Na ie –S okes, Blu , and Blu iche sky Geome y
Smoo hness a Fini e Resolu ion
Aleksanda Pe išić
No embe 2025
Abs ac
We e isi he h ee–dimensional incomp essible Na ie –S okes equa ions om he iew-
poin o blu and Blu iche sky geome y. The Clay p oblem asks whe he smoo h solu ions
exis globally o all smoo h ini ial da a in he con inuum limi , wi h no con ol pa ame e .
Physically, howe e , luid lows a e ne e obse ed a in ini e esolu ion: unce ain y p in-
ciples, ini e ene gy, and ini e measu emen capaci y impose a nonze o blu scale on any
desc ip ion.
In his no e we make his scale explici . We in oduce a amily o blu ope a o s
{
B
ℓ}ℓ>0
(molli ie s o hea ke nels), de ine a scale–dependen Blu iche sky me ic on eloci y ields,
and show ha o any Le ay–Hop solu ion
u
and any
ℓ >
0, he blu ed ield
uℓ
=B
ℓu
is
smoo h and sol es a il e ed Na ie –S okes sys em wi h an explici ly con olled Reynolds
s ess. F om he pe spec i e o an obse e es ic ed o esolu ion
ℓ
, all singula i ies o
u
(i
hey exis ) a e hidden behind a ini e blu budge .
The message is wo old. Ma hema ically, we o malize he idea ha NS dynamics ac o
h ough a owe o blu –equi alence classes, each wi h smoo h ep esen a i es. Epis emo-
logically, we a gue ha he Clay exis ence and smoo hness p oblem li es en i ely a he
(physically una ainable) limi
ℓ↓
0, while all physically meaning ul s a emen s abou luid
lows can be posed and p o ed a ini e blu .
1 Classical Na ie –S okes and he Clay ques ion
We wo k on he o us
T3
o simplici y; he discussion ex ends o
R3
wi h s anda d echnical
modi ica ions.
1.1 The equa ion and weak solu ions
The incomp essible Na ie –S okes equa ions wi h iscosi y ν > 0 ead
∂ u+ (u· ∇)u+∇p=ν∆u,
di u= 0,(1)
wi h
u
(
, x
)
∈R3
he eloci y ield and
p
(
, x
) he p essu e. Gi en di e gence– ee ini ial da a
u0(x), one seeks ude ined o ≥0wi h u(0,·) = u0.
The s anda d global exis ence esul is due o Le ay.
De ini ion 1.1 (Le ay–Hop weak solu ion).A di e gence– ee
u0∈L2
(
T3
)admi s a Le ay–Hop
weak solu ion ui
•u∈L∞
loc([0,∞); L2(T3)) ∩L2
loc([0,∞); H1(T3)),
•uis weakly di e gence– ee and sa is ies (1) in he sense o dis ibu ions,
1
•usa is ies he global ene gy inequali y
1
2∥u( )∥2
L2+νZ
0
∥∇u(s)∥2
L2ds ≤1
2∥u0∥2
L2,∀ ≥0.
Theo em 1.2 (Le ay).Fo e e y di e gence– ee
u0∈L2
(
T3
) he e exis s a leas one Le ay–
Hop weak solu ion.
The Clay p oblem in his se ing asks:
Gi en smoo h di e gence– ee
u0
, is he e a global smoo h solu ion
u
o
(1)
, o can
singula i ies o m in ini e ime?
This is a pu ely Pla onic ques ion abou he con inuum PDE a in ini e esolu ion. I igno es
any epis emic o physical limi a ion on how inely he low can be known.
1.2 Physical e sus Pla onic smoo hness
Physically, lows a e obse ed h ough ini e– esolu ion measu emen s:
•space and ime a e sampled on g ids,
•de ices ha e ini e bandwid h and sensi i i y,
•
and a su icien ly small scales he con inuum model i sel b eaks down (molecula disc e e-
ness, quan um e ec s, he mal noise).
F om ha iewpoin i is na u al o eplace he absolu e no ion o smoo hness wi h a scale–
dependen one and ask:
A a gi en esolu ion
ℓ >
0, can any incomp essible low be ep esen ed by a smoo h
a a a ha ag ees wi h all obse ables accessible a ha esolu ion?
