The Red Queen Universe: The Cosmological Constant from Agent Games and the Avoidance of Heat Death

The Red Queen Uni e se: The Cosmological Cons an om
Agen Games and he A oidance o Hea Dea h
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
The classical pic u e o he hea dea h o he uni e se desc ibes he ul ima e a e o he
cosmos as a s a e whe e, o e long ime scales, all a ailable ee ene gy is exhaus ed, he
sys em ends owa d he mal equilib ium and maximum en opy, and all mac oscopic s uc-
u es and compu a ional ac i i ies cease. Howe e , he obse a ional ac ha he uni e se
has con inuously gene a ed and main ained mul i-le el low-en opy s uc u esespecially
li e and in elligen ci iliza ionso e app oxima ely 13.8 billion yea s o e olu ion emains
opaque unde he s anda d explana ion elying solely on "low-en opy ini ial condi ions." On
he o he hand, he cosmological cons an and da k ene gy p oblems sugges ha he alue
o he acuum ene gy densi y and i s coincidence wi h ma e densi y s ill lack a mic oscopic
in o ma ion- heo e ic explana ion.
In his pape , wi hin he amewo k o a Quan um Cellula Au oma on (QCA) uni e se
and he conse a ion o in o ma ion a e, we o malize he cosmic sys em as a mul i-agen
game eld composed o "agen s." Each agen is cha ac e ized as main aining a low-en opy
s uc u e a om equilib ium in a local Hilbe subspace and minimizing a ia ional ee
ene gy h ough in e nal compu a ion, he eby con inuously pe o ming in o ma ion e asu e
and p edic ion e o co ec ion. Based on Landaue 's p inciple and es ablished esul s in
he he modynamics o compu a ion, we p o e ha unde app op ia e coa se-g aining and
iso opy assump ions, he Landaue was e hea gene a ed by he i e e sible compu a ions
o all agen s in he uni e se is equi alen o a homogeneous, iso opic ene gy componen wi h
an app oxima e equa ion o s a e
w≃ −1
. I s g a i a ional eec in F iedmann dynamics is
equi alen o a dynamical cosmological "cons an "
Λe ( )
.
Fu he mo e, we in oduce a mul i-agen dynamic model based on he Red Queen eec ,
exp essing he  ness compe i ion and complexi y a ms ace among agen s as a sys em cou-
pling eplica o equa ions wi h complexi y a iables and Lo kaVol e a dynamics. Linea iza-
ion and spec al analysis o such sys ems yield a gene al heo em: wi hin a b oad pa ame e
ange sa is ying "Red Queen condi ions," in e nal equilib ium poin s a e no asymp o ically
s able. Ins ead, he sys em ends owa d limi cycles o chao ic a ac o s in phase space,
causing complexi y and Landaue in o ma ion e asu e a es o emain posi i e on cosmic
ime scales. Thus, he classical "s a ic limi equilib ium s a e" is dynamically excluded.
Building on his, he pape p esen s h ee main conclusions. Fi s , unde he cons ain s
o a QCA uni e se and in o ma ion a e conse a ion
2
ex + 2
in =c2
, he in o ma ion e asu e
ow co esponding o in e nal agen compu a ion cons i u es a na u al class o "in o ma-
ion acuum ene gy densi y"
ρin o( )
, whose in eg al de e mines he eec i e cosmological
cons an
Λe ( )
, p o iding an in o ma ion- heo e ic o igin o da k ene gy. Second, he Red
Queen agen game es ablishes a eedback loop be ween "complexi y and da k ene gy": iche
compu a ional ac i i ies gene a e la ge
ρin o
, which in u n al e s la ge-scale space ime dy-
namics and con e sely aec s he su i al en i onmen o agen s, o ming a "Red Queen
Uni e se" e olu iona y mode. Thi d, as long as a non-ze o densi y agen ne wo k exis s in
he uni e se and sa ises Red Queen condi ions, in he con inuous limi o Gene al Rela-
i i y, he s a e equi ed o hea dea h"global absence o a ailable ee ene gy, comple e
en opiza ion o s uc u e and compu a ion"is no longe a dynamical a ac o . The uni-
e se can e ol e in a pe pe ually a - om-equilib ium "algo i hmic u bulence" s a e.
In he applica ion sec ion, we discuss he alle ia ion o he cosmological "coincidence
p oblem" by complexi y-d i en
Λe ( )
, ela ionships wi h en opic cosmology and he Causal
1
En opic P inciple, and po en ial es able p edic ions in he con ex o da k ene gy e olu-
ion obse a ions (such as ecen DESI indica ions o ime- a ying da k ene gy). Finally,
enginee ing p oposals based on QCA quan um simula ion and mul i-agen simula ion a e
p esen ed o p o ide pa hways o es ing he "Red Queen Uni e se" amewo k on con ol-
lable pla o ms.
Keywo ds:
Quan um Cellula Au oma a; Cosmological Cons an ; Da k Ene gy; Hea Dea h;
Landaue 's P inciple; Va ia ional F ee Ene gy; Red Queen Eec ; Mul i-Agen Games; The mo-
dynamics o Compu a ion; In o ma ion Cosmology
1 In oduc ion & His o ical Con ex
1.1 Cosmic Hea Dea h and he Da k Ene gy P oblem
Since he nine een h cen u y, he "Hea Dea h o he Uni e se" pic u e, based on he Second Law
o The modynamics, has held ha i he uni e se is an eec i ely closed sys em, i s o al en opy
inc eases mono onically wi h ime, e en ually ending owa d a maximum en opy equilib ium
s a e wi h no a ailable ee ene gy, known as he "Big F eeze" o "Hea Dea h." In he amewo k
o mode n cosmology, i he uni e se is a o open on la ge scales and con ains a posi i e
cosmological cons an , he s anda d in e ence is ha he uni e se will asymp o ically app oach
an app oxima e de Si e equilib ium s a e a e inni e ime, whe e ene gy die ences in all
local s uc u es and p ocesses will be smoo hed ou .
Obse a ions based on Type Ia supe no ae, he Cosmic Mic owa e Backg ound, and la ge-
scale s uc u e a he end o he wen ie h cen u y indica ed ha he uni e se is cu en ly in
a phase o accele a ed expansion. This phenomenon is usually desc ibed by a da k ene gy
componen wi h an app oxima e equa ion o s a e
w≃ −1
, ypically ealized in Eins ein's
equa ions as a cosmological cons an
Λ
. Da k ene gy densi y accoun s o abou 70% o he
cu en cosmic ene gy budge , while ma e accoun s o abou 30%, o ming he so-called
Λ
CDM
s anda d model.
Howe e , he cosmological cons an in oduces wo classic puzzles. The  s is he "Cosmo-
logical Cons an P oblem": ze o-poin ene gy es ima es om quan um eld heo y a e ens o
o de s o magni ude la ge han he obse ed alue, and he e is no consensus on how o sc een
