Fini e-In o ma ion Uni e se
and Pa ame e Vec o Θ:
En opy Bounds, Axioma iza ion
and Quan um Cellula Au oma on Sou ce Code
Leng h
Haobo Ma1Wenlin Zhang2
1Independen Resea che
2Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
S anda d pic u e o con inuous space ime and quan um ield heo y sugges s:
ma hema ically speci ying a comple e “uni e se” objec seems o equi e in ini ely
much in o ma ion—in ini ely many space ime poin s, in ini ely many deg ees o
eedom a each poin , ini ial condi ions as in ini e-p ecision eal- alued unc ions.
This “in ini e-in o ma ion uni e se” pic u e aces undamen al di icul ies bo h phys-
ically and in o ma ion- heo e ically. On o he hand, black hole en opy bounds,
holog aphic en opy bounds and quan um compu a ion limi s join ly s ongly sug-
ges : wi hin ini e ene gy and ini e space ime egion, physically dis inguishable
in o ma ion amoun has ini e uppe bound.
Building on solid ounda ion gi en by Bekens ein en opy bound, Bousso holo-
g aphic bound and Lloyd compu a ion limi , his pape in oduces “ ini e in o ma-
ion capaci y” axiom: exis s ini e cons an Imax <∞such ha physically dis in-
guishable o al in o ma ion amoun o en i e obse able uni e se does no exceed
Imax. Unde his axiom, we p o e: “uni e se” can be iewed as objec comple ely
speci ied by ini e bi s ing Θ, gi e sys ema ic pa ame e ec o decomposi ion
Θ = (Θs ,Θdyn,Θini),
espec i ely desc ibing space ime/la ice/ opological s uc u e, quan um cellu-
la au oma on (QCA) dynamics ules and ini ial quan um s a e.
In conc e e Di ac- ype QCA uni e se model, we cons uc i ely gi e in o ma ion
complexi y uppe bound: unde s ong ansla ion symme y, ixed ga e se and
ini e-p ecision disc e iza ion assump ions, encoding leng hs o s uc u al pa am-
e e s, dynamical pa ame e s, ini ial s a e pa ame e s can be con olled a o de
O(102), O(103), O(102–103) bi s espec i ely, hus ob aining ypical sou ce code
in o ma ion amoun es ima e
Ipa am(Θ) = |Θs |+|Θdyn|+|Θini| ∼ 103bi s,
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a smalle han maximum en opy Smax(Θ) ∼1090–122 bi s es ima ed om
uni e se ho izon a ea. We u he p o e in o ma ion–en opy inequali y
Ipa am(Θ) + Smax(Θ) ≤Imax,
showing “small sou ce code” and “gian en opy uni e se” compa ible unde
ini e in o ma ion capaci y axiom.
To add ess in ui i e ques ion “why can ex emely small ini ial da a e ol e in o
ex emely high complexi y uni e se”, his pape analyzes e olu iona y s uc u e
o QCA uni e se om dynamical sys em and quan um supe posi ion pe spec i e:
gi en sho pa ame e ec o Θ, en i e uni e se his o y iewable as ini e p og am
linea uni a y e olu ion in high-dimensional Hilbe combina o ial space; quan um
supe posi ion is no “b u e o ce enume a ion o all pe mu a ion combina ions”, bu
ealizes ampli ude and phase uni ied upda e and in e e ence on “linea en elope
o all classical combina ions”. This explains why “sou ce code uni e se” wi h ini e
algo i hmic complexi y can mac oscopically p esen complex s uc u e app oaching
maximum en opy.
Finally, we discuss s uc u e on pa ame e space MΘ, ini e in o ma ion inequal-
i y cons ain s on la ice numbe and uni Hilbe dimension, and how an h opic
p inciple and physical cons ain s comp ess possible uni e se se o ex emely small
ealizable subse unde gi en ini e Imax and pa ame e ec o Θ.
Keywo ds: Fini e in o ma ion; En opy bounds; Pa ame e ec o ; Quan um cellula
au oma on; Sou ce code leng h; Bekens ein bound; Holog aphic p inciple; Kolmogo o
complexi y
1 In oduc ion: F om “In ini e-In o ma ion Uni e se”
o “Fini e-In o ma ion Uni e se”
In s anda d con inuous space ime and quan um ield heo y pic u e, comple e “uni e se
s a e” usually iewed as ollowing objec :
1. Lo en z me ic gµν(x) on ou -dimensional mani old M, gi ing me ic enso a each
space ime poin x;
2. Quan um ield ˆ
Φa(x) on M, ini ial condi ion o each mode needs o speci y one eal
unc ion;
3. Vacuum/ he mal s a e/ luc ua ion con igu a ion on some ini ial Cauchy hype su -
ace.
