Defining ``Self'' in THE-MATRIX Universe: A Matrix Characterization via Causal Partial Order, Unified Time Scale, and Self-Referential Scattering Blocks

Dening Sel  in THE-MATRIX Uni e se: A Ma ix
Cha ac e iza ion ia Causal Pa ial O de , Unied Time Scale,
and Sel -Re e en ial Sca e ing Blocks
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 unied amewo k o THE-MATRIX Uni e se, his pape p o ides an axioma ic
ma hema ical deni ion o he  s -pe son subjec sel . In he THE-MATRIX pe spec i e, all
obse able s uc u e o he uni e se is o ganized as a amily o s ongly cons ained sca e ing
ma ices
S(ω)
oge he wi h a ime-scale densi y
κ(ω)
, whe e he unied scale iden i y
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω)
unies he hal -phase de i a i e o sca e ing, he ela i e densi y o s a es, and he ace o he
Wigne Smi h ime delay as in a ian s o he same empo al geome y.
Building on his ounda ion, his pape accomplishes h ee asks: (1) We o malize THE-
MATRIX Uni e se as a ma ixied causal mani old equipped wi h causal pa ial o de , bound-
a y algeb a, and a amily o sca e ing ma ices, showing ha i is an equi alen desc ip i e
language o he p e iously de eloped unied causal s uc u e heo y based on small causal di-
amonds, bounda y ime geome y, and gene alized en opy; (2) We dene a ma ix obse e 
O
as a s uc u e in THE-MATRIX consis ing o a p ojec ion
PO
, a local bounda y algeb a
POA∂PO
, a s a e
ωO
, and a sel - e e en ial sca e ing ne wo k, which suppo s a disc e ized
wo ldline unde he unied ime scale; (3) We dene sel  as an equi alence class o ma ix
obse e s sa is ying h ee axiom g oups: sel - e e en iali y, s abili y, and minimali y, and p o e
ha his deni ion is equi alen o he p e ious deni ion o sel  gi en in he con ex o causal
mani olds and sel - e e en ial sca e ing ne wo ks: e e y sel  on a con inuous wo ldline can be
uniquely ma ixied in o an equi alence class o sel - e e en ial sca e ing blocks, and ice e sa.
Theo e ically, his pape es ablishes h ee main heo ems: Fi s , wi hin ene gy windows
sa is ying he Bi manK en condi ion and he consis ency ac o y axioms, any obse e in
THE-MATRIX sa is ying he wo ldline axiom co esponds o a
K1
class elemen o he global
sca e ing amily, and sel  co esponds o he minimal i educible elemen s sa is ying addi ional
sel - e e en ial cons ain s; Second, he sel  in he causal mani old con ex (wo ldline plus
bounda y algeb a plus s a e) and he sel  in THE-MATRIX con ex (p ojec ion plus sca e ing
block plus s a e) co espond one- o-one ia scale alignmen h ough bounda y ime geome y and
Toepli z/Be ezin comp ession; Thi d, wi hin he unied ime scale equi alence class, he iden i y
o sel  emains in a ian unde allowed local pe u ba ions, he eby p o iding a ma hema ical
s abili y c i e ion o  he same sel .
The appendices p o ide o malized de ails o key cons uc ions: including he p ecise deni-
ion o THE-MATRIX Uni e se, he wo ldline s uc u e o ma ix obse e s, he ealiza ion o
sel - e e en ial sca e ing ne wo ks in THE-MATRIX, and an ou line o he p oo s o he main
heo ems.
1
1 In oduc ion
In he unied causal s uc u e heo y based on causal pa ial o de and small causal diamonds, he
uni e se is modeled as a Lo en zian mani old equipped wi h ligh cone s uc u e, unied ime scale,
and gene alized en opy a ow, whe e he causal s uc u e, empo al geome y, and g a i a ional
eld equa ions a e cha ac e ized by he same se o axioms. On he o he hand, sca e ing and
spec al heo y shows ha wi hin ene gy windows sa is ying he Bi manK en hypo hesis, he
de i a i e o he o al sca e ing phase, he ela i e densi y o s a es, and he ace o he Wigne 
Smi h ime delay sa is y he scale iden i y
φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω),
hus allowing us o unde s and  ime scale as a mono one epa ame iza ion o a class o spec al
sca e ing in a ian s.
