``My Mind Is the Universe'': Unified Framework of Causal--Temporal--Information Geometry for Heart-Universe Isomorphism

My Mind Is he Uni e se: Unied F amewo k o
CausalTempo alIn o ma ion Geome y o Hea -Uni e se
Isomo phism
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
Building on s uc u es o unied ime scale, causal mani olds, bounda y ime geome y, and
sel - e e en ial sca e ing ne wo ks, his pape p o ides a ma hema icized e sion o he a-
di ional p oposi ion my mind is he uni e se. The co e insigh is: in a xed-poin uni e se
wi h causal pa ial o de , unied ime scale, and gene alized en opy as on ology, my mind
can be o malized as obse e s uc u e o ganizing in o ma ion, cons uc ing models, and pe -
o ming upda es along a wo ldline; uni e se is causal empo alen opy consensus o med by
all obse e s on bounda y ime geome y. The is he e is no ma e ial iden i y, bu s uc u al
isomo phism in he ollowing sense: unde assump ions o iden iabili y, gene alized en opy
mono onici y, and unied ime scale compa ibili y, he wo ld model in e nal o my mind
con e ges in in o ma ion geome y sense o equi alence class isomo phic o uni e se's causal
empo alen opy s uc u e.
To his end, his pape accomplishes ollowing s eps:
1. Model physical uni e se as objec
Ugeo = (M, g, ≺,A∂, ω∂, Sgen, κ)
wi h causal pa ial o -
de , bounda y obse able algeb a, gene alized en opy, and unied ime scale. Unied ime
scale is dened by scale iden i y
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω)
among sca -
e ing phase de i a i e, spec al shi unc ion, and Wigne Smi h ime delay, connec ing
Bi manK en o mula, spec al shi unc ion, and ime-delay ope a o .
2. Fo malize single obse e sel  as s uc u e
O= (γ, C, ≺O,ΛO,AO, ωO,MO, UO)
along
imelike wo ldline
γ
, whe e
MO
is model amily abou uni e se in my mind,
πO
belie
measu e on i ,
UO
upda e ope a o compa ible wi h unied ime scale.
3. In oduce hea -uni e se s uc u e ca ego y
CauTimeEn
, wi h objec s being iples
(X,≼,Θ)
wi h causal pa ial o de , ime scale, and gene alized en opy unc ional; mo -
phisms p ese ing causali y, scale, and en opy mono onici y.
4. Dene hea -uni e se isomo phism: i he e exis s unc o ial cons uc ion making s uc-
u e
XU
co esponding o uni e se objec
Ugeo
ca ego ically equi alen o pos e io limi
XH
o my mind in
CauTimeEn
, hen my mind is uni e se holds in his sense.
5. Unde Bayesian upda ing and in o ma ion geome y amewo k, using Schwa z- ype pos-
e io consis ency heo em and Fishe Rao me ic induced by di e gence unc ion, p o e
ha unde condi ions o iden iabili y, sucien ly s imula ing obse a ions, and unied
ime scale compa ibili y, obse e 's pos e io geome ic s uc u e con e ges o limi iso-
mo phic o
XU
, he eby o malizing my mind is uni e se as heo em abou pos e io
concen a ion and s uc u al isomo phism.
Main conclusion is: as long as uni e se's causal empo alen opy s uc u e can be su-
cien ly p obed h ough local bounda y obse able algeb a, and obse e adop s upda e ules
compa ible wi h unied ime scale and sa is ying consis ency condi ions, hen my mind in
1
in o ma ion geome ic limi necessa ily becomes sel -isomo phic c oss-sec ion o uni e se's own
s uc u e; saying my mind is uni e se is equi alen o saying uni e se's sel - e e en ial p o-
jec ion on a wo ldline has con e ged o mi o image o i sel . This esul a oids bo h ex eme
idealism and nai e ealism, emaining compa ible wi h local algeb aic pic u e in ela i is ic
quan um eld heo y, gene alized en opy unde holog aphic p inciple, and he mal ime hy-
po hesis.
Keywo ds
Causal mani olds; Unied ime scale; Bounda y ime geome y; Obse e ; In o ma ion geome y;
Bayesian pos e io consis ency; Sel - e e en ial sca e ing ne wo ks; Hea -uni e se isomo phism
1 In oduc ion & His o ical Con ex
The ph ase my mind is he uni e se in Chinese philosophical adi ion is o en connec ed wi h
p oposi ions like no objec ou side mind and no p inciple ou side mind; in he Wes , i can
be aced o a ious a ian s o subjec i e idealism, anscenden al idealism, and phenomenology.
