Uni ied Time Scale and Con inuous Complexi y
Geome y
o Compu a ional Uni e ses:
Sca e ing Mo he Scale, Con ol Mani old, and
Cons uc ion o Me ic G
Haobo Ma1Wenlin Zhang2
1Independen Resea che
2Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
In p e ious wo ks, we axioma ized he “compu a ional uni e se” as disc e e
objec Ucomp = (X, T,C,I), and sepa a ely cons uc ed disc e e complexi y geom-
e y and disc e e in o ma ion geome y on i . Howe e , in ha amewo k, he
single-s ep cos unc ion C emained abs ac ly assigned, wi h i s connec ion o
eal physical ime scales no ye sys ema ically cha ac e ized. This pape , based on
he uni ied ime scale sca e ing mo he scale
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω),
in oduces “con ol mani old” Mand sca e ing amily S(ω;θ), sys ema ically
embedding he cos o disc e e s eps in compu a ional uni e se in o a Riemannian-
ype me ic Ginduced by κ(ω), he eby cons uc ing con inuous complexi y geom-
e y consis en wi h physical ime scales.
Speci ically, we i s iew each physically ealizable compu a ional uni e se
Ucomp as combina ion o some con ollable sca e ing sys em: con igu a ion up-
da es a e d i en by con ol pa ame e θ∈ M, sca e ing ma ix S(ω;θ) desc ibes
physical esponse in equency domain, Wigne –Smi h g oup delay ma ix Q(ω;θ)
gi es local esponse o uni ied ime scale densi y. Subsequen ly, we de ine me ic
Gab(θ) = ZΩ
w(ω) ∂aQ(ω;θ)∂bQ(ω;θ)dω
and p o e: unde na u al egula i y assump ions, Gis posi i e de ini e wi h
good co a iance unde con ol coo dina e ans o ma ions and in e nal gauge ans-
o ma ions; u he mo e, o any su icien ly smoo h con ol pa h θ( ), i s leng h
induced by G
LG[θ] = ZT
0qGab(θ( )) ˙
θa( )˙
θb( ) d
1
in app op ia e disc e e limi is equi alen o con inuous e sion o disc e e com-
plexi y dis ance.
We also p o e: o amily o compu a ional uni e ses {U(h)
comp} e ined a dis-
c e e scale h→0, i hei single-s ep cos s a e cons uc ed om uni ied ime scale
sca e ing esponse, hen con igu a ion g aph dis ance d(h)con e ges in G omo –
Hausdo sense o geodesic dis ance dGon con ol mani old. This gi es igo ous
b idge om comple ely disc e e compu a ional uni e se o con inuous complexi y
geome y.
Finally, we discuss na u ali y o his con inuous complexi y geome y in ca ego -
ical sense: aking con ol mani old and i s me ic Gas geome ic image o “com-
pu a ional uni e se objec s,” we can cons uc ca ego y C lSca wi h con ol–
sca e ing pai s (M, S) as objec s, p o ing exis ence o unc o s uc u e be ween
disc e e compu a ional uni e se ca ego y CompUni and C lSca p ese ing
complexi y dis ance. This es ablishes con inuous geome ic ounda ion o subse-
quen ly es ablishing ca ego ical equi alence be ween “physical uni e se ca ego y ↔
compu a ional uni e se ca ego y.”
Keywo ds: Compu a ional uni e se; Uni ied ime scale; Sca e ing mo he scale; Con-
ol mani old; Complexi y geome y; Riemannian me ic; Wigne –Smi h delay; G omo –
Hausdo con e gence; Ca ego y heo y
1 In oduc ion
In he “compu a ional uni e se” scheme, he en i e uni e se is iewed as disc e e dy-
namical sys em on coun able con igu a ion se X: one-s ep upda e ela ion T⊂X×X
de e mines eachabili y om one con igu a ion o ano he , single-s ep cos unc ion C:
X×X→[0,∞] assigns ime/ene gy and o he esou ce cos s o each upda e. P e ious
wo k has p o en: unde axiom assump ions o ini e in o ma ion densi y and local up-
da e, one can iew (X, T,C) as weigh ed g aph, cons uc ing geome ic objec s such as
complexi y dis ance d(x, y), complexi y ball olume, complexi y dimension, and disc e e
Ricci cu a u e, es ablishing “disc e e complexi y geome y.” Simul aneously, h ough
obse a ion ope a o amilies and ask-awa e ela i e en opy, we cons uc ed “disc e e
in o ma ion geome y” on con igu a ion space, making in o ma ion dimension and com-
plexi y dimension ha e na u al inequali y ela ions.
