Unified Characterization of ``My Mind Is the Universe'' in Matrix Universe THE-MATRIX

Unied Cha ac e iza ion o My Mind Is he Uni e se in Ma ix
Uni e se THE-MATRIX
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, causal mani olds, and ma-
ix uni e se THE-MATRIX, his pape p o ides axioma izable, heo em-p o able ma hema ical
cha ac e iza ion o philosophical p oposi ion my mind is he uni e se. Fi s , physical uni e se
is cha ac e ized on one hand as Lo en z causal mani old
Ugeo
wi h small causal diamond gene al-
ized en opy s uc u e and bounda y ime geome y; on he o he hand as ma ix uni e se
Uma
con olled by sca e ing ma ix amily
S(ω)
, Wigne Smi h ime-delay ma ix
Q(ω)
, and spec-
al shi unc ion. Th ough Bi manK en o mula and Wigne Smi h heo y, we dene unied
ime scale densi y
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω)
and align i wi h gene alized en-
opy a ia ion and modula ow ime. Second, indi idual sel  is o malized as obse e iple
I= (γ, AO,{ω(τ)
O}τ∈R)
along imelike wo ldline, whe e mind is modeled as amily o s a is ical
models o uni e se pa ame e s
θ∈Θ
and hei pos e io ajec o y
{πτ}
unde Bayesian up-
da e, ca ying Fishe Rao me ic
gFishe
in EguchiAma i in o ma ion geome y sense. Unde
app op ia e assump ions o iden iabili y, egula i y, and Bayesian consis ency, we cons uc
physical me ic
gphys
induced om ma ix uni e se spec al da a, p o ing
gphys =gFishe
, hus
in o ma ion geome y o my mind and uni e se's pa ame e geome y a e isome ic in limi ing
sense. Main heo em shows: when unied ime scale equi alence class
[τ]
unies sca e ing
ime, modula ime, geome ic ime wi h obse e 's p ope ime and cogni i e ime, and pos-
e io
πτ⇒δθ∗
, ma ix uni e se s uc u e o uni e se is isomo phic o model mani old o my
mind in in o ma ion geome y, allowing s ic s a emen my mind is he uni e se. Appen-
dices p o ide ca ego ical equi alence ou line be ween ma ix uni e se and geome ic uni e se,
echnical p oo o in o ma ion geome y and spec al da a alignmen , and oy model example
o one-dimensional
δ
po en ial ing wi h Aha ono Bohm ux.
Keywo ds
Ma ix uni e se THE-MATRIX; Unied ime scale; Bounda y ime geome y; Gene alized en opy
and QNEC; Tomi aTakesaki modula heo y; The mal ime hypo hesis; Wigne Smi h ime delay;
In o ma ion geome y; Bayesian consis ency; Obse e and mind
1 In oduc ion & His o ical Con ex
P oposi ion my mind is he uni e se in Eas Asian philosophical adi ion commonly exp esses
iden i y and insepa abili y o subjec and wo ld. Howe e , lacking explici s uc u al language, i
dicul ly in e aces di ec ly wi h con empo a y ma hema ical physical heo ies o space ime, quan-
um elds, and in o ma ion. On o he hand, mode n physics g adually cons uc s s uc u al pic u es
1
like s a ic uni e se, bounda y p io i y,  ime as s a e-dependen pa ame e  ac oss mul iple in e -
wining ou es: e.g., block uni e se in gene al ela i i y, holog aphic g a i y and gene alized en opy,
Tomi aTakesaki modula heo y in ope a o algeb as and ConnesRo elli he mal ime hypo h-
esis, and ime eadings cha ac e ized by Bi manK en o mula and Wigne Smi h ime delay in
sca e ing heo y.