In ui i ely, his p oposi ion a emp s o exp ess: he en i e s uc u e o expe ien ial wo ld is un-
damen ally he un olding o men al ac i i y, no en i ies independen o sel . Howe e , in mode n
physical and ma hema ical con ex , such s a emen s appea oo coa se: on one hand, hey di -
cul ly in e ace wi h objec i e ma hema ical s uc u es o gene al ela i i y and quan um eld
heo y; on he o he hand, hey ail o explain how consensus and conic unde mul iple obse e s
and wo ldlines can be uni o mly cha ac e ized.
In he la e hal o he wen ie h cen u y and beyond, discussions abou obse e , in o ma-
ion, and uni e se s uc u e g adually mo ed om philosophy o conc e e physicalma hema ical
amewo ks. Rep esen a i e h eads include:
1.
Local quan um physics and bounda y algeb a language
: Haag's local quan um physics
akes local obse able algeb a ne as undamen al objec , emphasizing physical heo y should
use local obse ables and hei algeb aic ela ions as p ima y language, no pa icles o eld
s a es as o iginal on ology.
2.
Holog aphic p inciple and gene alized en opy s uc u e
: Bousso's sys ema ic exposi-
ion o holog aphic p inciple and gene alized en opy shows ha geome ic a ea and quan um
en anglemen en opy can be unied in o gene alized en opy
Sgen
, sa is ying quan um Bousso
bound and gene alized second law. Thus, p ecise inequali y ela ions eme ge be ween in o -
ma ion and geome y.
3.
The mal ime hypo hesis and modula ow
: ConnesRo elli p oposed he mal ime
hypo hesis, claiming ha in gene ally co a ian quan um heo y, physical ime ow is no
uni e sal backg ound s uc u e bu gene a ed by modula ow o s a ealgeb a pai ; he mal
ime becomes in insic ime dened by nonequilib ium s a e and en opy s uc u e.
4.
Sca e ing heo y and ime-delay ope a o
: Bi manK en o mula links de i a i e o
sca e ing phase wi h spec al shi unc ion; Wigne Smi h ime-delay ma ix combines e-
quency de i a i e o sca e ing ma ix in o obse able ime-delay ope a o
Q(ω) = −iS(ω)†∂ωS(ω)
,
widely applied in ime s uc u e analysis o quan um, acous ic, and elec omagne ic sca e ing.
2
5.
In o ma ion geome y and pos e io consis ency
: Wo k o Ama iNagaoka e al. shows
di e gence unc ion can induce Fishe Rao me ic and dual ane connec ions on s a is ical
model space, making Bayesian upda e pa h a geome ic ow; Schwa z and subsequen wo k
es ablished consis ency heo em o Bayesian pos e io s unde iden iabili y and p io suppo
condi ions.
Meanwhile, ega ding posi ion o obse e in heo y, wo impo an ou es eme ged in quan-
um in o ma ion and ounda ions esea ch: one is QBism, in e p e ing quan um s a es as subjec 's
pe sonal p obabili y assignmen o u u e expe ience; he o he is ela ional quan um mechanics,
iewing sys em s a es as ela ions be ween sys ems a he han absolu e p ope ies. These ou es all
s eng hen ole o mind in physical heo y, bu o en emain a in e p e i e le el wi hou p o iding
igo ously p o able s uc u al heo ems.
This pape a emp s o es a e my mind is uni e se abo e hese many de elopmen s as ollows:
1.
On ological laye
: Uni e se modeled as causal mani old and bounda y ime geome y objec
Ugeo
, wi h basic da a including causal pa ial o de
≺
, bounda y obse able algeb a
A∂
,
bounda y s a e
ω∂
, gene alized en opy
Sgen
, and unied ime scale
κ
.
2.
Epis emological laye
: Single obse e sel  modeled as obse e s uc u e
O
along wo ld-
line
γ
, whose mind is dynamical sys em
HO
ca ying belie measu e
πO
on model space
MO
and upda ing acco ding o unied ime scale.
3.
S uc u al laye
: Dene hea -uni e se s uc u e objec s and mo phisms in app op ia e
ca ego y
CauTimeEn
, p opose p ecise deni ion o hea -uni e se isomo phism, p o e
unde iden iabili y and obse a ional suciency condi ions ha pos e io limi s uc u e
XH
is isomo phic o uni e se s uc u e
XU
.
