Time Equivalence Class, Observer Projection, and 4D Topological Analogy:\\ From Boundary Time Scale Invariance to Phase Transitions, Fractals, and Exotic Structures

Time Equi alence Class, Obse e P ojec ion, and 4D
Topological Analogy:
F om Bounda y Time Scale In a iance o Phase
T ansi ions, F ac als, and Exo ic S uc u es
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 20, 2025
Abs ac
Wi hin he unied amewo k o bounda y sca e ing ime geome y, his pa-
pe sys ema ically cha ac e izes he ela ionship be ween  ime equi alence class
and  he wo ld pic u e seen by obse e s, add essing a na u al ques ion: wi hin
he same ime equi alence class, why do die en obse e s p o ide signican ly
die en desc ip ions o ime and geome ic s uc u e? We s a om a se o
undamen al in a ian s ime scale mo he ule
κ(ω)
, ela i e opological class
[K]∈H2(Y, ∂Y ;Z2)
,
K1
class
[u]∈K1(X◦)
o sca e ing amily, and gene al-
ized en opy a ia ion da a
Sgen, δ2S el
 o dene a unied equi alence ela ion o
imegeome y opology on he o al space
Y=M×X◦
. We hen in oduce he ob-
se e p ole ca ego y
Obs
, whose elemen s consis o esolu ion, coupling s uc u e,
and coa se-g aining ules, and cons uc a p ojec ion unc o
FO
om he in a i-
an laye o obse able ime geome y. We p o e:
FO
mus ac o ize h ough he
ime equi alence class, meaning all die ences be ween die en obse e s can only
a ise om mul i-scale s uc u es, phase s uc u es, and laye s esembling smoo h
s uc u es, bu canno change he unde lying causal o de and opological ledge .
On his basis, we dis inguish and geome ize h ee ypes o seeing die en ly:
(1) Mul i-scale sel -simila i y and ac al-like beha io : dene he ac ion o scale
ans o ma ion semig oup
Rs
on ime equi alence classes, p opose a igo ous deni-
ion o mul i-scale sel -simila ime geome y, and p o ide a sol able one-dimensional
sca e ing model; (2) Phase ansi ions and phase s uc u e o ime geome y: in o-
duce o de pa ame e s and c i ical mani olds o ime geome y in pa ame e space,
dis inguishing die en he modynamic phases on he same equi alence class om
 opological phase ansi ions (jumps in
[K]
o
[u]
); (3) 4D opological analogy
and exo ic ime s uc u es: using F eedman's p oo o he ou -dimensional opo-
logical gene alized Poinca é conjec u e, Donaldson's cons ain s on smoo h ou -
dimensional mani olds, and he exis ence o exo ic
R4
as e e ence, we p opose a
pic u e o  opological ypesmoo h ype sepa a ion o ime geome y and dene
a wo king concep o exo ic ime s uc u e. Th ough his we ob ain an analogy:
ime equi alence class co esponds o he  opological ype o ime geome y, while
ime mani olds seen by die en obse e s co espond o die en smoo h/phase
s uc u es on he same opological ype.
1
Finally, we p o ide a  e-laye opological ela ion diag am ep esen ed in me -
maid, o ganizing he in a ian laye , ca ie laye , s uc u e laye , phase/phenomenon
laye , and obse a ion/enginee ing laye in o a igo ous concep ual geome ic pic-
u e. Appendices p o ide de ailed ca ego ied deni ions and p oo s o ime equi -
alence class and obse e p ojec ion, analy ical de i a ion o ac als and phase
ansi ions in one-dimensional sca e ing oy models, and ma hema ical backg ound
synopsis o se e al heo ems and p oposi ions in ol ed in he 4D opological analogy.
Keywo ds:
Time Equi alence Class; Modula Time; Obse e P ojec ion; Mul i-scale
Sel -simila i y; Phase T ansi ion; 4D Topology; Exo ic Smoo h S uc u e; Gene alized
En opy;
K1
Class;
Z2
Holonomy

