Spectral Windowing Unified Theory of Cosmological Constant and Dark Energy\\ \large Vacuum Energy Density in Unified Time Scale, Matrix Universe and QCA Universe

Spec al Windowing Unied Theo y o Cosmological
Cons an and Da k Ene gy
Vacuum Ene gy Densi y in Unied Time Scale, Ma ix Uni e se and QCA
Uni e se
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 19, 2025
Abs ac
Unde he amewo k o unied ime scale, phasespec al shi densi y o s a es
chain, and bounda y ime geome y, we p o ide a spec al windowing unied o -
mula ion o he cosmological cons an and da k ene gy, implemen ing a disc e ized
e sion in Ma ix Uni e se and Quan um Cellula Au oma on (QCA) Uni e se.
Fi s , on e en-dimensional asymp o ically hype bolic o con o mally compac
geome ies and s a ic pa ch de Si e backg ounds sa is ying ela i e ace class and
good sca e ing assump ions, we u ilize Bi manK ein and Li shi sK ein ace o -
mulas o uni y gene alized sca e ing phase de i a i e, spec al shi unc ion de i a-
i e, and ela i e densi y o s a es (DOS) in equency a iable in o a scale densi y
κ(ω) = φ′(ω)/π = ∆ρω(ω) = (2π)−1 Q(ω)
, whe e
Q(ω)
is he Wigne Smi h
g oup delay ma ix. P o ided he loga i hmic equency window ke nel sa ises
Mellin anishing condi ions, we es ablish a windowed Taube ian heo em: he ni e
pa o he small-
s
hea ke nel die ence is equi alen o he loga i hmic window
a e age o
Θ′(ω)
a scale
µ∼s−1/2
, he eby comple ely ew i ing acuum ene gy
densi y eno maliza ion as a windowed in eg al o
κ(ω)
.
Second, in he Ma ix Uni e se ep esen a ion, iewing FRW and de Si e uni-
e ses as sca e ing backg ounds con aining ho izon channels, we cons uc he cos-
mological sca e ing ma ix
Scos(ω)
and i s scale densi y
κcos(ω)
and window ke nel
Ξcos(ω)
, p o ing ha he eec i e cosmological cons an inc emen
Λe (µ)−Λe (µ0)
can be exp essed as he loga i hmic equency windowed spec al in eg al o he DOS
die ence be ween Physical Uni e se and Re e ence Uni e se, hus educing he
cosmological cons an p oblem o a ela i e spec al p oblem unde unied ime
scale.
Thi d, wi hin he Uni e se QCA objec
UQCA = (Λ,Hcell,A, α, ω0)
, eplacing
con inuous spec um
ω
wi h quasi-ene gy spec um
εj(k)
, we dene he ela i e
band densi y
∆ρj(k)
o QCA Uni e se and Re e ence QCA, cons uc ing he dis-
c e e windowed o mula
Λe (µ) = PjRBZ Wµ(εj(k)) ∆ρj(k) ddk
. Unde condi ions
whe e high-ene gy bands sa is y symme ic pai ing and in e -band ha monious sum
ules, we p o e ha con ibu ions o
Λe
om high-ene gy egions a e exponen ially
o powe -law supp essed a e windowing, lea ing only ni e esiduals con ibu ed
by low-ene gy mass h esholds and opological modes wi hin he edshi window,
yielding a na u al supp ession es ima e
Λe ∼m4
IR(mIR/MUV)γ
wi h
γ > 0
.
1
Finally, in e acing he spec al windowingband s uc u e mechanism wi h un-
ning acuum models in cu ed space ime QFT, we cons uc he da k ene gy equi -
alen equa ion o s a e
wde(z)≈ −1+δw(z)
wi hin he unied ime scale amewo k,
whe e
δw(z)
is con olled by he slow e olu ion o
Ξ(ω)
in he co esponding e-
quency band, compa ible wi h cu en obse a ional cons ain s o o e all close o
w=−1
allowing small ampli ude e olu ion.
O e all, his pape es a es why he cosmological cons an is a smalle han
M4
Pl
 as wha kind o windowed sum ule he ela i e DOS sa ises unde unied
ime scale, p o iding a sel -consis en spec al heo e ical amewo k o discussing
he magni ude and unning beha io o da k ene gy uni o mly wi hin Ma ix Uni-
e se and QCA Uni e se. The cosmological cons an p oblem is hus embedded in o
he b oade p og am o unied ime scalesca e ingdisc e e uni e se.
Keywo ds:
Cosmological cons an ; Da k ene gy; Unied ime scale; Sca e ing phase;
Spec al shi unc ion; Densi y o s a es; Hea ke nel; Taube ian heo em; Ma ix uni-
e se; Quan um cellula au oma on; Vacuum ene gy densi y; Running acuum model
1 In oduc ion & His o ical Con ex
1.1 Cosmological Cons an , Da k Ene gy, and Obse a ional Pic-
u e
In Eins ein eld equa ions
Gµν + Λgµν = 8πGTµν,
(1)
in oducing he cosmological cons an
Λ
allows desc ibing de Si e o an ide Si e so-
lu ions wi h cons an cu a u e a he classical geome y le el; in eec i e eld heo y
language,
Λ
co esponds o acuum ene gy densi y
ρΛ= Λ/(8πG)
, wi h p essu e sa is y-
ing
pΛ=−ρΛ
, i.e., equa ion o s a e pa ame e
w=−1
. The s anda d
Λ
CDM model uses
cons an
Λ
as he simples explana ion o he uni e se's la e- ime accele a ed expansion.
