Observer Properties and Consensus Geometry on Causal Networks:\\ Unified Formalization of Local Partial Orders, Information States, and Update Operators

Abs ac
In a wo ld iew based on causal pa ial o de s, any single obse e possesses only
a local agmen : pa ial e en s a ini e esolu ion, pa ial causal ela ions, and pa -
ial in o ma ion s a es on locally obse able algeb as. Mul iple obse e s a emp o
achie e “consensus” on he same uni e se causal ne wo k h ough communica ion
and upda es, he eby econs uc ing a consis en wo ld desc ip ion. In he abs ac
causal ne wo k amewo k, his pape o malizes obse e s as mul i-componen ob-
jec s equipped wi h geome ic domains, local pa ial o de s, esolu ion scales, ob-
se able algeb as, bounda y s a es, model amilies, s a e upda e ope a o s, and
u ili y unc ions, es ablishing a uni ied heo y o “consensus geome y.”
A he geome ic le el, gi en a amily o local causal agmen s {(Ci,≺i)}i∈I
co e ing e en se X, i local pa ial o de s sa is y ˇ
Cech- ype consis ency condi ions
in o e lapping egions, hen he e exis s a unique global pa ial o de (X, ≺) as
causal consensus ex ension; o he wise causal consensus exis s a mos a coa se
esolu ion le els. Resolu ion is cha ac e ized by e en pa i ions Piand obse able
algeb as Ai; he ichness o common e inemen P∗and algeb a in e sec ion Acom =
TiAide e mines he ineness o achie able consensus.
A he in o ma ion and dynamical le el, conside s a e amilies {ω( )
i}on com-
mon obse able algeb a Acom, communica ion channels Tij, and weigh ma ix
W= (wij). Based on Umegaki ela i e en opy
D(ρ∥σ) = ρ(log ρ−log σ)
and i s da a p ocessing inequali y, we cons uc weigh ed o al de ia ion unc-
ion
Φ( )=X
i∈I
λiDω( )
i∥ω∗,
p o ing ha when channels sa is y da a p ocessing inequali y, communica ion
g aph is s ongly connec ed, weigh ma ix is p imi i e, and common ixed poin ω∗
exis s, Φ( )is a s ic ly mono one non-inc easing Lyapuno unc ion, making s a e
i e a ion con e ge o unique s a e consensus ωcons =ω∗. This s uc u e simul ane-
ously encompasses classical a e age consensus algo i hms and con ac i e lows on
quan um channels.
A he model le el, iewing candida e causal dynamical models as elemen s o
compac space M, unde app op ia e iden i iabili y and la ge de ia ion condi ions,
we p o e ha as obse a ion da a inc eases, he in e sec ion o accep able model
se s M(T)
io each obse e con ac s wi h p obabili y one o he unique ue model
M∗, achie ing model consensus.
The abo e geome ic, in o ma ional, and model s uc u es a e uni ied in o a
“consensus easible egion” Ocons in obse e p ope y space O. This pape p o ides
se e al necessa y o su icien condi ions o causal consensus, s a e consensus, and
model consensus, p oposing a se o quan i a i e indica o s including geome ic
o e lap deg ee, esolu ion compa ibili y, algeb a in e sec ion dimension, ela i e
en opy de ia ion, and communica ion g aph connec i i y, demons a ing how o
sys ema ically analyze “how mul iple obse e s wea e he same causal wo ld” om
he causal ne wo k pe spec i e.
Keywo ds: Causal ne wo k; Pa ial o de ; Obse e ; Resolu ion; Obse able algeb a;
Rela i e en opy; Quan um channels; Dis ibu ed consensus; ˇ
Cech consis ency; Model
selec ion
1
1 In oduc ion and His o ical Con ex
In he mains eam pic u e o ela i i y and quan um ield heo y, space ime causal s uc-
u e can be abs ac ed as pa ial o de ≺on e en se M, e.g., he causal se app oach
models Lo en zian space ime as a locally ini e pa ially o de ed se (M, ≺). Wi hin
his s uc u e, a single obse e collec s local in o ma ion along i s wo ldline and o ms
a subjec i e model o he “wo ld” a ini e esolu ion and ini e bandwid h. On he
o he hand, in dis ibu ed sys ems and mul i-agen con ol, ex ensi e wo k s udies how
mul iple nodes achie e a e age consensus o consis en es ima ion unde communica ion
cons ain s h ough i e a i e upda es.
Al hough hese wo applica ion con ex s a e as ly di e en , hey sha e a common
abs ac co e: mul iple “obse e s” wi h local pe spec i es and local in o ma ion s a es
loca ed on he same unde lying causal s uc u e, a emp ing o cons uc a consis en
“global desc ip ion” h ough communica ion and upda es. In opology and shea he-
o y, his p oblem appea s in he o m o “can local da a be glued in o global objec s,”
wi h igo ous ools being shea locali y and gluing condi ions, and ˇ
Cech cohomology. In
causal and s uc u al lea ning on ie s, people a e beginning o use pa ial o de s and
in o ma ion geome y o cha ac e ize mo e gene al causal s uc u es and hei iden i iable
con en .
On he o he hand, ela i e en opy and i s mono onici y cons i u e an impo an
co ne s one o quan um and classical in o ma ion heo y. The join con exi y and da a
p ocessing inequali y o Umegaki ela i e en opy play co e oles in quan um channel
analysis, he modynamic inequali ies, and in o ma ion geome y. These esul s gua an ee
ha unde comple ely posi i e ace-p ese ing maps, dis inguishabili y be ween s a es
does no inc ease, na u ally p o iding Lyapuno unc ion candida es o “s a e consensus”
con e gence.
Based on he abo e backg ound, his wo k a emp s o gi e gene al answe s o he
ollowing ques ions a an abs ac le el:
1. How o uni o mly desc ibe he geome ic, algeb aic, and in o ma ional p ope ies
o “obse e s” in causal ne wo k language?
2. Unde wha condi ions can local pa ial o de s be glued in o a single global causal
ne wo k, achie ing causal consensus?
3. Unde wha condi ions can obse e s a e i e a ions on common obse able algeb as
con e ge o uni ied s a e consensus?
4. Unde wha iden i iabili y condi ions will he in e sec ion o obse e model amilies
almos su ely con ac o he unique ue model as da a accumula es, o ming model
consensus?
In exis ing li e a u e, econs uc ion o causal s uc u e mos ly ocuses on “ eco e -
ing opology and me ic om global causal pa ial o de ,” while dis ibu ed consensus
mos ly assumes unde lying sys em dynamics is known. This pape s a s om he op-
posi e di ec ion: assuming wha is gi en is mul iple obse e s’ local causal agmen s
and in o ma ion s a es, s udying unde wha condi ions a common causal ne wo k and
consensus s a e can be eco e ed om his local da a.
The main con ibu ions o his pape can be summa ized as:
2