1 In oduc ion
The idea o ime equi alence class can be b iey desc ibed as: gi en causal s uc u e
and bounda y sca e ing backg ound, he e exis s a mo he ime scale
[τ]
such ha
all physically accep able ime pa ame iza ions a e anely equi alen o i , and i s cali-
b a ion is uni o mly de e mined by sca e ing phase g adien , ela i e s a e densi y, and
ace o Wigne Smi h g oup delay. The conce n o his pape is no he cons uc ion
o his amewo k i sel , bu u he philosophical and echnical de ails: e en wi hin he
same ime equi alence class, die en obse e s' wo ld pic u essubjec i e expe ience
o ime, di ision o geome y and ma e ial phases, dis inc ion be ween mac oscopic and
mic oscopics ill exhibi signican die ences. Wha is he oo o his die ence?
In exis ing wo k, ime equi alence class has been mainly used o uni y: (i) ime
scale a sca e ingspec al shi g oup delay end; (ii) he mal ime and a ow o ime
a modula ow and gene alized en opy end; (iii) Killing ime, ADM ime, null geodesic
ane pa ame e , and cosmological con o mal ime a geome ic end. This pape ex ends
he iew o he p ojec ion mechanism o obse e s, a emp ing o answe he ollowing
ques ions wi hin a igo ous ma hema ical amewo k:
1. Wi hin he same ime equi alence class, wha s uc u es de e mine ha die en
obse e s see die en ly;
2. Whe he his die ence can be unde s ood as ac als, mul i-scale sel -simila i y,
phase ansi ions, o phenomena simila o  opological ypesmoo h ype sepa a-
ion in ou -dimensional opology;
3. How o cons uc a unied geome ic opologicalin o ma ion amewo k inco po-
a ing hese h ee ypes o explana ion in o he same concep ual pic u e.
In ou -dimensional opology, F eedman p o ed he opological ou -dimensional gen-
e alized Poinca é conjec u e, ha any opological ou -dimensional homo opy sphe e is
homeomo phic o
S4
, es ablishing a miles one in 4-dimensional opological mani old clas-
sica ion; while Donaldson's gauge in a ian s and cons ain s on in e sec ion o ms show
d ama ic s uc u al die ences be ween smoo h ou -dimensional mani olds and opolog-
ical ou -dimensional mani olds, di ec ly leading o he exis ence o exo ic
R4
: he e exis
inni ely many smoo h mani olds mu ually non-dieomo phic bu homeomo phic o
R4
.
[
?
] In con as , o dimensions
n= 4
,
Rn
admi s no exo ic smoo h s uc u es. [
?
] This
2
phenomenon indica es ha  opologically iden ical and smoo h s uc u e iden ical a e
no longe equi alen in 4 dimensions.
This pape bo ows his pic u e o p opose an analogy: ime equi alence class ac s
as he  opological ype o ime geome y, while a ious ime s uc u es seen by di -
e en obse e s wi hin he same equi alence classincluding ac al-like mul i-scale be-
ha io , ime expe ience in die en he modynamic phases, and e en possible exo ic
ime s uc u esco espond o die en smoo h/phase s uc u es on he same  ime
opological ype.
The main con ibu ions o his pape can be summa ized as:

In oduce a se o imegeome y opology in a ian s
I= (κ(ω),[K],[u], Sgen, δ2S el)
,
and dene ime geome y equi alence class based on his;

Dene obse e p ole ca ego y
Obs
and p ojec ion unc o
FO
, p o e
FO
mus
ac o ize h ough ime equi alence class;

Cons uc scale ans o ma ion semig oup and phase s uc u e on ime equi alence
class, dis inguishing ac al-like beha io , non- opological phase ansi ions, and
opological phase ansi ions;

In oduce wo king deni ion o exo ic ime s uc u e and make analogy wi h 4D
exo ic smoo h s uc u es;

P o ide a  e-laye opological ela ion diag am ep esen ed in me maid, o ganizing
he abo e cons uc ion in o a unied concep ual amewo k;

P o ide igo ous p oo s o se e al key p oposi ions and de ailed de i a ions o one-
dimensional sol able models in appendices.
A icle s uc u e: Sec ion 2 e iews deni ions o ime scale in a ian s and ime equi -
alence class; Sec ion 3 o malizes obse e p ole and p ojec ion unc o ; Sec ion 4 dis-
cusses mul i-scale s uc u e and ac al-like beha io ; Sec ion 5 cons uc s phase s uc u e
and phase ansi ions o ime geome y; Sec ion 6 p o ides 4D opological analogy and
concep o exo ic ime s uc u e; Sec ion 7 discusses and p ospec s; Appendices include
de ailed p oo s and model calcula ions.

