Boundary-Driven Fractal Generation Mechanism: From Time Scale Mother Ruler to Feedback Scattering and Critical Interfaces

Bounda y-D i en F ac al Gene a ion Mechanism:
F om Time Scale Mo he Rule o Feedback
Sca e ing and C i ical In e aces
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
Wi hin unied bounda y imesca e ing amewo k, analyze widesp ead  ac-
als g ow on bounda ies phenomenon in na u e. P o ide igo ous cha ac e iza ion
om h ee complemen a y h eads:
(i) On o al space
Y=M×X◦
o mani old wi h bounda y and pa ame e space,
ake ime scale mo he ule
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω)
, ela i e
opological class
[K]∈H2(Y, ∂Y ;Z2)
, sca e ing amily
K1
class
[u]∈K1(X◦)
as undamen al in a ian s; dene bounda y as codimension-one in e ace simul a-
neously bea ing bounda y condi ions, ux, en opy eadings; p o e all obse able
imes and bounda y ac als ul ima ely exp essible as unc ions o bounda y da a.
(ii) Wi hin his amewo k cons uc h ee ypical bounda y ac al gene a-
ion mechanisms: eedback- ype bounda ies (sel - e e en ial sca e ing ne wo ks),
c i ical- ype bounda ies (scale- ee phase in e aces), Laplace- ype bounda ies (diusion/po en ial-
d i en g ow h on s). Gi e heo ems o one-dimensional sel -simila sca e ing oy
model, scaling equa ions o c i ical in e aces, Laplacian g ow h equa ion
n∝ |∇u|
;
p o e unde condi ions no in insic cha ac e is ic scale + local ules + bounda y
eedback, non- i ial xed poin s and limi se s o scale ans o ma ion semig oups
possess sel -simila i y wi h ac al geome ic and spec al measu es.
(iii) In e ace abo e bounda y ac al s uc u es wi h ime scale mo he ule
κ(ω)
; in oduce igo ous deni ion o  ac al ime delay; p o e in mul i-laye sel -
simila sca e ing ne wo ks, c i ical in e aces, Laplacian g ow h on s, g oup delay
dis ibu ion
τ(ω)
and ela i e s a e densi y
ρ el(ω)
possess sel -simila i y unde log-
a i hmic equency scaling, gi ing  empo al ac al ex u e as spec al empo al
p ojec ion o bounda y ac als.
Fu he p o e: gi en xed ime geome y equi alence class
[τ]
, bounda y ac als
do no al e causal o de ing o ime equi alence class i sel , only changing empo al
ex u e and s a is ical p ope ies a die en scales.
Unied conclusion: ac als no subjec i ely c ea ed by obse e s bu na u ally
gene a ed on high-ux, high- eedback bounda ies unde coupling o scale- ee bulk
dynamics and bounda y condi ions; bounda y is s age whe e ac als mos easily
gene a ed and mos easily ead ou .
P o ide ealizable sca e ing and g ow h expe imen al schemes; appendices p o e
sel - e e en ial sca e ing oy model ecu sion and sel -simila i y, de i e c i ical in-
1
e ace scaling laws, o malize ela ionship be ween ac al ime scales and ime
equi alence classes.
Keywo ds:
Bounda y Geome y; F ac al In e aces; Wigne Smi h Time Delay; Bi man
K ein Fo mula; Laplacian G ow h; C i ical Phenomena; Sch ammLoewne E olu ion;
Time Scale Mo he Rule

1 In oduc ion and His o ical Con ex
Na u al s uc u es wi h ac al cha ac e is ics almos wi hou excep ion appea a bound-
a ies: coas line i e ne wo k bounda ies, cloudcon ec i e laye shea su aces, discha ge
channelmedium in e aces, ee b anchdend i e g ow h on s, u bulen o ex sepa a-
ion laye s, condensed ma e and s a is ical physics c i ical clus e s and phase in e aces.