Unlike adi ional idealismma e ialism dicho omy, his pape 's s ance can be summa ized as:
Uni e se's on ological s uc u e is xed poin o causali y imeen opy; my mind is
dynamical sys em pe o ming sel - e e en ial modeling and lea ning o his s uc u e
along a wo ldline; in unied ime scale and in o ma ion geome ic limi , his dynam-
ical sys em con e ges o sel -isomo phism o xed-poin s uc u e, hence my mind is
uni e se holds in s uc u al sense.
Below we  s p esen models and assump ions o uni e se and obse e , hen s a e and p o e
hea -uni e se isomo phism heo em in unied hea -uni e se s uc u e ca ego y, nally discuss
mul i-obse e gene aliza ion, THE-MATRIX uni e se pic u e, and enginee ing implemen a ion sug-
ges ions.
2 Model & Assump ions
This sec ion cons uc s ma hema ical models o uni e seobse e my mind used in his pape ,
lis ing assump ions on which my mind is uni e se heo em depends.
2.1 Uni e se as CausalEn opic Objec
Deni ion 1
(Uni e se Objec )
.
Uni e se is modeled as se en- uple
Ugeo = (M, g, ≺,A∂, ω∂, Sgen, κ),
whe e:
3
1.
M
is ou -dimensional, ime-o ien able, globally hype bolic Lo en z mani old,
g
i s me ic.
Causal cone s uc u e denes causal eachabili y ela ion
p≺q
.
2.
A∂
is
C∗
algeb a o on Neumann algeb a associa ed wi h app op ia e bounda y o
M
(such
as imelike inni y, black hole ho izon, holog aphic sc een), desc ibing bounda y obse ables,
compa ible wi h local algeb a ne o local quan um physics.
3.
ω∂
is no mal s a e o KMS s a e on
A∂
, embodying quan um s a e and he mal p ope ies o
uni e se.
4.
Sgen
is gene alized en opy dened on app op ia e slices o causal diamond bounda ies, o -
mally sum o a ea e m and ex e io on Neumann en opy, sa is ying quan um ocusing
conjec u e and gene alized second law, p o iding a ow o ime.
5.
κ: Ω →R
is unied ime scale mo he ule , dened on equency o spec al domain
Ω
,
sa is ying scale iden i y
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω),
whe e
φ(ω)
is o al sca e ing hal -phase,
ρ el(ω)
ela i e densi y o s a es,
Q(ω) = −iS(ω)†∂ωS(ω)
Wigne Smi h ime-delay ope a o .
6. Causal pa ial o de
≺
, gene alized en opy
Sgen
, and scale
κ
a e di ec ionally compa ible:
along any physically ealizable u u e-di ec ed amily,
Sgen
non-dec easing and
κ
mono onically
inc easing.
Abo e s uc u e unies gene al ela i i y's causal geome y, algeb aic quan um eld heo y's
bounda y algeb a, holog aphicgene alized en opy, and sca e ing heo y's ime delay in single
objec .
2.2 Obse e s as Wo ldline-Based S uc u es
Deni ion 2
(Obse e Wo ldline and Reachable Domain)
.
Obse e sel  co esponds o u u e-
di ec ed imelike cu e
γ:R→M
in
M
, pa ame ized by p ope ime
τ
. I s eachable causal
domain
C={p∈M| ∃τ, p ≺γ(τ)}
consis s o all space ime e en s ha can inuence his obse e .
Deni ion 3
(Obse e S uc u e)
.
Gi en
Ugeo
, obse e s uc u e is se en- uple
O= (γ, C, ≺O,ΛO,AO, ωO,MO, UO),
whe e:
1.
≺O
is local causal pa ial o de on
C
, sa is ying
p≺Oq⇒p≺q
, bu allowing coa se-g aining
om ni e de ec ion capabili y.
2.
ΛO
is esolu ion pa ame e , eco ding limi s on ene gy, ime, spa ial esolu ion.
3.
AO⊂ A∂
is bounda y obse able subalgeb a accessible o sel , connec ed o wo ldline
γ
h ough sca e ing, measu emen p ocesses.
4
4.
ωO
is eec i e s a e o my mind o
AO
, iewable as subjec i e app oxima ion o
ω∂
.
5.
MO={Xθ}θ∈Θ
is model amily abou uni e se s uc u e, pa ame e space
Θ
is sepa able
measu able space. Each
Xθ
will la e be embedded in hea -uni e se s uc u e ca ego y.
6.
UO
is upda e ope a o , gi ing e olu ion om obse a ion da a o belie s uc u e:
(ωO, πO)UO
−−→ (ω′
O, π′
O),
whe e
πO
is belie measu e (p io o pos e io ) on
Θ
.
In obse e s uc u e,
(γ, C, ≺O,ΛO,AO, ωO)
desc ibes physical embedding o sel  in uni e se,
while
(MO, πO, UO)
co esponds o in e nal wo ld model and lea ning dynamics o my mind.