Howe e , o uly uni y compu a ional uni e se wi h physical uni e se, abs ac cos
unc ion Calone is insu icien . We need o answe wo key ques ions:
1. How does single-s ep cos C(x, y) ela e o physical ime scale?
2. Can disc e e complexi y dis ance d(x, y) be app oxima ed by geodesic dis ance dG
o some con inuous me ic Ginduced by physical ime scale?
Fo his pu pose, his pape in oduces uni ied ime scale sca e ing mo he scale. Fo
physical sca e ing sys em, le i s sca e ing ma ix be S(ω), hen o al sca e ing phase
φ(ω), spec al shi unc ion de i a i e ρ el(ω), and Wigne –Smi h g oup delay ma ix
Q(ω) sa is y mo he o mula
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω),
2
iewing uni ied ime scale densi y κ(ω) as “ ime uni on each equency band.” When
embedding compu a ional uni e se in o physical sys em composed o con ollable sca e -
ing p ocesses, each s ep upda e can be unde s ood as con ol ope a ion on some sca e ing
ma ix amily S(ω;θ); hus, single-s ep cos can na u ally be cons uc ed om esponse
in eg al o g oup delay ma ix Q(ω;θ).
The main pu pose o his pape is o sys ema ically comple e his cons uc ion and
p o e consis ency be ween esul ing me ic Gab(θ) and disc e e complexi y geome y.
Full ex s uc u e: Sec ion 2 e iews uni ied ime scale sca e ing mo he scale, in o-
ducing sca e ing amily unde con ol pa ame iza ion. Sec ion 3 cons uc s complexi y
me ic Gon con ol mani old, discussing i s basic p ope ies and gauge in a iance. Sec-
ion 4 p o es disc e e complexi y dis ance consis en wi h geodesic dis ance dGin con ol
mani old limi , gi ing ep esen a i e examples. Sec ion 5 in oduces con ol–sca e ing
objec ca ego y C lSca , discussing unc o ela ions wi h compu a ional uni e se ca e-
go y CompUni . Appendices p o ide de ailed p oo s o main p oposi ions and heo ems.
2 Uni ied Time Scale and Con ol Sca e ing Family
This sec ion e iews uni ied ime scale sca e ing mo he scale, in oducing sca e ing
amily S(ω;θ) unde con ol pa ame iza ion.
2.1 Re iew o Uni ied Time Scale Sca e ing Mo he Scale
Le H0, H be pai o sel -adjoin ope a o s sa is ying app op ia e aceable pe u ba ion
condi ions, making wa e ope a o s exis and comple e. Co esponding sca e ing ope a o
is
S=W∗
+W−,
in equency domain ep esen a ion can be w i en as equency- esol ed sca e ing
ma ix amily S(ω). Le o al sca e ing phase
φ(ω) = a g de S(ω)
and spec al shi unc ion ξ(ω) sa is y Bi man–K ein o mula
de S(ω) = exp −2πiξ(ω).
Wigne –Smi h g oup delay ma ix de ined as
Q(ω) = −iS(ω)†∂ωS(ω).
Unde egula assump ions, he e exis s uni ied ime scale densi y
κ(ω) = φ′(ω)/π =ξ′(ω) = ρ el(ω) = (2π)−1 Q(ω),
whe e ρ el is ela i e s a e densi y unc ion, Q(ω) is ace o g oup delay ma ix.
This mo he o mula shows ha sca e ing o al phase de i a i e, spec al shi unc ion
de i a i e, and g oup delay ace a e consis en modulo cons an s, allowing κ(ω) o be
iewed as “ ime scale densi y on equency domain.”
3
2.2 Con ol Mani old and Sca e ing Family S(ω;θ)
In compu a ional uni e se, we conside physically ealizable compu a ional sys em whose
con ollable pa ame e s o m ini e-dimensional mani old M, wi h coo dina es deno ed
θ= (θ1, . . . , θd).
De ini ion 2.1 (Con ol Mani old and Sca e ing Family).A con ol–sca e ing sys em
consis s o ollowing da a:
1. Con ol mani old M, being d-dimensional di e en iable mani old;
2. Fo each θ∈ M and equency ω, assigning uni a y sca e ing ma ix S(ω;θ),
di e en iable in θ, ω, sa is ying mo he o mula condi ion in ω;
3. Fo each θ, de ining g oup delay ma ix
Q(ω;θ) = −iS(ω;θ)†∂ωS(ω;θ).
