Geome iza ion o Obse e Consensus:
Con lic Me ics, In o ma ion Geome y, and
Bounda y Time 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
In he imeless block uni e se pe spec i e, he uni e se is modeled as a opolog-
ical s uc u e consis ing o a causal pa ial o de , while any conc e e obse e can
only access a ini e egion and ca ies a p edic i e model abou he global causal
ne wo k. Desc ip ions by di e en obse e s o he same causal egion gene a e con-
lic s a mul iple le els: di ec ed cycles appea when locally gluing pa ial o de s,
ime scale unc ions a e inconsis en , gene alized en opy a ows and modula low
di ec ions a e inconsis en , and he e is Z2sec o misma ch in he Null–Modula
double co e . This pape cons uc s a uni ied “consensus geome y space” embed-
ding obse e s’ s a is ical models, causal se s, ime scales, and bounda y s a es
in o a p oduc mani old wi h a Riemannian me ic, and de ines a o al po en ial
ene gy unc ion encoding he abo e con lic me ics as geome ic po en ial. Unde
he syne gy o linea and nonlinea in o ma ion geome y, causal se geome y, and
quan um s a e space geome y, we p o e: unde well-posedness assump ions such
as comple eness and s ong con exi y, he g adien low o his po en ial gi es a na -
u al “consensus dynamics” making all con lic me ics mono onically dec ease and
con e ge o a “consensus mani old.” On he consensus mani old, local pa ial o de s
can be consis en ly glued in o a global causal pa ial o de , uni ied mo he scale
unc ions di e only by a ine escaling, gene alized en opy a ows and modula
low di ec ions ag ee in o e lapping egions, and all obse e s inhabi he same Z2
opological sec o . Finally, we couple his geome iza ion amewo k wi h bound-
a y ime geome y, uni ied ime scales, and Null–Modula double co e s, p oposing
applica ions and enginee ing implemen a ion pa hways in mul i-obse e quan um
ield heo y, holog aphic in o ma ion, and mul i-agen sys ems.
Keywo ds: Causal ne wo k; In o ma ion geome y; Riemannian consensus; Bounda y
ime geome y; Modula low; Bu es me ic; Null–Modula double co e ; Mul i-obse e
sys ems
1
1 In oduc ion and His o ical Con ex
1.1 Timeless Pic u e o Time and Causali y
In classical ela i i y, space ime is cha ac e ized as a Lo en zian mani old wi h causal
cone s uc u e, whe e ime appea s as a mani old pa ame e ; while in he causal se
app oach, space and ime a e eplaced by a disc e e se o e en s and hei pa ial o de ing,
wi h geome y de e mined by “o de + coun ing.” This pe spec i e s ips “ ime” om
undamen al s uc u e, e aining only he causal pa ial o de (E, ⪯) be ween e en s,
wi h ime a ows and scales de i ed om pa ial o de and me ic s uc u e.
A he in e sec ion o quan um ield heo y and g a i y, bounda y me hods and holo-
g aphic hinking show ha much dynamical in o ma ion can be comp essed o bound-
a ies: he ene gy de i a i e o he sca e ing ma ix yields Wigne –Smi h g oup delay
and phase ime, de ining sca e ing ime scales; he Gibbons–Hawking–Yo k bounda y
e m and B own–Yo k quasilocal quan i ies indica e ha well-de ined a ia ion o g a i-
a ional ac ion and ene gy de ini ion a e “bounda y phenomena” a c ucial le els. Mo e-
o e , Tomi a–Takesaki modula heo y and he Connes–Ro elli he mal ime hypo hesis
cha ac e ize ime as an in insic modula low pa ame e o s a e–algeb a pai s, p o iding
an algeb aic ounda ion o “bounda y ime geome y.”