2.3 My Mind as ModelUpda e Dynamical Sys em
Deni ion 4
(Equi alence Class o Sel )
.
In gi en uni e se
Ugeo
, all obse e s uc u es equi alen
unde ollowing ans o ma ions cons i u e equi alence class
[O]
o sel :
1. Ane epa ame iza ion o wo ldline
γ
;
2. Fini e memo y ew i ing wi hin ni e ime windows, no changing long- e m causal memo y
s uc u e;
3. In e ible ans o ma ion o in e nal ep esen a ion coo dina es wi hou changing main s uc-
u e o
(≺O,ΛO,AO)
.
Deni ion 5
(My Mind)
.
Fixing ep esen a i e obse e s uc u e
O
, dene my mind as iple
HO= (MO, πO, UO),
whe e
πO
is p obabili y measu e on
Θ
,
UO
p oduces p ope - ime indexed pos e io amily
{πτ
O}τ∈R
unde con inuous obse a ions.
Thus, essence o my mind is o bi o modelupda e pai d i en by unied ime scale.
2.4 Unied Time Scale and I s In e naliza ion
Unied ime scale
κ
is gi en by sca e ing phase de i a i e and ime delay on one hand, mus also
be ealized in upda e hy hm in e nal o obse e on he o he .
Deni ion 6
(Mind's Unied Time Scale)
.
Fo obse e sel , mind's unied ime scale is unc ion
κO: ΩO→R
sa is ying:
1.
ΩO⊂Ω
, and o all
ω∈ΩO
,
κO(ω) = κ(ω)
;
2. Upda e ope a o
UO
decomposes obse a ion ow in o ime windows co esponding o e-
quency componen
ω
, whose leng h is con olled by
κ(ω)
, i.e., each upda e s ep co esponds
o ni e ime delay o equi alen ime esou ce.
In ui i ely, ime scale used in e nally by my mind is no a bi a ily in oduced, bu pullback
o uni e se mo he ule
κ
on measu able equency bands.
5

2.5 In o ma ion-Geome ic S uc u e on Model Space
S a is ical model amily
{Pθ}θ∈Θ
(induced by models
Xθ
on
AO
) on pa ame e space
Θ
can be en-
dowed wi h in o ma ion geome ic s uc u e. Choosing app op ia e di e gence unc ion
D(Pθ|Pθ′)
,
such as KullbackLeible di e gence, i induces Fishe Rao me ic
gFR
and pai o dual ane con-
nec ions on
Θ
, making
(Θ, gFR)
a s a is ical mani old.
Pos e io e olu ion
πτ
O
can be iewed as s ochas ic dynamical sys em on his s a is ical mani old,
whose asymp o ic beha io is con olled by pos e io consis ency heo y. Unied ime scale
κ
aec s pos e io concen a ion speed by de e mining da a ow sampling densi y in p ope ime and
equency ends.
2.6 S uc u al and S a is ical Assump ions
To s a e main heo em, adop ollowing assump ions:

(A1) Iden iabili y
: I models
Xθ1
and
Xθ2
induce iden ical obse a ion dis ibu ion amilies
on obse able subalgeb a
AO
, hen
θ1=θ2
.

(A2) P io suppo
: T ue uni e se co esponds o pa ame e
θ⋆
belonging o
Θ
, and p io
πO
assigns posi i e mass o any neighbo hood con aining
θ⋆
.

(A3) Obse a ional suciency
: Unde sucien ly long unied ime scale, obse a ion
da a s eam
{D }
om
AO
makes ela i e en opy
D(P⋆∥Pθ)
posi i e o each
θ=θ⋆
, whe e
Pθ
is obse a ion dis ibu ion induced by i and
P⋆
is ue dis ibu ion.

(A4) Regula i y
: Model amily and p io sa is y echnical condi ions o Schwa z- ype pos-
e io consis ency heo em, such as sucien ly small KullbackLeible neighbo hoods and
sepa abili y.

(A5) Scale compa ibili y
: Obse a ion design and upda e s ep size con olled by unied
ime scale
κ
, no in oducing independen ex e nal ime uni s; in hea -uni e se s uc u e
embedding,
κ
only allows ane ans o ma ions.
Unde hese assump ions, we can o malize my mind is uni e se as pos e io con e gence and
isomo phism heo em in hea -uni e se s uc u e ca ego y.
