Time C ys als and Null–Modula Z2Holonomy
unde Uni ied Time Scale:
Floque –QCA Time C ys als, Topological Pa i y,
and Enginee ing Implemen a ion in Compu a ional
Uni e se
Haobo Ma1Wenlin Zhang2
1Independen Resea che
2Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
On ounda ion o compu a ional uni e se axioma ic amewo k Ucomp = (X, T,C,I)
and uni ied ime scale mas e scale
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω)
his pape cons uc s ully disc e e ime c ys al heo y, uni ying i wi h Null–
Modula Z2holonomy and ime–in o ma ion–complexi y join geome ic s uc u e.
We i s in oduce, on compu a ional uni e se implemen a ion o e e sible
quan um cellula au oma on (QCA), Floque –QCA objec UFQCA = (X, UF,CT,I),
whe e UFis local Floque e olu ion ope a o wi h pe iod T,CTis uni ied ime scale
cos o one Floque s ep. We gi e compu a ional uni e se sense de ini ions o dis-
c e e ime ansla ion symme y and spon aneous b eaking, and om complexi y
geome y and in o ma ion geome y pe spec i es, cha ac e ize ime c ys al phase:
on any ini ial s a e amily sa is ying local obse abili y and bounded ene gy den-
si y assump ions, exis s local obse able Owhose expec a ion alue exhibi s s ic
pe iod mT a he han Tin long- ime e olu ion, whe e m≥2 is in ege .
Subsequen ly, on p e iously cons uc ed causal diamond chain and Null–Modula
double co e s uc u e, we in oduce cyclic chain o Floque –QCA ime c ys als:
each Floque pe iod co esponds o one causal diamond, o ming diamond chain
{♢k}k∈Z. On his chain, we de ine o each pe iod modulo-2 ime phase label
ϵk∈Z2induced by sca e ing phase inc emen , cons uc Null–Modula double
co e e
D→Do diamond chain. We p o e: exis ence o pe iod-doubling ime c ys-
al (m= 2) co esponds p ecisely o non i ial Z2holonomy o Floque con ol
loop on Null–Modula double co e , i.e., closed Floque con ol loop has no closed
li ed pa h on double co e , hus gi ing exac co espondence be ween ime c ys al
pa i y and Null–Modula holonomy.
A enginee ing le el, we conside ime c ys al eadou and obus ness unde i-
ni e complexi y budge . By combining uni ied ime scale equency domain wi h
1
spec al windowing e o con ol heo y (PSWF/DPSS), we cons uc class o “ ini e-
o de window unc ion obse a ion ope a o s” o ime c ys al eadou , p o e: unde
condi ions ha Floque gap exceeds ce ain h eshold and local noise sa is ies ini e
co ela ion leng h assump ion, sampling ime c ys al signal wi h DPSS ype ead-
ou window in ini e s eps can obus ly disc imina e pe iod-doubling pa i y wi h
complexi y budge N=O(∆−2log(1/ε)) while e o p obabili y no exceeding ε,
whe e ∆ is Floque quasiene gy gap.
Finally, we iew ime c ys als as “disc e e phase locke s” o uni ied ime scale:
on con ol mani old (M, G), ime c ys al phase co esponds o class o Floque
con ol loops wi h Z2holonomy, gi ing special minimal wo ldline amily in ime–
in o ma ion–complexi y join a ia ional p inciple. We discuss po en ial expe imen-
al ole o ime c ys als as local s anda ds o uni ied ime scale, and complemen a y
ela ionship wi h FRB phase me ology and δ– ing–AB sca e ing me ology.
Keywo ds: Compu a ional uni e se; Uni ied ime scale; Quan um cellula au oma on;
Floque ime c ys al; Null-Modula double co e ; Z2holonomy; Spec al windowing ead-
ou ; DPSS
1 In oduc ion
Time c ys als ini ially p oposed as phase spon aneously b eaking ime ansla ion symme-
y: sys em’s g ound s a e o s eady s a e exhibi s non i ial pe iodic s uc u e in ime.
Al hough o iginal “con inuous ime c ys al” idea cons ained in s ic equilib ium, in pe i-
odically d i en sys ems (Floque sys ems), Floque ime c ys als spon aneously b eaking
disc e e ime ansla ion symme y ac ually ealized. In hese sys ems, ime ansla ion
g oup Zsymme y spon aneously b oken o mZ, mani es ed as obse able esponse o
comple e Floque pe iod Tha ing supe pe iod mT, commonly m= 2 pe iod-doubling
ime c ys als.