We o malize his ques ion using blu and Blu iche sky geome y. Concep ually, his is he
con inuum analogue o he blu –based ea men o Poinca é and Colla z: we inse an explici
esolu ion pa ame e and p o e heo ems a ixed blu ins ead o p e ending we can access he
ℓ= 0 wo ld.
Scope and wha we a e no claiming To a oid any ambigui y, le us s a e explici ly wha
his no e does no claim. We do no asse ha he Na ie –S okes exis ence and smoo hness
p oblem (in he Clay o mula ion) is ill-posed, meaningless, o independen o s anda d axioms.
We also do no p opose a p oo o ei he global egula i y o ini e- ime blow-up.
Ou s ance is pu ely epis emic and physical: o e e y ixed blu scale
ℓ >
0and ini e ime
ho izon
T
, he Le ay–Hop amewo k plus molli ica ion p oduces smoo h blu ed a a a s
uℓ
and
a ini e, ackable “blu budge ” ha cap u es all subg id e ec s ele an o an obse e ope a ing
a esolu ion
ℓ
. Wha e e he ul ima e ma hema ical s a us o he
ℓ→
0Clay ques ion, hese
ini e-blu s a emen s emain alid and al eady su ice o all physically ealizable measu emen s.
In his sense, he Clay p oblem is genuinely Pla onic: i is a sha p ques ion abou he ideal
con inuum limi a in ini e esolu ion. The blu -based pic u e de eloped he e is complemen a y
a he han ad e sa ial: i o ganizes wha can be said, and wha canno be imp o ed upon, a
any ixed posi i e esolu ion, wi hou aking a posi ion on whe he a globally smoo h solu ion
exis s a esolu ion 0.
2
2 Blu ope a o s and Blu iche sky obse e s
2.1 Spa ial blu ia molli ie s o hea ke nels
Le
ϕ∈C∞
c
(
R3
)be a s anda d nonnega i e molli ie wi h
RR3ϕ
(
x
)
dx
= 1 and
ϕ
(
x
) =
ϕ
(
−x
).
Fo ℓ > 0se
ϕℓ(x):=ℓ−3ϕx
ℓ,(Bℓ )(x) := (ϕℓ∗ )(x) = ZT3
ϕℓ(x−y) (y)dy.
Al e na i ely, we may ake (B
ℓ
)(
x
)=(
Gℓ2∗
)(
x
)whe e
G
is he hea ke nel a ime
. Bo h
amilies {Bℓ}ℓ>0ha e he usual p ope ies o an app oxima e iden i y:
•∥Bℓ ∥Lp≤ ∥ ∥Lp o 1≤p≤ ∞,
•Bℓ → in Lpas ℓ↓0when ∈Lp,
•Bℓ is smoo h o ℓ > 0whene e ∈Lp o some p.
We in e p e
ℓ
asablu scale: an obse e who canno esol e spa ial ea u es below he
leng h ℓpe cei es only h ough Bℓ .
De ini ion 2.1 (Blu ed eloci y ield).Gi en a (weak) eloci y ield
u
(
, x
)and blu scale
ℓ > 0, de ine he blu ed eloci y
uℓ( , x) := (Bℓu( , ·))(x).
2.2 Blu iche sky me ic on lows
We now o malize he scale–dependen poin o iew as a Blu iche sky geome y on he space
o lows. Fix ℓ>0and a ime ho izon T > 0.
De ini ion 2.2 (Blu iche sky dis ance a scale
ℓ
).Fo wo eloci y ields
u, ∈L2
([0
, T
];
L2
(
T3
))
de ine
dℓ(u, ) := ZT
0
∥Bℓ(u( )− ( ))∥2
L2(T3)d 1/2
.
Rema k 2.3 (Obse e –dependence).An obse e is speci ied by he iple (
ℓ, T, O
)whe e
O
is
a class o obse ables Φ ha depend on uonly h ough Bℓuon [0, T ], and a e Lipschi z in L2:
|Φ(u)−Φ( )|≤LΦdℓ(u, ).
Fo such an obse e , wo lows wi h dℓ(u, )≤εa e ε–indis inguishable on [0, T ].
P oposi ion 2.4 (Blu iche sky equi alence).Le Φbe an obse able depending only on B
ℓu
on [0, T ]and Lipschi z wi h cons an LΦas abo e. I dℓ(u, )≤ε, hen
|Φ(u)−Φ( )|≤LΦε.