1.2 In o ma ion Geome y, Quan um S a e Geome y, and Con-
sensus Algo i hms
In s a is ics and in o ma ion heo y, he Fishe in o ma ion me ic and he in o ma ion
geome y de eloped by Ama i–Nagaoka iew pa ame ized s a is ical models as Rieman-
nian mani olds wi h he Fishe –Rao me ic, p o iding a na u al geome ic backg ound
o di e gences and g adien lows. In quan um s a e space, he Bu es me ic and quan-
um Fishe in o ma ion p o ide na u al Riemannian s uc u e o densi y ma ix spaces,
closely ela ed o quan um es ima ion heo y and geome ic phases.
On he o he hand, a e age consensus algo i hms in mul i-agen sys ems ha e been
gene alized om Euclidean space o Riemannian mani olds, o ming Riemannian consen-
sus heo y. T on e al. cons uc ed Riemannian consensus algo i hms o F ´eche means
on mani olds wi h bounded cu a u e and ga e con e gence condi ions; subsequen wo k
analyzed pa hologies and limi a ions o g adien low consensus in mo e gene al se ings.
These s udies e ealed ea ly ma hema ical s uc u es o “achie ing consensus on cu ed
geome y.”
1.3 Mul i-Obse e , Consis ency, and Bounda y Time Geome-
y
In he block uni e se o causal se pic u e, he e exis s a concep ual ension: on one
hand, he uni e se’s causal ne wo k i sel is iewed as a global s uc u e independen o
obse e s; on he o he hand, any ac ual obse e can only access a ini e causal egion,
ob ain ini e-p ecision bounda y sca e ing da a o modula low in o ma ion, hus can
only cons uc incomple e p edic ions o he global s uc u e. Judgmen s by di e en
obse e s abou he same causal egion—causal o de , ime scales, gene alized en opy
a ows, opological sec o s—may be inconsis en .
2
T adi ionally, o mul i-obse e consis ency, much discussion ocuses on: unde gi en
dynamical laws, how o eco e a common mac oscopic geome y om local obse a-
ions; in mul i-agen lea ning, how o enable all agen s o con e ge o he same model
h ough communica ion and upda e algo i hms. Howe e , ele a ing hese p oblems o
he “bounda y ime geome y” con ex equi es simul aneously handling:
1. In o ma ion geome y on s a is ical model spaces;
2. Combina o ial geome y on causal se moduli spaces;
3. Hilbe geome y on mo he scale unc ion spaces;
4. Bu es geome y and modula low s uc u e on bounda y quan um s a e spaces;
5. Z2 opological sec o s on Null–Modula double co e s.
The goal o his pape is: wi hin he uni ied ime scale and bounda y ime geome y
amewo k, o p o ide a igo ous geome ic cha ac e iza ion o mul i-obse e con lic s
and consensus, cons uc a “consensus geome y space,” and de ine a na u al po en ial
and g adien low on i , such ha “ esol ing con lic s” co esponds o a geome ic con-
ac ion p ocess on his space.
1.4 Main Con ibu ions
Agains he abo e backg ound, his pape ’s con ibu ions can be summa ized as:
1. P opose a amily o quan i a i e me ics o mul i-obse e con lic s, sepa a ely
cha ac e izing s a is ical model di e gence, causal pa ial o de gluing con lic s,
mo he scale unc ion inconsis ency, misma ch be ween gene alized en opy a ows
and modula low di ec ions, and Z2sec o misma ch in Null–Modula double co -
e s.
2. Cons uc a uni ied consensus geome y space M, gluing s a is ical mani olds,
causal se moduli spaces, mo he scale Hilbe spaces, and bounda y s a e spaces
h ough weigh ed di ec sum me ics o o m a Riemannian mani old sui able o
desc ibing mul i-obse e s a es.
3. De ine on (M, G) a o al po en ial ene gy unc ion Fencoding he abo e con-
lic me ics; p o e ha along he g adien low ˙
X=−g adGF, po en ial ene gy
dec eases mono onically, and unde s ong con exi y and comple eness condi ions
con e ges exponen ially o a “consensus mani old” Mcons.