Since Mandelb o sys ema ically p oposed  ac al geome y, people ecognized uni-
ed scale- ee geome ic and s a is ical s uc u es behind hese pa e ns. Bu ques ion
o why ac als always appea a bounda ies s ill lacks comple e amewo k simul ane-
ously conside ing geome y, dynamics, ime scales.
In non-equilib ium g ow h eld, Wi enSande diusion-limi ed agg ega ion (DLA)
model gi es ypical bounda y g ow h ac al scena io: pa icles andom walk in bulk,
i e e sibly adhe e upon eaching clus e bounda y, p oducing b anching ac al clus e s
wi h ac al dimension be ween Euclidean and bounda y dimensions.
On o he hand, c i ical phenomena and in e ace heo y show phase in e aces in c i -
ical sys ems hemsel es p esen ac al geome y: wo-dimensional c i ical pe cola ion,
Ising model, andom eld Ising model, igid pe cola ion model in e aces uni o mly de-
sc ibable by Sch ammLoewne E olu ion (SLE), wi h in e ace Hausdo  dimension and
SLE pa ame e
κ
ha ing exac ela ion
d= 1 + κ/8
; c i ical pe cola ion ou e bounda y
co esponds o
κ= 6
, in e ace dimension
4/3
.
In pa allel, quan um and wa e sca e ing heo y, Wigne Smi h in oduced ime delay
ope a o and ma ix connec ing sca e ing ma ix
S(ω)
wi h equency de i a i e, cha ac-
e izing g oup delay o sca e ed sys em on inciden wa e packe s; ex ended o quan um,
acous ic, elec omagne ic sca e ing sys ems, o ming spec al ime scale amewo k cen-
e ed on Wigne Smi h ime delay ma ix.
Simul aneously, Bi manK ein o mula and Li shi sK ein spec al shi heo y gi e
exac connec ion be ween sca e ing phase, spec al shi unc ion, ela i e s a e densi y,
making o al sca e ing phase de i a i e equal ela i e s a e densi y and Wigne Smi h
ime delay ace unde app op ia e condi ions, ma hema ically uni ying phasedensi y
ime delay h ee ime scale p oxies.
Exis ing discussions o  ac al bounda ies mos ly ocus on geome ic dimensions,
coa se-g ained s a is ics, uni e sali y classes, a ely di ec ly unied wi h ime scales,
sca e ing, in o ma ion geome y. On o he hand, discussions o  ime essence o en
e ol e a ound obse e and measu emen p oblems, ei he conside ing ime and s uc-
u e s ongly dependen on obse e p ojec ion, o ea ing bounda y me ely as echnical
bounda y condi ion, igno ing i s cen al ole in ux and in o ma ion.
This pape 's s ance summa izable as:
1. Physical s uc u es no subjec i ely c ea ed by obse e s bu join ly de e mined
by bulk dynamics and bounda y condi ions; obse e s me ely couple a ni e esolu ion
on ce ain bounda ies, eading unc ions o bounda y ope a o s.
2
2. Bounda ies no only bea geome ic in e ace ole bu also dynamical closu e
bounda y condi ions, ux and ene gyphaseen opy eading con e sion, opology and
ime scale ancho ing.
3. Time scale uniable on sca e ing ma ix and spec al shi unc ion as ime scale
mo he ule
κ(ω)
, classiable wi h modula ime and geome ic ime in o same ime
equi alence class.
4. F ac al gene a ion mainly occu s a bounda ies, appea ing as non- i ial xed
poin s o limi se s o ime scale mo he ule and bounda y ope a o s a mul iple scales;
so-called  empo al ac al ex u e is p ecisely such bounda y ac als' p ojec ion in
spec al empo al domain.
In p e ious unied bounda y imesca e ing and bounda y ime geome y wo k, we
cons uc ed unied amewo k o ime scale mo he ule
κ(ω)
, ime equi alence class
[τ]
,
bounda y ope a o algeb a.