Fo malizing obse e as mul i-componen s uc u e
Oi= (Ci,≺i,Λi,Ai, ωi,Mi, Ui, ui,{Cij}j∈I),
sepa a ely cha ac e izing geome ic domain, local pa ial o de , esolu ion, obse -
able algeb a, s a e, model amily, upda e ule, u ili y unc ion, and communica ion
s uc u e, he eby uni ying abs ac desc ip ions o physical obse e s and compu-
a ional nodes.

A he geome ic and pa ial o de le el, gi ing su icien condi ions o local causal
agmen s {(Ci,≺i)} o be glued in o unique global pa ial o de (X, ≺), wi h co e
being co e age, ini e o e lap, and ˇ
Cech- ype consis ency; and poin ing ou ha i
his condi ion b eaks, s ong- o m causal consensus is una ainable.

A he in o ma ion le el, cons uc ing common obse able algeb a Acom =TiAi
and s a e amily {ω( )
i}, p o ing ha when channels a e comple ely posi i e ace-
p ese ing maps sa is ying da a p ocessing inequali y, communica ion g aph is s ongly
connec ed, and common ixed poin ω∗exis s, weigh ed ela i e en opy
Φ( )=X
i
λiDω( )
i∥ω∗
is a Lyapuno unc ion, gua an eeing s a e consensus con e gence. This s uc u e
simul aneously encompasses classical a e age consensus, dis ibu ed il e ing, and
quan um ne wo k symme iza ion and s eady s a e design.

A he model le el, gi ing iden i iabili y assump ions based on la ge de ia ions
and Kullback–Leible di e gence, p o ing ha as obse a ion ime T→ ∞, he
in e sec ion o h eshold-sc eened model se s M(T)
io each obse e con ac s wi h
p obabili y one o unique ue model M∗.