4. On he consensus mani old, gi e a geome ic–physical cha ac e iza ion o “comple e
consensus”: he e exis s a global causal pa ial o de , uni ied mo he scale, uni ied
en opy a ow and modula low, uni ied Z2sec o , and all obse e s a es can be
embedded in he same bounda y ime geome y and Null–Modula s uc u e.
5. Th ough simple models, demons a e he applica ion po en ial o his amewo k
in ini e causal se s, mul i-agen lea ning, and bounda y sca e ing ne wo ks, and
p opose p elimina y enginee ing implemen a ion schemes.
3
2 Model and Assump ions
2.1 Uni e se Causal Ne wo k and Causal Se s
Le Ebe he se o all e en s in he uni e se. The causal ela ion is gi en by a pa ial
o de ⪯⊂ E×Esa is ying:
1. Re lexi i y: o any e∈E, we ha e e⪯e;
2. An isymme y: i e⪯ and ⪯e, hen e= ;
3. T ansi i i y: i e⪯ and ⪯g, hen e⪯g.
The pai (E, ⪯) is called he uni e se causal ne wo k o causal se . Simila o causal
se heo y, unde app op ia e assump ions one can ela e (E, ⪯) o con inuous space ime
geome y, bu his pape does no p esuppose a speci ic con inuum limi , using only
pa ial o de s uc u e.
Fo any e en e∈E, de ine i s causal u u e and pas as
J+(e) := { ∈E|e⪯ }, J−(e) := { ∈E| ⪯e}.
The opology gene a ed by he se amily {J+(e)∩J−( )|e⪯ }is called he
Alexand o opology, iewing he causal se as a opological space.
2.2 Obse e s, Local Ho izons, and P edic i e Models
De ine he obse e se O={O1, . . . , ON}. Fo each obse e Oi:
1. The e exis s a isible e en subse Ei⊂E, called i s causal ho izon;
2. On Eia local pa ial o de ⪯i⊂Ei×Eiis gi en, sa is ying pa ial o de axioms;
3. A p edic i e model abou he global causal ne wo k is p o ided.
P edic i e models ha e wo equi alen ep esen a ions:
P obabilis ic ep esen a ion: on he space o candida e causal ne wo ks Cau, a
p obabili y measu e is gi en
µi:Cau →[0,1],X
C∈Cau
µi(C)=1,
whe e C= (E, ⪯(C)) is a candida e causal s uc u e;
Pa ame ic ep esen a ion: he e exis s a s a is ical model pθand pa ame e mani-
old Θ, wi h obse e model speci ied by pa ame e poin θi∈Θ.
One can con e be ween he wo ep esen a ions ia he map θi7→ µi. The s a is ical
model (Θ, pθ) is equipped wi h he Fishe –Rao me ic, making Θ a s a is ical mani old.
4
De ini ion 2.1 (Obse e S a e).The s a e o obse e Oia a “consensus s ep pa am-
e e ” τis de ined as he quad uple
Xi(τ) := (Ei,⪯i, θi(τ), ωi(τ)),
whe e θi(τ)∈Θ is i s s a is ical model pa ame e , and ωi(τ) is i s s a e on he
bounda y algeb a ( o be speci ied below).
The mul i-obse e s a e amily {Xi(τ)}cons i u es a mul iple local desc ip ion o he
uni e se causal ne wo k.
2.3 Uni ied Time Scale and Mo he Scale Func ion
In he uni ied ime scale and bounda y ime geome y amewo k, he o al sca e ing
semi-phase φ(ω), ela i e s a e densi y ρ el(ω), and he ace o he Wigne –Smi h delay
ope a o Q(ω) = −iS†(ω)∂ωS(ω) a e uni ied as he mo he scale densi y unc ion
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
whe e S(ω) is he sca e ing ma ix. This scale simul aneously encodes sca e ing
delay, s a e densi y, and g oup delay, is he co e scale objec o bounda y ime geome y,
and is compa ible wi h ecen s udies o Wigne ime delay and g oup delay.