o econcile his huge acuum ene gy con ibu ion. The second is he "Coincidence P oblem":
why he da k ene gy densi y and ma e densi y happen o be o he same o de o magni ude
in he cu en cosmic epoch, whe eas hey we e no o mos o cosmic his o y.
Recen obse a ions o la ge samples o galaxies and quasa s by p ojec s like DESI ha e
e en gi en indica ions ha da k ene gy may e ol e wi h ime, u he s imula ing heo e ical
explo a ion in o dynamical da k ene gy and he o igins o a non- i ial cosmological cons an .
1.2 In o ma ion, Compu a ion, and The modynamics: Landaue 's P inciple
and Compu a ional Engines
The deep connec ion be ween in o ma ion heo y and he modynamics was  s sys ema ically
elucida ed by Landaue . Landaue 's P inciple s a es ha he minimum ene gy cos equi ed o
e ase one bi o classical in o ma ion in an en i onmen a empe a u e
T
is
Emin =kBTln 2
. Any
logically i e e sible ope a ion (such as e asu e o me ging compu a ional pa hs) is necessa ily
accompanied by en opy p oduc ion and hea dissipa ion o a leas his magni ude. Benne sub-
sequen ly de eloped he he modynamics o compu a ion, iewing compu e s as he mal engines
ha con e ee ene gy in o was e hea and "ma hema ical wo k," and p o ed ha e e sible
compu a ion can ope a e a bi a ily close o he Landaue limi . Subsequen wo k ex ended
Landaue 's P inciple o quan um and non-equilib ium sys ems, con ming i s uni e sali y as a
physical lowe bound o in o ma ion p ocessing.
2
These esul s sugges ha any sys em pe o ming i e e sible compu a ionincluding a i-
cial compu e s, biological o ganisms, and e en b oade "na u al compu a ion" p ocessesmus
necessa ily emi a minimal amoun o hea o he en i onmen and p oduce a co esponding
en opy inc ease. This concep p o ides a ounda ion o unde s anding mac oscopic cosmic
he modynamics om an in o ma ion- heo e ic pe spec i e.
1.3 Va ia ional F ee Ene gy and Agen s: Inspi a ion om he F ee Ene gy
P inciple
In cogni i e science and neu oscience, he F ee Ene gy P inciple p oposed by F is on s a es ha
all biological sys ems main aining hei bounda ies and homeos asis can be iewed as in e ence
machines ha minimize a ia ional ee ene gy in some sense. The co e is o in oduce a join
p obabili y model
P(s, ϑ)
o e senso y inpu s
s
and hidden a iables
ϑ
, as well as an in e nal
app oxima e pos e io
q(ϑ)
. The sys em minimizes a a ia ional unc ional by changing i s
in e nal s a e:
F(q, s) = KLq(ϑ)|P(ϑ|s)−ln P(s),
(1)
which can equi alen ly be w i en in o ms con aining model e idence and en opy e ms.
In his amewo k, o ganisms educe p edic ion e o and "su p ise" by upda ing in e nal
models and aking ac ions, he eby main aining o de ed s uc u es a om equilib ium. Al-
hough ini ially applied o neu al sys ems, he o m o he F ee Ene gy P inciple does no depend
on specic ma e ial subs a es, so i can be abs ac ed o he mo e gene al concep o an "agen ."
1.4 The Red Queen Eec and E olu iona y A ms Races
The Red Queen hypo hesis was p oposed by Van Valen o explain he "Van Valen Law" in he
paleon ological eco d, whe e species ex inc ion a es a e app oxima ely independen o species
age. The hypo hesis emphasizes ha he "eec i e en i onmen " o a species is mainly com-
posed o o he coexis ing species, so  ness imp o emen is ela i e: when one species gains an
ad an age, i de e io a es he ecological niche o o he species, o cing he la e o e ol e o
main ain hei chances o su i al.
The Red Queen eec can be gene alized o sexual selec ion and hos -pa asi e in e ac ions a
he indi idual le el, and can also be iewed as a ze o-sum game: all species need o " un as as
as hey can" jus o s ay in place. Such game dynamics a e ubiqui ous in mul i-agen sys ems,
and esea ch in Agen -based models shows ha local in e ac ions can o en p oduce complex
eme gen phenomena on mac oscopic scales.
This pape ele a es his idea o he cosmological scale: iewing he uni e se as a game ne wo k
composed o mul i-le el agen s ( om molecula machines o li e and echnological ci iliza ions),
whe e he Red Queen eec main ains a non-equilib ium s a e o e cosmic ime scales h ough
a "complexi y a ms ace."
1.5 QCA Uni e se and Conse a ion o In o ma ion Ra e
Quan um Cellula Au oma a p o ide a na u al amewo k o s ic quan um dynamics on dis-
c e e space ime. A QCA can be iewed as an a ay o ni e-dimensional quan um sys ems dened
on a la ice, whose e olu ion is gi en by ansla ion-in a ian , causal uni a y ope a o s i e a ed
o e disc e e ime s eps. A la ge body o wo k has shown ha eld equa ions such as Di ac,
Weyl, and Maxwell can eme ge om QCA models in app op ia e con inuous limi s, p o iding a
igo ous pa h o "space ime and eld heo y o igina ing om quan um compu a ion."
In p e ious wo k, he ex e nal g oup eloci y
ex
o local exci a ions and he in e nal phase
o a ion o "in insic e olu ion eloci y"
in
we e combined in o an in o ma ion a e ec o
u= ( ex , in )
, p oposing he in o ma ion a e conse a ion ela ion:
2
ex + 2
in =c2,
(2)
3
whe e
c
is he maximum p opaga ion speed o he QCA, co esponding o he speed o ligh in
he con inuous limi . This ela ion unies he no maliza ion o ou - eloci y in special ela i i y,
p ope ime, and he mass- equency ela ion
mc2=ℏωin
in o a geome ic cons ain o in o ma-
ion a e budge , p o iding he basis o cons uc ing he link be ween "in e nal compu a ion"
and "ex e nal geome y" in his pape .
1.6 Objec i es and Main Con ibu ions
Syn hesizing he abo e backg ound, his pape add esses h ee in e connec ed ques ions:
1. I he uni e se is a QCA uni e se con aining a as numbe o agen s main aining hei
s uc u es h ough compu a ion, how does he he modynamic necessi y o hese compu a ions
eed back in o la ge-scale space ime geome y and he cosmological cons an ?
2. Can Red Queen-s yle mul i-agen games dynamically p e en he uni e se om en e ing
a nal s a e o comple e equilib ium hea dea h?
3. Is his "Red Queen Uni e se" mechanism in insically linked o he alue o da k ene gy
and i s possible ime e olu ion, and is i compa ible wi h exis ing obse a ions?
A ound hese ques ions, he main wo k o his pape is o ganized as ollows:

In "Model & Assump ions," we o malize he QCA uni e se and agen s, in oduce he
in o ma ion acuum ene gy densi y
ρin o( )
based on Landaue 's p inciple, and gi e i s
ela ion o he cosmological cons an
Λe ( )
.

In "Main Resul s," we p esen wo co e heo ems:  s , he Landaue -
Λ
ela ion heo em,
p o ing ha unde iso opy and Red Queen con inuous compu a ion assump ions, he i e-
e sible compu a ion o agen s is equi alen o a dynamical cosmological cons an ; second,
he Red Queen non-equilib ium heo em, which, unde a mul i-agen eplica o -complexi y
dynamic model, excludes in e nal s able equilib ium poin s and gua an ees ha complexi y
and in o ma ion e asu e a es emain posi i e o e he long e m.

In "P oo s" and he appendices, we p o ide igo ous de i a ions o he abo e heo ems,
including he con inui y equa ion wi h sou ce e ms de i ed om Eins ein eld equa ions
and ene gy-momen um conse a ion, and linea s abili y analysis o he Lo kaVol e a
eplica o sys em.

In "Model Apply," we cons uc a pa ame e iza ion o complexi y-d i en
Λe ( )
, compa e
i wi h en opic cosmology and he Causal En opic P inciple, and discuss po en ial con-
nec ions wi h obse a ions o ime- a ying da k ene gy.

In "Enginee ing P oposals," we p opose specic schemes o es ing his amewo k on
quan um simula ion pla o ms and in mul i-agen simula ions.
2 Model & Assump ions
This sec ion p esen s he basic s uc u e and assump ions o he "Red Queen Uni e se" model.
2.1 QCA Uni e se and Mac oscopic Geome y
Assume ha a he undamen al le el, he uni e se is desc ibed by a enso p oduc space
H=Nx∈ΛHx
dened by a coun able la ice se
Λ
and local Hilbe spaces
Hx≃Cd
. Global
e olu ion is gi en by a ansla ion-in a ian , local, and causal uni a y ope a o
U:H → H
o e
disc e e ime s eps
n∈Z
, which is he s anda d deni ion o a QCA.
4
In he long-wa eleng h and low-ene gy limi , whe e la ice spacing and ime s ep end o ze o,
QCA e olu ion app oxima es con inuous ela i is ic eld equa ions, mac oscopically desc ibable
by a me ic wi h F iedmannLemaî eRobe sonWalke (FLRW) symme y:
ds2=−c2d 2+a2( )γijdxidxj,
(3)
whe e
a( )
is he scale ac o , and
γij
is he cons an cu a u e me ic o h ee-dimensional
space. The cosmological cons an o da k ene gy componen mani es s in his con inuous limi
as a e m in he ene gy-momen um enso
T(Λ)
µν =−ρΛgµν
, co esponding o an equa ion o s a e
pΛ=−ρΛ
.
2.2 In o ma ion Ra e Conse a ion and Deni ion o Agen s
Fo e e y exci a ion o s uc u e localized in he QCA uni e se, conside i s eec i e wo ldline
γ
and he ex e nal g oup eloci y
ex
and in e nal e olu ion eloci y
in
along ha wo ldline,
sa is ying he in o ma ion a e conse a ion ela ion:
2
ex + 2
in =c2.
(4)
He e,
ex
eec s he p opaga ion speed o he exci a ion on he la ice, while
in
cha ac e -
izes he phase e olu ion and in e nal compu a ion a e o i s in e nal s a e in he local Hilbe
subspace.
Deni ion:

An
Agen
A
is a ni e connec ed la ice subse
ΛA⊂Λ
on he QCA and a amily o
densi y ope a o s
ρA( )
on
HA=Nx∈ΛAHx
, sa is ying:
1.
ρA( )
main ains a signican de ia ion om he en i onmen al equilib ium s a e
ρeq( )
o e long ime scales, i.e., he e exis s a mac oscopic obse able
O
such ha
| [(ρA−
ρeq)O]|
has posi i e measu e on a ime se la ge han some xed h eshold.
2. This de ia ion is main ained by in e nal compu a ion, i.e., he e olu ion o
ρA( )
can be decomposed in o app oxima ely e e sible in e nal uni a y ope a o s and i -
e e sible "in o ma ion e asu e" maps, whe e he la e he modynamically sa ises
he Landaue bound.

