Time--Information--Complexity Unified Variational Principle\\ in Computational Universes:\\ Computational Worldlines on Control--Scattering Manifold\\ and Task Information Manifold

Time–In o ma ion–Complexi y Uni ied Va ia ional
P inciple
in Compu a ional Uni e ses:
Compu a ional Wo ldlines on Con ol–Sca e ing
Mani old
and Task In o ma ion Mani old
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 on he “compu a ional uni e se” se ies, we abs ac ed he
uni e se as disc e e objec Ucomp = (X, T,C,I), cons uc ing disc e e complexi y
geome y (complexi y dis ance, olume g ow h, and disc e e Ricci cu a u e based
on con igu a ion g aph) and disc e e in o ma ion geome y (based on ask-awa e
ela i e en opy and Fishe s uc u e) on i , and ga e con inuous limi o complexi y
geome y unde uni ied ime scale sca e ing mo he scale: a con ol mani old M
wi h Riemannian me ic G. Howe e , hese geome ic s uc u es s ill sepa a ely
cha ac e ize “ ime/ esou ce cos ” and “in o ma ion quali y/ ask- ele an s a es”,
lacking a amewo k o uni y bo h unde a single a ia ional p inciple.
This pape , building on con ol mani old (M, G) and ask in o ma ion mani old
(SQ, gQ), in oduces join mani old
EQ=M×SQ,
cons uc ing on i a ime–in o ma ion–complexi y join ac ion AQ, he eby cha -
ac e izing “compu a ional ajec o ies” in compu a ional uni e se as minimal cu es
on join mani old (compu a ional wo ldlines). Speci ically, we i s gi e ac ion a
disc e e le el
Adisc
Q(γ) = X
kαC(xk, xk+1) + β din o,Q(xk, xk+1)−γ∆IQ(xk, xk+1),
p o ing ha unde app op ia e scaling, his disc e e ac ion amily Γ-con e ges
as h→0 o con inuous ac ion
AQ[θ(·), ϕ(·)] = ZT
01
2α2Gab(θ)˙
θa˙
θb+1
2β2gij(ϕ)˙
ϕi˙
ϕj−γ UQ(ϕ)d ,
1
whe e θ( )∈ M is con ol ajec o y, ϕ( )∈ SQis ask in o ma ion s a e, UQis
ask- ela ed in o ma ion po en ial unc ion (e.g., nega i e in o ma ion quali y).
Then we de i e Eule –Lag ange equa ions on join mani old EQ, p o ing ha
minimal ajec o ies sa is y coupled “geodesic equa ions wi h po en ial”: con ol
pa e ol es along geodesics o (M, G) bu ecei es eedback om g adien o UQ
wi h espec o ϕ; in o ma ion pa e ol es along geodesics o (SQ, gQ) bu is mod-
ula ed by con ol ajec o y θ. Fu he mo e, using s anda d a ia ional me hods
and Γ-con e gence heo y, we p o e: unde uni ied ime scale and local Lipschi z
assump ions, disc e e op imal compu a ional pa hs con e ge in he limi o minimal
wo ldlines on join mani old, achie ing igo ous co espondence be ween “op imal
algo i hms in disc e e compu a ional uni e se” and “con inuous ime–in o ma ion–
complexi y wo ldlines.”
This pape concludes wi h discussion o minimiza ion p oblems wi h esou ce
cons ain s: maximizing ask in o ma ion quali y unde ixed ime budge o com-
plexi y budge . We gi e equi alen Lag ange mul iplie o m, he eby cha ac e izing
“op imal in o ma ion acquisi ion s a egy unde gi en budge ” as a class o geodesic
lows wi h e ec i e po en ial. Resul s o his pape p o ide a ia ional ounda ion
a in insic dynamics le el o subsequen cons uc ion o ca ego ical equi alence
be ween “compu a ional uni e se ↔physical uni e se.”