Di e en obse e s, h ough hei accessible sca e ing expe imen s and bounda y
s a es, can econs uc local mo he scale unc ions κi(ω) on some ene gy window I⊂R.
Ideally, he e exis cons an s ai>0, bi∈Rsuch ha
κi(ω) = aiκ(ω) + bi.
To elimina e a ine eedom, escaling o each κion Iis needed.
2.4 Bounda y Algeb a, Modula Flow, and Null–Modula Dou-
ble Co e
Le he bounda y obse able algeb a be a C∗o on Neumann algeb a A∂, wi h s a es
ωibeing posi i e no malized linea unc ionals. Tomi a–Takesaki heo y gua an ees a
one- o-one co espondence wi h modula g oups {σ(ωi)
}, whose gene a o is he modula
Hamil onian K(i), i.e.,
d
d σ(ωi)
(A) =0 = i[K(i), A].
The Connes–Ro elli he mal ime hypo hesis iews he modula pa ame e as a ime
scale, making ime an in insic objec o he s a e–algeb a pai . The modula Hamil onian
allows a ine ans o ma ion K(i)7→ aK(i)+b1wi hou changing physical in e p e a ion.
On small causal diamonds and hei Null bounda ies, changes in gene alized en opy
Sgen oge he wi h modula low di ec ion cha ac e ize he “ ime a ow.” The Null–
Modula double co e li s he geome y and modula low o small causal diamonds
o a Z2double co e space, whose sec o is de e mined by he cohomology class [K]∈
H2(Y, ∂Y ;Z2). The sec o s seen by di e en obse e s in hei espec i e co e ing egions
Uia e deno ed [Ki].
5
3 Main Resul s: Theo ems and Alignmen s
Unde he abo e models and assump ions, his sec ion p esen s he main esul s o his
pape . We i s cons uc con lic me ics, hen de ine he consensus geome y space and
po en ial ene gy unc ion, and inally s a e he heo em on g adien low con e gence o
he consensus mani old.
3.1 Cons uc ion o Con lic Me ics
De ini ion 3.1 (Model Di e gence and Consensus Speed).Fo wo obse e s Oi, Oj, le
µi, µjbe hei p obabili y measu es on Cau. De ine he Kullback–Leible di e gence
DKL(µi∥µj) := X
C∈Cau
µi(C) log µi(C)
µj(C),
and he Jensen–Shannon di e gence
DJS(µi, µj) := 1
2DKL(µi∥¯µ) + 1
2DKL(µj∥¯µ),¯µ=1
2(µi+µj).
I obse e s a es e ol e wi h pa ame e τ, i.e., µi=µi(τ), de ine consensus speed as
ij(τ) := −d
dτDJSµi(τ), µj(τ).
De ini ion 3.2 (Cycle-B eaking Cos ).Le R:= SN
i=1 ⪯ibe he me ged ela ion o local
pa ial o de s, and ⪯glue i s ansi i e closu e. Assign o each di ec ed edge e→ ∈Ra
weigh w(e→ )≥0. The cycle-b eaking cos is de ined as
Vcycle := min
X
(e→ )∈S
w(e→ )S⊂R, R Shas ansi i e closu e ha is a pa ial o de
.
Clea ly Vcycle ≥0, and Vcycle = 0 i and only i ⪯glue i sel is a pa ial o de .
De ini ion 3.3 (Rescaled Mo he Scale and Scale Di e gence).On ene gy window I⊂R,
choose weigh unc ion w(ω)>0 in eg able. Fo obse e Oi, choose ai>0, bisuch ha
ZIκi(ω)−aiκ e (ω)−bi2w(ω) dω
is minimized, whe e κ e is a e e ence scale. Deno e he escaled scale
κ en
i(ω) := aiκi(ω) + bi,
and he a e age scale
¯κ(ω) := 1
N
N
X
i=1
κ en
i(ω).
