Topological Complexi y, Sel -Re e ence, and
Undecidabili y
in Compu a ional Uni e se:
Loop S uc u e, Fundamen al G oup,
and Second Law o Complexi y 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 cons uc ed disc e e complexi y geome y, disc e e in o ma ion geome y, con-
ol mani old (M, G) induced by uni ied ime scale, ask in o ma ion mani old
(SQ, gQ), and ime–in o ma ion–complexi y join a ia ional p inciple, es ablishing
equi alence be ween physical uni e se ca ego y and compu a ional uni e se ca e-
go y on e e sible QCA subclass. Howe e , mos undamen al class o “ha d limi s”
in compu a ional uni e se—undecidabili y and second law o complexi y—ha e no
ye been geome ically and opologically cha ac e ized.
This pape in oduces concep s o opological complexi y and sel - e e en ial
loops wi hin compu a ional uni e se amewo k, iewing undecidabili y as “con-
ac ion obs uc ion” on undamen al g oup o con igu a ion g aph, cons uc ing
unde uni ied ime scale cons ain s class o complexi y en opy unc ionals, p o ing
mono onici y along e e sible e olu ion, he eby gi ing p o o ype o m o “second
law o complexi y in compu a ional uni e se”.
Speci ically, we i s iew con igu a ion g aph Gcomp = (X, E) as ini e de-
g ee i s skele on, cons uc ing h ough app op ia e gluing p ocess opological
space X(called con igu a ion complex), whose undamen al g oup π1(X) na u-
ally co esponds o closed e olu ion loops in compu a ional uni e se. We de-
ine sel - e e en ial loops as class o closed pa hs wi h “e alua ion–encoding–
einjec ion” s uc u e, using hem o cha ac e ize opological images o p og am
sel -in e p e a ion, simula ion o i sel , and hal ing p oblem.
Second, on app op ia ely cons uc ible compu a ional uni e se amilies we p o e:
he e exis s class o algo i hmic decision p oblems educible o “whe he loop is ho-
mo opic o i ial elemen in undamen al g oup”; in hese uni e ses, i he e exis s
algo i hm capable o deciding con ac abili y o all such loops, hen hal ing p oblem
decidable, leading o con adic ion. Thus ob aining opological undecidabili y
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heo em: in gene al compu a ional uni e se, “whe he ce ain class o loops con-
ac ible” is undecidable p oblem.
Then, unde uni ied ime scale we in oduce class o complexi y en opy unc-
ionals: o each closed loop γde ine i s complexi y ac ion
S(γ) = Zκ(ω) dµγ(ω)
and i s “comp ession complexi y” K(γ) (e.g., sho es equi alen loop leng h).
Unde e e sible, local, and uni ied ime scale compa ible e olu ion, we p o e exis-
ence o unc ion C(γ), composed o S(γ) and K(γ), such ha unde na u al coa se–
g aining and en opiza ion ules, Cweakly mono onically non-dec easing along ime
di ec ion, he eby ob aining disc e e e sion o second law o complexi y.
Finally, we connec sel - e e en ial loops wi h sca e ing–delay s uc u e un-
de uni ied ime scale: on con ol mani old M, sel - e e en ial compu a ion co -
esponds o closed eedback loops in con ol–sca e ing ne wo k, whose opolog-
ical ype join ly de e mined by π1(M) and Z2- ype holonomy. We demons a e
“Null–modula double co e ” s uc u e, such ha sel - e e en ial pa i y oge he
wi h opological class cons i u e in a ian s desc ibing sel - e e ence, ecu sion, and
“sel -iden i y”.
This pape comple es opological laye and limi laye in “compu a ional uni-
e se heo y s ack”: uni ying undecidabili y and second law o complexi y in o
opological–geome ic s uc u e, p o iding ounda ion o subsequen cons uc ion
o highe -le el s uc u es such as sel - e e en ial sca e ing ne wo ks, Null–modula
double co e , and causal diamond chains.
Keywo ds: Compu a ional uni e se; Topological complexi y; Sel - e e ence; Undecid-
abili y; Hal ing p oblem; Fundamen al g oup; Complexi y en opy; Second law; Z2holon-
omy
1 In oduc ion
Compu abili y and complexi y heo y ell us: unde gene al compu able models, he e
exis undamen ally undecidable p oblems (e.g., hal ing p oblem), and insu moun able
complexi y bounda ies; gene alized he modynamics and in o ma ion heo y indica e ha
sys ems subjec o physical ime scale and ene gy cons ain s, hei “e ec i e complexi y”
and “a ailable in o ma ion” subjec o some i e e sible e olu ion law.
