P ime Theo ies, Umb ella Gene a o s, and a Ze a “Theo y o
E e y hing”
A Toy Model o Cascading Ma hema ical S uc u e
Aleksanda Pe išić
Sep embe 2025
Abs ac
We i s de elop a minimal, addi ion–only pic u e in which me e s a e successo clocks
kN
,
o de ed by e inemen and closed unde in e sec ion and gene a ed union. In his successo
la ice, p imes appea as he indecomposable (“pe ec ”) me e s; composi es a e cohe en
o e laps o p ime me e s. F om in e sec ions and a single ie–b eake ( i s coincidence) we
eco e
gcd
/
lcm
; wi h a wo–way ebasing and he same ie–b eake we eco e o dina y
mul iplica ion. We hen exhibi a whole amily o compa ible p oduc s pa ame e ized on he
p ime exponen s, wi h Eule –p oduc a a a s, and commen on he s uc u al ie be ween
a pa icula mul iplica i e choice and he c i ical line o
ζ
, in he sense o a condi ional
chain h ough scale–neu al blu and midline uni a i y. Companion no es elabo a ing he
mul iplica i e choice and he p ime– heo y umb ella a e e e enced inline.
We hen build a didac ic oy model o heo ies s a ing om he i ial 1- heo y, hen
he single-p ime heo ies
p
- heo ies whose only heo ems asse ha “
pk
is a powe o
p
,”
and hen hei combina ions. We in oduce a compac gene a o o malism ha oggles
p ime heo ies on/o and includes/excludes selec ed p ime powe s, bo h as a o mal monoid
polynomial and as a Di ichle se ies. Tu ning e e y swi ch “on” yields he Eule p oduc — he
Riemann ze a unc ion—which plays he ole o a closu e o “ heo y o e e y hing.” Finally,
a e allowing
s∈C
, analy ic con inua ion leads o he amilia landscape whe e ze a’s
non i ial ze os ac as me a- heo ies ha couple all p ime heo ies a once. The goal is
illus a ion a he han p oo : o show how simple a oms (p ime-powe s a emen s) o e lap
and in e wine in o ich global s uc u e.
1 Seed: 0, successo , and closu e
Peano’s seed is: a null symbol 0, a successo ope a ion
succ
, and he closu e ha p oduces
N
.
He e we keep only he successo iewpoin bu e use o p i ilege a single, absolu e “me e ” a
p io i.
Two minimal ques ions abou successo
Q1.
Repea abili y. I we can each 2, can we (pe haps a e an ex ac ion) each a cohe en “3”?
We assume we can—bu we do no assume a unique mechanism beyond cohe ence o he
nex objec .
Q2.
Uni o m me e . Does
succ
ac wi h iden ical me e independen o base poin ? We do no
assume his globally. Ins ead we ke nelize: we pai objec s wi h a ixed ma ke
suc
and
ega d nume als as nes ed pai ings (
n, suc
); any local a ia ion is elega ed o he ancilla y
slo .
The ou come is an “equally spaced” açade o
N
su icien o index posi ions, wi hou
commi ing o a unique conc e e me e unde nea h.
1
2 Me e s as successo clocks and hei la ice
Fo k≥1, de ine he k–successo (clock) as he submonoid
kN={k, 2k, 3k, . . . }⊂N.
O de clocks by e inemen :
kN⪯mN⇐⇒ kN⊇mN( ine clock si s highe ).
The amily {kN:k≥1}is closed unde :
mee : kN∧mN:= kN∩mN,join: kN∨mN:= ⟨kN∪mN⟩,
hence o ms a dis ibu i e la ice pu ely inside addi ion.
De ini ion 2.1 (P ime/Pe ec me e s).A me e
R
is indecomposable i
R
=
X∧Y
implies
X=Ro Y=R.
P oposi ion 2.2 (P imes a e he pe ec me e s).The indecomposable clocks a e p ecisely
{pN:pp ime}. Mo eo e e e y kNhas a unique e inemen
kN=^
p
p p(k)N,
a mee o p ime–powe clocks ( ini e p oduc since p(k)=0 o all bu ini ely many p).