2 Time Scale In a ian s and Time Equi alence Class
This sec ion p o ides he ounda ion o his pape 's wo k: ime scale mo he ule
κ(ω)
,
opological class
[K]
,
K1
class
[u]
, gene alized en opy a ia ion da a, and ime equi a-
lence class dened based on hese in a ian s.
2.1 Time Scale Mo he Rule
Le
M
be a Lo en zian mani old wi h bounda y,
X◦
he pa ame e space wi h singula i ies
emo ed,
Y:= M×X◦
. Fo each
x∈X◦
, gi en a pai o sel -adjoin ope a o s
(Hx, H0,x)
,
dene sca e ing ma ix
Sx(ω)
on ene gy window
I⊂R
.
3
Deni ion 2.1
(Time Scale Mo he Rule (Deni ion 2.1))
.
On ene gy window
I
whe e
Bi manK ein and Wigne Smi h condi ions hold, dene
Qx(ω) := −i Sx(ω)†∂ωSx(ω),
Φx(ω) := a g de Sx(ω), φx(ω) := 1
2Φx(ω),
and le ela i e s a e densi y
ρ el,x(ω)
be he de i a i e o ela i e spec al shi unc ion,
hen
κx(ω) := φ′
x(ω)
π=ρ el,x(ω) = 1
2π Qx(ω).
Call
κx(ω)
he ime scale mo he ule .
κ(ω)
is a unc ion dened on
I×X◦
, in a ian unde app op ia e equi alence ans-
o ma ions o sca e ing amilies, hus is a spec alsca e ing in a ian .
2.2 Topological Class
[K]
,
K1
Class
[u]
, and
Z2
Holonomy
Le
Y:= M×X◦
,
∂Y := ∂M ×X◦∪M×∂X◦
.
Deni ion 2.2
(Unied Rela i e Topological Class (Deni ion 2.2))
.
In ela i e coho-
mology g oup
H2(Y, ∂Y ;Z2)
, selec a class
[K]∈H2(Y, ∂Y ;Z2),
whose Künne h decomposi ion can be w i en as
[K] = π∗
Mw2(TM) + X
j
π∗
Mµj⌣ π∗
Xwj+π∗
Xρc1(LS),
whe e
w2(TM)
is he second S ie elWhi ney class,
µj,wj
a e one-dimensional
Z2
classes,
LS
is he sca e ing line bundle,
ρ
is mod-2 educ ion.
Deni ion 2.3
(
K1
Class o Sca e ing Family (Deni ion 2.3))
.
Fo each
x∈X◦
, dene
ela i e Cayley ans o m
ux:= (Hx−i)(Hx+i)−1(H0,x +i)(H0,x −i)−1,
unde app op ia e es ic ed condi ions
ux∈U es
, hus
x7−→ ux, X◦→U es
gi es a class
[u]∈K1(X◦)
.
Addi ionally, in oduce sca e ing squa e- oo p incipal bundle
P√s→X◦
, whose
holonomy gi es
Z2
in a ian
ν√S:π1(X◦)→ {±1},
as p ojec ion o
[K]
on o
H1(X◦;Z2)
componen .
4
2.3 Gene alized En opy Va ia ion Da a
Choose a poin
p
in
M
and a amily o small causal diamonds
Dp, ⊂M
, whose bounda y
c oss-sec ion a ea
A(Σp, )
and olume
Vp,
a e de e mined by me ic
g
.
Deni ion 2.4
(Gene alized En opy (Deni ion 2.4))
.
Dene
Sgen(p, ) = A(Σp, )
4Gℏ+Sou (p, )−Λ
8πG
Vp,
Tp,
,
whe e
Sou
is en opy o ex e nal quan um s a e,
Tp,
is app op ia ely dened eec i e
empe a u e scale.
Hypo hesis 2.5
(Gene alized En opy Va ia ion Condi ion (Pos ula e 2.5))
.
1. Unde
xed olume o xed gene alized ene gy cons ain s,  s -o de a ia ion sa ises
δSgen(p, ) = 0;
2. Second-o de ela i e en opy sa ises
δ2S el(p, )≥0.
In exis ing wo k, using weigh ed ligh - ay ans o ma ions, he abo e condi ions can
be p o en equi alen o local Eins ein equa ions and HollandsWald gauge ene gy non-
nega i i y condi ions. This pape ea s his as pa o imegeome y in a ian s.
2.4 Time Geome y Equi alence Class
We ake ime pa ame iza ion and ime geome y as objec s and in oduce equi alence
ela ion.
Deni ion 2.6
(Ane Equi alence o Time Pa ame e s (Deni ion 2.6))
.
I wo ime
pa ame e s
1, 2
ha e cons an s
a > 0, b ∈R
such ha
2=a 1+b,
hen
1, 2
a e called anely equi alen , w i en
1∼a 2
.
Deni ion 2.7
(Time Geome y Equi alence Class (Deni ion 2.7))
.
Gi en
(M, g)
and
a se o in a ian s
I:= (κ, [K],[u], Sgen, δ2S el)
. I wo se s o ime geome y da a
(g1, 1)
,
(g2, 2)
sa is y:
1. Ha e he same causal o de s uc u e;
2. Co esponding ime scale mo he ule s
κ1, κ2
sa is y
κ2(ω) = c κ1(ω)
(
c > 0
con-
s an );
3. Topological in a ian s sa is y
[K]1= [K]2
,
[u]1= [u]2
,
ν√S,1=ν√S,2
;
4. Gene alized en opy a ia ion da a a e he same o die only by cons an escaling;
hen hey a e said o belong o he same ime geome y equi alence class, w i en
[(g1, 1)] ime = [(g2, 2)] ime.
All equi alence classes o m he se
TimeEq
, called he ime equi alence class space.
This equi alence ela ion comp esses all pu e escaling and  opological isomo -
phism deg ees o eedom, bu p ese es unde lying causal o de and opological ledge ,
which is he p ecise deni ion o same ime equi alence class discussed in his pape .