This pape specically ocuses on bounda y-d i en ac al gene a ion mechanism,
sys ema ically answe ing h ee ques ions:
1. Can one say in p o able sense  ac als p oduced om bounda ies? 2. In uni-
ed bounda y imesca e ing amewo k, h ough wha specic mechanisms do ac als
gene a e a bounda ies? 3. Why in many physical sys ems do bounda y geome y o
spec al s uc u es na u ally p esen ac als while bulk desc ip ion i sel emains simple
and local?

2 Model and Assump ions
2.1 To al Space, Bounda y, Sca e ing Sys em
Le
M
be Lo en zian mani old wi h bounda y
∂M
ep esen ing bulk space ime,
X◦
be
pa ame e space emo ing singula i ies (d i ing equency, ex e nal eld s eng h, geo-
me ic de o ma ion con ol pa ame e s); dene o al space
Y:= M×X◦
wi h bounda y
∂Y =∂M ×X◦∪M×∂X◦
.
Fo each
x∈X◦
, gi en sel -adjoin ope a o pai
(Hx, H0,x)
ep esen ing in e ac ing
sys em and e e ence sys em, sa is ying s anda d sca e ing heo y assump ions:
1. Die ence
Hx−H0,x
is ace class o sucien ly as -decaying bounded pe u -
ba ion; 2. Wa e ope a o s and uni a y sca e ing ope a o
Sx
exis , ene gy-shell de-
composi ion gi es xed- equency sca e ing ma ix
Sx(ω)
; 3. Fo a.c. spec um almos
e e ywhe e,
Sx(ω)
is uni a y ma ix on ni e channels.
Unde abo e assump ions, dene Wigne Smi h ime delay ma ix
Qx(ω) := −iS†
x(ω)∂ωSx(ω),
whose ace gi es sum o g oup delays.
On o he hand, Bi manK ein o mula and spec al shi heo y show: o sui able
Sch ödinge - ype ope a o s, spec al shi unc ion
ξx(ω)
exis s making ela i e s a e den-
si y
ρ el,x(ω) = ∂ωξx(ω)
and sca e ing phase
Φx(ω) = a g de Sx(ω)
sa is y
ξx(ω) = −1
2πiln de Sx(ω), ρ el,x(ω) = ∂ωΦx(ω)
2π.
3
2.2 Time Scale Mo he Rule and Time Equi alence Class
In unied sca e ing ime scale amewo k, dene ime scale mo he ule o each pa-
ame e
x∈X◦
:
κx(ω) := φ′
x(ω)
π=ρ el,x(ω) = 1
2π Qx(ω),
whe e
is ace o e channel space. This equali y holds unde s anda d condi ions sa -
is ying Bi manK ein o mula and Wigne Smi h deni ion, ep esen ing same objec 's
die en p ojec ions: phase g adien , ela i e s a e densi y, g oup delay ace.
In b oade bounda y ime geome y amewo k, also exis modula ime scale om
Tomi aTakesaki modula ow and geome ic ime scale gene a ed by bounda y Hamil-
onian. Unde app op ia e ma ching condi ions, in oduce ime scale equi alence class
[τ]
making sca e ing ime
τsca
, modula ime
τmod
, geome ic ime
τgeom
mac oscopically
ela ed o ep esen a i e ime unc ion
τ
ia ane escalings:
τsca =a1τ+b1, τmod =a2τ+b2, τgeom =a3τ+b3,
whe e
ai>0
,
bi∈R
slowly a ying in gi en physical window. Time geome y essen ial
da a join ly de e mined by equi alence class
[τ]
and causal s uc u e, no by specic scale
choice.
2.3 Physical Bounda y and Bounda y Ope a o s
Deni ion 2.1
(Physical Bounda y)
.
