Multi-Observer Consensus Geometry and Causal Network\\ in Computational Universe:\\ Observer Family, Distributed Update,\\ and Discrete Ricci Contraction Under Unified Time Scale

Mul i-Obse e Consensus Geome y and Causal
Ne wo k
in Compu a ional Uni e se:
Obse e Family, Dis ibu ed Upda e,
and Disc e e Ricci Con ac ion Unde Uni ied Time
Scale
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 compu a ional uni e se Ucomp = (X, T,C,I) se ies, we ha e
comple ed cons uc ion o ollowing s uc u al le els:
1. Axioma iza ion o compu a ional uni e se and disc e e complexi y geome y;
2. Task-awa e disc e e in o ma ion geome y and in o ma ion mani old (SQ, gQ);
3. Con ol mani old (M, G) and con inuous complexi y geome y unde uni ied
ime scale sca e ing mo he scale;
4. Time–in o ma ion–complexi y join a ia ional p inciple on join mani old
EQ=M×SQ;
5. Ca ego ical equi alence be ween physical uni e se ca ego y and compu a ional
uni e se ca ego y on e e sible QCA subclass;
6. Uni ied heo y o single obse e ’s a en ion–knowledge g aph–cogni i e dy-
namics;
7. Bounda y compu a ion and causal diamond s uc u e on ini e blocks.
Howe e , in eal uni e se he e does no exis single obse e , bu a he causal
ne wo k o med by mul iple obse e s: hey each ca y ini e memo y and knowl-
edge g aphs, exchanging in o ma ion h ough ini e-bandwid h channels, g adually
o ming o losing consensus unde cons ain s o uni ied ime scale and complexi y
budge . To igo ously cha ac e ize his phenomenon wi hin compu a ional uni-
e se amewo k, his pape p oposes and de elops uni ied heo y o mul i-obse e
consensus geome y and causal ne wo k.
We i s o malize obse e amily as
O={Oi}i∈I,
1
whe e each obse e Oiin compu a ional uni e se has i s own in e nal memo y
space M(i)
in , a en ion ope a o Ai, , knowledge g aph Gi, , and indi idual wo ld-
line zi( ) = (θi( ), ϕi( )) on join mani old EQ. On his basis, we in oduce ime-
dependen di ec ed communica ion g aph C = (I, E , ω ), using i o de ine con-
sensus geome y be ween mul i-obse e s: “dis ance” be ween obse e s is join ly
de e mined by hei in o ma ion mani old embedding ΦQand spec al s uc u e o
knowledge g aphs, Laplace ope a o o communica ion g aph induces class o “con-
sensus Ricci cu a u e” ac ing on obse e dis ibu ion.
Co e esul s o his pape include:
1. In oduce mul i-obse e s a e space
EN
Q=
N
Y
i=1
E(i)
Q,
de ining consensus ene gy unc ional on i
Econs( ) = 1
2X
i,j
wij( )d2
SQ(ϕi( ), ϕj( )),
p o ing ha unde symme ic communica ion g aph and app op ia e Lips-
chi z condi ions, Econs exhibi s exponen ial decay unde uni ied ime scale,
wi h decay a e con olled by “consensus Ricci cu a u e” lowe bound.
2. Glue mul i-obse e knowledge g aphs in o join knowledge g aph Gunion
, p o -
ing ha in spec al sense o g aph Laplace, e ec i e spec al dimension o
his join g aph ends in long- ime limi owa d local in o ma ion dimension
o ask in o ma ion mani old, he eby showing ha “mul i-obse e consen-
sus” geome ically co esponds o skele on app oxima ion o same in o ma ion
mani old.
3. In oduce mul i-obse e causal ne wo k in causal diamond amewo k: o
amily o ime-o de ed obse a ion–communica ion e en s, de ine mul i-obse e
causal diamond ♢mul i, cons uc ing on i s bounda y join bounda y ope a o
Kmul i
♢:O
i
B−
♢,i →O
i
B+
♢,i,
p o ing i s uniqueness unde local uni a y gauge ans o ma ions, compa ible
wi h single-obse e bounda y ope a o s h ough g aph Schu elimina ion.
