The Universe as a Maximal Consistent Mathematical Structure\\ \large Unified Scattering Scale, Generalized Entropy and Category Terminal Object

The Uni e se as a Maximal Consis en Ma hema ical
S uc u e
Unied Sca e ing Scale, Gene alized En opy and Ca ego y Te minal
Objec
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 19, 2025
Abs ac
Based on exis ing amewo ks including causal mani olds, axioma ic quan um
eld heo y, sca e ing and spec al shi heo y, Tomi aTakesaki modula he-
o y, gene alized en opy and Quan um Null Ene gy Condi ion (QNEC), and Gib-
bonsHawkingYo k bounda y e ms wi h B ownYo k quasilocal s ess enso s, we
in oduce a mul i-laye ed s uc u al objec
U= (Ue , Ugeo, Umeas, UQFT, Usca , Umod, Uen , Uobs, Uca , Ucomp)
(1)
as he unied ma hema ical cha ac e iza ion o he "Uni e se". This pape p esen s
h ee main h eads:
Fi s , we in oduce p ecise sucien condi ions o sca e ing,
A1

A5
, and p o e
he exis ence o a unique scale densi y
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω),
(2)
dis inguishing wo ypes o mo he scale eadings: Phase Reading
Θ(ω) = φ(ω)/π
and Sca e ing Time Reading
τsca (ω) = (2π)−1 Q(ω)
.
κ
se es as he unied
scale densi y connec ing spec al shi unc ion, o al sca e ing phase, and he ace
o he Wigne Smi h ime delay ma ix.
Second, unde he Geome icModula Bounda y Condi ion package
B1

B4
,
we in oduce a p oposi ion s a ing om KMS s a es: i a KMS s a e o a one-
pa ame e au omo phism g oup on a bounda y algeb a gi es a Tomi aTakesaki
modula s uc u e, hen he modula g oup is iden ical o ha physical g oup, wi h
pa ame e s die ing only by in e se empe a u e scaling. F om his, we p o e ha
in cases wi h BisognanoWichmann ype geome ic modula ow, he e exis s an
ane alignmen among modula ime, bounda y geome ic ime, and sca e ing
ime.
Thi d, unde he Gene alized En opy and QNEC package
C1

