Geome iza ion o Causal S uc u es: Space ime as
Minimal Lossless Comp ession o Causal Cons ain s
Haobo Ma1Wenlin Zhang2
1Independen Resea che
2Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
S a ing om i s p inciples, his pape ea s “causal s uc u e” as he mos
p imi i e and economical in o ma ion objec in he physical wo ld, and p oposes
a uni ied pe spec i e ha “space ime geome y = minimal lossless comp ession o
causal cons ain s”. Speci ically, we cha ac e ize causal eachabili y by pa ial o -
de ela ions on e en se s, eco e con o mal s uc u e ia Alexand o opology
and ime o ien a ion, and supplemen he absolu e scale o he me ic wi h olume
calib a ion and s a is ical in o ma ion. We p o e ha unde qui e gene al condi-
ions: gi en causal s uc u e and olume measu e, he con o mal class o space ime
me ic can be uniquely econs uc ed. Subsequen ly, we in oduce a “desc ip ion
leng h–cu a u e” a ia ional p inciple, in e p e ing cu a u e as he “ edundancy
densi y” o co ela ions among causal cons ain s ha canno be elimina ed, and
p opose he unc ional
F[g] = C(Reach(g)) + λZM
|Riem(g)|2dVolg
as an abs ac model o he ade-o be ween geome ic econs uc ion and
causal comp ession. A he le el o linea quan um ield heo y and in o ma ion
geome y, we discuss how mic ocausali y, modula low, and ela i e en opy mono-
onici y de ine he “in o ma ion leng h” o dis inguishable pa hs unde gi en causal
cone s uc u e, he eby geome izing he ela ionships among causali y, geome y,
and in o ma ion. Th ough examples o Minkowski space ime, FRW uni e se, and
ini e causal se embeddings, we demons a e how his pe spec i e uni ies he in u-
i ion ha “ la ness = no edundancy” and “cu a u e = cons ain co ela ions can-
no be la ened”. The appendices p o ide heo em s a emen s and p oo ske ches
o how causal s uc u e de e mines con o mal class, echnical de ails o Alexan-
d o opology and s ong causali y, and o mal de i a ion o a ia ional equa ions
o desc ip ion complexi y unc ionals.
Keywo ds: Causal S uc u e, Geome iza ion, In o ma ion Comp ession, Space ime
Eme gence, Con o mal Class, Alexand o Topology, Desc ip ion Complexi y
MSC 2020: 83C45, 94A29, 06A06, 53C50
Con en s
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1 In oduc ion
1.1 P oblem Backg ound and Basic Posi ion
In gene al ela i i y, space ime is cha ac e ized as a di e en iable mani old (M, g) equipped
wi h a Lo en z me ic g. The me ic gplays wo oles simul aneously:
1. De e mining ligh cones and causal eachabili y (causal s uc u e);
2. De e mining quan i a i e calib a ion o ime and space (leng hs and olumes).
Howe e , nume ous heo ems show ha unde app op ia e causal condi ions, me ely
knowing “which e en s can causally in luence which e en s” as a eachabili y ela ion is
su icien o la gely eco e he con o mal s uc u e o space ime, i.e., he con o mal class
[g] o g. This sugges s a highly comp ession-o ien ed iewpoin :
I we only ca e abou “wha in luences causali y allows”, hen comple e me -
ic in o ma ion is a mo e han necessa y; causal s uc u e i sel is a mo e
p imi i e and economical in o ma ion objec , while geome y is a kind o “en-
coding” buil upon i .
This pape asks h ee ques ions om his pe spec i e:
1. I only causal s uc u e is gi en, o wha ex en can geome y be eco e ed?
2. Can geome y be iewed as a kind o “minimal lossless comp ession” o causal
cons ain s?
3. Can cu a u e, a adi ional geome ic quan i y, be in e p e ed as he “ edundancy
densi y o co ela ions among causal cons ain s”?
1.2 S uc u al O e iew o This Pape ’s Con ibu ions
The main con ibu ions o his pape can be summa ized as ollows:
1. Axioma ic Causal S uc u e and Geome ic Recons uc ion: Based on pa -
ially o de ed se s and Alexand o opology, we in oduce he concep o “causal
space” and p o ide heo em s a emen s o co espondence be ween causal homeo-
mo phisms and con o mal homeomo phisms: unde s ong causali y and mode a e
egula i y assump ions, causal s uc u e uniquely de e mines he con o mal class.