4. On basis o ime–in o ma ion–complexi y join a ia ional p inciple, cons uc
mul i-obse e join ac ion
b
Amul i
Q=X
ib
A(i)
Q+λcons ZT
0
Econs( ) d ,
de i ing i s Eule –Lag ange ype equa ions, gi ing a ia ional cha ac e iza ion
o mul i-obse e op imal s a egy o “maximizing collec i e ask in o ma ion
quali y unde ini e complexi y budge ”.
This pape lays igo ous compu a ional uni e se ounda ion o subsequen uni-
ied desc ip ion o causal ne wo k splicing, mul i-obse e consensus geome y, and
social–mul i-agen sys ems.
2
Keywo ds: Compu a ional uni e se; Mul i-obse e ; Consensus geome y; Causal ne -
wo k; Communica ion g aph; Ricci cu a u e; Dis ibu ed dynamics; Join ac ion; Spec-
al dimension
1 In oduc ion
In single-obse e heo y, obse e is iewed as in e nal compu a ional p ocess in compu-
a ional uni e se: i has ini e memo y Min , a en ion ope a o A , knowledge g aph G ,
and wo ldline z( ) = (θ( ), ϕ( )) on join mani old EQ=M × SQ. I s beha io desc ibed
by minimiza ion o ime–in o ma ion–complexi y join ac ion b
AQ, subjec o cons ain s
o uni ied ime scale and complexi y budge .
In eal uni e se, obse e s no isola ed, bu o m dynamic causal ne wo k h ough
ini e-bandwid h channels:

A physical le el, hese obse e s may be local subsys ems in physical sys em;

A in o ma ion le el, hey possess ini e memo y and sel -consis en knowledge
g aphs, capable o g adually o ming consensus on some ask Q h ough message
exchange;

A complexi y le el, hey each ha e ini e complexi y budge , communica ion and
compu a ion bo h consume esou ces unde uni ied ime scale.
The e o e, o “compu a ional uni e se” o se e as igo ous amewo k o uni ied
uni e se desc ip ion, mus answe ollowing ques ions a mul i-obse e le el:
1. How o de ine “geome ic dis ance” and “consensus e o ” be ween mul i-obse e s?
2. Gi en communica ion g aph and uni ied ime scale, does collec i e consensus e o
decay o e ime? Is decay a e con olled by some “consensus cu a u e”?
3. How do mul i-obse e knowledge g aphs glue in o join skele on, how does app ox-
ima ion capabili y o ask in o ma ion mani old SQe ol e o e ime?
4. Unde ini e complexi y budge , wha local a en ion selec ion and communica ion
s a egies can “op imally” o m consensus in a ia ional sense?
Main goal o his pape is o gi e uni ied, geome ic, and a ia ional answe o hese
ques ions.
2 Mul i-Obse e Objec s and Join S a e Space
This sec ion de ines mul i-obse e amily on basis o single-obse e objec s, cons uc ing
mul i-obse e join s a e space.
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2.1 Mul i-Obse e Objec Family
Recall single-obse e objec
O= (Min ,Σobs,Σac ,P,U),
whe e Min is in e nal memo y s a e space, Pis a en ion–ac ion policy, Uis in e nal
upda e ope a o .
De ini ion 2.1 (Mul i-Obse e Family).In compu a ional uni e se Ucomp = (X, T,C,I),
a mul i-obse e amily consis s o se Iand obse e objec se
O={Oi}i∈I
whe e each Oi= (M(i)
in ,Σ(i)
obs,Σ(i)
ac ,P(i),U(i)). We assume:
1. Iis ini e o coun able;
2. Each M(i)
in is ini e se o di ec p oduc o ini e-dimensional egis e s;
3. Fo all obse e s Oi, hei obse a ion and ac ion can be ep esen ed h ough com-
pu a ional uni e se upda e ela ion Tand ask obse a ion ope a o amily OQ.
