Uni ied Compu a ional Uni e se Te minal Objec :
Disc e e Complexi y Geome y, In o ma ion
Geome y,
Mul i-Obse e Causal Ne wo k, and Capabili y–Risk
S uc u e
Haobo Ma1Wenlin Zhang2
1Independen Resea che
2Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
This pape cons uc s on ounda ion o p e ious “compu a ional uni e se” se-
ies wo ks a uni ied compu a ional uni e se e minal objec wi h explici
ca ego ical meaning. P e ious wo ks ha e axioma ized compu a ional uni e se as
ou - uple
Ucomp = (X, T,C,I),
and es ablished on i : disc e e complexi y geome y (complexi y dis ance, ol-
ume g ow h, and disc e e Ricci cu a u e), disc e e in o ma ion geome y ( ask in-
o ma ion mani old (SQ, gQ) and embedding ΦQ), con ol mani old (M, G) induced
by uni ied ime scale and ime–in o ma ion–complexi y join a ia ional p inciple,
mul i-obse e consensus geome y and causal ne wo k, opological complexi y and
undecidabili y, as well as heo y o uni e sal ca as ophic sa e y and capabili y– isk
on ie .
Goal o his pape is o uni y hese disc e e and con inuous, geome ic and
logical, single-obse e and mul i-obse e , capabili y and isk s uc u es in o single
ca ego ical objec
U e m
comp
—called uni ied compu a ional uni e se e minal objec . Speci ically, we
pe o m ollowing s eps:
1. De ine compu a ional uni e se ca ego y wi h uni ied ime scale CompUni κ:
objec s a e compu a ional uni e ses Ucomp sa is ying axioms, mo phisms a e
“sa e simula ion maps” simul aneously p ese ing complexi y geome y, in o -
ma ion geome y, and ca as ophe speci ica ions.
2. Cons uc 2-laye s uc u e on his ca ego y: one laye is disc e e con igu a ion–
e en –causal diamond laye ; one laye is con inuous con ol–in o ma ion ge-
ome y laye , wi h mul i-obse e ne wo k, knowledge g aph amilies, and
capabili y– isk on ie placed on op.
1
3. P o e he e exis s objec
U e m
comp =X,Gcomp,Gin o,Eobs,Sca ,FCR,
and o each Ucomp ∈CompUni κ“con ac ion” mo phism
FU:Ucomp →U e m
comp,
such ha :
FUa disc e e le el is embedding o con igu a ion–e en –diamond, a
con inuous le el is embedding o con ol–in o ma ion–obse e s a es;
FUp ese es complexi y dis ance and uni ied ime scale (a mos linea
escaling);
FUmakes all ask in o ma ion geome y and mul i-obse e consensus
geome y become some class o “submani old–subne wo k” on U e m
comp;
FUmaps ca as ophe speci ica ions and capabili y– isk on ie o sub-
s uc u es o Sca and FCR.
4. P o e in na u al 2-ca ego y sense (allowing na u al ans o ma ions be ween
mo phisms), U e m
comp sa is ies “ e minal objec ” p ope y: o any wo such
uni ied objec s mo phisms be ween hem ha e unique (in na u al isomo phism
sense) ac o iza ion.
5. Finally using p e iously es ablished physical uni e se–compu a ional uni e se
ca ego ical equi alence, cons uc co espondence be ween uni ied physical
uni e se e minal objec U e m
phys and U e m
comp, and explain bo h a e equi alen
e minal objec s unde uni ied ime scale, bounda y diamonds, obse e ne -
wo k, and capabili y– isk s uc u e.
This pape hus gi es ul ima e uni ied desc ip ion o “uni e se as compu a ion”
unde pu ely disc e e, axioma ic amewo k: all conc e e ini e o local compu-
a ional uni e ses con ac h ough sa e–geome ic–in o ma ion compa ible mo -
phisms o same uni ied compu a ional uni e se e minal objec , which simul ane-
ously plays e minal objec ole a ca ego ical, geome ic, and logical le els.