A in e sec ion o g a i y and quan um in o ma ion, esea ch h ough gene alized en opy
Sgen
and quan um ene gy condi ions (such as quan um null ene gy condi ion QNEC, quan um ocusing
conjec u e QFC) shows deep connec ion be ween local geome ic dynamics and second a ia ion o
bounda y quan um en opy; a ow o ime can be in insically ex ac ed om en opy mono onici y
wi hou adding absolu e ime.
Meanwhile, in o ma ion geome y om wo k o Eguchi, Ama i e al. de elops iewpoin o
iewing s a is ical models as mani olds wi h na u al me ic and connec ion, whe e Umegaki ela i e
en opy and b oade di e gence amilies p o ide unied sou ce o Fishe Rao me ic and
α
connec-
ions. In his pe spec i e,  a ional obse e 's mind is na u ally modeled as poin on pa ame e
mani old and i s ajec o y d i en by obse a ions. Bayesian pos e io consis ency heo ems gua -
an ee ha unde iden iabili y and mode a e egula i y condi ions, pos e io dis ibu ions con e ge
almos su ely o ue pa ame e poin , hus obse e 's in e nal model lea ns ue wo ld in limi ing
sense.
This pape 's co e claim is: in uni e se wi h bounda y ime geome y and ma ix uni e se THE-
MATRIX unied scale, one can es a e my mind is he uni e se in s ic ma hema ical sense as
ma ix s uc u e o uni e se and model mani old o obse e 's mind a e in o ma ion-geome ically
isome ic unde unied ime scale. Specically:
1. Uni e se's
ex e nal cha ac e iza ion
: on one hand Lo en z mani old
Ugeo
wi h gene alized
en opy and causal pa ial o de ; on o he hand ma ix uni e se
Uma
con olled by sca e ing
ma ix amily
S(ω)
, Wigne Smi h ime delay
Q(ω)
, and spec al shi unc ion
ξ(ω)
; bo h
equi alen in app op ia e ca ego y h ough Bi manK en o mula and hea ke nelspec al
ow ools.
2. Sel 's
in e nal cha ac e iza ion
: obse e along imelike wo ldline
γ
, whose mind is
modeled as p obabili y dis ibu ion
πτ
on pa ame e space
Θ
, using Bayesian upda e when
obse ing ma ix uni e se sca e ing da a. Rela i e en opy
D(θ∥θ0)
induces Fishe Rao
me ic
gFishe
, hus mind i sel ca ies in o ma ion geome ic s uc u e.
3. Is's
s uc u al meaning
: unde unied ime scale, me ic
gphys
cons uc ed om ma ix
uni e se spec al da a coincides wi h
gFishe
ob ained om ela i e en opy Hessian, and pos-
e io
πτ⇒δθ∗
. In his limi , local geome y o my mind's model mani old comple ely
coincides wi h uni e se's pa ame e geome y, allowing bo h o be iewed as same geome ic
objec wi h wo coo dina e sys ems.
This pape 's goal is no making me aphysical decla a ions, bu p o iding explici ma hema ical
amewo k whe e my mind is he uni e se becomes se o heo ems ha can be s a ed, analyzed,
e en es ed in simplied models. Below we gi e co esponding axioma ic se ing, main heo ems
and p oo s, demons a ing ope abili y o his amewo k in one-dimensional sca e ing oy models
and mul i-po elec omagne ic sca e ing ne wo k enginee ing p oposals.
2
2 Model & Assump ions
This sec ion p esen s unied model o uni e se, ma ix uni e se, and my mind, lis ing key assump-
ions needed o subsequen heo ems.