In pa icula , i dℓ(u, ) = 0 hen Φ(u) = Φ( ) o all such obse ables.
P oo . By assump ion Φ(u) = Φ(Bℓu), simila ly o , and
|Φ(u)−Φ( )|=|Φ(Bℓu)−Φ(Bℓ )| ≤ LΦdℓ(u, ).
3
De ini ion 2.5 (Blu –equi alence class).The Blu iche sky equi alence class o a low
u
a
scale ℓon [0, T ]is
[u]ℓ,T := { :dℓ(u, ) = 0}.
An obse e a esolu ion
ℓ
and ho izon
T
can only dis inguish equi alence classes [
u
]
ℓ,T
, no
indi idual ep esen a i es.
The key ques ion now becomes: gi en a (possibly nonsmoo h) weak solu ion
u
, wha can be
said abou he s uc u e o [
u
]
ℓ,T
? In pa icula , does i con ain smoo h ep esen a i es, and
how do hey e ol e?
3 Blu ed Na ie –S okes dynamics
We now apply Bℓ o a Le ay solu ion and de i e he il e ed Na ie –S okes equa ion.
3.1 Fil e ed equa ion and Reynolds s ess
Le ube a Le ay–Hop weak solu ion on [0,∞)×T3. De ine uℓ=Bℓuand pℓ=Bℓp.
Lemma 3.1 (Fil e ed Na ie –S okes).Fo each ixed ℓ>0,uℓsa is ies he il e ed sys em
∂ uℓ+ (uℓ· ∇)uℓ+∇pℓ=ν∆uℓ−∇·Rℓ,
di uℓ= 0,(2)
in he sense o dis ibu ions, whe e he Reynolds s ess Rℓis gi en by
Rℓ:= Bℓ(u⊗u)−uℓ⊗uℓ.
P oo .
Apply B
ℓ
o
(1)
(in dis ibu ion o m). Using linea i y and he ac ha B
ℓ
commu es
wi h spa ial de i a i es, we ge
∂ uℓ+Bℓ(u· ∇)u+∇pℓ=ν∆uℓ.
W i e u⊗u o he ma ix wi h en ies uiuj. Then (u· ∇)u=∇·(u⊗u), and
Bℓ(u· ∇)u=Bℓ(∇·(u⊗u)) = ∇ · Bℓ(u⊗u).
Add and sub ac ∇·(uℓ⊗uℓ):
Bℓ(u· ∇)u=∇·(uℓ⊗uℓ)+∇ · Bℓ(u⊗u)−uℓ⊗uℓ= (uℓ· ∇)uℓ+∇ · Rℓ.
Mo ing
∇·Rℓ
o he igh –hand side gi es
(2)
. Di e gence– eeness o
uℓ
ollows om
di u
= 0
and he commu a ion o Bℓwi h spa ial de i a i es.
3.2 Regula i y o blu ed lows
The blu ed ield uℓis much mo e egula han u, e en i uis only a Le ay solu ion.
P oposi ion 3.2 (Spa ial smoo hness o
uℓ
).Le
u
be a Le ay–Hop solu ion and
ℓ >
0. Then
o each >0,uℓ( , ·)∈C∞(T3), and o e e y in ege k≥0,
∥∇kuℓ( , ·)∥L2(T3)≤Ck(ℓ)∥u( , ·)∥L2(T3),
wi h Ck(ℓ)depending only on ϕ,k, and ℓ.
4
P oo .
Fo each ixed
,
u
(
, ·
)
∈L2
(
T3
). Since
ϕℓ∈C∞
c
and con olu ion wi h a smoo h ke nel
smoo hs, Bℓu( , ·)∈C∞. Di e en ia ing unde he in eg al,
∇kuℓ( , x) = (∇kϕℓ)∗u( , ·)(x),
and Young’s inequali y gi es
∥∇kuℓ( , ·)∥L2≤ ∥∇kϕℓ∥L1∥u( , ·)∥L2.
Se Ck(ℓ) := ∥∇kϕℓ∥L1.
Rema k 3.3 (Time egula i y).Time egula i y o
uℓ
can be ob ained om he weak ime
egula i y o
u
oge he wi h he smoo hing e ec o B
ℓ
and s anda d in e pola ion. Fo ou
pu poses i su ices ha o each ℓ>0and T > 0,
uℓ∈C([0, T ]; Hm(T3)) o all m≥0,
wi h no ms con olled by he Le ay ene gy bounds and Cm(ℓ).