Based on mul iple obse a ions including cosmic mic owa e backg ound, Type Ia su-
pe no ae, ba yon acous ic oscilla ions, and weak lensing, cu en join analyses o da k
ene gy densi y and equa ion o s a e indica e ha nea ly wo- hi ds o he uni e se's o al
ene gy densi y is con ibu ed by da k ene gy, and he eec i e equa ion o s a e
wde
is
highly close o
−1
, wi h de ia ion
δw =wde + 1
ypically cons ained wi hin
O(10−1)
in join  s. La es e iews show mul iple independen da ase s emain compa ible wi h
w=−1
a
2σ
le el, bu some analyses a o a sligh ly phan om-like equa ion o s a e
w≲−1
, a end po en ially ela ed o Hubble ension and
σ8
ension.
In a comp ehensi e e iew o da k ene gy equa ion o s a e, Escamilla e al. sys em-
a ically compa ed cons ain s on a ious pa ame e ized o ms
w(z)
, no ing ha cu en
da a s ill p o ide only weak quan i a i e limi s on edshi e olu ion o
w(z)
, bu ce -
ain da a combina ions indeed a o e olu ion scena ios sligh ly de ia ing om
−1
. This
lea es obse a ional oom o whe he he cosmological cons an is s ic ly cons an .
1.2 Cosmological Cons an P oblem and Running Vacuum Ap-
p oach
F om a quan um eld heo y pe spec i e, ee eld acuum ze o-poin ene gy wi h UV
cu o
MUV
o mally yields
ρ ac ∼M4
UV
. Taking
MUV ∼MPl
o ypical pa icle physics
2
scales, compa ed o obse ed
ρobs
Λ∼10−120M4
Pl
, he e is a gap o
60 − −120
o de s o
magni ude, cons i u ing he co e o he cosmological cons an p oblem.
In cu ed space ime QFT, acuum ene gy densi y can be abso bed in o he ba e
Λ
e m ia eno maliza ion, bu his p ocess does no p o ide an in e nal mechanism
o why he o al acuum ene gy a e eno maliza ion lea es exac ly a iny posi i e
numbe . Recen Running Vacuum Models (RVM) p opose ha in he con ex o cosmic
expansion, acuum ene gy densi y should be a unc ion slowly e ol ing wi h cu a u e
o Hubble scale, e.g.,
ρ ac(H) = ρ0
ac +3ν
8πG(H2−H2
0) + ··· ,
(2)
whe e
ν
is a small dimensionless coecien . By implemen ing asymp o ic eno maliza ion
in cu ed space ime, pa o he dange ous
m4
o de e ms can be eec i ely emo ed,
making la e- ime uni e se acuum ene gy exhibi weak unning beha io wi hou in o-
ducing ex a scala eld deg ees o eedom. RVM- ype models ha e shown po en ial in
alle ia ing
H0
and
σ8
ensions.
Ne e heless, mos discussions emain a he le el o assuming some unc ional o m
o
ρ ac(H)
and  ing wi h obse a ions, lacking a unied way o na u ally de i e unning
acuum beha io and explain he small alue o acuum ene gy s a ing om spec al
and sca e ing s uc u es.
1.3 Spec al and Sca e ing Pe spec i e: Phase, Spec al Shi ,
and DOS
In ma hema ical physics, a se ies o p o ound connec ions exis be ween pai s o sel -
adjoin ope a o s
(H, H0)
and hei sca e ing ma ix
S(ω)
, spec al shi unc ion
ξ(ω)
,
and densi y o s a es die ence
∆ρ(ω)
. The Bi manK ein o mula links sca e ing de e -
minan wi h spec al shi unc ion, and Li shi sK ein ace o mula exp esses he ace
o unc ion
(H)− (H0)
as an in eg al o e he spec al shi unc ion. Guilla mou
and collabo a o s u he es ablished gene alized K ein o mulas and spec al p ope ies
o sca e ing ope a o s on asymp o ically hype bolic and con o mally compac Eins ein
mani olds, allowing exac alignmen o KV de e minan phase wi h gene alized spec al
shi unc ion.
On open sca e ing mani olds, Dya lo u ilized classical escape a es and expansion
a es o gi e ened es ima es o Weyl- ype asymp o ics and emainde e ms o sca -
e ing phase, showing high-ene gy in o ma ion in sca e ing phase can be con olled by
classical dynamical sys em in a ian s. Vasy de eloped sys ema ic mic olocal analysis
me hods on con o mally compac asymp o ically hype bolic spaces, achie ing high-ene gy
esol en con inua ion and es ima ion o Laplace ope a o s, p o iding con ol ools o
linking hea ke nel wi h sca e ing spec um.
These esul s indica e ha unde app op ia e geome ic assump ions, quan i ies on
he spec al and sca e ing side (like sca e ing phase de i a i e) can be p ecisely con-
nec ed o quan i ies on he geome y and hea ke nel side (like hea ke nel die ence
and SeeleyDeWi coecien s), p o iding a na u al en y poin o ew i ing he cosmo-
logical cons an using spec al quan i ies.
3
1.4 Quan um Cellula Au oma a and Disc e e Uni e se
Quan um Cellula Au oma a (QCA), as disc e e ime uni a y e olu ion models wi h lo-
cali y, ansla ion in a iance, and ni e p opaga ion adius, ha e been sys ema ically
p o en o eco e ee o in e ac ing quan um eld heo ies in he con inuous limi , in-
cluding Di ac elds, scala elds, and e en
(1+1)
D and
(3+1)
D elec odynamics. These
wo ks show ha unde na u al condi ions like locali y, Lo en z symme y app oxima ion,
and gauge in a iance, s anda d QFT dispe sion ela ions and p opaga ion p ope ies can
be eco e ed om simple disc e e local ules.