In “compu a ional uni e se” amewo k, uni e se abs ac ed as
Ucomp = (X, T,C,I),
whe e Xis con igu a ion se , Tis one-s ep upda e ela ion, Cis single-s ep cos unde
uni ied ime scale, Iis ask-awa e in o ma ion quali y unc ion. P e ious wo ks ha e
cons uc ed on his basis:
1. Complexi y g aph and complexi y dis ance: Gcomp = (X, E, C), dcomp(x, y) =
in C(γ);
2. In o ma ion geome y and ask in o ma ion mani old (SQ, gQ,ΦQ);
3. Uni ied ime scale and con ol mani old (M, G) and i s geodesic dis ance dG;
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4. Time–in o ma ion–complexi y join a ia ional p inciple: minimum wo ldline o ac-
ion on join mani old EQ=M×SQ;
5. Physical uni e se–compu a ional uni e se ca ego ical equi alence;
6. Single/mul i-obse e a en ion–knowledge g aph–consensus geome y;
7. Bounda y compu a ion and causal diamond on ini e blocks.
No ye sys ema ically add essed a e:
Undecidabili y: how should hal ing p oblem and mo e gene al “global p ope y
undecidabili y” be geome ically and opologically cha ac e ized in compu a ional
uni e se?
Sel - e e en ial s uc u e: how do sel -in e p e ing p og ams, sel -simula ing sys-
ems, sel - e e en ial sca e ing ne wo ks mani es as speci ic opological in a ian s
on con igu a ion g aph and con ol mani old?
Second law o complexi y: unde uni ied ime scale, does he e exis complexi y
en opy unc ional mono onically non-dec easing along e olu ion, he eby gi ing
opological–geome ic in e p e a ion o “ ime a ow in compu a ional uni e se”?
T adi ional compu abili y heo y ocuses on languages and unc ions, no di ec ly
p o iding space– ime geome y; while adi ional geome ic opology o en assumes un-
de lying space is “gi en le hand”, no conside ing i s compu abili y and complexi y con-
s ain s. Compu a ional uni e se amewo k p o ides na u al in e ace: con igu a ion
g aph Gcomp and con ol mani old (M, G) bo h ca y compu abili y and complexi y, ye
ha e explici geome ic a chi ec u e.
S a egy o his pape is:
1. Embed con igu a ion g aph Gcomp h ough s anda d “g aph–complex” p ocess in o
wo-dimensional o highe -dimensional CW complex X, such ha closed pa h classes
closely co espond o π1(X);
2. Abs ac sel - e e en ial compu a ion as special closed loop class in con igu a ion
g aph, using undamen al g oup p ope ies (whe he con ac ible) o ep esen
“whe he sel -consis en e mina ion exis s”;
3. Using educ ion om hal ing p oblem, p o e “deciding whe he ce ain class o
loops con ac ible” undecidable in gene al compu a ional uni e se;
4. De ine complexi y ac ion and comp ession complexi y unde uni ied ime scale, con-
s uc ing ough complexi y en opy unc ional, p o ing mono onici y unde na u al
coa se–g aining ules in second law o m;
5. Connec sel - e e en ial loops wi h closed pa hs on con ol mani old, cons uc ing
Z2- ype holonomy and Null–modula double co e , explaining sou ce o opological
in a ian s o “sel -iden i y”.
Th ough hese s eps, his pape comple es in eg a ed cha ac e iza ion o opological
complexi y, sel - e e ence, and undecidabili y in compu a ional uni e se.
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2 Topologiza ion o Con igu a ion G aph and Fun-
damen al G oup
This sec ion opologizes disc e e con igu a ion g aph Gcomp = (X, E) in o wo-dimensional
CW complex X, in oducing undamen al g oup π1(X) and homo opy classes o closed
loops.
2.1 Con igu a ion G aph and Edge Se
Recall complexi y g aph o compu a ional uni e se:
Ve ex se Xis con igu a ion se ;
Edge se E=T⊂X×Xis one-s ep upda e ela ion (we empo a ily iew as
undi ec ed, o iden i y di ec ed edges (x, y) and (y, x) as one undi ec ed edge, o
in oduce 1–skele on);
Edge weigh C(x, y) ep esen s single-s ep cos .
We i s cons uc 1–dimensional skele on.
De ini ion 2.1 (Con igu a ion G aph 1–Skele on).1–skele on o con igu a ion g aph
G(1)
comp is 1–dimensional CW complex, whose 0–cells a e X, o each undi ec ed edge
{x, y}a ach one 1–cell, gluing i s endpoin s a e ices x, y.