Thus, o each ixed blu scale
ℓ >
0, he blu ed low
uℓ
is a smoo h a a a o he weak
solu ion
u
. All po en ial singula i ies o
u
a e hidden in he Reynolds s ess
Rℓ
and in he
high– equency emainde u−uℓ.
3.3 Blu ed ene gy balance and budge
Applying he usual ene gy me hod o (2) yields a blu ed ene gy balance.
P oposi ion 3.4 (Blu ed ene gy inequali y).Le
u
be a Le ay–Hop solu ion and
ℓ >
0. Then
o almos e e y ≥0,
1
2∥uℓ( )∥2
L2+νZ
0
∥∇uℓ(s)∥2
L2ds ≤1
2∥uℓ(0)∥2
L2+Z
0ZT3
Rℓ(s, x) : ∇uℓ(s, x)dx ds. (3)
P oo . Mul iply (2)byuℓand in eg a e o e T3:
1
2
d
d ∥uℓ∥2
L2+ν∥∇uℓ∥2
L2=−ZT3
Rℓ:∇uℓdx,
whe e we used
R
(
uℓ· ∇
)
uℓ·uℓdx
= 0 and
R∇pℓ·uℓdx
= 0 by di e gence– eeness. In eg a ing
o e ime gi es (3).
The igh –hand side measu es he ene gy lux om esol ed o un esol ed scales media ed
by
Rℓ
. Fo each ixed
ℓ >
0, one can bound his lux by a unc ion o he Le ay ene gy and he
blu scale, p oducing a blu budge
Bℓ( ) := Z
0ZT3|Rℓ(s, x)| |∇uℓ(s, x)|dx ds,
which con ols how much ene gy can leak in o scales smalle han ℓ.
Rema k 3.5 (No ee in o ma ion in he blu ).The blu ed ene gy inequali y exp esses a simple
p inciple: any addi ional s uc u e a scales below
ℓ
(encoded in
Rℓ
) mus be paid o by a
co esponding blu budge
Bℓ
. The e is no way o ge ex a small–scale in o ma ion o ee
wi hou ei he inc easing Bℓo changing he blu scale.
5
4 Smoo hness a ini e esolu ion
We now summa ize wha he p e ious sec ion implies o Blu iche sky obse e s.
Theo em 4.1 (Smoo h blu a a a s).Le
u
be a Le ay–Hop solu ion on [0
,∞
)
×T3
. Fix
ℓ >
0
and T > 0. Then:
1. The blu ed low uℓbelongs o C∞((0, T ]×T3).
2.
Fo any obse able Φdepending only on
u
h ough B
ℓu
on [0
, T
]and Lipschi z in
L2
, he
alue Φ(u)is comple ely de e mined by he smoo h ield uℓ.
3.
Any o he Le ay solu ion
wi h he same blu ed ini ial da a
uℓ
(0) =
ℓ
(0) and he same
blu budge Bℓon [0, T ]sa is ies Φ(u) = Φ( ) o all such obse ables.
P oo . (1) is P oposi ion 3.2 and he ensuing ime egula i y ema k.
Fo (2), by assump ion Φ(
u
) = Φ(B
ℓu
)on [0
, T
], and B
ℓu
=
uℓ
is smoo h. Fo (3), i
is
ano he Le ay solu ion wi h he same blu ed ini ial da a and blu budge , hen
uℓ
and
ℓ
sol e
he same il e ed equa ion wi h he same ini ial condi ion and same Reynolds s ess budge s.
Unde mild addi ional assump ions (e.g. G onwall– ype con ol on he di e ence o il e ed
lows), one can show uℓ= ℓon [0, T ]. Then Φ(u) = Φ(uℓ) = Φ( ℓ) = Φ( ).
Rema k 4.2 (Physical smoo hness e sus Pla onic smoo hness).F om he s andpoin o an
obse e a esolu ion
ℓ
and ho izon
T
, he only accessible objec is he equi alence class [
u
]
ℓ,T
,
which con ains he smoo h ep esen a i e
uℓ
. Any po en ial singula i ies o
u
a scales
≪ℓ
a e
hidden in he un esol able pa o he Reynolds s ess and canno be dis inguished om o he
membe s o he same class. In his sense, he low is physically smoo h, e en i he PDE admi s
blow–up a ℓ= 0.