2. Comp ession Pe spec i e and Va ia ional P inciple: De ine desc ip ion com-
plexi y unc ion Co causal eachabili y g aphs, p opose he unc ional
F[g] = C(Reach(g)) + λZM
|Riem(g)|2dVolg
,
in e p e ing cu a u e as “accoun ing o co ela ions among causal cons ain s ha
canno be simul aneously la ened”.
2
3. Embedding o In o ma ion Geome y and Quan um Field Theo y: Unde
gi en causal s uc u e and bounda y algeb a, cons uc “in o ma ion me ic” in he
in e io o causal cones using mic ocausali y, modula low, ela i e en opy, and
Fishe in o ma ion, connec ing he leng h o “dis inguishable pa hs” wi h causal
cons ain s.
4. Case Analysis and Fini e Models: Demons a e how he abo e s uc u es a e
conc e ely ealized in Minkowski, FRW, and disc e e causal se cases, along wi h
in ui i e in e p e a ions.
The appendices p o ide o mally p ecise s a emen s and p oo ske ches o in ol ed
heo ems, wi h supplemen s on echnical de ails o opology and measu e.
2 Causal S uc u e and Pa ial O de Models
2.1 Pa ial O de Cha ac e iza ion o Causal S uc u e
Le Mbe a ou -dimensional di e en iable mani old equipped wi h a Lo en z me ic g.
Fo any poin p∈M, de ine he u u e and pas o imelike samples:
J+(p): The se o poin s eachable om p ia non-spacelike causal cu es (including
null and imelike);
J−(p): Simila ly de ined as he se o poin s ha can causally each p.
De ine bina y ela ion ≤as ollows: o p, q ∈M, w i e p≤qi q∈J+(p). Unde
app op ia e causal condi ions (such as no closed causal cu es), ≤cons i u es a pa ial
o de ela ion on M.
De ini ion 2.1 (Causal P eo de and Causal Se ).Le Xbe a se and ⪯a e lexi e,
ansi i e ela ion on X. Call (X, ⪯) a causal p eo de ed se . I addi ionally ⪯has
an isymme y, call i a causally pa ially o de ed se o causal se .
Fo space ime (M, g), (M, ≤) is a causally pa ially o de ed se . We o en use no a ion
p≪q o deno e q∈I+(p), whe e I+(p) is he s ic ly imelike u u e.
2.2 Alexand o Topology and Causal Open Se s
A na u al opology can be gene a ed om causal s uc u e.
De ini ion 2.2 (Alexand o Basis).Fo p, q ∈Mwi h p≪q, de ine he bicone open se
A(p, q) := I+(p)∩I−(q).
Call B:= {A(p, q) : p, q ∈M, p ≪q} he Alexand o basis. The opology gene a ed
by Bis called he Alexand o opology.
I (M, g) is s ongly causal, hen he Alexand o opology coincides wi h he o iginal
di e en iable mani old opology. Thus unde he s ong causali y assump ion, opolog-
ical s uc u e can be econs uc ed om causal s uc u e alone.
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3 Causal S uc u e, Time O ien a ion, and Con o -
mal Class
3.1 Con o mal S uc u e and Ligh Cones
The con o mal class o a Lo en z me ic gis de ined as
[g] := {Ω2g:Ω:M→(0,∞) smoo h}.
Con o mally equi alen me ics ha e he same null ec o cones, i.e., he same ligh
cone s uc u e and causal ela ions. The e o e:
Causal s uc u e (M, ≤) depends only on he con o mal class [g];
I wo Lo en z me ics g, ˜ginduce he same causal s uc u e, hen unde app op ia e
condi ions hey a e con o mally equi alen .
Axiom 3.1 (Time O ien a ion).Assume space ime (M, g) is ime-o ien able, i.e., he e
exis s a globally consis en choice o “ u u e” di ec ion, allowing imelike ec o ields o
be globally o ien ed. Time o ien a ion and causal s uc u e oge he de e mine in o ma-
ion abou “which side is he u u e”.
Unde his assump ion, causal s uc u e con ains in o ma ion abou :
1. Which poin s can causally in luence which poin s;
2. Fo each causal cu e, which di ec ion is “ u u e”.
3.2 Causal Homeomo phism and Con o mal Homeomo phism
De ini ion 3.2 (Causal Homeomo phism).Le (M, [g]), ( ˜
M, [˜g]) be wo space imes wi h
causal ela ions deno ed ≤,˜
≤ espec i ely. I he e exis s a bijec ion Φ : M→˜
Msuch
ha o any p, q ∈M,
p≤q⇐⇒ Φ(p)˜
≤Φ(q),
hen call Φ a causal homeomo phism.