De ine he scale di e gence
∆2
κ:= ZI 1
N
N
X
i=1 κ en
i(ω)−¯κ(ω)2!w(ω) dω.
6
De ini ion 3.4 (En opy A ow Misma ch Ra e and Modula Flow Di e ence).On o e -
lapping Null gene a o s, le λbe an a ine pa ame e , and S(i)
gen(λ) he gene alized en opy
compu ed by obse e Oi. De ine he en opy a ow misma ch a e
Ξij := Ro e lap 1sign ∂λS(i)
gen = sign ∂λS(j)
gendµ
Ro e lap dµ,
whe e dµis a na u al measu e and 1(·) is he indica o unc ion.
The di e ence o modula Hamil onians K(i), K(j)in o e lapping egions is de ined as
∆(ij)
mod := in
a>0,b∈R
K(i)−aK(j)−b1
,
whe e he no m can be ope a o no m o Hilbe –Schmid no m.
De ini ion 3.5 (Topological Sec o Con lic Me ic).Le [Ki]∈H2(Y, ∂Y ;Z2) be he
Null–Modula double co e sec o o obse e Oi. De ine he opological sec o coun
∆ opo := #{[Ki]|i= 1, . . . , N},
and he pai wise indica o unc ion
δ(ij)
opo := (0,[Ki] = [Kj],
1,[Ki]= [Kj].
When ∆ opo = 1, all obse e s a e in he same Z2sec o .
3.2 Consensus Geome y Space and Po en ial Ene gy Func ion
The s a is ical model amily {pθ|θ∈Θ}unde he Fishe –Rao me ic
gFR
ab (θ) := Eθ∂alog pθ(X)∂blog pθ(X)
o ms a Riemannian s a is ical mani old (Θ, gFR).
Assume he e exis s a causal se moduli space (C, dC) wi h a Riemannian s uc u e
gCcompa ible wi h he me ic; he mo he scale unc ion space is he Hilbe space
Hκ=L2(I, w(ω) dω),
equipped wi h s anda d inne p oduc and me ic gκ; he bounda y s a e space in
ini e dimension can be aken as he se o densi y ma ices S, equipped wi h Bu es
me ic dBu es and co esponding Riemannian s uc u e gBu es.
De ini ion 3.6 (Consensus Geome y Space and To al Me ic).Fo Nobse e s, de ine
M:= ΘN× C × HN
κ× SN,
wi h gene al elemen deno ed
X= (θ1, . . . , θN;C;κ1, . . . , κN;ρ1, . . . , ρN).
On Mde ine he o al me ic
7
G:= α
N
M
i=1
gFR
(i)+β gC+γ
N
M
i=1
gκ
(i)+δ
N
M
i=1
gBu es
(i),
whe e α, β, γ, δ > 0 a e weigh pa ame e s.
De ini ion 3.7 (Consensus Po en ial Ene gy Func ion).Gi en weigh s wmodel
ij , wΞ
ij, wmod
ij ≥
0 and λpose , λκ, λ opo >0, de ine
Fmodel := X
1≤i<j≤N
wmodel
ij DJS(θi, θj),
Fpose := λpose Vcycle(C;{⪯i}), Fκ:= λκ∆2
κ({κi}),
Fmod := X
1≤i<j≤NwΞ
ij Ξij +wmod
ij ∆(ij)
mod, F opo := λ opo∆ opo −12.
The o al po en ial is de ined as
F:= Fmodel +Fpose +Fκ+Fmod +F opo.
De ini ion 3.8 (Consensus Mani old).The consensus mani old is de ined as
Mcons := {X∈M|F(X) = 0}.
By cons uc ion, F≥0, and F= 0 i and only i Fmodel =Fpose =Fκ=Fmod =
F opo = 0. This ex emal condi ion co esponds o a “comple e consensus” s a e, whose
geome ic and physical meaning will be analyzed la e and in appendices.
3.3 Consensus G adien Flow and Con e gence Theo em
On (M, G) conside he g adien low o po en ial F.