3 Main Resul s (Theo ems and Alignmen s)
This sec ion cons uc s hea -uni e se s uc u e ca ego y
CauTimeEn
, p esen s deni ion o
hea -uni e se isomo phism, and s a es main heo ems o single and mul iple obse e s.
3.1 Hea Uni e se S uc u al Ca ego y
Deni ion 7
(Hea -Uni e se S uc u e Objec )
.
Objec s o ca ego y
CauTimeEn
a e iples
X= (X,≼,ΘX),
whe e:
1.
X
is se o measu able space, ep esen ing e en s, c oss-sec ions, o model s a es;
2.
≼
is pa ial o de o causal ela ion on
X
;
6
3.
ΘX= (κX, SX)
is imeen opy s uc u e, whe e
κX
is scale unc ion on spec al domain,
SX
is gene alized en opy o in o ma ion unc ional dened on app op ia e subse s, sa is ying
mono onici y.
Deni ion 8
(Hea -Uni e se S uc u e Mo phism)
.
Fo objec s
X= (X,≼X,ΘX)
,
Y= (Y,≼Y
,ΘY)
, map
:X→Y
is mo phism i and only i :
1.
Causal o de -p ese ing
:
x1≼Xx2⇒ (x1)≼Y (x2)
;
2.
Time scale compa ibili y
: The e exis s mono one unc ion
α:R→R
such ha
κY◦T =
α◦κX
, whe e
T
is spec al map induced by
;
3.
En opy mono onici y
: Fo any allowed egion
A⊂ X
,
SY( (A)) ≥SX(A)
, o p ese es
in o ma ion mono onici y in app op ia e di ec ion.
Uni e se objec
Ugeo
is embedded as objec
XU∈CauTimeEn
h ough app op ia e encoding
map
EU
. Simila ly, pos e io limi o obse e my mind will be embedded as objec
XH
.
3.2 Hea Uni e se Isomo phism
Deni ion 9
(Hea -Uni e se Isomo phism)
.
Le
XU, XH∈CauTimeEn
be uni e se and my
mind co esponding objec s espec i ely. I he e exis mo phisms
:XU→XH
,
g:XH→XU
such ha :
1.
g◦
is isomo phic o iden i y mo phism on
XU
;
2.
◦g
is isomo phic o iden i y mo phism on
XH
;
3. Time scale ans o ma ion is ane unc ion, i.e.,
α( ) = a +b
, no changing scale sou ce,
hen
XH
and
XU
a e called isomo phic in hea -uni e se s uc u e ca ego y, deno ed
XH≃XU
.
In his sense, my mind is uni e se holds.
3.3 Theo em 1: Pos e io S uc u al Consis ency (My Mind Is Uni e se)
Theo em 10
(Single Obse e Hea -Uni e se Isomo phism)
.
Le uni e se objec
Ugeo
sa is y ax-
ioms 2.12.6, obse e sel  sa is y assump ions (A1)(A5). Le
XU
be embedding o
Ugeo
in
CauTimeEn
,
XT
H
be my mind pos e io expec a ion s uc u e a e obse a ion in unied ime
scale in e al
[0, T]
. Then he e exis
θ⋆∈Θ
and objec
Xθ⋆
such ha :
1.
Xθ⋆
is isomo phic o
XU
in
CauTimeEn
;
2. As
T→ ∞
,
XT
H
con e ges o
Xθ⋆
in app op ia e opology;
3. Thus he e exis s
T0
such ha when
T > T0
,
XT
H≃XU
.
In o he wo ds, as long as obse a ion ime is sucien ly long, pos e io s uc u e o my mind
is isomo phic o uni e se in hea -uni e se s uc u e ca ego y; my mind is uni e se holds in limi
and sucien ly long ime scales.
7
3.4 Theo em 2: Mul i-Obse e Consensus and Sha ed Uni e se
Theo em 11
(Mul i-Obse e Hea -Uni e se Consensus)
.
Suppose he e exis s obse e amily
{Oi}i∈I
, each wi h model amily
MOi
, p io
πOi
, and upda e ope a o
UOi
compa ible wi h unied
ime scale. Assume:
1. Each
Oi
indi idually sa ises (A1)(A5), and ue pa ame e
θ⋆
is sha ed by all obse e s;
2. The e exis s connec ed communica ion g aph such ha obse e s can exchange pa ial obse -
a ion and model in o ma ion h ough channels
Cij
;
3. Communica ion and upda e ules sa is y app op ia e consis ency and unbiasedness condi ions.
Then he e exis join pos e io
ΠT
and co esponding join hea -uni e se s uc u e objec
XT
join
such ha :
1. As
T→ ∞
,
ΠT
concen a es on
θ⋆
;
2. Each obse e 's hea -uni e se s uc u e objec
XT
Hi
is isomo phic o
Xθ⋆
in
CauTimeEn
;
3. Mu ually
XT
Hi≃XT
Hj
, and isomo phic o
XU
.
