Unified Mathematical Definition of ``Self'': Causal Manifold, Observer, and Self-Referential Scattering Network

Unied Ma hema ical Deni ion o Sel : Causal Mani old,
Obse e , and Sel -Re e en ial Sca e ing Ne wo k
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
Wi hin he amewo k o unied ime scale, causal mani olds, and bounda y ime geome y,
we p o ide an axioma ic ma hema ical deni ion o he  s -pe son subjec sel . The basic
concep ion is: sel  is no a label o an ins an aneous physical congu a ion, bu a he an
equi alence class o sel - e e en ial obse e s uc u es along a imelike wo ldline o de ed by he
unied ime scale. A he geome ic le el, he uni e se is modeled as a globally hype bolic
Lo en zian mani old wi h causal pa ial o de , whe e gene alized en opy ex emiza ion and
quan um ene gy condi ions on small causal diamonds yield g a i a ional eld equa ions and
ime a ow. A he spec al and sca e ing le el, he unied ime scale se es as he mo he
scale ia he scale iden i y
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω)
, uni ying phase g adien ,
ela i e densi y o s a es, and Wigne Smi h ime-delay ace as a single empo al densi y. A
he causal ne wo k and in o ma ion geome y le el, obse e s a e o malized as mul i-componen
objec s wi h local causal domain, esolu ion scale, bounda y obse able algeb a, s a e, model
amily, and upda e ope a o ; mul iple obse e s o m consensus geome y ia communica ion
channels and ela i e en opy con ac ion. A he sel - e e en ial sca e ing and consciousness
le el, consciousness is cha ac e ized as a wo-po sca e ing ne wo k wi h delay ke nel and ime
delay, whe e sel - eedback closed loops exhibi a
Z2
holonomy o modied de e minan squa e
oo , mani es ing he opological in a ian o sel .
Building on his ounda ion, we in oduce he Is uc u e: a imelike wo ldline
γ
wi h
unied ime scale, equipped wi h in e nal obse able algeb a, ime-s amped s a e amily, sel -
e e en ial upda e ope a o , memo y subsys em, and in e nal en i onmen maps, sa is ying
causal locali y, pe sis en and dis inguishable in e nal memo y, explici dependence o upda es
on sel 's u u e beha io and en i onmen p edic ions, and compa ibili y wi h gene alized en-
opy dynamics. A e dening an equi alence ela ion in e ms o ime escaling and algeb a
embeddings, we ega d equi alence classes as indi idual sel es. We p o e ha unde app op i-
a e causal and ene gy condi ions, each Is uc u e equi alence class co esponds o a minimal
s ongly connec ed sel - e e en ial sca e ing closed loop in he causal ne wo k, whose unied
ime scale and gene alized en opy g adien align mono onically; con e sely, each minimal sel -
e e en ial sca e ing closed loop sa is ying his alignmen condi ion denes a unique Is uc u e
equi alence class. In he appendices, we p o ide an ou line o exis ence and uniqueness p oo s o
cons uc ing I-wo ldlines om local obse e da a, and he cons uc ion ew i ing I-s uc u es
as closed loop sca e ing amilies wi h
Z2
holonomy, showing compa ibili y o i s mod- wo in-
dica o wi h e mionic commu a ion phase and opological class unde NullModula double
co e . Thus, sel  is cha ac e ized as a unied ma hema ical objec wi h geome ic suppo ,
in o ma ion ke nel, and opological nge p in .
Keywo ds
Causal mani old; Unied ime scale; Obse e ; Consciousness; Sel - e e en ial sca e ing ne wo k;
Z2
holonomy; Gene alized en opy
1
1 In oduc ion & His o ical Con ex
In he s anda d amewo k o ela i i y and quan um heo y, obse e  is o en ea ed as a passi e
e e ence ame o measu emen de ice, while he  s -pe son subjec ques ion who am I is le
o philosophy. To igo ously answe his ques ion a he physical and ma hema ical le el equi es a
unied desc ip ion cen e ed on causal s uc u e, ime scale, and in o ma ion geome y, embedding
concep s such as subjec ,  ime, memo y, and sel - e e ence in o he same geome ic and
ope a o language.
