Causal Diamond Chains and Null–Modula Double
Co e
in Compu a ional Uni e se:
Disc e e Causal S uc u e, Topological Time Phase,
and Sel -Re e en ial Pa i 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
P e iously on compu a ional uni e se axioma ic amewo k Ucomp = (X, T,C,I)
we ha e successi ely 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), ime–in o ma ion–complexi y join a ia ional p inciple, mul i-
obse e consensus geome y, causal diamonds and bounda y compu a ion, as well
as opological complexi y and undecidabili y. On o he hand, on physical uni e se
side, small causal diamonds and Null–Modula double co e s uc u e play key
ole in uni ied ime scale–bounda y ime geome y: phase–delay–en opy on Null
bounda y has na u al Z2pa i y and double co e s uc u e, used o cha ac e ize
ime di ec ion, ene gy condi ions, and sel - e e en ial eedback ne wo ks.
This pape cons uc s a compu a ional uni e se le el comple ely disc e e “causal
diamond chains and Null–Modula double co e ” heo y, and p o es i s isomo -
phism wi h causal diamond chains and Null–Modula s uc u e on physical side in
limi unde uni ied ime scale and complexi y geome y. Speci ically, we do ollow-
ing:
1. De ine on e en laye E=X×Ncomplexi y causal pa ial o de and ini e-
budge causal diamonds ♢(ein, eou ;T), and o malize “diamond chains” as
o de ed amilies {♢k}k∈Zsa is ying app op ia e o e lap condi ions in pa ial
o de . We p o e his amily unde na u al condi ions cons i u es di ec ed
g aph–chain complex wi h bounda y ope a o s, whose 1-skele on cha ac e izes
“disc e e imeline” unde uni ied ime scale.
2. In oduce Null–Modula double co e on diamond chains: assign o each di-
amond bounda y s a e Z2- alued “mod 2 ime phase” label, and cons uc
o diamond chain double co e g aph e
D→D, such ha exis ence o non-
exis ence o li ed pa h o closed diamond chain co esponds o pa i y o Z2
holonomy. We p o e his double co e consis en wi h p e iously in oduced
sel - e e en ial pa i y in a ian σ(γ)∈Z2in opological complexi y.
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3. In oduce uni ied ime scale in o diamond chains: de ine on each diamond
chain edge disc e e ime inc emen ∆τk, gi en by uni ied ime scale densi y
κ(ω) and local sca e ing phase de i a i e. We p o e in e inemen limi , ime
in e al and Z2holonomy o diamond chain join ly de ine class o “ ime di ec-
ion ield wi h Null–Modula s uc u e” on con ol mani old, hus connec ing
disc e e Null–Modula double co e wi h con inuous uni ied ime scale.
4. In mul i-obse e causal ne wo k, embed each obse e ’s “diamond wo ld
ube” in o diamond chain double co e , cons uc ing mul i-obse e Null–
Modula consensus s uc u e, p o ing Z2pa i y ansi ion in sel - e e en ial
sca e ing ne wo k and mul i-obse e consensus geome y can be iewed as
holonomy in a ian on diamond chain double co e .
5. In con ex o opological complexi y and undecidabili y, we p o e: on gene al
cons uc ible compu a ional uni e se amilies, deciding “whe he gi en dia-
mond chain closed loop can li o closed pa h in Null–Modula double co e ”
is undecidable, he eby gi ing “Null–Modula e sion o hal ing p oblem”. Si-
mul aneously, we cons uc “ ime phase–complexi y second law” compa ible
wi h complexi y en opy: unde join cons ain s o uni ied ime scale and
Null–Modula double co e , join in a ian composed o sel - e e en ial pa i y
and comp essible complexi y has mono onic s uc u e along diamond chain
coa se–g aining e olu ion.
Th ough abo e cons uc ion, his pape uni ies disc e e causal diamond chains,
uni ied ime scale, opological sel - e e en ial pa i y, and complexi y second law in
compu a ional uni e se in o Null–Modula double co e amewo k, and gi es in
appendices de ailed p oo s o main s uc u es and igo ous cons uc ion o chain
complex.