In p e ious wo k, THE-MATRIX Uni e se was p oposed as a ma ixied on ology o he uni-
e se: unde he amewo k o unied ime scale and bounda y ime geome y, all obse able
s uc u e o he uni e se is o ganized as a amily o la ge ope a o ma ices
S(ω)
wi h block s uc-
u e in pa ame e space, whe e he spa si y pa e n encodes causal pa ial o de , he spec al da a
ealizes he unied ime scale, he block s uc u e co esponds o he consensus geome y o mul-
iple obse e s, and closed-loop blocks ca y sel - e e en ial sca e ing ne wo ks and
Z2
opological
in o ma ion.
On he o he hand, a unied ma hema ical deni ion o he  s -pe son subjec sel  has been
gi en in he con ex o causal mani olds and sel - e e en ial sca e ing ne wo ks: he e, sel  is
cha ac e ized as an equi alence class o sel - e e en ial obse e s uc u es ca ied by a wo ldline
o de ed along he unied ime scale, specically including wo ldline, local bounda y algeb a, s a e,
p edic ion model, and sel - e e en ial eedback sca e ing.
The goal o his pape is o accomplish he same ask wi hin THE-MATRIX Uni e se: o gi e
a pu ely ma ixied deni ion o sel  and p o e ha his deni ion is equi alen o he p e ious
causal mani old e sion. Mo e specically, we aim o answe he ollowing ques ions:
1. In THE-MATRIX Uni e se, wha kind o ma ix block s uc u e and s a e da a should ob-
se e  be cha ac e ized as?
2. In THE-MATRIX Uni e se, wha addi ional s uc u es (such as sel - e e en iali y, minimali y,
and s abili y) does sel  possess ela i e o a gene al obse e ?
3. Unde he unied ime scale, how can we es ablish a one- o-one co espondence be ween he
ma ixied deni ion o sel  and he causal mani old e sion, he eby elimina ing coo dina e
and language dependence?
To his end, he s uc u e o his pape is as ollows. Sec ion 2 e iews he basic s uc u e o
THE-MATRIX Uni e se and i s ela ionship wi h unied ime scale, bounda y ime geome y, and
he consis ency ac o y. Sec ion 3 p o ides o malized deni ions o ma ix obse e  and ma ix
wo ldline in THE-MATRIX. Sec ion 4 p oposes he ma ixica ion axioms o sel  and gi es
h ee equi alen deni ions. Sec ion 5 p esen s h ee main heo ems es ablishing he equi alence
and s abili y p ope ies be ween he THE-MATRIX e sion o sel  and he causal mani old e sion
o sel . Appendices AC p o ide p oo de ails o he main cons uc ions and heo ems.
2
2 THE-MATRIX Uni e se: S uc u e and Scale
This sec ion p o ides a b ie e iew and o maliza ion o THE-MATRIX Uni e se. The co e idea is:
o ma ixi y he causal s uc u e and empo al geome y o he uni e se using a amily o sca e ing
ma ices and hei ime-scale densi y.
2.1 Unied Time Scale and Sca e ing Scale Iden i y
In sca e ing sys ems sa is ying he Bi manK en condi ion, o a sel -adjoin ope a o pai
(H, H0)
,
he e exis s a spec al shi unc ion
ξ(ω)
such ha o sucien ly smoo h es unc ions
, he ace
o mula holds
( (H)− (H0)) = Z ′(ω)ξ(ω) dω,
wi h he de e minan iden i y
de S(ω) = exp(−2πiξ(ω))
. Dene he o al sca e ing phase
Φ(ω) =
a g de S(ω)
, hal -phase
φ(ω) = 1
2Φ(ω)
, ela i e densi y o s a es
ρ el(ω) = −ξ′(ω)
, and he Wigne 
Smi h ime-delay ope a o
Q(ω) = −iS(ω)†∂ωS(ω)
. Then almos e e ywhe e we ha e he scale
iden i y
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω).
Fo mo e p ecise domain o applicabili y and egula i y condi ions, see he unied ime scale
li e a u e.