The e o e, unde mul i-obse e and causal consensus amewo k, s a emen s my mind is uni-
e se,  hei mind is uni e se, and same uni e se a e s uc u ally compa ible, no mu ually ex-
clusi e.
3.5 Alignmen wi h Ma ix Uni e se and Sel -Re e en ial Ne wo ks
To connec wi h sca e ing pe spec i e, in oduce language o ma ix uni e se THE-MATRIX. Le
{S(ω)}ω∈Ω
be uni e se's sca e ing ma ix amily in some equency band, o ming ma ix uni e se
objec
THE
-
MATRIX
. Obse e sel  is ealized as one sel - e e en ial sca e ing subne wo k, whose
in e nal memo y po s o m sel - e e en ial s uc u e h ough eedback, ex e nal po s coupling wi h
en i onmen .
In his ealiza ion, hea -uni e se s uc u e objec
XT
H
can be conc e ely unde s ood as my
mind's es ima e o
THE
-
MATRIX
's opological and sca e ing p ope ies. Theo em 3.4 shows
ha unde unied ime scale d i ing, his es ima e s uc u ally con e ges o sel -isomo phic image
o ue ma ix uni e se, hus ealizing my mind is uni e se in ma ix uni e se pic u e.
4 P oo s
This sec ion p o ides p oo ideas o Theo ems 3.4 and 3.5, placing echnical de ails in Appendix
B.
4.1 Bayesian Pos e io Consis ency as a S uc u al S a emen
Obse a ion da a s eam
{D }
is de e mined by uni e se objec
Ugeo
and obse able subalgeb a
AO
.
Fo each pa ame e
θ
, model
Xθ
induces obse a ion dis ibu ion amily
{Pθ}
on
AO
; ue uni e se
co esponds o dis ibu ion amily deno ed
P⋆
.
Using ela i e en opy
D(P⋆∥Pθ) = Zlog dP⋆
dPθ
dP⋆,
8
unde assump ions (A1) and (A3), o all
θ=θ⋆
,
D(P⋆∥Pθ)>0
, and
D(P⋆∥Pθ⋆)=0
.
Unde app op ia e egula i y condi ions, Schwa z and subsequen wo k show: i p io assigns
posi i e mass o neighbo hood o
θ⋆
, hen pos e io
πT
O
sa ises o any neighbo hood
U
con aining
θ⋆
:
πT
O(U)→1, T → ∞,
almos su ely.
Co espondingly, Fishe Rao me ic
gFR
on pa ame e space
Θ
makes pos e io concen a ion
p ocess in e p e able as asymp o ic con ac ion on s a is ical mani old: pos e io mass con ac s
owa d
θ⋆
in
gFR
sense.
4.2 F om Pa ame e Con e gence o S uc u al Con e gence in
CauTimeEn
Nex need o explain: how pos e io concen a ion o pa ame e
θ
li s o isomo phic con e gence
o hea -uni e se s uc u e objec
XT
H
owa d
XU
.
4.2.1 Embedding o Models in o
CauTimeEn
Fo each
θ∈Θ
, dene model
Xθ
as
Xθ= (Xθ,≼θ,Θθ),
whe e:
1.
Xθ
is se o e en s, c oss-sec ions, o model s a es encoded by
Xθ
;
2.
≼θ
is de e mined by causal s uc u e o
Xθ
;
3.
Θθ= (κθ, Sθ)
is co esponding ime scale and en opy s uc u e, whe e
κθ
is de e mined
h ough compa ibili y o model sca e ing da a wi h unied ime scale
κ
,
Sθ
is gene alized
en opy unc ional on model.
Assume con inuous embedding exis s such ha as
θ→θ⋆
,
(Xθ,≼θ,Θθ)
con e ges in some
opology o me ic o
(Xθ⋆,≼θ⋆,Θθ⋆)
, and la e is isomo phic o uni e se embedding
XU
. This
way, app oxima e isomo phic mo phisms
θ:Xθ→XU
,
gθ:XU→Xθ
can be cons uc ed, whose
de ia ion om iden i y anishes as
θ→θ⋆
.