We conside changes in θas co esponding o con ol ope a ions on compu a ional
sys em, such as adjus ing ga e pa ame e s, coupling s eng hs, o ex e nal ields; while
S(ω;θ) is physical sca e ing s uc u e a ached o his con ol poin .
2.3 Connec ion wi h Compu a ional Uni e se
Fo gi en compu a ional uni e se Ucomp = (X, T,C,I), i each s ep upda e can be ealized
h ough con ol pa h o some con ol–sca e ing sys em, he e exis s ollowing connec ion
s uc u e:
1. Family o con ol pa hs θk( ) (e.g., piecewise cons an ), each co esponding o a
class o disc e e upda e sequences xk→xk+1;
2. Fo each s ep (xk, xk+1), he e exis con ol pa ame e θk∈ M and physical ime
window [ k, k+1], such ha in his window sys em’s sca e ing ma ix is gi en by
S(ω;θk);
3. Single-s ep cos C(xk, xk+1) can be exp essed using some in eg al unc ion o uni ied
ime scale densi y κ(ω) and Q(ω;θk).
Typical cons uc ion is: o each s ep (x, y), single-s ep physical ime cos is
τ(x, y) = ZΩx,y
κ(ω) dµx,y(ω),
whe e Ωx,y is equency band in ol ed in his s ep upda e, µx,y is co esponding spec-
al measu e. I we de ine C(x, y) as app op ia e scaling o τ(x, y), hen disc e e com-
plexi y dis ance can be iewed as physical ime o al unde uni ied ime scale.
The goal o his pape is o go beyond s ep-by-s ep in eg a ion le el, di ec ly con-
s uc ing me ic Gon con ol mani old such ha geome ic leng h along con ol pa h
ag ees wi h disc e e complexi y dis ance in app op ia e limi .
4
3 Complexi y Me ic GInduced by Uni ied Time
Scale
This sec ion cons uc s me ic Gon con ol mani old Mand analyzes i s basic p ope ies.
3.1 Cons uc ion o Me ic
Deno e ∂a=∂/∂θa. Fo each θ∈ M and equency ω, g oup delay ma ix Q(ω;θ)
is ini e-dimensional He mi ian ma ix. Conside i s de i a i e wi h espec o con ol
pa ame e
∂aQ(ω;θ).
To scale impo ance o di e en equency bands, in oduce non-nega i e weigh unc-
ion w(ω) sa is ying
ZΩ
w(ω) dω < ∞,
whe e Ω is equency band se o in e es .
De ini ion 3.1 (Me ic Induced by Uni ied Time Scale).On con ol mani old Mde ine
second-o de enso
Gab(θ) = ZΩ
w(ω) ∂aQ(ω;θ)∂bQ(ω;θ)dω.
I Gab(θ) is posi i e de ini e a e e y poin , hen Gis Riemannian me ic on M, called
complexi y me ic induced by uni ied ime scale.
In ui i ely, Gab(θ) measu es “quad a ic change in ensi y o uni ied ime scale esponse
when making in ini esimal changes in con ol di ec ions aand b.”
3.2 Posi i e De ini eness and Degene a e Di ec ions
P oposi ion 3.2 (Posi i e De ini eness Condi ion).I o any nonze o angen ec o
= a∂a∈TθM, he e exis s se o equencies ω∈Ωsuch ha
∂ Q(ω;θ) = a∂aQ(ω;θ)= 0,
and ace inne p oduc o ∂ Q(ω;θ)o e his equency band
ZΩ
w(ω) ∂ Q(ω;θ)∂ Q(ω;θ)dω > 0,
hen Gab(θ)is posi i e de ini e a θ.
This p oposi ion s a es ha me ic posi i e de ini eness depends on whe he con ol
di ec ion is “obse able” unde uni ied ime scale: i o some di ec ion , g oup delay
ma ix Q(ω;θ) is insensi i e ac oss all equency bands, i.e., ∂ Q≡0, hen his di ec ion
con ibu es no hing o ime scale, co esponding o me ic degene acy; con e sely, i he e
exis s nonze o esponse and weigh w(ω) does no cancel his equency band, hen Gis
posi i e in his di ec ion.
5
In ac ual modeling, one can quo ien ou pu e gauge di ec ions (con ol deg ees o ee-
dom ine ec i e o ime scale) h ough quo ien space ope a ion on con ol coo dina es,
ob aining non-degene a e me ic.