Rema k 2.3 (Co–sie e: p imes as a ke nel).Conside he amily
F
=
{kN
:
k≥
2
}
on he e ex
se
N≥2
. Keep only he e ices o incidence deg ee 1(poin s ha belong o exac ly one clock in
F
). The su i o s a e p ecisely he p imes: a composi e si s in a leas wo clocks (a p ope
di iso ’s and i s own), a p ime
p
only in
pN
. This one–sho , pa allel cu is he “co–E a os henes”
ke nel.
3 How addi ion bi hs mul iplica i e s uc u e
Th ee ope a ions om clocks
All a ise om in e sec ions plus a single ie–b eake ( i s coincidence). Exp essed on p ime
exponen s p(·):
Ope a ion on clocks On in ege s On exponen s
las –always–hi (mee ) gcd(a, b) min( p(a), p(b))
i s coincidence (join) lcm(a, b) max( p(a), p(b))
mu ual ebase + i s hi a·b p(a)+ p(b)
Conc e ely, p oduc wi hou naming gcd/lcm: de ine
a⋆b:= i s hi o aNmeasu ed in b– icks and bNmeasu ed in a– icks.
Since bo h ebased clocks equal (ab)N, he i s hi is ab, and exponen s add.
Rema k 3.1 (The lcm/gcd semi ing).The pai (
gcd,lcm
)is an idempo en , commu a i e semi ing
on
N≥1
; in alua ions i is jus (
min,max
)coo dina ewise. Ou “ i s coincidence” algeb a is
exac ly his semi ing; o dina y p oduc adds a hi d ope a ion co esponding o +on exponen s.
2
4 A amily o compa ible p oduc s
Once p imes a e ecognized as he a omic me e s, any p oduc ule o he o m
a⊙ b:= Y
p
p p p(a), p(b)
is de e mined by a p imewise bi a ia e law
p
:
N×N→N
(all bu ini ely many ze o inpu s).
Canonical ins ances:
p(i, j) = min(i, j)⇝gcd,
p(i, j) = max(i, j)⇝lcm,
p(i, j) = i+j⇝o dina y mul iplica ion,
p(i, j) = 1{i>0}1{j>0}⇝“suppo –only” mixe (dis inc p ime coun ).
Each choice has an Eule –p oduc a a a
F (s) = Y
pX
k≥0
cp,k p−ks,
wi h coe icien s cp,k e lec ing which p ime–powe s ands a e allowed locally.
5 Closu e, Eule p oduc s, and he me a–laye
F om p ime me e s o he ze a closu e
Swi ching e e y p ime ladde “on” (and all hei powe s) o mally yields
Y
p1+p−s+p−2s+···=Y
p
1
1−p−s=ζ(s) (ℜs>1),
and, ia analy ic con inua ion, he amilia global landscape. In his eading,
ζ
is a closu e
ha agg ega es all p ime me e s; i s non i ial ze os ac as me a–modes coupling all local p ime
ladde s a once (s anda d explici – o mula heu is ics). See he companion “umb ella” no e o a
didac ic ende ing o hese cascades and hei me a–s uc u e.
On he s uc u al ie o he c i ical line
I one ixes mul iplica ion by h ee neu al cons ain s—powe –compa ibili y (“ epea hen
o ge ”), uni a y Mellin on he midline, and scale–neu al blu on he log–line— hen a single
bounda y posi i i y (Fejé –Mellin) deli e s a compac condi ional chain o he c i ical line o
ζ
(loca ion o ze os). This is an implica ion inside ha egime, no a s and–alone p oo ; i makes
p ecise he sense in which “ he choice o mul iplica ion al eady seals he deal.”
6 P obabilis ic shadow: why p imes look hec ic
Ou co–sie e selec s p imes as deg ee–1poin s in he mul iples hype g aph. Nea
x
, su i ing
(being p ime) means dodging all ea lie p ime clocks, p oducing he Me ens p oduc heu is ic
P (p ime nea x)≈Y
p≤x
1−1
p∼e−γ
log x,
hence mean gap
∼log x
wi h isible local i egula i y: many small, almos independen aps
supe impose a noisy su ace on a slow end.