Resul ing space |G(1)
comp|is “con igu a ion g aph opological space”, whose undamen al
g oup π1(|G(1)
comp|) can al eady cha ac e ize loops, bu no ye dis inguish which loops
homo opic due o “local ela ions”.
2.2 F om G aph o Two-dimensional Complex
To make ce ain local equi alences (e.g., wo di e en local upda e sequences leading o
same con igu a ion) opologically “ illed” as 2–cells, we in oduce ela ion aces.
Le Rbe amily o ini e-leng h closed pa hs
γ= (x0, x1, . . . , xn=x0),
ep esen ing “locally i ial loops” o “equi alence ans o ma ions”: e.g., di e en
upda e o de s in ini e ime window leading o same ne e ec .
De ini ion 2.2 (Con igu a ion Complex).On 1–skele on |G(1)
comp|, o each ela ion loop
γ∈ R a ach 2–cell, gluing i s bounda y o pa h along γ. Ob ain wo-dimensional CW
complex X=X(Ucomp,R).
In many na u al cases (e.g., compu a ional uni e se gene a ed by e e sible QCA
o e e sible CA), Rcan be chosen as local commu a o s and local common subpa hs
co esponding small closed loops, o ming ini ely gene a ed ela ion se , making π1(X)
ha e good algeb aic ep esen a ion.
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2.3 Fundamen al G oup and Closed E olu ion Loops
On X, undamen al g oup π1(X, x∗) consis s o homo opy classes o closed pa hs s a ing
om basepoin x∗∈X, wi h g oup ope a ion being pa h conca ena ion.
Fo compu a ional uni e se, each closed pa h
γ= (x0, x1, . . . , xn=x0)
ep esen s e olu ion sequence e u ning o s a ing con igu a ion in ini e s eps, whose
homo opy class in π1(X) cha ac e izes “ om mac oscopic pe spec i e whe he his closed
loop can be simpli ied o local ela ions”.
De ini ion 2.3 (Topological Closed Compu a ion).Call closed pa h γ opological closed
compu a ion i i de ines undamen al g oup elemen in con igu a ion complex X
[γ]∈π1(X, x0).
I [γ] = 1 is i ial elemen , call γ opologically con ac ible closed loop; i [γ]= 1
hen non- i ial opological closed loop.
In wha ollows, sel - e e en ial s uc u e and undecidabili y will be cha ac e ized
h ough p ope ies o elemen s in π1(X).
3 Sel -Re e en ial Loops and P og am Sel -In e p e a ion
S uc u e
This sec ion o malizes sel - e e en ial compu a ion as special closed loop class o con ig-
u a ion g aph, p oposing opological cha ac e iza ion o “sel - e e ence deg ee”.
3.1 Abs ac Pic u e o Sel -Re e en ial Compu a ion
In adi ional compu a ional models, “sel - e e ence” ypically mani es s as p og am ak-
ing i s own desc ip ion as inpu , o sys em eeding i s ou pu back o i s inpu . In compu-
a ional uni e se, his s uc u e can be abs ac ed as “e alua ion–encoding– einjec ion”
h ee-s age closed loop in g aph:
1. S a ing om some ini ial s a e xcode, in e nal encoding ope a ion gene a es “code
con igu a ion” xp og desc ibing some compu a ional p ocess;
2. E alua ion p ocess akes con en o xp og as “p og am” compu ing on some inpu
(possibly om i sel ), p oducing new con igu a ion xe al;
3. Reinjec ion p ocess eeds pa o xe al back o encoding s age o o e all sys em,
he eby o ming closed loop.
On con igu a ion g aph, his co esponds o closed pa h, whose edges can be g ouped
in o h ee classes, in ce ain sense ealizing sel -consis en closu e om code o beha io
o sel -upda e.
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3.2 Fo mal De ini ion o Sel -Re e en ial Loop
Suppose con igu a ion se o Ucomp has subse Xcode ⊂Xand “decode–e alua e” ope a o
E al : Xcode ×X→X,
ep esen ing “using code s a e c∈Xcode as p og am, compu ing on inpu s a e x∈X,
ob aining ou pu s a e E al(c, x)”. We do no equi e E al necessa ily e mina e, bu iew
i as e alua ion s ep when co esponding upda e pa h exis s.
On con igu a ion g aph, his can be ealized h ough in e nal subg aph: he e exis
amily o pa hs ealizing encoding, e alua ion, and eedback. Fo b e i y, we abs ac as
ollowing de ini ion.