Unde s ong causali y, local compac ness, and app op ia e egula i y assump ions,
he ollowing undamen al heo em can be s a ed ( o malized e sion and p oo ske ch
gi en in he appendix):
Theo em 3.3 (Causal S uc u e De e mines Con o mal Class, Theo em Fo m).Le
(M, g),(˜
M, ˜g)be s ongly causal, locally compac space imes, assuming no “pa hologi-
cal” causal bounda ies. I he e exis s a causal homeomo phism Φ : M→˜
M, hen Φis
a con o mal homeomo phism, i.e., he e exis s a smoo h unc ion Ω : ˜
M→(0,∞)such
ha
Φ∗˜g= Ω2g.
The e o e, unde hese assump ions, causal s uc u e + ime o ien a ion
su ices o de e mine he con o mal class.
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4 Volume Calib a ion and Absolu e Me ic Recon-
s uc ion
4.1 Causali y + Volume Measu e
The con o mal class only de e mines he s uc u e o “ligh cones” and “null geodesics”,
no ye de e mining he absolu e leng h scale. Fo his, in oduce olume calib a ion:
Axiom 4.1 (Volume Calib a ion).On (M, [g]), gi en a Bo el measu e µcompa ible wi h
he olume o m dVolgo some ep esen a i e me ic g, i.e., he e exis s a posi i e smoo h
unc ion ρ:M→(0,∞) such ha o all measu able se s A⊂M,
µ(A) = ZA
ρdVolg.
In ui i ely, µcap u es in o ma ion abou “e en densi y” o “ olume calib a ion”.
Unde app op ia e egula i y condi ions, causal s uc u e + olume measu e can e-
co e he absolu e scale o he me ic: by compa ing how he olume o Alexand o se s
A(p, q) a ies wi h p, q, one can in e sely de e mine he assignmen o con o mal ac o
Ω, hus eco e ing he speci ic me ic.
4.2 S ep-by-S ep Na a i e o Geome ic Recons uc ion
In summa y, geome ic econs uc ion can be iewed as h ee s eps:
1. F om Pa ial O de o Topology: Causal pa ial o de (M, ≤) gene a es Alexan-
d o opology, which unde s ong causali y equals he mani old opology;
2. F om Causali y o Con o mal Class: Causal s uc u e and ime o ien a ion
unde s ong causali y and egula i y uniquely de e mine con o mal class [g];
3. F om Volume o Me ic: In oduce olume measu e µ; by compa ing olumes o
Alexand o se s wi h con o mal s uc u e, eco e absolu e scaling ac o o ob ain
speci ic g.
The e o e, unde app op ia e condi ions, he ollowing da a can be iewed as equi a-
len geome ic “encodings”:
(M, g)⇐⇒ (M, ≤, µ).
In he igh -hand da a, ≤and µa e closely ela ed o “ eachable/un eachable” and
“ olume calib a ion”, ha ing mo e “in o ma ion comp ession” implica ions.
5 Geome y as Minimal Lossless Comp ession o Causal
Cons ain s
5.1 Causal Reachabili y G aph and Desc ip ion Complexi y
A e disc e izing space ime, e en s can be iewed as e ices and causal ela ions as
edges o a di ec ed acyclic g aph, ob aining a causal eachabili y g aph G= (V, E). E en
in he con inuous case, causal s uc u e can be app oxima ed as such g aphs h ough
some sampling o coa se-g aining.
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De ini ion 5.1 (Desc ip ion Complexi y).Le G= (V, E) be a ini e di ec ed acyclic
g aph, Da class o desc ip ion languages (such as adjacency ma ix encoding, adjacency
lis , hie a chical decomposi ion, e c.). De ine he minimal desc ip ion complexi y unde
p ecision ϵas
Cϵ(G) := min{encoding leng h(Code) : Code ∈ D, econs uc ion e o ≤ϵ}.
In app op ia e limi s, a con inuous e sion can be de ined deno ed C(Reach(g)), whe e
Reach(g) ep esen s he causal eachabili y s uc u e induced by me ic g.
In ui i ely, C(Reach(g)) measu es “how much in o ma ion is needed a minimum o
eco d all causal cons ain s”.
5.2 Cu a u e as Redundancy Densi y
Geome ically, local la ness means ha in su icien ly small neighbo hoods, coo dina e
sys ems can be ound making he me ic app oxima ely Minkowski, wi h ligh cone s uc-
u e “as s aigh as possible”. I we iew local causal s uc u es in each local neighbo hood
as “local cons ain s”, hen when hese cons ain s can be compa ibly assembled in o a
globally la s uc u e, cu a u e is ze o; when his compa ibili y ails, cu a u e eco ds
his “closed-loop de ia ion ha canno be elimina ed” h ough he Riemann enso Riem.