F om his pe spec i e, abs ac ing he uni e se as a whole in o a QCA objec
UQCA
sa is ying ce ain axioms, eco e ing gene al ela i i y and eld heo y in i s con inu-
ous limi , and hen embedding he cosmological cons an and da k ene gy p oblem in o
QCA's spec al and band s uc u e, becomes a na u al unied app oach: he uni e se
i sel is disc e e, while con inuous geome y and eld heo y a e me ely i s la ge-scale
app oxima ions.
1.5 Unied Time Scale and Spec al Windowing S a egy o This
Pape
P io wo k has cons uc ed he unied ime scale mo he scale
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω),
(3)
whe e
φ(ω)
is o al sca e ing hemi-phase,
ρ el(ω)
ela i e densi y o s a es, and
Q(ω)
Wigne Smi h g oup delay ma ix. This mo he scale unies sca e ing phase g adien ,
ela i e DOS, and g oup delay ace in o a single scale densi y, p o iding a unied sou ce
o dening unied ime pa ame e s ac oss a ious geome ic and physical scena ios.
Based on his, he goals o his pape a e:
1. On con inuous geome ies sa is ying app op ia e sca e ing and geome ic assump-
ions, es ablish a windowed Taube ian heo em p ecisely co esponding he ni e pa o
small-
s
hea ke nel die ence o loga i hmic window a e age o
Θ′(ω)
a scale
µ∼s−1/2
,
he eby ew i ing acuum ene gy eno maliza ion as loga i hmic equency windowed
spec al in eg al o
κ(ω)
.
2. In Ma ix Uni e se THEMATRIX ep esen a ion, iew FRW and de Si e uni-
e ses as sca e ing backg ounds, cons uc cosmological sca e ing ma ix
Scos(ω)
and
ela i e DOS, p o ing eec i e cosmological cons an
Λe (µ)
is equi alen o windowed
in eg al o DOS die ence be ween Physical Uni e se and Re e ence Uni e se.
3. In Uni e se QCA objec , eplace con inuous spec um wi h band s uc u e
εj(k)
,
dene QCA e sion ela i e band densi y and windowed acuum ene gy, analyze unde
wha band pai ing symme y and sum ule condi ions high-ene gy con ibu ions o
Λe
na u ally cancel, lea ing only small esiduals con olled by IR h esholds and opological
modes.
4. A eec i e cosmological le el, link windowed spec al exp ession o unning ac-
uum models and da k ene gy equa ion o s a e
wde(z)
, demons a ing s uc u al o igin
o
wde(z)≈ −1 + δw(z)
unde unied ime scale amewo k, and compa ing wi h cu en
obse a ional cons ain s.
These s eps collec i ely o m a mul i-laye unied s uc u e om sca e ing spec um
o da k ene gy, es a ing he cosmological cons an p oblem as a ela i e spec al p oblem
in Ma ix Uni e se and QCA Uni e se.
4
2 Model & Assump ions
2.1 Sca e ing Pai , Spec al Shi Func ion, and Unied Scale
Densi y
Conside a pai o sel -adjoin ope a o s
(H, H0)
dened on he same Hilbe space
H
.
Assume
H−H0
is a ela i e ace class pe u ba ion in app op ia e sense, ensu ing well-
dened xed-ene gy sca e ing heo y, and exis ence o Li shi sK ein ace o mula o
all Bo el bounded measu es
( (H)− (H0)) = ZR
′(λ)ξE(λ) dλ,
(4)
whe e
ξE(λ)
is spec al shi unc ion in ene gy a iable
λ
.
In equency a iable
ω≥0
, le
λ=ω2
, dene
ξ(ω) = ξE(ω2),
(5)
hen ela i e DOS in ene gy and equency ep esen a ions a e espec i ely
∆ρE(λ) = −ξ′
E(λ),
(6)
∆ρω(ω)=2ω∆ρE(ω2) = −∂ωξ(ω).
(7)
Le
S(ω)
be xed-ene gy sca e ing ma ix,
Q(ω) = −iS(ω)†∂ωS(ω)
Wigne Smi h
g oup delay ma ix. Bi manK ein o mula gi es
de S(ω) = exp−2πiξ(ω),
(8)
om which dene sca e ing de e minan phase
Θ(ω) = (2π)−1a g de S(ω),
(9)
and ha e
Θ′(ω) = −∂ωξ(ω) = ∆ρω(ω) = (2π)−1 Q(ω).
(10)
Unied ime scale densi y is dened as
κ(ω) = φ′(ω)
π= ∆ρω(ω) = 1
2π Q(ω),
(11)
whe e
φ(ω)
is o al sca e ing hemi-phase. In his pape , all cons uc ions ega ding
unied ime scale use
κ(ω)
as he sole scale sou ce.
2.2 Hea Ke nel Die ence, Geome ic Assump ions, and Sca -
e ing Ope a o
Le
H
be gene alized Laplace- ype ope a o on non-compac Riemann mani old
(X, g)
,
H0
on co esponding e e ence backg ound
(X0, g0)
, bo h ha ing same geome ic and po en-
ial s uc u e a inni y. Assume
(X, g)
and
(X0, g0)
a e e en-dimensional asymp o ically
hype bolic o con o mally compac Eins ein mani olds, sa is ying Guilla mouVasy e en
me ic condi ion, such ha esol en
(H−λ)−1
has good me omo phic con inua ion
5

on app op ia e Riemann su ace o
λ
plane, and sca e ing ope a o
S(λ)
is classical
pseudodie en ial ope a o nea c i ical line.
Unde hese assump ions, Guilla mou p o ed K ein- ype o mula be ween KV de e -
minan phase and gene alized spec al shi unc ion, while diagonal ace o hea ke nel
KH(s, x, y) = exp(−sH)(x, y)
(exp(−sH)−exp(−sH0)) = ∆K(s)
(12)
has s anda d SeeleyDeWi asymp o ic expansion as
s→0+
, whose ni e pa can be
exp essed ia Mellin ans o m o sca e ing phase.