Abs ac ing he abo e condi ions in o a “consensus easible egion” Ocons in ob-
se e p ope y space O, and demons a ing h ough se e al ini e examples he
mechanism o causal consensus ailu e and eco e y o weak consensus h ough
coa se-g aining.
The subsequen s uc u e is a anged as ollows: i s gi ing models and assump ions;
hen s a ing main heo ems and hei p oo amewo ks; hen discussing se e al appli-
ca ions and enginee ing sugges ions; inally summa izing and gi ing de ailed p oo s and
examples in appendices.
2 Model and Assump ions
2.1 E en Se and Local Causal F agmen s
Le Xbe he e en se . We do no p esuppose a global causal ela ion on X, bu a he
assume obse a ions appea only in he o m o local pa ial o de s.
3
De ini ion 2.1 (Local Causal F agmen ).A local causal agmen is a pai (C, ≺C),
whe e C⊆Xis an e en subse and ≺Cis a pa ial o de ela ion on C, i.e., sa is ying
o all x, y, z ∈C:
1. Re lexi i y: x⪯Cx;
2. An isymme y: i x⪯Cy, y ⪯Cx hen x=y;
3. T ansi i i y: i x⪯Cy, y ⪯Cz hen x⪯Cz.
We cus oma ily use x≺Cy o deno e x⪯Cyand x=y.
In a amily o obse e s {Oi}i∈I, each obse e Oiis associa ed wi h a local causal
agmen (Ci,≺i). Assume co e age condi ion
[
i∈I
Ci=X,
meaning each e en is accessed by a leas some obse e .
To a oid pa hological cases, we u he assume ini e o e lap condi ion: o any x∈X,
he se {i∈I:x∈Ci}is ini e.
2.2 Obse e P ope y Vec o s
De ini ion 2.2 (Obse e ).Obse e Oiis he ollowing mul i-componen objec :
Oi=Ci,≺i,Λi,Ai, ωi,Mi, Ui, ui,{Cij}j∈I,
whe e:
1. Ci⊆X: eachable causal domain, gi ing he se o e en s ha obse e can di ec ly
obse e o in luence.
2. ≺i: local causal pa ial o de de ined on Ci.
3. Λi: esolu ion scale, can be iewed as coa se-g aining map om ideal ine e en
space X ine o Ci, o equi alen ly as a pa i ion Pi={Bi,α}α∈Ii; highe esolu ion
co esponds o ine pa i ion.
4. Ai: obse able algeb a, ypically a C∗subalgeb a o bounded ope a o algeb a on
some Hilbe space, con aining measu able and con ollable quan i ies.
5. ωi:Ai→C: s a e, a posi i e no malized linea unc ional, cha ac e izing obse e ’s
belie on Ai; in ini e dimension co esponds o densi y ma ix ρi.
6. Mi⊆ M: candida e model amily, whe e Mis compac space o causal dynamical
models, e.g., causal Ma ko ne wo ks, Lag angians, o ansi ion ke nels.
7. Ui: s a e upda e ope a o
Ui: (ωi, d)7→ ω′
i,
mapping da a dand cu en s a e o new s a e; specialized o linea a e aging o m
in consensus i e a ion below.
4
8. ui: u ili y unc ion o p e e ence unc ion, de ined on ac ion space Ho model space
M, used o decision-making.
9. Cij: s uc u al pa ame e s o communica ion channel, desc ibing bandwid h, la-
ency, noise, us weigh s, e c., om Oj o Oi, inducing comple ely posi i e ace-
p ese ing map Tij a in o ma ion le el.
This pape mainly ocuses on he impac o he i s se en componen s and commu-
nica ion g aph s uc u e on consensus exis ence and con e gence.
2.3 Communica ion G aph and Channel Model
Le Gcomm = (I, Ecomm) be he communica ion g aph, wi h e ex se being obse e
indices I, and edge se
Ecomm ={i, j}: he e exis s a leas one di ec ion o nonze o bandwid h be ween i, j.
On common obse able algeb a, communica ion channels a e ep esen ed by com-
ple ely posi i e ace-p ese ing maps:
Assump ion 1 (Communica ion Channels).1. Fo each di ec ed edge j→i, he e
exis s comple ely posi i e ace-p ese ing map (CPTP map) Tij :S(Acom)→
S(Acom), whe e S(·) deno es s a e space; i no edge, hen Tij is ze o ope a o .
2. The e exis s weigh ma ix W= (wij)i,j∈Isa is ying wij ≥0,Pjwij = 1, and
wij >0 only when he e exis s j→idi ec ion communica ion.
The common obse able algeb a is de ined as
Acom :=
i∈I
Ai,
assuming Acom is non i ial excep scala mul iples o he iden i y elemen .
2.4 Models and P obabilis ic S uc u e
Le obse a ion da a sequence D= (d(1), . . . , d(T)) be gene a ed unde ue model M†∈
M. Each obse e Oihas likelihood unc ion Li(M;D) o pos e io densi y πi(M| D).
The accep able model se a e h eshold sc eening is de ined as
M(T)
i:= M∈ Mi:Li(M;D)≥ϵi(T),
whe e h eshold ϵi(T) a ies wi h da a olume.
In model consensus discussion we adop he ollowing iden i iabili y and consis ency
assump ions, wi h speci ic s a emen s in heo ems la e .
3 Main Resul s: Theo ems and Alignmen s
This sec ion p esen s he main heo ems and s uc u al conclusions o his pape . P oo
de ails a e concen a ed in subsequen P oo s sec ion and appendices.
5