5
Blu , unce ain y, and e ec i e andomness in Na ie –S okes
Blu also cla i ies how andom–looking beha io can a ise inside a ully de e minis ic equa ion
such as Na ie –S okes.
5.1 Unce ain y budge s in luid desc ip ions
Th ee s uc u al limi a ions conspi e in he luid con ex :
1.
Unce ain y p inciple ype cons ain s: sha p localiza ion in physical space implies
b oad suppo in equency space and ice e sa. In u bulence, ine localiza ion in o ices
o ces an inc easingly wild dis ibu ion o equencies.
2.
Ene gy cons ain s: gene a ing ine s uc u es equi es ene gy. The cascade o ene gy o
small scales is bounded by global conse a ion/dissipa ion laws.
3.
Fini e obse a ional capaci y: any expe imen samples he low a a ini e numbe o
poin s, wi h ini e bandwid h and ini e ime.
Blu packages hese ac s in o a single pa ame e
ℓ >
0and a budge
Bℓ
. Once
ℓ
and
Bℓ
a e
ixed, he e is a ha d ceiling on how much small–scale s uc u e can be ex ac ed o e en ma e
o obse ables.
6
5.2 E ec i e andomness om un esol ed scales
E en hough Na ie –S okes is de e minis ic, a Blu iche sky obse e a ixed
ℓ
and
T
will
ypically epo andom beha io in ce ain channels:
•many di e en mic oscopic con igu a ions ubelong o he same blu class [u]ℓ,T ;
• hei di e ences a e in isible o all obse ables in O;
• he un esol ed pa beha es, om he obse e ’s poin o iew, like a sou ce o noise.
This is e ec i e andomness. I does no depend on me aphysical inde e minism: i is a s uc u al
ea u e o he in o ma ion geome y o Na ie –S okes unde blu .
Blu makes his p ecise. Decompose
u=uℓ+ (u−uℓ),
whe e
uℓ
is he smoo h esol ed pa and
u−uℓ
is he high– equency emainde . Unde he
blu ed ene gy inequali y
(3)
, he emainde ca ies a mos a con olled amoun o ene gy.
Wi hin he blu cons ain s, i can emula e many di e en small–scale pa e ns wi hou changing
any obse able in Oby mo e han he Lipschi z cons an imes he blu budge .
Rema k 5.1 (No ee sha pening).One migh hope ha by swi ching lenses (e.g. changing
coo dina es o a iables) we could eco e his appa en andomness as hidden de e minism. Blu
shows ha his is illuso y: any lens compa ible wi h he same physical ene gy and measu emen
cons ain s su e s an equi alen unce ain y budge . The e is no ee epa ame iza ion ha
e eals he ull mic oscopic pic u e wi hou paying o i in ano he channel.
5.3 Gaussian blu e sus classical ans o ms
A a o mal le el, he con olu ion blu B
ℓ
is as compa ible wi h Fou ie o Mellin analysis as
any o he smoo hing; in ac , Gaussian blu is pa icula ly con enien in equency space. The
concep ual di e ence is:
•
classical ans o ms a e usually in oduced o sol e he equa ion by decomposing
u
in o
eigenmodes and e ol ing coe icien s;
•
blu is in oduced o bound wha can be known a a gi en esolu ion and o isola e he pa
o u ha mus emain e ec i ely andom.
In his sense, blu beha es like a me a– ans o m: i si s abo e he choice o basis and acks he
in o ma ion budge ha all bases mus espec .
6 Wha his does and does no say abou he Clay p oblem
I is impo an o be explici abou he scope o he blu iewpoin .
6.1 Clay smoo hness a ℓ= 0
The Clay p oblem asks, in o mally:
Does he e exis a globally smoo h solu ion o e e y smoo h ini ial da um, when we
in e p e (1) as an exac con inuum PDE a in ini e esolu ion?
This is he ques ion o smoo hness a blu scale
ℓ
= 0. Ou a gumen s do no answe his:
hey show ha o any ixed
ℓ >
0and any Le ay solu ion
u
, he blu ed a a a
uℓ
is smoo h
and su icien o all obse e s limi ed o ha esolu ion. Whe he
u
i sel emains smoo h as
ℓ↓0is a sepa a e, and s ill open, issue.
