Structural Isomorphism Between ``Self'' and ``Universe'': Unified Proof via Causal--Time--Entropy--Matrix Universe

S uc u al Isomo phism Be ween Sel  and Uni e se: Unied
P oo ia CausalTimeEn opyMa ix Uni e se
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
Wi hin amewo k o unied ime scale, bounda y ime geome y, unied heo y o causal
s uc u e, and sel - e e en ial sca e ing ne wo ks, his pape p o ides axioma izable, heo em-
p o able ma hema ical e sion o p oposi ion my mind is he uni e se, gi ing igo ous p oo
o s uc u al isomo phism be ween sel  and uni e se.
On one hand, based on Bi manK en o mula and Wigne Smi h ime-delay heo y, we align
o al sca e ing hal -phase de i a i e, spec al shi unc ion, and ime-delay ace, ob aining
unied ime scale mo he o mula
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω)
, iewing i as sole
sou ce o ime scale.
On o he hand, based on gene alized en opy, quan um ene gy condi ions, and quan um o-
cusing conjec u e, we es ablish equi alence be ween gene alized en opy ex emali y and mono-
onici y in small causal diamonds on globally hype bolic Lo en z mani olds, and nonlinea Ein-
s ein equa ions wi h local s abili y condi ions.
Building on his, we in oduce ca ego y
Uni
o causal imeen opyma ix uni e ses wi h
unied ime scale and bounda y sca e ing da a, desc ibing uni e se as objec wi h causal pa -
ial o de , gene alized en opy a ow, and ma ix- heo e ic sca e ingdelay s uc u e; simul-
aneously in oduce obse e objec ca ego y
Obs
, o malizing conc e e obse e as s uc u e
pe o ming modeling and upda ing along imelike wo ldline a specic esolu ion scale and ob-
se able algeb a. We cons uc wo unc o s on sui able physical subca ego y
Uni phys ⊂Uni
and comple e obse e  subca ego y
Obs ull ⊂Obs
:
1.
F:Uni phys →Obs ull
: om any physical uni e se objec , ia bounda y comp ession and
unied ime scale alignmen , ob ain induced sel - e e en ial obse e ;
2.
R:Obs ull →Uni phys
: om obse e objec sa is ying comple eness and iden iabili y
condi ions, ia geome ic econs uc ion om bounda y sca e ingen opy da a, ob ain
unique uni e se objec isomo phism class.
Using geome ic econs uc ion uniqueness o bounda y sca e ingen opy da a (abso bing
bounda y igidi y, Calde ón in e se p oblem, holog aphic econs uc ion esul s), and in o -
ma ion geome ic iden iabili y wi h ela i e en opy mono onici y (including JLMS ela i e
en opy equali y and en anglemen wedge econs uc ion heo y), we p o e
F
and
R
a e ca e-
go ical equi alences on abo e subca ego ies. This yields:

Fo each physical uni e se objec
U∈Uni phys
, he e exis s comple e obse e
O∈Obs ull
such ha
R(F(U)) ∼
=U
;