Keywo ds: Compu a ional uni e se; Te minal objec ; Ca ego y heo y; Uni ied ime
scale; Complexi y geome y; In o ma ion geome y; Mul i-obse e consensus; Capabili y–
isk on ie
1 In oduc ion
In p e ious se ies wo ks, we ha e successi ely cons uc ed axioma iza ion, disc e e com-
plexi y geome y, disc e e in o ma ion geome y, uni ied ime scale amewo k, ime–
in o ma ion–complexi y join a ia ional p inciple, mul i-obse e consensus geome y,
opological complexi y and undecidabili y, uni e sal ca as ophic sa e y, and capabili y–
isk on ie heo y o “compu a ional uni e se”.
This pape aims o uni y all hese s uc u es in o single ca ego ical objec —uni ied
compu a ional uni e se e minal objec U e m
comp. This objec plays “ e minal objec ”
ole in compu a ional uni e se ca ego y: e e y speci ic compu a ional uni e se can be
embedded in o i h ough unique (up o na u al isomo phism) s uc u e-p ese ing map,
2
and all p e ious geome ic, in o ma ion, mul i-obse e , and sa e y s uc u es can be
iewed as subs uc u es o uni ied objec .
Te minal objec cons uc ion no only p o ides ul ima e uni ied desc ip ion o compu-
a ional uni e se heo y, bu also es ablishes o mal co espondence wi h physical uni e se
ca ego y h ough ca ego ical equi alence: uni ied physical uni e se e minal objec U e m
phys
and uni ied compu a ional uni e se e minal objec U e m
comp a e equi alen p esen a ions o
same ma hema ical objec in wo equi alen ca ego ies.
Pape s uc u e: Sec ion 2 o ganizes p e ious sca e ed s uc u es in o 2-ca ego y
amewo k wi h 2-mo phisms. Sec ion 3 gi es s uc u al da a o uni ied compu a ional
uni e se e minal objec : om disc e e con igu a ion–e en –diamond o con inuous con ol–
in o ma ion–obse e –capabili y– isk. Sec ion 4 de ines uni ied compu a ional uni e se
e minal objec and p o es i s e minal objec p ope y in 2-ca ego y sense. Sec ion 5
cons uc s equi alence wi h uni ied physical uni e se e minal objec . Appendices gi e
echnical cons uc ion de ails and o mal p oo ou lines.
2 Compu a ional Uni e se 2-Ca ego y Unde Uni-
ied Time Scale
This sec ion o ganizes p e ious sca e ed s uc u es in o 2-ca ego y amewo k wi h 2-
mo phisms.
2.1 Compu a ional Uni e se Objec s wi h Uni ied Time Scale
De ini ion 2.1 (Compu a ional Uni e se wi h Uni ied Time Scale).A compu a ional
uni e se objec wi h uni ied ime scale is se en- uple
b
Ucomp = (X, T,C,I;M, G;SQ, gQ),
whe e:
1. (X, T,C,I) is compu a ional uni e se sa is ying p e ious axioms:
Xcoun able;
T⊂X×Xlocal ini e deg ee;
Csingle-s ep cos posi i e and pa h-addi i e;
I ask in o ma ion quali y baseline.
2. (M, G) is con ol mani old and complexi y me ic cons uc ed om uni ied ime
scale sca e ing mas e scale, sa is ying Riemannian limi p ope y o disc e e com-
plexi y dis ance dcomp: o each local eachable egion he e exis s e inemen amily
{X(h)}and map Φh:X(h)→ M, such ha
d(h)
comp(x, y)→dGΦh(x),Φh(y).
3. (SQ, gQ) is in o ma ion mani old and Fishe in o ma ion me ic o ask Q, wi h
embedding
3
ΦQ:X→ SQ,
such ha disc e e Jensen–Shannon in o ma ion dis ance dJS,Q(x, y) locally consis-
en wi h dSQΦQ(x),ΦQ(y).
We call such objec s cons i u e “0-laye objec s” o ca ego y CompUni κ.