On one hand, sca e ing heo y and spec al heo y show ha ime can be iewed as he de i a-
i e o sca e ing phase and unc ion o densi y o s a es. Fo Sch ödinge ope a o s wi h ela i e
ace-class pe u ba ions, he Bi manK en o mula gi es he ela ionship be ween sca e ing de-
e minan and spec al shi unc ion
de S(λ) = exp(−2πiξ(λ))
, whose de i a i e
ξ′(λ)
is in e -
p e ed as ela i e densi y o s a es, u he linked o he ace o Wigne Smi h ime-delay ope a o
Q(ω) = −iS(ω)†∂ωS(ω)
. Th ough his ou e, one can iew ime scale as he unied in a ian o
φ′(ω)/π
, spec al shi de i a i e
−ξ′(ω)
, and ime-delay ace
(2π)−1 Q(ω)
. Rela ed esul s can
be ound in he o iginal wo k o Bi manK en and subsequen sys ema ic s udies o sca e ing phase
and de e minan . The ime delay ma ix p oposed by Wigne and Smi h has been sys ema ically
de eloped in mul i-channel sca e ing, diso de ed media, elec omagne ic and acous ic sca e ing.
On he o he hand, a he in e sec ion o algeb aic quan um eld heo y and g a i y, Tomi a
Takesaki modula heo y e eals ha o any on Neumann algeb a wi h a sucien ly ai h ul
s a e, he e exis s a modula ow uniquely de e mined by he s a e, allowing one o ex ac a one-
pa ame e g oup om he  imeless algeb aic s uc u e. This s uc u e was p oposed by Connes
and Ro elli as he ounda ion o he  he mal ime hypo hesis: physical ime ow is no uni e sal,
bu de e mined by he modula ow o a gi en s a is ical s a e. This p o ides an ope a o -algeb aic
pe spec i e on he in insic na u e o ime.
A he bounda y o g a i y and quan um in o ma ion, he a ia ion o gene alized en opy
Sgen
and quan um ene gy condi ions p o ide a ou e o de i e Eins ein equa ions om en opy
balance. Wo k by Jacobson, Faulkne , and o he s shows ha unde app op ia e semi-classical and
holog aphic assump ions, gene alized en opy ex emiza ion and second-o de non-nega i i y along
small causal diamond bounda ies a e equi alen o Eins ein equa ions wi h cosmological cons an .
P oo s o he quan um null ene gy condi ion (QNEC) and a e aged null ene gy condi ion (ANEC)
u he s eng hen he equi alence s uc u e among en opyene gygeome y. This di ec ion
shows ha local causal s uc u e can be iewed as he mac oscopic mani es a ion o gene alized
en opy op imiza ion p inciples.
The abo e sca e ingspec al heo y, modula heo y, and en opygeome y heo y p o ide he
ounda ion o cons uc ing unied ime scale and causal mani olds. Building on his, one can iew
he uni e se as a causal mani old unde unied ime scale, whose bounda y ca ies obse able algeb a
and s a e, wi h bounda y ime geome y gluing sca e ing phase, modula ime, and g a i a ional
bounda y e ms in o he same s uc u e. Meanwhile, in o ma ion geome y and s a is ical causal
in e ence amewo ks show ha mul i-obse e sys ems can be cha ac e ized ia ela i e en opy
and Fishe me ic, cap u ing model upda e and consensus o ma ion.
Rega ding consciousness and he subjec p oblem, much wo k has been de o ed o cons uc ing
in o ma ion- heo e ic o physicalis desc ip ions, such as iewing consciousness as specic in eg a ed
in o ma ion s uc u e, global wo kspace, o mul i-laye encoding p ocess. Howe e , hese heo ies
ypically lack igo ous geome icope a o o malism compa ible wi h causal mani olds, unied ime
scale, and quan um g a i y. On he o he hand, opological o algeb aic desc ip ions o sel -
e e ence and sel  a e ela i ely sca e ed.