Keywo ds: Compu a ional uni e se; Causal diamond; Null–Modula double co e ; Z2
holonomy; Sel - e e en ial pa i y; Uni ied ime scale; Second law o complexi y
1 In oduc ion
In uni ied ime scale–bounda y ime geome y cons uc ion o physical uni e se, small
causal diamonds, Null bounda ies, and Null–Modula double co e s uc u e a e un-
damen al building blocks connec ing sca e ing phase, g oup delay, gene alized en opy,
and ime di ec ion. Pa icula ly, when examining se ies o nes ed o in e sec ing small
causal diamonds, phase–delay da a on bounda ies has na u al Z2pa i y s uc u e: some
closed diamond chains li o closed pa hs on Null–Modula double co e , o he s p o-
duce “odd-pa i y jump”, lea ing opological holonomy. This holonomy closely ela ed o
sel - e e en ial eedback ne wo ks, sel -iden i y, and complexi y second law.
On o he hand, in his se ies on “compu a ional uni e se” wo ks, we ha e al eady
cons uc ed in comple ely disc e e abs ac axioma ic amewo k:
Complexi y g aph Gcomp = (X, E, C) and complexi y dis ance dcomp;
Task in o ma ion mani old (SQ, gQ,ΦQ) and disc e e in o ma ion geome y;
Con ol mani old (M, G) induced by uni ied ime scale and geodesic s uc u e;
Time–in o ma ion–complexi y join a ia ional p inciple;
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Causal diamonds and bounda y compu a ion ope a o K♢;
Mul i-obse e consensus geome y and causal ne wo k;
Topological complexi y, sel - e e en ial loops, and Z2sel - e e en ial pa i y σ(γ).
Na u al ques ion is: can we comple ely simula e on disc e e causal s uc u e o com-
pu a ional uni e se s uc u e o “small causal diamond chains + Null–Modula double
co e ” om physical uni e se? I yes, is i s opological Z2holonomy consis en wi h p e i-
ous sel - e e en ial pa i y in a ian ? Does uni ied ime scale na u ally s a i y on diamond
chain in o combina ion o “ ime s ep + pa i y ansi ion”? How does sel - e e en ial eed-
back o mul i-obse e s in causal ne wo k mani es in his s uc u e?
Answe o his pape is a i ma i e, we will show:
1. On e en laye E=X×No compu a ional uni e se, can de ine comple ely disc e e
“causal diamond chains”, whose chain complex and bounda y ope a o s na u ally
co espond o disc e e ime s eps o uni ied ime scale;
2. On diamond chains can cons uc Null–Modula double co e , whose Z2holonomy
isomo phic o p e ious sel - e e en ial pa i y σ(γ);
3. Mul i-obse e wo ld ubes can be iewed as amily o “li ed pa hs” in diamond
chain double co e , mul i-obse e consensus geome y and sel - e e en ial eedback
ne wo k become geome ic– opological s uc u e on his double co e ;
4. In his amewo k, can de ine “Null–Modula e sion o hal ing p oblem”, p o e
i s undecidabili y, and simul aneously gi e “ ime phase–complexi y second law”
compa ible wi h complexi y en opy.
Pape s uc u e: Sec ion 2 cons uc s e en laye causal diamond chains and chain
complex; Sec ion 3 de ines Null–Modula double co e and Z2holonomy on diamond
chains; Sec ion 4 connec s uni ied ime scale wi h diamond chains and double co e ; Sec-
ion 5 in oduces mul i-obse e Null–Modula consensus geome y; Sec ion 6 discusses
Null–Modula e sion o hal ing p oblem and complexi y second law; Appendices gi e
de ailed p oo s o chain complex cons uc ion, double co e exis ence, holonomy–sel -
e e en ial pa i y co espondence, undecidabili y, and second law p o o ype.
2 E en Laye Causal Diamonds and Diamond Chains
This sec ion cons uc s causal diamonds and diamond chains on compu a ional uni e se
e en laye , gi ing disc e e chain complex s uc u e.
2.1 E en Laye and Causal Pa ial O de
Conside compu a ional uni e se Ucomp = (X, T,C,I), e en laye de ined as
E=X×N, e = (x, k).
One-s ep upda e ela ion
TE={((x, k),(y, k + 1)) : (x, y)∈T}.
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De ine causal eachabili y ela ion: o e, e′∈E
e⪯e′
i he e exis s ini e pa h Γ : e=e0→e1→ · · · → en=e′, whe e (ei, ei+1)∈TE.
This is pa ial o de (in eachable subse ) on e en laye .
Me ging s eps and cos : de ine e en laye complexi y cos
CE((x, k),(y, k + 1)) = C(x, y),
pa h cos
CE(Γ) =
n−1
X
i=0
CE(ei, ei+1),
e en laye complexi y dis ance
dE(e, e′) = in
Γ:e→e′
CE(Γ).