3.1 Causal Consensus: Gluing Theo em o Local Pa ial O -
de s
Fi s we gi e he de ini ion o s ong- o m causal consensus.
De ini ion 3.1 (Causal Consensus).The obse e amily {Oi}i∈Iachie es causal con-
sensus i he e exis s pa ially o de ed se (X, ≺) and injec i e maps
ei:Ci,→X
sa is ying:
1. Fo any x, y ∈Ci,
x≺iy⇐⇒ ei(x)≺ei(y).
2. Fo any x∈Ci∩Cj, we ha e ei(x) = ej(x).
In his case, (X, ≺) is called he causal consensus ex ension o local causal agmen s.
Consis ency o local pa ial o de s in o e lapping egions is cha ac e ized by ˇ
Cech- ype
condi ions.
De ini ion 3.2 (ˇ
Cech- ype Consis ency).Fo any ini e subse J⊆I, deno e
CJ:=
j∈J
Cj.
I he e exis s pa ial o de ≺Jde ined on CJsuch ha o all j∈Jand x, y ∈CJ,
x≺Jy⇐⇒ x≺jy,
hen he local pa ial o de amily {≺i}is consis en on CJ. I his holds o all ini e
J, he amily {≺i}sa is ies ˇ
Cech- ype consis ency.
Unde co e age and ini e o e lap, we ha e he ollowing gluing heo em.
Theo em 3.3 (Causal Ne wo k Gluing Theo em).Le {(Ci,≺i)}i∈Ibe a amily o local
causal agmen s on Xsa is ying:
1. Co e age: SiCi=X;
2. Fini e o e lap: o any x∈X, he se {i:x∈Ci}is ini e;
3. ˇ
Cech- ype consis ency: as in De ini ion 2.2.
Then he e exis s unique pa ial o de ≺such ha :
1. (X, ≺)is a pa ially o de ed se ;
2. Fo each i, he es ic ion o ≺ o Ciequals ≺i.
In o he wo ds, (X, ≺)is a causal consensus ex ension, unique up o isomo phism.
This heo em shows: he key o s ong- o m causal consensus exis ence is ha he
co e o med by local causal agmen s sa is ies ˇ
Cech- ype consis ency. O he wise, s ong
consensus is una ainable; weak consensus can only be discussed a coa se esolu ion
le els.
6
3.2 Resolu ion, Common Re inemen , and Consensus Limi
Resolu ion s uc u e is cha ac e ized by e en pa i ions.
De ini ion 3.4 (Pa i ion and Common Re inemen ).Le X ine be he ideal ine e en
se . In a amily o pa i ions {Pi}i∈I, each
Pi={Bi,α}α
is a pa i ion o X ine. I he e exis s pa i ion P∗such ha o each i,Piis a coa sening
o P∗, i.e., o any B∈Pi he e exis s B′∈P∗wi h B⊆B′, hen P∗is called a common
e inemen .
Common e inemen exis ence can be desc ibed by equi alence ela ions. Pa i ion Pi
co esponds o equi alence ela ion
x∼iy⇐⇒ ∃B∈Pi:x, y ∈B.
Common e inemen exis s i and only i he in e sec ion ela ion
R:=
i
∼i
is an equi alence ela ion.
In ini e cases, he condi ion o common e inemen exis ence can be es a ed as “no
in e laced block con lic s,” wi h speci ic s a emen s in Appendix B.
The limi s uc u e o esolu ion consensus is he quo ien space a e e en s a e
comp essed by equi alence classes o R. Highe esolu ion and ine common e inemen
enable iche causal s uc u es ha consensus can dis inguish.
3.3 Obse able Algeb a In e sec ion and S a e Consensus
The common obse able algeb a is de ined as
Acom :=
i∈I
Ai.
Assume Acom is non i ial excep scala mul iples o he iden i y elemen .
Le ω( )
i∈ S(Acom) be obse e Oi’s s a e es ima e on common algeb a a ime , wi h
upda e ule being linea a e aging ype
ω( +1)
i=X
j∈I
wijTijω( )
j,
whe e Tij a e CPTP maps and W= (wij) is ow s ochas ic ma ix.
Rela i e en opy akes Umegaki o m: in ini e dimension, i ω=ωρ, ω′=ωσco e-
spond o densi y ma ices ρ, σ, hen
D(ω∥ω′) := D(ρ∥σ) = ρ(log ρ−log σ).
P oposi ion 3.5 (Single-s ep Con ac ion o Rela i e En opy).Le ω∗∈ S(Acom)be
some ixed s a e. Assume:
7
1. Each Tij sa is ies da a p ocessing inequali y, i.e., o all s a es ω, ω′,
DTij(ω)∥Tij(ω′)≤D(ω∥ω′);
2. Weigh ma ix Wis ow s ochas ic, and he e exis s weigh λi>0sa is ying Piλi=
1and λ⊤W=λ⊤.
De ine o al de ia ion unc ion
Φ( ):= X
i∈I
λiDω( )
i∥ω∗.
Then o any ,
Φ( +1) ≤Φ( ).
This p oposi ion shows: unde na u al weigh ing condi ions, ela i e en opy is mono-
one non-inc easing unde consensus i e a ion, p o iding Lyapuno unc ion candida e o
consensus con e gence.
I we u he assume common ixed poin exis s and communica ion g aph has su i-
cien mixing, we ob ain s a e consensus con e gence heo em.
Assump ion 2 (Common Fixed Poin and P imi i i y).1. Communica ion g aph Gcomm
is s ongly connec ed.
2. Weigh ma ix Wis p imi i e, i.e., he e exis s k∈Nsuch ha Wkhas all posi i e
elemen s.
3. The e exis s s a e ω∗∈ S(Acom) sa is ying o all i, j
Tij(ω∗) = ω∗.
Theo em 3.6 (Con e gence o S a e Consensus).Unde Assump ion 2, o any ini ial
s a e amily {ω(0)
i}i∈I, he i e a ion
ω( +1)
i=X
j
wijTij(ω( )
j)
con e ges o uni ied s a e ω∗, i.e.,
lim
→∞ ω( )
i=ω∗,∀i∈I.
In he classical case, he abo e esul educes o s anda d conclusions o linea a e age
consensus; in he quan um case, i co esponds o a class o quan um Ma ko chains wi h
common ixed poin con e ging o unique s eady s a e.
8
3.4 Model Consensus and Almos Su e Iden i ica ion o T ue
Model
Le model space Mbe a compac me ic space, wi h ue model M†∈ M. Fo each
M∈ M, deno e PMas i s induced da a dis ibu ion.
Assump ion 3 (Iden i iabili y and La ge De ia ions).1. The e exis s unique M∗∈
Msuch ha o any M=M∗, KL di e gence sa is ies
DPM∗∥PM>0.
2. Fo each obse e i, h eshold ϵi(T) can be chosen as unc ion o da a leng h Tsuch
ha unde ue model M∗, as T→ ∞
PM∗M∗∈ M(T)
i→1,PM∗∃M=M∗, M ∈ M(T)
i→0.
This condi ion can be e i ied h ough law o la ge numbe s and Sano - ype la ge
de ia ion esul s.
Theo em 3.7 (Almos Su e Con ac ion o Model Consensus).Unde Assump ion 3, o
any δ > 0, he e exis s T0such ha when T≥T0,
PM∗
i∈I
M(T)
i={M∗}≥1−δ.
In o he wo ds, as obse a ion ime ends o in ini y, he in e sec ion o accep able
model se s o all obse e s con ac s wi h p obabili y one o unique ue model M∗, achie -
ing s ong- o m model consensus.
3.5 Consensus Geome y and Indica o Sys em
Syn hesizing he abo e esul s, in obse e p ope y space
O:= Y
i∈I
P(X)×Pose s ×Res ×Alg × S × Models ×Upda es×Comm,
de ine subse Ocons ⊆ O as all sa is ying:
1. The e exis s causal consensus ex ension (X, ≺);
2. The e exis s s a e consensus ωcons;
3. The e exis s nonemp y model consensus se Mcons (single poin in s ong o m).
Call Ocons he easible egion o consensus geome y.
Based on his, in oduce he ollowing indica o s:

Geome ic o e lap deg ee:
θij := µ(Ci∩Cj)
µ(Ci∪Cj),
whe e µis coun ing measu e o olume measu e.
9

In shea and shea s uc u e aspec s, ˇ
Cech consis ency o local pa ial o de s
and common e inemen p oblem can be iewed as simpli ied e sion o gluing
p oblems in noncommu a i e geome y and shea heo y, p omising u he
de elopmen in o “causal shea ” pe spec i e.
5. Po en ial Risks and Ex ension Di ec ions

This pape does no explici ly handle malicious o Byzan ine obse e s; in
p esence o nodes in en ionally h owing inco ec pa ial o de s o s a es,
causal consensus and s a e consensus may ail, equi ing in oduc ion o obus
consensus and aul ole ance mechanisms.

Fo sys ems wi h sel - e e ence o ci cula in o ma ion s uc u e, he e may be
si ua ions o “local consensus sel -consis en bu globally non-embeddable,”
ela ed o ecen esea ch on ci cula in o ma ion s uc u e and non-classical
causal models.
In summa y, consensus geome y amewo k p o ides uni ied con ex o s uc u al
analysis o mul i-obse e causal wo ld, bu ex ensions o in ini e dimension, s ong noise,
and ad e sa ial en i onmen s s ill equi e u he esea ch.
8 Conclusion
This pape sys ema ically cons uc s heo e ical amewo k o “obse e p ope ies and
consensus geome y on causal ne wo ks” a in e sec ion o abs ac causal ne wo ks and
in o ma ion geome y. By o malizing obse e s as mul i-componen objec s wi h geo-
me ic domains, local pa ial o de s, esolu ion scales, obse able algeb as, in o ma ion
s a es, model amilies, and upda e ope a o s, he ollowing main conclusions a e ob ained:
1. Unde co e age, ini e o e lap, and ˇ
Cech- ype consis ency condi ions, local causal
agmen s can be glued in o unique global pa ial o de , achie ing s ong- o m
causal consensus; o he wise s ong consensus is una ainable, weak consensus can
only be discussed a coa se-g aining le els.
2. Resolu ion s uc u e and obse able algeb a in e sec ion de e mine ineness o con-
sensus; dimension o common e inemen and common algeb a a e na u al indica o s
o “consensus esolu ion.”
3. On common algeb a, Lyapuno unc ion measu ed by Umegaki ela i e en opy can
cha ac e ize mono one con e gence o s a e consensus i e a ion; unde condi ions
o s ongly connec ed communica ion g aph, p imi i e weigh ma ix, and common
ixed poin exis ence, s a es necessa ily con e ge o uni ied consensus s a e.
4. Unde app op ia e iden i iabili y and la ge de ia ion condi ions, in e sec ion o
h eshold-sc eened model se s o each obse e con ac s wi h p obabili y one o
unique ue model, achie ing s ong- o m model consensus.
5. By in oducing indica o s such as geome ic o e lap deg ee, esolu ion compa ibili y,
algeb a in e sec ion dimension, and ela i e en opy de ia ion, cons uc consensus
easible egion in obse e p ope y space, p o iding quan i a i e ools o analyzing
“whe he consensus occu s easily.”
16