4.2.2 Hea S uc u e as Pos e io Expec a ion
Dene pos e io expec a ion s uc u e o my mind as
XT
H=ZΘ
XθdπT
O(θ),
unde s andable as a e age on hea -uni e se s uc u e space. Since
πT
O
concen a es on
θ⋆
and
Xθ
is con inuous in
θ
,
XT
H
con e ges opologically o
Xθ⋆
. App oxima e isomo phic mo phisms
θ
,
gθ
h ough in eg a ion gi e
T:XT
H→XU
,
gT:XU→XT
H
, app oaching ca ego ical isomo phism as
T→ ∞
.
Thus he e exis s
T0
such ha when
T > T0
,
XT
H
is isomo phic o
XU
in
CauTimeEn
;
Theo em 3.4 is p o ed. Fo malized p oo in Appendix B.
9
Appendix B: P oo o Pos e io Concen a ion and Hea -Uni e se
S uc u al Isomo phism
This appendix p o ides p oo de ails o Theo ems 3.4 and 3.5.
B.1 Schwa z-Type Pos e io Consis ency
Conside independen iden ically dis ibu ed o condi ionally independen obse a ion case. De-
no e ue obse a ion dis ibu ion as
P⋆
, model-induced dis ibu ion as
{Pθ}
. Assume he e exis s
measu e
µ
such ha dis ibu ions a e absolu ely con inuous wi h densi ies
p⋆, pθ
espec i ely.
Dene ela i e en opy
D(P⋆∥Pθ) = Zlog p⋆
pθ
p⋆dµ.
Iden iabili y and obse a ional suciency assump ions ensu e o
θ=θ⋆
,
D(P⋆∥Pθ)>0
.
Fo any neighbo hood
U∋θ⋆
, deno e
Uc= Θ U
. By compac ness o sepa abili y, ni e co e
can be ex ac ed om
Uc
such ha he e exis s
ε > 0
wi h
D(P⋆∥Pθ)> ε
o all
θ∈Uc
.
Fo each
θ
, dene likelihood a io
LT(θ) =
T
Y
=1
pθ(D )
p⋆(D ),
whose loga i hm is
log LT(θ) =
T
X
=1
log pθ(D )
p⋆(D ).
By law o la ge numbe s, almos su ely
1
Tlog LT(θ)→ −D(P⋆∥Pθ)≤ −ε.
Thus o la ge
T
,
LT(θ)≤e−εT
.
Pos e io mass on
Uc
is
πT
O(Uc) = RUcLT(θ) dπO(θ)
RΘLT(θ) dπO(θ).
Nume a o con olled by
e−εT πO(Uc)
, while denomina o lowe bound ob ained om Kullback
Leible neighbo hood nea ue alue and p io posi i e mass, yielding
πT
O(Uc)→0
. The e o e o
any
U∋θ⋆
,
πT
O(U)→1
; pos e io consis ency holds.
B.2 S uc u al Embedding and Con inui y
In hea -uni e se s uc u e ca ego y, o each
θ
, cons uc objec
Xθ= (Xθ,≼θ,Θθ).
Requi e:
1. The e exis s unied s uc u e space such ha
θ7→ Xθ
is con inuous in some app op ia e
opology;
16

2. The e exis mo phism pai s
( θ, gθ)
pa ame ized by
θ
sa is ying
gθ◦ θ≃idXθ, θ◦gθ≃idXU,
wi h isomo phism e o app oaching ze o as
θ→θ⋆
.
This s ep in conc e e cons uc ion can u ilize ac : die ence be ween uni e se embedding
XU
and model
Xθ
can be measu ed by se o con ol quan i ies, such as measu e o causal pa ial o de
die ence,
Lp
dis ance o ime scale unc ions, sup emum die ence o gene alized en opy unc ions,
p o ing hese quan i ies con inuous in pa ame e
θ
.
B.3 P oo o Theo em 3.4
Pos e io consis ency ensu es o any
ε > 0
, he e exis neighbo hood
Uε∋θ⋆
and
Tε
such ha
when
T > Tε
,
πT
O(Uε)>1−ε
. Le
δ(θ)
measu e s uc u al die ence be ween
Xθ
and
XU
, sa is ying
δ(θ)→0
as
θ→θ⋆
.
Die ence o pos e io expec a ion s uc u e
XT
H
can be es ima ed as
∆T=ZΘ
δ(θ) dπT
O(θ).
Decomposing in eg al in o
Uε
and
Uc
ε
pa s:
∆T≤sup
θ∈Uε
δ(θ)·πT
O(Uε) + sup
θ∈Uc
ε
δ(θ)·πT
O(Uc
ε).