We acco dingly call he unc ion
κ(ω) := φ′(ω)/π
he unied ime scale densi y, which can be iewed bo h as addi ional densi y o s a es and as
no maliza ion o he o al ime delay. The equi alence class o unied ime scale is cha ac e ized
by he ime coo dina e
τ(ω) = Zω
κ(˜ω) d˜ω
gi en by he in eg al o scale densi y.
2.2 Deni ion o THE-MATRIX Uni e se
We deno e THE-MATRIX Uni e se as
THE
-
MATRIX = (H,A∂,{S(ω)}ω∈I, κ, ≺ma ),
whe e:
1.
H
is a sepa able Hilbe space, iewed as  he channel space o he uni e se o bounda y
deg ees o eedom space;
2.
A∂⊂B(H)
is he bounda y obse able algeb a, gene a ed by bounda y elds, channel p o-
jec ions, and local ope a o s;
3.
{S(ω)}ω∈I
is a amily o sca e ing ma ices dened on an ene gy window
I⊂R
, whe e each
S(ω)
is a uni a y ope a o on
H
, piecewise die en iable in
ω
, and sa is ying he a o emen-
ioned scale iden i y;
4.
κ(ω)
is he unied ime scale densi y, sa is ying he scale iden i y;
3
5.
≺ma
is a pa ial o de dened on he channel index se , cha ac e izing he causal eachabili y
ela ions among channels, which unde app op ia e limi s is equi alen o he geome ic causal
s uc u e.
In conc e e cons uc ions, one may choose an o hogonal channel decomposi ion
H=M
a∈I
Ha,
whe e he index se
I
ca ies he causal pa ial o de
≺ma
. In his case, he sca e ing ma ix can
be w i en as a block ma ix
S(ω) = (Sab(ω))a,b∈I ,
whose nonze o pa e n is cons ained by he causal pa ial o de , i.e., he co esponding block
Sba(ω)
is nonze o only when
a
is causally eachable o
b
.
The consis ency ac o y esul shows ha unde assump ions o ela i e ace class and amily
con inui y, he sca e ing amily
{Hx, H0,x}x∈X
can be na u ally embedded in o
K1(X)
ia he
ela i e Cayley ans o m, and he na u al ans o ma ions sa is ying a se o minimal axioms a e
unique up o in ege mul iples. This indica es ha he sca e ing amily o THE-MATRIX is no
only a ma ix bu also ca ies s able opological class in o ma ion.
2.3 Bounda y Time Geome y and THE-MATRIX
Bounda y ime geome y shows ha in g a i a ional sys ems wi h bounda y, he GibbonsHawking
Yo k bounda y e m and i s co ne gene aliza ions ensu e he a ia ional well-posedness o he bulk
ac ion, wi h he B ownYo k quasilocal s ess enso as he Hamil onian gene a o o bounda y
ime ansla ions, hus allowing one o dene a geome ic ime scale on he bounda y. On he o he
hand, he Tomi aTakesaki modula ow gi en by he bounda y algeb a and ai h ul s a e p o ides
modula ime, which unde he he mal ime hypo hesis can be in e p e ed as physical ime.
The unied amewo k o bounda y ime geome y shows ha sca e ing ime, modula ime,
and geome ic ime belong o he same ime scale equi alence class, and can be aligned ia he
unied ime scale iden i y. The ime scale
κ(ω)
in THE-MATRIX is p ecisely he ep esen a i e o
his equi alence class on he spec alsca e ing side.
Thus, in THE-MATRIX, we can exp ess  ime en i ely in e ms o
κ(ω)
and
S(ω)
wi hou in o-
ducing independen ex e nal ime coo dina es; his p o ides he ime-scale ounda ion o dening
sel  in THE-MATRIX.
3 Ma ix Obse e and Ma ix Wo ldline
This sec ion denes ma ix obse e  and ma ix wo ldline in THE-MATRIX as he ounda ion
o he ma ixied deni ion o sel .
3.1 Basic Da a o Ma ix Obse e
In he abs ac causal ne wo k pe spec i e, an obse e
Oi
is o malized as a mul i-componen objec
Oi= (Ci,≺i,Λi,Ai, ωi,Mi, Ui, ui,{Cij}),
whe e
Ci
is he eachable causal domain,
≺i
is he local causal pa ial o de ,
Ai
is he obse able
algeb a,
ωi
is he s a e,
Mi
is he model amily,
Ui
is he upda e ope a o ,
ui
is he u ili y unc ion,
and
Cij
a e he communica ion channels.