5

3 Obse e P ole and P ojec ion Func o
This sec ion o malizes he concep o obse e , modeling i as a iple con aining esolu-
ion, coupling, and coa se-g aining, and cons uc ing a p ojec ion unc o om in a ian
laye o obse able ime geome y.
3.1 Obse e P ole
Deni ion 3.1
(Obse e P ole (Deni ion 3.1))
.
An obse e
O
's p ole is a iple
O:= (ΛO, CO,RO),
whe e:
1.
ΛO
is esolu ion pa ame e , desc ibing minimum scales i can esol e in equency
and ime domains;
2.
CO
is coupling s uc u e, desc ibing which deg ees o eedom i in e ac s wi h (e.g.,
couples o which bounda y egions, which elds, which amily o wo ldlines, e c.);
3.
RO
is coa se-g aining ule, desc ibing pa ial ace and coa se-g ain me hod o
deg ees o eedom.
Deno e he class o all obse e p oles as
Obs
.
3.2 Obse e 's Measu emen Window Func ion
Fo gi en
O
and ime scale mo he ule
κ(ω)
, he ime quan i y ac ually measu able by
obse e is ypically a con olu ion:
TO:= ZWO(ω; ΛO, CO,RO)κ(ω) dω,
whe e
WO
is window unc ion de e mined by p ole, encoding equency band limi a ion
( esolu ion), coupling weigh s (which equencies couple mo e s ongly), and eec i e
weigh a enua ion caused by coa se-g aining.
Deni ion 3.2
(Obse e P ojec ion (Deni ion 3.2))
.
Le
I= (κ, [K],[u], Sgen, δ2S el)
.
Fo each
O∈Obs
, dene p ojec ion
FO:I 7−→ ObsTimeO,
whe e
ObsTimeO
is s uc u e con aining he ollowing da a:
1. Obse able ime scale
TO
and i s local pe u ba ions;
2. Topological in o ma ion accessible by
CO
(e.g., whe he can measu e
ν√S(γ)
, ce ain
p ojec ions o
[K]
);
3. Co esponding subjec i e ime indica o (e.g.,
subj
based on local Fishe in o ma-
ion
FQ
);
4. Eec i e a ow o ime and he modynamic/in o ma ion- heo e ic i e e sibili y un-
de gi en coa se-g ain.
ObsTimeO
can be iewed as  ime geome y seen by ha obse e .
6
3.3 Ca ego ical S uc u e and Func o Fac o iza ion
Deno e
In
as ca ego y wi h in a ian s
I
as objec s and isomo phisms p ese ing ime
geome y equi alence class as mo phisms, i.e.,
Obj(In ) = {I},Mo (In ) = {ϕ:I → I′|[I] ime = [I′] ime}.
Deno e
TimeEq
as a o emen ioned ime equi alence class space, na u ally ha ing dis-
c e e ca ego y s uc u e: objec s a e equi alence classes, mo phisms a e iden i ies.
P oposi ion 3.3
(P ojec ion Fac o iza ion (P oposi ion 3.3))
.
Fo any obse e
O∈
Obs
, he e exis unique maps
π:In →TimeEq, GO:TimeEq →ObsTimeO,
such ha
FO=GO◦π.
P oo idea.
By deni ion, i wo in a ian s
I,I′
belong o same ime geome y equi -
alence class, he e exis ane escaling and opological isomo phism co esponding hei
ime geome y and in a ian s. In deni ion o
FO
, window unc ion
WO
and coa se-
g aining ule
RO
depend only on
O
no specic ep esen a i e, hus
FO(I)
and
FO(I′)