These objec s ma hema ically o en ely on uncoun ably in ini e-dimensional unc ion
spaces, whose “comple e p ecise speci ica ion” seems o equi e in ini ely many numbe s,
i.e., in ini e in o ma ion amoun . This b ings se e al di icul ies:
Physically, ha d o explain how objec equi ing in ini e in o ma ion o speci y can
be compa ible wi h eal uni e se o ini e ene gy and ini e olume;
In o ma ion- heo e ically, “uni e se” ha canno be encoded in any ini e-leng h
bi s ing ha d o econcile wi h compu a ional ealizabili y;
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Philosophically, whe he “un ep esen able” uni e se has es able physical meaning
is ques ionable.
Meanwhile, black hole he modynamics and holog aphic en opy bounds gi e s ongly
opposi e sugges ion: wi hin ini e adius and ene gy egion, con ainable en opy has s ic
uppe bound; holog aphic bound u he poin s ou , o “ligh shee ” associa ed wi h some
closed spa ial su ace Σ, o al en opy ha can be ans e ed h ough any means does
no exceed A(Σ)/4Gℏ, whe e A(Σ) is a ea o Σ. These esul s join ly poin o simple
conclusion: unde ini e olume and ini e ene gy, physically dis inguishable
o al in o ma ion amoun is ini e.
This pape adds one axiom on his physical in ui ion: physically dis inguishable o al
in o ma ion amoun o en i e obse able uni e se has ini e uppe bound Imax. Unde his
axiom, “in ini e-in o ma ion uni e se” pic u e eplaced by “ ini e-in o ma ion uni e se”:
uni e se can be comple ely speci ied by some ini e-leng h bi s ing Θ, we call i “uni e se
pa ame e ec o ” o “sou ce code”.
To conc e ize his idea, we choose quan um cellula au oma on (QCA) as disc e e
uni e se model ca ie , cons uc pa ame e ized uni e se objec U(Θ), sys ema ically
cha ac e ize s uc u e, dynamics and ini ial s a e iple decomposi ion o Θ, gi e uppe
bound es ima e o sou ce code in o ma ion amoun Ipa am(Θ), de i e in o ma ion–en opy
inequali y Ipa am(Θ) + Smax(Θ) ≤Imax, and explain na u alness o “small sou ce code,
la ge complexi y” in quan um supe posi ion and dynamical sys em language.
2 Physical En opy Bounds and Fini e In o ma ion
Capaci y Axiom
2.1 Uni ied View o Physical En opy Bounds
Conside sphe ical egion o adius R, ene gy E, con aining some physical sys em. Beken-
s ein en opy bound gi es uppe bound on en opy s o able in his egion
S≤2πRE
ℏc,
showing unde ini e ene gy and ini e linea dimension, en opy canno inc ease inde -
ini ely. Fo si ua ion including g a i y, i egion o ms black hole, Bekens ein–Hawking
o mula shows black hole en opy
SBH =Aho
4Gℏ,
connec ing en opy wi h ho izon a ea. Holog aphic en opy bound u he poin s ou ,
o any spa ial su ace Σ, along con ac ing null ligh shee di ec ion, en opy lux Sligh shee
passing h ough his ligh shee sa is ies
Sligh shee ≤A(Σ)
4Gℏ.
On o he hand, Lloyd and Ma golus–Le i in ype compu a ion limi heo em poin s
ou : quan um sys em suppo ed by ene gy E, maximum numbe o logical ope a ions
Nops execu able wi hin ime in e al Thas uppe bound
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Nops ≤2ET
πℏ,
showing wi hin ini e ene gy and ini e ime, compu a ional esou ces also ini e.
Al hough hese inequali ies ha e di e en o ms and applicabili y condi ions, in his
wo k we only ex ac hei common co e: ini e ene gy, ini e space ime egion and
ini e e olu ion ime ⇒dis inguishable in o ma ion ini e.