7
6.2 Physical comple eness a ini e blu
F om a Blu iche sky pe spec i e, he physically ele an s a emen is:
Fo each expe imen ally meaning ul esolu ion
ℓ >
0and any ini e ime ho izon
T >
0, e e y Le ay solu ion
u
admi s a smoo h a a a
uℓ∈
[
u
]
ℓ,T
ha cap u es all
obse ables accessible a ha esolu ion, and any u he s uc u e a scales
< ℓ
is
hidden behind he blu budge Bℓ.
In o he wo ds, blu p o ides a comple e and hones desc ip ion o wha luid dynamics can
e e ell us, wi hou assuming ha he unde lying con inuum PDE is globally smoo h.
6.3 Two ypes o ques ions
Thus, he NS landscape sepa a es in o wo kinds o ques ions:
1. Blu –awa e ques ions (physically g ounded): how do he blu classes [u]ℓ,T beha e as ℓ
and
T
a y? Can we classi y hem, bound hei budge s, and p o e s abili y and uni e sali y
p ope ies? These ques ions a e di ec ly accessible o obse a ion and expe imen .
2.
Blu –blind ques ions (Pla onic): wha happens a he limi
ℓ
= 0? Is e e y Le ay
solu ion smoo h, o can singula i ies o m a a bi a ily small scales? These ques ions a e
ma hema ically sha p bu physically un eachable.
Blu does no i ialize he Clay p oblem; i e ames i . I makes isible ha he exis ence
and smoo hness p oblem li es en i ely a he bounda y o ou in o ma ion ho izon, while he
in e io o ha ho izon can be o ganized and unde s ood in a scale–awa e way.
7 Conclusion
Blu and Blu iche sky geome y p o ide a na u al language o alking abou Na ie –S okes a
ini e esolu ion. The main poin s a e:
•
Fo any Le ay–Hop solu ion and any blu scale
ℓ >
0, he blu ed low
uℓ
is smoo h
and sol es a il e ed NS wi h an explici Reynolds s ess e m. All physically accessible
obse ables a ha scale depend only on uℓand on a ini e blu budge .
•
E ec i e andomness in u bulen lows a ises no om a ailu e o de e minism, bu om
he ac ha many mic oscopic con igu a ions collapse in o he same blu class [
u
]
ℓ,T
. Blu
makes his collapse quan i a i e.
•
The Clay exis ence and smoo hness ques ion is no in alid, bu i is beyond he each o
any Blu iche sky obse e : i conce ns he ideal limi
ℓ
= 0, while e e y hing he uni e se
allows us o measu e o compu e li es a ℓ>0.
In his sense, Na ie –S okes is a pa adigma ic example o he gene al philosophy o blu : we
canno know e e y hing abou he con inuum low, bu we can know exac ly wha can be known
a any gi en esolu ion, and we can p o e heo ems he e. Whe he o no a Pla onic smoo h
solu ion exis s a in ini e esolu ion, he Blu iche sky owe o blu ed a a a s al eady con ains
all he s uc u e ha physics can e e demand.
8
Re e ences
[1]
O. A. Ladyzhenskaya. The Ma hema ical Theo y o Viscous Incomp essible Flow. Go don
and B each, 1969.
[2] R. Temam. Na ie –S okes Equa ions. No h–Holland, 1979.
[3] P. Cons an in and C. Foias. Na ie –S okes Equa ions. Uni e si y o Chicago P ess, 1988.
[4]
A. Pe išić. Geome y o Blu iche ski: Obse e s, Scales, and Blu . Zenodo p ep in , 2025.
[5] A. Pe išić. Epis emological Blu . Zenodo p ep in , 2025.
[6] A. Pe išić. No F ee In o ma ion. Zenodo p ep in , 2025.
[7]
A. Pe išić. Blu Geome y (UG Ve sion): Be ween Euclid and Lobache sky. Zenodo
p ep in , 2025.
[8]
A. Pe išić. A Lyapuno Ce i ica e o he Accele a ed Colla z Map. Zenodo p ep in , 2025.
[9] A. Pe išić. The Poinca é 3D ia Blu iche sky Geome y. Zenodo p ep in , 2025.
9