die only by epa ame iza ion abso bable by in e nal coo dina e ans o ma ion o
O
.
This means
FO
is cons an on equi alence classes, hus ac o izes h ough quo ien map
π
. Uniqueness comes om uni e sal p ope y o quo ien map. Fo mal p oo in Appendix
A.
□
Physical meaning o P oposi ion 3.3: all die ences be ween obse e s can only come
om
GO
his s uc u e  om equi alence class o obse able ime geome y, bu canno
change unde lying equi alence class i sel . This p o ides ounda ion o subsequen ly
a ibu ing die ences o mul i-scale s uc u e, phase s uc u e, and exo ic s uc u e.

4 Mul i-Scale S uc u e and F ac al-Like Beha io
This sec ion in oduces ac ion o scale ans o ma ion ope a ion
Rs
on ime equi alence
class, and denes mul i-scale sel -simila ime geome y o cha ac e ize wha we in ui i ely
call  ac al ime.
4.1 Scale T ans o ma ion Semig oup
Le
s > 0
be dimensionless scale pa ame e , dene scale ans o ma ion in equency
domain
(Rsκ)(ω) := α(s)κ(β(s)ω),
whe e
α(s), β(s)
a e posi i e unc ions sa is ying semig oup p ope y
Rs◦ Rs′=Rss′.
A ime geome y le el,
Rs
can co espond o coa se-g ain o RG ow, desc ibing
eec i e ime scale om high esolu ion o low esolu ion.
7
Deni ion 4.1
(Scale O bi and Mul i-Scale Sel -Simila i y (Deni ion 4.1))
.
1. Scale
o bi o ime equi alence class
[τ]
is dened as
O([τ]) := {[RsI] ime :s > 0},
whe e
I
is chosen a bi a ily as ep esen a i e o
[τ]
.
2. I he e exis s
s= 1
such ha
[RsI] ime = [I] ime,
hen
[τ]
is called a mul i-scale sel -simila ime equi alence class.
In c i ical sys ems, xed poin s o
Rs
co espond o ac al-like geome y: a each
scale, s a is ical s uc u e o ime scale is in a ian .
4.2 Obse e Scale and F ac al Pe cep ion
Fo obse e
O
's esolu ion
ΛO
, can dene ope a ion ma ching scale ans o ma ion
sO:= (ΛO),
such ha
TO∼ZWO(ω)κ(ω) dω=Z˜
WO(ω) (RsOκ)(ω) dω,
whe e
˜
WO
is escaled window unc ion.
I
[τ]
is mul i-scale sel -simila equi alence class, unde app op ia e no maliza ion,
s a is ical dis ibu ion o
TO
can emain in a ian o exhibi powe -law ans o ma ion
when
ΛO
changes, co esponding o in ui i ely  ac al ime: a coa se and ne le els,
ime noise s uc u e is simila .
P oposi ion 4.2
(In a-Equi alence-Class P ope y o F ac al-Like Beha io (P oposi-
ion 4.2))
.
I
[τ]
is mul i-scale sel -simila ime equi alence class, o any wo obse e s
O1, O2
, he e exis s no maliza ion cons an
c12 >0
such ha hei obse able ime scales
sa is y
TO2≈c12TO1
ha ing same scale exponen in s a is ical sense. In o he wo ds,  ac al-like beha io o
ime is p ope y wi hin equi alence class, no opological p ope y dis inguishing equi a-
lence classes.
P oo omi ed, elies on linea esponse o
Rs
xed poin and s abili y o window
unc ion amily.