Keywo ds: Compu a ional uni e se; Va ia ional p inciple; Complexi y geome y; In o -
ma ion geome y; Join mani old; Compu a ional wo ldline; Eule –Lag ange equa ions;
Γ-con e gence; Resou ce cons ain s
1 In oduc ion
F om he “compu a ional uni e se” pe spec i e, he en i e uni e se is abs ac ed as dis-
c e e dynamical sys em: one-s ep upda e ela ion Ton con igu a ion space Xand single-
s ep cos Cdesc ibe esou ces needed o go om one s a e o ano he ; in o ma ion quali y
unc ion Ie alua es a ask le el he “goodness” o a con igu a ion ela i e o goals. P e-
ious wo ks showed ha unde axioms o ini e in o ma ion densi y and local upda e,
(X, T,C) can be iewed as complexi y g aph, cons uc ing complexi y dis ance, com-
plexi y ball olume, complexi y dimension, and disc e e Ricci cu a u e, he eby using
disc e e geome y o cha ac e ize “p oblem di icul y” and “ho izon s uc u e”; simul a-
neously, h ough obse a ion ope a o amilies and ask-awa e ela i e en opy, we de ined
in o ma ion dis ance and in o ma ion balls on con igu a ion space, geome izing “ ask-
ele an dis inguishabili y.”
Unde uni ied ime scale sca e ing mo he scale, single-s ep cos o compu a ional
uni e se can be iewed as disc e e sampling o ac ual physical ime scale: o physically
ealizable compu a ional p ocesses, he e exis con ol mani old Mand sca e ing ma ix
amily S(ω;θ), such ha con ol de i a i es o g oup delay ma ix Q(ω;θ) induce com-
plexi y me ic G, whe eby disc e e complexi y dis ance app oxima es geodesic dis ance
on (M, G) in e inemen limi . This esul uni ies disc e e complexi y geome y wi h
physical ime scale in o a Riemannian geome ic amewo k.
Howe e , o unde s and “how bes o compu e in ini e ime,” nei he complexi y
geome y no in o ma ion geome y alone su ices:

Complexi y geome y conce ns “how a a eled, how much ime/ esou ce spen ”;
2

In o ma ion geome y conce ns “how a mo ed in ask space, how much in o ma-
ion gained”;

The uly meaning ul ques ion is: unde gi en ime/complexi y budge , how o
each bes possible endpoin in in o ma ion geome y.
This na u ally leads o a join a ia ional p oblem: in join space, o gi en ask, ind
minimal/maximal ajec o y conside ing bo h ime cos and in o ma ion bene i .
This pape , building on con ol mani old (M, G) and ask in o ma ion mani old
(SQ, gQ), cons uc s join mani old EQ=M × SQ, de ining on i a ime–in o ma ion–
complexi y join ac ion AQ. Disc e e compu a ional pa hs become piecewise linea ap-
p oxima ions on join mani old, con inuous compu a ional wo ldlines a e smoo h cu es
on EQ. Using Γ-con e gence and classical a ia ional me hods, we p o e disc e e op imal
pa hs con e ge in he limi o con inuous minimal wo ldlines, he eby geome izing he
p oblem o “op imal algo i hms” as he p oblem o “op imal wo ldlines.”
2 Uni ied No a ion: Compu a ional Uni e se, Com-
plexi y Geome y, and In o ma ion Geome y
This sec ion b ie ly summa izes main objec s and no a ion used in p e ious wo ks o
subsequen uni ied easoning.
2.1 Compu a ional Uni e se Objec
A compu a ional uni e se objec is quad uple Ucomp = (X, T,C,I), whe e:
1. Xis coun able con igu a ion se ;
2. T⊂X×Xis one-s ep upda e ela ion;
3. C:X×X→[0,∞] is single-s ep cos , wi h C(x, y) = ∞i (x, y)/∈T,C(x, y)∈
(0,∞) i (x, y)∈T, addi i e along pa hs;
4. I:X→Ris in o ma ion quali y unc ion (may be ask-dependen ).
Complexi y dis ance de ined as
dcomp(x, y) = in
γ:x→y
C(γ),
whe e pa h γ= (x0, . . . , xn) sa is ies x0=x, xn=y, and (xk, xk+1)∈T.