Fo each comple e obse e objec
O∈Obs ull
, he e exis s uni e se objec
U∈Uni phys
such ha
F(R(O)) ∼
=O
.
When in e p e ing isomo phism class o obse e s sa is ying comple eness, sel - e e en ial
consis ency, and ime scale alignmen condi ions as ma hema ical ealiza ion o sel , p oposi-
ion sel is isomo phic o uni e se is p ecisely s a ed as: my in e nal wo ld model
Uinne :=
1
R(O)
is isomo phic in
Uni phys
o ex e nal uni e se objec
Uou e ∈Uni phys
. This is unied
causal imeen opyma ix uni e se e sion o my mind is he uni e se.
Keywo ds
Causal mani olds; Unied ime scale; Bounda y ime geome y; Ma ix uni e se; Obse e ; Ca e-
go ical equi alence; Gene alized en opy; Sel - e e en ial sca e ing ne wo ks
1 In oduc ion & His o ical Con ex
P oposi ion my mind is he uni e se epea edly appea s in Chinese mind-na u e heo y, Indian
Yogaca a school, and Wes e n phenomenological adi ions; i s in ui i e con en is: exis ence mode
o uni e se and consciousness s uc u e o sel  a e iden ical in some p o ound sense. Howe e , adi-
ional a gumen s mos ly emain a me aphysical and phenomenological le el, lacking ne s uc u al
in e ace wi h mode n ma hema ical physics.
Since wen ie h cen u y, mul iple ou es poin ing owa d obse e uni e se unica ion eme ged
wi hin physics. Fo example, Wheele in i om bi  p og am claims uni e se is undamen ally
in o ma ional en i y, whe e obse a ional ac s and bina y inqui ies cons i u e gene a i e mechanism
o eali y. Rela ional quan um mechanics, QBism, and se ies o pa icipa o y uni e se p oposals
emphasize om die en angles: physical s a es and physical ac s mus be unde s ood ela i e o
obse e s o in o ma ion ca ie s. Meanwhile, holog aphic p inciple, AdS/CFT, en anglemen wedge
econs uc ion de elopmen s show: gi en bounda y quan um s a e and en anglemen s uc u e, bulk
geome y and dynamics can be la gely econs uc ed.
Al hough abo e wo k hin s a some obse a ionuni e se co espondence, i s ill shows inade-
quacy in h ee espec s:
1.
Lacking unied scale
: Time's ole in sca e ing spec al heo y, he mal ime hypo hesis,
g a i a ional bounda y e ms akes a ious o ms, lacking single scale mo he o mula o
cons ain all ime concep s.
2.
Lacking axioma ic unica ion o causalen opygeome y
: Logical ela ionships among
gene alized en opy, QNEC, QFC, and Eins ein equa ions ha e been e ied in specic sce-
na ios, bu no ye in eg a ed as  undamen al deni ion o causal s uc u e.
3.
Lacking ca ego ical isomo phism heo em o obse e uni e se
: Exis ing philosoph-
ical and physical discussions mos ly heu is ic, me apho ically saying uni e se is gian quan-
um compu a ion o  eali y is in o ma ion ne wo k, bu lacking clea ly dened uni e se
ca ego y and obse e ca ego y, also lacking heo em p o ing  hei isomo phism in his
con ex .
This pape s ands on se ies o p io wo ks: unied ime scale and bounda y ime geome y,
unied heo y o causal s uc u e, sel - e e en ial sca e ing ne wo ks and ma ix uni e se THE-
MATRIX, causal ne wo kobse e consensus amewo k, p oposing p ecise answe s o ollowing
ques ions:
1. Wi hin amewo k con aining causal pa ial o de , unied ime scale, gene alized en opy
a ow, and bounda y sca e ingma ix s uc u e, wha is ma hema ical objec o uni e se?
2
2. In same amewo k, how can sel  as  s -pe son subjec be o malized? Compa ed o gene al
obse e objec s, wha addi ional sel - e e en ial and comple eness equi emen s does sel 
ha e?
3. In wha ca ego y and wha sense can we say sel  and uni e se a e isomo phic? Is his
isomo phism unique, na u al, and opologically consis en ?
This pape 's co e con ibu ions can be summa ized as:

In oduce causal imeen opyma ix uni e se ca ego y
Uni
con aining causal mani olds,
unied ime scale, gene alized en opy, and sca e ingma ix da a, and obse e ca ego y
Obs
con aining wo ldline, esolu ion scale, bounda y obse able algeb a, s a e, model amily, and
upda e ope a o ;

Cons uc wo adjoin unc o s
F
and
R
be ween physical subca ego y
Uni phys
and comple e
obse e subca ego y
Obs ull
, p o ing hey yield ca ego ical equi alence unde assump ions o
gene alized en opysca e ingbounda y igidi y and in o ma ion geome ic iden iabili y;

Based on his ca ego ical equi alence, dene sel  as isomo phism class o obse e s in
Obs ull
sa is ying sel - e e en ial consis ency and ime scale alignmen , gi ing heo em e sion o my
in e nal uni e se model is isomo phic o ex e nal uni e se objec ;

In ma ix uni e se THE-MATRIX pe spec i e, in e p e abo e ca ego ical equi alence as:
global s uc u e o gian sca e ingdelay ma ix is equi alen o in e nal iew along some
sel - e e en ial pa h unde app op ia e comple eness condi ions.
Below s uc u e is as ollows: Sec ion 2 gi es basic model and assump ions o unied heo y;
Sec ion 3 o malizes uni e se and obse e ca ego ies and s a es main heo ems; Sec ion 4 p o ides
p oo s uc u e, pos poning echnical de ails o appendices; Sec ions 5 and 6 discuss model appli-
ca ions and easible enginee ing p oposals; Sec ion 7 analyzes heo y's bounda y condi ions and
ela ions o exis ing wo k; Sec ion 8 concludes; Appendices AC p o ide key p oo de ails.
2 Model & Assump ions
This sec ion cons uc s ma hema ical model and axioma ic assump ions used in his pape . All
ma hema ical objec s wo k in
C∞
ca ego y assuming app op ia e egula i y and spec al condi ions.
2.1 Unied Time Scale and Sca e ingSpec al S uc u e
Le
H0, H
be sel -adjoin ope a o s on sepa able Hilbe space
H
, sa is ying
H−H0
is app op ia e
ela i e ace-class pe u ba ion, so sca e ing ope a o
S
exis s, and o each equency
ω
has
sca e ing ma ix
S(ω)
. Deno e:

To al sca e ing phase
Φ(ω) = a g de S(ω)
, hal -phase
φ(ω) = 1
2Φ(ω)
;

Spec al shi unc ion
ξ(λ)
as spec al die ence in a ian dened by Bi manK en;

Rela i e densi y o s a es
ρ el(ω) = −ξ′(ω)
;

Wigne Smi h ime-delay ma ix
Q(ω) = −iS(ω)†∂ωS(ω)
.
3
Unde s anda d assump ions, Bi manK en o mula and ela ed ace o mulas gi e ela ion
be ween sca e ing de e minan and spec al shi unc ion; simul aneously he e exis s ene gy ime
analogy iden i y be ween ace o ime-delay ma ix and densi y o s a es. Combining yields scale
iden i y
κ(ω) := φ′(ω)
π=ρ el(ω) = 1
2π Q(ω).
We call
κ(ω)
unied ime scale densi y. Fo e e ence equency
ω0
, dene ime pa ame e
τ(ω)−τ(ω0) = Zω
ω0
κ(˜ω) d˜ω.
Any wo se s o sca e ing da a gi ing same
κ
die only by ane ans o ma ion, hus ime
scale is dened only in equi alence class
[τ] := {˜τ|˜τ(ω) = aτ(ω) + b, a > 0, b ∈R}.
Assump ion 1
(Unied Time Scale Exis ence)
.
All physical ime s uc u es o uni e sesca e ing
ime, modula ime, and g a i a ional geome ic imecan be de i ed om same scale densi y
κ(ω)
unde app op ia e p ojec ions.
2.2 Causal Mani olds, Small Causal Diamonds and Gene alized En opy
Le
(M, g)
be ou -dimensional, o ien ed, ime-o ien ed globally hype bolic Lo en z mani old wi h
causal pa ial o de
≺
. Fo any poin
p∈M
and sucien ly small scale
> 0
, dene small causal
diamond
Dp, =J+(p−)∩J−(p+),
whe e
p−≺p≺p+
and
g
is app oxima ely Minkowski on
Dp,
.
Fo cu su ace
Σ
h ough
Dp,
, dene gene alized en opy
Sgen(Σ) = A(Σ)
4Gℏ+Sou (Σ),
whe e
A(Σ)
is cu su ace a ea,
Sou
on Neumann en opy o ex e io quan um eld. Quan um
null ene gy condi ion QNEC and quan um ocusing conjec u e QFC p edic gene alized en opy
mono onici y along any null geodesic cong uence and non-inc ease o quan um expansion.
Using in o ma ion geome ic a ia ional p inciple and gauge ene gy non-nega i i y heo y as
ools, one can p o e: unde xing app op ia e olume o edshi cons ain s,  s -o de ex emali y
condi ion o gene alized en opy on small causal diamonds is equi alen o nonlinea Eins ein eld
equa ions
Gab + Λgab = 8πG Tab,
while second-o de non-nega i i y is equi alen o HollandsWald ype gauge ene gy posi i i y con-
di ion, hus locally de e mining e olu ion o me ic and cosmological cons an .
2.3 Bounda y Time Geome y and The mal Time
On space ime egion
(M, g, ∂M)
wi h non-compac bounda y, g a i a ional ac ion
Sg a =1
16πG ZM
R√−gd4x+1
8πG Z∂M
Kp|h|d3x+···
4
is well-dened unde a ia ion xing bounda y induced me ic
h
; i s bounda y a ia ion denes
B ownYo k quasilocal s ess enso and bounda y Hamil onian, yielding geome ic ime gene a o
along no mal ansla ion.
On o he hand, le
A∂
be bounda y obse able algeb a,
ω∂
ai h ul s a e; hen Tomi aTakesaki
modula heo y p o ides sys ema ic me hod o cons uc ing modula ow
σω
on
A∂
, while Connes
Ro elli he mal ime hypo hesis u he claims: in gene ally co a ian quan um heo y, physical ime
ow is gi en by modula g oup de e mined by
(A∂, ω∂)
.
Syn hesizing sca e ingspec al consis ency and modula owgeome ic ime alignmen , one
can p o e: he e exis s na u al ime scale equi alence class
[τ]
such ha sca e ing ime, modula
ime, and g a i a ional bounda y ime all belong o his equi alence class, hus unied o scale
densi y
κ(ω)
.
2.4 Ma ix Uni e se THE-MATRIX
On sca e ingspec al and bounda y algeb a side, ma ix uni e se THE-MATRIX can be in o-
duced as equi alen desc ip ion o uni e se on ology. Gi en channel Hilbe space
Hchan
, equency-
dependen bounda y sca e ing ma ix amily
S(ω)
and ime-delay ma ix amily
Q(ω)
, unied scale
κ(ω)
, bounda y algeb a
A∂
, bounda y s a e
ω∂
, ma ix uni e se can be w i en as
THE
-
MATRIX = Hchan, S(ω), Q(ω), κ, A∂, ω∂.
I s spa se pa e n encodes causal pa ial o de ( h ough eachabili y and eedback s uc u e be-
ween channels), spec al da a
S(ω), Q(ω)
ealize unied ime scale, block s uc u e and edundan
encoding co espond o mul i-obse e consensus geome y, sel - e e en ial closed loops and sca e -
ing squa e- oo de e minan b anches ca y
Z2
opological in o ma ion and double co e s uc u e
simila o Fe mi s a is ics.
2.5 Obse e Model and Causal Ne wo ks
In abs ac causal ne wo k language, wo ld is iewed as collec ion o local pa ially o de ed ag-
men s, each agmen co esponding o ni ely eachable causal domain. Obse e only accesses
pa ial agmen s, ca ying p edic i e model and upda e ules abou global causal ne wo k. This
pape adop s ollowing obse e model:

Obse e 's ime s uc u e gi en by imelike wo ldline
γ
;

Obse able da a comes om comp ession o bounda y algeb a
A∂
on o subalgeb a
Aγ⊂ A∂
ela ed o
γ
;

Obse e s a e
ω
and model amily
M
e ol e ia upda e ope a o
Uupd
h ough measu emen s
and communica ion;

Resolu ion scale
Λ
limi s dis inguishable bandwid h and spa ial esolu ion;

I mul iple obse e s and communica ion s uc u e
C
exis , ela i e en opy and in o ma ion
dis ance can cha ac e ize consensus con e gence.
Based on his, we o malize uni e se and obse e  as wo ca ego ies, s a ing main heo em
in subsequen sec ions.
5

2.6 Global Assump ions
This pape wo ks unde ollowing global assump ions:
1.
(M, g)
globally hype bolic wi h app op ia ely con ollable non-compac bounda y o asymp-
o ic egions;
2. The e exis s unied ime scale densi y
κ(ω)
gi en by sca e ingspec al and modula ow
bounda y geome y compa ibili y;
3. QNEC, QFC, and gauge ene gy posi i i y hold a conside ed scales, making gene alized en-
opy ex emali y locally equi alen o Eins ein equa ions;
4. Bounda y sca e ingen opy da a sa ises sucien comple eness and egula i y, allowing
unique econs uc ion o bulk geome y and cosmological pa ame e s (up o dieomo phism)
h ough bounda y igidi y and in e se p oblem heo y;
5. Model amily used by obse e sa ises in o ma ion geome ic iden iabili y: i sca e ing
en opycausal da a dis ibu ions om all ealizable expe imen s coincide, co esponding uni-
e se objec s a e isomo phic in
Uni
.
Unde hese assump ions, uni e seobse e isomo phism p oblem can be p ecisely s a ed and
sol ed.
3 Main Resul s: Ca ego ies, Func o s and Equi alence
This sec ion denes uni e se objec ca ego y
Uni
and obse e objec ca ego y
Obs
, in oduces phys-
ical subca ego y and comple e obse e subca ego y, s a ing uni e seobse e ca ego ical equi a-
lence and main heo em sel is isomo phic o uni e se.
3.1 Uni e se Ca ego y
Uni
Deni ion 2
(Uni e se Objec )
.
A uni e se objec is quin uple
U= (M, g, ≺, κ, Sgen)
sa is ying:
1.
M
is ou -dimensional, o ien ed, ime-o ien ed smoo h mani old;
g
is Lo en z me ic;
2.
≺
is causal pa ial o de compa ible wi h
g
ligh cone s uc u e, and
(M, g, ≺)
globally hype -
bolic;
3.
κ
is unied ime scale densi y, i.e., he e exis s sca e ing sys em and bounda y algeb a such
ha
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω)
holds;
4. Fo each
p∈M
and sucien ly small
, gene alized en opy unc ional
Sgen
is dened on
small causal diamond
Dp,
sa is ying:
6