Fu u e di ec ions include: de eloping unc ional analysis e sion o causal consensus in
in ini e dimension and con inuous ield heo y; es a ing consensus geome y in shea and
highe ca ego y amewo ks; in oducing obus and aul - ole an s uc u es agains ma-
licious obse e s; and embedding his amewo k in o b oade “ alue–causali y–in o ma ion”
uni ied sys em o explo e ela ionship be ween ee choice and causal consensus.
Acknowledgemen s, Code A ailabili y
This wo k is based on public li e a u e and heo e ical ools o de i a ion and cons uc-
ion.
This esea ch did no use any specialized nume ical code o simula ion p og ams, so
no publicly a ailable code implemen a ion exis s.
Re e ences
[1] S. Su ya, “The causal se app oach o quan um g a i y,” Li ing Re . Rela i i y, 22,
5 (2019).
[2] K. Ma in, P. Panangaden, “Space ime geome y om causal s uc u e and a mea-
su emen ,” P oc. Symp. Appl. Ma h., 68, 191–206 (2010).
[3] M. M. Ansanelli e al., “E e y hing ha can be lea ned abou a causal s uc u e,”
a Xi :2407.01686 (2024).
[4] V. Vilasini, E. Po mann, A. J. P. Ga ne , “Embedding cyclic in o ma ion- heo e ic
s uc u es in acyclic causal models,” Phys. Re . A 110, 022227 (2024).
[5] P. Schapi a, “An In oduc ion o Ca ego ies and Shea es,” Lec u e No es (2022).
[6] The S acks P ojec , Tag 04TP, “Glueing shea es” (online monog aph).
[7] U. Sch eibe , Z. ˇ
Skoda, “Ca ego i ied symme ies,” Lec u e No es (2008).
[8] L. Xiao, S. Boyd, S.-J. Kim, “Dis ibu ed a e age consensus wi h leas -mean-squa e
de ia ion,” P oc. MTNS (2006).
[9] R. Me ched, “On Dis ibu ed A e age Consensus Algo i hms,” a Xi :2502.16200
(2025).
[10] G. Pa langeli, A. D’Innocenzo, “A dis ibu ed algo i hm o eaching a e age con-
sensus wi h unbalanced edge weigh s,” Elec onics 13, 4114 (2024).
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um 5, 387 (2021).
[12] E. E e e al., “Equali y condi ions o da a p ocessing inequali y o α–z ela i e
en opies,” J. Ma h. Phys. 61, 102201 (2020).
[13] E. A. Ca len, E. H. Lieb, M. Loss, “Mono onici y e sions o Eps ein’s conca i y
heo em and ela ed inequali ies,” Linea Algeb a Appl. 645, 100–134 (2022).
17
[14] S. Ma heus, “On he mono onici y o ela i e en opy,” En opy 27, 954 (2025).
[15] M. Mosonyi, “Con exi y p ope ies o he quan um R´enyi di e gences,” Re . Ma h.
Phys. 26, 1450001 (2014).
[16] M. B. Ruskai, S. Sza ek, E. We ne , “An analysis o comple ely posi i e ace-
p ese ing maps on M2,” Linea Algeb a Appl. 347, 159–187 (2002).
[17] J. Guo e al., “Designing open quan um sys ems wi h known s eady s a es,” Quan-
um 9, 1612 (2025).
[18] F. Ticozzi, L. Viola, “S abilizing en angled s a es wi h quasi-local quan um dynam-
ical semig oups,” Phil. T ans. R. Soc. A 370, 5259–5269 (2012).
[19] A. Mon ana i, S. S. Sangha i, “Dis ibu ed consensus by belie p opaga ion,” IEEE
T ans. In . Theo y 56, 476–488 (2010).
[20] D. Ghion e al., “Robus dis ibu ed Kalman il e ing wi h e en - igge ed commu-
nica ion,” J. F anklin Ins . 360, 13564–13590 (2023).
[21] T. Ban e al., “Di e en iable s uc u e lea ning wi h pa ial o de s,” Ad . Neu al
In . P ocess. Sys . 37 (2024).
[22] E. Ande son, “Spaces o spaces,” a Xi :1412.0239 (2014).
[23] A. Zaghi, “Rela ional quan um dynamics as a opological and cohomological ame-
wo k,” P ep in (2023).
[24] F. Bullo, “Lec u es on Ne wo k Sys ems,” Ve sion 1.6 (2022).
A Rigo ous P oo o Causal Ne wo k Gluing Theo-
em
Theo em A.1 (Res a emen o Theo em 3.3).Le {(Ci,≺i)}i∈Ibe a amily o local causal
agmen s on Xsa is ying:
1. SiCi=X;
2. Fo any x∈X, he se {i:x∈Ci}is ini e;
3. Fo any ini e J⊆I, he e exis s pa ial o de ≺Jde ined on CJ=Tj∈JCjsuch
ha o all j∈Jand x, y ∈CJ,
x≺Jy⇐⇒ x≺jy.
Then he e exis s unique pa ial o de ≺such ha o each i, es ic ion o ≺ o Ci
equals ≺i.
18
P oo . (1) De ine global ela ion
De ine bina y ela ion on X
xRy ⇐⇒ ∃i∈Isuch ha x, y ∈Ci, x ≺iy.
By pa ial o de p ope ies, Ris clea ly i e lexi e (i.e., no xRx), bu ansi i i y is
no ye known.
(2) An isymme y and no local con adic ion
I he e exis x, y such ha xRy and yRx simul aneously hold, hen he e exis i, j
such ha