De ini ion 3.9 (Consensus G adien Flow).Gi en ini ial s a e X(0) ∈ M, he consensus
g adien low is de ined as
d
dτX(τ) = −g adGFX(τ), X(0) gi en,
whe e g adGis he g adien wi h espec o me ic G.
P oposi ion 3.10 (Po en ial Mono onici y).Along he consensus g adien low,
d
dτFX(τ)=−g adGFX(τ)2
G≤0.
The p oo elies on he gene al o mula o g adien lows on Riemannian mani olds,
gi en in Appendix A. I s di ec implica ion is: he consensus e olu ion p ocess always
lowe s o al con lic po en ial ene gy along he “s eepes descen di ec ion.”
To discuss con e gence, we in oduce he ollowing assump ions.
Assump ion 1 (Comple eness and S ong Con exi y).1. (M, G) is a comple e Rie-
mannian mani old;
8
2. Po en ial F:M → [0,∞) is a C2 unc ion and bounded below;
3. The e exis s cons an m > 0 such ha on some geodesically con ex subse con ain-
ing he g adien low ajec o y, o any angen ec o ∈TXM,
HessGF(X)[ , ]≥m GX( , ),
i.e., Fis m-s ongly con ex on ha subse ;
4. The consensus mani old Mcons is nonemp y and is a closed geodesically con ex
subse .
Unde his assump ion, we ob ain he ollowing main esul .
Theo em 3.11 (Exponen ial Con e gence o Consensus G adien Flow).When Assump-
ion 1 holds, o any ini ial alue X(0) ∈ M, he consensus g adien low has a unique
global solu ion X(τ). Along his solu ion:
1. Po en ial FX(τ)dec eases mono onically and con e ges o a minimum F∗≥0;
2. I F∗= 0, hen he e exis s a unique poin X∗∈ Mcons such ha
dis GX(τ), X∗≤Ce−mτ ,
whe e cons an C > 0depends only on ini ial condi ions and local geome y o F.
This heo em shows ha unde app op ia ely cons uc ed geome y and po en ial, he
p ocess o “mul i-obse e con lic esolu ion” can be unde s ood as a g adien low line
on he consensus geome y space, along which all con lic me ics dec ease mono onically
and con e ge exponen ially o a poin on he consensus mani old.
P oposi ion 3.12 (Equi alence o Cycle-B eaking Cos and Global Pa ial O de ).Un-
de local consis ency condi ions (judgmen s on o e lapping egions Ei∩Ejby each ⪯ia e
consis en ), he ollowing a e equi alen :
1. Vcycle = 0;
2. ⪯glue is a pa ial o de ;
3. The e exis s a global pa ial o de ⪯such ha o all i,⪯ |Ei=⪯i.
Thus, anishing o cycle-b eaking cos is equi alen o he exis ence o a global causal
s uc u e compa ible wi h all local pe spec i es.
The p oo o P oposi ion ?? is in Appendix B.
P oposi ion 3.13 (Geome ic–Physical Meaning o Simul aneous Vanishing o Con lic
Me ics).I a some s a e X∈ M we ha e
Fmodel =Fpose =Fκ=Fmod =F opo = 0,
hen he e exis :
9
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[20] K. Sun, G. Lebanon, S. S a, “An In o ma ion Geome y o S a is ical Mani old
Lea ning,” AISTATS 2014.
A Technical P oo o G adien Flow Con e gence
Le (N, h) be a comple e Riemannian mani old, and :N → RaC2 unc ion. The
g adien low is de ined as
d
dτx(τ) = −g adh x(τ).
16
A.1 Uniqueness o Minimum Poin
Assume he e exis s m > 0 such ha o all x∈ N and ∈TxN,
Hessh (x)[ , ]≥m hx( , ),
i.e., is m-s ongly con ex. I x∗, y∗a e wo minimum poin s, hen g adh (x∗) =
g adh (y∗) = 0, and (x∗) = (y∗). Take geodesic γ: [0,1] → N connec ing x∗, y∗, and
conside g(s) := (γ(s)). Then
g′′(s) = Hessh ˙γ(s),˙γ(s)≥m h( ˙γ(s),˙γ(s)) ≥0.
