Equivalence Characterization of Causal Structure and Potential Observer Category: Unified Time Scale, No-Local-Observer Limit, and Universe Ontology

Equi alence Cha ac e iza ion o Causal S uc u e and Po en ial
Obse e Ca ego y: Unied Time Scale, No-Local-Obse e Limi ,
and Uni e se On ology
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
Wi hin he amewo k o unied ime scale, bounda y ime geome y, and GLS uni e se
objec s, his pape p o ides a igo ously o malized answe o he ques ion: Can causali y be
iewed as a p oduc o obse e s? The co e conclusions a e:
1. Gi en a geome icdynamical uni e se objec
Ugeo = (X, ⪯, M, g, ˆ
H, . . . )
, we can cons uc
a ca ego y
Obspo
consis ing o all po en ial obse e s, whose objec s a e imelike wo ld-
lines wi h memo y sys ems, and whose mo phisms a e coa se-g aining maps p ese ing
causal and in o ma ional consis ency.
2. Using he  eachable memo y s uc u e o
Obspo
, we can econs uc a pa ial o de
⪯obs
wi hou explici ly e e encing
(X, ⪯)
, ep esen ing e en p ecedence ela ions ha can be
s ably eco ded in memo y by some po en ial obse e .
3. Unde na u al physical assump ions o locali y, in o ma ion eachabili y, and decohe ence
s abili y, we p o e ha
⪯obs=⪯
. The e o e, he causal s uc u e o he uni e se and he
po en ial obse e ca ego y a e mu ually equi alen desc ip ions in a na u al sense.
Based on his, we dis inguish be ween he po en ial obse e ca ego y and he ac ual
obse e subse 
A ⊂ Obj(Obspo )
. The so-called no-obse e limi  does no comple ely e ase
all obse a ional s uc u e, bu me ely se s
A=∅
(no local obse e s), while he po en ial
obse e ca ego y and geome icsca e ing laye s ill exis . The global pu e s a e o he uni e se
ρglobal( )
and he unied ime scale
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω)
can be in e p e ed as he in e nal memo y and ime scale o he uni e se sel - e e en ial supe -
obse e , he eby dissol ing he on ological ension o whe he a no-obse e uni e se s ill has
causali y.
The main heo em o his pape shows ha unde he GLS uni e se objec axiom sys em,
causal s uc u e as memo y eachabili y ela ion o po en ial obse e s is a igo ously p o able
equi alence, while ac ual obse e ne wo ks a e me ely ac i a ed sub amilies wi hin he po-
en ial obse e ca ego y. The appendices p o ide o malized ca ego y cons uc ions, deni ions
o eachable pa ial o de s, and consis ency p oo s be ween unied ime scale and decohe ence
quan um Da winism.
Keywo ds
Causal s uc u e; Po en ial obse e ca ego y; Unied ime scale; GLS uni e se objec s; No-local-
obse e limi ; Memo y eachable pa ial o de ; Quan um Da winism
1
1 In oduc ion
Obse e s play a dual ole in con empo a y discussions o quan um g a i y, quan um in o ma ion,
and cosmological on ology:
On he one hand, gene al ela i i y and quan um eld heo y, gi en
M, gµν
and local Hamil o-
nian
ˆ
H
, seem capable o dening causal s uc u e, ime e olu ion, and sca e ing p ocesses wi hou
any men ion o obse e s; on he o he hand, quan um measu emen , decohe ence, quan um Da -
winism, and ela ional quan um mechanics emphasize ha all physically accessible e en s and
causali y a e always ealized h ough some class o in o ma ion ca ie s and memo y sys ems (i.e.,
gene alized obse e s).
This pape a emp s o p o ide a consis en ma hema ical answe o he ollowing ques ions:
1. Can he causal pa ial o de
(X, ⪯)
o he uni e se be equi alen ly unde s ood as he agg ega e
s uc u e o memo y eachabili y ela ions o all po en ial obse e s?
2. In he no-obse e limi  (especially in he ea ly uni e se), how do causal s uc u e and unied
ime scale main ain hei on ological exis ence in he absence o local obse e s?
3. Wi hin he amewo k o unied ime scale
κ(ω)
, bounda y ime geome y, and GLS ca ego ical
uni e se objec s, how can global s a e objec i i y, local s a e ela ionali y be compa ible wi h
causalobse e equi alence?
Exis ing discussions o en emain a he in e p e a ional le el: e.g., whe he he moon exis s
when no one obse es i , whe he causali y depends on obse e s, e c. The goal o his pape is
no o p opose a new philosophical posi ion, bu o p o ide a o malizable, heo emizable answe
wi hin he al eady cons uc ed amewo k o unied ime scale and GLS uni e se objec s, such ha :