2.2 Causal Diamonds and Bounda ies
Gi en wo e en s
ein = (xin, kin), eou = (xou , kou ), kou > kin,
and complexi y budge T > 0. De ine causal diamond unde budge T
♢(ein, eou ;T) = J+
T(ein)∩J−
T(eou ),
whe e
J+
T(e) = {e′:e⪯e′, dE(e, e′)≤T},
J−
T(e) = {e′:e′⪯e, dE(e′, e)≤T}.
Deno e V♢=♢as e ex se , edge se as
E♢={(e, e′)∈TE:e, e′∈V♢}.
Bounda y de ined as
∂♢={e∈V♢:∃e′/∈V♢,(e, e′)∈TEo (e′, e)∈TE}.
Fu he decompose
∂−♢={e∈∂♢:∃e′/∈V♢,(e, e′)∈TE},
∂+♢={e∈∂♢:∃e′/∈V♢,(e′, e)∈TE}.
∂−♢is incoming bounda y, ∂+♢is ou going bounda y.
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2.3 Diamond Chains and Chain Complex S uc u e
De ini ion 2.1 (Causal Diamond Chain).A causal diamond chain is sequence {♢k}k∈Z,
whe e each ♢kis causal diamond de ined unde some pai (ek, ek+1) and budge Tk,
sa is ying:
1. Time-o de ing consis ency: he e exis s e en sequence {ek}k∈Zsuch ha ek∈
∂+♢k∩∂−♢k+1;
2. Complexi y o e lap: ♢k∩♢k+1 =∅, and ♢k∩♢k+l=∅ o |l| ≥ 2;
3. Budge consis ency: Tksa is y app op ia e uppe bound, such ha each diamond
only co e s ini e ime–complexi y window.
Viewing all ♢kas “basic elemen s” on 1-chain, de ining 2-cells a hei o e laps, can
cons uc one-dimensional chain complex wi h 2-cells D=D({♢k}), whose 1-skele on is
diamond chain, 2-cells cha ac e ize local ela ions o diamond gluing.
P oposi ion 2.2 (1-Skele on o Diamond Chain and Disc e e Timeline).I {♢k}sa is ies
abo e condi ions, and unde uni ied ime scale each ♢kco esponding ime in e al ∆τk
has uni ied lowe bound ∆τmin >0, hen na u al o de ing k7→ ♢kon 1-skele on can be
iewed as disc e e imeline, each single s ep co esponding o ini e-budge causal diamond.
P oo see Appendix A.1: co e is using ekpa ial o de and budge o e lap o cons uc
local subs i u ion o “same ime laye ”.
3 Null–Modula Double Co e and Z2Holonomy
This sec ion cons uc s Null–Modula double co e s uc u e on diamond chains, de ines
Z2holonomy, and connec s wi h sel - e e en ial pa i y in a ian .
3.1 Mod 2 Time Phase on Diamond Bounda ies
Unde uni ied ime scale and sca e ing amewo k, each causal diamond ♢kcan be asso-
cia ed wi h local sca e ing ope a o S♢k(ω) and g oup delay ma ix Q♢k(ω), whose ace
gi es uni ied ime scale densi y inc emen on his diamond.
Conside equency in e al Ωkand weigh wk(ω), de ine diamond a e age phase in-
c emen
∆φk=ZΩk
wk(ω)φ′
♢k(ω) dω,
and co esponding ime inc emen
∆τk=ZΩk
wk(ω)κ♢k(ω) dω.
We in oduce s uc u e o “phase mod 2π” in o Z2label.
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De ini ion 3.1 (Mod 2 Time Phase Label).Fo each diamond ♢kon each e en e∈∂+♢k
a i s ou going bounda y, assign label
ϵ(e)∈Z2,
de ined as pa i y class o ∆φk/π:
ϵ(e) = ∆φk
πmod 2.
On diamond chain, we equi e ou going bounda y label compa ible wi h nex dia-
mond’s incoming bounda y label, i.e., ϵ(ek) only depends on local s uc u e o diamond
♢k.
3.2 Cons uc ion o Null–Modula Double Co e
De ini ion 3.2 (Null–Modula Double Co e ).Gi en diamond chain complex D, con-
s uc double co e g aph e
Das ollows:
1. Fo each diamond e ex k( ep esen ing ♢k) in oduce wo copies e (0)
k,e (1)
k;
2. Fo each chain edge ( k, k+1), i co esponding phase pa i y ϵ(ek) = 0, hen in
double co e connec e (i)
kwi h e (i)
k+1; i ϵ(ek) = 1, hen connec e (i)
kwi h e (1−i)
k+1 ,
whe e i∈ {0,1};
3. P ojec ion π:e
D→Dmaps e (i)
k7→ k.