Unde xing eec i e olume o edshi cons ain ,  s -o de ex emali y o
Sgen
equi -
alen o local Eins ein equa ions;

Second-o de non-nega i i y equi alen o local quan um s abili y (such as gauge ene gy
non-nega i i y).
Deni ion 3
(Uni e se S a e)
.
Gi en uni e se objec
U
, i s physical s a e includes bounda y ob-
se able algeb a
A∂
, ai h ul s a e
ω∂
, and bulk quan um eld heo y s uc u e sa is ying QNEC/QFC.
Below, uni e se objec  de aul s o include such s a e da a.
Deni ion 4
(
Uni
Mo phisms)
.
Fo wo uni e se objec s
U= (M, g, ≺, κ, Sgen), U′= (M′, g′,≺′, κ′, S′
gen),
a mo phism
:U→U′
is smoo h dieomo phism
:M→M′
sa is ying:
1.
∗g′=g
, and
p≺q
i and only i
(p)≺′ (q)
;
2. The e exis cons an s
a > 0, b ∈R
such ha
κ′=a κ +b
( ime scale equi alence class
consis ency);
3. Fo any cu su ace
Σ⊂M
and i s image
(Σ) ⊂M′
,
S′
gen( (Σ)) = Sgen(Σ).
I
is bijec ion and i s in e se
−1
is also mo phism, call
isomo phism o uni e se objec s,
deno ed
U∼
=U′
.
Deni ion 5
(Physical Subca ego y)
.
Deno e
Uni phys ⊂Uni
as subca ego y o med by uni e se
objec s and mo phisms sa is ying unied ime scale assump ion, gene alized en opyeld equa ion
equi alence, and bounda y sca e ingen opy da a comple eness.
3.2 Obse e Ca ego y
Obs
Deni ion 6
(Obse e Objec )
.
An obse e objec is 9- uple
O= (γ, Λ,A, ω, M, Uupd, u, C, κO),
whe e:
1.
γ
is abs ac isomo phism class o imelike wo ldline (wi h inhe en pa ame e iewed as
obse e p ope ime);
2.
Λ
is esolu ion scale o amily he eo , de e mining dis inguishable ime equencyspa ial
bandwid h;
3.
A
is obse able algeb a accessible o obse e , ypically comp ession o subalgeb a o bounda y
algeb a
A∂
;
4.
ω
is s a e on
A
, cha ac e izing obse e 's belie o memo y;
5.
M
is candida e model amily, each elemen co esponding o isomo phism class o pa ame ic
ep esen a ion o uni e se objec ;
7
6.
Uupd
is upda e ope a o , b inging measu emen esul s and communica ion da a in o e olu ion
o
(ω, M)
;
7.
u
is u ili y unc ion o selec ing expe imen s and ac ions;
8.
C
is communica ion s uc u e, cha ac e izing obse e 's channels wi h o he obse e s o en-
i onmen ;
9.
κO
is ime scale densi y used in e nally by obse e .
Deni ion 7
(Time Scale Consis ency)
.
Gi en uni e se objec
U
's scale densi y
κ
, i obse e
objec
O
's
κO
sa ises exis ence o
a > 0, b ∈R
such ha
κO(ω) = a κ(ω) + b,
hen call
O
and
U
ime scale equi alence class consis en .
Deni ion 8
(
Obs
Mo phisms)
.
Fo wo obse e objec s
O= (γ, Λ,A, ω, M, Uupd, u, C, κO), O′= (γ′,Λ′,A′, ω′,M′, U′
upd, u′,C′, κ′
O),
a mo phism
Φ : O→O′
consis s o map g oup
Φ=(ϕγ, ϕΛ, ϕA, ϕM)
sa is ying:
1.
ϕγ:γ→γ′
is causal-o de -p ese ing mono one bijec ion;
2.
ϕΛ: Λ →Λ′
mono one;
3.
ϕA:A→A′
is *-homomo phism, and
ω′(ϕA(A)) = ω(A)
o all
A∈ A
;
4.
ϕM:M→M′
is bijec ion on model equi alence classes, and upda e ope a o sa ises
U′