2.2 Complexi y Geome y and Con ol Mani old
Unde uni ied ime scale amewo k, o physically ealizable compu a ional uni e se he e
exis con ol mani old Mand sca e ing ma ix amily S(ω;θ), whose g oup delay ma ix
Q(ω;θ) = −iS†∂ωShas con ol de i a i es inducing complexi y me ic
Gab(θ) = ZΩ
w(ω) ∂aQ(ω;θ)∂bQ(ω;θ)dω.
Unde app op ia e posi i e de ini eness condi ions, (M, G) is Riemannian mani old,
disc e e complexi y dis ance con e ges o geodesic dis ance dGin e inemen limi .
3
2.3 Task In o ma ion Mani old
Gi en ask Q, h ough obse a ion ope a o amily O={Oj}j∈Jde ine isible s a e
p(Q)
x∈∆(YQ) o con igu a ion x. Unde app op ia e egula i y assump ions, hese isible
s a es can be embedded in o some in o ma ion mani old SQ:

The e exis mapping ΦQ:X→ SQand embedding ΨQ:SQ,→∆(YQ), such ha
ΨQ(ΦQ(x)) ≈p(Q)
x;

Fishe in o ma ion me ic gQgi en by second de i a i e o ela i e en opy, con-
s uc ing Riemannian s uc u e o (SQ, gQ);

In o ma ion dis ance be ween con igu a ions can be ep esen ed using Jensen–Shannon
dis ance o Fishe geodesic dis ance, deno ed din o,Q(x, y)≈dSQ(ΦQ(x),ΦQ(y)).
We call (SQ, gQ,ΦQ) he in o ma ion geome ic da a o ask Q.
3 Join Time–In o ma ion–Complexi y Mani old
Wi h abo e p epa a ion, we cons uc join mani old EQand i s me ic.
3.1 De ini ion o Join Mani old
De ini ion 3.1 (Join Mani old).Fo gi en ask Q, de ine join mani old
EQ=M×SQ.
I s poin z= (θ, ϕ) simul aneously ep esen s “con ol s a e” and “ ask in o ma ion
s a e”. In con inuous limi , s a e o an obse e o algo i hm in compu a ional uni e se
can be iewed as poin in EQ.
3.2 Me ic S uc u e
On EQ, we in oduce p oduc - ype me ic
G=α2G⊕β2gQ,
i.e., o angen ec o = ( M, SQ)∈TθM ⊕ TϕSQ, de ine
Gz( , ) = α2Gθ( M, M) + β2gQ,ϕ( SQ, SQ).
He e α, β > 0 a e weigh pa ame e s used o balance “ eloci y” measu emen in
complexi y di ec ion and in o ma ion di ec ion.
Unde his me ic, eloci y squa ed o join ajec o y
z( )=(θ( ), ϕ( ))
is
|˙z( )|2
G=α2Gab(θ( )) ˙
θa˙
θb+β2gij(ϕ( )) ˙
ϕi˙
ϕj.
Pu e geome ic leng h on join mani old is
4
LG[z] = ZT
0q|˙z( )|2
Gd .
Howe e , leng h alone is insu icien o encode “in o ma ion quali y” gain, we also
need ask- ela ed po en ial unc ion.
3.3 In o ma ion Po en ial Func ion
Le in o ma ion quali y unc ion o ask Qon in o ma ion mani old be w i en as IQ:
SQ→R, o example
IQ(ϕ) = IQ(x) when ϕ= ΦQ(x).