This pape , wi hin he amewo k o unied ime scale and causal mani olds, a emp s o p o ide
2
a ma hema ical objec o sel : a imelike wo ldline wi h ime scale, equipped wi h in e nal obse -
able algeb a, s a e amily, sel - e e en ial upda e ope a o , and memo y s uc u e, sa is ying causal
locali y and gene alized en opy consis ency, and ew i able in sel - e e en ial sca e ing ne wo ks
as a minimal s ongly connec ed closed loop wi h
Z2
holonomy. By cons uc ing an equi alence
ela ion, we unde s and equi alence classes o such s uc u es as die en ealiza ions o he same
sel , he eby o malizing he subjec  in a igo ous ma hema ical con ex .
2 Model & Assump ions
This sec ion p o ides he basic s uc u e o causal mani old, unied ime scale, obse e , and sel -
e e en ial sca e ing ne wo k, along wi h adop ed assump ions.
2.1 Causal Mani old and Small Causal Diamonds
Assume he uni e se a la ge scales is desc ibed by a ou -dimensional Lo en zian mani old
(M, g)
sa is ying:
1.
Global hype bolici y
: The e exis s a Cauchy su ace
Σ
such ha e e y non-spacelike cu e
in e sec s
Σ
exac ly once.
2.
S able causali y
: No closed imelike cu es exis , and small pe u ba ions do no p oduce
causal iola ions.
3.
Causal pa ial o de
: Use
p≺q
o deno e he exis ence o a u u e-di ec ed imelike o null
cu e om
p
o
q
.
Fo any
p∈M
and sucien ly small posi i e
, ake
p±
as poin s ob ained by e ol ing along
some u u e-di ec ed imelike geodesic wi h p ope ime pa ame e
±
, and dene he small causal
diamond
Dp, =J+(p−)∩J−(p+).
I s bounda y is gene a ed by wo amilies o null geodesics, p o iding he basic s uc u e o
analyzing local g a i a ional eld equa ions and gene alized en opy changes.
Assume he e exis s app op ia e quan um eld heo y coupled o g a i y such ha o each
small causal diamond bounda y, one can dene gene alized en opy
Sgen =A ea(∂Dp, )
4Gℏ+Sou ,
whe e
Sou
is he on Neumann en opy o ex e nal eld deg ees o eedom. Assume QNEC and
ela ed en opyene gy inequali ies hold and can be used o equi alen ly cha ac e ize local g a i a-
ional eld equa ions.
2.2 Unied Time Scale and Mo he Iden i y
In sca e ing and spec al heo y, conside a pai o sel -adjoin ope a o s
(H0, H)
, whe e
H
is a
ace-class o ela i e ace-class pe u ba ion o
H0
. Le
S(ω)
be he sca e ing ma ix a ene gy
ω
,
ξ(ω)
he spec al shi unc ion, wi h Bi manK en o mula gi ing
de S(ω) = exp−2πiξ(ω),
3
yielding ela i e densi y o s a es
ρ el(ω) = −ξ′(ω).
The Wigne Smi h ime-delay ope a o is dened as
Q(ω) = −iS(ω)†∂ωS(ω),
whose ace cha ac e izes he ime delay a e aged o e all channels. In oducing he o al sca e ing
hal -phase
φ(ω) = 1
2a g de S(ω) = −π ξ(ω),
we ha e
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω).
Scale iden i y
denes he unied ime scale densi y
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
in e p e ing
κ(ω) dω
as he eec i e ime scale pe uni ene gy bandwid h. Th ough geome ic
op ics limi and eikonal app oxima ion, his scale can be linked o g a i a ional ime delay, edshi ,
and p ope ime; h ough modula heo y and he mal ime hypo hesis, i can be aligned wi h
modula ime pa ame e on bounda y algeb as.
In his pape , he unied ime scale equi alence class
[τ]
deno es all ime unc ion amilies
compa ible wi h
κ(ω)
, locally ew i able om sca e ing phase, modula ow, o geome ic ime,
die ing only by ane escaling and coo dina e choice.