In o al space
Y=M×X◦
, physical bounda y
B
is codimension-one submani old sa is ying:
1. Simul aneously bea s bulk eld bounda y condi ions; 2. Ca ies measu able ux
(ene gy, pa icle numbe , en opy, in o ma ion ux); 3. Suppo s bounda y obse able
algeb a
AB
making bounda y s a es ia GNS cons uc ion ealizable as Hilbe space
ec o s; 4. Admi s well-dened ime scale eadings ia
κx(ω)
on
B
.
2.4 Bounda y F ac al Deni ion
Deni ion 2.2
(Bounda y F ac al)
.
Subse
F⊂ B
called bounda y ac al i sa ises:
1.
Geome ic sel -simila i y
: Exis s scale ans o ma ion semig oup
{Tλ}λ>0
ac ing
on
F
such ha o some disc e e o con inuous scale se
Λ⊂(0,∞)
,
TλF
and
F
locally
isomo phic o s a is ically equi alen ;
2.
F ac ional Hausdo  dimension
: Hausdo  dimension
dH(F)
is non-in ege
and
d op < dH(F)< dembed
, whe e
d op
opological dimension,
dembed
embedding space
dimension;
3.
Spec al sel -simila i y
: I
F
as sca e ing sys em bounda y has well-dened
κx(ω)
, hen exis s escaling
ω7→ λω
making
κx(λω)
and
κx(ω)
unc ionally ela ed ia
powe law o loga i hmic scaling.

3 Main Resul s (Th ee Bounda y F ac al Gene a ion
Mechanisms)
This sec ion s a es h ee main heo ems co esponding o h ee bounda y ac al gene -
a ion ypes. P oo s gi en in Sec ion 4 and appendices.
4
Theo em 3.1
(Sel -Simila Feedback Sca e ing Ne wo k)
.
Conside one-dimensional
sel - e e en ial sca e ing sys em: a each s age
n
, uni cell has in e nal s uc u e de-
e mined by p e ious s age cell
Un−1
plus sel -simila eedback, o al sca e ing ma ix
sa is ying ecu sion
Sn(ω) = F[Sn−1(ω), ω],
whe e
F
is unc ional p ese ing uni a i y and sa is ying ce ain scaling symme y.
I ini ial cell
S0(ω)
and unc ional
F
sa is y:
(i) Scale in a iance:
F[S(λω), λω] = S(ω)
o
λ∈Λ
(disc e e scale se );
(ii) Feedback non- i iali y:
F
no iden i y map, in oducing phase and ampli ude mod-
ica ions;
(iii) Con e gence: Sequence
{Sn}
con e ges in ope a o no m sense o limi
S∞(ω)
;
Then:
(a) Limi sca e ing ma ix
S∞(ω)
possesses sel -simila s uc u e:
S∞(λω) = UλS∞(ω)U†
λ
whe e
Uλ
is uni a y;
(b) G oup delay dis ibu ion
τ(ω) = (2π)−1 Q∞(ω)
sa ises scaling ela ion
τ(λω) =
λ−ατ(ω) + cons
o
α∈(0,1)
;
(c) F equency suppo se o
τ(ω)
is Can o -like se wi h Hausdo  dimension
dH=
1−α
.
Thus sel - e e en ial eedback gene a es spec al empo al ac al a bounda y.
Theo em 3.2
(C i ical In e ace F ac als)
.
Conside wo-dimensional phase sepa a ion
sys em a c i icali y, in e ace
Γ
be ween phases sa is ying con o mal in a iance and de-
sc ibed by Sch ammLoewne E olu ion SLE
κ
.
Unde con o mal eld heo y and SLE amewo k assump ions, i in e ace
Γ
simul a-
neously se es as sca e ing sys em bounda y (e.g., elec omagne ic o quan um sca e ing
wi h in e ace as po en ial/conduc i i y discon inui y), hen:
(a) In e ace Hausdo  dimension
dH= 1 + κ/8
whe e
κ
is SLE pa ame e ;
(b) Bounda y sca e ing phase
φ(ω)
a e aged o e in e ace ensemble sa ises mul i-
ac al spec um, singula i y s eng h dis ibu ion unc ion
(α)
non- i ial;
(c) Time scale mo he ule
κx(ω)
uc ua ions on in e ace possess scale- ee co e-
la ion s uc u e, co ela ion unc ion decaying as powe law
C( )∼ −η
whe e
η
ela ed o
κ
ia con o mal dimension ela ion.