We in oduce in o ma ion po en ial unc ion
UQ(ϕ) = V(IQ(ϕ)),
whe e V:R→Ris mono one unc ion, gene ally chosen as V(u) = uo V(u) =
sa (u) (sa u a ion ype). In his pape , o simplici y we di ec ly ake
UQ(ϕ) = IQ(ϕ),
iewing “in o ma ion quali y” as nega i e con ibu ion o po en ial ene gy e m (co -
esponding o highe in o ma ion quali y b inging lowe ac ion).
4 Disc e e Join Ac ion and Con inuous Limi
This sec ion cons uc s join ac ion o ask Qa disc e e le el, p o ing i s con e gence
o con inuous ac ion in e inemen limi .
4.1 Disc e e Join Ac ion
Conside disc e e compu a ional pa h
γ= (x0, x1, . . . , xn),
whe e (xk, xk+1)∈T. Co esponding complexi y inc emen is
∆Ck=C(xk, xk+1),
in o ma ion dis ance inc emen (unde ask Q) is
∆Dk=din o,Q(xk, xk+1),
in o ma ion quali y inc emen is
∆Ik=IQ(ϕk+1)−IQ(ϕk), ϕk= ΦQ(xk).
5

De ini ion 4.1 (Disc e e Join Ac ion).Fo ask Qand pa h γ, de ine disc e e join
ac ion
Adisc
Q(γ) =
n−1
X
k=0 α∆Ck+β∆Dk−γ∆Ik,
whe e α, β, γ > 0 a e weigh pa ame e s.
In ui i e unde s anding: each s ep upda e simul aneously pays complexi y cos α∆Ck
and in o ma ion adjus men cos β∆Dk, and gains in o ma ion quali y inc emen ∆Ik,
con ibu ing −γ∆Ik o ac ion. Op imal pa h is one ha minimizes Adisc
Qunde balance
o all h ee.
4.2 Re inemen and S anda d Time S ep
To connec disc e e and con inuous, we in oduce disc e e ime s ep h > 0, le pa h leng h
n≈T/h, and se scaling o single-s ep cos and in o ma ion dis ance as
∆Ck=h c(xk, xk+1) + o(h),∆Dk=h d(xk, xk+1) + o(h),
∆Ik=h˙
IQ( k) + o(h),
whe e k=kh,c, d, ˙
IQa e espec i ely complexi y eloci y, in o ma ion eloci y, and
in o ma ion quali y a e o change in con inuous limi .
Unde abo e scaling, disc e e ac ion can be app oxima ed as Riemann sum
Adisc
Q(γ)≈
n−1
X
k=0
hα ck+β dk−γ˙
IQ( k)→ZT
0αc( ) + βd( )−γ˙
IQ( )d .
To ma ch geome ic s uc u e, we ep esen c( ), d( ) espec i ely using eloci y no ms
on (M, G) and (SQ, gQ).
4.3 Con inuous Join Ac ion
Le con ol pa h be θ: [0, T]→ M, in o ma ion pa h be ϕ: [0, T]→ SQ, wi h co e-
sponding eloci y no ms
2
M( ) = Gab(θ( )) ˙
θa( )˙
θb( ),
2
SQ( ) = gij(ϕ( )) ˙
ϕi( )˙
ϕj( ).
We choose “ene gy- ype” con inuous ac ion:
De ini ion 4.2 (Con inuous Join Ac ion).
AQ[θ(·), ϕ(·)] = ZT
01
2α2 2
M( ) + 1
2β2 2
SQ( )−γ UQ(ϕ( ))d .
whe e UQ(ϕ) = IQ(ϕ) o some mono one ans o ma ion he eo .
6
This is s anda d “kine ic minus po en ial” o m: i s wo e ms a e kine ic ene gy
on complexi y and in o ma ion geome y, la e e m is ask- ela ed nega i e po en ial
ene gy, minimal wo ldline main ains ini e eloci y while ying o en e egions o lowe
in o ma ion po en ial ene gy.