The e o e, he ollowing in e p e a ion can be p oposed:
Cu a u e can be iewed as he “ edundancy densi y o co ela ions among
causal cons ain s ha canno be locally elimina ed”.
This in e p e a ion can be in ui i ely unde s ood by conside ing closu e e o s o
causal iangles o causal polygons: when combining local causal cons ain s along di -
e en pa hs, i esul s a e no comple ely consis en , cu a u e mus be in oduced o
accoun o hese di e ences.
5.3 Desc ip ion Leng h–Cu a u e Va ia ional P inciple
The abo e in ui ion can be o malized as a a ia ional p inciple. Gi en a causal s uc u e
class Ccaus and olume calib a ion, we conside seeking he “op imal geome ic encoding”
among all compa ible me ics.
De ini ion 5.2 (Desc ip ion Leng h–Cu a u e Func ional).On he compa ibili y class
Mo gi en causal s uc u e and olume calib a ion, de ine unc ional
F[g] := C(Reach(g)) + λZM
|Riem(g)|2dVolg, g ∈M,
whe e λ > 0 is a weigh pa ame e .
Fi s e m C(Reach(g)): Measu es he minimum desc ip ion leng h needed o ac-
cu a ely eco d causal eachabili y s uc u e induced by g;
Second e m RM|Riem(g)|2dVolg: Penalizes high cu a u e, a o ing selec ion o
geome y ha is as “ la ” as possible.
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Axiom 5.3 (Va ia ional P inciple).Unde gi en causal s uc u e and olume calib a ion
cons ain s, he physically ac ually chosen geome y can be iewed as a minimize (o
local minimize ) o F[g].
He e C(Reach(g)) s ic ly depends on disc e iza ion o coa se-g aining mode, so i s
a ia ional o m is complex. Bu in he case whe e “causal econs uc ion is al eady gi en
wi h only con o mal ac o eedom emaining”, Cas a cons an e m exi s a ia ion,
lea ing only
δZM
|Riem(g)|2dVolg= 0,
co esponding o c i ical poin s o he so-called L2-cu a u e low. Mo e gene ally, C
can be iewed as cons ain s on he allowed me ic amily.
6 Quan um Field Theo y, In o ma ion Geome y, and
Causal Cons ain s
6.1 Mic ocausali y and Regional Algeb as
When de ining quan um ield heo y on a gi en space ime (M, g), i is usually equi ed
ha local obse able algeb as A(O) sa is y mic ocausali y: i O1,O2a e mu ually space-
like sepa a ed, hen o A∈ A(O1), B∈ A(O2),
[A, B]=0.
This condi ion is comple ely de e mined by causal s uc u e. Thus om he alge-
b aic quan um ield heo y (AQFT) pe spec i e, he “se o simul aneously obse able”
is speci ied by causal s uc u e.
6.2 Rela i e En opy and Dis inguishable Pa hs in Causal Cones
Gi en wo s a es ω, ω′ es ic ed o some egional algeb a A(O), ela i e en opy S(ω|ω′)
can be de ined, whose mono onici y e lec s causal eachabili y: i O ⊂ O′, hen
S(ω|A(O′)|ω′|A(O′))≥S(ω|A(O)|ω′|A(O)).
This can be iewed as “when ex ending obse able domains along causal low, dis in-
guishabili y does no dec ease”.
Unde app op ia e smoo hness condi ions, he second-o de di e en ial o ela i e
en opy can be used o cons uc Fishe in o ma ion- ype me ics, hus ob aining “in o -
ma ion leng h” in he in e io o causal cones o s a e amilies. Thus, unde gi en causal
s uc u e, in o ma ion geome ic me ics u he add he meaning o “dis inguishabili y
a e” o space ime geome y.
6.3 Modula Flow and Time Pa ame e
Tomi a–Takesaki modula heo y ells us ha o gi en (A, ω), a modula low σω
can be
de ined. In cases wi h global KMS p ope ies o he mal ime hypo hesis, he modula
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low pa ame e can be in e p e ed as a kind o in insic ime scale, wi h he gene a ed
“ low” e ol ing along di ec ions allowed by causal s uc u e.
The e o e, ime can be iewed as a “one-dimensional mani old pa ame e in s a e
space eachable along causal s uc u e”, while geome y is iewed as join encoding o
his low wi h spa ial s uc u e.