2.1 Geome ic Uni e se and Bounda y Time Geome y
Le
(M, g)
be ou -dimensional globally hype bolic Lo en z mani old,
≺
causal pa ial o de induced
by ligh cone s uc u e. Fo each poin
p∈M
and small scale pa ame e
, choose small causal
diamond
Dp,
whose bounda y is gene a ed by wo amilies o null geodesics. Choosing ane
pa ame e
λ
in one null di ec ion, le
Σλ
be c oss-sec ion; dene gene alized en opy o each c oss-
sec ion:
Sgen(λ) = A(Σλ)
4Gℏ+Sou (λ),
whe e
A
is c oss-sec ion a ea,
Sou
on Neumann en opy o ex e io quan um eld. Quan um null
ene gy condi ion (QNEC) gi es lowe bound on s ess enso along null di ec ion, exp essed in e ms
o second de i a i e o
Sou (λ)
; unde app op ia e assump ions, his condi ion can be iewed as local
p ojec ion o quan um ocusing conjec u e (QFC), which links gene alized en opy a ia ion wi h
Eins ein-like equa ions.
In cases wi h bounda y
∂M
, g a i a ional ac ion is
I[g, Ψ] = 1
16πG ZM
R√−gd4x+1
8πG Z∂M
Kp|h|d3x+Ima e [Ψ, g]+
(co ne & null-like bounda y e ms)
,
whe e
K
is ex insic cu a u e ace,
h
induced bounda y me ic. GibbonsHawkingYo k bounda y
e m ensu es eld equa ions well-posed unde a ia ion xing bounda y geome ic da a; B own
Yo k quasilocal s ess enso
TBY
ab
is Hamil onian gene a o o bounda y ime ansla ion, yielding
geome ic ime pa ame e
τgeom
a e choosing amily o ime ansla ion ec o elds on bounda y.
Syn hesizing abo e s uc u es, we dene
geome ic uni e se
as
Ugeo = (M, g, ≺,A∂, ω∂, Sgen, κ),
whe e
A∂
is bounda y obse able algeb a,
ω∂
bounda y s a e,
κ
unied ime scale densi y in oduced
om sca e ing heo y sho ly and aligned wi h
τgeom
.
2.2 Ma ix Uni e se THE-MATRIX and Unied Time Scale
A spec alsca e ing end, conside pai o sel -adjoin ope a o s
(H, H0)
sa is ying ela i e ace-
class pe u ba ion condi ion and Bi manK en assump ions. Deno e sca e ing ma ix on abso-
lu ely con inuous spec um as
S(ω)
, o al sca e ing de e minan
de S(ω)=eiΦ(ω), φ(ω) = 1
2Φ(ω),
and spec al shi unc ion
ξ(ω)
. Bi manK en o mula gi es
de S(ω) = exp−2πiξ(ω),
hus
Φ′(ω) = −2πξ′(ω)
; dene ela i e densi y o s a es
ρ el(ω) = −ξ′(ω) = Φ′(ω)
2π=φ′(ω)
π.
3
On o he hand, Wigne Smi h ime-delay ma ix is dened as
Q(ω) = −iS(ω)†∂ωS(ω),
whose ace's eal pa unde app op ia e no maliza ion gi es sum o ime delays; in mul i-channel
sca e ing sys ems, eigen alues o
Q(ω)
a e so-called p ope delay imes.
Unde s anda d condi ions,
1
2π Q(ω) = ρ el(ω) = φ′(ω)
π,
hus we can in oduce unied ime scale densi y
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω).
Fo e e ence equency
ω0
, dene ime scale
τsca (ω)−τsca (ω0) = Zω
ω0
κ(˜ω) d˜ω.
We call
Uma =Hchan, S(ω), Q(ω), κ(ω),A∂, ω∂
a
ma ix uni e se THE-MATRIX
i and only i :
1.
Hchan
is di ec sum o Hilbe spaces o all in/ou  bounda y channels;
S(ω)
is uni a y and
sucien ly die en iable ma ix- alued unc ion on each ene gy window;
2. Fo each small causal diamond
Dp,
, sca e ing da a o i s bounda y obse able subalgeb a
can be embedded in ni e-dimensional ma ix block o
S(ω)
, wi h such embedding compa ible
wi h co esponding
K1
class a amily le el;
3. Unied ime scale densi y
κ(ω)
aligns wi h
τgeom
a geome ic end, s a ed mo e p ecisely in
Sec ion 3.1 sho ly.