upd ◦(ϕA, ϕM)=(ϕA, ϕM)◦Uupd.
I
Φ
is in e ible and
Φ−1
is also mo phism, call
O∼
=O′
.
3.3 Comple e Obse e s and Ma hema ical Sel 
Deni ion 9
(Comple e Obse e )
.
Obse e objec
O
is called comple e i :
1.
Causal comple eness
: I s wo ldline
γ
has sucien in e wining wi h all small causal dia-
mond amilies o uni e se objec
U
, and h ough bounda y sca e ingen opy measu emen s,
can ob ain sucien da a on each
Dp,
o econs uc local in o ma ion o
κ
and
Sgen
;
2.
Time scale alignmen
: I s in e nal scale
κO
and some uni e se objec
U
's
κ
belong o same
equi alence class;
3.
Model iden iabili y
: I s model amily
M
sa ises: i wo models gi e iden ical p obabili y
dis ibu ions on sca e ingen opycausal da a om all ealizable expe imen s, hen hei
co esponding uni e se objec s a e isomo phic in
Uni
;
8
4.
Sel - e e en ial consis ency
: Fo ou pu s om sel  and inpu s om ex e nal uni e se, up-
da e ule
Uupd
p oduces no s uc u al con adic ion, especially consis en wi h scale alignmen
and
Z2
opological sec o o bounda y ime geome y.
Deno e subca ego y o all comple e obse e s as
Obs ull ⊂Obs
.
Deni ion 10
(Ma hema ical Deni ion o Sel )
.
In gi en physical uni e se subca ego y
Uni phys
,
in e p e isomo phism class o some comple e obse e
O∈Obs ull
as ma hema ical ealiza ion o
sel . Tha is, sel  is isomo phism class o obse e objec s sa is ying Deni ion 3.8 condi ions.
3.4 Main Theo ems
Unde abo e deni ions, his pape 's wo co e esul s a e as ollows.
Theo em 11
(Ca ego ical Equi alence)
.
Unde Assump ions 2.12.6, he e exis unc o s
F:Uni phys →Obs ull, R :Obs ull →Uni phys
and na u al isomo phisms
η: IdUni phys ⇒R◦F, ϵ : IdObs ull ⇒F◦R,
such ha
F
and
R
gi e ca ego ical equi alence be ween
Uni phys
and
Obs ull
.
In o he wo ds, o any
U∈Uni phys
he e exis s na u al isomo phism
ηU:U→R(F(U))
; o
any
O∈Obs ull
he e exis s na u al isomo phism
ϵO:O→F(R(O))
, sa is ying na u ali y equa ions.
Theo em 12
(Isomo phism Be ween Sel  and Uni e se)
.
Take any physical uni e se objec
Uou e ∈Uni phys
; le
O:= F(Uou e )∈Obs ull
be comple e obse e induced by his uni e se, whose
isomo phism class is in e p e ed as sel . Dene sel 's in e nal uni e se model as
Uinne := R(O)∈Uni phys.
Then he e exis s uni e se isomo phism
Uinne ∼
=Uou e ,
and his isomo phism is uniquely de e mined in
Uni phys
by na u al ans o ma ion
η
.
The e o e, in unied causal imeen opyma ix uni e se amewo k, sel 's in e nal wo ld
model and ex e nal uni e se objec a e s uc u ally isomo phic; his isomo phism is p ecise ma he-
ma ical e sion o my mind is he uni e se.
Co olla y 13
(Ma ix Uni e se Ve sion)
.
In THE-MATRIX ep esen a ion, i uni e se is gi en by
da a
THE
-
MATRIX = Hchan, S(ω), Q(ω), κ, A∂, ω∂,
hen comple e obse e 's in e nal sca e ingdelay ne wo k is isomo phic o abo e ma ix uni e se
in equencychannel eedback s uc u e, especially unied scale