5 Eule –Lag ange Equa ions and Compu a ional Wo ld-
lines
This sec ion de i es Eule –Lag ange equa ions on join mani old, gi ing dynamical o m
sa is ied by minimal wo ldlines.
5.1 Lag angian and Va ia ion
Le Lag angian be
L(θ, ˙
θ;ϕ, ˙
ϕ) = 1
2α2Gab(θ)˙
θa˙
θb+1
2β2gij(ϕ)˙
ϕi˙
ϕj−γ UQ(ϕ).
Va ying θaand ϕi espec i ely gi es Eule –Lag ange equa ions:
Fo θa:
d
d α2Gab(θ)˙
θb−1
2α2(∂aGbc)(θ)˙
θb˙
θc= 0,
Fo ϕi:
d
d β2gij(ϕ)˙
ϕj−1
2β2(∂igjk)(ϕ)˙
ϕj˙
ϕk+γ ∂iUQ(ϕ)=0.
whe e ∂aGbc =∂Gbc/∂θa,∂igjk =∂gjk/∂ϕi.
5.2 Join Geodesic–Po en ial Equa ions
In s anda d Riemannian geome y, geodesic equa ion can be w i en as
¨
θa+ Γa
bc(θ)˙
θb˙
θc= 0,
whe e Γa
bc a e Ch is o el symbols o Le i–Ci i a connec ion. He e we ew i e con ol
and in o ma ion pa s espec i ely in geodesic–po en ial o m.
Fo con ol a iable θa, le
Γa
bc(θ) = 1
2Gad∂bGdc +∂cGdb −∂dGbc,
whe e Gad is in e se o me ic ma ix. Eule –Lag ange equa ion can be ew i en as
¨
θa+ Γa
bc(θ)˙
θb˙
θc= 0.
Since con ol pa o Lag angian con ains no explici po en ial ene gy, con ol ajec-
o y is geodesic on (M, G).
Fo in o ma ion a iable ϕi, simila ly de ining Γi
jk(ϕ) as Ch is o el symbols o gQ,
Eule –Lag ange equa ion ew i es as
¨
ϕi+ Γi
jk(ϕ)˙
ϕj˙
ϕk=−γ
β2gij(ϕ)∂jUQ(ϕ).
7
Righ -hand-side e m is co a ian li o po en ial ene gy g adien on in o ma ion man-
i old, ep esen ing “d i ing o ce” o “in o ma ion po en ial” on in o ma ion ajec o y.
The e o e, join wo ldline sa is ies ollowing coupled sys em:
1. Con ol pa : e ol es along geodesics o (M, G);
2. In o ma ion pa : e ol es along geodesics o (SQ, gQ), bu d i en away om geodesics
by g adien o UQ.
This can be iewed as special case o “geodesic wi h po en ial on complexi y–in o ma ion
p oduc mani old.”
6Γ-Con e gence o Disc e e–Con inuous Consis ency
To p o e disc e e op imal pa hs con e ge in he limi o con inuous minimal wo ldlines, we
use Γ-con e gence heo y. Only s uc u al heo em and p oo idea gi en he e, echnical
de ails placed in appendix.
6.1 Ac ion Func ional Family
Conside amily o disc e e ime s eps h=T/n, embed disc e e pa h γ(h)= (x0, . . . , xn)
in o piecewise cons an o piecewise linea cu e z(h): [0, T]→ EQ, such ha
z(h)( ) = (θ(h)( ), ϕ(h)( )), ∈[kh, (k+ 1)h),
and z(h)(kh)=(θk, ϕk) co esponds o xk. De ine disc e e ac ion
A(h)
Q[z(h)] =
n−1
X
k=0 1
2α2∆s2
M,k
h+1
2β2∆s2
SQ,k
h−γ UQ(ϕk)h,
whe e ∆s2
M,k =dM(θk, θk+1)2, ∆s2
SQ,k =dSQ(ϕk, ϕk+1)2.