Thus c i ical phase in e aces na u ally ca y ac al empo al ex u e.
Theo em 3.3
(Laplacian G ow h F on F ac als)
.
Conside Laplacian g ow h sys em
(e.g., DLA, iscous nge ing, elec odeposi ion) whe e g ow h on
∂Ω( )
sa ises
n∝ |∇u|,
whe e
u
sa ises Laplace equa ion
∇2u= 0
in ex e io
Ωc( )
wi h app op ia e bounda y
condi ions.
I ini ial clus e
Ω(0)
is compac and g ow h p ocess sa ises:
5

(i) No cha ac e is ic leng h scale in bulk:
u
only depends on geome y, no in insic
leng h;
(ii) Local g ow h ule:
n
only depends on local
|∇u|
;
(iii) Sc eening: In e io poin s do no pa icipa e in g ow h a e being su ounded;
Then:
(a) G ow h on
∂Ω( )
a long imes possesses ac al s uc u e wi h s a is ical ac al
dimension
d ∈(d, d + 1)
(
d
spa ial dimension);
(b) T ea ing g ow h on as ime-dependen sca e ing bounda y, cumula i e ime delay
in eg a ed o e g ow h his o y exhibi s sel -simila s uc u e;
(c) Ac i e g ow h ip dis ibu ion on on sa ises mul i ac al measu e, singula i y
spec um non- i ial.
Thus Laplacian g ow h mechanism gene a es ac als on mo ing bounda ies.
Co olla y 3.4
(F ac al Time Does No Al e Time Equi alence Class)
.
Unde Theo-
ems 3.13.3 se ings, gi en xed ime equi alence class
[τ]
(i.e., causal s uc u e and
mac oscopic ime ow di ec ion de e mined), bounda y ac als gene a ed by abo e h ee
mechanisms:
(a) Do no al e ime equi alence class
[τ]
i sel , i.e., sca e ing ime, modula ime,
geome ic ime emain anely ela ed;
(b) Only al e s a is ical dis ibu ion and co ela ion s uc u e o ime scale eadings
κx(ω)
a die en equency scales;
(c) On coa se-g ained scales, can dene eec i e ime scale
¯κ(ω)
ia ensemble o win-
dow a e aging, making
¯κ
sa is y same ane ela ions as o iginal
κ
.
Thus bounda y ac als only in oduce ne ime ex u e, no changing ime di ec ion
o causali y.

4 P oo s
This sec ion p o ides p oo ou lines o main heo ems; de ailed calcula ions in appendices.
4.1 P oo Ske ch o Theo em 3.1
Conside one-dimensional sel -simila sca e ing ne wo k. A s age
n
, uni cell sca e ing
ma ix
Sn(ω)
sa ises ecu sion
Sn=F[Sn−1, ω]
.
S ep 1: Scale in a iance implies unc ional equa ion
Condi ion (i) gi es
F[S(λω), λω] = S(ω)
o
λ∈Λ
. Fo disc e e scale se
Λ = {2k:
k∈Z}
, aking
λ= 2
yields
F[S(2ω),2ω] = S(ω).
6
I e a ing his ela ion, i sequence con e ges o
S∞
, hen
S∞
mus sa is y
S∞(2ω) = U2S∞(ω)U†
2,
whe e
U2
de e mined by unc ional
F
's uni a y pa .
S ep 2: G oup delay scaling
F om
Q(ω) = −iS†∂ωS
, scaling ela ion o
S
implies
Q∞(2ω)=2−1U2Q∞(ω)U†
2+
(bounda y e ms)
.