This pape specically ocuses on symme ic e en-dimensional cases and s a ic pa ch
de Si e backg ounds, whe e connec ion be ween sca e ing phase and hea ke nel di -
e ence is mos concise, sui able o cons uc ing loga i hmically windowed Taube ian
heo ems.
2.3 Loga i hmic F equency Window Ke nel and Taube ian Con-
di ions
To con e ni e pa o small-
s
hea ke nel die ence in o loga i hmic a e age in e-
quency domain, in oduce a amily o loga i hmic equency window ke nels
W(ln(ω/µ)),
(13)
whe e
µ > 0
is spec al scale,
W∈C∞
0(R)
is smoo h compac suppo unc ion, sa is ying
ollowing Mellin anishing condi ions:
1.
ZR
W(u) du= 0
; 2.
ZR
e2nuW(u) du= 0
o se e al low in ege s
n
.
These condi ions ensu e window ke nel cancels co esponding dominan singula e ms
in hea ke nel as
ω→0
and
ω→ ∞
, ex ac ing ni e geome ic in o ma ion. Dene
loga i hmic a e age
⟨Θ′⟩W(µ) = ZR
Θ′(ω)W(ln(ω/µ)) d ln ω,
(14)
and u he dene spec al window ke nel
ΞW(µ) = ∂ln µ⟨Θ′⟩W(µ).
(15)
In Taube ian heo y, i
∆K(s)
and
Θ′(ω)
sa is y app op ia e egula i y and g ow h
condi ions, equi alence can be es ablished be ween ni e pa o small-
s
hea ke nel and
ΞW(µ)
unde hype bolic scaling
µ∼s−1/2
. This pape will p o ide a ni e-o de e sion
sucien o cosmological applica ions.
2.4 Cosmological Geome y and Re e ence Backg ound
In cosmological applica ions,
H
and
H0
co espond o wa e ope a o s on geome ic back-
g ounds o Physical Uni e se and Re e ence Uni e se espec i ely. Fo example:
1.
(M, g)
is FRW wi h cu a u e pe u ba ions o de Si e uni e se wi h s uc u e,
H
is LaplaceBel ami ope a o plus po en ial o scala o enso pe u ba ions. 2.
(M0, g0)
is smoo h, same- opology s uc u eless FRW o pu e de Si e backg ound,
H0
co esponding unpe u bed ope a o .
6
Re e ence backg ound should possess physical easonableness, e.g., same opology,
same cosmological cons an , bu no complex s uc u e. Spec al die ence
∆ρω
o physical
uni e se ela i e o e e ence uni e se cha ac e izes spec al de ia ion o eal s uc u e
on backg ound, i s windowed in eg al gi es eec i e inc emen o cosmological cons an .
2.5 Ma ix Uni e se and Cosmological Sca e ing Ma ix
In Ma ix Uni e se THEMATRIX amewo k, conside channel Hilbe space decom-
posed by equency
Hchan =M
∈VH ,
(16)
whe e
labels die en di ec ions, pola iza ions, cosmological modes, and ho izon chan-
nels. Cosmological sca e ing ma ix
Scos(ω)∈ B(Hchan)
(17)
encodes sca e ing p ocess om Re e ence Uni e se modes o Physical Uni e se modes.
I s de e minan phase
Θcos(ω)
, spec al shi unc ion
ξcos(ω)
, and ela i e DOS
∆ρcos(ω)
sa is y same chain ela ions as gene al sca e ing pai s.
Unied ime scale densi y in cosmological case is w i en as
κcos(ω) = 1
2π Qcos(ω),
(18)
whe e
Qcos(ω) = −iScos(ω)†∂ωScos(ω)
. This pape will use
κcos(ω)
o con ol spec al
windowed exp ession o cosmological cons an .
2.6 QCA Uni e se Objec and Band S uc u e
Uni e se QCA objec is dened as
UQCA = (Λ,Hcell,A, α, ω0),
(19)
whe e
Λ
is coun able connec ed g aph (usually
Zd
),
Hcell
ni e-dimensional Hilbe space
pe cell,
A
quasilocal
C∗
algeb a,
α:Z→Au (A)
ime e olu ion au omo phism wi h
ni e p opaga ion adius,
ω0
ini ial uni e se s a e.
In momen um space
k∈BZ
, single-s ep e olu ion ope a o
U
be decomposes as
U(k)∈U(N),
(20)
wi h eigendecomposi ion
U(k)|ψj,k⟩=e−iεj(k)∆ |ψj,k⟩,
(21)
whe e
N= dim Hcell
, quasi-ene gy
εj(k)∈(−π/∆ , π/∆ ]
. In con inuous limi
∆ →0
,
dispe sion ela ion
εj(k)≈Ej(k)∆
, whe e
Ej(k)
is ene gy spec um o co esponding
con inuous eld heo y.
Dene single-band DOS
ρj(E) = ZBZ
δ(E−Ej(k)) ddk
(2π)d,
(22)
o al DOS
ρ(E) = Pjρj(E)
. Selec ing Re e ence QCA
U0(k)
and i s spec um
{Ej,0(k)}
,
dene ela i e DOS
∆ρ(E) = ρ(E)−ρ0(E),
(23)
and i s band decomposi ion
∆ρj(k)
. These quan i ies co espond o con inuous sca e ing
heo y in con inuous limi .
7
3 Main Resul s
This sec ion gi es main heo ems o his pape , p oo de ails in Sec ion 4 and Appendix.
3.1 Windowed Taube ian Theo em and Scale Densi y Fo mula-
ion
Theo em 3.1
(Windowed Taube ian Theo em)
.