geome icsca e ing causal s uc u e and

memo y eachabili y o po en ial obse e ca ego y
mu ually econs uc each o he ma hema ically, he eby ealizing a igo ous e sion o causali y
as obse e  in he sense o ca ego ical equi alence.
The main h ead o his pape is as ollows: Sec ion 2 e iews he co e componen s o GLS
uni e se objec s and unied ime scale; Sec ion 3 denes he po en ial obse e ca ego y
Obspo
;
Sec ion 4 in oduces he pa ial o de
⪯obs
induced by he memo y s uc u e o po en ial obse e s,
and p o es
⪯obs=⪯
unde app op ia e assump ions; Sec ion 5 discusses he no-local-obse e limi
and he on ological s a us o he uni e se supe -obse e ; Sec ion 6 analyzes he compa ibili y
o unied ime scale, decohe ence, and quan um Da winism wi hin his amewo k. Appendices
p o ide o malized p oo de ails and se e al echnical lemmas.
2 GLS Uni e se Objec s and Geome icSca e ing Causal S uc-
u e
2.1 Geome icDynamical Uni e se Objec
We adop a class o abs ac GLS uni e se objec s, whose geome icdynamical laye p o ides he
ollowing da a:
2

E en se
X
, unde s ood as localizable e en s on space ime mani old
M
wi h some disc e e
o con inuous indexing;

Causal pa ial o de
⪯⊂ X×X
, sa is ying eexi i y, an isymme y, and ansi i i y, com-
pa ible wi h he ligh cone s uc u e o
M, gµν
;

Me ic geome y
(M, g)
, o i s disc e e subs i u e in QCA uni e se, such as causal ne wo ks
wi h bounded deg ee, la ices, e c.;

Local dynamics: ei he Hamil onian
ˆ
H
wi h co esponding uni a y e olu ion
U( ) = exp(−iˆ
H )
,
o disc e e- ime quan um cellula au oma on upda e ope a o
U
, whose local suppo espec s
causal s uc u e.
Deni ion 1
(Geome icDynamical GLS Uni e se Objec )
.
A geome icdynamical GLS uni e se
objec is
Ugeo = (X, ⪯, M, g, ˆ
H, . . . ),
whe e
(X, ⪯)
ep esen s he causal pa ial o de ,
(M, g)
is he con inuous geome ic backg ound (o
QCA subs i u e),
ˆ
H
o
U
is he local dynamics, sa is ying mic o-causali y and ene gy condi ions
among s anda d assump ions.
A his le el, causal s uc u e is iewed as a geome icdynamical on ological s uc u e, inde-
penden o he exis ence o ac ual obse e s.
2.2 Unied Time Scale and Sca e ing Causali y
The unied ime scale is p o ided by sca e ing heo y. Conside a class o sca e ing sys ems
sa is ying ace-class pe u ba ion and wa e ope a o comple eness condi ions. The o al phase
φ(ω)
o he sca e ing ma ix
S(ω)
, he spec al shi unc ion densi y
ρ el(ω)
, and he Wigne 
Smi h ime-delay ma ix
Q(ω) = −iS(ω)†∂ωS(ω)
sa is y he unied scale iden i y:
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω).
He e
κ(ω)
can be in e p e ed as he mo he ime scale densi y dened in he equency domain,
whose in eg al gi es eec i e a i al ime, delay accumula ion, e c.
The ela ionship be ween unied ime scale and causal s uc u e can be oughly unde s ood as:

The geome icdynamical laye de e mines which e en s can inuence each o he (causal pa -
ial o de );

The sca e ingscale laye p o ides he  empo al weigh and esolu ion o hese causal ela-
ions in he equency/ene gy domain.
In his pape , we assume ha he unied ime scale is cons uc ed comple ely wi hin he GLS
amewo k; he ocus is on how o embed he obse e laye on his ounda ion and o mally ealize
causalobse e equi alence.
3
3 Cons uc ion o Po en ial Obse e Ca ego y
This sec ion denes he concep o po en ial obse e , making i depend only on he geome ic
dynamical uni e se objec
Ugeo
, independen o whe he ac ual obse e s exis in any pa icula
uni e se his o y.
3.1 Wo ldline and Memo y Sys em o Po en ial Obse e
In ui i ely, an obse e equi es a leas :
1. A wo ldline espec ing he causal pa ial o de ( imelike ajec o y);
2. An in e nal memo y sys em e ol ing along he wo ldline;
3. Abili y o in e ac wi h local obse able algeb as and upda e memo y.
Deni ion 2
(Po en ial Obse e Objec )
.
Gi en a GLS uni e se objec
Ugeo
, he da a o a po en ial
obse e
O
includes:
1. A causal chain
LO⊂X
, i.e., o any
x, y ∈LO
, ei he
x⪯y
o
y⪯x
;
2. A amily o memo y Hilbe spaces and s a es
(HO, µ(τ))
e ol ing wi h pa ame e
τ
;
3. Fo each
x∈LO
, he e exis s a local obse able algeb a
A(x)⊂ B(Hx)
and CPTP channel
Φx:B(Hx)⊗ B(HO)→ B(HO),
desc ibing he p ocess o  eading in o ma ion om local sys em and w i ing o memo y.
We call
O
a po en ial obse e when i s
(HO, µ(τ),Φx)
is compa ible wi h he local dynamics
ˆ
H
o QCA ules o
Ugeo
, i.e., does no iola e mic o-causali y and ene gy cons ain s.
Deni ion 3
(Po en ial Obse e Ca ego y)
.
Le
Obspo
be he ca ego y whe e:

Objec s a e all po en ial obse e s
O
;