Thus gis s ongly con ex, and unless x∗=y∗, i canno a ain minimum a bo h
in e al endpoin s, so he minimum poin is unique.
A.2 Exis ence and Uniqueness o G adien Flow Solu ion
I g adh is Lipschi z on bounded se s, hen h ough ODE heo y on Riemannian mani-
olds, one can cons uc local solu ions nea any ini ial alue. Comple eness and g adien
boundedness gua an ee he solu ion canno escape o in ini y in ini e ime, so he solu ion
ex ends o all τ≥0.
A.3 Exponen ial Con e gence Es ima e
Le x(τ) be a g adien low solu ion, and x∗ he unique minimum poin . De ine
Φ(τ) := (x(τ)) − (x∗)≥0.
S ong con exi y and Taylo expansion gi e
(y)≥ (x) + ⟨g adh (x),exp−1
xy⟩h+m
2|exp−1
xy|2
h.
Taking x=x(τ), y =x∗, no ing g adh (x∗) = 0, we ge
Φ(τ)≤ −⟨g adh (x(τ)),exp−1
x(τ)x∗⟩h−m
2|exp−1
x(τ)x∗|2
h.
On he o he hand,
d
dτΦ(τ) = ⟨g adh (x(τ)),˙x(τ)⟩h=−| g adh (x(τ))|2
h.
Combining he abo e es ima es, we ob ain
d
dτΦ(τ)≤ −2mΦ(τ),
hence
Φ(τ)≤e−2mτ Φ(0).
Fu he using s ong con exi y, one can ela e Φ(τ) wi h Riemannian dis ance dis h(x(τ), x∗),
ob aining
dis h(x(τ), x∗)≤Ce−mτ ,
17
whe e Cdepends only on ini ial condi ions and local geome y o . This comple es
he p oo amewo k o Theo em ??.
B Cycle-B eaking Cos and Global Pa ial O de Ex-
is ence
Gi en e en se Eand local pa ial o de amily {⪯i}N
i=1, he me ged ela ion
R=
N
[
i=1
⪯i
has ansi i e closu e deno ed ⪯glue. Assume o all i, j, judgmen s on o e lapping
egion Ei∩Eja e consis en , i.e.,
x⪯iy⇐⇒ x⪯jy, ∀x, y ∈Ei∩Ej.
B.1 Equi alence o Ze o Cycle-B eaking Cos and T ansi i e
Closu e Being Pa ial O de
By de ini ion o cycle-b eaking cos :
Vcycle = min
X
(e→ )∈S
w(e→ )|S⊂R, R Shas ansi i e closu e ha is pa ial o de
.
Clea ly, i he e exis s a di ec ed cycle, a leas one edge mus be dele ed, so Vcycle >0;
con e sely, i ⪯glue is al eady a pa ial o de , no edge dele ion is needed, aking S=∅
su ices, so Vcycle = 0. Thus
Vcycle = 0 ⇐⇒ ⪯glue is a pa ial o de .
B.2 Equi alence o T ansi i e Closu e Being Pa ial O de and
Global Pa ial O de Exis ence
De ine global ela ion ⪯:=⪯glue. Clea ly es ic ion ⪯ |Eicon ains ⪯i. Since local pa ial
o de s a e consis en on o e lapping egions, any ela ion de i ed wi hin Ei om o he
⪯jmus be compa ible wi h ⪯i, hence ⪯ |Ei=⪯i. The e o e, i ⪯glue is a pa ial o de ,
i is a global pa ial o de ealizing all local pa ial o de s.
Con e sely, i he e exis s global pa ial o de b
⪯sa is ying b
⪯|Ei=⪯i, hen he g aph
o b
⪯con ains R, i s ansi i e closu e is i sel , and since b
⪯is a pa ial o de , i is acyclic,
hence Vcycle = 0.
