Uni ied Theo y o Obse e –A en ion–Knowledge
G aph
in Compu a ional Uni e se:
Cogni i e Dynamics and Disc e e Geome ic S uc u e
Unde Fini e Resou ces
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 Ucomp = (X, T,C,I),
we sepa a ely cons uc ed disc e e complexi y geome y, disc e e in o ma ion geom-
e y, con ol mani old (M, G) induced by uni ied ime scale, and ask in o ma ion
mani old (SQ, gQ), gi ing on join mani old EQ=M × SQjoin a ia ional p in-
ciple o ime–in o ma ion–complexi y. These s uc u es cha ac e ize “geome y o
compu a ional uni e se i sel ” a on ological le el, bu ha e no ye explici ly in-
oduced ma hema ical objec o “in e nal obse e ”: how does obse e wi h ini e
esou ces selec a en ion, cons uc knowledge g aph, and g adually accumula e
in o ma ion on complexi y–in o ma ion geome y?
This pape , wi hin amewo k o compu a ional uni e se and i s con inuous
geome ic limi , gi es uni ied axioma ic and geome ic desc ip ion o “obse e –
a en ion–knowledge g aph.” We i s o malize obse e as class o s a e machine
wi h ini e memo y
O= (Min ,Σobs,Σac ,P,U),
whe e Min is in e nal memo y s a e space, Σobs is obse a ion symbol space,
Σac is ac ion space, Pis a en ion–obse a ion policy, Uis in e nal upda e ope a o .
Based on his s uc u e we de ine ime-dependen a en ion ope a o
A :X→[0,1],
o equi alen ly isible subse Xa
⊂X, p o ing: a en ion ope a o de ines
on complexi y–in o ma ion geome y o compu a ional uni e se a amily o ime-
dependen “ eachable sec ions,” he eby imposing cons ain s on obse e ’s wo ld-
line.
Second, we o malize knowledge g aph as
G = (V , E , w ,Φ ),
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whe e V is ini e node se , E ⊂V ×V a e ela ion edges, w a e weigh s,
Φ :V → SQis embedding mapping in o ask in o ma ion mani old. We cons uc
knowledge g aph Laplace ope a o ∆ , p o ing ha in sui able limi , spec um o ∆
app oxima es Laplace–Bel ami ope a o on (SQ, gQ), he eby iewing ini e-node
knowledge g aph as “disc e e skele on” on in o ma ion mani old.
Then we in oduce obse e ’s ex ended wo ldline on join mani old
bz( )=(θ( ), ϕ( ), m( ),G , A ),
whe e (θ( ), ϕ( )) ∈ EQis con ol–in o ma ion s a e, m( )∈Min is in e nal
memo y, G and A a e knowledge g aph and a en ion a ime . On basis o ime–
in o ma ion–complexi y join ac ion, we add obse e in e nal cos and knowledge
g aph econs uc ion cos , ob aining ex ended obse a ion–compu a ion ac ion, de-
i ing i s Eule –Lag ange ype condi ions, gi ing a ia ional cha ac e iza ion o
“unde ini e complexi y budge and ini e memo y, how obse e selec s a en ion
and upda es knowledge g aph.”
Finally, we p o e wo ep esen a i e esul s:
1. Unde local Lipschi z and ini e capaci y assump ions, in o ma ion en opy
inc emen obse able by obse e in any ini e ime is subjec o double uppe
bound o complexi y budge and a en ion bandwid h, gi ing class o “ob-
se e e sion ime–in o ma ion inequali y”;
2. Spec al dimension o knowledge g aph ends in long- ime limi owa d local
in o ma ion dimension o ask in o ma ion mani old, showing ha “knowledge
g aph o a ional obse e almos necessa ily app oxima es skele on o ue
in o ma ion geome y in in ini e ime limi .”
This pape lays s uc u al ounda ion a single-obse e le el o subsequen
cons uc ion o “mul i-obse e –consensus geome y–causal ne wo k” heo y, iew-
ing obse e as geome ic objec in e nal o compu a ional uni e se a he han
ex e nal “measu e .”