Mo phisms
:O → O′
a e maps p ese ing wo ldline causal o de ing and memo y in o ma-
ion eachabili y, including epa ame iza ion, memo y coa se-g aining, and in e nal encoding
ans o ma ions, such ha
does no in oduce supe luminal in o ma ion ow.
In ui i ely,
Obspo
desc ibes all obse e wo ldlines and memo y s uc u es ha a e physically
possible unde a gi en geome icdynamical uni e se objec , wi hou equi ing hem o be ac ually
ac i a ed in any pa icula uni e se his o y.
3.2 Ac ual Obse e Subse
In a conc e e uni e se his o y, he obse e s ha uly eme ge occupy only a small ac ion o he
po en ial obse e se .
Deni ion 4
(Ac ual Obse e Se )
.
The ac ual obse e se
A
is dened as a subse o
Obspo
:
A ⊂ Obj(Obspo ),
ep esen ing obse e s whose memo y sys ems a e ac ually ac i a ed and pa icipa e in in o ma ion
s o age in a gi en uni e se e olu ion his o y.
4
Thus, he no-local-obse e limi  should be mo e p ecisely s a ed as:
|A| = 0
a he han
Obspo
being emp y. The po en ial obse e ca ego y s ill exis s as de e mined by
Ugeo
.
4 Equi alence o Causal Pa ial O de and Obse e Memo y Reach-
able Pa ial O de
This sec ion cons uc s a pa ial o de
⪯obs
de e mined solely by he po en ial obse e ca ego y,
and p o es unde easonable assump ions ha i coincides wi h he geome icdynamical causal
pa ial o de
⪯
.
4.1 Dening Reachable Pa ial O de om Po en ial Obse e s
Deni ion 5
(Memo y Reachabili y Rela ion)
.
Fo any
x, y ∈X
, dene he ela ion
x⪯obs y
as:
The e exis s some po en ial obse e
O ∈ Obspo
, and pa ame e s
τx< τy
, such ha :
1.
x, y ∈LO
and co espond o wo ldline poin s a imes
τx, τy
;
2. In he memo y s a e
µ(τy)
a ime
τy
, in o ma ion abou e en
x
can s ill be eco e ed h ough
some obse able means (possibly coa se-g ained).
Deno ed as:
x⪯obs y⇐⇒ ∃O, τx< τy:x, y ∈LO,In ox,→µ(τy).
He e
In ox,→µ(τy)
indica es ha he e exis s some obse able algeb a elemen , POVM, o
pos -p ocessing p ocedu e ha can eco e s a is ical in o ma ion abou
x
om
µ(τy)
.
P oposi ion 6.
Unde he abo e deni ion,
⪯obs
is a pa ial o de ela ion on
X
.
P oo .
Reexi i y is ob ained by aking
x=y
wi hou e ol ing memo y; an isymme y and
ansi i i y depend on he causali y o obse e wo ldlines and mono onici y o memo y upda es;
see Appendix A.1 o de ails.
4.2 Geome ic Causali y Implies Memo y Reachable Pa ial O de
Theo em 7
(Geome ic Causali y Implies Memo y Reachabili y)
.
I
x⪯y
(geome icdynamical
causal pa ial o de ), hen unde locali y and obse abili y assump ions, we mus ha e
x⪯obs y
.
P oo (Ske ch).
F om
x⪯y
and he mic o-causali y o
Ugeo
, he e exis s a causal cu e ( imelike
o null) ex ending om
x
o
y
. Along his cu e, cons uc a po en ial obse e
O
whose wo ldline
LO
con ains
x, y
, and a ime
x
in e ac s wi h he local obse able algeb a
A(x)
, w i ing some
in o ma ion abou
x
in o memo y
µ(τx)
.
Since local dynamics and ene gy condi ions gua an ee ha in o ma ion is no comple ely an-
nihila ed in ni e ime, he e exis s
τy> τx
such ha
µ(τy)
s ill e ains eco e able
In ox
. Thus
x⪯obs y
. Fo malized p oo in Appendix A.2.
This heo em shows ha e e y causal ela ion gi en by he geome icdynamical laye can be
ealized by he memo y ajec o y o some po en ial obse e , hence
⪯⊆⪯obs
.
5