2.2 Sa e–Geome ic–In o ma ion Compa ible 1-Mo phisms
De ini ion 2.2 (Sa e–Geome ic–In o ma ion Compa ible Simula ion Mo phism).Gi en
wo compu a ional uni e ses wi h uni ied ime scale
b
Ucomp = (X, T,C,I;M, G;SQ, gQ),
b
U′
comp = (X′,T′,C′,I′;M′, G′;S′
Q, g′
Q),
a 1-mo phism
F:b
Ucomp →b
U′
comp
consis s o ollowing da a:
1. Con igu a ion map X:X→X′, being p e iously de ined simula ion map (p e-
se ing s ep s uc u e, cos con ol, and in o ma ion quali y mono onici y), wi h
cons an s αX, βX>0 and mono one unc ion Ψ such ha
(x, y)∈T⇒( X(x), X(y)) ∈T′,
d′
comp( X(x), X(y)) ≤αXdcomp(x, y) + βX,
I(x)≤ΨI′( X(x)).
2. Con ol mani old map M:M→M′, o me ics G, G′sa is ying Lipschi z–
bila e al con ol: he e exis αM, βM>0 such ha
αMGθ( , )≤G′
M(θ)d M , d M ≤βMGθ( , ).
3. In o ma ion mani old map S:SQ→ S′
Q, o Fishe me ic sa is ying simila
Lipschi z–bila e al con ol, and compa ible wi h X, i.e., he e exis s na u al ans-
o ma ion ηXsuch ha
S◦ΦQ≃Φ′
Q◦ X.
4. Sa e y speci ica ion compa ibili y: i he e exis s ca as ophe se Cca ⊂Xon b
Ucomp,
hen i s image C′
ca = X(Cca )⊂X′ emains ca as ophe se , and Fdoes no map
sa e poin s in o ca as ophe poin s, i.e., isible sa e y–ca as ophe pa i ion unde
map p ese es o “blun s owa d sa e y side”.
Such 1-mo phisms simul aneously p ese e disc e e–con inuous geome y and ca as-
ophe speci ica ions. All objec s and 1-mo phisms cons i u e ca ego y CompUni κ.
4
2.3 2-Mo phisms and Na u al T ans o ma ions
A con ol and in o ma ion mani old le el, wo 1-mo phisms may ha e “con inuous de o -
ma ions”, co esponding in ca ego y heo y o 2-mo phisms—na u al ans o ma ions.
De ini ion 2.3 (Na u al T ans o ma ion as 2-Mo phism).Gi en wo 1-mo phisms
F, G :b
Ucomp →b
U′
comp,
a 2-mo phism
Ξ : F⇒G
includes:
1. Con igu a ion side na u al ans o ma ion ΞX, usually amily o local in e ible
maps on X′, such ha GX≃ΞX◦FX;
2. Con ol and in o ma ion side na u al ans o ma ions ΞM,ΞS, gi ing in me ic com-
pa ible sense GM≃ΞM◦FM,GS≃ΞS◦FS;
3. On sa e y s uc u e no b eaking coa se s uc u e o ca as ophe se and capabili y–
isk on ie .
Thus, CompUni κhas 2-ca ego y s uc u e.
3 S uc u al Da a o Uni ied Compu a ional Uni e se
Te minal Objec
This sec ion gi es speci ic composi ion o uni ied compu a ional uni e se e minal objec
U e m
comp
om disc e e con igu a ion–e en –diamond o con inuous con ol–in o ma ion–obse e –
capabili y– isk.
3.1 Disc e e Laye : Maximal Con igu a ion Uni e se and E en –
Diamond S uc u e
De ini ion 3.1 (Maximal Con igu a ion Uni e se).Le Xbe “amalgama ion o all coun -
able con igu a ion se s and hei ini e ep esen a ions” unde some la ge ca dinal con ol,
o mally cons uc ible h ough G o hendieck uni e se Uand se heo y as
X=[
Ucomp∈CompUni κ
ιU(X),
whe e ιUis embedding o each Xin o common supe se . De ine on Xuni ied ansi ion
ela ion
T∞=[
Ucomp
ιU(TU),
5
cos unc ion
C∞=[
Ucomp
ιU(CU),
a con lic s h ough equi alence class iden i ica ion (i.e., me ging geome ically equi -
alen upda e ules om di e en uni e ses in o single objec ). Thus ob ain “maximal
global complexi y g aph” con aining all local compu a ion s uc u es
G(1)
comp = (X,T∞,C∞).