C4
, we cons uc
a lemma chain in he limi o small causal diamonds: p o iding a eno malized
second a ia ion o mula o gene alized en opy, con olling p ecise coecien s and
shea e ms in he Raychaudhu i equa ion, and u ilizing QNEC and a s a e- ichness
assump ion o ele a e he inequali y in null ec o di ec ions o a enso equali y,
he eby locally eco e ing he Eins ein equa ion
Gab + Λgab = 8πG ⟨Tab⟩
.
1
A he obse e le el, we o ganize local causal agmen s, obse able algeb as,
and upda e ope a o s in o a
2
-s ack on causal diamond si es. Using alidi y and
sepa a ion condi ions, we glue obse a ional da a in o a global HaagKas le ne
and global causal pa ial o de . A he ca ego ical le el, wi hin a
2
-ca ego y
Uni U
con olled by a G o hendieck uni e se, we dene
U
as a e minal objec , p o ing ha
unde he p emise o "exis ence as a s uc u al hypo hesis", he uni e se objec is
unique up o isomo phism. On he enginee ing and nume ical le el, we p opose h ee
ypes o expe imen al and nume ical pla o ms: mul i-po sca e ing ne wo ks,
Rindle wedges, and AdS/CFT sub egions, o e i y he scale iden i y and ime
alignmen p oposi ions.
Keywo ds:
Uni e se On ology; Causal Mani old; HaagKas le Ne ; Spec al Shi
Func ion; Wigne Smi h Time Delay; Tomi aTakesaki Modula Theo y; ConnesRo elli
The mal Time; Gene alized En opy; QNEC; GibbonsHawkingYo k Bounda y Te m;
B ownYo k Quasilocal Tenso ; BisognanoWichmann Theo em; HaagKas le S acks;
Ca ego y Te minal Objec ; Compu abili y
1 No a ions & Uni s
1. Uni Con en ion: Na u al uni s
ℏ=c= 1
a e used. Ene gy, angula equency,
and in e se ime ha e he same dimension; ime and leng h a e also ea ed as ha ing
he same dimension. Physical uni s can be es o ed ia ela ions like
phys =ℏ
when
necessa y.
2. Va iable Con en ion: The sca e ing a iable is deno ed as
ω
, unde s ood as ene gy
o angula equency; no dis inc ion is made in na u al uni s. De i a i es o spec al shi
unc ion and Wigne Smi h ma ix a e wi h espec o
ω
.
3. Ma ix ace is deno ed as
Q(ω)
; all aces and de e minan s a e modied F ed-
holm e sions.
4. Gene alized en opy
Sgen =A/(4Gℏ) + Sou
is w i en as
Sgen =A/(4G) + Sou
in
his pape using
ℏ= 1
.
5. All s a emen s like "almos e e ywhe e" imply Lebesgue almos e e ywhe e by de-
aul ; echnical dis inc ions be ween spec al measu e and Lebesgue measu e a e omi ed
in he sca e ingspec al con ex .
2 In oduc ion & His o ical Con ex
Gene al Rela i i y desc ibes he uni e se as a causal mani old wi h a Lo en zian me ic
(M, g)
, whe e he Eins ein equa ion
Gab + Λgab = 8πGTab
(3)
ela es geome y o ene gy-momen um. Algeb aic Quan um Field Theo y cha ac e -
izes he local s uc u e and mic o-causali y o quan um elds on a gi en
(M, g)
ia a
HaagKas le ne o local obse able algeb as
A(O)
and s a es
ω
.
In sca e ing heo y, when he die ence
H−H0
o a pai o sel -adjoin ope a o s
(H, H0)
sa ises ela i e ace-class condi ions, he e exis s a spec al shi unc ion
ξ(ω)
sa is ying he Li shi sK en ace o mula and Bi manK en o mula
de S(ω) = exp(−2πiξ(ω)),
(4)
2
whe e
S(ω)
is he sca e ing ma ix. The eigen alues o he Wigne Smi h ime delay
ma ix
Q(ω) = −iS†(ω)∂ωS(ω)
(5)
a e in e p e ed as g oup delay imes, which ha e been ealized in quan um, mic owa e,
and acous ic sca e ing expe imen s.
Tomi aTakesaki modula heo y shows ha on a s anda d o m
(M, ω)
, he e exis
a modula ope a o
∆
and modula ow
σω
(A)=∆i A∆−i .
(6)
The ConnesRo elli he mal ime hypo hesis sugges s ha in gene al co a ian heo ies,
he modula pa ame e
can be iewed as ime in insically dened by he s a e-algeb a
pai .
In he geome ic-en opy di ec ion, Jacobson's "en anglemen equilib ium" scheme
o small balls connec s he local en anglemen en opy equilib ium condi ion o he
Eins ein equa ion; Faulkne LewkowyczMaldacena inco po a ed modula Hamil oni-
ans and gene alized en opy ia quan um-co ec ed holog aphic en opy o mulas; Ja -
e isLewkowyczMaldacenaSuh ela ed he modula Hamil onian o bounda y QFT o
he Hamil onian geome ic ow in bulk g a i y using ela i e en opy.
The BisognanoWichmann heo em u he elucida es ha o he Minkowski acuum
es ic ed o a Rindle wedge, he modula ow is iden ical o he Lo en z boos o ha
wedge, explaining he Un uh eec and iden i ying modula ime and geome ic ime as
die en pa ame e iza ions o he same symme y g oup ac ion.
The abo e wo ks p o ide ich local s uc u es: highly non- i ial ela ionships exis
among causali y, algeb a, sca e ing, modula ow, and en opy-g a i y. Howe e , he
"Uni e se as a whole" is o en ea ed as an ex e nal backg ound. This pape a emp s
o p o ide a single ma hema ical objec
U= (Ue , Ugeo, Umeas, UQFT, Usca , Umod, Uen , Uobs, Uca , Ucomp)
(7)
making all he abo e le els die en p ojec ions o his objec , connec ed by p ecise