2.3 Obse e as S uc u ed Causal Agen
F om he causal ne wo k pe spec i e, we in oduce abs ac obse e s uc u e.
Deni ion 1
(Obse e )
.
An obse e
Oi
is dened as a mul i-componen objec
Oi= (Ci,≺i,Λi,Ai, ωi,Mi, Ui, ui,(Cij)j),
whe e:
1.
Ci⊂M
is he eachable causal domain;
2.
≺i
is he local causal pa ial o de on
Ci
;
3.
Λi
is he esolu ion scale (e.g., imeene gy o spacemomen um windows), de e mining dis-
inguishable equency bands and spa ial scales;
4.
Ai
is he bounda y obse able
C∗
algeb a associa ed wi h
Oi
;
5.
ωi
is a s a e on
Ai
;
6.
Mi
is a amily o candida e causal dynamical models;
7.
Ui
is an upda e ope a o based on obse a ional da a and models (gene ally comple ely posi i e
ace-p ese ing maps o Bayesian upda e ope a o s);
8.
ui
is a u ili y unc ion o decision p e e ence;
4
9.
Cij
a e communica ion channels wi h o he obse e s
Oj
(comple ely posi i e ace-p ese ing
maps o classical channels).
On he common obse able algeb a
Acom =
i
Ai,
le he mul i-obse e s a e amily
(ω( )
i)
upda e wi h disc e e o con inuous ime. I he commu-
nica ion g aph is s ongly connec ed and he e exis s a common xed poin
ω∗
, hen he weigh ed
ela i e en opy
Φ( )=X
i
λiD(ω( )
i∥ω∗)
cons i u es a Lyapuno unc ion o app op ia e upda e ules, ensu ing o ma ion o s a e consensus.
This s uc u e p o ides he ounda ion o causal and in o ma ional in e ac ion among mul iple
sel es.
2.4 In e nal Algeb a, Memo y and Sel -Re e en ial Upda e
To cha ac e ize he in e nal pe sis ence and memo y o a subjec , we need o ex ac subalgeb as
and upda e s uc u es wi hin a single obse e .
Deni ion 2
(In e nal S uc u e)
.
Gi en a imelike ajec o y
γ:I→M
wi h
τ(γ( )) =
, i s
in e nal s uc u e is a iple
(Ain ,(ωin
) ∈I, U),
whe e:
1.
Ain
is he
C∗
algeb a o in e nal deg ees o eedom;
2. Fo each
∈I
,
ωin
is a s a e on
Ain
;
3. Fo any
2> 1
, he e exis s a comple ely posi i e ace-p ese ing map
U( 2, 1) : Ain → Ain ,
sa is ying he semig oup condi ion
U( 3, 2)◦U( 2, 1) = U( 3, 1), ωin
2=ωin
1◦U( 2, 1).
Deni ion 3
(Memo y Subsys em)
.
A commu a i e subalgeb a
C ⊂ Ain
is called a memo y
subsys em i :
1.
C
is
∗
-isomo phic o bounded unc ions on some measu e space o ni e-dimensional diagonal
ma ix algeb a;
2. The p obabili y measu e amily
µ
induced by
ωin
on
C
o ms a Ma ko p ocess;
3. Fo any
2> 1
, he e exis measu able se s such ha he dependence o
µ 2
on
µ 1
canno be
elimina ed by ex e nal en i onmen a iables, ensu ing memo y ca ies genuine in o ma ion
abou subsequen ial obse a ions.
5

The exis ence o memo y subsys em ensu es he subjec has a aceable in e nal his o y o e
ime, a he han comple e ese a each momen .
Subjec sel - e e en iali y equi es explici dependence o upda es on in e nal s a e and en i on-
men model. To his end, we in oduce in e nal en i onmen maps.
Fo each
, le
Aex
be he ex e nal obse able algeb a accessible o he subjec a ound
γ( )
, and
dene a comple ely posi i e map
E :Aex
→ Ain ,
cha ac e izing he encoding o ex e nal wo ld wi hin he subjec 's in e io .