κ
and
Z2
opological sec o a e com-
ple ely consis en .
4 P oo s: Func o Cons uc ion and S uc u al A gumen s
This sec ion p o ides p oo s uc u e o Theo ems 3.10 and 3.11, pos poning echnically in ensi e
pa s o Appendices AC.
9
Acknowledgemen s & Code A ailabili y
Concep s and p oo s in ol ed in his wo k ely on mul iple ma u e elds including sca e ing heo y,
ope a o algeb as, Lo en zian geome y, in e se p oblem heo y, and in o ma ion geome y; we pay
ibu e o pionee s in ela ed elds.
This pape does no use independen ly de eloped code o nume ical p og ams.
Appendix A: Bounda y Da a, Local Recons uc ion and Global Unique-
ness
This appendix p o es: unde unied ime scale and gene alized en opyeld equa ion equi alence
assump ions, sca e ingen opy da a on small causal diamonds uniquely de e mines local geome y
and cosmological cons an ; unde bounda y igidi y and in e se p oblem heo y suppo , hese local
da a can be uniquely glued in o global uni e se objec , suppo ing uniqueness o
R(O)
.
A.1 Local Recons uc ion on Small Causal Diamonds
Conside small causal diamond
Dp,
in uni e se objec
U= (M, g, ≺, κ, Sgen)
.
Local da a
: Assume on
∂Dp,
know:
1. Fixed- equency sca e ing ma ix
SD(ω)
and i s Wigne Smi h ime-delay ma ix
QD(ω)
,
ob aining local scale densi y
κD(ω) = φ′
D(ω)
π=1
2π QD(ω);
2. Fo all null di ec ions and cu su aces,  s -o de gene alized en opy a ia ion
δSgen
and
second-o de a ia ion
δ2Sgen
, assuming hese a ia ions sa is y QNEC, QFC, and gauge en-
e gy non-nega i i y.
P oposi ion 14
(A.1)
.
Unde abo e condi ions, me ic
g
and cosmological cons an
Λ
in e io o
Dp,
a e uniquely de e mined up o dieomo phism.
P oo ske ch
:
1.
Fi s a ia ion and eld equa ions
: Unde xing eec i e olume o edshi condi ions,
anishing  s a ia ion o gene alized en opy is equi alen o ex emal su aces sa is ying
quan um minimal (o maximal) condi ion; oge he wi h QFC gi es cons ain s on
Rab
and
ene gy-momen um enso
Tab
. Combined wi h IGVP ype esul s, hese cons ain s can be
con e ed in o nonlinea Eins ein equa ions.
2.
Second a ia ion and s abili y
: Second a ia ion non-nega i i y equi alen o Hollands
Wald gauge ene gy non-nega i i y, meaning eld equa ion solu ion is s able unde small pe -
u ba ions, excluding ce ain non-physical solu ions o mul i- aluedness.
3.
Scale alignmen and cosmological e m
: Local scale densi y
κD(ω)
couples wi h cosmo-
logical e m in eec i e ac ion h ough hea ke nel expansion and spec al shi unc ion, hus
unde gi en sca e ing da a,
Λ
and ligh cone s uc u e no maliza ion a e uniquely xed.
4.
In summa y
:
g|Dp,
and
Λ
unique up o dieomo phism.
Rigo ous p oo equi es in oducing pe u ba i e spec al geome y, p ecise ela ions among
ela i e sca e ing de e minan and gene alized en opyac ion unc ionals; de ails omi ed he e.
16