2.2 Physically Dis inguishable In o ma ion and Equi alence Classes
To o malize “dis inguishable in o ma ion”, we in oduce ollowing concep . Le Sbe se
o all possible s a es o some physical sys em, each s a e ep esen ed by densi y ope a o
ρ. Gi en amily o easible measu emen s {Aα}and measu emen p ecision lowe bound
ϵ, we say ρ1, ρ2∈ S physically dis inguishable i and only i exis s some Aαsuch ha
(ρ1Aα)− (ρ2Aα)
> ϵ.
Fu he de ine equi alence ela ion ρ1∼ρ2i o all easible measu emen s and p e-
cision equi emen s, bo h indis inguishable. Thus ull se Spa i ioned in o physically
dis inguishable s a e equi alence classes, deno ed S/∼. We de ine “physically dis in-
guishable in o ma ion amoun ” o his sys em as
Iphys := log2
S/∼
.
Unde physical p emises o Bekens ein and Bousso en opy bounds, co esponding
Iphys o a ious conc e e sys ems (such as ini e- adius sphe e, ini e ho izon uni e se)
necessa ily ini e. We ele a e his ini eness o uni e se-le el axiom.
2.3 Fini e In o ma ion Capaci y Axiom and Encoding Map
Axiom 2.1 (Fini e In o ma ion Capaci y).Exis s ini e cons an Imax <∞such ha
physically dis inguishable in o ma ion amoun o en i e obse able uni e se sa is ies
Iphys(Uni e se)≤Imax.
F om his axiom, can immedia ely in oduce abs ac encoding map
Enc : Uphys → {0,1}≤Imax ,
whe e Uphys ep esen s all physically dis inguishable “uni e se objec s”, {0,1}≤Imax se
o bi s ings o leng h no exceeding Imax. We deno e encoding o some conc e e uni e se
as
Θ := Enc(U)∈ {0,1}≤Imax ,
call Θ “pa ame e ec o ” o “sou ce code” o his uni e se.
Con e sely, exis s some decoding map Dec such ha Dec(Θ) gi es uni e se physically
equi alen o U. Thus, unde ini e in o ma ion capaci y axiom, “uni e se” can be iewed
as decoding esul o some ini e bi s ing.
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3 Pa ame e ized Rep esen a ion o Uni e se and QCA
Uni e se Model
3.1 Abs ac S uc u e o Uni e se Objec
Unde mode n ield heo y and ope a o algeb a amewo k, uni e se can be abs ac ed
as ca ego ical s uc u e U, including space ime mani old, me ic, quan um ields, local
algeb a, base s a e, e c. This wo k does no need his comple e complexi y, bu only
e ains componen s di ec ly ela ed o in o ma ion complexi y.
Fo his, we conside class o disc e ized uni e se models, whose basic composi ion as
ollows:
1. La ice si e se Λ, can be ini e o locally ini e;
2. Each la ice si e x∈Λ has uni Hilbe space Hcell, dimension dcell;
3. Global Hilbe space H=Nx∈ΛHcell;
4. Quasi-local C∗algeb a A ⊂ B(H) gene a ed by local ope a o s;
5. Time e olu ion au omo phism g oup α:Z→Au (A), desc ibing disc e e ime
upda e;
6. Ini ial s a e ω0, i.e., posi i e no malized linea unc ional on A.
We deno e such objec as
U= (Λ,Hcell,A, α, ω0).
3.2 Quan um Cellula Au oma on as Unde lying Model
To conc e ize α, we adop quan um cellula au oma on (QCA) amewo k: ime s ep
n→n+ 1 ealized by some local uni a y ope a o U, i.e., o all local ope a o s A∈ A
ha e
α(A) = U†AU.
Locali y equi emen means U’s ac ion has ini e p opaga ion wi hin causal ligh cone,
i.e., exis s ini e adius such ha ope a o Aac ing only on some egion Rexpands
only o neighbo hood o Ra e one e olu ion s ep.
In his wo k, we do no discuss speci ic classi ica ion o QCA, bu only use ollowing
ac : gi en app op ia e local ga e se and la ice s uc u e, QCA can app oxima e ee
Di ac ield and mo e gene al low-ene gy ield heo y in con inuous limi . Thus, na u al
o ep esen uni e se objec as pa ame e ized QCA uni e se
U(Θ) = Λ(Θ),Hcell(Θ),A(Θ), αΘ, ωΘ
0,
whe e all componen s de e mined by pa ame e ec o Θ.