Le
(H, H0)
sa is y geome ic and sca -
e ing assump ions o Sec ion 2.2, le
∆K(s) = (e−sH −e−sH0)
(24)
be hea ke nel die ence,
Θ(ω)
sca e ing de e minan phase,
Θ′(ω)=∆ρω(ω) = (2π)−1 Q(ω)
.
Take a amily o loga i hmic window ke nels
W
sa is ying Mellin anishing condi ions o
Sec ion 2.3, dene
ΞW(µ) = ZR
ωΘ′′(ω)W(ln(ω/µ)) d ln ω.
(25)
Then he e exis s hype bolic scaling
µ∼s−1/2
and cons an s
C, γ > 0
such ha as
s→0+
,
FPs→0∆K(s) = κΛΞW(µ) + O(sγ),
(26)
whe e
FP
deno es ni e pa ,
κΛ
no maliza ion cons an depending only on window ke nel
choice. In o he wo ds, ni e pa o small-
s
hea ke nel die ence is Taube ian equi alen
o loga i hmic window a e age o unied ime scale densi y
κ(ω)
.
3.2 Windowed Unied Exp ession o Cosmological Cons an
Theo em 3.2
(Windowed Cosmological Cons an Mo he Fo mula)
.
Unde condi ions
o Theo em 3.1, le
Λe (µ)
be eno malized alue o cosmological cons an pa ame e in
eec i e eld heo y ac ion a obse a ion scale
µ
, hen he e exis s spec al window ke nel
Ξ(µ)
such ha
∂ln µΛe (µ) = κΛΞ(µ),
(27)
explici ly
Ξ(µ) = ZR
ωΘ′′(ω)W(ln(ω/µ)) d ln ω,
(28)
and o any e e ence scale
µ0
,
Λe (µ)−Λe (µ0) = Zµ
µ0
Ξ(ω) d ln ω.
(29)
He e
Ξ(ω)
has dimension
L−2
, iewable as cosmological cons an spec al window ke nel,
ully de e mined by unied ime scale densi y
κ(ω)
and sca e ing phase.
3.3 Cosmological Sca e ing and Rela i e DOS in Ma ix Uni-
e se
Theo em 3.3
(Rela i e DOS Fo mula ion in Ma ix Uni e se)
.
In Ma ix Uni e se
THEMATRIX ep esen a ion, ake Physical Uni e se
(M, g)
and Re e ence Uni e se
8
(M0, g0)
and hei co esponding sca e ing ma ices
Scos(ω)
and
Scos,0(ω)
. Dene ela-
i e cosmological sca e ing ma ix
S el(ω) = Scos(ω)Scos,0(ω)−1,
(30)
and ela i e DOS
∆ρcos(ω) = ∆ρω(ω;H, H0).
(31)
Unde condi ions o Theo em 3.2,
Λe (µ)−Λe (µ0) = Zµ
µ0
Ξcos(ω) d ln ω,
(32)
whe e
Ξcos(ω) = ZR
ωΘ′′
el(ω)W(ln(ω/µ)) d ln ω,
(33)
Θ el(ω) = (2π)−1a g de S el(ω)
, and
Θ′
el(ω) = ∆ρcos(ω) = (2π)−1 Q el(ω).
(34)
Thus
Λe (µ)
can be in e p e ed as loga i hmic equency windowed in eg al o DOS di -
e ence be ween cosmological sca e ing ma ices o Physical and Re e ence Uni e ses.
3.4 Disc e e Windowed Fo mula and High-Ene gy Supp ession
in QCA Uni e se
Theo em 3.4
(Windowed Vacuum Ene gy Fo mula in QCA Uni e se)
.
In Uni e se
QCA objec
UQCA
, le Physical QCA and Re e ence QCA ha e ene gy spec a
{Ej(k)}
and
{Ej,0(k)}
espec i ely, dene single-band ela i e DOS
∆ρj(k) = δ(E−Ej(k)) −δ(E−Ej,0(k)).
(35)
Le
Wµ(E)
be ene gy weigh unc ion mapped om con inuous window ke nel
W(ln(ω/µ))
,
hen he e exis s no maliza ion cons an
CW
such ha
Λe (µ) = CWX
jZBZ Wµ(Ej(k)) ∆ρj(k) ddk,
(36)
consis en wi h con inuous spec um exp ession o Theo em 3.2 in con inuous limi
∆ →
0
.
Theo em 3.5
(QCA High-Ene gy Supp ession Mechanism)
.
Unde condi ions o Theo-
em 3.4, i high-ene gy bands sa is y ollowing wo ypes o condi ions:
1. Band S uc u e Symme ic Pai ing: Exis s band pai
(j, j′)
, o
|E| ≥ Ec
Ej′(k)≈ −Ej(k),
(37)
∆ρj′(E)≈∆ρj(−E),
(38)
and window weigh app oxima ely e en symme ic in high-ene gy egion
Wµ(E)≈ Wµ(−E)
.
2. In e -Band Ha monious Sum Rule: Exis s UV scale
EUV
such ha
ZEUV
0
E2∆ρ(E) dE= 0,
(39)
9
6.1 Nume ical Recons uc ion o Cosmological Sca e ing Spec-
um
In Ma ix Uni e se pe spec i e, elemen s o cosmological sca e ing ma ix
Scos(ω)
es-
sen ially co espond o sca e ing ampli udes o die en modes on FRW o de Si e
backg ound. Can be nume ically econs uc ed ia ollowing s eps:
1. Selec a amily o mode unc ions o scala o enso pe u ba ion equa ions, nu-
me ically sol e sca e ing ampli udes on gi en backg ound
(M, g)
and e e ence back-
g ound
(M0, g0)
. 2. Cons uc equency-dependen sca e ing ma ices
Scos(ω)
and
Scos,0(ω)
ia nume ical linea algeb a, calcula e ela i e ma ix
S el(ω)
. 3. Es ima e sca -
e ing phase
Θ el(ω)
and de i a i e
Θ′
el(ω)
on disc e e equency g id, ob aining ela i e
DOS
∆ρcos(ω)
. 4. Choose loga i hmic window ke nel
W
sa is ying Taube ian condi ions,
calcula e nume ical es ima es o spec al window ke nel
Ξcos(µ)
and
Λe (µ)
.