x, y ∈Ciand x≺iy;

x, y ∈Cjand y≺jx.
Taking J={i, j}, we ha e x, y ∈CJ, and ˇ
Cech consis ency equi es ≺Jon CJ o be
consis en wi h ≺i,≺j, o cing x≺Jyand y≺Jx o hold simul aneously, con adic ing
≺Jbeing pa ial o de . The e o e Ris an isymme ic.
(3) T ansi i e closu e and cycle exclusion
De ine ≺as ansi i e closu e o R, i.e., x≺yi and only i he e exis s ini e chain
x=x0Rx1R· · · Rxn=y.
Need o p o e ≺is an isymme ic. I he e exis s x=ywi h x≺yand y≺x, splicing
chains yields non i ial closed cycle
x=x0Rx1R· · · Rxn=x,
whe e n≥1.
Fo each ela ion xkRxk+1, he e exis s ikwi h xk, xk+1 ∈Cikand xk≺ikxk+1. Fini e
o e lap p ope y gua an ees se {ik}n−1
k=0 is ini e; aking
J={i0, . . . , in−1},
all poin s on cycle {xk}belong o
[
i∈J
Ci,
and o each adjacen pai (xk, xk+1), he e exis s ik∈Jmaking i ha e xk≺ikxk+1
in Cik.
By ˇ
Cech consis ency, uni ied pa ial o de ≺Jexis s on CJ=Ti∈JCi, wi h each
local pa ial o de consis en wi h ≺Jin o e laps. Th ough ini e-s ep expansion and
ansi i i y, we ge in CJ
x≺Jx,
con adic ing pa ial o de de ini ion. The e o e ≺is an isymme ic.
Re lexi i y can be ob ained by de ining non-s ic ela ion ⪯as
x⪯y⇐⇒ x=yo x≺y.
T ansi i i y is gua an eed by ansi i e closu e de ini ion.
19
(4) Local consis ency
Fo any iand x, y ∈Ci, i x≺iy, hen clea ly xRy, hence x≺y, ob aining ha ≺
ex ends ≺ion Ci.
Con e sely, i x, y ∈Ciand x≺y, hen he e exis s ini e chain
x=x0Rx1R· · · Rxn=y.
Each s ep xkRxk+1 is p oduced ia some Cik. Using ˇ
Cech consis ency, on
CJ′:=
k
Cik∩Ci
he e exis s uni ied pa ial o de ≺J′, wi h o each k
xk≺J′xk+1.
By ansi i i y we ge x≺J′y; hen by ≺J′being consis en wi h ≺ion CJ′, we ob ain
x≺iy.
The e o e es ic ion o ≺ o each Ciis consis en wi h ≺i.
(5) Uniqueness
I he e exis s ano he pa ial o de ≺′sa is ying same condi ions, hen o each i,≺′
|Ci=≺i=≺ |Ci. By co e age p ope y, o any x, y ∈X, i x≺y, hen he e exis s chain
segmen ally alling in each Ci; hese ela ions mus also hold in ≺′, and ice e sa, hence
≺=≺′. In sense allowing bijec i e elabeling o X, we ob ain isomo phism uniqueness.
B P oo o Common Re inemen P oposi ion
P oposi ion B.1 (Equi alen Cha ac e iza ion o Common Re inemen Exis ence).Le
X ine be ini e se , {Pi}i∈Ia amily o pa i ions on i . Fo each i, de ine equi alence
ela ion
x∼iy⇐⇒ ∃B∈Pi:x, y ∈B.
Le
R:=
i∈I
∼i.
Then he ollowing wo a e equi alen :
1. The e exis s common e inemen P∗such ha o each i,Piis coa sening o P∗;
2. Rela ion Ris equi alence ela ion (i.e., e lexi e, symme ic, and ansi i e).
P oo . (1) I common e inemen P∗exis s, hen co esponding equi alence ela ion ∼∗
sa is ies ∼∗⊆∼i o all i, he e o e ∼∗⊆R. On he o he hand, o any x, y i x∼∗y, hen
x, y mus belong o same P∗block, and P∗is ines pa i ion, so when any equi alence
ela ion Ris con ained in all ∼i, necessa ily ∼∗=R. The e o e Ris equi alence ela ion.
(2) I Ris equi alence ela ion, hen i s equi alence class se
PR:= {[x]R:x∈X ine}
20
is a pa i ion, and om R⊆∼iwe know Piis coa sening o PR. The e o e PRis
common e inemen .
When ansi i i y is absen , no equi alence ela ion con ained in all ∼iexis s, hence
no common e inemen exis s. Speci ic “in e laced block con lic ” cons uc ion and de ails
see main ex and discussion; no epea ed he e.
C Con e gence o Rela i e En opy- ype Consensus
P ocess
C.1 De ailed P oo o P oposi ion 3.5
P oposi ion C.1 (Single-s ep Con ac ion o Rela i e En opy).Unde Assump ion 2.5
condi ions, o any ∈N,
Φ( +1) ≤Φ( ).
P oo . Fo ixed i,
ω( +1)
i=X
j
wijTij(ω( )
j).
By join con exi y o ela i e en opy,
Dω( +1)
i∥ω∗≤X
j
wijDTij(ω( )
j)∥Tij(ω∗).
By da a p ocessing inequali y,
DTij(ω( )
j)∥Tij(ω∗)≤Dω( )
j∥ω∗.
Combining yields
Dω( +1)
i∥ω∗≤X
j
wijDω( )
j∥ω∗.
Mul iplying bo h sides by λiand summing o e i,
Φ( +1) =X
i
λiDω( +1)
i∥ω∗
≤X
i
λiX
j
wijDω( )
j∥ω∗
=X
j
X
i
λiwijDω( )
j∥ω∗.
Unde λ⊤W=λ⊤condi ion, Piλiwij =λj, hence
Φ( +1) ≤X
j
λjDω( )
j∥ω∗= Φ( ).
P oposi ion is p o ed.
21