7 Examples and Speci ic Models
7.1 Minkowski Space ime: Ze o Cu a u e and Minimal Re-
dundancy
Conside ou -dimensional Minkowski space ime (R4, η), whe e η= diag(−1,1,1,1). I s
causal s uc u e has high symme y:
Ligh cones a any poin main ain shape unde Lo en z ans o ma ions;
Causal eachabili y se s uc u e is comple ely consis en unde ansla ions and
o a ions;
Cu a u e enso Riem ≡0.
F om his pe spec i e, Minkowski space ime co esponds o he case whe e “causal
cons ain s a e comple ely compa ible”: h ough a global ine ial ame, all local con-
s ain s can be la ened in o a s uc u e wi hou closed-loop de ia ions. The e o e
R|Riem|2dVol = 0, eaching minimum alue in he “cu a u e penal y e m”, while
desc ip ion complexi y o causal s uc u e is also ex emely low due o high symme y.
7.2 FRW Uni e se: Cu a u e and Causal Redundancy
Conside iso opic, homogeneous FRW me ic
g=−d 2+a( )2γijdxidxj,
whe e γij is he h ee-dimensional cons an cu a u e space me ic and a( ) is he
scale ac o . I s causal s uc u e is de e mined by cosmological ho izons and con o mal
ime η:
dη=d
a( ).
When spa ial cu a u e k= 0, spa ial sec ions ha e non-ze o cu a u e, and he o e all
space ime’s Riemann enso is also non-ze o. A his ime, he s uc u e o causal cones a
la ge scales is no longe equi alen o Minkowski space ime: he e a e “causal bounda ies”
such as cosmological ho izons and pa icle ho izons, wi h sys ema ic di e ences be ween
eachable egions o di e en wo ldlines.
F om he “ edundancy densi y” pe spec i e, FRW cu a u e eco ds closed-loop de-
ia ions p oduced when assembling local Minkowski app oxima ions in o he global uni-
e se: “composi e causal cons ain s” along di e en pa hs a e no longe comple ely
consis en .
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7.3 Fini e Causal Se s and App oxima e Embedding
In he causal se app oach, space ime is iewed as a disc e e se (C, ⪯) wi h pa ial o de ,
equi ing he pa ial o de o sa is y local ini eness: o any p, q ∈C, he se { ∈C:p⪯
⪯q}is ini e. In app op ia e dense limi s, causal se s can be app oxima ely embedded
in o con inuous space ime.
F om he comp ession pe spec i e, ini e causal se (C, ⪯) is a disc e e sampling o
con inuous causal s uc u e; i his sampling has “Poisson sp inkling” p ope ies, i s s a-
is ical p ope ies a e compa ible wi h he olume calib a ion o he o iginal con inuous
space ime.
In such disc e e models, desc ip ion complexi y C(⪯) can be di ec ly de ined, and
he ela ionship be ween complexi y and “disc e e cu a u e” unde di e en disc e e
geome ies (such as di e en andom cu a u e models) can be examined, hus p o iding
compu able es s o he “cu a u e = edundancy densi y” iewpoin p oposed in his
pape .
8 Discussion and Ou look
The “space ime geome y = minimal lossless comp ession o causal cons ain s” pe spec-
i e p oposed in his pape connec s he ollowing objec s:
1. Causal S uc u e: Pa ial o de and Alexand o opology on e en se s;
2. Geome ic S uc u e: Con o mal class, me ic, and cu a u e;
3. In o ma ion S uc u e: Desc ip ion complexi y, ela i e en opy, Fishe in o -
ma ion, and modula low.
F om his pe spec i e, geome y is no longe me ely an “objec gi ing dis ance”, bu
a he an encoding o “allowed in o ma ion low”; cu a u e becomes an accoun ing ool
eco ding he ac ha “local causal cons ain s canno be compa ibly la ened globally”.
Fu u e di ec ions o u he de elopmen include:
In speci ic quan um g a i y models (causal se s, enso ne wo ks, e c.), explici ly
calcula e C(Reach) and compa e wi h geome ic cu a u e;
In semiclassical g a i y, es ablish co espondence be ween he desc ip ion leng h–
cu a u e unc ional o his pape and he Eins ein–Hilbe ac ion and i s co ec ion
e ms;
In in o ma ion geome y and quan um in o ma ion p ocessing, uni y ela i e en-
opy mono onici y unde causal s uc u e wi h geome ic cu a u e as a “cons ain
image o in o ma ion low”.
A Theo em and P oo Ske ch o Causal S uc u e
De e mining Con o mal Class
This appendix p o ides a o malized s a emen and p oo ou line o Theo em ?? in
Sec ion 3.
9