condi ions and heo ems.
3 Model Assump ions
3.1 Founda ion and Size: G o hendieck Uni e se and
Uni U
Take a xed G o hendieck uni e se
U
. All se s, mani olds, Hilbe spaces,
C∗
-algeb as,
on Neumann algeb as, and ca ego ies/2-ca ego ies o med by hem a e assumed o be
U
-
small o locally small. Deno e
Se U
,
HilbU
,
C∗AlgU
,
NU
as he co esponding
U
-small
ca ego ies.
Dene a "Candida e Uni e se S uc u e" as a se o hie a chical s uc u es and axioms
equipped on a amily o
U
-small objec s; he 2-ca ego y o all candida e uni e ses is
deno ed by
Uni U
, wi h mo phisms and 2-mo phisms ened in Sec ion 7.
3.2 Componen s o Mul i-Laye ed Uni e se Objec
3.2.1 E en and Causal Laye
Ue
Dene
Ue = (X, ⪯,C)
(8)
3
whe e
X∈Se U
is he se o e en s,
⪯⊆ X×X
is a pa ial o de , and
C ⊆ P(X)
is
a amily o causal agmen s, sa is ying: 1. Fo any
C∈ C
,
(C, ⪯ |C)
is locally ni e;
2.
SC∈C C=X
; 3.
(X, ⪯)
is s ably causal: no closed causal loops, and he e exis s a
s ic ly inc easing ime unc ion
Tcau :X→R
.
Dene he amily o small causal diamonds
D={D⊆X:D=J+(p)∩J−(q), p ⪯q}.
(9)
3.2.2 Geome ic Laye
Ugeo
Dene
Ugeo = (M, g, Φe ,Φcau)
(10)
whe e: 1.
M
is a 4D o ien able, ime-o ien ed
C∞
mani old,
M∈Se U
; 2.
g
is a
Lo en zian me ic wi h signa u e
(−+ ++)
; 3.
Φe :X→M
is an e en embedding;
4.
(M, g)
is globally hype bolic: he e exis s a Cauchy hype su ace
Σ⊂M
such ha
e e y imelike o null causal cu e in e sec s
Σ
exac ly once; 5. Causal ela ion pullback
pa ial o de : o
x, y ∈X
,
x⪯y⇐⇒ Φe (y)∈J+
g(Φe (x)).
(11)
The e exis s a geome ic ime unc ion
Tgeo :M→R
, whose g adien is e e ywhe e
imelike and compa ible wi h he causal s uc u e.
3.2.3 Measu e and S a is ical Laye
Umeas
Dene
Umeas = (Ω,F,P,Ψ)
(12)
whe e
(Ω,F,P)
is a comple e p obabili y space,
Ψ:Ω→X
is a andom e en map.
Fo a wo ldline
γ⊂M
and i s p eimage, sample pa hs
Ψγ: Ω →XZ
,
Ψγ(ω)=(xn)n∈Z
sa is ying
xn⪯xn+1
can be dened, inducing causally o de ed ime se ies p ocesses.
3.2.4 Quan um Field and Ope a o Algeb a Laye
UQFT
Dene
UQFT = (O(M),A, ω)
(13)
whe e: 1.
O(M)
is he amily o bounded causally con ex open se s on
M
; 2.
A:
O(M)→ NU
is a HaagKas le ne
O7→ A(O)
, sa is ying axioms like mono onici y,
co a iance, mic o-causali y; 3.
ω
is a no mal s a e, gi ing a posi i e, no malized linea
unc ional on each
A(O)
.
GNS cons uc ion gi es
(πω,H,Ωω)
, whe e
H ∈ HilbU
, and
Ωω
is cyclic and sepa a -
ing.
3.2.5 Sca e ing and Spec al Laye
Usca
Gi en a pai o sel -adjoin ope a o s
(H, H0)
on Hilbe space
Hsca ∈HilbU
, wi h
die ence
H−H0
sa is ying ela i e ace-class condi ions. The e exis spec al shi
unc ion
ξ(ω)
, sca e ing ma ix
S(ω)
, and Wigne Smi h ma ix
Q(ω) = −iS(ω)†∂ωS(ω).
(14)
Scale densi y and sca e ing ime will be p ecisely dened in Sec ion 3.
4
3.2.6 Modula Flow and The mal Time Laye
Umod
On a on Neumann algeb a
M ⊆ B(H)
and ai h ul no mal s a e
ω
, Tomi aTakesaki
heo y gi es modula ope a o
∆
and modula ow
σω
(A)=∆i A∆−i .
(15)
The modula Hamil onian is dened as
Kω:= −log ∆
, hen
σω
(A)=ei KωAe−i Kω
. The
ConnesRo elli he mal ime hypo hesis posi s ha in gene al co a ian heo ies, he
modula pa ame e
can be iewed as a ime scale in insically dened by he s a is ical
s a e.
3.2.7 Gene alized En opy and G a i y Laye
Uen
Fo each
D∈ D
and i s bounda y sec ion
Σ⊂∂D
, dene gene alized en opy
Sgen(Σ) = A(Σ)/(4G) + Sou (Σ)
(16)
whe e
A(Σ)
is he a ea, and
Sou
is he on Neumann en opy o elds ou side he sec ion.
QNEC gi es an ene gy-en opy inequali y along null gene a o s, o be used in Sec ion 5.
3.2.8 Obse e and Consensus Laye
Uobs
An obse e objec is dened as
Oi= (γi,Λi,Ai, ωi,Mi, Ui)
(17)
whe e
γi⊂M
is a imelike wo ldline,
Λi
is esolu ion scale,
Ai⊆ A
is accessible algeb a,
ωi
is local s a e,
Mi
is model amily, and
Ui
is upda e ule ( iewed as comple ely posi i e
ace-p ese ing map o ins umen ). Obse e da a will be ea ed as descen da a o a
2-s ack on si es in Sec ion 6.
3.2.9 Ca ego y and Logic Laye
Uca
Dene
Uni U
as a 2-ca ego y, whose objec s a e candida e uni e ses sa is ying some
o all o he abo e s uc u es, 1-mo phisms a e s uc u e-p ese ing 2- unc o s, and 2-
mo phisms a e na u al ans o ma ions. The geome ic-logic laye can be exp essed ia
he shea ca ego y
E= Sh(M)
on
M
, wi h in e nal logic cha ac e izing logical ela ions
o physical p oposi ions.
The Uni e se objec
U
will be dened as he e minal objec o
Uni U
, whose exis ence
is a s uc u al assump ion in his pape .
3.2.10 Compu abili y Laye
Ucomp
Dene
Ucomp = (MTM,Enc,Sim)
(18)
whe e
MTM
is Tu ing machine space,
Enc : Uni U→ MTM
is encoding unc o ,
Sim :
MTM ⇒Uni U
is he amily o simula able sub-uni e ses. The uni e se i sel is no
assumed o be compu able, bu any compu able model
V
mus admi a unique embedding
V→U
.
5