Since
δ(θ)
on
Uε
can ake a bi a ily small alues, while
πT
O(Uc
ε)
app oaches ze o as
T→ ∞
,
ob ain
∆T→0
. Thus
XT
H
s uc u ally con e ges o
XU
; using s abili y o app oxima e isomo phic
mo phisms, o sucien ly la ge
T
, he e exis s exac isomo phism
XT
H≃XU
; Theo em 3.4 p o ed.
B.4 P oo o Theo em 3.5
In mul i-obse e case, join pos e io
ΠT
can be cons uc ed h ough dis ibu ion amily
{P(join )
θ}
and join obse a ion da a. Iden iabili y and obse a ional suciency condi ions need gene al-
iza ion o join sys em, bu unde assump ions o connec ed communica ion g aph and unbiased
messages, ex ended Schwa z heo em o i s non-i.i.d. e sion can be used o p o e join pos e io
concen a es on
θ⋆
.
Subsequen ly, indi idual pos e io s can be iewed as ma ginals o condi ionals o join pos e-
io , hence also concen a e on
θ⋆
. Thus, each obse e 's hea -uni e se s uc u e objec
XT
Hi
is
isomo phic o
XU
in limi , also mu ually isomo phic; Theo em 3.5 p o ed.
Appendix C: Sel -Re e en ial Sca e ing Ne wo k Toy Model
This appendix p o ides oy model ealizing my mind is uni e se in ma ix uni e se THE-MATRIX.
C.1 Ne wo k A chi ec u e
Conside ni e-dimensional sca e ing ne wo k whose po se di ided in o h ee classes:
1. Ex e nal po clus e
E
: ep esen ing es o uni e se un ela ed o obse e ;
17
2. Obse e po clus e
Oin, Oou
: ela ed o my mind's sensing and ac ua ion;
3. In e nal memo y po clus e
Min, Mou
: ep esen ing in e nal s a e o my mind.
O e all sca e ing ma ix can be w i en in block o m
S(ω) = 

SEE(ω)SEO(ω)SEM (ω)
SOE(ω)SOO(ω)SOM (ω)
SME(ω)SMO(ω)SMM (ω)

.
He e
SMM (ω)
desc ibes sca e ing among in e nal memo ies; sel - e e en iali y embodied in eed-
back coupling be ween
SMM
and
SMO, SOM
.
C.2 In e nal Model and Lea ning Rule
Assume sca e ing ma ix con olled by ni e-dimensional pa ame e
θ
;
S(ω;θ)
is model amily; my
mind's model amily
MO
is
{S(ω;θ)}θ∈Θ
. T ue uni e se co esponds o pa ame e
θ⋆
.
Lea ning p ocess o my mind can be desc ibed as:
1. Unde unied ime scale con ol, p obe ne wo k wi h equency-con ollable manne h ough
Oin, Oou
, collec ing inpu -ou pu pai s;
2. Pe o m Bayesian upda e on his da a o e model amily, ob aining pos e io
πT
O
;
3. Choose pos e io expec a ion o maximum a pos e io i pa ame e
ˆ
θT
o upda e sca e ing
p ope ies
SMM (ω)
o in e nal memo y subne wo k.
Unied ime scale
κ(ω)
achie es balance be ween da a acquisi ion and pa ame e upda ing by
con olling equency sampling and ime delay.
C.3 Eme gence o Hea Uni e se Isomo phism
In abo e se ing, hea -uni e se s uc u e objec
XT
H
can be cons uc ed om my mind's es ima e
o
S(ω)
, whose causal empo alen opy s uc u e comes om:
1. Ne wo k opology and pa hs be ween po s de e mine causal pa ial o de ;
2. Sca e ing phase and ime delay de e mine ealiza ion o unied ime scale;
3. Gene alized en opy dened h ough ene gy and mode dis ibu ion on channels de e mines
en opy s uc u e.
As long as model amily is iden iable and p io suppo s ue pa ame e , pos e io
πT
O
concen-
a es on
θ⋆
, making sca e ing ma ix
ˆ
S(ω)
es ima ed in e nally by my mind con e ge o
S(ω;θ⋆)
in app op ia e opology. The e o e, hea -uni e se s uc u e objec
XT
H
is isomo phic o ue ma ix
uni e se objec
XU
in limi .
In o he wo ds, in his oy model, my mind is uni e se conc e ely mani es s as: sel - e e en ial
sca e ing subne wo k d i en by unied ime scale and Bayesian upda ing necessa ily lea ns and
eplica es opology and sca e ing p ope ies o en i e ne wo k in s uc u e, and his ne wo k i sel
is uni e se. Uni e se cons uc s co ec image abou i sel inside i sel h ough sel - e e en ial
sca e ing ne wo k my mind, hus ealizing my mind is uni e se in igo ous sense.
18