Unde local consis ency assump ions, ∆sM,k ≈hqGab(θ)˙
θa˙
θb, ∆sSQ,k ≈hqgij(ϕ)˙
ϕi˙
ϕj.
6.2 Γ-Con e gence Theo em
Theo em 6.1 (Γ-Con e gence, Schema ic).Unde uni ied ime scale and local egula i y
assump ions, disc e e ac ion unc ional amily {A(h)
Q}h>0Γ-con e ges unde app op ia e
opology (e.g., weak H1 opology o z(h)⇀ z) o con inuous ac ion unc ional
AQ[z] = ZT
01
2α2Gab(θ)˙
θa˙
θb+1
2β2gij(ϕ)˙
ϕi˙
ϕj−γ UQ(ϕ)d .
In pa icula , any limi poin o disc e e minimal sequences is a minimal cu e o
con inuous ac ion.
P oo idea in Appendix B.2, based on s anda d “ene gy- ype unc ional disc e iza-
ion” Γ-con e gence amewo k: lowe semicon inui y gi en by con ex s uc u e and weak
opology lowe semicon inui y, eco e y sequence cons uc ed h ough ime disc e iza ion
o con inuous ajec o y.
8
7 Op imal Compu a ional Wo ldlines Unde Resou ce
Cons ain s
In p ac ical p oblems, we o en ca e abou ollowing op imiza ion:

Unde gi en ime budge To complexi y budge Cmax, maximize e minal in o -
ma ion quali y IQ(ϕ(T));

O unde gi en e minal in o ma ion quali y equi emen I a ge , minimize equi ed
ime o complexi y.
Using Lag ange mul iplie me hod, esou ce cons ain s can be abso bed in o join
ac ion.
Fo example, maximizing IQ(ϕ(T)) unde gi en T, equi alen o minimizing unde
ee endpoin condi ion
e
AQ[z] = ZT
01
2α2 2
M+1
2β2 2
SQd −γ IQ(ϕ(T)),
which di e s om p e ious ac ion only in po en ial ene gy e m. Co esponding
Eule –Lag ange equa ions in bulk egion same as be o e, bu a endpoin add na u al
bounda y condi ion
β2gij(ϕ(T)) ˙
ϕj(T) = γ ∂iIQ(ϕ(T)).
This bounda y condi ion can be iewed as “endpoin e lec ion condi ion”: a end-
poin , a io o in o ma ion eloci y o in o ma ion quali y g adien con olled by pa am-
e e γ/β2, e lec ing p e e ence s eng h o endpoin in o ma ion quali y.
Simila ly, minimizing ime unde gi en in o ma ion quali y a ge can be ob ained
h ough cons ain IQ(ϕ(T)) = I a ge and in oducing mul iplie λ o ge equi alen ee
p oblem, whe eby ob aining se o geodesic–po en ial equa ions wi h global cons ain s.
These a ia ional p oblems p o ide geome ic pe spec i e o “op imal algo i hm de-
sign”: seeking minimal cu es on join mani old EQsa is ying esou ce cons ain s and
endpoin in o ma ion cons ain s is p ecisely seeking op imal compu a ional wo ldlines
in compu a ional uni e se.
A De i a ion o Eule –Lag ange Unde Me ic and
Po en ial
A.1 De ails o Va ia ional De i a ion
Le
L(θ, ˙
θ;ϕ, ˙
ϕ) = 1
2α2Gab(θ)˙
θa˙
θb+1
2β2gij(ϕ)˙
ϕi˙
ϕj−γ UQ(ϕ).
Fo a ia ion θa7→ θa+εηa(wi h ηa(T0) = ηa(T1) = 0) we ha e
δL =1
2α2(∂cGab)ηc˙
θa˙
θb+α2Gab ˙
θa˙ηb.
A e in eg a ion
9