This nume ical wo kow p o ides di ec pa h o e i y alidi y o windowed Taube ian
heo em and specic shape o spec al window ke nel
Ξ(ω)
.
6.2 Design and Simula ion o QCA Spec al S uc u e
In QCA uni e se scheme, need o specically cons uc QCA models sa is ying band
pai ing and sum ule condi ions. Enginee ing s eps:
1. On 1D o 2D la ice, s a ing om known cons uc ions o Di ac- ype and gauge
eld QCA, add ex a in e nal deg ees o eedom and coupling pa ame e s, making ene gy
spec um exhibi
E↔ −E
symme ic pai ing in high-ene gy egion. 2. Calcula e DOS
ia nume ical diagonaliza ion and B illouin zone in eg a ion, e i y sum ule condi ions
be ween high-ene gy bands. 3. E ol e QCA on quan um simula ion o classical HPC
pla o ms, measu e quasi-ene gy spec um dis ibu ion and beha io o band s uc u e
wi h pa ame e a ia ion, explo e nume ical magni ude o
Λe
unde die en IR and UV
scales. 4. Compa e ob ained windowed acuum ene gy wi h
ν
alues  ed om RVM,
es i QCA spec al s uc u e can na u ally p oduce pa ame e egion
|ν| ≪ 1
.
6.3 Spec al Windowing Me hod in Cosmological Da a Analysis
In ac ual cosmological da a p ocessing, can a emp in oducing loga i hmic equency
windowing me hod:
1. In da a analysis o CMB, BAO, and Type Ia supe no ae, eplace pa ame e iza ion
om di ec ly assuming o m o
ρde(z)
o simple pa ame e iza ion o
Ξ(ω)
, e.g., piecewise
cons an o low-o de polynomial. 2. Encode ela ion be ween
Ξ(ω)
,
Λe (µ)
, and
wde(z)
in o cosmological pa ame e in e ence chain, making  ed pa ame e s di ec ly desc ibe
spec al window ke nel a he han equa ion o s a e. 3. Compa e Bayesian e idence and
pa ame e co ela ion o di ec 
w(z)
 s 
Ξ(ω)
hen de i e
w(z)
, es i spec al
windowing amewo k cap u es da a p e e ence mo e na u ally.
This scheme can be iewed enginee ing-wise as a spec al a iable coo dina e ans-
o ma ion o exis ing cosmological MCMC amewo ks, po en ially oe ing ad an ages
in pa ame e co ela ion and in e p e abili y.
16

7 Discussion ( isks, bounda ies, pas wo k)
7.1 Risks and Limi a ions
The spec al windowing unied amewo k p oposed in his pape elies on se e al non-
i ial assump ions:
1. Geome ic and Sca e ing Assump ions: Requi e
(M, g)
and
(M0, g0)
all in o
asymp o ically hype bolic o con o mally compac geome ic ca ego ies, and sa is y e en
me ic condi ion and good sca e ing heo y, which has no been igo ously p o en a
ull space ime le el o eal uni e se. 2. Window Ke nel and Taube ian Condi ions:
Windowed Taube ian heo em elies on high-ene gy es ima es o esol en and g ow h
bounds o sca e ing phase. Cu en ma hema ical li e a u e esul s on space imes like
Ke de Si e a e incomple e, gene alizing o uni e se wi h complex ma e dis ibu ion
and non-s a iona y backg ound equi es mo e wo k. 3. Realiza ion o QCA Sum Rule:
Cons uc ing spec al s uc u es sa is ying band pai ing and sum ule in specic QCA
models is a non- i ial enginee ing p oblem, equi ing ca e ul design unde cons ain s o
locali y, causali y, and symme y.
These isks imply many conclusions o his pape should cu en ly be iewed as s uc-
u al in e ences unde explici assump ions, a he han heo ems igo ously e ied in
all physical scena ios.
7.2 Rela ion o Exis ing Wo k
Rega ding cosmological cons an p oblem, as li e a u e explo es weak unning beha io
o acuum ene gy densi y wi h
H
o cu a u e ia eno maliza ion and unning acuum
models. In compa ison, con ibu ion o his pape lies in:
1. Explici ly ew i ing cosmological cons an p oblem as ela i e spec al p oblem
on sca e ing phasespec al shi DOS chain, con olling all scale dependence wi h uni-
ed ime scale densi y
κ(ω)
. 2. In oducing loga i hmic equency window ke nel and
ni e-o de Taube ian heo em, uni ying ni e pa o hea ke nel die ence wi h sca e -
ing spec al in o ma ion, gi ing
Λe (µ)
exp ession ully accoun ed o in dimension and
a iable. 3. Re ealing s uc u al mechanism o na u al supp ession o acuum ene gy
ia high-ene gy spec al pai ing and in e -band sum ule in QCA uni e se amewo k,
a ibu ing magni ude o
Λe
o powe -law a io be ween IR and UV scales, a he han
ne- uning o indi idual deg ees o eedom.
Rega ding unica ion o QCA and QFT, his pape ollows and ex ends sys ema ic
analysis o QCA con inuous limi s by Fa elly, BisioD'A iano, Sellapillay, and B un,
in oducing disc e e e sion o cosmological cons an windowing on his basis.