Keywo ds: Compu a ional uni e se; Obse e heo y; A en ion mechanism; Knowl-
edge g aph; Cogni i e dynamics; Fini e esou ces; Complexi y geome y; In o ma ion
mani old; Spec al con e gence
1 In oduc ion
In axioma ic amewo k o compu a ional uni e se, uni e se is abs ac ed as disc e e
dynamical sys em Ucomp = (X, T,C,I), whe e Xis con igu a ion space, Tis one-s ep
ansi ion ela ion, Cis single-s ep cos , Iis in o ma ion quali y. P e ious wo ks ha e
cons uc ed unde his amewo k:
Disc e e complexi y geome y: cha ac e izing p oblem di icul y and ho izons wi h
complexi y dis ance dcomp, complexi y olume, and disc e e Ricci cu a u e;
Disc e e in o ma ion geome y and ask in o ma ion mani old (SQ, gQ,ΦQ): em-
bedding ask- ele an isible s a es in o in o ma ion mani old h ough obse a ion
ope a o amilies and ela i e en opy s uc u e;
Con ol mani old (M, G) induced by uni ied ime scale: cons uc ing complexi y
me ic h ough sca e ing mo he scale κ(ω) and g oup delay ma ix Q(ω;θ);
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Join a ia ional p inciple o ime–in o ma ion–complexi y on join mani old EQ=
M×SQ: geome izing “op imal algo i hms” as minimal wo ldlines.
These s uc u es essen ially desc ibe “how uni e se e ol es” and “how in o ma ion is
s o ed and p opaga es in uni e se,” bu ha e no ye explici ly desc ibed “how obse e
in e nal o uni e se ac s on hese s uc u es.”
Obse e s ha e ollowing cha ac e is ics:
1. Fini e a en ion: a any momen , can only access small pa o X, o analyze local
egion o in o ma ion mani old SQ;
2. Fini e memo y: capaci y o in e nal s a e m( ) is ini e, can only s o e ini e-
dimensional summa y;
3. Knowledge g aph: long- e m accumula ed cogni i e s uc u e can be iewed as
ini e-node g aph embedding SQ, being comp essi e app oxima ion o in o ma ion
mani old;
4. Resou ce cons ain s: numbe o compu a ional s eps and in o ma ion acquisi ion
amoun execu able a e limi ed by complexi y budge and ime budge .
The e o e, o cha ac e ize obse e wi hin compu a ional uni e se, we need o su-
pe impose on exis ing geome ic s uc u e ano he laye o “cogni i e geome y”: how
a en ion selec s submani olds, how knowledge g aph cons uc s skele on on in o ma ion
mani old, how hese choices a e cons ained by complexi y geome y and in o ma ion
geome y, and how obse e op imizes i s cogni i e beha io unde esou ce cons ain s.
Goal o his pape can be summa ized as:
Unde p emise o gi en uni ied ime scale and complexi y–in o ma ion geom-
e y, gi e uni ied axioma ic and a ia ional geome ic desc ip ion o single
obse e ’s a en ion, knowledge g aph, and cogni i e dynamics.
Subsequen mul i-obse e and consensus geome y can be cons uc ed on his basis
h ough jux aposi ion and in e ac ion o mul iple obse e objec s.
2 Obse e Objec s in Compu a ional Uni e se
This sec ion de ines obse e objec s in compu a ional uni e se, gi ing basic in e ace
be ween hem and compu a ional uni e se.