In ui i ely,
Uma
comp esses all obse able causal and empo al s uc u es o uni e se in o amily
o sca e ing ma ices a ying wi h equency, and ime delay and spec al shi da a de i ed om
hem.
2.3 Modula Flow and The mal Time
A ope a o algeb a end, Tomi aTakesaki heo y shows: gi en on Neumann algeb a
M
wi h
ai h ul s a e
ω
, one can cons uc modula ope a o
∆
and one-pa ame e g oup
σω
o modula
au omo phisms
σω
(A) = ∆i A∆−i , A ∈ M,
his modula ow sa ises KMS condi ion and plays ole o  ime e olu ion in many quan um eld
heo ies and cu ed space ime backg ounds.
ConnesRo elli he mal ime hypo hesis p oposes: o gi en physical s a e
ω
, i s modula ow
pa ame e can be in e p e ed as in insic ime unde ha s a e; in con ex o gene ally co a ian
quan iza ion o gene al ela i i y, he mal ime p o ides scheme o ex ac ing ime pa ame e in
Hamil onian cons ain sys em wi h no o e all ime.
This pape u ilizes his idea o align modula ime
τmod
wi h sca e ing ime
τsca
and geome ic
ime
τgeom
on bounda y algeb a
A∂
, cons uc ing unied ime scale equi alence class
[τ]
.
4
2.4 Fo maliza ion o Sel  and Mind
2.4.1 Sel  as Wo ldline Comp ession
Gi en geome ic uni e se
Ugeo
, dene:
Deni ion 1
(Sel )
.
A sel  is iple
I= (γ, AO,{ω(τ)
O}τ∈R),
whe e:
1.
γ:R→M
is imelike wo ldline, whose pa ame e
τ
akes alue in p ope ime and unied
ime scale equi alence class
[τ]
;
2.
AO⊂ A∂
is subalgeb a ob ained h ough some comple ely posi i e comp ession map
Φ :
A∂→ AO
, ep esen ing bounda y obse ables accessible o sel ;
3.
ω(τ)
O
is in e nal s a e o sel  a scale
τ
, sa is ying exis ence o comple ely posi i e map
Ψτ:
AO→ A∂
such ha
ω(τ)
O(A) = ω∂(Ψτ(A)), A ∈ AO.
This ensu es sel 's in e nal s a e is compa ible wi h uni e se bounda y s a e.
2.4.2 Mind as In o ma ion Geome ic Model Mani old
Mind is no single ins an aneous s a e, bu amily o lea nable models abou uni e se and hei
upda e ajec o y.
Le
Θ⊂Rd
be smoo h pa ame e mani old,
{Pθ}θ∈Θ
s a is ical model amily om imple-
men able expe imen s in ma ix uni e se (such as mul i- equency sca e ing expe imen s), each
Pθ
p obabili y measu e on some ou come space. Umegaki ela i e en opy is dened as
D(θ∥θ0) = Zlog dPθ
dPθ0
dPθ,
unde app op ia e die en iabili y and con exi y condi ions, Eguchi's in o ma ion geome y heo y
shows i s Hessian
gij(θ0) = ∂2
∂θi∂θjD(θ∥θ0)θ=θ0
gi es Fishe Rao me ic, while hi d-o de de i a i es dene Ama iChen so enso , yielding amily
o
α
connec ions.
Deni ion 2
(Mind's Model Space)
.
Mind's model space is mani old wi h me ic and connec ion
Uhea = (Θ, gFishe ,∇(α),{πτ}τ∈R),
whe e
π0
is p io dis ibu ion,
πτ
pos e io o pa ame e
θ
a scale
τ
.
2.4.3 Obse a ion and Upda e
Unde unied ime scale, sel  pe o ms se ies o expe imen s in ma ix uni e se, ob aining obse -
a ion da a s eam
D[0,τ]
. Wi h likelihood unc ion
L(D[0,τ]|θ)
sucien ly egula in
θ
, pos e io
upda e is
πτ(θ)∝π0(θ)L(D[0,τ]|θ).