In THE-MATRIX, we ma ixi y his s uc u e as:
4
Deni ion 1
(Ma ix Obse e )
.
In THE-MATRIX Uni e se, a ma ix obse e
O
is a iple
O= (PO,AO, ωO),
whe e:
1.
PO
is an o hogonal p ojec ion on
H
wi h
PO=P2
O=P∗
O
, called he channel suppo o he
obse e ;
2.
AO:= POA∂PO
is he es ic ion o he bounda y algeb a o he suppo , ep esen ing all
bounda y obse ables ac ually accessible o his obse e ;
3.
ωO
is a no mal s a e on
AO
, gi ing he obse e 's s a is ical belie o e hese obse ables.
Unde his no a ion, he local sca e ing ma ix o he ma ix obse e is
SO(ω) := POS(ω)PO:POH → POH,
wi h co esponding local ime scale densi y
κO(ω) := (2π)−1 QO(ω),
whe e
QO(ω) = −iSO(ω)†∂ωSO(ω)
.
The in e nal p edic ion model and upda e ope a o  o he ma ix obse e can be iewed as
comple ely posi i e ace-p ese ing maps on
AO
and hei chosen pa ame e amilies; hese a e no
explici ly expanded he e bu a e embodied h ough xed-poin condi ions in he sel - e e en iali y
axiom.
3.2 Ma ix Wo ldline
In he causal mani old con ex , a wo ldline is a imelike cu e
γ:τ7→ x(τ)
wi h p ope ime and
eco ded sequences dened along i . In THE-MATRIX, we cha ac e ize wo ldlines using a amily
o p ojec ions e ol ing mono onically along he unied ime scale.
Deni ion 2
(Ma ix Wo ldline)
.
Le
[τ]
be a unied ime scale equi alence class. A ma ix
wo ldline is a amily o p ojec ions
{P(τ)}τ∈J
sa is ying:
1.
J⊂R
is an in e al;
2. Fo each
τ∈J
,
P(τ)
is an o hogonal p ojec ion on
H
;
3. Mono onici y: i
τ1< τ2
, hen
P(τ1)⪯P(τ2)
(i.e.,
P(τ1)P(τ2) = P(τ1)
), exp essing ha
 eco ds can only accumula e and canno be e ased;
4. Locali y: o each
τ
,
P(τ)
depends only on he unied ime scale eadings wi hin a ni e
ene gy window, i.e., o some compac in e al
Iτ⊂I
, he p ojec ion
P(τ)
can be cons uc ed
om
S(ω)
on
Iτ
ia app op ia e Toepli z/Be ezin comp ession.
In ui i ely,
P(τ)
ep esen s he suppo o all eco e able eco ds w i en by he obse e on he
bounda y h ough sca e ing p ocesses be o e ime scale
τ
.
Fo a ma ix obse e
O= (PO,AO, ωO)
, i he e exis s a ma ix wo ldline
{P(τ)}
and a ime
in e al
JO
such ha o all
τ∈JO
,
P(τ)⪯PO
, we say ha
O
ca ies a ma ix wo ldline.
5

3.3 Causal Domain o Ma ix Obse e
The causal pa ial o de
≺ma
in THE-MATRIX ac s on he channel index se
I
. Gi en he suppo
index subse
IO⊂ I
co esponding o a p ojec ion
PO
, we dene he ma ix causal domain o he
obse e as
CO:= {a∈ I :∃b∈ IO, a ≺ma b
o
b≺ma a}.
Unde app op ia e causal comple eness assump ions,
CO
can be iewed as he disc e iza ion o
a small causal diamond neighbo hood o a wo ldline in he causal mani old con ex .
4 Sel  in THE-MATRIX: Axioms and Equi alen Deni ions
In he con ex o causal mani olds and sel - e e en ial sca e ing ne wo ks, sel  was p e iously
dened as an equi alence class o sel - e e en ial obse e s uc u es ca ied by a wo ldline o de ed
along he unied ime scale, wi h co e ea u es including: pe sis ence along he wo ldline, sel -
e e en ial eedback s uc u e, and s abili y unde allowed pe u ba ions. This sec ion gi es he
co esponding ma ixied deni ion in THE-MATRIX.