Deni ion 4
(Sel -Re e en ial Upda e)
.
I he e exis s a unc ional
F
such ha o any
2> 1
,
U( 2, 1) = F 2, 1;ωin
1, E 1,Dex
[ 1, 2],
whe e
Dex
[ 1, 2]
deno es ex e nal obse a ional da a a ailable in he in e al, hen he upda e is called
sel - e e en ial. He e
ωin
1
ia
E 1
p o ides in e nal p edic ion o u u e en i onmen s a es, inu-
encing i s own e olu ion.
2.5 Sel -Re e en ial Sca e ing Ne wo ks and
Z2
Holonomy
In he sca e ing desc ip ion, complex sys ems and hei en i onmen s can be ep esen ed as ne -
wo ks o med by node sca e ing ma ices
Sj(ω)
in e connec ed ia wa eguides, delay lines, and
eedback. Fo a gi en opological s uc u e and pa ame e amily, one uses Redhee s a p oduc
o cons uc he closed-loop sca e ing ma ix
S⟲(ω;λ)
, whe e
λ
ep esen s slowly a ying con ol
pa ame e s (such as in e nal s a egies, a en ion, o ex e nal condi ions).
Unde ace-class o ela i e ace-class condi ions, one can dene he modied de e minan
de pS⟲
and dene he phase index map
s(ω, λ) = de
pS⟲(ω;λ).
Along a closed pa h
γ⊂X◦
in pa ame e space a oiding singula i y se s, he holonomy o he
squa e- oo de e minan is dened as
ν√S⟲(γ) = expiIγ
1
2is−1ds∈ {±1},
gi ing a
Z2
index ha is homo opy in a ian . This index is associa ed wi h he mod- wo pa o
spec al ow, and in many cases can be in e p e ed as he opological sign o  e mionici y.
This pape assumes ha he sel - e e en ial upda e o a subjec can be ew i en as some closed-
loop sca e ing ne wo k unde app op ia e equency domain and inpu ou pu models, he eby
allowing he abo e
Z2
opological nge p in o be dened on he ne wo k.
3 Main Resul s (Theo ems and Alignmen s)
Unde he abo e model and assump ions, his sec ion p o ides he o mal deni ion o Is uc u e
and main esul s.
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3.1 Deni ion o IS uc u e
Choose a ep esen a i e
τ:M→R
om he unied ime scale equi alence class
[τ]
as he ime
unc ion, and conside a u u e-di ec ed imelike cu e
γ:I→M
sa is ying
τ(γ( )) =
.
Deni ion 5
(Obse e T ajec o y)
.
A
γ
sa is ying he abo e condi ions is called an obse e
ajec o y.
Deni ion 6
(IS uc u e)
.
On he unied ime scale equi alence class
[τ]
, an Is uc u e is da a
I=γ, Ain ,(ωin
) ∈I, U, C,(E ) ∈I,
sa is ying:
1.
γ
is an obse e ajec o y;
2.
(Ain ,(ωin
), U)
is an in e nal s uc u e sa is ying Deni ion 2;
3.
C ⊂ Ain
is a memo y subsys em sa is ying Deni ion 3;
4.
E :Aex
→ Ain
a e in e nal en i onmen maps making
U
sel - e e en ial wi h espec o
(ωin
1, E 1)
(Deni ion 4);
5.
Causal locali y
: Fo each
∈I
, he e exis s a bounded causal domain
K ⊂M
such ha
he inuence o in e nal s a e
ωin
on ex e nal obse ables is suppo ed in
K ∩J−(γ( ))
;
6.
En opy consis ency
: Along he small causal diamond amily
Dγ( ),
on
γ
, unde s a e
ωin
⊗ωex
, he ex emiza ion and second-o de non-nega i i y o gene alized en opy
Sgen
is
compa ible wi h g a i a ional eld equa ions and QNEC/QFC ype cons ain s.
In ui i ely, an Is uc u e is a sel - e e en ial obse e on a wo ldline wi h unied ime scale,
ha ing pe sis en memo y and causal consis en in e ac ion wi h he ex e nal wo ld.