In classical case his is s anda d Bayesian upda e; in quan um case gene alized measu emen s
and quan um ope a ions can be used, bu his pape ocuses on s a is ical le el abs ac ed by
Pθ
.
5

2.5 Key Assump ions
Subsequen main heo ems ely on ollowing assump ions:
1.
Unied ime scale assump ion
: The e exis s unied ime scale equi alence class
[τ]
such
ha sca e ing ime
τsca
, geome ic ime
τgeom
, modula ime
τmod
, and obse e 's p ope
ime and cogni i e ime all belong o
[τ]
.
2.
Iden iabili y
: S a is ical model amily
{Pθ}
is iden iable, i.e., i
Pθ1=Pθ2
o all imple-
men able expe imen s, hen
θ1=θ2
.
3.
Model comple eness
: T ue ma ix uni e se co esponds o some
θ∗∈Θ
.
4.
Bayesian consis ency
: P io
π0
has posi i e mass on
θ∗
; obse a ion p ocess sa ises s an-
da d condi ions o heo ems like DoobBa on, hus pos e io con e ges almos su ely o
δθ∗
.
5.
Eguchi egula i y
: Rela i e en opy
D(θ∥θ0)
has sucien die en iabili y and s ic con-
exi y in pa ame e s, hus in o ma ion geome ic s uc u e is well-beha ed.
Unde hese assump ions, we can p ecisely s a e and p o e main heo em o my mind is he
uni e se.
3 Main Resul s (Theo ems and Alignmen s)
This sec ion s a es h ee main esul s: unied ime scale heo em, in o ma ion geome ic isome y
heo em, and my mind is he uni e se heo em. Rigo ous p oo s gi en in Sec ion 4 and appendices.
3.1 Unied Time-Scale Theo em
Theo em 3
(Unied Time Scale)
.
Le
Ugeo
sa is y gene alized en opy mono onici y condi ion gi en
by QNEC/QFC,
Uma
sa is y Bi manK en and Wigne Smi h assump ions; bounda y algeb a
A∂
has Tomi aTakesaki modula s uc u e and co esponding he mal ime ow. Le
I
be obse e
along imelike wo ldline
γ
wi h p ope ime
τp op
; cogni i e ime
τcog
dened as pa ame e making
ela i e en opy inc emen
D(ω(τ+∆τ)
O∥ω(τ)
O)
ake xed alue. Then he e exis s unied ime scale
equi alence class
[τ]
such ha
τsca ∼τgeom ∼τmod ∼τp op ∼τcog,
whe e 
∼
 deno es ane equi alence ela ion.
This heo em shows ha unde app op ia e physical condi ions, sca e ing ime, geome ic ime,
modula ime, and obse e 's in e nal ime can be unied in o same scale equi alence class, p o iding
ounda ion o subsequen alignmen o mind wi h uni e se geome y.
3.2 In o ma ion-Geome ic Isome y Theo em
To in oduce isome y be ween in o ma ion geome y and physical geome y, need o dene me ic
on uni e se end.
6
Deni ion 4
(Physical Pa ame e Me ic)
.
Conside pa ame ized sca e ing amily
S(ω;θ)
in
ma ix uni e se, wi h spec al shi unc ion
ξ(ω;θ)
and ela i e DOS
ρ el(ω;θ)
. Fo ene gy window
I⊂R
and weigh unc ion
W(ω)≥0
, dene
gphys
ij (θ) = ZI
W(ω)∂ilog ρ el(ω;θ)∂jlog ρ el(ω;θ) dω.
Unde ni e-o de Eule Maclau in and Poisson sum dis ibu ion heo y amewo k,
gphys
can
be unde s ood as spec al exp ession o second-o de de o ma ion o ela i e en opy.
Theo em 5
(In o ma ion Geome ic Isome y)
.