3.3 Geome ic Leng h and Physical Time o Con ol Pa h
Unde me ic G, leng h o di e en iable con ol pa h θ: [0, T]→ M de ined as
LG[θ] = ZT
0qGab(θ( )) ˙
θa( )˙
θb( ) d .
P oposi ion 3.3 (Rela ion Be ween Leng h and Uni ied Time Scale).Unde app op ia e
egula i y and sepa a ion o a iables assump ions, complexi y leng h LG[θ]induced by
con ol pa h θ( )has p opo ional ela ion wi h physical ime scale in eg al accumula ed
along his pa h, i.e., he e exis s cons an α > 0such ha
LG[θ] = αZT
0
d ZΩ
w(ω)κ(ω;θ( ))2dω,
whe e κ(ω;θ) = (2π)−1 Q(ω;θ)is uni ied ime scale densi y.
Key he e is no ing aceable ela ionship be ween
∂aQ(ω;θ)
and
∂aκ(ω;θ),
and h ough app op ia e no maliza ion, pa h leng h can be in e p e ed as squa e oo
o m o “ene gy- ype in eg al” o uni ied ime scale densi y on con ol mani old.
4 Con inuous Limi o Disc e e Complexi y Dis ance
This sec ion connec s uni ied ime scale induced me ic Gwi h p e ious disc e e com-
plexi y geome y, p o ing ha in app op ia e e inemen limi , complexi y dis ance on
con igu a ion g aph con e ges o geodesic dis ance on con ol mani old.
4.1 Disc e e Con ol G id and Con igu a ion G aph
Conside amily o compu a ional uni e ses {U(h)
comp}labeled by h > 0, whose con ol
deg ees o eedom a e disc e ized in o g id M(h)⊂ M, e.g.,
M(h)={θ∈ M :θ=θ0+hn, n ∈Zd} ∩ K,
whe e K⊂ M is compac se . Fo each θ∈ M(h), compu a ional uni e se U(h)
comp
locally ealizes upda es h ough con ol pa ame e θ, de ining i s con igu a ion g aph
complexi y dis ance d(h).
We assume single-s ep cos C(h)(x, y) on con ol g id has uni ied ime scale in e p e-
a ion: when con ol changes om θ o adjacen poin θ+hea, co esponding single-s ep
cos is
6
C(h)(θ, θ +hea) = c(θ)h+o(h),
whe e
c(θ) = X
a,b
Gab(θ) a b1/2
o some uni ec o gi es local eloci y.
4.2 Gene al Theo em o Dis ance Con e gence
Theo em 4.1 (Riemannian Limi o Complexi y Dis ance).Le (M, G)be con ol mani-
old induced by uni ied ime scale, {U(h)
comp} amily o compu a ional uni e ses wi h con ol
g id M(h), wi h co esponding complexi y dis ance d(h). Assume:
1. Fo any θ∈K⊂ M, he e exis s poin θ(h)∈ M(h)such ha θ(h)→θ;
2. Single-s ep cos C(h)(θ(h), θ(h)+hea) = pGaa(θ)h+o(h), wi h simila consis ency
o o he di ec ions;
3. Reachabili y s uc u e o con igu a ion g aph consis en wi h adjacency ela ion o
con ol g id, no “jumping” ex a edges.
Then as h→0, o any θ1, θ2∈K,
lim
h→0d(h)(θ(h)
1, θ(h)
2) = dG(θ1, θ2),
whe e dGis geodesic dis ance o Riemannian me ic G.
This heo em is high-dimensional gene aliza ion o one-dimensional esul om p e-
ious pape . I shows: as long as disc e e con ol s ep size consis en wi h local eloci y
induced by uni ied ime scale, disc e e complexi y dis ance app oxima es Riemannian
geodesic dis ance in e inemen limi .
4.3 Rep esen a i e Example: One-dimensional Two-po Sca -
e ing Ne wo k
Conside one-dimensional wo-po sca e ing ne wo k wi h sca e ing ma ix
S(ω;θ) = (ω;θ) ′(ω;θ)
(ω;θ) ′(ω;θ),
whe e θ∈[θmin, θmax]⊂Ris some con ol pa ame e (e.g., po en ial well dep h o
phase shi ). Fo each θ, g oup delay ma ix
Q(ω;θ) = −iS(ω;θ)†∂ωS(ω;θ)
is 2 ×2 He mi ian ma ix.