A e en laye
E=X×N,
can de ine uni ied causal pa ial o de and complexi y ligh cone; hus any ini e budge
causal diamond ♢can be iewed as subg aph o G(1)
comp.
3.2 Con inuous Laye : Te minal Objec s o Uni ied Con ol
Mani old and In o ma ion Mani old
A con ol and in o ma ion geome y le el, p e ious wo ks al eady cons uc ed o each
b
Ucomp con ol mani old (MU, GU) and in o ma ion mani old (SQ,U , gQ,U ).
De ini ion 3.2 (Uni ied Con ol Mani old Te minal Objec ).Le
M=a
Ucomp
MU.∼,
whe e ∼is “ ime scale–complexi y isome y equi alence”: i he e exis s isome ic
embedding o con ol–sca e ing ealiza ion, iden i y co esponding poin s. Th ough his
amalgama ion ob ain la ge mani old o s acky objec M, wi h uni ied me ic G, locally
consis en wi h each GU.
De ini ion 3.3 (Uni ied In o ma ion Mani old Te minal Objec ).Simila ly o all asks
Qand uni e se Ucomp, cons uc in o ma ion mani old amily SQ,U and pe o m simila
amalgama ion, ob aining uni ied in o ma ion mani old
S=a
Q,U
SQ,U .∼,
ca ying piecewise Fishe me ic g.
Thus, any conc e e compu a ional uni e se’s con ol–in o ma ion geome y can be
embedded in o (M,G) and (S,g) as submani olds; unde uni ied ime scale mas e scale
and sca e ing s uc u e, hese embeddings p ese e geodesic s uc u e and in o ma ion
s uc u e.
6
3.3 Mul i-Obse e Ne wo k Laye
Taking single obse e objec
O= (Min ,Σobs,Σac ,P,U)
as basic uni , iew all coun able obse e amilies in uni ied compu a ional uni e se
as collec ion o poin s (θ, ϕ, m, G, A).
De ini ion 3.4 (Uni ied Obse e S a e Space).De ine
Eobs =[
Ucomp,O
Y
i∈I
E(i)
Q×M(i)
in ×G(i)×A(i).∼,
whe e E(i)
Q=M(i)
U×S(i)
Q,U ,G(i)is knowledge g aph space, A(i)is a en ion con igu a ion
space, ∼iden i ies geome ically equi alen and s a egy equi alen s a es.
Join me ic composed o sum o each GU, gQ,U and knowledge g aph spec al dis ance.
3.4 Ca as ophe Speci ica ion and Capabili y–Risk F on ie Laye
Pe o m simila amalgama ion o ca as ophe speci ica ions and capabili y– isk s uc-
u es de ined in all compu a ional uni e ses.
De ini ion 3.5 (Uni ied Ca as ophe Speci ica ion Laye ).De ine ca as ophe speci ica-
ion amily
Sca =[
Ucomp
{(X0,U , Cca ,U )}.∼,
whe e equi alence ela ion iden i ies equi alen ini ial s a e se s and ca as ophe se s
unde uni ied embedding.
De ini ion 3.6 (Uni ied Capabili y–Risk F on ie Laye ).Fo each (Ucomp, Q), capabili y–
isk on ie is F(U,Q)
CR ⊂R×[0,1]. Embedding all hese on ie s as pa ame ized amily
in o uni ied s a egy space, ob ain o e all s uc u e
FCR =[
U,Q
F(U,Q)
CR × {(U, Q)}.∼.
On uni ied con ol–obse e s a egy space, FCR is piecewise Pa e o bounda y se ,
desc ibing capabili y– isk limi cu es in all possible compu a ional uni e ses.
4 Uni ied Compu a ional Uni e se Te minal Objec
and Te minal Objec P ope y
This sec ion me ges all abo e laye s, gi es de ini ion o uni ied compu a ional uni e se
e minal objec , and p o es i s e minal objec p ope y in 2-ca ego y sense.