Taking ace gi es
τ(2ω) = 2−1τ(ω) + cons ,
i.e.,
α= 1
case. Fo gene al
λ= 21/β
, simila ly ob ain
α= 1 −1/β
.
S ep 3: Can o se s uc u e
Sel -simila ecu sion wi h gaps gene a es hie a chical equency suppo : each s age
emo es ce ain equency in e als; limi suppo is Can o -like se . Box-coun ing di-
mension calculable ia
N(ϵ)∼ϵ−dH
whe e
N(ϵ)
is
ϵ
-co e numbe , gi ing
dH= 1 −α
.
De ailed cons uc ion in Appendix A.
4.2 P oo Ske ch o Theo em 3.2
Fo c i ical in e ace desc ibed by SLE
κ
:
Hausdo  dimension
: S anda d SLE heo y gi es
dH= 1 + κ/8
.
Sca e ing phase mul i ac ali y
: In e ace as sca e ing bounda y con ibu es
phase accumula ion
φ∼RΓϕ ds
whe e
ϕ
is local phase densi y. In e ace sel -simila i y
implies
ϕ
is mul i ac al measu e; Legend e ans o m o pa i ion unc ion gi es singu-
la i y spec um
(α)
. Explici calcula ions o SLE
6
(c i ical pe cola ion) in Appendix
B.
Co ela ion decay
: Time scale
κx(ω)
uc ua ions ela ed o con o mal eld bound-
a y ope a o co ela ions; powe -law decay
η
ela ed o
κ
ia
η= 2 −dH
o simila
con o mal dimension ela ion.
4.3 P oo Ske ch o Theo em 3.3
Fo Laplacian g ow h:
F ac al dimension
: DLA clus e s' ac al dimension
d ≈1.71
(2D),
d ≈2.5
(3D)
om ex ensi e nume ics; heo e ical bounds ia sc eening a gumen s.
Time delay sel -simila i y
: G ow h on as ime-dependen bounda y; cumula i e
delay
∆τ∼R −1
nds
. Since
n∝ |∇u|
and
|∇u|
scale- ee,
∆τ
in eg al exhibi s sel -simila
uc ua ions.
Mul i ac al g ow h measu e
: Ac i e ips whe e
|∇u|
la ge concen a e on ac al
subse ; g ow h measu e
dµ ∝ |∇u|ds
is mul i ac al. Singula i y spec um compu a ion
ia he modynamic o malism in Appendix C.
4.4 P oo Ske ch o Co olla y 3.4
Key obse a ion: Time equi alence class
[τ]
de e mined by ane ela ions among
τsca
,
τmod
,
τgeom
a mac oscopic scales.
7
Bounda y ac als in oduce ne s uc u e a small equency/leng h scales bu p e-
se e ensemble-a e aged o coa se-g ained ime scales. Fo mally:
¯κ(ω) := Z|ω′−ω|<∆
κ(ω′)w(ω′−ω)dω′,
whe e
w
is window unc ion and
∆
is coa se-g aining scale.
Unde Theo ems 3.13.3 condi ions,
¯κ
sa ises same Bi manK ein and Wigne Smi h
ela ions as o iginal
κ
up o co ec ions
O(∆/ω)
. Thus ime equi alence class p ese ed
a esolu ions coa se han
∆
.
Causal o de ing: F ac als a spa ial bounda y don' al e bulk causal s uc u e (ligh
cones, imelike cu es); hus causal pa ial o de in a ian unde bounda y ac al gene -
a ion. De ailed p oo in Appendix D.

5 Model Applica ions
5.1 Mic owa e F ac al An ennas and G oup Delay Measu emen s
Cons uc sel -simila ac al an enna (e.g., Sie pinski gaske , Koch cu e) as sca e ing
bounda y. Measu e
S(ω)
ia ec o ne wo k analyze ; compu e
Q(ω)
and
κ(ω)
nume i-
cally.