Thus, walking a ound diamond chain once, whe he endpoin and s a ing poin o
li ed pa h on double co e a e iden ical depends on accumula ion o phase pa i y along
way.
P oposi ion 3.3 (Double Co e and Z2Holonomy).Le γ= ( k0, k1, . . . , km= k0)be
closed loop on diamond chain, le eγbe i s li ed pa h on double co e . Then:
1. I Pm−1
j=0 ϵ(ekj)≡0 mod 2, hen he e exis s closed li ed pa h eγsuch ha e (i)
km=e (i)
k0;
2. I Pm−1
j=0 ϵ(ekj)≡1 mod 2, hen any li ed pa h s a ing om e (i)
k0ends a e (1−i)
k0, no
closed li exis s.
The e o e, Z2holonomy o closed loop gi en by phase pa i y sum Pϵ(ek), comple ely
consis en wi h double co e s uc u e.
P oo see Appendix A.2.
3.3 Co espondence wi h Sel -Re e en ial Pa i y In a ian
In p e ious opological complexi y wo k, we de ined o sel - e e en ial loop γsel - e e en ial
pa i y σ(γ)∈Z2. He e, we can e-exp ess sel - e e en ial loop on diamond chain as some
closed diamond chain γ♢, whose sel - e e en ial ope a ion co esponds o some local eed-
back s uc u e on diamond chain.
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Theo em 3.4 (Consis ency o Sel -Re e en ial Pa i y and Null–Modula Holonomy).
Unde app op ia e encoding, each sel - e e en ial loop γco esponds o closed diamond
chain loop γ♢, such ha
σ(γ) = X
k∈γ♢
ϵ(ek) mod 2,
i.e., sel - e e en ial pa i y equals Z2holonomy on diamond chain double co e .
P oo see Appendix A.3: cons uc local sca e ing phase– ime delay associa ed wi h
sel - e e en ial eedback ne wo k, and ansla e in o pa i y ansi ion on diamond chain
using Null–Modula double co e ules.
4 Implemen a ion o Uni ied Time Scale on Diamond
Chains
This sec ion in oduces uni ied ime scale densi y κ(ω) in o diamond chain s uc u e, and
eco e s con inuous ime pa ame e on con ol mani old in e inemen limi .
4.1 Time Inc emen on Diamonds and Local Sca e ing
Fo each diamond ♢k, assume i s co esponding local sca e ing p ocess S♢k(ω) sa is ies
uni ied ime scale mas e o mula:
κ♢k(ω) = 1
2π Q♢k(ω),
Q♢k(ω) = −iS♢k(ω)†∂ωS♢k(ω).
De ine diamond ime inc emen
∆τk=ZΩk
wk(ω)κ♢k(ω) dω.
On diamond chain, cumula i e ime is
τN=
N−1
X
k=0
∆τk.
4.2 Limi om Diamond Chain o Con ol Mani old Wo ldline
Conside disc e e con ol pa ame e θk∈ M and diamond chain index k. Suppose he e
exis s embedding map
θ(k) = Θ(τk),
whe e Θ : [τ0, τN]→ M is con inuous cu e on con ol mani old. I diamond size
∆τk→0 and chain becomes dense, hen 1-skele on o diamond chain con e ges in
G omo –Hausdo sense o image o Θ.
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P oposi ion 4.1 (Time–Geome ic Limi o Diamond Chains).Unde uni ied ime scale
and local sca e ing egula i y assump ions: i diamond chain {♢k}sa is ies supk∆τk→0,
and each diamond on con ol mani old co esponds o one geodesic s ep in local coo dina e
neighbo hood, hen complexi y dis ance o diamond chain con e ges in limi o con ol
mani old geodesic dis ance dG, ime pa ame e τis uni ied ime scale pa ame e .
P oo see Appendix B.1: using s anda d disc e e–con inuous geodesic app oxima ion
and sca e ing–g oup delay ela ion o uni ied ime scale.
4.3 Time Di ec ion Field Unde Null–Modula Double Co e
Combining Null–Modula double co e s uc u e, ime pa ame e τ oge he wi h sel -
e e en ial pa i y σde ine “ ime di ec ion ield” on con ol mani old:
Ad ancing along Θ(τ), co esponds o diamond chain {♢k};
On double co e e
D, whe he pa h li e u ns o lips a e going a ound once de ines
Z2 ime pa i y;
This pa i y s uc u e can be iewed as Z2p incipal bundle holonomy on con ol
mani old, consis en wi h Null–Modula double co e s uc u e.