Unde assump ions o Sec ion 2.5, he e exis
weigh unc ion
W(ω)
and ene gy window
I
such ha :
1. Me ic
gphys
cons uc ed om ma ix uni e se spec al da a equals Fishe Rao me ic
gFishe
in Eguchi in o ma ion geome y;
2. This equali y holds in neighbo hood a ound
θ∗
.
In o he wo ds, nea ue pa ame e , in o ma ion geome y o mind and spec alsca e ing
geome y o uni e se a e isome ic.
3.3 My Mind Is he Uni e se Theo em
Based on unied ime scale and in o ma ion geome ic isome y heo ems, we can gi e igo ous
e sion o my mind is he uni e se.
Deni ion 6
(My Mind Is he Uni e se)
.
In gi en uni e se
(Ugeo, Uma )
and obse e mind s uc-
u e
I, Uhea
, i sa is ying:
1.
Unied ime scale
: The e exis s
[τ]
such ha all physical and cogni i e ime scales belong
o one equi alence class;
2.
Iden iabili y and Bayesian consis ency
: T ue pa ame e
θ∗
has posi i e p io mass,
pos e io
πτ⇒δθ∗
;
3.
Geome ic isome y
: In neighbo hood o
θ∗
,
gphys =gFishe
;
hen my mind is he uni e se is said o hold be ween his obse e and his uni e se.
Theo em 7
(My Mind Is he Uni e se)
.
Unde assump ions o Sec ion 2.5, o ma ix uni-
e se THE-MATRIX and obse e mind s uc u e along wo ldline
γ
, he e exis unied ime scale
equi alence class
[τ]
and pa ame e mani old
(Θ, g)
such ha in Bayesian consis ency limi , model
mani old o my mind is isome ic o uni e se's pa ame e geome y, hus sa is ying all condi ions
o Deni ion 3.2. In o he wo ds, in his sense
My mind
≃
Uni e se
,
whe e
≃
deno es s uc u al isomo phism unde in o ma ion geome y and unied ime scale.
4 P oo s
This sec ion p o ides p oo ideas and key s eps o main heo ems; comple e echnical de ails and
pa ial echnical cons uc ions placed in appendices.
7
4.1 P oo o Theo em 3.1 (Unied Time-Scale)
Unied ime scale heo em needs o align ou ypes o ime: sca e ing ime
τsca
, geome ic ime
τgeom
, modula ime
τmod
, and obse e 's p ope ime
τp op
and cogni i e ime
τcog
.
1.
Sca e ing ime and DOS ime
By Bi manK en o mula,
ξ(ω)
linked o phase
Φ(ω)
o
de S(ω)
, hus ela i e DOS
ρ el(ω) =
−ξ′(ω)=Φ′(ω)/(2π)
. On o he hand, ace o Wigne Smi h ime-delay ma ix sa ises
Q(ω) = 2πρ el(ω)
. The e o e unied ime scale densi y
κ(ω) = ρ el(ω) = 1
2π Q(ω).
Dene
τsca
as in eg al o
κ(ω)
; hen
τsca
is unique up o ane escaling wi hin gi en ene gy
window (igno ing in ege spec al ow and bounda y e ms).
2.
Geome ic ime and gene alized en opy
On small causal diamonds, QNEC and QFC show gene alized en opy second a ia ion ela ed
o ene gy densi y along null di ec ion and geome ic con ac ion a e. When g a i a ional
ac ion includes GHY bounda y e m, bounda y ime ansla ion Hamil onian de e mined
by B ownYo k enso ; equi ing gene alized en opy ow consis en wi h ADM/Bondi ime
pa ame e yields geome ic ime scale
τgeom
. In holog aphic o semiclassical window, bounda y
sca e ing da a and DOS die ence can be econs uc ed om bulk geome y, so
τgeom
and
τsca
die a mos by ane ans o ma ion.
3.