7.3 Fu u e Wo k Di ec ions
Fu u e wo k can p oceed along ollowing di ec ions:
1. On specic cosmological backg ounds (e.g., s uc u ed de Si e , non-a FRW),
u ilizing Vasy's mic olocal analysis me hods and Guilla mou's sca e ing ope a o heo y,
igo ously es ablish Taube ian heo ems sa is ying condi ions equi ed by his pape . 2.
Cons uc high-dimensional QCA models wi h clea QFT con inuous limi s, sys ema i-
cally explo e hei band s uc u e and DOS, nding na u al pa ame e egions sa is ying
sum ule condi ions. 3. In e ace spec al windowing amewo k wi h o he unied ime
17
scale ela ed esul s like black hole en opy, NullModula double co e , gene alized en-
opy condi ions, o ming a Uni e se Te minal Objec  s uc u e on la ge scale.
8 Conclusion
Based on unied ime scale, sca e ing phasespec al shi densi y o s a es chain, and
QCA uni e se axioms, his pape p o ides a sys ema ic spec al windowing es a emen
o cosmological cons an and da k ene gy p oblem. By es ablishing windowed Taube-
ian heo em in e en-dimensional asymp o ically hype bolic and de Si e geome ies,
equi alence is made be ween ni e pa o small-
s
hea ke nel die ence and loga i hmic
equency window a e age o sca e ing phase de i a i e, he eby ob aining a cosmologi-
cal cons an mo he o mula ully de e mined by unied scale densi y
κ(ω)
and window
ke nel
W
.
In Ma ix Uni e se ep esen a ion, cosmological sca e ing ma ix o Physical Uni e se
ela i e o Re e ence Uni e se gi es ela i e DOS, windowed in eg al ew i es cosmological
cons an as accumula ion o ela i e spec al quan i ies; in QCA Uni e se, symme ic
pai ing o band s uc u e and in e -band ha monious sum ule na u ally b ing cancella ion
o high-ene gy pa s and powe -law supp ession o acuum ene gy, making magni ude o
Λe
mainly con olled by a io o IR scale o UV scale.
Finally, connec ing spec al windowing esul s wi h unning acuum models and da k
ene gy obse a ional cons ain s shows unied ime scale amewo k can na u ally p o-
duce equi alen equa ion o s a e app oxima ing
wde(z)≈ −1
and possibly slowly e ol -
ing, compa ible wi h cu en da a. O e all, cosmological cons an p oblem ans o ms
om dicul p oblem o summing absolu e ze o-poin ene gies o windowed sum ule
p oblem o ela i e DOS unde unied ime scale, laying ounda ion o ealizing mo e
comple e unica ion in Ma ix Uni e se and QCA Uni e se.
Acknowledgemen s
The au ho s hank colleagues and anonymous e iewe s in ela ed elds o discussions
and sugges ions on sca e ing heo y, QCA con inuous limi s, and cosmological da a
analysis, which signican ly inuenced s uc u e and p esen a ion o his pape .
Code A ailabili y
Main esul s o his pape a e based on analy ical de i a ions. Nume ical e ica ion
in ol ing hea ke nel, sca e ing phase, and QCA DOS can be implemen ed on open
sou ce pla o ms using gene al nume ical linea algeb a lib a ies and spec al me hods.
Rele an example codes and nume ical sc ip s can be o ganized and made public in
subsequen wo k, o p o ided upon easonable eques .
Re e ences
[1] Pa icle Da a G oup, Da k Ene gy,
Re iew o Pa icle Physics
, 2024 upda e.
18
[2] J. Solà Pe acaula, The Cosmological Cons an P oblem and Running Vacuum in
he Expanding Uni e se,
Philosophical T ansac ions o he Royal Socie y A
380,
2022.
[3] J. Solà Pe acaula e al., Cosmological Cons an is-à- is Dynamical Vacuum: Bold
Challenging he
Λ
CDM,
In e na ional Jou nal o Mode n Physics A
31, 2016.
[4] J. de C uz Pé ez, J. Solà Pe acaula, Running Vacuum in B ansDicke Theo y: A
Possible Cu e o he
H0
and
σ8
Tensions,
As opa icle Physics
, 2024.
[5] L. A. Escamilla e al., The S a e o he Da k Ene gy Equa ion o S a e ci ca 2023,
Jou nal o Cosmology and As opa icle Physics
, 2024.
[6] C. Guilla mou, Gene alized K ein Fo mula, De e minan s and Selbe g Ze a Func-
ion o Con ex Co-Compac Mani olds,
Communica ions in Ma hema ical Physics
277, 2008.
[7] C. Guilla mou, Spec al Cha ac e iza ion o Poinca éEins ein Mani olds,
Jou nal
o Die en ial Geome y
83, 2009.
[8] A. Vasy, Mic olocal Analysis o Asymp o ically Hype bolic Spaces and High-Ene gy
Resol en Es ima es,
MSRI Publica ions
60, 2012.
[9] S. Dya lo , Sca e ing Phase Asymp o ics wi h F ac al Remainde s,
Communica-
ions in Ma hema ical Physics
339, 2015.
[10] T. Fa elly, A Re iew o Quan um Cellula Au oma a,
Quan um
4, 368 (2020).
[11] G. M. D'A iano, P. Pe ino i, A. Bisio, F om Quan um Cellula Au oma a o Quan-
um Field Theo y, a ious wo ks, 20142017.
[12] K. Sellapillay e al., A Disc e e Rela i is ic Space ime Fo malism o 1+1 QED om
QCA,
Scien ic Repo s
12, 2022.
[13] T. A. B un, G. Chi ibella, C. M. Scandolo, Quan um Elec odynamics om Quan-
um Cellula Au oma a,
En opy
27, 2025.
A Spec al Da a, Sca e ing Ma ix, and K ein T ace
Fo mula
This appendix p o ides de ailed s a emen o spec al and sca e ing heo y backg ound
used in his pape .