This modeling choice no necessa ily de i ed om ini e in o ma ion axiom, bu as
ep esen a i e class o disc e e uni e se scheme. Fini e in o ma ion axiom equi es exis-
ence o some class o ini e-dimensional, ini ely-gene a ed s uc u es; QCA model has
good locali y and compu a ional ealizabili y in his class o s uc u es, hus na u al
candida e.
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4 T iple Decomposi ion o Pa ame e Vec o Θ
4.1 S uc u e o Pa ame e Space
A e gi ing QCA uni e se amewo k, ole o pa ame e ec o Θ can be na u ally dis-
inguished in o h ee ca ego ies:
1. “S uc u al pa ame e s” de e mining la ice s uc u e and opology;
2. “Dynamical pa ame e s” de e mining local e olu ion ules;
3. “Ini ial s a e pa ame e s” de e mining ini ial quan um s a e s uc u e.
Fo mally w i e
Θ = (Θs ,Θdyn,Θini),
whe e each componen has independen in o ma ion con en .
4.2 S uc u al Pa ame e s Θs
S uc u al pa ame e s speci y la ice opology and local dimension:
La ice dimension dla (e.g., 1D, 2D, 3D, 4D);
La ice size Nsi es (o cha ac e is ic scale unde pe iodic bounda y condi ions);
Uni Hilbe dimension dcell;
Topological da a (e.g., bounda y condi ion ype, wis , de ec );
Symme y g oup labels.
Unde s ong ansla ion symme y assump ion, s uc u al pa ame e s can be ex-
emely compac . Fo example, h ee-dimensional cubic la ice wi h pe iodic bounda y,
need only speci y:
Θs = (dla = 3, Nlinea , dcell,bounda y ype).
I Nlinea ∼1026 (cosmological scale la ice spacing), encode as 90-bi in ege ; dcell ∼10
encode as 4 bi s; bounda y condi ion ype se e al bi s. To al
|Θs |∼O(102) bi s.
4.3 Dynamical Pa ame e s Θdyn
Dynamical pa ame e s speci y QCA local ga es. Typical se ing: choose ini e uni e sal
ga e se G={G1, . . . , GK}, each ga e ac s on limi ed la ice si es. Upda e ope a o U
w i en as
U=Y
laye s
Y
x∈laye
Gix(x),
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whe e ix∈ {1, . . . , K}ga e index a posi ion x.
I ga e se G ixed (analogous o “elemen a y pa icles and in e ac ion ypes” gi en
by S anda d Model), hen dynamical pa ame e s only need speci y ga e sequence a -
angemen . Unde s ong symme y ( ansla ion in a iance, gauge symme y), numbe
o independen pa ame e s d as ically educed.
Es ima e: i ga e lib a y size K∼10, need log2K∼4 bi s pe ga e; i upda e equi es
L∼102laye s, o al independen ga es ∼103, hen
|Θdyn| ∼ O(103) bi s.
Addi ionally conside ing ga e pa ame e ine- uning (e.g., coupling cons an angles),
i each pa ame e needs 10-bi p ecision, s ill wi hin kilobi o de .
4.4 Ini ial S a e Pa ame e s Θini
Ini ial s a e ω0can be pu e s a e o mixed s a e. Simples case, pu e s a e
|ψ0⟩∈H=O
x∈Λ
Cdcell .
Dimension o His dNsi es
cell , nai ely seems o need exponen ially many pa ame e s o
speci y |ψ0⟩. Howe e , unde ollowing assump ions, ini ial s a e encoding leng h can be
con olled:
Ini ial s a e has simple enso p oduc s uc u e o low en anglemen (e.g., acuum
s a e, cohe en s a e);
Ini ial luc ua ion desc ibable by ini e pa ame e amily (e.g., Gaussian andom
ield wi h limi ed modes);
Ce ain conse a ion laws o symme y cons ain s educe e ec i e deg ees o ee-
dom.
Conc e e es ima e: i ini ial s a e is enso p oduc o single-si e s a es plus small
pe u ba ion, pe u ba ion expansion o ini e o de , numbe o independen pa ame e s
∼102–103. I each pa ame e 10-bi p ecision,
|Θini| ∼ O(102–103) bi s.
Fo highly en angled ini ial s a es, encoding leng h may be la ge , bu unde ini eness
axiom, mus emain wi hin ini e Imax ange.