In p e ious wo ks o his se ies, we cons uc ed “uni ied ime scale–compu a ional
uni e se” heo y a highe le el, including:
1. Compu a ional uni e se axioma ic sys em Ucomp = (X, T,C,I), iewing uni e se as
e e sible e olu ion on disc e e complexi y g aph;
2. Uni ied ime scale mas e scale κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω), uni ying
sca e ing phase de i a i e, spec al shi densi y, and g oup delay ace as single
ime scale densi y;
3. Con ol mani old (M, G) induced by uni ied ime scale and complexi y geome y;
4. Causal diamonds, bounda y compu a ion ope a o s, and causal diamond chains
{♢k};
5. Null–Modula double co e and Z2holonomy cons uc ed on diamond chains, sel -
e e ence pa i y and opological complexi y;
6. Time–in o ma ion–complexi y join a ia ional p inciple and mul i-obse e con-
sensus geome y.
2
In his amewo k, ime no longe ex e nal pa ame e , bu embodimen o uni ied ime
scale in sca e ing–complexi y geome y; ime di ec ion, ime pa i y, and sel - e e ence
s uc u e embodied h ough Null–Modula double co e and Z2holonomy.
Co e ques ions o his pape :
1. How o igo ously de ine Floque –QCA ime c ys als in pu ely disc e e amewo k
o compu a ional uni e se–uni ied ime scale, gi ing hei geome ic– opological
cha ac e iza ion?
2. How does pe iod-doubling pa i y o ime c ys als ela e o Null–Modula Z2holon-
omy?
3. How o pe o m s able eadou and enginee ing implemen a ion o ime c ys als
unde ini e complexi y budge ?
We will see ime c ys als na u ally ealized as class o phases in Floque –QCA in
compu a ional uni e se, Null–Modula double co e p o ides in insic Z2 opological in-
a ian , spec al windowing eadou p o ides op imal solu ion o hei obse a ion unde
ini e complexi y budge .
2 P elimina ies: Compu a ional Uni e se, Uni ied
Time Scale, and Floque –QCA
2.1 Compu a ional Uni e se and QCA Implemen a ion
Recall compu a ional uni e se objec
Ucomp = (X, T,C,I),
whe e Xcoun able con igu a ion se , T⊂X×Xone-s ep upda e ela ion, Csingle-
s ep cos , I ask in o ma ion quali y unc ion. S anda d abs ac ion o e e sible QCA:
o la ice si e se Λ and ini e-dimensional Hilbe space Hxa each si e, global Hilbe
space H=Nx∈ΛHx, e e sible QCA is local uni a y ope a o U:H → H sa is ying
local causali y cons ain s.
In compu a ional uni e se, iew con igu a ion x∈Xas label o some no malized
basis ec o |x⟩ ∈ H, one-s ep upda e ela ion de ined by
(x, y)∈T⇐⇒ ⟨y|U|x⟩ = 0
single-s ep cos C(x, y) gi en by physical ime equi ed o execu e Uo i s local de-
composi ion once unde uni ied ime scale.
2.2 Uni ied Time Scale and Floque E olu ion
On physical side, conside pe iodically d i en sys em wi h ime-dependen Hamil onian
H( +T) = H( ), co esponding Floque e olu ion ope a o
UF=Texp −iZT
0
H( ) d ,
3
whose eigen alues e−iεαT,εαa e quasiene gies.
In uni ied ime scale–sca e ing amewo k, iew UF(ω) as equency-domain sca e ing–
e olu ion ope a o , equency ωdependence embodied h ough d i e spec um and sys em
esponse. Fo each Floque pe iod, de ine local g oup delay ma ix QF(ω) = −iUF(ω)†∂ωUF(ω),
whose ace gi es local uni ied ime scale densi y inc emen
κF(ω) = (2π)−1 QF(ω).
In compu a ional uni e se, we conce n “each Floque pe iod as one causal diamond”
disc e e e sion, cons uc ed in Sec ion 3.