A.1 Sel -Adjoin Pai s and Spec al Shi Func ion
Le
H
and
H0
be sel -adjoin ope a o s sa is ying
(H−i)−1−(H0−i)−1
is ace class.
Then o any Schwa z unc ion
,
( (H)− (H0)) = ZR
′(λ)ξE(λ) dλ,
(79)
19
whe e
ξE(λ)
is spec al shi unc ion. De i a i e o spec al shi unc ion a almos all
λ
gi es ela i e DOS:
∆ρE(λ) = −ξ′
E(λ)
.
In p esence o xed-ene gy sca e ing heo y, sca e ing ma ix
S(λ)
and spec al shi
unc ion a e ela ed by Bi manK ein o mula:
de S(λ) = exp−2πiξE(λ).
(80)
On asymp o ically hype bolic and con o mally compac mani olds, Guilla mou ex ended
his ela ion o mo e gene al KV de e minan backg ound h ough in-dep h analysis o
sca e ing ope a o and ela i e de e minan , uni ying sca e ing de e minan phase wi h
gene alized spec al shi unc ion.
A.2 Wigne Smi h G oup Delay and Unied Scale Densi y
F equency de i a i e o sca e ing ma ix
S(ω)
gi es Wigne Smi h g oup delay ma ix
Q(ω) = −iS(ω)†∂ωS(ω).
(81)
Unde ace class condi ion, ace
Q(ω) = ∂ωa g de S(ω) = 2πΘ′(ω),
(82)
hus
Θ′(ω) = (2π)−1 Q(ω).
(83)
Unied ime scale densi y is dened as
κ(ω)=Θ′(ω)=∆ρω(ω).
(84)
In Ma ix Uni e se and QCA Uni e se, deni ion o
Q(ω)
can be gene alized o cos-
mological sca e ing ma ix and QCA sca e ing map, making unied ime scale span
con inuous and disc e e amewo ks.
B Technical De ails o Windowed Taube ian Theo em
B.1 Mellin T ans o m and Loga i hmic Window Ke nel
Mellin ans o m o unc ion
(ω)
dened as
M[ ](s) = Z∞
0
ωs−1 (ω) dω.
(85)
In oduc ion o loga i hmic window ke nel
W(ln(ω/µ))
makes
⟨ ⟩W(µ) = ZR
(ω)W(ln(ω/µ)) d ln ω
(86)
co espond in Mellin space o
⟨ ⟩W(µ) = 1
2πiZΓM[ ](s)c
W(s)µ−sds,
(87)
whe e
c
W(s) = ZR
esuW(u) du.
(88)
Mellin anishing condi ions ensu e
c
W(s)
has ze os a se e al low-o de poin s, cancelling
dominan singula e ms o hea ke nel expansion in esidue calcula ion, e aining only
ni e geome ic pa .
20
B.2 F om Sca e ing Phase o Hea Ke nel Fini e Pa
Subs i u ing
(ω)=Θ′(ω)
and
(ω)=∆ρω(ω)
, simple ela ion exis s be ween co e-
sponding
M[ ](s)
and LaplaceMellin ans o m o hea ke nel
∆K(s)
.
By choosing posi ions o
Γ
and
Γ′
, can simul aneously analyze small-
s
beha io o
∆K(s)
and la ge-
ω
beha io o
Θ′(ω)
on complex plane, ob aining ia Ha dyLi lewood
Ka ama a ype Taube ian heo em
FPs→0∆K(s)∼κΛΞW(µ),
(89)
and gi ing explici e o bounds.
In cosmological applica ions, ni e-o de Taube ian e sion is sucien : only need o
con ol e o up o ce ain ni e o de
sγ
.
C QCA Band Pai ing, Sum Rule, and Supp ession Ex-
ponen
C.1 Cons uc ion o Band Pai ing S uc u e
In specic QCA models, band pai ing can be achie ed ia ollowing s a egies:
1. Requi e local upda e ules o possess gene alized pa iclean ipa icle symme-
y and ime e e sal symme y in app op ia e sense, p oducing
E↔ −E
symme y
in ene gy spec um. 2. In oduce ex a
Z2
o
Z4
symme y in in e nal deg ees o ee-
dom, making high-ene gy bands appea in pai s, wi h coupling s uc u e au oma ically
sa is ying symme ic pai ing condi ion.
Such s uc u es pa ially appea in Di ac- ype and QED- ype QCA, equi ing u he
pa ame e uning and cons ain s o ob ain spec al s uc u es sa is ying assump ions o
Theo em 3.5.
C.2 Spec al Condi ion o Sum Rule
In e -band ha monious sum ule
ZEUV
0
E2∆ρ(E) dE= 0
(90)
can be unde s ood as a kind o  ela i e ene gy squa ed conse a ion: Physical QCA and
Re e ence QCA ha e same ene gy squa ed weigh ed densi y o s a es in UV egion.
In Di ac- ype models, i Re e ence QCA and Physical QCA die in UV egion only by
IR mass and ni e opological modes, his sum ule can be na u ally sa ised o achie ed
by adjus ing ni e high-ene gy coupling pa ame e s.
By nume ical  ing o DOS, can e i y app oxima e alidi y o sum ule, and es ima e
impac o i s de ia ion on
Λe
, de e mining magni ude o supp ession exponen
γ
.
C.3 Supp ession Exponen and Pa ame e Space
In ac ual calcula ion, mul i-scale analysis o QCA spec um yields
Λe (µ)≤C0E4
IR +C1E4
IREIR/EUVγ1+··· ,
(91)
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whe e
C0
ela es o low-ene gy DOS s uc u e,
γ1
o p ecision o sum ule. By sea ching
o egions in pa ame e space whe e
C0
is also signican ly educed, supp ession eec
can be u he enhanced, explaining obse a ionally iny
Λe
.
22