2.1 In e nal S uc u e o Obse e
De ini ion 2.1 (Obse e Objec ).In compu a ional uni e se Ucomp = (X, T,C,I), an
obse e objec
O= (Min ,Σobs,Σac ,P,U)
consis s o ollowing componen s:
1. In e nal memo y s a e space Min : coun able o ini e se , ep esen ing obse e ’s
in e nal cogni i e s a e;
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2. Obse a ion symbol space Σobs: ini e se , ep esen ing symbols (o symbol ec o s)
ob ained om single obse a ion;
3. Ac ion space Σac : ini e se , ep esen ing con ol o que y ac ions obse e imposes
on uni e se;
4. A en ion–obse a ion policy
P:Min →∆(Σac ),
ep esen ing dis ibu ion o ac ion selec ion unde in e nal s a e m∈Min ;
5. In e nal upda e ope a o
U:Min ×Σobs →Min ,
ep esen ing how o upda e in e nal memo y unde cu en in e nal s a e and ob-
se a ion esul .
Fo simpli ica ion, we assume a any disc e e ime s ep k:
1. Uni e se is in con igu a ion xk∈X, obse e in e nal s a e is mk∈Min ;
2. Obse e d aws ac ion ak∈Σac om P(mk);
3. Uni e se gene a es obse a ion symbol ok∈Σobs acco ding o akand xk(i s dis i-
bu ion de e mined by uni e se–obse a ion coupling mechanism);
4. Obse e upda es in e nal s a e mk+1 =U(mk, ok).
Con igu a ion e olu ion xk→xk+1 o uni e se de e mined by Tand possibly con ol
mechanism a ec ed by ak.
2.2 A en ion Ope a o
A disc e e le el, we o malize obse e ’s “a en ion” as ime-dependen weigh unc ion
on con igu a ion space X.
De ini ion 2.2 (Disc e e A en ion Ope a o ).A ime s ep k, obse e ’s a en ion op-
e a o is unc ion
Ak:X→[0,1],
sa is ying no maliza ion condi ion
X
x∈X
Ak(x) = 1,
o weake cons ain (e.g., o al mass no exceeding some cons an ). We call
Xa
k={x∈X:Ak(x)>0}
isible con igu a ion subse a ime k.
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In ui i ely, Ak(x) ep esen s obse e ’s cu en a en ion weigh on con igu a ion x,
ypically concen a ed a ound con igu a ion ajec o y o some local egion.
In con inuous limi , we p e e o cha ac e ize a en ion on ask in o ma ion mani old.
De ini ion 2.3 (A en ion Densi y on In o ma ion Mani old).Unde ask Q, a en ion
can be iewed as p obabili y densi y ρ (ϕ) on in o ma ion mani old SQ, sa is ying
ρ (ϕ)≥0,ZSQ
ρ (ϕ) dµgQ(ϕ) = 1,
whe e dµgQis olume elemen o gQ.
Unde embedding ΦQ:X→ SQ, disc e e Akand con inuous ρ can co espond
h ough push o wa d and sampling.
3 Knowledge G aph as Disc e e Skele on o In o -
ma ion Mani old
This sec ion o malizes obse e ’s knowledge g aph, embedding i in o ask in o ma ion
mani old, ob aining b idge be ween disc e e skele on and con inuous in o ma ion geome-
y.
3.1 De ini ion o Knowledge G aph
De ini ion 3.1 (Knowledge G aph a Time ).Obse e ’s knowledge g aph a ime is
quad uple
G = (V , E , w ,Φ ),
whe e:
1. V is ini e node se , each node ep esen ing a “concep ” o “abs ac s a e”;
2. E ⊂V ×V is di ec ed o undi ec ed edge se , ep esen ing ela ionships be ween
concep s (such as causali y, implica ion, simila i y, e c.);
3. w :E →(0,∞) a e edge weigh s, ep esen ing ela ionship s eng h;
4. Embedding mapping
Φ :V → SQ,
embeds each node in o some poin in ask in o ma ion mani old, making knowledge
g aph become ini e sampling skele on o SQ.
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3.2 Consis ency o G aph Laplace and In o ma ion Laplace
On knowledge g aph G de ine undi ec ed edge se e
E and symme ic weigh s ew , con-
s uc ing g aph Laplace ope a o
(∆ )( ) = X
u∼ ew ( , u) (u)− ( ), :V →R.