2.3 Basic De ini ion o Floque Time C ys als
In gene al Floque sys em, ime ansla ion g oup Zac ion n7→ n+ 1, co esponding o
i e a ion Un
F. Time c ys al is spon aneous b eaking o his symme y:
De ini ion 2.1 (Floque Time C ys al, Physical Side).In pe iodically d i en sys em,
i exis s local obse able Oand ini ial s a e amily {ρ0}such ha o almos all ρ0,
expec a ion sequence
⟨O⟩n= (ρ0U†n
FOUn
F)
exhibi s s ic pe iod m > 1 in long- ime limi , i.e.,
⟨O⟩n+m=⟨O⟩n,
and sa is ies no sho e pe iod, sys em called in Floque ime c ys al phase o pe iod
mT. Typical case m= 2 ime c ys al.
We e o mula e his concep in QCA–compu a ional uni e se amewo k.
3 Floque –QCA Time C ys als in Compu a ional Uni-
e se
3.1 Floque –QCA Objec
De ini ion 3.1 (Floque –QCA Compu a ional Uni e se).A Floque –QCA compu a-
ional uni e se objec is quad uple
UFQCA = (X, UF,CT,I),
whe e:
1. Xcon igu a ion se , as no malized basis ec o labels o global Hilbe space H;
2. UF:H → H local Floque e olu ion ope a o co esponding o d i e pe iod T;
3. CT:X×X→[0,∞] complexi y cos o one Floque s ep, sa is ying CT(x, y)>0
i ⟨y|UF|x⟩ = 0;
4. I:X→R ask in o ma ion quali y unc ion.
4
One Floque e olu ion s ep ep esen ed on e en laye E=X×Zas
(x, n)7→ (y, n + 1),⟨y|UF|x⟩ = 0.
Complexi y cos iewable as in eg al o uni ied ime scale o e single pe iod.
3.2 Disc e e Time T ansla ion Symme y and B eaking
In compu a ional uni e se, iew UFas gene a o o “ ime ansla ion one s ep”. Fo ob-
se able O(e.g., local ope a o Oxac ing only on ini e egion), i s disc e e ime e olu ion
O(n) = U†n
FOUn
F.
Fo ini ial s a e ρ0( iewable as densi y ope a o ), obse a ion sequence
⟨O⟩n= (ρ0O(n)).
De ini ion 3.2 (Floque Time C ys al in Compu a ional Uni e se).In Floque –QCA
compu a ional uni e se, i exis s local obse able O, in ege m≥2, and ini ial s a e
amily R0(sa is ying ini e densi y and ini e co ela ion leng h condi ions) such ha :
1. Fo almos all ρ0∈ R0, exis s su icien ly la ge n0such ha o all n≥n0
⟨O⟩n+m=⟨O⟩n,
2. No 1 ≤m′< m exis s making same condi ion hold,
hen UFQCA called in ime c ys al phase o pe iod mT.
In pa icula , when m= 2, called pe iod-doubling ime c ys al.
3.3 Floque Spec um and Quasiene gy Band S uc u e
Unde ini e olume o app op ia e bounda y condi ions, UFhas eigendecomposi ion
UF|ψα⟩= e−iεαT|ψα⟩,
whe e εα∈(−π/T, π/T] a e quasiene gies.
Time c ys al exis ence closely ela ed o “symme y spli ing s uc u e” in quasiene gy
band s uc u e: e.g., in m= 2 case, exis s wo bands wi h quasiene gies di e ing by π/T,
making cohe en supe posi ion in e olu ion unde go sign lip e e y wo pe iods.
Fo mally, can adop s uc u e p ojec ing o subspaces HA,HBsa is ying
U2
F|ψ⟩ ≈ e−i2εT |ψ⟩,
and UFexchanges HAwi h HB.
Mo e impo an ly, in compu a ional uni e se–complexi y geome y, we can ansla e
phase s uc u e o Floque spec um o Null–Modula Z2holonomy on causal diamond
chains, de eloped in nex sec ion.
5
4 Null–Modula Z2Holonomy and Time C ys al Pa -
i y
This sec ion cons uc s Floque –QCA ime c ys al implemen a ion on causal diamond
chains and Null–Modula double co e , p o es co espondence be ween pe iod pa i y and
Z2holonomy.
4.1 Floque Pe iod as Causal Diamond Chain
View single-pe iod Floque e olu ion as one causal diamond ♢F:
Diamond in e io e ices a e e en se om some ini ial s a e laye o nex laye
wi hin complexi y budge T;
Diamond bounda y a e pe iod ini ial/ inal e en s;
Diamond olume e olu ion gi en by local decomposi ion o UF;
Bounda y ope a o K♢Fisomo phic o UFac ion on bounda y.