3
7 A e wo d: Me e s a e no i ial
Abandoning he dogma o a unique, indi isible me e and s aying wi hin successo alone, we
ind a e inemen la ice wi h genuine a oms. “Being a me e ” is no p imi i e; i is a s uc u ed
p ope y. P imes a e hose pe ec me e s. In e sec ions + simple ie–b eake s u n addi i e
clocks in o he ull a i hme ic zoo: mee /join, p oduc , and a whole spec um o al e na i e
p oduc s, each wi h a clean p imewise seman ics and an Eule –p oduc a a a . When he
ull closu e is swi ched on, he ze a landscape and i s me a–modes appea ; unde a speci ic
mul iplica i e choice he midline becomes s uc u ally singled ou .
Mini–dic iona y (one line each).
•Clocks kN= addi i e g ids (successo s e e y k). Re inemen o de = inclusion.
•P imes = indecomposable clocks (canno be w i en as non i ial o e laps).
•Co–sie e = keep poin s in exac ly one clock ⇒p imes.
•Fi s coincidence = join/lcm (max on exponen s). Las –always–hi = mee /gcd (min).
•Mu ual ebase + i s hi = o dina y p oduc (sum o exponen s).
•Family o p oduc s = choose pp imewise; Eule a a a = QpPkcp,kp−ks.
8 Mo i a ion o e heo y g ow h
Now we ix mul iplica ion in o de o show how hese assump ions we a e awa e o o no and
ob ain a s ikingly simple p o o ype o a ma hema ical heo y as he ollowing: ix a p ime
p
.
Conside he single p edica e “
x
is a powe o
p
.” The only heo ems a e
p, p2, p3, . . .
. The e is
no di isibili y ela ion, no addi ion—jus an in ini e chain o s a emen s o he o m “
pk
is a
powe o p.”
This no e shows how:
1. s a ing om he i ial 1- heo y, hen 2-, 3-, 5- heo ies, we ge in ini e cascades;
2.
combining p ime heo ies p oduces a new wo ld o mixed s a emen s (e.g. numbe s like
6,12,18,24 om mixing 2and 3);
3.
a compac umb ella gene a o u ns p ime heo ies and p ime powe s on/o , gi ing a amily
ha anges om spa se subse s o ull closu e;
4.
swi ching e e y hing on eco e s
ζ
(
s
), and once we allow
s∈C
, analy ic con inua ion e eals
me a-s uc u e—ze os as “me a- heo ies” whose ain oscilla ions ouch e e y p ime heo y a
once.
9 The i ial 1- heo y and he single-p ime p- heo y
De ini ion 9.1 (Language and p edica es).We use a language wi h cons an s 1and each p ime
p
, mul iplica ion “
·
,” and, o each p ime
p
, a una y p edica e
Powp
(
x
) o be ead as “
x
is a
powe o p.” No o he symbols a e used.
De ini ion 9.2 (The 1- heo y).The 1- heo y has he single axiom
Pow1
(1) and no in e ence
ules. I p o es only he i ial s a emen 1.
De ini ion 9.3 (The single-p ime heo y Tp).Tphas he wo schema a:
4
BasepPowp(p).
S eppF om Powp(x)in e Powp(p·x).
P oposi ion 9.4 (Comple eness o T
p
).The p o able sen ences o T
p
a e exac ly
{Powp
(
pk
) :
k≥1}.
P oo .
Base
p
gi es
k
= 1. Applying S ep
pk−
1 imes yields
pk
. Con e sely, no e m o he han
pkcan be o med by epea ed mul iplica ion by ps a ing om p.
Example 9.5 (The 2- heo y cascade).Pow2(2),Pow2(4),Pow2(8),Pow2(16),....
10 Combining p ime heo ies: mix u es and s uc u e
Le
S
be a ini e se o p imes. We o m a combined heo y T
S
ha con ains all axioms and
ules o Tp o p∈Sand addi ionally allows us o mul iply powe s om di e en p imes.
De ini ion 10.1 (Combined heo y TS).The language ex ends Theo em 9.1; o each q∈S:
Baseq: PowS(q),S epq: PowS(x)⇒PowS(q·x).
Add he p oduc ule: om PowS(x)and PowS(y)in e PowS(xy).
P oposi ion 10.2 (Wha T
S
p o es).The heo ems o T
S
a e p ecisely he s a emen s
PowS
(
x
)
whe e
x=Y
q∈S
qeq, eq∈N,no all 0.
P oo .