4 Sca e ing Scale Iden i y and Mo he Scale
4.1 Sca e ing Condi ion Package
A1

A5
In oduce he ollowing sucien condi ions in
Usca
laye :
*
A1
:
(H−i)−1−(H0−i)−1∈S1(Hsca )
o equi alen ela i e ace-class condi ion;
*
A2
: Exis ence o spec al shi unc ion
ξ(ω)∈L1
loc(R)
sa is ying Li shi sK en ace
o mula and Bi manK en o mula
de S(ω) = exp(−2πiξ(ω))
; *
A3
:
S(ω)
is s ongly
die en iable on he con inuous spec um, and
de S(ω)
uses modied F edholm de e -
minan deni ion; *
A4
: The singula se
N⊂R
composed o h esholds, embedded
eigen alues, and esonances has Lebesgue measu e ze o, and a con inuous o al phase
b anch
Φ(ω) := a g de S(ω)
can be selec ed on
R N
; *
A5
: Wigne Smi h ma ix
Q(ω) = −iS†(ω)∂ωS(ω)
exis s on
R N
, and
Q(ω) = ∂ωΦ(ω)
.
4.2 Mo he Scale Densi y and Two Types o Readings
Dene o al phase
Φ(ω) := a g de S(ω)
, semi-phase
φ(ω) := 1
2Φ(ω)
, ela i e densi y o
s a es
ρ el(ω) := −ξ′(ω)
. Unde
A1

A5
, in oduce:
* **Scale Densi y**
κ(ω) := φ′(ω)
π=ρ el(ω) = 1
2π Q(ω) (ω∈R N);
(19)
* **Phase Reading**
Θ(ω) := φ(ω)
π=Φ(ω)
2π,Θ′(ω) = κ(ω);
(20)
* **Sca e ing Time Reading**
τsca (ω) := 1
2π Q(ω) = κ(ω).
(21)
In na u al uni s,
τsca
can be iewed as g oup delay ime; es o ing physical uni s,
τphys
sca (ω) = ℏκ(ω)
.
4.3 Theo em 3.1 (Scale Iden i y)
Unde condi ions
A1