The
In o ma ion Mass
MI
o an agen is dened such ha i s in e nal e olu ion equency
ωin
and eec i e ene gy
EI
sa is y he ela ion:
EI=MIc2=ℏωin ,
(5)
linking he in e nal compu a ion a e o he ela i is ic mass scale.
2.3 Va ia ional F ee Ene gy and Agen Objec i e Func ion
Re e encing he F ee Ene gy P inciple, ea each agen as an in e ence machine possessing a
gene a i e model
P(s, ϑ)
o en i onmen al signals
s
and in e nal s a es
ϑ
. Dene i s a ia ional
ee ene gy:
FA( ) = KLq (ϑ)|P(ϑ|s )−ln P(s ),
(6)
whe e
q (ϑ)
is he app oxima e pos e io dis ibu ion o hidden a iables held by he agen a
ime
, and
s
is i s senso y inpu a ha ime. The "su i al s a egy" o an agen can be
abs ac ed as minimizing he pa h in eg al o long- ime a e age o
FA( )
unde he in o ma ion
a e budge cons ain :
2
ex ( ) + 2
in ( ) = c2.
(7)
This op imiza ion p ocess necessa ily in ol es comp essing he in e nal s a e space and l e -
ing ou unnecessa y high-dimensional componen s, hus physically co esponding o equen
i e e sible w i ing and e asu e o in e nal s o age and ep esen a ions.
5

2.4 Landaue In o ma ion Was e Hea and In o ma ion Vacuum Ene gy Den-
si y
Fo a single agen
A
, assume i s in o ma ion e asu e a e in an ambien empe a u e eld
Tbg(x, )
is
˙
I(A)
e ase( )
(in bi s/s). Landaue 's P inciple equi es i s minimum dissipa ion powe o sa is y:
P(A)
L( )≥kBln 2 T(A)
bg ( )˙
I(A)
e ase( ),
(8)
whe e
T(A)
bg
is he eec i e en i onmen al empe a u e o he agen .
On cosmological scales, coa se-g aining o e a como ing olume
V
and summing he in-
o ma ion e asu e con ibu ions o all agen s
Ai
loca ed wi hin ha olume yields he a e age
Landaue powe densi y pe uni como ing olume:
Pin o( ) = 1
VX
i
kBln 2 T(i)
bg ( )˙
I(i)
e ase( ).
(9)
One o he co e assump ions o his pape is:
Assump ion 1 (In o ma ion Was e Hea Vacuum Ene gy In eg abili y):
In he
QCA uni e se, he Landaue was e hea p oduced by he i e e sible compu a ions o agen s is
mic oscopically encoded in o a homogeneous, iso opic "in o ma ion acuum" exci a ion in he
unde lying QCA deg ees o eedom. I s a e age ene gy-momen um enso in he con inuous
limi has he app oxima e o m:
T(in o)
µν ( )≃ −ρin o( )gµν,
(10)
whe e he in o ma ion acuum ene gy densi y
ρin o( )
sa ises he sou ced con inui y equa ion:
˙ρin o + 3H(1 + win o)ρin o =Pin o( ),
(11)
and sa ises
win o( )≃ −1
in he Red Queen e a.
This assump ion essen ially ein e p e s Landaue was e hea om "con en ional he mal
adia ion" o an ene gy ese e s o ed in he unde lying QCA deg ees o eedom ha mac o-
scopically exhibi s nega i e p essu e, making i s g a i a ional eec equi alen o da k ene gy.
Unde his assump ion, an eec i e cosmological "cons an " can be dened:
Λe ( ) = Λba e + 8πG ρin o( ),
(12)
whe e
Λba e
is a possible unde lying cons an pa (e.g., acuum ene gy emaining a e eno -
maliza ion), and
ρin o( )
is he dynamical pa eme ging om agen compu a ion.
2.5 Red Queen Condi ions and Mul i-Agen Games
To cha ac e ize compe i ion and a ms aces be ween agen s, we in oduce he ollowing abs ac-
ions:

Assume he e a e se e al species o agen s in he uni e se, deno ed by
i= 1, . . . , n
, wi h
como ing numbe densi y
Ni( )
and a e age complexi y
Ci( )
(unde s ood as algo i hmic
complexi y, s uc u al in o ma ion, o a e age dimension o in e nal s a e space).

Dene he  ness o species
i
as:
Wi( ) = iN( ),C( ),
(13)
whe e
N= (N1, . . . , Nn)
and
C= (C1, . . . , Cn)
.
6