Modula ime and he mal ime
Tomi aTakesaki heo y endows bounda y algeb a
A∂
wi h modula ow
σω
. In he mal ime
hypo hesis, choosing some physical s a e
ω∂
as e e ence, modula pa ame e
is in e p e ed
as physical ime unde ha s a e. In equilib ium s a es sa is ying KMS condi ion, modula
ime and Hamil onian ime ela ed by empe a u e ac o ; in gene al cu a u e backg ounds,
need mode a e assump ions ensu ing compa ibili y o modula ow wi h geome ic ime. This
pape assumes cons an s
a, b
exis such ha
τmod =a τgeom +b,
hus modula ime belongs o same equi alence class as geome ic ime.
4.
P ope ime and cogni i e ime
P ope ime
τp op
along imelike wo ldline
γ
dened h ough line elemen o
gµν
, usually
equi alen locally o app op ia ely chosen geome ic ime pa ame e . Cogni i e ime
τcog
dened as ime scale when ela i e en opy inc emen is xed: choosing cons an
∆D > 0
,
equi e
Dω(τ+∆τ)
O∥ω(τ)
O= ∆D,
hen
∆τ
denes uni cogni i e ime. In unied amewo k o en opyene gy ime, second-
o de de o ma ion o
D
con olled by ene gys ess enso and geome ic ime e olu ion, hus
τcog
and
τmod
can be p o ed o die by cons an ac o .
In summa y, he e exis s unied ime scale equi alence class
[τ]
such ha all ime scales can
be con e ed o each o he h ough ane ans o ma ions; Theo em 3.1 p o ed. Mo e de ailed
cons uc ion and echnical condi ions gi en in Appendix A.
8
4.2 P oo o Theo em 3.2 (In o ma ion-Geome ic Isome y)
Co e o in o ma ion geome ic isome y heo em is p o ing
gphys
cons uc ed om spec al da a
coincides wi h
gFishe
cons uc ed om ela i e en opy Hessian in neighbo hood o
θ∗
.
1.
Rela i e en opy and DOS exp ession
Fo implemen able expe imen s in ma ix uni e se (e.g., s a is ics o mul i- equency sca e ing
phase and ime delay), likelihood
Pθ
can be cons uc ed. In app op ia e windowed limi ,
ela i e en opy
D(θ∥θ0)
can be ew i en in in eg al exp ession o m using spec al shi
unc ion and DOS die ence:
D(θ∥θ0)≈ZI
Fρ el(ω;θ), ρ el(ω;θ0)dω,
whe e
F
is some local unc ion sa is ying egula i y condi ions,
I
ene gy window. This ew i -
ing elies on s anda d connec ions among hea ke nelspec al shi  ela i e de e minan .
2.
Hessian and Fishe Rao me ic
In Eguchi heo y, Fishe Rao me ic gi en by second de i a i e o
D
wi h espec o pa am-
e e s. Subs i u ing abo e in eg al exp ession, ob ain
gFishe
ij (θ0) = ZI
W(ω)∂ilog ρ el(ω;θ0)∂jlog ρ el(ω;θ0) dω,
whe e weigh unc ion
W
comes om second-o de expansion o
F
a e e ence poin . Com-
pa ing wi h
gphys
in Deni ion 3.1, bo h equal when
W
chosen app op ia ely.
3.
Regula i y and locali y
Eguchi egula i y ensu es exis ence and posi i e-deni eness o Hessian. In neighbo hood o
θ∗
, DOS die ence a ies smoo hly wi h pa ame e , and
ρ el(ω;θ∗)
is nonze o, hus loga i h-
mic de i a i e well-beha ed. Weigh unc ion
W
can be chosen as expe imen ally ealizable
equency window, e.g., smoo h compac ly suppo ed unc ion, hus
gphys
also well-dened.
4.
Conclusion
The e o e in neighbo hood o
θ∗
,
gphys =gFishe
; Theo em 3.2 p o ed. De ailed unc ional
analysis and windowed Taube ian a gumen s in Appendix B.