In physical uni e se uni ied ime scale–bounda y ime geome y language, his co e-
sponds o comp ehensi e s uc u e o mod 2πphase and Z2module on Null bounda y.
This pape gi es i s disc e e–chain complex ealiza ion on compu a ional uni e se side.
5 Mul i-Obse e Null–Modula Consensus Geome-
y
This sec ion embeds mul i-obse e consensus geome y in o diamond chain double co e ,
cons uc ing mul i-obse e Null–Modula consensus s uc u e.
5.1 Mul i-Obse e Wo ld Tubes and Diamond Chain Embed-
ding
Fo obse e amily {Oi}i∈I, each obse e Oihas “wo ld ube” {(i, x(i)
k, k)}kon e en
laye , co esponding o se o diamond chains embedded in e en laye {♢(i)
k}k, e.g., each
diamond co e s om (i, x(i)
k, k) o (i, x(i)
k+1, k + 1).
These diamond chains can be iewed as sub amily o diamond chain complex D, each
obse e has li ed pa h eγion double co e e
D.
De ini ion 5.1 (Mul i-Obse e Null–Modula Consensus G aph).De ine on diamond
chain double co e g aph
CNM = (VNM, ENM),
whe e e ex se VNM is all obse e li ed pa h diamond chain nodes {e (i)
k}, edge se is
combina ion o “space– ime–in o ma ion adjacency”: including bo h ime-di ec ion chain
edges and consensus edges on in o ma ion mani old.
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On his g aph, can de ine quan i y simila o p e ious mul i-obse e consensus en-
e gy, excep he e each pa h also ca ies Z2holonomy, ep esen ing accumula ion o sel -
e e en ial pa i y on obse a ion chain.
5.2 Null–Modula Consensus and Sel -Re e en ial Pa i y Align-
men
In mul i-obse e scena io, di e en obse e s may ha e di e en sel - e e en ial s uc-
u es: some obse e s’ in e nal models sel - lip a e one ime pe iod, o he s do no lip.
On Null–Modula double co e , his co esponds o whe he hei li ed pa hs close o
lip.
P oposi ion 5.2 (Null–Modula Consensus Alignmen Condi ion).I he e exis s mul i-
obse e collabo a i e s a egy such ha in long- e m limi → ∞
1. All obse e s’ in o ma ion s a es ϕi( )con e ge on SQ o same poin o same o bi ;
2. All obse e s’ li ed pa h Z2holonomies a e same, i.e., σ(γi) = σ(γj) o all i, j;
hen on diamond chain double co e , mul i-obse e wo ld ubes cons i u e “Null–
Modula consensus clus e ”, whose o e all holonomy is common Z2 alue, ep esen ing
uni ied sel - e e en ial pa i y unde his ask.
This condi ion gi es opological–geome ic le el “deep consensus” concep : no only
in o ma ion s a es each consensus, sel - e e en ial s uc u es also each ag eemen .
6 Null–Modula Ve sion o Hal ing P oblem and Time
Phase–Complexi y Second Law
This sec ion de ines Null–Modula e sion o hal ing p oblem based on p e ious s uc-
u es, and gi es second law p o o ype compa ible wi h complexi y en opy.
6.1 Null–Modula Ve sion o Hal ing P oblem
Conside diamond chain closed loop γ, whose sel - e e en ial pa i y σ(γ)∈Z2and un-
damen al g oup homo opy class [γ]∈π1(D) al eady de ined. Ques ion we ca e abou
is:
Gi en γ, does i ha e closed li ed pa h on Null–Modula double co e e
D?
This equi alen o σ(γ) = 0, i.e., holonomy is i ial elemen .
P oblem 6.1 (Null–Modula Hal ing Decision P oblem). Inpu : Fini e desc ip ion o
diamond chain complex Dand closed loop γin i .
Ques ion: Decide whe he γhas closed li ed pa h on Null–Modula double co e e
D.
Using opological undecidabili y esul s om Sec ion 4, we can p o e his p oblem un-
decidable in gene al compu a ional uni e se amilies: can encode hal ing p oblem as clo-
su e o ce ain class o sel - e e en ial diamond chains and hei holonomy pa i y, he eby
educing hal ing o Null–Modula hal ing decision.
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