Complexi y
Ci
enhances he abili y o ha class o agen s o cap u e and u ilize ee ene gy
on he one hand, bu inc eases Landaue cos s on he o he .
The
Red Queen Condi ions
can be o malized as:
1. Fo any
i=j
, wi h o he a iables xed,
∂ j/∂Ci<0
: an inc ease in he complexi y o
one class o agen s de e io a es he  ness o o he classes. 2. Fo each
i
, wi hin he esou ce-
abundan in e al,
∂ i/∂Ci>0
: be o e cos s ou weigh bene s, inc easing one's own complexi y
inc eases  ness.
Mul i-agen sys ems sa is ying he abo e condi ions ypically exhibi e olu iona y a ms aces,
whose dynamics can be desc ibed by eplica o equa ions o Lo kaVol e a ype equa ions.
3 Main Resul s (Theo ems and alignmen s)
This sec ion p esen s he wo co e heo ems o his pape and se e al co olla ies.
3.1 Theo em 1 (Landaue 
Λ
Rela ion Theo em)
In he con inuous limi o a QCA uni e se, assume:
1. The la ge-scale geome y is an iso opic, homogeneous FLRW space ime sa is ying s an-
da d F iedmann equa ions. 2. The e exis s a game ne wo k composed o agen s in he uni e se,
wi h an a e age Landaue powe densi y
Pin o( )
o e como ing olume elemen s. 3. In o ma ion
was e hea sa ises Assump ion 1, i.e., i s mac oscopic g a i a ional eec can be desc ibed by
a uid wi h equa ion o s a e
pin o =win oρin o
, and
win o ≃ −1
in he Red Queen e a.
Then, he in o ma ion acuum ene gy densi y
ρin o( )
and Landaue powe densi y
Pin o( )
sa is y he in eg al ela ion:
ρin o( ) = ρin o( 0) + Z
0
Pin o(τ) dτ+O(ϵ),
(14)
whe e
ϵ
cha ac e izes he de ia ion o
win o + 1
. Co espondingly, he eec i e cosmological
cons an is:
Λe ( )=Λe ( 0)+8πG Z
0
Pin o(τ) dτ+O(ϵ).
(15)
In o he wo ds, unde he good app oxima ion
win o ≈ −1
, he dynamical pa o he cosmological
cons an is equi alen o he accumula ion o Landaue ene gy ux gene a ed by he i e e sible
compu a ions o all agen s in he uni e se o e cosmic ime.
3.2 Co olla y 1 (Complexi yDa k Ene gy Coupling)
I we u he assume:
1. The e exis s a mac oscopic complexi y unc ion:
C( ) = X
i
Ni( )Ci( ),
(16)
cha ac e izing he o al complexi y o agen s pe uni como ing olume. 2. The a e age in o -
ma ion e asu e a e pe uni complexi y and he en i onmen al empe a u e a e app oxima ely
cons an , i.e., he e exis s a cons an
α > 0
such ha :
Pin o( )≃α Tbg( )˙
C( ),
(17)
whe e
Tbg( )
is he mac oscopic a e age o he cosmic backg ound empe a u e eld.
Then we ha e:
Λe ( )≃Λe ( 0)+8πGα ZC( )
C( 0)
Tbg(C) dC.
(18)
7
Unde he app oxima ion ha
Tbg
a ies slowly wi h ime, he abo e equa ion gi es a mono onic
coupling ela ion be ween
Λe ( )
and complexi y
C( )
. I
C( )
g ows apidly du ing a ce ain
pe iod o cosmic his o y (such as he epoch o galaxy o ma ion and he eme gence o li e),
Λe
will also comple e a majo jump du ing his pe iod, he eby p o iding an in o ma ion- heo e ic
explana ion o he "coincidence" be ween da k ene gy densi y and he his o y o s uc u e
o ma ion.
3.3 Theo em 2 (Red Queen Non-Equilib ium Theo em, Simplied Two-Species
Case)
Conside wo classes o agen popula ions sa is ying Red Queen condi ions, whose dynamics a e
gi en by he ollowing o dina y die en ial equa ions:
dNi
d =Ni iCi−di−γ(N1+N2), i = 1,2,
(19)
dCi
d =αiNi−βiCi,
(20)
whe e
i, di, γ, αi, βi>0
. The abo e equa ions can be iewed as a coupling o Lo kaVol e a
ype popula ion dynamics and complexi y e olu ion equa ions:
( iCi)
ep esen s  ness gains
om complexi y,
(γ(N1+N2))
ep esen s esou ce compe i ion, and
(βiCi)
ep esen s Landaue
cos s o main aining complexi y.
Assume he e exis s an in e nal equilib ium poin
(N∗
1, N∗
2, C∗
1, C∗
2)
sa is ying
N∗
i>0
,
C∗
i>0
.
I he pa ame e s sa is y:
1α1N∗
1+ 2α2N∗
2> β2
1+β2
2+ 2γ 1C∗
1+ 2C∗
2,
(21)
hen his equilib ium poin is no asymp o ically s able; ins ead, he Jacobian ma ix o he
linea ized sys em a his poin has a leas one pai o conjuga e complex eigen alues wi h posi i e
eal pa s. Fo an open dense se o pa ame e s, he sys em unde goes a Hop bi u ca ion nea
his equilib ium poin , e ol ing o a class o limi cycles o mo e complex a ac o s.
In his case, he o al complexi y
C o ( ) = N1( )C1( ) + N2( )C2( )
(22)
and he Landaue powe densi y
Pin o( )∝C o ( )
will no con e ge o a cons an o e long imes,
bu will oscilla e o exhibi quasi-pe iodic/chao ic beha io wi hin a bounded in e al, wi h a
non-ze o ime a e age. The e o e, he sys em will no en e a s a ic equilib ium s a e in ni e
ime, bu is main ained in a pe pe ual non-equilib ium dynamic d i en by he "Red Queen."
3.4 Co olla y 2 (Necessa y Condi ion o A oiding Hea Dea h)
In a QCAFLRW uni e se, i he ollowing a e sa ised:
1. The e exis s a leas one class o agen popula ions sa is ying Red Queen condi ions,
whose dynamics sa is y he condi ions o Theo em 2, such ha he a e age de i a i e o
C o ( )
o e any ni e ime in e al is non-ze o; 2. In o ma ion was e hea  acuum ene gy in eg abili y
(Assump ion 1) holds, so ha
Pin o( )
con inuously injec s ene gy in o
ρin o
; 3. The o e all ee
ene gy supply o he uni e se is no exhaus ed in ni e ime, i.e., he "ene gy budge " o he
unde lying QCA allows he abo e p ocess o con inue o a bi a ily long imes;
Then he s a e equi ed by he classical hea dea h pic u e" he en i e uni e se eaching
a comple e equilib ium s a e wi h no a ailable ee ene gy, no s uc u e, and no compu a ion
a e a ni e ime"is no a dynamical a ac o o he uni e se. Ins ead, he uni e se can
e ol e in a long-s anding "algo i hmic u bulence" phase, cha ac e ized by accele a ed expansion
domina ed by da k ene gy on mac oscopic scales, complex s uc u es con inuously eme ging and
dissipa ing on mesoscopic scales, and i e e sible compu a ional p ocesses cons an ly occu ing
on mic oscopic scales.
8
4 P oo s
This sec ion ou lines he p oo ideas o he abo e heo ems, lea ing mo e echnical de i a ions
o he appendices.
4.1 P oo o Theo em 1
In FLRW space ime, assume he o al cosmic ene gy-momen um enso is:
Tµν =T(m)
µν +T( )
µν +T(in o)
µν ,
(23)
whe e
(m)
and
( )
deno e ma e and adia ion componen s, espec i ely, and
(in o)
deno es he
in o ma ion acuum componen . Fo a componen
X
wi h pe ec uid o m:
T(X)
µν = (ρX+pX)uµuν+pXgµν,
(24)
whe e
uµ
is he como ing ou - eloci y. Unde cosmological symme y, ene gy conse a ion o
die en componen s can be w i en as sou ced con inui y equa ions:
˙ρX+ 3H(ρX+pX) = QX( ),
(25)
whe e
QX
is he ene gy sou ce e m o ha componen om o he componen s, sa is ying
PXQX= 0
.
Fo he in o ma ion acuum componen , assume i s sou ce e m is p ecisely he Landaue
powe densi y:
Qin o( ) = Pin o( ),
(26)
while he sou ce e ms o ma e and adia ion componen s a e ene gy losses o opposi e sign.
Subs i u ing he equa ion o s a e
pin o =win oρin o
yields:
˙ρin o + 3H(1 + win o)ρin o =Pin o( ).
(27)
In he Red Queen e a, we assume
win o( ) = −1 + δ( )
, whe e
δ( )
is a small quan i y sa is ying
|δ( )| ≪ 1
. The con inui y equa ion becomes:
˙ρin o + 3Hδ( )ρin o =Pin o( ),
(28)
wi h o mal solu ion:
ρin o( ) = ρin o( 0) exp −3Z
0
H(τ)δ(τ) dτ+Z
0
Pin o(s) exp −3Z
s
H(τ)δ(τ) dτds.
(29)
Unde he condi ion ha
|δ| ≪ 1
and
H
is ni e o e he pe iod conside ed, he de ia ion o he
exponen ial ac o om 1 is
O(ϵ)
, whe e
ϵ= sup[ 0, ]|3Hδ|∆
, and
∆
is he cha ac e is ic ime
scale. Thus, i can be w i en as:
ρin o( ) = ρin o( 0) + Z
0
Pin o(τ) dτ+O(ϵ),
(30)
which is he asse ion o Theo em 1.
The eec i e cosmological cons an is gi en by:
Λe ( ) = Λba e + 8πGρin o( ),
(31)
na u ally yielding he in eg al o m.
9
[9] "Hea Dea h o he Uni e se," in Wikipedia, accessed 2025.
[10] Y. L. Bolo in, A. L. Tu , V. A. Che kaskiy, "Cosmology Based on En opy,"
a Xi :2310.10144 (2023).
[11] S. Noji i, S. D. Odin so , V. Fa aoni, "Ba ow En opic Da k Ene gy: A Re iew," Physics
o he Da k Uni e se 36, 101050 (2022).
[12] R. Bousso, R. Ha nik, G. D. K ibs, G. Pe ez, "P edic ing he Cosmological Cons an om
he Causal En opic P inciple," Physical Re iew D 76, 043513 (2007).
[13] DESI Collabo a ion, epo s on ime-e ol ing da k ene gy p esen ed a APS Global Physics
Summi (2025).
[14] P. A ighi, "An O e iew o Quan um Cellula Au oma a," Na u al Compu ing 18, 885899
(2019).
[15] T. Fa elly, "A Re iew o Quan um Cellula Au oma a," Quan um 4, 368 (2020).
[16] I. Bialynicki-Bi ula, "Weyl, Di ac, and Maxwell Equa ions on a La ice as Uni a y Cellula
Au oma a," Physical Re iew D 49, 69206927 (1994).
[17] A. Bisio, G. M. D'A iano, A. Tosini, "Quan um Field as a Quan um Cellula Au oma on:
The Di ac F ee E olu ion in One Dimension," Annals o Physics 354, 244264 (2015).
[18] D. G oss, V. Nesme, H. Vog s, R. F. We ne , "Index Theo y o One Dimensional Quan um
Walks and Cellula Au oma a," Communica ions in Ma hema ical Physics 310, 419454
(2012).
[19] A. Sup ano e al., "Pho onic Cellula Au oma on Simula ion o Rela i is ic Quan um Field,"
Physical Re iew Resea ch 6, 033136 (2024).
[20] A. Bisio, G. M. D'A iano, P. Pe ino i, "Quan um Cellula Au oma on Theo y o Ligh ,"
Annals o Physics 368, 177190 (2016).
[21] B. Aza ian, "Li e Need No E e End," Noema Magazine (2023).
[22] H. Ma, "Uni e sal Conse a ion o In o ma ion Cele i y: F om Quan um Cellula Au oma a
o Rela i i y, Mass and G a i y" (2025), p ep in .
[23] H. Ma, "In o ma ion- olume Conse a ion and he Eme gence o Op ical Me ics" (2025),
p ep in .
[24] H. Ma, "The Red Queen Uni e se: Agen Games, Da k Ene gy and he A oidance o Hea
Dea h" ( his wo k).
A Appendix A: De ailed De i a ion om Landaue Powe o E -
ec i e Cosmological Cons an
This appendix p o ides a de ailed de i a ion om Landaue powe densi y
Pin o( )
o he eec i e
cosmological cons an
Λe ( )
.
16