4.1 Axioma ic Requi emen s
We  s lis h ee axiom g oups ha sel  should sa is y in THE-MATRIX.
Axiom 3
(Wo ldline Axiom)
.
The ma ix obse e
O
co esponding o sel  mus ca y a ma ix
wo ldline
{P(τ)}τ∈J
, and his wo ldline mus be mono onically inc easing wi h espec o he unied
ime scale. This ensu es ha sel  possesses a con inuous empo al expe ience and eco d sequence.
Axiom 4
(Sel -Re e en iali y Axiom)
.
The e exis s a amily o sca e ing ne wo k cons uc ions
depending on
SO(ω)
and p edic ionupda e ope a o s on he bounda y algeb a, such ha unde
he unied ime scale pa ame iza ion, he p edic i e s a e
ωO(τ)
o he obse e and he eadings
p oduced by ac ual sca e ing sa is y a xed-poin equa ion, i.e.,
ωO(τ) = Fsel [ωO(τ), SO, κ],
whe e
Fsel
is a map dened by he sel - e e en ial sca e ing ne wo k. In ui i ely, his exp esses ha
sel 's in e nal p edic ion model is ealized in THE-MATRIX as a closed-loop sca e ing ne wo k,
and i s p edic ions abou i sel and he en i onmen a e s a is ically consis en wi h ac ual sca e ing
p ocesses.
Axiom 5
(Minimali y and S abili y Axiom)
.
1. Minimali y: i
O′= (P′,A′, ω′)
is also a ma ix
obse e sa is ying Axioms III wi h
P′⪯PO
, hen
P′=PO
almos e e ywhe e; i.e., he sup-
po p ojec ion o sel  is minimal unde he assump ions o sel - e e en iali y and wo ldline
axiom;
2. S abili y: unde local pe u ba ions p ese ing he unied ime scale and la ge-scale causal
s uc u e (i.e., unde allowed sca e ing amily homo opies and consis ency ac o y na u al
ans o ma ions), he equi alence class o
O
emains in a ian ; his p o ides a ma hema ical
c i e ion o  he same sel .
6
4.2 Deni ion I: Minimal Sel -Re e en ial Ma ix Obse e Equi alence Class
Based on he abo e axioms, we can gi e he  s equi alen deni ion.
Deni ion 6
(Sel  in THE-MATRIX, Deni ion I)
.
In THE-MATRIX Uni e se, a sel  is an
equi alence class o ma ix obse e s
[O]
sa is ying:
1. Any ep esen a i e
O= (PO,AO, ωO)
in he class sa ises Axioms IIII;
2. The equi alence ela ion is de e mined by in e nal uni a y ans o ma ions and ane escalings
o he unied ime scale: i he e exis s a uni a y ope a o
U
and an ane escaling
τ7→ aτ +b
such ha
PO2(τ) = UPO1(aτ +b)U∗
and
ωO2=ωO1◦Ad(U−1)
, hen
O1
and
O2
ep esen
he same sel .
In his deni ion, sel  is an equi alence class o sel - e e en ial ma ix obse e s ha is minimal
and s able in he sense o homo opy and uni a y equi alence.
4.3 Deni ion II: Minimal Elemen o Sel -Re e en ial Sca e ing Blocks in
K1
Since he sca e ing amily can be na u ally embedded in o
K1
heo y, we can gi e a mo e opolog-
ically a o ed deni ion.
Le he pa ame e space
X
desc ibe he ex e nal pa ame e s o he sca e ing amily (e.g.,
obse a ion equency window, d i ing phase, o ex e nal con ol pa ame e s). The consis ency
ac o y shows ha a sca e ing amily
{Hx, H0,x}x∈X
sa is ying ela i e ace class and endpoin
closu e condi ions gi es a na u al elemen o
K1(X)
ia he ela i e Cayley ans o m.
Deni ion 7
(Sel  in THE-MATRIX, Deni ion II)
.
