσ(λ, x⊥)Tσσ(λ, x⊥) dλdd−2x⊥,
whe e Tσσ is s ess–ene gy enso componen in null di ec ion, gσweigh unc ion
de e mined by geome y.
Axiom 3 (Modula Flow Localiza ion Axiom).1. Fo small causal diamond Dp, , abo e
local modula Hamil onian exp ession exis s wi hin semiclassical–holog aphic win-
dow;
2. Modula ime pa ame e mod is mono onically co-o ien ed wi h null di ec ion a ine
pa ame e λ.
5
2.4 GHY Bounda y Te m, B own–Yo k S ess, and Geome ic
Time
In g a i a ional sys em wi h codimension-one bounda y ∂M, Eins ein–Hilbe ac ion
SEH =1
16πG ZM
R√−gd4x
p oduces no mal de i a i e e ms on bounda y du ing a ia ion. To ensu e well-
de ined a ia ion unde ixed bounda y induced me ic hab, mus add Gibbons–Hawking–
Yo k bounda y e m
SGHY =1
8πG Z∂M
Kp|h|d3x,
plus gene aliza ions o null and co ne e ms. B own–Yo k quasilocal s ess enso
de ined as
Tab
BY =2
p|h|
δS
δhab
=1
8πG(Kab −Khab) + ··· ,
whe e ··· deno es null and co ne co ec ions. Fo Killing ec o aalong bounda y
ime ansla ion, co esponding Hamil onian is
H∂=ZΣ
Tab
BY anbdd−1x,
whe e Σ is spa ial slice on bounda y, nai s no mal. Geome ic ime τgeom can be
iewed as pa ame e gene a ed by H∂, ela ed o bounda y ac ion ia Hamil on–Jacobi
ela ion.
Axiom 4 (Bounda y Va ia ion Axiom).1. Ac ion S=SEH +SGHY +··· has well-
de ined a ia ion unde ixed bounda y geome ic da a;
2. Tab
BY is bounded, and H∂is well-de ined on selec ed bounda y shee amily;
3. Bounda y ime ansla ion g oup {Φτgeom }can be con o mally aligned wi h modula
low wi hin semiclassical window.
2.5 Gene alized En opy and Local Quan um Condi ions
Fo cu su ace Σ inside causal diamond Dp, , de ine gene alized en opy
Sgen(Σ) = A(Σ)
4Gℏ+Sou (Σ),
whe e Sou is on Neumann en opy o quan um ield ou side cu su ace. Jacob-
son’s “en anglemen equilib ium” and subsequen wo k show ha unde app op ia e con-
s ain s, ex emum condi ion o Sgen on small balls o small causal diamonds is equi alen
o local Eins ein equa ions.
Quan um Null Ene gy Condi ion gi es local inequali y in null di ec ion:
⟨Tkk(x)⟩ψ≥ℏ
2π
d2Sou
dλ2(x),
whe e kais null ec o , λi s a ine pa ame e . This inequali y has been igo ously
p o en in b oad CFT classes, closely ela ed o local modula Hamil onian de o ma ion.
6
Axiom 5 (Gene alized En opy–Ene gy Axiom).1. Fo each small causal diamond
Dp, , unde ixed app op ia e “ olume” o equi alen local conse a ion cons ain ,
Sgen a ains i s -o de ex emum a a e e ence cu su ace;
2. Fo all null di ec ion de o ma ions, second-o de a ia ion o Sgen sa is ies QNEC/QFC-
ype inequali ies;
3. Rela i e en opy is independen o Cauchy slice olia ion, equi alen o Iye –Wald
canonical ene gy.
2.6 Topology and Z2Sec o Assump ion
In sys ems wi h sel - e e en ial sca e ing ne wo ks o non i ial opology, squa e oo
o sca e ing semi-phase gi es p incipal Z2bundle whose holonomy ν√S(γ)∈ {±1}can
be iewed as opological indica o on loops. Co esponding bulk BF heo y sec o class
[K]∈H2(Y, ∂Y ;Z2) desc ibes possible opological anomalies.
Axiom 6 (Topological Non-anomaly Axiom).In small causal diamond limi and ini e
egions glued om hem, sa is ies [K] = 0, equi alen ly, o all app op ia e closed loops
γ,ν√S(γ) = +1.
3 Main Resul s: Theo ems and Alignmen s
Unde abo e axioma ic sys em, his pape p oposes and a gues ollowing main esul s.