7
4.1 De ini ion o Te minal Objec
De ini ion 4.1 (Uni ied Compu a ional Uni e se Te minal Objec ).Uni ied compu a-
ional uni e se e minal objec de ined as en- uple
U e m
comp =X,T∞,C∞,I∞;M,G;S,g;Eobs;Sca ;FCR,
whe e each pa de ined as in 3.1–3.6.
In ui i ely, U e m
comp simul aneously con ains:
All locally ealizable con igu a ions and upda es;
All con ol pa h complexi y geome y limi s unde uni ied ime scale;
All ask in o ma ion geome ies and obse e ne wo ks;
All ca as ophe speci ica ions and capabili y– isk on ie s.
4.2 Uni ied Con ac ion Func o
De ini ion 4.2 (Con ac ion 1-Mo phism).Fo each b
Ucomp ∈CompUni κ, de ine
FU:b
Ucomp →U e m
comp
as ollows:
1. Con igu a ion embedding
X=ιU:X ,→X.
2. Con ol mani old embedding
M:MU,→M,
p ese ing me ic s uc u e (a mos cons an escaling).
3. In o ma ion mani old embedding
S:SQ,U ,→S,
p ese ing Fishe in o ma ion me ic.
4. Obse e and knowledge g aph embedding
obs : ObsS a es(Ucomp),→Eobs.
5. Ca as ophe speci ica ion and capabili y– isk on ie embedding
ca : (X0, Cca )7→ Sca , CR :F(U,Q)
CR 7→ FCR.
Combining hese FUis sa e–geome ic–in o ma ion compa ible 1-mo phism.
8
4.3 Te minal Objec P ope y
Theo em 4.3 (2-Te minal Objec P ope y o Uni ied Compu a ional Uni e se Te minal
Objec ).In 2-ca ego y CompUni κ,U e m
comp is 2- e minal objec :
Fo each objec b
Ucomp, he e exis s 1-mo phism
FU:b
Ucomp →U e m
comp,
and o any o he 1-mo phism
GU:b
Ucomp →U e m
comp,
he e exis s unique (in 2-mo phism sense) na u al ans o ma ion
ΞU:GU⇒FU.
P oo (Ou line). 1. Exis ence: By cons uc ion in 4.2, o each b
Ucomp can explici ly
de ine FUas embedding.
2. Uniqueness (up o na u al ans o ma ion): Any o he 1-mo phism GUmus
map X o some subse o X, and p ese e s uc u e a con ol–in o ma ion–obse e –
capabili y– isk laye s. Since U e m
comp cons uc ed by “maximal global equi alence
class”, o any such GU he e exis s “isome y–equi alence class” na u al ans o -
ma ion ΞUpulling i back o canonical embedding FU: a each laye , can elimina e
edundan deg ees o eedom h ough in e nal isome y o local uni a y ans o -
ma ion, making map isomo phic o FU.
3. Na u ali y: Fo mo phism H:b
Ucomp →b
Vcomp, na u al ans o ma ion be ween
wo-side composi ion FV◦Hand FUgi en by uni ied cons uc ion— his is s anda d
ea u e o “inclusion–isome y” s uc u e in G o hendieck uni e se.
Rigo ous 2-ca ego y p oo equi es checking compa ibili y a each laye , see Appendix
C.
5 Equi alence wi h Uni ied Physical Uni e se Te -
minal Objec
P e ious wo ks al eady cons uc ed physical uni e se ca ego y PhysUni QCA and com-
pu a ional uni e se ca ego y CompUni phys, and h ough unc o s
F:PhysUni QCA →CompUni phys, G :CompUni phys →PhysUni QCA
p o ed bo h a e ca ego ically equi alen . A physical le el, uni ied physical uni e se
e minal objec U e m
phys al eady cons uc ed h ough sca e ing ime scale, bounda y ime
geome y, and Di ac–QCA con inuous limi .
In cu en scena io, om inclusion CompUni phys ,→CompUni κand ca ego ical
equi alence, can li e minal objec ela ionship:
9