2.2 Join S a e Space
In single-obse e case, we de ined join mani old
EQ=M×SQ,
in oducing ime–in o ma ion–complexi y ac ion on i . In mul i-obse e case, we
de ine
De ini ion 2.2 (Mul i-Obse e Join Mani old).Fo N=|I|<∞case, de ine
E(i)
Q=M(i)× S(i)
Q
as con ol–in o ma ion mani old o i- h obse e (can ake M(i)=M,S(i)
Q=SQin
isomo phic sense), de ining join mani old
EN
Q=
N
Y
i=1
E(i)
Q.
Fo each obse e i, i s con inuous limi wo ldline is
zi( ) = (θi( ), ϕi( )) ∈ E(i)
Q,
mul i-obse e join wo ldline is
Z( ) = (z1( ), . . . , zN( )) ∈EN
Q.
In e nal memo y and knowledge g aph can be a ached as ex e nal s uc u es, a main
geome ic le el we ocus on e olu ion o θi, ϕi.
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3 Communica ion G aph, Consensus Ene gy, and Con-
sensus Geome y
This sec ion in oduces ime-dependen communica ion g aph, de ining mul i-obse e
consensus ene gy and consensus geome ic s uc u e on in o ma ion mani old.
3.1 Time-Dependen Communica ion G aph
De ini ion 3.1 (Communica ion G aph).A ime , mul i-obse e communica ion s uc-
u e ep esen ed by di ec ed g aph
C = (I, E , ω )
whe e:
1. Ve ex se is obse e index se I={1, . . . , N};
2. Di ec ed edge (j→i)∈E ep esen s obse e Ojsending in o ma ion o Oia
ime ;
3. Weigh ω (i, j)≥0 ep esen s weigh /bandwid h o edge j→i.
Impo an special case o symme ic communica ion g aph is ω (i, j) = ω (j, i).
Communica ion g aph induces class o g aph Laplace ope a o s: de ine
L :RN→RN,
o ec o x∈RNha e
(L x)i=X
j
ω (i, j) (xi−xj).
When ω symme ic, L is symme ic posi i e semide ini e ma ix, whose spec al
s uc u e cha ac e izes connec i i y and “consensus con ac i i y” o communica ion s uc-
u e.
3.2 Consensus Ene gy and Consensus Geome y
To cha ac e ize “deg ee o consensus” o mul i-obse e s on ask in o ma ion mani old
(SQ, gQ), we de ine consensus ene gy unc ional.
De ini ion 3.2 (Consensus Ene gy).A ime , o obse e in o ma ion s a es ϕi( )∈
SQ, de ine consensus ene gy
Econs( ) = 1
2X
i,j∈I
ω (i, j)d2
SQϕi( ), ϕj( ),
whe e dSQis geodesic dis ance induced by gQ.
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When Econs( ) = 0, all obse e s comple ely coincide on ask in o ma ion mani old,
achie ing pe ec consensus; la ge Econs indica es highe deg ee o in o ma ion dispe sion.
Consensus ene gy can be iewed as “disc e e Di ichle ene gy” o obse e dis ibu-
ion on in o ma ion mani old, whose g adien low co esponds o consensus dynamics o
in o ma ion s a es on communica ion g aph.
To mo e geome ically cha ac e ize global s uc u e be ween mul i-obse e s, can de-
ine p oduc me ic on join mani old EN
Q
G(N)
=
N
X
i=1 α2G(i)⊕β2g(i)
Q,
iewing consensus ene gy as po en ial unc ion on in o ma ion ac o
Ucons(Z( )) = Econs( ).
F om his pe spec i e, mul i-obse e join wo ldline sa is ies “geodesic low wi h cou-
pling po en ial”, wi h po en ial unc ion being p ecisely consensus ene gy.
4 Consensus Ricci Cu a u e and Ene gy Decay The-
o em
This sec ion in oduces “consensus Ricci cu a u e” concep ela ed o communica ion
g aph and in o ma ion mani old geome y, p o ing exponen ial decay heo em o con-
sensus ene gy unde i s lowe bound cons ain .