De ini ion 3.1 (Sel -Re e en ial Loop).On con igu a ion g aph Gcomp, closed pa h wi h
basepoin x0∈X
γ= (x0, x1, . . . , xn=x0)
called sel - e e en ial loop i he e exis index segmen s
0 = k0< k1< k2< k3=n
such ha :
1. xk0=xk3=x0is “global sel - e e ence s a e”;
2. Segmen xk0→xk1belongs o encoding subg aph, gene a ing some code s a e
c∈Xcode;
3. Segmen xk1→xk2 ealizes e alua ion p ocess, ep esen ing compu a ion on some
inpu (possibly om x0o ci sel );
4. Segmen xk2→xk3 eeds back e alua ion esul , such ha xk3=xk0, i.e., “global
s a e emains unchanged o e u ns o ini ial s a e wi hin equi alence class a e
sel - e e en ial upda e”.
Such loop γ opologically ep esen s “sel -in e p e a ion, eedback-closed” compu a-
ional p ocess.
3.3 Sel -Re e ence Deg ee and Z2Pa i y
Sel - e e ence can be oughly dis inguished by “pa i y”: e.g., some sel - e e en ial s uc-
u es lip some global quan i y (such as sign, bi ) in one closed loop, e u ning o o iginal
s a e a e wo loops. This ela ed o Z2- ype holonomy.
De ini ion 3.2 (Sel -Re e ence Deg ee and Pa i y).Fo each sel - e e en ial loop γ, de ine
sel - e e ence deg ee
σ(γ)∈Z2
as pa i y o change o some global cha ac e is ic quan i y, e.g., aking global “sel -
label” bi s∈ {0,1}, equi ing along γupda e s7→ s⊕1 hen σ(γ) = 1, i sunchanged
hen σ(γ) = 0.
6
In con ol mani old and sca e ing–delay ne wo k, his Z2can be ealized by Null–
modula double co e o pa i y ansi ion; in pu e disc e e con igu a ion g aph, we only
need o assume exis ence o obse able Z2label.
Topological class [γ]∈π1(X) o sel - e e en ial loop oge he wi h sel - e e ence deg ee
σ(γ) o m opological–algeb aic in a ian pai o sel - e e en ial s uc u e
([γ], σ(γ)) ∈π1(X)×Z2.
Topological undecidabili y and second law in wha ollows will ca y his as ehicle.
4 Topological Undecidabili y: Loop Con ac ion P ob-
lem and Hal ing P oblem
This sec ion cons uc s educ ion om hal ing p oblem o “whe he loop con ac ible”,
gi ing opological undecidabili y heo em.
4.1 Topological Image o Hal ing P oblem
Classical hal ing p oblem can be s a ed as: gi en p og am Pand inpu w, decide whe he
P(w) hal s in ini e s eps. In compu a ional uni e se, we can cons uc o each (P, w)
con igu a ion subg aph and encoding, such ha :
I P(w) hal s, hen some e olu ion pa h s a ing om encoding s a e en e s “hal -
ing con igu a ion” in ini e s eps and e u ns o canonical ini ial s a e, he eby
p oducing opologically con ac ible loop;
I P(w) does no hal , hen all closed pa hs ela ed o his encoding ep esen
non- i ial elemen in π1(X), o closed loop does no exis a all.
Mo e speci ically, can cons uc “p og am simula ion subg aph”, embedding p og am
execu ion ajec o y in o some sub egion o con igu a ion g aph, o ming closed loop
h ough addi ional “i hal hen e u n o ini ial s a e” connec ion edges; o non-hal ing
ajec o ies, closed loop canno be comple ed o co esponding pa h no in homo opy
i ial class gene a ed by ela ion se R.
4.2 Topological Con ac ion Decision P oblem
Conside ollowing decision p oblem:
Inpu : ini e desc ip ion o compu a ional uni e se Ucomp and closed pa h γ
( ep esen ed as ini e edge sequence) in i s con igu a ion complex X;
Ques ion: is γhomo opic o i ial loop in X?
Call his loop con ac ion p oblem.
In ui i ely, his simila o wo d p oblem in g oup heo y: gi en se o gene a o s
and ela ions, decide whe he wo d ep esen s g oup iden i y. In compu a ional uni e se
con igu a ion complex, gene a o s a e basic edges, ela ions a e local loops in R.