4.3 Memo y Reachable Pa ial O de Implies Geome ic Causali y
The mo e cons aining hal is he con e se:
Theo em 8
(Memo y Reachabili y Implies Geome ic Causali y)
.
In GLS uni e se objec s assum-
ing no supe luminal in o ma ion ow, i
x⪯obs y
, hen we mus ha e
x⪯y
.
P oo (Ske ch).
Assume
x⪯ y
and
y⪯ x
, i.e.,
x, y
a e spacelike sepa a ed o incompa able. I
he e exis s a po en ial obse e
O
whose memo y
µ(τy)
a ime
τy
s ill e ains eco e able
In ox
,
his implies an in o ma ion ow channel om
x
o
y
passing h ough he obse e 's in e nal deg ees
o eedom.
Bu unde GLS axioms, he obse e is also jus a physical subsys em o he uni e se, whose
in e nal p opaga ion obeys he same causal pa ial o de . The e o e, memo y eachabili y om
x
o
y
implies he exis ence o some imelike cu e om
x
leading o
y
, con adic ing he spacelike
sepa a ion assump ion.
Fo mally, he obse e can be iewed as a local sys em embedded in
M
, whose in e nal p op-
aga ion cone is con ained wi hin he backg ound ligh cone. I he memo y eachabili y s uc u e
can ansmi in o ma ion be ween
x, y
, hen necessa ily
x⪯y
. See Appendix A.3 o de ails.
F om Theo ems 4.3 and 4.4, we immedia ely ob ain:
Co olla y 9.
Unde he locali y and mic o-causali y assump ions o GLS uni e se objec s,
⪯obs=⪯.
This shows ha he geome icdynamical causal pa ial o de o he uni e se is comple ely
equi alen o he pa ial o de o memo y eachabili y ela ions o all po en ial obse e s. In o he
wo ds, causal s uc u e and po en ial obse e ca ego y a e mu ually equi alen p esen a ions o he
same uni e sal e minal objec in a na u al sense.
We can he e o e unde s and causal s uc u e as memo y s uc u e o po en ial obse e ne -
wo ks wi hou loss o gene ali y, and ice e sa. This is he co e ma hema ical esul o his pape .
5 No-Local-Obse e Limi and Uni e se Supe -Obse e 
5.1 Rein e p e a ion o No-Local-Obse e Limi
In p e ious discussions o no-obse e uni e se, we conside he limi
|A| → 0
. Combined wi h he
equi alence heo em o he p e ious sec ion, we can p o ide a mo e p ecise o mula ion.
Deni ion 10
(No-Local-Obse e Limi )
.
In GLS uni e se objec s, he no-obse e limi  should
be s ic ly unde s ood as he no-local-ac ual-obse e limi :
|A| = 0,Obspo =∅.
A his poin , he geome icdynamical causal pa ial o de
(X, ⪯)
and po en ial obse e ca e-
go y
Obspo
s ill exis , he memo y eachable pa ial o de
⪯obs
and
⪯
s ill sa is y he equi alence
heo em; only no subsys em is ac ually ac i a ed as a local obse e .
Physically, his co esponds o he ea ly uni e se o s uc u eless uni e se: geome y, elds,
sca e ing, and unied ime scale al eady exis , causal s uc u e exis s as an on ological objec , bu
no complex subsys em has ye o med a s able memo y ca ie .
A his poin , causali y as po en ial obse e ne wo k s ill holds, only all obse e s a e in he
po en ial s a e.
6
5.2 Uni e se as a Whole as Supe -Obse e 
In he GLS amewo k, he global pu e s a e o he uni e se
ρglobal( )
can be iewed as a sel -
e e en ial supe -obse e memo y.
Deni ion 11
(Uni e se Supe -Obse e )
.
Dene a special po en ial obse e
Ouni
:
1.
LOuni =X
, i.e., i s wo ldline  a e ses all e en s in an abs ac sense;
2. The memo y Hilbe space is he global Hilbe space
Hglobal
, wi h memo y s a e
ρglobal( )
;
3. Memo y upda e is gi en by uni a y e olu ion
U( ) = exp(−iˆ
H )
, i.e.,
ρglobal( ) = U( )ρglobal(0)U( )†.
Al hough his supe -obse e  is dicul o conc e ize as any local physical en i y, on ologically
i can be iewed as an EBOC-s yle e e nal block obse e : i s memo y is he global pu e s a e o
he uni e se, eco ding he s a is ical s uc u e o all possible e en s in he mos de ailed manne .
P oposi ion 12.
In he no-local-obse e limi
|A| = 0
, he uni e se s ill has a unique supe -
obse e
Ouni
, whose memo y s a e
ρglobal( )
and co esponding causalsca e ing s uc u e com-
ple ely de e mine he causal pa ial o de
⪯
and po en ial obse e ca ego y
Obspo
.
The e o e, no-obse e uni e se in he s ic GLS sense should be unde s ood as no-local-
obse e uni e se, no no-obse e -wha soe e uni e se. The uni e se as a whole can always be
iewed as i s own supe -obse e , whose sel - e e en ial memo y s uc u e oge he wi h he unied
ime scale denes he uni e se on ology.
6 Compa ibili y o Unied Time Scale, Decohe ence, and Quan um
Da winism
This sec ion explains how unied ime scale and decohe encequan um Da winism na u ally embed
in o he causalobse e equi alence amewo k, esol ing he con o e sy o whe he wa e unc-
ion collapses due o obse e s.
6.1 Decohe ence as In o ma ion Diusion in Po en ial Obse e Ne wo k
Conside he s anda d decohe ence model o sys em
S
and en i onmen
E
. Ini ial pu e s a e
|Ψ(0)⟩=|ψS⟩⊗|0E⟩
e ol es unde in e ac ion Hamil onian o en angled s a e
|Ψ( )⟩=X
i
ci( )|iS⟩⊗|ϕE
i( )⟩.
I
|ϕE
i( )⟩
a e app oxima ely o hogonal, he educed sys em s a e
ρS( )≈X
i
|ci( )|2|iS⟩ ⟨iS|
appea s as a diagonal classical mix u e. This p ocess can be iewed as: a la ge numbe o po-
en ial obse e s in he en i onmen (such as local deg ees o eedom o en i onmen subblocks
7
Ek
) edundan ly eco d poin e s a e in o ma ion abou
S
in hei espec i e memo y deg ees o
eedom.
In he amewo k o his pape , his co esponds o:

The e exis many objec s in he po en ial obse e ca ego y
Obspo
whose wo ldlines pass
h ough e en s whe e
S
occu s, s o ing edundan in o ma ion abou
|iS⟩
in memo y s a es;

Poin e s a es a e he encodings mos s able and edundan in he memo ies o hese po en ial
obse e s.
Quan um Da winism u he emphasizes: classical objec i i y a ises om edundan copying
o in o ma ion in he en i onmen ; in ou language, his is equi alen o: ce ain e en se s lea e
consis en aces in he memo ies o many objec s in
Obspo
, he eby o ming causal s uc u e wi h
b oad consensus on
⪯obs
.
6.2 Unied Time Scale as Time Pa ame e o Supe -Obse e
The unied ime scale
κ(ω)
in his amewo k is na u ally in e p e ed as he  equency-domain
ime coo dina e o he supe -obse e .

Fo he supe -obse e
Ouni
,
κ(ω)
p o ides a consis en scale o sca e ing phase de i a i e,
spec al shi densi y, and Wigne Smi h delay ace, se ing as he mo he scale o he
uni e se's o e all equency-domain causal s uc u e.

Fo any local po en ial obse e
O
, i s local empo al expe ience can be iewed as some
sampling and coa se-g aining o
κ(ω)
, e.g., measu ing he local delay spec um h ough local
sca e ing p ocesses o dene i s own ime.
The causalobse e equi alence heo em gua an ees ha whe he om he geome icsca e ing
side ( ia
κ(ω)
and
⪯
) o om he obse e ne wo k side ( ia po en ial obse e memo y and
⪯obs
),
he esul ing ime a ow and causal o de ing a e consis en .
6.3 GLS Ve sion o Wa e unc ion On ology
In his amewo k, he wa e unc ion (o mo e p ecisely, he global densi y ope a o ) plays a dual
ole:
1. As he memo y s a e
ρglobal
o he supe -obse e
Ouni
, i is pa o he uni e se's on ological
s uc u e, independen o any specic local obse e ;
2. As he educed s a e
ρα= Cα(ρglobal)
o local obse e
Oα
, i is a ela ional objec ela i e
o i s causal agmen
Cα
.
Decohe ence and quan um Da winism explain: he s abili y o local educed s a es in he po-
en ial obse e ne wo k de e mines he eme gence o classical eali y, no he in oduc ion o
consciousness o specic ypes o obse e s. The causalobse e equi alence u he shows ha
hese s able s uc u es can be comple ely ew i en in e ms o po en ial obse e memo y eacha-
bili y ela ions, equi alen o he geome icsca e ing causal pa ial o de .
8
7 Discussion and P ospec s
The causalpo en ial obse e ca ego y equi alence cons uc ion p esen ed in his pape p o ides
a sel -consis en on ological pic u e o he ela ionships among unied ime scale, GLS uni e se
objec s, and no-local-obse e limi :