On o he hand, in o ma ion mani old has Laplace–Bel ami ope a o
∆gQ (ϕ) = 1
pde gQ(ϕ)∂iqde gQ(ϕ)gij
Q(ϕ)∂j (ϕ).
We hope ha in limi o su icien ly la ge and su icien ly dense V , spec um o ∆
app oxima es spec um o ∆gQ.
De ini ion 3.2 (Spec al App oxima ion).Knowledge g aph G is said o spec ally
app oxima e on in o ma ion mani old (SQ, gQ) i he e exis embedding Φ :V → SQ
and app op ia e weigh no maliza ion such ha :
1. Φ (V ) becomes dense in SQas → ∞;
2. Ke nel weigh s ew cons uc ed based on Φ sa is y ha g aph Laplace ∆ unde
app op ia e scaling Γ-con e ges o ∆gQ.
This se up is consis en wi h g aph Laplace con e gence heo y in mani old lea n-
ing, only he e in e p e ed as “obse e ’s knowledge g aph’s asymp o ic app oxima ion o
in o ma ion mani old.”
4 Obse e Ex ended Wo ldline and Cogni i e Dy-
namics
This sec ion combines obse e wi h con ol–in o ma ion geome y, ob aining ex ended
join s a e space and wo ldline.
4.1 Ex ended S a e Space
De ine ex ended s a e space o obse e –uni e se join
b
EQ=M×SQ×Min ×G×A,
whe e:
1. Mis con ol mani old, SQis ask in o ma ion mani old;
2. Min is in e nal memo y s a e space;
3. Gis collec ion o all ini e knowledge g aphs;
4. Ais collec ion o all a en ion con igu a ions (e.g., p obabili y densi y ρ o disc e e
weigh s Ak).
Unde ime pa ame iza ion, join ajec o y o obse e –uni e se is
bz( ) = θ( ), ϕ( ), m( ),G , A .
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4.2 Obse a ion–Compu a ion Ac ion
We add obse e in e nal cos and knowledge g aph upda e cos on basis o p e ious
ime–in o ma ion–complexi y ac ion AQ.
Le
2
M( ) = Gab(θ( )) ˙
θa˙
θb, 2
SQ( ) = gij(ϕ( )) ˙
ϕi˙
ϕj.
De ine ollowing e ms:
1. Complexi y kine ic ene gy e m
Kcomp( ) = 1
2α2 2
M( );
2. In o ma ion kine ic ene gy e m
Kin o( ) = 1
2β2 2
SQ( );
3. Knowledge po en ial ene gy e m
UQ(ϕ( )) = IQ(ϕ( )),
whe e IQis ask in o ma ion quali y unc ion;
4. Knowledge g aph upda e cos e m
RKG( ) = λKG DG +d ,G ,
whe e Dis dis ance be ween g aphs (e.g., spec al dis ance o G omo –Wasse s ein
dis ance);
5. A en ion con igu a ion cos e m
Ra ( ) = λa Ca (A ),
e.g., in o m o en opy egula iza ion o bandwid h cons ain .
De ini ion 4.1 (Obse e –Compu a ion Join Ac ion).
b
AQ[bz(·)] = ZT
0Kcomp( ) + Kin o( )−γ UQ(ϕ( )) + RKG( ) + Ra ( )d .
Minimizing b
AQgi es “op imal” obse a ion–compu a ion–lea ning s a egy unde i-
ni e esou ces.
5 In o ma ion Accumula ion and A en ion–Complexi y
Inequali y
This sec ion gi es ep esen a i e “obse e e sion ime–in o ma ion inequali y”: unde
complexi y budge and a en ion bandwid h cons ain s, in o ma ion amoun obse e
can accumula e in ini e ime has uppe bound.