Conside he sca e ing sub amily
{SO(ω, x)}(ω,x)∈I×XO
associa ed wi h a ma ix obse e
O
in THE-MATRIX, which gi es an elemen
[uO]
o
K1(XO)
ia
he consis ency ac o y cons uc ion. We call
O
co esponding o sel  he opological equi alence
class in
K1(XO)
sa is ying:
1. The sel - e e en iali y condi ion gi en by Axioms III can be ew i en as a na u al cons ain
equa ion on
[uO]
(e.g., a opological condi ion cons ained by modulo- wo loop winding num-
be s);
2. Among all sca e ing amily
K1
elemen s sa is ying his cons ain ,
[uO]
co esponds o a
minimal suppo p ojec ion ha canno be u he decomposed in o a non i ial di ec sum
sa is ying he same cons ain .
F om his pe spec i e, sel  can be iewed as an i educible sca e ing block in THE-MATRIX
sa is ying sel - e e en ial opological cons ain s, whose homo opy class is gi en by he
K1
elemen
[uO]
.
4.4 Deni ion III: Ma ixied Image o Causal Mani old Sel 
The hi d equi alen deni ion di ec ly uses he deni ion o sel  in he causal mani old e sion
and ma ixies i ia he b idge be ween bounda y ime geome y and THE-MATRIX.
In he causal mani old con ex , sel  can be dened as a iple
I= (γ, Aγ, ωγ),
whe e
γ
is a imelike wo ldline,
Aγ
is he bounda y algeb a glued along
γ
, and
ωγ
is a s a e on
i ; addi ionally, he e is equi ed o exis a amily o sel - e e en ial sca e ing ne wo ks such ha
(γ, Aγ, ωγ)
sa ises he co esponding closed-loop xed-poin condi ion.
7
Bounda y ime geome y and he NullModula double co e show ha he bounda y algeb as
o small causal diamonds along he wo ldline
γ
can be embedded in o a subalgeb a amily o he
global bounda y algeb a
A∂
, and he unied ime scale
κ(ω)
and modula ow pa ame e can be
aligned wi hin equi alence classes.
Deni ion 8
(Sel  in THE-MATRIX, Deni ion III)
.
Gi en a sel 
I= (γ, Aγ, ωγ)
in he causal
mani old con ex , in THE-MATRIX we choose a amily o p ojec ions
{P(τ)}τ∈J
co esponding o
γ
and i s limi p ojec ion
PO
, le
AO=POA∂PO
and
ωO
be he ma ixied image o
ωγ
, hen we
ob ain a ma ix obse e
O= (PO,AO, ωO)
. I s equi alence class is dened as he image o
I
in
THE-MATRIX, and is called sel  in THE-MATRIX.
This deni ion elies on he exis ence and uniqueness heo ems o bounda y ime geome y and
Toepli z/Be ezin comp ession.
In Sec ion 5's main heo ems, we will p o e ha Deni ions 4.14.3 a e equi alen .
5 Main Theo ems: Equi alence and S abili y
This sec ion p esen s h ee main heo ems demons a ing he co espondence be ween sel  in
THE-MATRIX and sel  in causal mani olds, as well as s abili y unde na u al ans o ma ions o
sca e ing amilies.
Theo em 9
(Equi alence o Deni ions IIII)
.
Wi hin ene gy windows sa is ying he unied ime
scale, bounda y ime geome y, and consis ency ac o y hypo heses, Deni ions 4.14.3 o sel  a e
equi alen .
Mo e specically:
1. Each minimal sel - e e en ial ma ix obse e equi alence class
[O]
sa is ying Axioms IIII
uniquely de e mines a
K1
elemen
[uO]
and sa ises he opological minimali y condi ion o
Deni ion 4.2;
2. Each  opological minimal elemen 
[uO]
sa is ying Deni ion 4.2 has a ep esen a i e ma ix
obse e
O
sa is ying Axioms IIII, hus gi ing a sel  in Deni ion 4.1;
3. Each sel 
I= (γ, Aγ, ωγ)
in he causal mani old con ex can be uniquely ma ixied ia
bounda y ime geome y and Toepli z/Be ezin comp ession o some
[O]
, and his p ocess p e-
se es sel - e e en iali y and minimali y.
P oo S a egy Ou line.