Geome icsca e ing side: desc ibing he causal and empo al s uc u e o he uni e se ia
(X, ⪯, M, g, ˆ
H)
and
κ(ω)
;

Obse e side: desc ibing all easible obse a ional s uc u es ia po en ial obse e ca ego y
Obspo
and memo y eachable pa ial o de
⪯obs
;

Equi alence heo em:
⪯obs=⪯
makes causali y and obse e ne wo k wo p esen a ions o
he same uni e sal e minal objec , na u ally ein e p e ing no-obse e uni e se discussions
as no-local-obse e bu wi h supe -obse e and po en ial obse e uni e se.
Fu u e wo k di ec ions include: in specic QCA uni e se models and THE-MATRIX Uni e se,
explici ly cons uc ing
Obspo
, and e i ying whe he causalobse e equi alence s ill holds unde
ni e size and ni e in o ma ion condi ions, and how i u he couples wi h black hole in o ma ion
pa adox, cosmological cons an p oblem, e c.
Appendix A: Fo malized P ope ies and P oo s o Memo y Reach-
able Pa ial O de
A.1 Pa ial O de P ope ies o Memo y Reachabili y Rela ion
In Deni ion 4.1, he memo y eachabili y ela ion
⪯obs
mus sa is y he h ee pa ial o de axioms.
Reexi i y.
Fo any
x∈X
, cons uc a degene a e po en ial obse e
Ox
whose wo ldline con ains
only e en
x
, wi h memo y sys em eading i s own s a e o ex e nal me ic in o ma ion a
x
and
immedia ely w i ing back. Then
x⪯obs x
.
An isymme y.
I
x⪯obs y
and
y⪯obs x
, hen he e exis po en ial obse e s
O1,O2
and pa ame-
e s
τx< τy
,
τ′
y< τ′
x
, such ha
O1
's memo y a
τy
con ains in o ma ion abou
x
, and
O2
's memo y
a
τ′
x
con ains in o ma ion abou
y
. This means in o ma ion ows exis om
x
o
y
and om
y
o
x
. Unde GLS axioms' no-closed-causal-cu e and no-supe luminal-p opaga ion condi ions,
his can only occu in he degene a e case
x=y
(o he wise cons uc ing closed imelike cu es o
supe luminal signals). Thus i
x⪯obs y, y ⪯obs x
hen
x=y
.
T ansi i i y.
I
x⪯obs y
and
y⪯obs z
, hen he e exis po en ial obse e s
O1
and
O2
whose
memo ies a app op ia e imes con ain
In ox
and
In oy
espec i ely. Cons uc a new po en ial
obse e
O3
a eling along a composi e causal pa h con aining
x, y, z
, inhe i ing memo y con en s
om
O1,O2
along he way ( his is allowed, as
Obspo
pe mi s coa se-g aining and me ging maps).
Then
O3
's memo y when passing h ough
z
can con ain
In ox
, hus
x⪯obs z
.
The e o e
⪯obs
is a pa ial o de .
A.2 De ailed P oo : Geome ic Causali y Implies Memo y Reachabili y
Gi en
x⪯y
, his means he e exis s a imelike o null cu e
γ: [0,1] →M
sa is ying
γ(0) =
x, γ(1) = y
, wi h
γ
always lying wi hin he u u e ligh cone o
x
and pas ligh cone o
y
.
9