Unde app op ia e egula i y, me ic
G(θ) = ZΩ
w(ω) ∂θQ(ω;θ)∂θQ(ω;θ)dω
7
de ines one-dimensional Riemannian me ic G(θ)dθ2. I we disc e ize con ol g id in o
poin se θn=θ0+nh wi h s ep size h, se ing single-s ep cos
C(h)(θn, θn+1) = pG(θn)h,
hen o any θ1, θ2,
lim
h→0d(h)(θ(h)
1, θ(h)
2) = Zθ2
θ1pG(θ) dθ
.
This gi es ins an ia ion o his pape ’s heo y in conc e e compu able model.
5 Func o S uc u e o Con ol–Sca e ing Ca ego y
and Compu a ional Uni e se Ca ego y
This sec ion examines na u ali y o con ol mani old and me ic G om ca ego ical pe -
spec i e, cons uc ing ca ego y wi h con ol–sca e ing objec s as objec s, es ablishing
unc o ela ions wi h p e ious compu a ional uni e se ca ego y CompUni .
5.1 Con ol–Sca e ing Ca ego y C lSca
De ini ion 5.1 (Con ol–Sca e ing Objec ).A con ol–sca e ing objec is iple
C= (M, G, S),
whe e (M, G) is con ol mani old wi h Riemannian me ic, S(ω;θ) is sca e ing amily
sa is ying uni ied ime scale mo he o mula.
De ini ion 5.2 (Con ol–Sca e ing Mo phism).Be ween wo con ol–sca e ing objec s
C= (M, G, S), C′= (M′, G′, S′), mo phism is mapping :M→M′sa is ying:
1. is smoo h mapping, locally di eomo phism almos e e ywhe e;
2. Me ic is con olled ans o ma ion unde , i.e., he e exis cons an s α, β > 0 such
ha o all angen ec o s ∈TθM,
α Gθ( , )≤G′
(θ)(d θ , d θ )≤β Gθ( , );
3. Sca e ing amily compa ible unde , i.e., S′(ω; (θ)) and S(ω;θ) equi alen in
uni ied ime scale mo he o mula sense.
Wi h con ol–sca e ing objec s as objec s and con ol–sca e ing mo phisms as mo -
phisms, we o m ca ego y C lSca .
8
5.2 Func o om Compu a ional Uni e se o Con ol–Sca e ing
Objec s
Le CompUni phys be subca ego y o compu a ional uni e se objec s sa is ying “ ealiz-
able by uni ied ime scale sca e ing.” We cons uc unc o
F:CompUni phys →C lSca
as ollows:
1. Objec le el: Gi en Ucomp = (X, T,C,I), cons uc con ol mani old M, me ic G,
and sca e ing amily S(ω;θ) om i s physical ealiza ion, se ing
F(Ucomp) = (M, G, S).
2. Mo phism le el: Gi en simula ion mapping :Ucomp ⇝U′
comp be ween compu a-
ional uni e ses, co esponding o physical le el con ol and sca e ing ans o ma-
ion M:M→M′, se
F( ) = M.
P oposi ion 5.3 (Func o iali y).The abo e F o ms co a ian unc o , i.e.:
1. F(idUcomp ) = idF(Ucomp);
2. I :Ucomp →U′
comp,g:U′
comp →U′′
comp a e simula ion mo phisms, hen
F(g◦ ) = F(g)◦F( ).
This unc o li s “ om disc e e compu a ional uni e se o con inuous con ol–sca e ing
geome y” a objec le el, p ese ing con olled a ia ion o complexi y dis ance a mo -
phism le el.
Unde app op ia e egula assump ions, one can u he p o e: he e exis s in e se
cons uc ion G:C lSca →CompUni phys such ha G◦Fand F◦Ga e na u-
ally isomo phic o iden i y unc o espec i ely, making wo ca ego ies equi alen on
“physically ealizable subclass.” Speci ic p oo in ol es disc e izing con inuous con ol–
sca e ing sys em in o QCA- ype uni e se and con olling complexi y o e head, le o
u u e dedica ed discussion.
6 Conclusion
Based on uni ied ime scale sca e ing mo he scale, his pape cons uc ed con inuous
complexi y geome y o compu a ional uni e se: by in oducing con ol mani old M
and sca e ing amily S(ω;θ), using con ol de i a i es o g oup delay ma ix Q(ω;θ) o
cons uc me ic
Gab(θ) = ZΩ
w(ω) ∂aQ(ω;θ)∂bQ(ω;θ)dω,
9