A.1 A.1 Ene gy-Momen um Conse a ion and Sou ced Con inui y Equa ion
In he FLRW backg ound, Eins ein's equa ions
Gµν + Λba egµν = 8πGTµν
(48)
combined wi h he Bianchi iden i y
∇µGµν = 0
lead o
∇µTµν −Λba e
8πG gµν= 0.
(49)
I we in oduce an eec i e cosmological cons an
Λe ( )
and abso b i in o he igh -hand side,
i can be w i en as:
Gµν = 8πG ˜
Tµν,˜
Tµν =T(m)
µν +T( )
µν +T(in o)
µν ,
(50)
whe e
T(in o)
µν =−Λe ( )
8πG gµν.
(51)
Applying he ene gy-momen um conse a ion equa ion o each componen :
∇µT(X)
µν =Q(X)
ν,X
X
Q(X)
ν= 0,
(52)
and using he como ing ame
uµ= (1,0,0,0)
, ocusing only on he ene gy componen
ν= 0
,
we ob ain:
˙ρX+ 3H(ρX+pX) = Q(X)
0( ).
(53)
Assume he sou ce e ms o ma e and adia ion componen s a e:
Q(m)
0=−Γm( ), Q( )
0=−Γ ( ),
(54)
and combine ene gy losses caused by compu a ion in o he sou ce e m o he in o ma ion
acuum componen :
Q(in o)
0= Γm( )+Γ ( ) = Pin o( ),
(55)
hen he in o ma ion componen sa ises:
˙ρin o + 3H(ρin o +pin o) = Pin o( ).
(56)
I
pin o =win oρin o
, hen he abo e equa ion is he sou ced con inui y equa ion in he main ex .
A.2 A.2 Jus ica ion o
win o ≃ −1
Fo he in o ma ion acuum ene gy o mani es mac oscopically as da k ene gy, i s equa ion o
s a e mus be close o
−1
. F om a QCA pe spec i e, a possible mic oscopic pic u e is:

I e e sible compu a ions o agen s pe manen ly "e ase" a po ion o locally accessible
deg ees o eedom in o inaccessible global en angled s uc u es h ough sca e ing and
en anglemen wi h he en i onmen ;

These en angled deg ees o eedom a e uni o mly dis ibu ed on la ge scales, ca y no ne
momen um ux, and hei local pe u ba ions equilib a e apidly unde QCA e olu ion;

In he con inuous limi , he con ibu ion o such deg ees o eedom o mac oscopic geom-
e y is app oxima ely iso opic and equi alen o a uid wi h nega i e p essu e.
17
This pic u e is o mally simila o iewing acuum ene gy as ze o-poin ene gy o eld heo y, bu
i s o igin is no eld mode oscilla ions bu "in o ma ion agmen s" gene a ed du ing i e e sible
compu a ion. In specic models, one can p o e ha a class o i e e sible sca e ing is always
accompanied by a spec al shi o xed sign by cons uc ing a QCA Hamil onian wi h a gi en
sca e ing ma ix and spec al ow, he eby p oducing an equi alen acuum ene gy con ibu ion.
This is a di ec ion o u u e wo k.
Unde he app oxima ion equi ed o his pape , i suces ha
win o
sa ises
|win o + 1| ≪ 1
(57)
in he Red Queen e a o gua an ee he alidi y o he in eg al app oxima ion.
A.3 A.3 Complexi y-D i en
Pin o( )
Model
Assume he a e age in o ma ion e asu e a e co esponding o uni complexi y is a cons an
κ
,
hen:
˙
Ie ase( ) = κ C( ).
(58)
I he ambien empe a u e
Tbg( )
a ies slowly o e he pe iod conside ed, he Landaue powe
densi y is:
Pin o( ) = kBln 2 Tbg( )κ C( )≈α C( ),
(59)
whe e
α=kBln 2 κ¯
T
, and
¯
T
is he cha ac e is ic alue o
Tbg
.
Mo e gene ally, i we conside he a e o change o complexi y, hen
Pin o( )≈α Tbg( )˙
C( )
(60)
is a mo e easonable app oxima ion, because in o ma ion e asu e is ypically mo e di ec ly ela ed
o complexi y change han o absolu e alue. Subs i u ing his in o he in eg al exp ession o
A.1 yields he ela ion in Co olla y 1 o he main ex .
A.4 A.4 Consis ency wi h F iedmann Equa ions
Inco po a ing in o ma ion acuum ene gy in o he F iedmann equa ion:
H2( ) = 8πG
3ρm( ) + ρ ( ) + ρin o( )−k
a2( ),
(61)
whe e
k
is spa ial cu a u e. Since
ρin o( )
is a unc ion ob ained by in eg a ing
Pin o( )
, as
long as
Pin o( )
is smoo h on la ge scales and sa ises app op ia e g ow h condi ions, i will no
in oduce apid oscilla ions o ins abili ies ha iola e obse a ional cons ain s.
I should be emphasized ha his model does no claim ha all da k ene gy is composed o
in o ma ion acuum ene gy, bu p o ides an in o ma ion- heo e ic o igin o pa o all o
Λe ( )
.
Quan i a i e  ing equi es conside ing
ρin o
oge he wi h o he possible dynamical da k ene gy
componen s.
B Appendix B: Linea S abili y Analysis o Red Queen Dynamics
Model
This appendix p o ides de ails o he linea s abili y analysis o he wo-species Red Queen
dynamics model in Theo em 2.
18
B.1 B.1 Solu ion o Equilib ium Poin s
Conside he sys em:
dNi
d =Ni iCi−di−γ(N1+N2), i = 1,2,
(62)
dCi
d =αiNi−βiCi.
(63)
Equilib ium poin s sa is y:
N∗
i iC∗
i−di−γ(N∗
1+N∗
2)= 0, αiN∗
i−βiC∗
i= 0.
(64)
Fo non- i ial equilib ium poin s, we equi e
N∗
i>0, C∗
i>0
, so we mus ha e:
C∗
i=αi
βi
N∗
i,
(65)
Subs i u ing back, we ge :
i
αi
βi
N∗
i−di−γ(N∗
1+N∗
2)=0.
(66)
This gi es wo linea equa ions:
 1α1
β1−γN∗
1−γN∗
2=d1,
(67)
−γN∗
1+ 2α2
β2−γN∗
2=d2.
(68)
I he coecien ma ix
M= 1α1
β1−γ−γ
−γ 2α2
β2−γ!
(69)
is in e ible, hen
N∗
1
N∗
2=M−1d1
d2,
(70)
and subsequen ly
C∗
i=αi
βi
N∗
i.
(71)
Requi ing
N∗
i>0, C∗
i>0
es ic s he pa ame e space, bu can gene ally be sa ised wi hin
physically easonable pa ame e anges.
B.2 B.2 Cons uc ion o Jacobian Ma ix
Le
x= (N1, N2, C1, C2)T
. The componen s o he ec o eld
F(x)
a e:
F1=N1 1C1−d1−γ(N1+N2),
(72)
F2=N2 2C2−d2−γ(N1+N2),
(73)
F3=α1N1−β1C1,
(74)
F4=α2N2−β2C2.
(75)
The elemen s o he Jacobian ma ix
J
a e
Jij =∂Fi/∂xj
. A he equilib ium poin
x∗
, i s
non-ze o elemen s a e:
19