P edic ion:
κ(ω)
exhibi s sel -simila s uc u e unde equency escaling consis en
wi h an enna's geome ic scaling. Measu e ac al dimension om g oup delay spec um;
compa e wi h an enna's geome ic dimension.
5.2 C i ical Pe cola ion Clus e s as Quan um Sca e ing Bound-
a ies
In 2D c i ical pe cola ion simula ions, ex ac clus e ou e bounda y as sca e ing po-
en ial (e.g.,
V( ) = V0
inside clus e ,
0
ou side). Nume ically sol e Sch ödinge equa ion
o ob ain
S(ω)
and
ξ(ω)
.
P edic ion: Spec al shi unc ion uc ua ions possess mul i ac al spec um consis-
en wi h SLE
6
in e ace dimension
dH= 4/3
.
5.3 Elec odeposi ion F ac als and Tempo al Tex u es
In elec odeposi ion expe imen s, deposi me al on elec ode su ace o ming DLA-like
ac als. Measu e impedance spec um
Z(ω)
encoding elec ode-elec oly e in e ace
p ope ies.
P edic ion: Real pa
Re[Z(ω)]
ela ed o dissipa ion exhibi s ac al uc ua ions.
Ex ac eec i e ime delay ia K ame sK onig ela ions; e i y sel -simila i y.

8
6 Enginee ing P oposals
6.1 Sel -Re e en ial Sca e ing Ne wo k Tes bed
Cons uc mul i-s age mic owa e ne wo k whe e each s age ou pu eeds back o modi y
nex s age sca e ing p ope ies. Implemen ia a ac o s, phase shi e s, p og ammable
a enua o s.
Sys ema ically a y eedback pa ame e s; measu e con e gence o sel -simila
S∞(ω)
.
Ex ac Can o se s uc u e om g oup delay spec um; e i y Theo em 3.1 p edic ions.
6.2 C i ical In e ace Analog ia Tunable Me ama e ials
Design me ama e ial in e ace whose pa ame e s (pe mi i i y, conduc i i y) unable o
c i icali y. Use pe cola ion composi es o liquid c ys al mix u es.
A c i icali y, measu e sca e ing; e i y mul i ac al phase spec um. Compa e wi h
SLE p edic ions o app op ia e uni e sali y class.
6.3 Laplacian G ow h Elec ochemis y Pla o m
Elec ochemical cell wi h con olled g ow h condi ions ( ol age, concen a ion). In-si u
impedance spec oscopy du ing g ow h.
Ex ac empo al ac al signa u es om ime- esol ed impedance. Co ela e wi h
ex-si u ac al dimension measu emen s ia mic oscopy.

7 Discussion (Risks, Bounda ies, Pas Wo k)
7.1 Assump ions and Limi a ions
Main assump ions:
•
Bi manK ein o mula alidi y equi es ace-class pe u ba ions;
may ail o long- ange in e ac ions;
•
SLE applicabili y limi ed o con o mally in a ian
c i ical sys ems;
•
Laplacian g ow h analysis assumes quasis a ic app oxima ion; may
b eak a high g ow h a es.
Thus amewo k applies o nice sys ems wi h well-con olled bounda ies; ex ension
o s ongly non-local o a - om-equilib ium cases equi es cau ion.
7.2 Rela ion o Exis ing F ac al Li e a u e
This pape 's no el y: Uni y ac al gene a ion wi h ime scale amewo k ia Wigne 
Smi hBi manK ein iden i y. P e ious wo k mos ly sepa a e geome ic ac als om
empo al/spec al aspec s.
Connec ion o mul i ac als, DLA, SLE well-es ablished in espec i e communi ies;
his pape p o ides unied bounda y ime language.
7.3 Philosophical Implica ions
F ac als om bounda ies suppo s iew ha :
•
Complexi y o en eme gen a in e -
aces, no in insic o bulk;
•
Obse e s na u ally couple o bounda ies (measu emen
9