3.1 Theo em 1: Uni ied Time Scale Equi alence Class
Theo em 3.1 (Uni ied Time Scale Equi alence Class).Wi hin semiclassical–holog aphic
window whe e Axioms ??,??, and ?? hold, he e exis s ime scale equi alence class [τ]
sa is ying:
1. Sca e ing ime scale τsca belongs o [τ];
2. Modula ime τmod belongs o [τ];
3. Geome ic ime τgeom belongs o [τ].
Mo e speci ically, he e exis cons an s a > 0, b ∈Rsuch ha on conside ed ene gy
window and causal diamond amily,
τsca =aτmod +b, τgeom =cτmod +d,
whe e c > 0, d ∈Ra e cons an s. Equi alence class [τ]is ep esen ed by any o abo e
scales, unique up o a ine ans o ma ion.
7
3.2 Theo em 2: Equi alen Cha ac e iza ions o Causal Pa ial
O de
Theo em 3.2 (Equi alen Cha ac e iza ions o Causal Pa ial O de ).In egion whe e
Axioms ??,??,??,?? hold, uni ied ime scale equi alence class [τ]gi es equi alen
cha ac e iza ions o causal pa ial o de :
Fo any p, q ∈M, ollowing a e equi alen :
1. Geome ic causali y: q∈J+(p), i.e., he e exis s u u e-di ec ed non-spacelike cu e
om p o q;
2. Time scale mono onici y: he e exis s τ∈[τ]such ha o all connec ions γ om
p o qalong imelike cu es, τ(p)≤τ(q), wi h s ic inequali y o some connec ing
cu e;
3. Gene alized en opy mono onici y: o each su icien ly small causal diamond chain
{Dj}con aining p, q, gene alized en opy Sgen is co-mono one wi h τin null di ec-
ion, gi ing non-nega i e “en opy dis ance” in p→qlimi .
Thus, causal pa ial o de can equi alen ly be iewed as “mono onici y on uni ied
ime scale” o “mono onic s uc u e o gene alized en opy low on small causal diamond
chains.”
3.3 Theo em 3: Gene alized En opy Va ia ional P inciple and
Local G a i a ional Equa ions
Theo em 3.3 (IGVP and Eins ein Equa ions).When Axioms ?? and ?? hold, assuming
ma e ields sa is y app op ia e local conse a ion condi ions and Hadama d s a e con-
di ions, gene alized en opy a ia ional condi ion on small causal diamonds is equi alen
o local Eins ein equa ions:
Fo any p∈Mand su icien ly small , i o all null di ec ion de o ma ions, unde
ixed app op ia e cons ain s Sgen a ains i s -o de ex emum a e e ence cu su ace
wi h non-nega i e second-o de a ia ion, hen a p
Gab + Λgab = 8πG Tab,
whe e Gab is Eins ein enso , Tab ma e s ess–ene gy enso , Λcons an .
Con e sely, i abo e g a i a ional equa ions hold and ma e s a e sa is ies obse ed
local equilib ium condi ion, hen gene alized en opy on small causal diamonds sa is ies
i s -o de ex emum and second-o de non-nega i i y equi ed by Axiom ??.
3.4 Theo em 4: Null–Modula Double Co e , Ma ko P op-
e y, and Causal Chains
Theo em 3.4 (Ma ko S uc u e on Causal Chains).When Axioms ??,??,?? hold in
con o mal ield heo y acuum o i s small de o ma ions, o egion amily on null plane
o causal diamond amily {Dj}as hei con o mal images, modula Hamil onians sa is y
inclusion-exclusion s uc u e
8
K∪jDj=X
k≥1
(−1)k−1X
j1<···<jk
KDj1∩···∩Djk,
wi h co esponding ela i e en opy sa is ying Ma ko p ope y and s ong subaddi i i y
sa u a ion. This yields:
1. In o ma ion p opaga ion on causal diamond chains is “no ex a memo y” Ma ko
p ocess;
2. Uni ied ime scale τon his chain ag ees wi h modula ime, co-mono one wi h null
di ec ion a ine pa ame e ;
3. Gene alized en opy a ow on small causal diamond chains ag ees wi h geome ic
causal a ow.
3.5 Theo em 5: Equi alence o Topological Non-anomaly and
Gauge Ene gy Non-nega i i y
Theo em 3.5 (Topological Non-anomaly).When Axiom ?? holds, in ini e egions glued
om small causal diamonds, ollowing a e equi alen :
1. Z2–BF bulk sec o class [K]=0;
2. Fo all physically allowed closed loops γ, holonomy o sca e ing semi-phase squa e
oo sa is ies ν√S(γ) = +1;
3. On ma e con igu a ions sa is ying g a i a ional ield equa ions and local quan um
condi ions, second-o de a ia ion o gauge ene gy on small causal diamonds is
non-nega i e.