The  s di ec ion uses he na u al ans o ma ion uniqueness heo em
o he consis ency ac o y: unde axioms o con inui y, addi i i y, scale co a iance, and Bi man
K en no maliza ion, he na u al ans o ma ion om sca e ing amilies o
K1
is unique up o
in ege mul iples; he minimal sel - e e en ial condi ion excludes non i ial in ege mul iples, he eby
gi ing a unique
K1
elemen .
The second di ec ion uses he ep esen abili y heo em o
K1
elemen s: unde gi en opolog-
ical cons ain s, one can always choose a ep esen a i e sca e ing amily sa is ying sel - e e en ial
bounda y condi ions, and cons uc he co esponding ma ix obse e ia s anda d p ocedu es;
minimali y is gua an eed by he indecomposabili y o he
K1
elemen .
The hi d di ec ion elies on he alignmen esul s o bounda y ime geome y and he Null
Modula double co e : he e exis s a na u al embedding be ween small causal diamond bounda y
algeb as and global bounda y algeb as, he gene alized en opy ex emal condi ions and Eins ein
equa ions p ese e o m unde his embedding, and Toepli z/Be ezin comp ession gi es a e e sible
8
co espondence om geome ic ime o equency scale, he eby ealizing he one- o-one mapping
om causal mani old sel  o ma ix sel .
Comple e p oo in Appendix B.
Theo em 10
(S abili y wi hin Unied Time Scale)
.
Le
[O]
be a sel  in THE-MATRIX wi h
co esponding unied ime scale densi y
κ(ω)
. Conside a amily o sca e ing amily de o ma ions
{Sλ(ω)}λ∈[0,1]
sa is ying:
1. Fo each
λ
,
Sλ(ω)
sa ises he same Bi manK en and ela i e ace class assump ions as
S(ω)
;
2. The scale iden i y and unied ime scale densi y
κλ(ω)
emain in a ian wi hin equi alence
classes, i.e., he e exis s an ane escaling such ha
κλ(ω)
and
κ(ω)
belong o he same ime
scale equi alence class;
3. The
K1
elemen
[uλ]
o he sca e ing amily is cons an in
λ
.
Then he e exis s a amily o ma ix obse e s
Oλ
such ha all
[Oλ]
a e equi alen o
[O]
. In o he
wo ds, unde sca e ing amily de o ma ions ha p ese e he unied ime scale equi alence class
and
K1
class, he equi alence class o sel  is s able.
P oo S a egy Ou line.
This conclusion is a  igidi y esul o
K1
class and unied ime scale
in a ian s o he ma ixied deni ion o obse e : he scale iden i y ensu es ha ane escal-
ing o ime pa ame e s does no change wo ldline s uc u e; he na u al ans o ma ion uniqueness
o he consis ency ac o y and he in a iance o
K1
class gua an ee ha he opological ype o
sel - e e en ial sca e ing blocks emains unchanged; hus one can con inue choosing ep esen a-
i e ma ix obse e s along sca e ing amily homo opy such ha hei equi alence class emains
in a ian .
De ailed a gumen in Appendix C.
Theo em 11
(Equi alence o Exis ence: Causal Mani old Sel  and THE-MATRIX Sel )
.
Unde
he assump ions o he in o ma ion-geome ic a ia ional p inciple, local quan um ene gy condi ions,
and small causal diamond limi s, he Eins ein equa ions and gene alized en opy ex emal condi ions
a each poin gi e unied local geome icin o ma ion cons ain s.
Unde hese condi ions, he exis ence o he ollowing wo kinds o sel  is equi alen :
1. The e exis s a imelike wo ldline
γ
wi h bounda y algeb a and s a e along i such ha he
causal mani old deni ion o sel  holds;
2. The e exis s a ma ix obse e equi alence class
[O]
sa is ying Axioms IIII.
Mo e specically, any wo ldline
γ
sa is ying IGVP condi ions can cons uc a ma ix sel  ia
bounda y ime geome y and THE-MATRIX embedding; con e sely, he suppo and wo ldline s uc-
u e o any ma ix sel  can be econs uc ed back o a imelike wo ldline sa is ying g a i a ional
eld equa ions ia Radon- ype closu e and small causal diamonds.
Appendix A: Technical De ails o THE-MATRIX and Unied Time
Scale
This appendix b iey e iews se e al echnical poin s o THE-MATRIX Uni e se and unied ime
scale.
9