C.2 P oo F amewo k o Theo em 3.6
To p o e ajec o y con e gence o ω∗, wo complemen a y pe spec i es can be adop ed:
1. View o e all s a e amily Ω( ):= (ω( )
i)i∈Ias poin on p oduc s a e space S(Acom)⊗I,
de ine o e all channel
T(Ω) = X
j
wijTij(ωj)i∈I.
Unde Assump ion 2condi ions, Tis p imi i e CPTP map wi h Ω∗:= (ω∗)i∈Ias
unique ixed poin . P imi i i y and compac ness gua an ee T (Ω(0))→Ω∗.
2. Using Lyapuno unc ion Φ( )mono one non-inc easing wi h lowe bound ze o, com-
bined wi h p imi i i y o T, non i ial limi cycles o ixed poin amilies can be
excluded, ul ima ely ob aining all componen s con e ging o ω∗.
Comple e echnical de ails can be es ablished using gene aliza ions o Pe on–F obenius
heo y on Banach spaces and quan um Ma ko chain con e gence heo ems; omi ed he e.
D P oo o Model Consensus Con ac ion Theo em
Theo em D.1 (Res a emen o Theo em 3.7).Unde iden i iabili y and la ge de ia ion
Assump ion 3condi ions, o any δ > 0, he e exis s T0such ha when T≥T0,
PM∗
i∈I
M(T)
i={M∗}≥1−δ.
P oo . (1) Single obse e consis ency
Fo ixed i, by Assump ion 3(2) he e exis s Ti(δ) such ha when T≥Ti(δ),
PM∗M∗∈ M(T)
i≥1−δ
2|I|,
and
PM∗∃M=M∗, M ∈ M(T)
i≤δ
2|I|.
(2) Join e en es ima e
Deno e e en s
E1:=
i∈I
{M∗∈ M(T)
i}, E2:=
i∈I
{∄M=M∗:M∈ M(T)
i}.
By union bound and abo e es ima es,
PM∗(E1)≥1−δ
2,PM∗(E2)≥1−δ
2.
Hence
PM∗(E1∩E2)≥1−δ.
22
(3) In e sec ion single-poin p ope y
On e en E1∩E2, o each i,M(T)
icon ains M∗and no o he model. The e o e
i∈I
M(T)
i={M∗}.
Taking T0= maxiTi(δ), when T≥T0 he conclusion holds.
E Fini e Example: Consensus Failu e and Coa se-
g aining Repai wi h Th ee Obse e s
Example E.1 (Th ee-node Causal Cycle).Le X={a, b, c}. Th ee obse e s:

O1:C1={a, b}, pa ial o de a≺1b;

O2:C2={b, c}, pa ial o de b≺2c;

O3:C3={c, a}, pa ial o de c≺3a.
Geome ically, {Ci}co e s Xwi h o e laps o ming ing s uc u e. Pa ial o de s
wi hin each o e lap egion a e in e nally sel -consis en , bu o e all combina ion o ms
cycle
a≺1b≺2c≺3a.
I he e exis s global pa ial o de ≺embedding each local pa ial o de , i mus
simul aneously sa is y
a≺b, b ≺c, c ≺a,
con adic ing an isymme y o pa ial o de . The e o e no s ong- o m causal consen-
sus ex ension exis s.
Coa se-g aining epai
I we allow in oducing equi alence ela ion ∼making a∼b∼c, hen quo ien se
˜
X:= X/∼={˜x}
con ains only single equi alence class. Then he only possible pa ial o de is ˜x⪯˜x; in
his ex emely coa se pe spec i e, all local pa ial o de s degene a e o e lexi e ela ions,
and causal consensus holds a his le el.
This example shows:
1. Geome ic connec i i y is necessa y bu no su icien o s ong- o m causal con-
sensus;
2. Consis ency o local pa ial o de s in o e lapping egions is key o elimina ing causal
cycles;
3. When s ong consensus b eaks, weak consensus can be eco e ed h ough e en
equi alence class comp ession, bu a cos o losing de ailed s uc u e.
This example p o ides simple disc e e model o “causal cycles” and “ esolu ion ade-
o s” phenomena in mo e complex causal ne wo ks.
23