Con e sely, i he e exis s [K]= 0 o loop wi h ν√S(γ) = −1, one can cons uc
con igu a ion iola ing gauge ene gy non-nega i i y, b eaking consis ency o gene alized
en opy mono onici y and causal a ow.
4 P oo s
This sec ion gi es p oo ideas and key s eps o main heo ems, placing echnical de ails
in appendices.
4.1 P oo Idea o Theo em ??: Uni ied Time Scale Equi alence
Class
S ep 1: Exis ence and a ine uniqueness o sca e ing ime scale
By Axiom ??, scale iden i y holds in ene gy window I, wi h ρ el(ω)≥0 almos
e e ywhe e, no iden ically ze o. De ine
τsca (ω)−τsca (ω0) = Zω
ω0
ρ el(˜ω) d˜ω,
9
2. Geome ic pa ial o de (M, ⪯), mono onici y on uni ied scale, and gene alized
en opy mono onici y on small causal diamond chains a e equi alen , gi ing h ee
complemen a y cha ac e iza ions o causali y;
3. Gene alized en opy a ia ional p inciple on small causal diamonds is equi alen o
local Eins ein equa ions, making g a i a ional ield equa ions exp ession o “how
en opy o ganizes on causal bounda y”;
4. Null–Modula double co e and Ma ko p ope y gua an ee in o ma ion p opaga-
ion on causal diamond chains has locali y and no ex a memo y s uc u e;
5. Topological non-anomaly o Z2–BF sec o is equi alen o gauge ene gy non-nega i i y,
opologically cons aining allowed sec o s o causal s uc u e.
In his pic u e, space ime me ic, sca e ing ma ix, modula low, and gene alized
en opy a e no longe independen objec s, bu mani es a ions o same causal s uc u e
in di e en p ojec ions. Time is unde s ood as s ic ly mono one scale coo dina e on
his s uc u e, whose a ow join ly de e mined by gene alized en opy mono onici y and
opological non-anomaly.
Acknowledgemen s
Au ho s hank esea ch wo k in ele an ields o ounda ions p o ided o his pape ,
including spec al shi unc ion and Bi man–K e˘ın heo y, quan um null ene gy condi ion
and local modula Hamil onian, holog aphic en anglemen en opy and g a i a ional ield
equa ions, and se ies o s udies on Null–Modula double co e and Ma ko p ope y.
Code A ailabili y
All esul s in his pape based on analy ical de i a ions and exis ing ma hema ical physics
heo ems, wi hou specialized nume ical code. I u u e wo k conduc s simula ions based
on sca e ing ne wo ks and nume ical ela i i y, co esponding code implemen a ions will
be made publicly a ailable sepa a ely.
Re e ences
[1] M. Sh. Bi man and M. G. K e˘ın, “On he heo y o wa e and sca e ing ope a o s,”
So ie Ma h. Dokl. 3, 740–744 (1962).
[2] K. B. Sinha, “Spec al shi unc ion and ace o mula,” P oc. Indian Acad. Sci.
(Ma h. Sci.) 104, 571–588 (1994).
[3] D. Bo hwick, “The Bi man–K e˘ın o mula and sca e ing phase,” a Xi :2110.06370
(2021).
[4] H. Casini, M. Hue a and R. C. Mye s, “Towa ds a de i a ion o holog aphic en an-
glemen en opy,” JHEP 05, 036 (2011).
16
[5] T. Jacobson, “En anglemen Equilib ium and he Eins ein Equa ion,” Phys. Re .
Le . 116, 201101 (2016).
[6] R. Bousso, Z. Fishe , S. Leichenaue and A. C. Wall, “P oo o he Quan um Null
Ene gy Condi ion,” Phys. Re . D 93, 024017 (2016).
[7] S. Balak ishnan, T. Faulkne , Z. U. Khandke and H. Wang, “A gene al p oo o he
quan um null ene gy condi ion,” JHEP 09, 020 (2019).
[8] J. Koelle and S. Leichenaue , “Local modula Hamil onians om he quan um null
ene gy condi ion,” Phys. Re . D 97, 065011 (2018).