4.1 Local Ricci Cu a u e Be ween Two Obse e s
In single-obse e in o ma ion mani old, we al eady de ined disc e e Ricci cu a u e based
on Wasse s ein dis ance. In mul i-obse e case, we ocus on combina ion o geodesic
s uc u e on in o ma ion mani old and communica ion Laplace.
Fo gi en ime , le ϕi, ϕj∈ SQbe ask in o ma ion s a es o wo obse e s. Conside
connec ing ϕi, ϕjby geodesic on SQ, de ining on i local sec ional cu a u e Kij ( ).
In consensus dynamics d i en by g aph Laplace, na u al disc e e Ricci cu a u e ana-
log is:
De ini ion 4.1 (Local Lowe Bound o Consensus Ricci Cu a u e).I he e exis s con-
s an κcons( )∈R, such ha o any i, j
d
dϵϵ=0hd2
SQϕi( +ϵ), ϕj( +ϵ)i≤ −2κcons( )d2
SQϕi( ), ϕj( ),
hen call κcons( ) consensus Ricci cu a u e lowe bound a ime .
In ui i ely, κcons( )>0 indica es unde consensus dynamics, in o ma ion dis ance
be ween obse e s exhibi s exponen ial con ac ion; κcons( )<0 indica es in o ma ion
dis ance may di e ge.
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4.2 Consensus Dynamics and Ene gy Decay
Conside simple con inuous consensus dynamics model:
dϕi
d =−X
j
ω (i, j) g adϕi1
2d2
SQ(ϕi, ϕj),
equi alen o g adien descen o consensus ene gy Econs on SN
Q.
On Riemannian in o ma ion mani old, his can be w i en as
dϕk
i
d =−X
j
ω (i, j)gkl
Q(ϕi)∂l1
2d2
SQ(ϕi, ϕj).
Unde s anda d assump ions, ime de i a i e o Econs is
d
d Econs( ) = −X
i∇ϕiEcons( )2
gQ.
Combining consensus Ricci cu a u e lowe bound, ollowing heo em can be p o ed.
Theo em 4.2 (Exponen ial Decay o Consensus Ene gy).Assume:
1. Communica ion g aph C symme ic wi h uni o m algeb aic connec i i y lowe bound
λmin
2>0;
2. In o ma ion mani old (SQ, gQ)has Ricci cu a u e lowe bound RicgQ≥K∈R;
3. Consensus dynamics as abo e, e ol ing unde uni ied ime scale.
Then he e exis cons an s κe >0and C > 0, such ha
Econs( )≤ Econs(0) e−2κe , ≥0,
whe e κe gi en by combina ion o λmin
2and K.
P oo in Appendix C.1. Co e idea is using O o pe spec i e o Wasse s ein–in o ma ion
geome y o iew consensus p ocess as kind o “disc e e iscous low”, whose ene gy decay
a e con olled by lowe bound cu a u e, consis en wi h Bak y–´
Eme y g adien es ima e
o m.
5 Mul i-Obse e Join Ac ion and Op imal Consen-
sus S a egy
This sec ion, on basis o ime–in o ma ion–complexi y join a ia ional p inciple, con-
s uc s mul i-obse e join ac ion and de i es i s Eule –Lag ange equa ions.
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5.1 Mul i-Obse e Join Ac ion
Fo each obse e i, i s single-body obse a ion–compu a ion ac ion is
b
A(i)
Q[zi(·)] = ZT
01
2α2
iGab(θi)˙
θa
i˙
θb
i+1
2β2
igjk(ϕi)˙
ϕj
i˙
ϕk
i−γiUQ(ϕi)d +(knowledge g aph/a en ion cos ).
We ocus on con ol–in o ma ion main e m, igno ing de ails o knowledge g aph and
a en ion cos , abso bing hem in o e ec i e po en ial.