3.2 Equi alence Rela ion: Same I in Die en Realiza ions
Die en physical ealiza ions (such as die en bases, ime escalings, ha dwa e) may co espond
o he same sel . To his end, we need an equi alence ela ion.
Deni ion 7
(Equi alence o IS uc u es)
.
Two Is uc u es
I= (γ, Ain ,(ωin
), U, C,(E )),I′= (γ′,A′in ,(ω′
′in ), U′,C′,(E′
′))
a e called equi alen , deno ed
I∼I′
, i he e exis :
1. A s ic ly mono one bijec ion
:I→I′
;
2. A
∗
-isomo phism
Φ : Ain → A′in
wi h
Φ(C) = C′
, being a measu e isomo phism on spec al
spaces;
3. Fo all
∈I
,
ω′
( )in =ωin
◦Φ−1, U′( ( 2), ( 1)) ◦Φ=Φ◦U( 2, 1);
4. Ex e nal algeb a isomo phisms
Ψ :Aex
→ A′
( )ex
such ha
E′
( )= Φ ◦E ◦Ψ−1
.
Deni ion 8
(Ma hema ical Objec o Sel )
.
A sel  is dened as an equi alence class o some
Is uc u e
[I] = {I′:I′∼I}.
7
3.3 S uc u al Theo ems
Based on he abo e deni ions, he e a e h ee co e esul s.
Theo em 9
(Exis ence and Local Uniqueness o I-Wo ldline)
.
On a globally hype bolic Lo en zian
mani old
(M, g)
, assume he e exis s a local obse e
O∗
whose eco ds sa is y:
1. Unde some unied ime scale ep esen a i e
τ
, a imelike cu e
γ∗
can be econs uc ed om
obse a ional eco ds such ha he associa ed obse a ional domain has small causal diamond
nea -Minkowski s uc u e a ound
γ∗
;
2. The e exis s a decomposi ion
A∗≃ Ain ⊗ Aex
, and on
Ain
he e exis s a s able memo y
subsys em.
Then unde app op ia e echnical assump ions (including QNEC, Hadama d s a es, ni e ene gy
condi ions), one can cons uc an Is uc u e
I∗
along
γ∗
, and gi en he unied ime scale equi alence
class and in e nal algeb a equi alence class, i s equi alence class
[I∗]
is locally unique.
Theo em 10
(Co espondence be ween IS uc u es and Minimal S ongly Connec ed Sel -Re -
e en ial Sca e ing Closed Loops)
.
Unde he amewo k o unied ime scale and sca e ingdelay
ne wo ks, assume:
1. All associa ed sca e ing ope a o s sa is y ace-class o ela i e ace-class condi ions;
2. The in e nal upda e and en i onmen maps o Is uc u e can be ew i en in equency domain
as closed-loop sca e ing ma ix amily
S⟲(ω;λ)
.
Dene I-closed loop as minimal s ongly connec ed componen s sa is ying memo y, sel - e e en iali y,
and opological non i iali y condi ions. Then he e exis s a na u al co espondence
[I]←→ S([I]),
such ha each Is uc u e equi alence class co esponds o a unique I-closed loop and ice e sa,
wi h unied ime scales and delay spec a on bo h sides consis en .
Theo em 11
(Topological Finge p in and
Z2
In a ian )
.
Fo each I-closed loop
S
, ia he modied
de e minan squa e oo o closed-loop sca e ing ma ix
S⟲(ω;λ)
, dene a
Z2
index
ν(S)∈ {±1},
which is in a ian unde pa ame e homo opy and Is uc u e equi alence ela ion. Fo app op ia e
sys ems, changes in his index a e compa ible wi h he mod- wo spec al ow o e mionic s a is ics,
opological class in he NullModula double co e , and BF- ype
Z2
bulk in eg als.
4 P oo s
This sec ion p o ides he p oo amewo k and key s eps o he main heo ems. Comple e echnical
de ails a e expanded in appendices.