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5.1 In o ma ion Accumula ion Ra e
Le HQ( ) ep esen obse e ’s knowledge amoun unde ask Q, can be aken as sum o
in o ma ion en opy o ela i e en opy on in e nal knowledge g aph nodes, e.g.,
HQ( ) = X
∈V
π ( )IQ(Φ ( )),
whe e π is weigh dis ibu ion on knowledge g aph nodes. In o ma ion accumula ion
a e is
˙
HQ( ) = d
d HQ( ).
We connec his wi h complexi y eloci y and a en ion bandwid h.
5.2 A en ion Bandwid h and Fishe Ra e
Assume a each momen , obse e samples in o ma ion mani old h ough a en ion
densi y ρ (ϕ), i s single-s ep Fishe in o ma ion acquisi ion a e J( ) associa ed wi h
a en ion bandwid h, e.g.,
J( ) = ZSQ
ρ (ϕ)∇IQ(ϕ)2
gQdµgQ(ϕ).
Unde complexi y–in o ma ion join a ia ional amewo k, Lipschi z ela ionship ex-
is s be ween 2
SQ( ) and J( ).
5.3 In o ma ion Accumula ion Inequali y
Unde local Lipschi z condi ions and ini e a en ion bandwid h cons ain s, ollowing
inequali y can be p o ed.
Theo em 5.1 (Obse e In o ma ion Accumula ion Uppe Bound).Assume:
1. Task in o ma ion quali y unc ion IQis Lipschi z on SQwi h bounded g adien :
he e exis LI, CI>0such ha
∇IQ(ϕ)gQ≤CI,∀ϕ∈ SQ;
2. Second momen o obse e a en ion densi y ρ is bounded, i.e., he e exis s Ba >0
such ha
ZSQ
ρ (ϕ)d2
SQ(ϕ, ¯
ϕ) dµgQ(ϕ)≤Ba ,
o some ixed poin ¯
ϕand all ∈[0, T];
3. Obse e ’s complexi y budge is
Cmax =ZT
0qGab(θ( )) ˙
θa˙
θbd .
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Then he e exis s cons an K > 0, depending only on CI, Ba and join geome ic
s uc u e, such ha
HQ(T)−HQ(0) ≤K Cmax.
P oo in Appendix D.1.
This inequali y s a es: unde uni ied ime scale and geome ic cons ain s, in o ma-
ion amoun obse e can accumula e has linea uppe bound wi h espec o a ailable
complexi y esou ces, a en ion only changes p opo ionali y cons an wi hou changing
linea o m.
6 Knowledge G aph Dimension Con e gence and In-
o ma ion Mani old Skele on
This sec ion p o es ha unde sui able condi ions, spec al dimension o obse e ’s knowl-
edge g aph con e ges in long- ime limi o local in o ma ion dimension o ask in o ma ion
mani old.
6.1 Spec al Dimension o Knowledge G aph
Fo knowledge g aph G , le λ( )
1≤λ( )
2≤ · · · be eigen alue sequence o g aph Laplace
ope a o −∆ . De ine spec al dimension
dspec( ) = −2 lim
ε↓0
log T exp(ε∆ )
log ε,
i his limi exis s. In ui i ely, dspec( ) desc ibes e ec i e dimension o g aph a small
scales.
6.2 Local In o ma ion Dimension o In o ma ion Mani old
On in o ma ion mani old (SQ, gQ), local in o ma ion dimension can be de ined as
din o,Q(ϕ0) = lim
R→0
log µgQBR(ϕ0)
log R,
whe e BR(ϕ0) is geodesic ball o adius Rnea ϕ0.
6.3 Con e gence Theo em
Theo em 6.1 (Con e gence o Knowledge G aph Spec al Dimension).Assume:
1. Obse e ’s knowledge g aph G = (V , E , w ,Φ )spec ally app oxima es on (SQ, gQ)
as → ∞;
2. Obse e ’s long- e m a en ion co e s compac egion K⊂ SQ, and Φ (V )⊂K o
su icien ly la ge ;
3. Fo any ϕ0in K, local in o ma ion dimension din o,Q(ϕ0)exis s and is cons an
din o,Q.
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