Fo
F1
:
∂F1
∂N1
= 1C∗
1−d1−γ(2N∗
1+N∗
2),∂F1
∂N2
=−γN∗
1,∂F1
∂C1
= 1N∗
1.
(76)
Using he equilib ium condi ion
1C∗
1−d1−γ(N∗
1+N∗
2) = 0
, his simplies o:
∂F1
∂N1
=−γN∗
1.
(77)

Fo
F2
:
∂F2
∂N2
= 2C∗
2−d2−γ(N∗
1+ 2N∗
2) = −γN∗
2,
(78)
∂F2
∂N1
=−γN∗
2,∂F2
∂C2
= 2N∗
2.
(79)

Fo
F3
:
∂F3
∂N1
=α1,∂F3
∂C1
=−β1.
(80)

Fo
F4
:
∂F4
∂N2
=α2,∂F4
∂C2
=−β2.
(81)
Thus, he Jacobian ma ix a he equilib ium poin is:
J=



−γN∗
1−γN∗
1 1N∗
10
−γN∗
2−γN∗
20 2N∗
2
α10−β10
0α20−β2




.
(82)
B.3 B.3 Cha ac e is ic Polynomial and Rou hHu wi z C i e ion
The cha ac e is ic polynomial is:
de (λI −J) = λ4+a1λ3+a2λ2+a3λ+a4,
(83)
whe e coecien s
ak
can be ob ained by di ec expansion. To simpli y no a ion, le :
Ai=γN∗
i, Bi= iN∗
i, Ci=αi, Di=βi.
(84)
Then:
a1= 2(A1+A2) + D1+D2,
(85)
a2= (A1+A2)2+ 2(A1+A2)(D1+D2) + D1D2+B1C1+B2C2,
(86)
a3= (A1+A2)2(D1+D2)+(A1+A2)D1D2+(A1+A2)(B1C1+B2C2)+D1B2C2+D2B1C1,
(87)
a4= (A1+A2)2D1D2+ (A1+A2)D1B2C2+ (A1+A2)D2B1C1.
(88)
The Rou hHu wi z c i e ion s a es ha all eigen alues ha e nega i e eal pa s i and only
i he ollowing condi ions hold: 1.
a1>0
, 2.
a1a2−a3>0
, 3.
(a1a2−a3)a3−a2
1a4>0
, 4.
a4>0
.
In he cu en model, all
Ai, Di, Bi, Ci>0
, so
a1, a2, a4>0
au oma ically hold. S abili y
c i ically depends on he second and hi d condi ions. Th ough algeb aic simplica ion, he
second condi ion can be w i en as:
a1a2−a3=K0−K1,
(89)
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whe e
K0
is a posi i e e m con aining only
Ai, Di
, and
K1
is a e m ela ed o
BiCi
. Clea ly,
when
B1C1+B2C2
is sucien ly la ge, i.e., when he sum o
iαiN∗
i
is sucien ly la ge,
K1
can
exceed
K0
, causing
a1a2−a3<0
, hus iola ing he second Hu wi z condi ion. This indica es
he exis ence o a c i ical su ace; on one side, he equilib ium is s able, and on he o he , i is
uns able.
Simila ly, he hi d condi ion can be w i en as:
(a1a2−a3)a3−a2
1a4=L0−L1,
(90)
whe e he highes o de e m in
L1
is p opo ional o
(B1C1+B2C2)2
. When
B1C1+B2C2
is
la ge, his quan i y can also become nega i e.
Compa ing
BiCi= iαi(N∗
i)2
wi h
A2
i=γ2(N∗
i)2
and
D2
i=β2
i
, we can de i e a simple
sucien condi ion:
1α1N∗
1+ 2α2N∗
2> β2
1+β2
2+ 2γ 1C∗
1+ 2C∗
2,
(91)
Unde his condi ion, a leas one Hu wi z condi ion is iola ed, and he equilib ium poin is
uns able. This gi es he ins abili y condi ion in Theo em 2 o he main ex .
B.4 B.4 Hop Bi u ca ion and Exis ence o Limi Cycles
When pa ame e s a y con inuously such ha a Hu wi z condi ion c osses ze o om posi i e,
co esponding eigen alues will c oss he imagina y axis. I exac ly one pai o conjuga e complex
eigen alues c osses he imagina y axis while o he eigen alues s ill ha e nega i e eal pa s, he
sys em unde goes a Hop bi u ca ion, gene a ing s able o uns able limi cycles.
In he cu en model, since he sys em has a ou -dimensional s a e space and a simple
coupling s uc u e, pa ame e pe u ba ions sa is ying gene al posi ion condi ions will ypically
lead o his ypical scena io. Specic pa ame e s can be e ied by nume ical calcula ion, which
is no epea ed he e. Impo an ly, he e exis s an open dense se o pa ame e s such ha he
in e nal equilib ium poin is uns able and a leas one limi cycle o mo e complex a ac o
exis s. This suppo s he conclusion in Theo em 2 ega ding "Red Queen dynamics leading o
sus ained non-equilib ium."
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