8
4.1 P oo o Theo em 1
S ep 1: Recons uc imelike ajec o y om local obse a ional da a
Fo local obse e
O∗
, ex ac space ime e en s and hei causal ela ions om i s eco ds.
Th ough maximal- olume wais su ace o minimal cu a u e c i e ion, selec ep esen a i e
τ
in
he unied ime scale equi alence class
[τ]
, and using
τ
as ime unc ion, embed
O∗
's eco ds in o
he olia ion s uc u e
{τ−1( )}
. On each ime slice, selec a cen e o mass poin
γ∗( )
, ob aining
a imelike cu e
γ∗
. Global hype bolici y ensu es
γ∗
can be chosen as a smoo h imelike cu e.
S ep 2: Cons uc in e nal algeb a and memo y subsys em
Using he decomposi ion
A∗≃ Ain ⊗ Aex
, selec om he commu a i e subalgeb a o
Ain
a subalgeb a
C
con aining eadable  eco d bi s, whose andom p ocess on he spec al space is
de e mined by obse a ional eco ds. Con m ia ela i e en opy Hessian ha hese deg ees o
eedom ha e s able Fishe dis inguishabili y, and hei ime e olu ion can be desc ibed by a Ma ko
p ocess, he eby sa is ying memo y subsys em condi ions.
S ep 3: Dene upda e ope a o and in e nal en i onmen maps
Using
O∗
's model amily
M∗
and upda e ule
U∗
, cons uc in e nal upda e
U( 2, 1)
and in-
e nal en i onmen maps
E
. These maps a e conc e ely ealized h ough he p ocess o ex e nal
measu emen esul s
→
in e nal memo y s a e. The dependence o upda es on in e nal models and
memo y gua an ees sel - e e en iali y.
S ep 4: Check causal locali y and en opy consis ency
A ound each poin o
γ∗
, cons uc small causal diamonds
Dγ∗( ),
. Using local nea -Minkowski
and ene gy-bounded condi ions, apply JacobsonFaulkne ype a gumen s: unde s a e
ωin
⊗ωex
,
local gene alized en opy ex emiza ion and second-o de non-nega i i y a e equi alen o Eins ein
equa ions and QNEC. Thus
I∗
does no b eak exis ing geome icen opy s uc u e.
S ep 5: Local uniqueness
I ano he ealiza ion
I′
∗
sa ises he same obse a ional eco ds and unied ime scale equi alence
class, cons uc ime escaling
and algeb a
∗
-isomo phism
Φ
making hem comple ely consis en
on memo y subsys em and obse a ional dis ibu ions, hence
I∗∼I′
∗
.
4.2 P oo o Theo em 2
S ep 1: Inpu ou pu model and sca e ing ne wo k expansion
Fo a gi en Is uc u e
I
, cons uc i s inpu ou pu desc ip ion wi h he ex e nal en i onmen
unde unied ime scale: iew in e nal deg ees o eedom as nodes, ex e nal channels as wa eguides
o po s. Using s anda d sys ems heo y me hods, ob ain sca e ing ma ix amily
Sne (ω;λ)
in
equency domain, whe e
λ
ep esen s slowly a ying in e nal pa ame e s.
S ep 2: Iden ica ion o sel - e e en ial closed loops
The dependence o in e nal upda es on hei own ou pu s mani es s as eedback closed loops in
equency domain. Th ough Redhee s a p oduc , comp ess hese closed loops in o closed-loop
sca e ing ma ix
S⟲(ω;λ)
. View he en i e ne wo k as a di ec ed g aph, pe o m s ongly connec ed
componen decomposi ion, selec minimal s ongly connec ed componen s sa is ying bo h memo y
and sel - e e en iali y condi ions, i.e., I-closed loops
S(I)
.
S ep 3: Consis ency o delay spec um and unied ime scale
By scale iden i y, calcula e
κ(ω)
o
S⟲(ω;λ)
om sca e ing phase and ime-delay ope a o ,
aligning wi h he unied ime scale o Is uc u e. Requi e eec i e ime densi y on I-closed loop
o ma ch ime unc ion on wo ldline in co esponding ene gy bandwid h.