[9] H. Casini, E. Tes e and G. To oba, “Modula Hamil onians on he null plane and
he Ma ko p ope y o he acuum s a e,” J. Phys. A: Ma h. Theo . 50, 364001
(2017).
[10] G. S´a osi and T. Ugajin, “Modula Hamil onians o exci ed s a es, OPE blocks and
eme gen bulk ields,” JHEP 01, 012 (2018).
[11] A. Shahbazi-Moghaddam, “Aspec s o Gene alized En opy and Quan um Null En-
e gy Condi ion,” PhD hesis, Uni e si y o Cali o nia, Be keley (2020).
[12] E. Oh, I.-Y. Pa k and S.-J. Sin, “Comple e Eins ein equa ions om he gene alized
i s law o en anglemen ,” Phys. Re . D 98, 026020 (2018).
A Scale Iden i y and Cons uc ion o Uni ied Time
Scale
This appendix gi es de i a ion amewo k o scale iden i y and echnical de ails o exis ence–
uniqueness o uni ied ime scale equi alence class.
A.1 Spec al Shi Func ion and Bi man–K e˘ın Fo mula
Le (H, H0) be sel -adjoin ope a o pai sa is ying di e ence is ace-class o esol en
di e ence is ace-class. By spec al shi unc ion heo y, he e exis s unique (modulo
cons an ) unc ion ξ(ω) such ha o any ∈C∞
0(R),
( (H)− (H0)) = ZR
ξ(ω) ′(ω) dω.
Choosing smoo h app oxima ion o (λ) = χ(−∞,ω](λ), can in e p e ξ(ω) as ace o
wo spec al p ojec ion di e ence, i.e.,
ξ(ω) = EH((−∞, ω]) −EH0((−∞, ω]).
Bi man–K e˘ın o mula gi es ela ion be ween sca e ing de e minan and spec al
shi unc ion:
de S(ω) = exp−2πiξ(ω).
Taking loga i hm and di e en ia ing yields
17
∂ωΦ(ω) = ∂ωa g de S(ω) = −2πξ′(ω),
hence de ine
ρ el(ω) = −ξ′(ω) = 1
2πΦ′(ω).
A.2 Wigne –Smi h G oup Delay T ace
Sca e ing ma ix S(ω) is uni a y ope a o amily on absolu ely con inuous spec um; i s
equency de i a i e gi es Wigne –Smi h g oup delay ope a o
Q(ω) = −iS(ω)†∂ωS(ω).
By uni a i y o S(ω), Q(ω) is sel -adjoin . Decomposing S(ω) in o eigenphases and
eigen ec o s, S(ω) = Pne2iδn(ω)|n(ω)⟩⟨n(ω)|, hen
Q(ω) = 2 X
n
∂δn(ω)
∂ω |n(ω)⟩⟨n(ω)|+ o -diagonal e ms,
hence
Q(ω)=2X
n
∂δn(ω)
∂ω =∂ωΦ(ω).
This yields scale iden i y
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω).
In non-local po en ial o dissipa i e sca e ing cases, can gene alize abo e o mula
using modi ied de e minan s and sel -adjoin ex ension heo y; o m emains unchanged,
only equi ing eno maliza ion o spec al shi unc ion.
A.3 Windowed Clock and A ine Uniqueness
To supp ess esonance and high- equency ail e ec s, in oduce posi i e de ini e window
unc ion h∆(e.g., Poisson ke nel)
h∆(ω) = ∆
π(ω2+ ∆2),
de ine con olu ion
Θ∆(ω) = (ρ el ∗h∆)(ω) = ZR
ρ el(˜ω)h∆(ω−˜ω) d˜ω.
I ρ el(ω)≥0 and no iden ically ze o in conside ed ene gy window, hen Θ∆(ω)>0
almos e e ywhe e. De ine windowed ime scale
∆(ω)− ∆(ω0) = Zω
ω0
Θ∆(˜ω) d˜ω,
hen ∆is s ic ly inc easing and con inuous.
18
Fo any unc ion ˜
∆sa is ying same window condi ion wi h de i a i e almos e e y-
whe e p opo ional o Θ∆, he e exis a > 0, b ∈Rsuch ha ˜
∆=a ∆+b. Thus on
ene gy window I, scale o “sca e ing clock” is unique up o a ine ans o ma ion.
B Local De i a ion o IGVP and Local Eins ein Equa-
ions
B.1 Small Causal Diamond Geome y and A ea Va ia ion
In Riemann no mal coo dina es a poin p, me ic expands as
gµν(x) = ηµν −1
3Rµανβ(p)xαxβ+O(|x|3).