4.3 P oo o Theo em 3.3 (My Mind Is he Uni e se)
Theo em 3.3 is di ec syn hesis o Theo ems 3.1 and 3.2 wi h Bayesian consis ency.
1. By Theo em 3.1, he e exis s unied ime scale equi alence class
[τ]
aligning all physical and
cogni i e imes a uni e se end and obse e end.
2. By Theo em 3.2, he e exis s me ic
g
such ha uni e se pa ame e geome y
(Θ, gphys)
and
mind's in o ma ion geome y
(Θ, gFishe )
a e isome ic in neighbo hood o
θ∗
.
3. By Bayesian consis ency, pos e io
πτ⇒δθ∗
; hus as
τ→ ∞
, model mani old egion ac u-
ally accessed by mind con ac s o neighbo hood o
θ∗
, whe e bo h geome ies comple ely
coincide.
4. Syn hesizing h ee poin s sa ises all condi ions o Deni ion 3.2, hus in unied ime scale
and in o ma ion geome ic sense, my mind is he uni e se holds.
9
Appendix C: A Toy Model o Sel Hea Uni e se Equi alence
This appendix p o ides simplied model o one-dimensional
δ
po en ial ing wi h Aha ono Bohm
ux, illus a ing conc e e ealiza ion o my mind is he uni e se heo em in ni e-dimensional case.
C.1 Model Deni ion
Conside ci cle o adius
L
wi h coo dina e
x∈[0,2πL)
; place po en ial
V(x) = αδ(x)
a
x= 0
,
h eading magne ic ux
Φ
h ough ci cle, co esponding dimensionless ux
θAB = 2πΦ/Φ0
whe e
Φ0
is ux quan um. Bounda y condi ion wi h ux is
ψ(x+ 2πL)=eiθAB ψ(x).
A ene gy scale
E=k2
, wa e unc ion sa ises ee equa ion excep a
x= 0
; a
x= 0
sa ises
jump condi ion
ψ′(0+)−ψ′(0−) = αψ(0).
Sol ing gi es eigenequa ion
cos θAB = cos(kL) + α
ksin(kL).
C.2 Ma ix Uni e se and Pa ame e Space
Sca e ing ma ix o his sys em iewable as
2×2
ma ix (co esponding o clockwise/coun e clockwise
wo channels); pa ame e space is
Θ = R×S1
wi h coo dina es
θ= (α, θAB)
. F om eigen alue equa-
ion, sca e ing phase and ime delay can be explici ly w i en, ob aining
ρ el(ω;θ)
and unied ime
scale densi y
κ(ω;θ)
.
C.3 Mind's Model Mani old and Fishe Rao Me ic
Assume sel  can measu e ansmission and eec ion p obabili ies a mul iple equency poin s,
cons uc ing likelihood
Pθ
. In case whe e log-likelihood app oxima es Gaussian, Fishe in o ma-
ion ma ix equi alen o squa ed a e age o sca e ing ampli udes a e pa ame ic die en ia ion,
allowing explici calcula ion o Fishe Rao me ic
gFishe (θ)
.
On o he hand, when cons uc ing
gphys(θ)
om DOS die ence and ime delay, can use spec al
exp ession gi en in p e ious sec ion. In neighbo hood o
θ∗
, wo me ics coincide, demons a ing
conc e e ealiza ion o Theo em 3.2.
C.4 Conc e iza ion o My Mind Is he Uni e se
In his model, uni e se is ci cula sca e ing sys em pa ame ized by
(α, θAB)
; my mind is p ob-
abili y dis ibu ion o hese wo pa ame e s and i s in o ma ion geome ic mani old. Th ough su -
cien ly many obse a ions, pos e io con e ges nea ue pa ame e poin ; in ha egion mind's
in o ma ion geome y coincides wi h uni e se's pa ame e geome y, conc e ely embodying s uc-
u al meaning o my mind is he uni e se in simplied case.
16