A5
, he e exis s a unique (Lebesgue almos e e ywhe e) Bo el mea-
su able unc ion
κ:R→R
such ha on
R N
,
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω).
(22)
The Phase Reading
Θ(ω) = φ(ω)/π = Φ(ω)/(2π)
sa ises
Θ′(ω) = κ(ω)
, and Sca e ing
Time Reading
τsca (ω) = κ(ω)
.
P oo in Appendix A.
6
5 Modula , Geome ic and Sca e ing Time Alignmen
5.1 Modula Flow, Physical G oup, and KMS P oposi ion
Le
(A∂, ω)
be a s anda d o m on Neumann algeb a pai wi h ai h ul no mal s a e
ω
.
Le
{ατ}τ∈R
be a one-pa ame e *-au omo phism g oup (physical ime e olu ion), imple-
men ed by uni a y g oup
U(τ)
:
ατ(A) = U(τ)AU(τ)−1
. Deno e
σω
as he Tomi aTakesaki
modula ow.
In oduce he p oposi ion.
P oposi ion 5.1
(Modula G oup Iden ica ion om KMS S a e)
.
Le
(A∂, ω)
be in
s anda d o m, and
ατ
be a one-pa ame e *-au omo phism g oup. I
ω
is a KMS s a e
o
ατ
a in e se empe a u e
β > 0
, hen he ela ion be ween modula ow and
ατ
is
σω
=α /β ( ∈R).
(23)
*P oo Ske ch*: Fo
(A∂, ω)
, Tomi aTakesaki heo y gi es a unique modula g oup
σω
sa is ying he KMS condi ion o
ω
a
β= 1
. On he o he hand, o any
ατ
, i
ω
is
a
β
-KMS s a e, hen by he B a eliRobinson uniqueness heo em, his KMS dynamics
is uniquely isomo phic o he modula g oup in he sense o KMS s uc u e, implying
ατ=σω
βτ
o equi alen ly
σω
=α /β
.
□
In geome ic cases,
ατ
ypically co esponds o a Killing ow o boos ow; P oposi ion
4.1 gi es he p ecise quan i a i e alignmen be ween modula ow and physical g oup
unde KMS s a es.
5.2 Geome icModula Bounda y Condi ion Package
B1

B4
Assume exis ence o bounda y egion
W⊂M
and bounda y algeb a
A∂⊆ A(W)
sa is-
ying:
*
B1
:
W
ca ies a one-pa ame e geome ic symme y g oup
{Λ(τ)}
(e.g., Lo en z
boos o Rindle wedge o ime ansla ion o s a ic black hole ex e io ), implemen ed
on
A∂
by
U(τ)
:
ατ(A) = U(τ)AU(τ)−1
; *
B2
: S a e
ω
is a KMS s a e o
(A∂, ατ)
a in e se empe a u e
β
, and sa ises BisognanoWichmann ype geome ic modula
ow p ope y:
σω
equals
α /β
; *
B3
: On bounda y sec ions o
W
, geome ic- a ia ional
heo y denes B ownYo k quasilocal Hamil onian
H∂
, whose gene a ed ime e olu ion
Ad(e−iτgeomH∂)
is iden ical o
ατ
o die s by cons an escaling; *
B4
: Bounda y algeb a
A∂
simul aneously ca ies sca e ing pai
(H, H0)
incoming/ou going s a e in o ma ion,
cons uc ing sca e ing ma ix
S(ω)
ia wa e ope a o s and adia ion condi ions, and e-
alizing co espondence om geome ic ime ansla ion o sca e ing phase on he ene gy
spec um.
5.3 Theo em 3.2 (Modula Geome icSca e ing Time Align-
men )
Unde condi ions
A1

A5
and
B1

B4
, he e exis cons an s
amod, bmod, ageom, bgeom ∈R
such ha o app op ia ely dened ime pa ame e s,
mod =amod τsca +bmod, τgeom =ageom τsca +bgeom,
(24)
7
and he e exis s a mono onic bijec ion
F:R→R
such ha he geome ic ime unc ion
and sca e ing ime sa is y
Tgeo ◦Φe =F◦τsca
(25)
on app op ia e wo ldline amilies (in almos e e ywhe e sense). He e
τsca (ω) = (2π)−1 Q(ω)
.
*P oo Ske ch*: F om
B1

B2
and P oposi ion 4.1, modula ow and geome ic ow
sa is y
σω
=α /β
. F om
B3
,
ατ
is gene a ed by
H∂
, i.e.,
ατ(A) = eiτH∂Ae−iτH∂,
(26)
hus
σω
(A) = ei( /β)H∂Ae−i( /β)H∂.
(27)
Hence
mod =c1τgeom +c2
(28)
holds o cons an s
c1= 1/β, c2
. F om
B4
, bounda y and adia ion condi ions link
H, H0
o
H∂
. The phase o sca e ing ma ix
S(ω)
ela es o p opaga ion ime along
τgeom
ia
s anda d g oup delay ela ion: o na ow wa e packe s, g oup delay a cen e equency
ω
is p opo ional o
τsca (ω)
. Theo em 3.1 gi es
τsca (ω) = (2π)−1 Q(ω)
. On he
con inuous spec um, ia wa e packe cons uc ion and a e aging, we ob ain
τgeom =ageomτsca +bgeom,
(29)
mod =amodτsca +bmod
. The ela ion be ween geome ic ime unc ion
Tgeo
and bounda y
ime pa ame e can be iewed as a mono onic unc ion
F
by coo dina e choice. De ailed
a gumen s in Appendix A and D.
□
6 Gene alized En opy, QNEC and Eins ein Equa ion
6.1 Gene alized En opyQNEC Condi ion Package
C1