Conside ing small ball o small causal diamond cen e ed a pwi h adius , expansion
o bounda y a ea and olume con ains Rµν con ibu ions. Fo example d-dimensional
small ball olume has
V(Bp, ) = Ωd−1
d dh1−R(p)
6(d+ 2) 2+O( 4)i.
Along null ec o kagene a ed geodesic amily, expansion θsa is ies Raychaudhu i
equa ion
dθ
dλ=−1
2θ2−σabσab −Rabkakb.
Fo su icien ly small and app op ia e ini ial condi ions, a ea second-o de a ia ion
can be exp essed as e m con aining Rkk plus non-nega i e con ibu ions like θ2, σ2; in
→0 limi , Rkk domina es a ea a ia ion.
B.2 Gene alized En opy Va ia ion and Local Fi s Law
Fi s -o de gene alized en opy a ia ion is
dSgen
dλ=1
4Gℏ
dA
dλ+dSou
dλ.
Fi s -o de ela i e en opy S(ρ||σ) a ia ion wi h espec o sou ce s a e ρgi es local
i s law
δSou =δ⟨Kmod⟩,
whe e Kmod is modula Hamil onian o e e ence s a e σ. Fo sphe ical egion o small
causal diamond, can localize Kmod as s ess–ene gy enso in eg al, yielding
dSou
dλ∝ZTkk dλdd−2x⊥.
Requi ing dSgen/dλ= 0 unde ixed olume o equi alen cons ain , aking →0,
using p opo ional ela ion be ween Rkk e m in a ea a ia ion and abo e Tkk e m, can
ob ain
19
Rkk = 8πG Tkk.
Holding o all null di ec ions, combined wi h Bianchi iden i y and ene gy–momen um
conse a ion, eco e s comple e Eins ein equa ions.
B.3 Second-o de Va ia ion and Gauge Ene gy Non-nega i i y
Second-o de gene alized en opy a ia ion ela i e o e e ence s a e can be w i en as
gauge ene gy
E=δ2S el,
whose non-nega i i y is equi alen o Hollands–Wald gauge ene gy non-nega i i y,
quan i ied in QNEC/QFC esul s. Gauge ene gy non-nega i i y ensu es IGVP ex emum
on small causal diamonds is s able ex emum, gua an eeing s abili y o local Eins ein
equa ions and consis ency o causal a ow.
C Null–Modula Double Co e and Ma ko S uc-
u e
C.1 Modula Hamil onian on Null Plane
On null plane Po Minkowski space, conside hal -space desc ibed by ligh like coo di-
na es (u, , x⊥), e.g., ≥ (x⊥). Casini–Tes e–To oba gi e local modula Hamil onian
exp ession in acuum s a e:
KA= 2πZA
(λ− (x⊥)) T (λ, x⊥) dλdd−2x⊥,
whe e Ais egion on null plane. This exp ession shows modula low ansla es poin s
in null di ec ion, modula ime linea ly ela ed o a ine pa ame e λ.
C.2 Inclusion-exclusion P ope y and Ma ko P ope y
Fo egion amily {Aj}on null plane, using linea supe posi ion o s ess–ene gy enso
and modula Hamil onian de ini ion, can p o e
K∪jAj=X
k≥1
(−1)k−1X
j1<···<jk
KAj1∩···∩Ajk.
Rela i e en opy S(ρA||σA) sa is ies s ong subaddi i i y unde enso p oduc s uc-
u e; combined wi h abo e modula Hamil onian inclusion-exclusion ela ion, acuum
s a e o null plane egion amily sa is ies Ma ko p ope y, i.e., condi ional mu ual in o -
ma ion ze o. This means in o ma ion p opaga ion in causal chain along null di ec ion
ca ies no ex a memo y, depending only on adjacen segmen in o ma ion.
20
C.3 Con o mal Mapping o Causal Diamond Families
Th ough con o mal ans o ma ion, can map null plane egion amilies o causal diamond
amilies {Dj}in Minkowski space o mo e gene al backg ounds. Con o mal in a iance
ensu es local s uc u e and inclusion-exclusion p ope ies o modula Hamil onians p e-
se ed, es ablishing Ma ko s uc u e on causal diamond chains. Uni ied ime scale τ
on his chain co-mono one wi h null di ec ion a ine pa ame e , uni ying causal a ow,
modula ime a ow, and gene alized en opy a ow.
21