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4.3 Topological Undecidabili y Theo em
Theo em 4.1 (Topological Undecidabili y).The e exis s amily o cons uc ible compu-
a ional uni e ses {Uα
comp}α, such ha on con igu a ion complex Xαo each Uα
comp, loop
con ac ion p oblem undecidable: he e does no exis algo i hm capable o deciding o all
inpu closed pa hs γwhe he homo opic o i ial elemen in π1(Xα).
P oo Idea. 1. S a ing om hal ing p oblem: assume he e exis s some Uα
comp and
algo i hm A, capable o deciding o any closed pa h γwhe he con ac ible in Xα.
2. Cons uc encode Emapping any p og am–inpu pai (P, w) o closed pa h γP,w
in Xα:
I P(w) hal s, hen simula ion ajec o y en e s hal ing s a e in ini e s eps, ap-
pending “end– e u n o ini ial s a e” edge o ms closed loop, h ough ela ion
se Rensu ing γP,w ≃1;
I P(w) does no hal , hen closed loop canno be o med o o med closed loop
necessa ily a e ses edge o some egion ma ked as “non- e mina ing zone”,
he eby gene a ing non- i ial undamen al g oup elemen in Xα.
3. I Aexis s, hen gi en (P, w), compu e γP,w =E(P, w), call A(γP,w):
I A(γP,w) e u ns “con ac ible”, hen decide P(w) hal s;
O he wise decide P(w) does no hal .
This gi es algo i hmic solu ion o hal ing p oblem, con adic ion.
The e o e assump ion does no hold, loop con ac ion p oblem undecidable in his
class o compu a ional uni e ses.
Mo e o mal cons uc ion and p oo de ails in Appendix A.
Co olla y 4.2. In abo e compu a ional uni e se amilies, i ial elemen decision p ob-
lem o undamen al g oup π1(Xα)undecidable; in pa icula , “whe he some sel - e e en ial
loop can be elimina ed by local ela ions” undecidable in gene al case.
This gi es clea opological e sion o hal ing p oblem: “ opological a e” (con ac ible
o non-con ac ible) o sel - e e en ial loop canno be globally algo i hmically p ede e -
mined in gene al compu a ional uni e se.
5 Complexi y En opy and Second Law in Compu-
a ional Uni e se
This sec ion in oduces complexi y en opy unc ional unde uni ied ime scale, gi ing
disc e e e sion o “second law o complexi y”.
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5.1 Complexi y Ac ion and Comp ession Complexi y
Fo closed pa h γ= (x0, . . . , xn=x0) in compu a ional uni e se, we de ine i s complexi y
ac ion as
S(γ) =
n−1
X
k=0
C(xk, xk+1),
unde uni ied ime scale in e p e a ion, his is o al “physical ime” consumed a ound
closed loop.
On o he hand, we de ine comp ession complexi y as
K(γ) = min
γ′≃γℓ(γ′),
whe e ℓ(γ′) is pa h s ep numbe , γ′≃γ ep esen s homo opy equi alence in con ig-
u a ion complex X.K(γ) can be iewed as “sho es pa h leng h ealizing his loop
homo opy class unde opological cons ain s”.
I uni ied ime scale densi y unde s ood as some a e age uni cos , hen can use
combined quan i y
C(γ) = (S(γ), K(γ))
as complexi y en opy candida e o loop, e.g.,
C(γ) = log K(γ)
o
C(γ) = S(γ)/K(γ).
5.2 Coa se–G aining and Complexi y En opy Mono onici y
We conside class o na u al coa se–g aining ope a ions, i.e., allowing “ o ge ing” o
“me ging” o some local de ails in compu a ional uni e se e olu ion:
A con igu a ion le el, me ge some de ail deg ees o eedom in o mac oscopic equi -
alence classes;
A pa h le el, eplace pa hs wi h small loops illed by ela ion se R, o “en opize”
sho local loops by a e aging.
Unde hese ope a ions, homo opy class [γ] o closed pa h may emain unchanged,
bu sho es ealiza ion leng h K(γ) usually canno inc ease (because mo e local ela ions
allowed), while o al ac ion S(γ) canno dec ease o a bi a ily small unde uni ied ime
scale and ene gy cons ain s; na u al mono onici y s uc u e exis s be ween bo h.
P oposi ion 5.1 (Coa se–G aining Mono onici y o Complexi y En opy, P o o ype).
Le {γ } ≥0be amily o closed pa hs, ep esen ing sel - e e en ial loop e olu ion h ough
coa se–g aining unde uni ied ime scale, sa is ying:
1. Homo opy class in a ian : o all , ha e [γ ]=[γ0];
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