I sys em epea edly d i en in ime, o ms Floque diamond chain on e en laye
{♢F,k}k∈Z,
whe e each ♢F,k co esponds o k- h Floque pe iod.
Fo each ♢F,k, de ine a e age uni ied ime scale inc emen
∆τk=ZΩF
wF(ω)κF(ω) dω,
in pe iodically s able case, ∆τk≡∆τp opo ional o physical pe iod T.
4.2 Modulo-2 Time Phase and Z2Holonomy
In hi d wo k on diamond chains and Null–Modula double co e , we de ined modulo-2
ime phase label ϵk∈Z2 o each diamond, de e mined by sca e ing phase inc emen
modulo 2π.
In Floque case, de ine e ec i e phase inc emen pe pe iod as
∆φF= a g de UF, ϵF=⌊∆φF/π⌋mod 2.
Fo ime c ys als, especially pe iod-doubling phase, key s uc u e no single-pe iod
phase bu wo-pe iod closed loop
U2
F,
and i s co esponding sca e ing phase and g oup delay.
When cons uc ing diamond chain double co e e
DF→DF, le edge label o each
Floque pe iod diamond be ϵF. To al pa i y o Npe iods on closed chain
ΣN=
N
X
k=1
ϵFmod 2 = NϵFmod 2.
6
Fo m= 2 ime c ys al, na u al mechanism makes closed loop o wo pe iods ha e
non i ial Z2holonomy: e.g., i ϵF= 1, a e each pe iod, index on double co e lips
once, a e wo pe iods lips wice e u ning o o iginal index, bu o e all opology o
closed pa h exhibi s non i ial holonomy.
Mo e p ecisely, conside closed loop o Floque con ol pa ame e pa h ΓF⊂ M
(e.g., closed a ia ion o d i e p o ocol pa ame e s in pe iodic d i ing), i s Null–Modula
double co e holonomy
holZ2(ΓF)∈Z2
closely ela ed o ime c ys al pe iod pa i y.
4.3 Time C ys al Pa i y and Null–Modula Holonomy Co e-
spondence
Theo em 4.1 (Pe iod-Doubling Time C ys al and Z2Holonomy).Le UFQCA be Floque –
QCA compu a ional uni e se objec sa is ying:
1. Exis s uni o m olume limi and ini e co ela ion leng h ini ial s a e amily R0;
2. Floque spec um has quasiene gy gap ∆F>0, exis s wo bands εα, εβsa is ying
εβ≈εα+π/T;
3. On co esponding con ol mani old closed loop ΓF, Null–Modula double co e holon-
omy non i ial, i.e.,
holZ2(ΓF) = 1.
Then UFQCA in ime c ys al phase o pe iod 2T; con e sely, unde abo e egula i y
condi ions, i UFQCA in obus pe iod-2T ime c ys al phase, co esponding Floque con ol
closed loop’s Null–Modula holonomy is non i ial elemen .
P oo ske ch. “I ” di ec ion: Non i ial holonomy means unde wo-pe iod closed loop
some global Z2quan i y lips odd imes, in Floque spec um co esponds o “pa i y
swi ching” s uc u e making Floque subspaces exchange in one pe iod, e u n o o iginal
posi ion in wo pe iods, causing expec a ion alue o exhibi pe iod-2T lip s uc u e.
Using g oup heo y and quasiene gy band s uc u e can p o e exis s local obse able O
sa is ying ime c ys al condi ion.
“Only i ” di ec ion: Pe iod-doubling o ime c ys al means on Floque –QCA wo ld-
line exis s sel - e e ence eedback condi ion making wo pe iods globally close. Th ough
p e ious co espondence be ween sel - e e ence pa i y and Null–Modula holonomy, can
p o e co esponding closed loop holonomy non i ial.
De ailed p oo in Appendix C.
5 Time C ys al Readou and Enginee ing Implemen-
a ion Unde Fini e Complexi y Budge
This sec ion discusses s able eadou o ime c ys als unde ini e complexi y budge ,
gi es DPSS-based obse a ion s a egy and e o uppe bound.
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5.1 Readou Model and Noise
Conside local obse able Oon local egion Λ0⊂Λ, de ine disc e e ime sequence
an= (ρ0U†n
FOUn
F), n = 0,1, . . . , N −1.