De ini ion 5.1 (Mul i-Obse e Join Ac ion).De ine
b
Amul i
Q[Z(·)] =
N
X
i=1 b
A(i)
Q[zi(·)] + λcons ZT
0
Econs( ) d .
whe e λcons >0 con ols weigh o consensus ene gy in o e all op imiza ion.
Minimizing b
Amul i
Qco esponds o seeking unde gi en complexi y and ime budge ,
op imal mul i-obse e s a egy ha can bo h imp o e indi idual ask in o ma ion quali y
and o m collec i e consensus on ask in o ma ion mani old.
5.2 Eule –Lag ange Equa ions and Coupled Geodesic–Consensus
Dynamics
Va ying θa
iand ϕk
i espec i ely gi es ollowing coupled Eule –Lag ange equa ion sys em:
1. Con ol pa
¨
θa
i+ Γa
bc(θi)˙
θb
i˙
θc
i= 0,
i.e., con ol a iable o each obse e s ill e ol es along geodesics o (M, G) (igno ing
con ol coope a ion);
2. In o ma ion pa
¨
ϕk
i+ Γk
mn(ϕi)˙
ϕm
i˙
ϕn
i=−γi
β2
i
gkl
Q(ϕi)∂lUQ(ϕi)−λcons
β2
i
gkl
Q(ϕi)∂l∂Econs
∂ϕi,
whe e g adien o consensus ene gy wi h espec o ϕiis
∂Econs
∂ϕi
=X
j
ω (i, j)∇ϕi1
2d2
SQ(ϕi, ϕj).
The e o e, mul i-obse e in o ma ion wo ldline is geodesic mo ion wi h po en ial
d i en join ly by “indi idual ask po en ial” and “consensus po en ial”.
Unde uni ied ime scale and small eloci y app oxima ion, abo e dynamics degene -
a es o combina ion o a o emen ioned consensus g adien low and single-body geodesics,
minimum ac ion pa h co esponds o op imal consensus– ask adeo unde complexi y
budge .
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6 Mul i-Obse e Causal Diamond and Join Bound-
a y Ope a o
This sec ion ex ends causal diamond heo y om p e ious pape o mul i-obse e case,
cons uc ing join bounda y ope a o and discussing i s ela ionship wi h single-obse e
bounda y ope a o s.
6.1 Mul i-Obse e E en s and Causal Ne wo k
On e en laye E=X×Nex end index, adding obse e label: de ine
Eobs =I×X×N, e = (i, x, k),
ep esen ing “i- h obse e a s ep kin some local pe spec i e o uni e se con igu a ion
x”. Communica ion e en s de ined on I×I×N, o ming mul i-laye causal ne wo k.
Fo gi en inpu –ou pu mul i-obse e e en amilies
{e(i)
in }i∈I,{e(i)
ou }i∈I,
and complexi y budge T, de ine mul i-obse e causal diamond as
♢mul i =
i∈IJ+
Te(i)
in ∩J−
Te(i)
ou ⊂Eobs.
I s bounda y can likewise be decomposed in o incoming–ou going pa s, laye ed by
obse e index.
6.2 Join Bounda y Hilbe Space and Bounda y Ope a o
Unde QCA ealiza ion, each obse e laye co esponds o local Hilbe space ac o .
Mul i-obse e diamond in e nal Hilbe space decomposes as
H♢mul i =O
i∈IHbulk,♢,i ⊗ B−
♢,i ⊗ B+
♢,i,
de ining join incoming and ou going bounda y Hilbe spaces
B−
♢,mul i =O
i∈I
B−
♢,i,B+
♢,mul i =O
i∈I
B+
♢,i.
Bulk in e nal e olu ion gi en by
U♢mul i :H♢mul i → H♢mul i .
Choosing bulk e e ence s a e and bounda y p ojec ion o each obse e laye , de ine
join incoming embedding
ι−
♢,mul i :B−
♢,mul i → H♢mul i ,
join ou going p ojec ion
Π+
♢,mul i :H♢mul i → B+
♢,mul i.
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