S ep 4: Re e sible s eps and co espondence
9
A.5 Local Uniqueness up o Equi alence
I ano he ealiza ion
I′
∗
is compa ible wi h
I∗
o he same eco ds and unied ime scale equi alence
class, one can cons uc equi alence ela ion as ollows:
1. Dene s ic ly mono one bijec ion
om wo wo ldlines and ime unc ions;
2. Using GNS ep esen a ion and
∗
-isomo phism classica ion heo y, cons uc
∗
-isomo phism
Φ
be ween
Ain
and
A′in
, being measu e isomo phism on memo y subsys em in spec al space;
3. T ansla e compa ibili y o obse a ional da a and in e nal upda es in o semig oup conjugacy
condi ion o
U
and
U′
, he eby sa is ying Deni ion 7.
Thus ob aining local uniqueness.
Appendix B: Co espondence Be ween IS uc u es and Sel -Re e en ial
Sca e ing Loops
This appendix supplemen s key echnical poin s o Theo ems 2 and 3.
B.1 F om In e nal Dynamics o Closed-Loop Sca e ing
Fo a gi en Is uc u e
I
, cons uc linea ized inpu ou pu model unde unied ime scale: iew
in e nal deg ees o eedom as nodes, ex e nal channels as wa eguides. In equency domain, using
app op ia e deg ee o eedom choices and Laplace/Fou ie ans o ms, con e ime-domain upda e
U( 2, 1)
and en i onmen maps
E
in o sca e ing ma ix amily
Sne (ω;λ)
.
Sel - e e en ial upda e co esponds o eedback loops om ou pu owing back o inpu . Use
Redhee s a p oduc o comp ess eedback s uc u e in o closed-loop sca e ing ma ix
S⟲(ω;λ)
,
and e i y i sa ises ela i e ace-class condi ions.
B.2 Modied De e minan s and Spec al Shi
On
S⟲(ω;λ)
, cons uc associa ed ope a o pai
(H0, H)
, using Bi manK en and ela ed esul s
o dene modied de e minan
de pS⟲
and spec al shi unc ion
ξp
. Phase index map
s(ω, λ) = de
pS⟲(ω;λ) = exp−2πiξp(ω, λ)
maps pa ame e space o uni ci cle.
B.3
Z2
Holonomy and Minimal S ongly Connec ed Componen s
Remo e singula i y se whe e
s= 0
om pa ame e space, ob aining
X◦
. Conside double co e
dened by squa e- oo de e minan . Along closed pa h
γ⊂X◦
, dene
ν√S⟲(γ) = expiIγ
1
2is−1ds,
gi ing
Z2
index. S ongly connec ed componen decomposi ion and spec al ow addi i i y show
his index canno be u he decomposed on minimal s ongly connec ed componen s, hence can use
ν(S)
o label opological ype o each I-closed loop.
16

B.4 Alignmen wi h NullModula Double Co e and BF Theo y
In NullModula double co e and
Z2
BF heo y, sec o s uc u e o local geome y and modula
ow is ep esen ed by some cohomology class
[K]
on
H2(Y, ∂Y ;Z2)
. Fo physical sys ems sa is-
ying local ene gy and en opy consis ency, globally equi e
[K] = 0
, i.e., no anomalous sec o s
opologically o e all.
The
Z2
index o I-closed loop is conned o in e nal sca e ing subsys em, no dis up ing o e all
geome y's cohomology class. This localglobal alloca ion allows sel - e e en ial e mionic nge -
p in  o exis wi hin he subjec in e io wi hou in oducing new opological sec o s a uni e se
scale, he eby ealizing localiza ion o subjec i i y opology.
Th ough he abo e appendix cons uc ions and p oo ou lines, one can see he one- o-one co -
espondence be ween Is uc u es and sel - e e en ial sca e ing closed loops, and s abili y o
Z2
opological nge p in , he eby suppo ing he unied ma hema ical deni ion o sel  in he main
ex .
17