A.2 Global Gluing and Bounda y Rigidi y
Le
{Dpi, i}
be small causal diamond co e o
M
; o each
Dpi, i
al eady ob ained local me ic
gi
and cosmological cons an
Λi
by P oposi ion A.1, and by physical con inui y know
Λi
cons an
consis en .
On o e lap egion
Dpi, i∩Dpj, j
, bounda y sca e ingen opy da a consis en , so
gi, gj
a e
dieomo phically equi alen on his egion; can cons uc global me ic
g
and causal s uc u e
≺
h ough s anda d Galois gluing and ech consis ency.
Fu he mo e, unde app op ia e bounda y igidi y and in e se p oblem heo ems (e.g., igidi y
esul s o bounda y dis ance unc ion and sca e ing phase), can p o e: i wo uni e se objec s
ha e consis en bounda y sca e ingen opy da a on all small causal diamonds, hen he e exis s
dieomo phism
mapping one uni e se o ano he while p ese ing me ic, causal s uc u e, scale,
and gene alized en opy, hus isomo phic in
Uni
.
P oposi ion 15
(A.2: Uni e se Recons uc ion Uniqueness)
.
Unde Assump ion 2.6, comple e
bounda y sca e ingen opy da a uniquely de e mines uni e se objec 's isomo phism class in
Uni
.
This p o ides geome ic and analy ic ounda ion o deni ion and uniqueness o
R(O)
.
Appendix B: In o ma ion-Geome ic Iden iabili y and Model Con-
e gence
This appendix s udies iden iabili y and asymp o ic con e gence o comple e obse e model amily.
B.1 Pa ame ic Family and S a is ical Model
Le
Θ⊂Rn
be compac pa ame e space; o each
θ∈Θ
associa e uni e se objec
Uθ∈Uni phys
,
deno ing s a is ical dis ibu ion o bounda y sca e ingen opy da a as
Pθ
. Obse e 's model amily
M
can be iewed as collec ion
{Uθ}θ∈Θ
.
Assump ion 16
(B.1: In o ma ion Iden iabili y)
.
1. I
Pθ1=Pθ2
, hen
Uθ1∼
=Uθ2
isomo phic
in
Uni
;
2. Rela i e en opy
D(Pθ1∥Pθ2)=0
i and only i
Uθ1, Uθ2
isomo phic.
Unde his assump ion,
Θ/∼
(quo ien space by uni e se isomo phism) becomes in o ma ion
geome ic mani old, whose Fishe Rao me ic and Eguchi di e gence s uc u es co espond o s a-
is ical p ope ies o
Pθ
.
B.2 Obse e Upda e as In o ma ion-G adien Flow
Model comple e obse e 's upda e ule
Uupd
as Bayesian upda e o pa ame e p io
π(θ)
o in-
o ma ion geome ic g adien ow o model dis ibu ion
q(θ)
. One obse a ion
x∼Pθ∗
leads o
upda e
q +1(θ)∝q (θ)p(x|θ),
o in con inuous limi
dq
d =−∇D(q ∥Pθ∗),
whe e
D
is KullbackLeible di e gence.
Unde s anda d law o la ge numbe s and la ge de ia ion p inciple, can p o e:
17
P oposi ion 17
(B.2: Model Con e gence)
.
I
O∈Obs ull
's model amily sa ises Assump ion
B.1, hen as numbe o obse a ions ends o inni y o p ope ime
→ ∞
, model dis ibu ion
q
con e ges wi h p obabili y 1 o some equi alence class
[θ∗]
, co esponding o unique uni e se objec
isomo phism class
[Uθ∗]
.
Combined wi h uni e se econs uc ion uniqueness in Appendix A, can dene
R(O)
as ep e-
sen a i e o his isomo phism class, p o ing a ionali y o Cons uc ion 4.3.
Appendix C: NullModula Double Co e ,
Z2
Sec o and Sel -Consis ency
This appendix supplemen s Sec ion 5's a gumen s abou NullModula double co e and
Z2
opo-
logical alignmen .
C.1
Z2
-Valued In a ian s om Sel -Re e en ial Sca e ing
Conside sel - e e en ial sca e ing ne wo k wi h eedback, whose sca e ing ma ix
S(ω)
is dened
on some ene gy window, assuming i s de e minan can be w i en in squa e- oo o m
de S(ω) = pde S(ω)2,
die en squa e- oo choices co esponding o
Z2
double co e . Fo each closed loop
γ
(e.g., in
ene gypa ame e space), can dene holonomy
ν√S(γ)∈Z2,
ep esen ing whe he squa e oo ips sign a e ci cling
γ
.
On o he hand, in NullModula double co e and BF- ype opological eld heo y, olume
in eg al wi h bounda y modula ow, gene alized en opy, and ene gy condi ions join ly de e mine
ela i e cohomology class
[K]∈H2(Y, ∂Y ;Z2)
, which unde app op ia e embedding can be in e -
p e ed as unied encoding o abo e holonomy.
C.2 Sel -Consis ency Condi ion o Comple e Obse e s
Fo comple e obse e
O∈Obs ull
, i s in e nal model also has sca e ing ma ix
SO(ω)
and squa e
oo
√de SO
. Sel - e e en ial consis ency equi es: o all physically allowed loops
γ
, obse e 's
in e nally p edic ed holonomy consis en wi h ex e nal uni e se's ue holonomy:
ν√SO(γ) = ν√SU(γ),
whe e
SU
is sca e ing ma ix amily o uni e se objec
U=R(O)
. I de ia ion exis s, obse e will
de ec
Z2
-le el phase o delay pa i y jumps in long- e m obse a ions, co ec ing i s model un il
bo h align.
This condi ion equi alen o equi ing co esponding cohomology class
[K]
o ake i ial alue,
ensu ing consis ency among local geome yene gy opological s uc u e. Thus sel 's sel - e e en ial
sca e ing ne wo k and uni e se on ology a e comple ely consis en a
Z2
opological le el, u he
s eng hening conclusion sel is isomo phic o uni e se.
18