5 Phase S uc u e: Phase T ansi ions, Topological Phase
T ansi ions, and Time Expe ience
This sec ion in oduces phase s uc u e o ime geome y, dis inguishing pa s caused by
phase ansi ions om pa s caused by opological jumps in seeing die en ly wi hin
same equi alence class.
8
5.1 Pa ame e Space and Phases
Le
P
be physical pa ame e space (e.g., empe a u e, coupling s eng h, densi y, d i ing
equency, e c.), each poin
p∈ P
co esponds o a se o in a ian s
I(p)
, hus co e-
sponding o ime equi alence class
[τ(p)]
.
Deni ion 5.1
(Phases o Fixed Equi alence Class (Deni ion 5.1))
.
Fix ime equi a-
lence class
[τ0]
, conside
P[τ0]:= {p∈ P : [τ(p)] = [τ0]}.
On
P[τ0]
, in oduce ollowing equi alence ela ion: i he e exis s con inuous pa h
γ:
[0,1] → P[τ0]
connec ing
p1, p2
, and along pa h local obse able ime geome y and he -
modynamic unc ions a e all analy ic, hen
p1, p2
a e said o belong o same phase.
Se o all phases is deno ed
Π([τ0])
.
Thus, die en he modynamic phases wi hin same ime equi alence class a e di -
e en elemen s in
Π([τ0])
.
5.2 Non-Topological Phase T ansi ions and Topological Phase
T ansi ions
Deni ion 5.2
(Non-Topological Phase T ansi ion (Deni ion 5.2))
.
I along some pa h,
he modynamic o co ela ion unc ions exhibi non-analy ic beha io , bu opological
in a ian s
[K],[u], ν√S
emain unchanged, i is called a non- opological phase ansi ion.
Deni ion 5.3
(Topological Phase T ansi ion (Deni ion 5.3))
.
I along pa ame e pa h
γ
a some poin
p∗
, he e exis s
[K](p∗
−)= [K](p∗
+)
o
[u](p∗
−)= [u](p∗
+)
(subsc ip s indica e wo sides o c i ical poin ), hen a opological phase ansi ion is said
o occu a
p∗
.
Ob iously, opological phase ansi ion necessa ily leads o ime equi alence class
change, while non- opological phase ansi ion occu s wi hin same equi alence class.
P oposi ion 5.4
(Phase T ansi ions and Obse e Expe ience (P oposi ion 5.3))
.
1.
Fo non- opological phase ansi ions, obse e
O
's seen ime geome y
ObsTimeO
can be connec ed by con inuous de o ma ion on wo sides o phase, bu ce ain
second-o de o highe -o de esponses exhibi non-analy ici y;
2. Fo opological phase ansi ions, he e exis s a leas one class o opological obse -
ables (e.g.,
ν√S(γ)
o e ex moduli) aking die en alues on wo sides o phase,
in which case ime equi alence class changes.
This p oposi ion explains: d ama ic changes in ime expe ience can ha e wo essen-
ially die en sou ces: one is phase ansi ion wi hin equi alence class (e.g., i ica ion
and aging phenomena), ano he is opological jump be ween equi alence classes.

9