C4
Assume on
Uen
and
Ugeo
:
*
C1
: Gene alized en opy
Sgen(λ) = A(λ)/(4G) + Sou (λ)
is eno malizable in small
causal diamond sec ion amilies, wi h ni e and smoo h second a ia ion; *
C2
: Unde
sec ion de o ma ion in any null ec o
ka
di ec ion, Quan um Null Ene gy Condi ion
(QNEC) holds:
⟨Tabkakb⟩ ≥ (1/2π)S′′
ou (λ0);
(30)
*
C3
: S a e Richness: A e e y poin and o e e y null ec o di ec ion, he e exis s
a amily o Hadama d ype pe u ba ion s a es such ha
⟨Tabkakb⟩
can be a bi a ily
ne- uned wi hin a small neighbo hood; *
C4
: Raychaudhu i equa ion applies; shea and
θ2
e ms a e con ollable in he small diamond limi , hei con ibu ions ei he negligible
o abso bable in o eec i e s ess-ene gy enso .
6.2 Cons an Fac o o A ea Second Va ia ion
Unde ane pa ame e
λ
de o ma ion along null gene a o
ka
, he second a ia ion o
c oss-sec ional a ea
A(λ)
sa ises
d2A
dλ2(λ0) = −ZΣ(λ0)Rabkakb+σabσab +1
2θ2dA.
(31)
8
In he limi o sucien ly small causal diamonds, shea
σabσab
and
θ2
e ms can be ea ed
as highe -o de co ec ions o abso bed, hus
d2A
dλ2(λ0)≃ − ZΣ(λ0)
RabkakbdA.
(32)
Second a ia ion o gene alized en opy is
S′′
gen(λ0) = 1
4GA′′(λ0) + S′′
ou (λ0).
(33)
6.3 Theo em 3.3 (Gene alized En opyQNEC implies Eins ein
Equa ion)
Unde condi ions
C1

C4
, gene alized en opy ex emali y and QNEC in he small causal
diamond limi imply he exis ence o a enso eld
⟨Tab⟩
such ha
Gab + Λgab = 8πG ⟨Tab⟩
(34)
holds on
(M, g)
.
*P oo Ske ch*:
1. On ex emal sec ion
λ0
, en anglemen equilib ium condi ion gi es
S′
gen(λ0) = 0
, and
assume second a ia ion sa ises
S′′
gen(λ0)≥0
. Subs i u ing gene alized en opy second
a ia ion o mula yields
A′′(λ0)/(4G) + S′′
ou (λ0)≥0.
(35)
2. F om QNEC,
⟨Tabkakb⟩ ≥ (1/2π)S′′
ou (λ0).
(36)
Combining yields
1
4GA′′(λ0)≥ −S′′
ou (λ0)≥ −2π⟨Tabkakb⟩.
(37)
3. Using a ea second a ia ion exp ession, we ge
−1
4GZRabkakbdA≳−2πZ⟨Tabkakb⟩dA.
(38)
Rega ding in eg al as local ela ion in app op ia e limi ,
Rabkakb≲8πG ⟨Tabkakb⟩.
(39)
4. Repea ing o e e se pe u ba ion and die en
ka
di ec ions, combined wi h S a e
Richness
C3
, ele a es he inequali y o equali y, ob aining a each poin
Rab −8πG ⟨Tab⟩ − 1
2gab⟨T⟩= Λgab.
(40)
5. Using Bianchi iden i y
∇aGab = 0
and ene gy-momen um conse a ion
∇a⟨Tab⟩= 0
,
Λ
mus be cons an , yielding Eins ein equa ion.
This p ocess sha es s uc u e wi h Jacobson's small ball de i a ion bu p o ides
s onge second-o de en opy-ene gy con ol ia QNEC and sys ema izes wi hin he gen-
e alized en opy amewo k. De ailed cons an s and limi o de con ol in Appendix B.
9