In ideal ime c ys al phase, anexhibi s pe iod-ms uc u e when n≫1, ypically
m= 2 al e na ing sequence. Wi h local noise and dissipa ion, w i able as
an=sn+ηn,
whe e snideal ime c ys al signal, ηnnoise, assuming ηnze o-mean, ini e co ela ion
leng h Gaussian p ocess.
5.2 DPSS Window Func ion Readou
To ex ac pe iod s uc u e wi hin ini e complexi y s eps N, cons uc windowed Fou ie
spec um
ba(ω) =
N−1
X
n=0
wnane−iωn,
whe e {wn}window unc ion sequence. Acco ding o p e ious spec al windowing
eadou esul s, DPSS maximizes ene gy concen a ion unde gi en leng h Nand e-
quency band W, minimizing wo s -case e o unde ini e sample numbe and equency
band cons ain s.
Fo m= 2 ime c ys al, ideal signal main equency a ω=π(no malized angula
equency). Can choose DPSS window unc ion wi h bandwid h W≪π, ocusing on
spec al ene gy nea ω≈π.
5.3 E o Uppe Bound and Complexi y Budge
Le DPSS window unc ion be w(0), co esponding eigen alue λ0≈1, hen unde ini e
samples, e o a iance o main equency ene gy es ima ion sa is ies
Va ba(π)≤σ2
η|w(0)|2,
whe e σ2
ηnoise a iance.
To dis inguish “wi h ime c ys al signal” and “wi hou ime c ys al signal”, main ain
ce ain signal- o-noise a io
|E[ba(π)]|2
Va (ba(π)) ≥c0,
ob aining sample numbe equi emen
N=O∆−2log(1/ε),
whe e ∆ Floque quasiene gy gap (con olling ime c ys al signal ampli ude and dis-
sipa ion ime), εe o p obabili y.
Theo em 5.1 (Sample Complexi y o Fini e Complexi y Time C ys al Disc imina ion).
Unde condi ions:
8
1. Floque –QCA ime c ys al has quasiene gy gap ∆F>0;
2. Noise p ocess {ηn}ze o-mean, ini e co ela ion leng h, bounded a iance;
3. Readou window unc ion DPSS basis sequence w(0) unde app op ia e bandwid h
W;
o disc imina e whe he pe iod-2T ime c ys al signal exis s wi h e o p obabili y no
exceeding ε, equi ed complexi y s eps Nsa is ies
N≥C∆−2
Flog(1/ε),
whe e Ccons an .
P oo ske ch. Combines DPSS ene gy concen a ion, Chebyshe inequali y, and la ge de-
ia ion es ima ion, see Appendix D o de ails.
6 Uni ied Pe spec i e: Time C ys als as Disc e e
Phase Locking o Uni ied Time Scale
F om uni ied ime scale–con ol mani old–causal diamond chain–Null–Modula double
co e global pe spec i e, ime c ys als unde s andable as special “disc e e phase locke s”:
1. Floque con ol closed loop ΓFon con ol mani old (M, G) gene a es pe iodic ime
inc emen ∆τ h ough uni ied ime scale densi y κ(ω), has Z2holonomy on Null–
Modula double co e ;
2. Modulo-2 ime phase labels ϵkon causal diamond chain {♢F,k}synch onize wi h
Floque con ol holonomy, o ming “ ime pa i y locking”;
3. Time c ys al phase exis ence means in ime–in o ma ion–complexi y join a ia-
ional p inciple exis s special minimal wo ldline amily, simul aneously s able in
“ ime di ec ion–phase–sel - e e ence pa i y” h ee dimensions.
A expe imen al le el, ime c ys als iewable as local s anda ds o uni ied ime scale:
compa ed o FRB and δ– ing–AB sca e ing “passi e measu emen s”, ime c ys als p o-
ide “ac i ely gene a ed ime scale phase s uc u e”. By join ly embedding ime c ys als,
FRB, and δ– ing sca e ing in phase– equency me ology uni e se, can pe o m consis-
ency es ing and join calib a ion o uni ied ime scale model ac oss scales (labo a o y–
in e s ella –cosmological) pla o ms.
A P o o ypical Exis ence Theo em o Floque –QCA
Time C ys als
This appendix gi es ypical cons uc ion scheme and p o o ypical exis ence esul o
ime c ys al phases in QCA models.
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