4.5 To al Sou ce Code Leng h Es ima e
Combining abo e h ee pa s, ob ain o al pa ame e in o ma ion amoun
Ipa am(Θ) = |Θs |+|Θdyn|+|Θini|∼O(103) bi s.
This is ex emely compac encoding: kilobi -le el in o ma ion can speci y en i e uni-
e se s uc u e, dynamics and ini ial s a e.
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5 In o ma ion–En opy Inequali y and Compa ibil-
i y
5.1 Maximum En opy Es ima e
Fo obse able uni e se, ho izon a ea Aho izon ∼(1026 m)2∼1052 m2, Planck a ea ℓ2
Pl ∼
10−70 m2, hus
Smax ∼Aho izon
4ℓ2
Pl
∼10122 bi s.
This is as onomically la ge en opy uppe bound.
5.2 In o ma ion–En opy Inequali y
P oposi ion 5.1 (In o ma ion–En opy Inequali y).Unde ini e in o ma ion capaci y
axiom, uni e se pa ame e in o ma ion amoun and maximum en opy sa is y
Ipa am(Θ) + Smax(Θ) ≤Imax.
P oo . To al physically dis inguishable in o ma ion o uni e se includes:
1. Sou ce code in o ma ion: Ipa am(Θ);
2. Mac os a e en opy: Smax(Θ), ep esen ing dis inguishable mic oscopic con igu a-
ions unde gi en s uc u e.
By axiom, o al in o ma ion ≤Imax, hus inequali y holds.
5.3 Compa ibili y o Small Sou ce Code and La ge En opy
P oposi ion ?? shows: e en i Ipa am(Θ) ∼103ex emely small, can s ill accommoda e
Smax(Θ) ∼10122, only equi es Imax ≥10122.
This esol es appa en pa adox: “How can small sou ce code gene a e high-en opy
uni e se?” Answe : sou ce code speci ies ules, no all de ails; en opy measu es s a e
space size, no ule complexi y. Quan um supe posi ion and dynamical e olu ion allow
ini e ules o gene a e exponen ially la ge s a e space.
6 Quan um Supe posi ion and E olu ion o Com-
plexi y
6.1 Dynamical Sys em Pe spec i e
Gi en pa ame e ec o Θ, uni e se his o y is ini e p og am execu ion:
|ψ(n)⟩=Un|ψ0⟩,
whe e Uand |ψ0⟩de e mined by Θ. Al hough p og am ini e, s a e space Hexpo-
nen ially la ge, allowing sys em o explo e high-complexi y egion.
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6.2 Quan um Supe posi ion s. Classical Enume a ion
Key di e ence: quan um e olu ion no “enume a ing all classical con igu a ions”, bu
“cohe en supe posi ion in linea space spanned by all con igu a ions”. S a e a ime n
|ψ(n)⟩=X
con igs
ccon ig(n)|con ig⟩,
whe e coe icien s ccon ig(n) compu ed by uni a y ma ix p oduc , no independen
enume a ion. This explains how ini e algo i hm can gene a e exponen ially many in e -
e ing ampli udes.
6.3 En opy G ow h and The maliza ion
Th ough local in e ac ions, ini ially low-en anglemen s a e g adually he malizes, en-
opy app oaches maximum Smax. This no con adic ion wi h ini e sou ce code, bu
esul o dynamical e olu ion: sou ce code speci ies e olu ion law, he maliza ion is con-
sequence o e olu ion.
7 Pa ame e Space S uc u e and An h opic Con-
s ain s
7.1 Pa ame e Space Mani old
All possible Θ o m pa ame e space
MΘ=Ms × Mdyn × Mini.
Dimension dim MΘ∼103. Howe e , only ex emely small subse physically ealiz-
able.
7.2 Physical Cons ain s
Va ious physical equi emen s cons ain pa ame e space:
Consis ency wi h obse ed low-ene gy physics (S anda d Model);
Cosmological e olu ion ma ching obse a ions;
S abili y and causali y cons ain s;
The modynamic consis ency.
These cons ain s d as ically educe iable pa ame e egion.
7.3 An h opic P inciple
Fu he , an h opic p inciple equi es uni e se pa ame e s allow eme gence o complex
s uc u es (obse e s). This adds addi ional selec ion mechanism, possibly educing e-
alizable uni e se se o coun able o e en unique.
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