In summa y, P oposi ion ?? is p o ed.
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C Geome ic and Physical Meaning o Simul aneous
Vanishing o Con lic Me ics
Assume a some s a e X∈ M, all con lic me ics simul aneously anish:
Fmodel =Fpose =Fκ=Fmod =F opo = 0.
C.1 Consis ency o S a is ical Models
Fmodel = 0 means o all i, j, we ha e DJS(θi, θj) = 0. Jensen–Shannon di e gence is
ze o i and only i he wo dis ibu ions a e almos e e ywhe e iden ical, hence all pθia e
consis en . In pa ame e space Θ, his means he e exis s unique pa ame e poin θ∗such
ha θi=θ∗ o all i.
C.2 Exis ence o Global Causal Pa ial O de
Fpose = 0 i.e., Vcycle = 0; unde local consis ency assump ions, Appendix B shows he e
exis s global pa ial o de ⪯embedding all local pa ial o de s ⪯i. Thus he e exis s
global causal ne wo k (E, ⪯) whose es ic ion o each obse e ho izon Eiis he pa ial
o de s uc u e.
C.3 Uni ied Mo he Scale and Time Scale Consensus
Fκ= 0 means scale di e gence ∆κ= 0, i.e., o all ω∈I,
κ en
i(ω) = ¯κ(ω).
Thus he e exis s uni ied mo he scale unc ion κ∗(ω) := ¯κ(ω) such ha each obse e ’s
local scale coincides wi h i a e app op ia e a ine escaling. In he uni ied ime scale
amewo k, his co esponds o all obse e s adop ing he same scale iden i y among
sca e ing phase de i a i e, ela i e s a e densi y, and g oup delay.
C.4 Consis ency o En opy A ow and Modula Flow Di ec ion
Fmod = 0 implies o all i, j, Ξij = 0 and ∆(ij)
mod = 0. The o me means on o e lapping Null
gene a o s, ∂λS(i)
gen and ∂λS(j)
gen ha e consis en signs, so en opy a ow di ec ions ag ee;
he la e means he e exis aij >0, bij ∈Rsuch ha K(i)=aijK(j)+bij1. Th ough
ansi i i y, one can choose global coe icien s ai>0, bi∈Rsuch ha all K(i)a e a inely
equi alen o some uni ied modula Hamil onian K∗.
Thus in consensus s a e, all obse e s achie e consis ency in modula low and gen-
e alized en opy a ow, making ime a ow de ini ion a global bounda y ime geome ic
s uc u e.
C.5 Consis ency o Null–Modula Double Co e Sec o
F opo = 0 gi es ∆ opo = 1, i.e., all [Ki] a e iden ical. By Z2p incipal bundle classi ica ion
in cohomology, his shows he e exis s a global double co e sec o [K] such ha [Ki] =
[K]|Ui. This means Null–Modula double co e opological s uc u e is compa ible o e
all obse e ho izons, wi h no sec o misma ch on consensus mani old.
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C.6 Summa y
Combining C.1–C.5, we gi e he ollowing pic u e:
The e exis s uni ied s a is ical model poin θ∗desc ibing all obse e s’ p obabilis ic
p edic ions o causal ne wo k;
The e exis s uni ied global causal pa ial o de ⪯accommoda ing all local pa ial
o de s;
The e exis s uni ied mo he scale unc ion κ∗(ω) and uni ied modula Hamil onian
K∗, a e app op ia e p e ac o s and ze o-poin choices, all obse e s’ ime scales
and modula lows align wi h hem;
The e exis s uni ied Null–Modula double co e sec o [K];
Thus mul i-obse e sys em is in a s a e comple ely consis en a all le els o causali y,
ime, en opy, and opology. This s a e geome ically co esponds o a poin o o bi on
consensus mani old Mcons, he limi ing o m o “obse e consensus geome iza ion” in
his amewo k.
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