Using Base
q
and S ep
q
we gene a e
qeq
o each
q∈S
; mul iplying hem yields any such
x. Con e sely, any de i a ion p oduces a p oduc o p ime powe s in S.
Example 10.3 (The i s mix u es: S={2,3}).We see h ee in e laced cascades:
2,4,8,16, . . .
| {z }
2-chain
,3,9,27, . . .
| {z }
3-chain
,6,12,18,24,36, . . .
| {z }
mix u es
.
Adding 5en iches he landscape o all 2n3m5kwi h in ege s n, m, k ≥0no all ze o.
11 An umb ella gene a o : oggling p imes and p ime powe s
The p eceding sec ions desc ibe which numbe s appea as heo ems. We now encode en i e
amilies a once ia a compac gene a o . The e a e wo pa allel encodings:
11.1 Algeb aic ( o mal monoid) encoding
Wo k in he ee commu a i e monoid algeb a
Z
[
⟨p imes⟩
]. Fo each p ime
p
and exponen
k≥
1
ix a selec o ap,k ∈ {0,1}. Fo each p ime p ix a swi ch bp∈ {0,1}. De ine he heo y p o ile
Ta,b := Y
p∈P
1 + X
k≥1
ap,k pkbp.(11.1)
By expansion, he coe icien o n=Qpepin Ta,b equals 1i o each p:
bp=0⇒ep= 0, bp=1⇒ ∃ a decomposi ion ep=X
j
kjwi h each ap,kj= 1,
i.e. we may only use hose p ime powe s whose
ap,k
swi ch is on, and only om p imes wi h
bp= 1. (I all ap,k ≡1 hen we ge all powe s pepa pwhen bp= 1.)
5
11.2 Analy ic (Di ichle se ies) encoding
Fo Re s > 1de ine
Fa,b(s) := X
n≥1
ca,b(n)
ns=Y
p∈P
1 + X
k≥1
ap,k p−ksbp.(11.2)
He e
ca,b
(
n
)
∈ {
0
,
1
}
is he membe ship indica o de e mined by he same combina o ics as
abo e. The analy ic encoding ca ies he same oggling seman ics bu li es whe e p oduc s
con e ge.
Rema k 11.1 (Pa ial sums and p oduc s).Swi ching on ini ely many p imes and ini ely many
powe s yields ini e polynomials/ ini e Eule p oduc s. Tu ning on in ini ely many bu wi h
con e gence cons ain s (e.g. ap,k ≡1,bp≡1and Re s>1) yields classical Di ichle se ies.
12 The c own: he ze a closu e
I we ully open e e y ga e by se ing
ap,k ≡
1and
bp≡
1 o all p imes, hen Equa ion (11.2)
becomes
Fa,b(s) = Y
p∈P
1+p−s+p−2s+· · · =Y
p∈P
1
1−p−s=ζ(s),Re s>1.(12.1)
In he oy-model sense,
ζ
is he closu e o “ heo y o e e y hing”: i agg ega es all p ime heo ies
and all hei powe s.
Rema k 12.1 (Complex
s
and analy ic con inua ion).The Eule p oduc
(12.1)
de ines
ζ
(
s
)on
Re s >
1. Analy ic con inua ion and a unc ional equa ion ex end
ζ
o a me omo phic unc ion
on
C
wi h a simple pole a
s
= 1. This s ep—impo ing complex analysis—is he momen
when he “ oy” gains hidden dep h: phenomena on he ex ended plane encode global a i hme ic
egula i ies.
13 Me a- heo ies: ze a’s ze os as global chi ps
In his na a i e, a me a- heo y is a global mode ha ouches e e y p ime heo y a once.
Non i ial ze os
ρ
o
ζ
play his ole. One way o eel his is h ough a smoo hed p ime-powe
sum (a s anda d explici - o mula a a a ):
X
n≥1
Λ(n)g(log n)
ns=b
g(s)−X
ρb
g(ρ)+( i ial/a chimedean e ms).(13.1)
He e
g
is a sho , well-beha ed blu in he log scale, and
b
g
is i s Laplace/Fou ie ans o m. The
igh -hand side shows ha each ze o
ρ
con ibu es a cohe en “chi p”
b
g
(
ρ
) ha is ed by all
p ime powe s on he le . Thus, while indi idual p ime heo ies a e a omis ic, ze os li e a he
me a-le el, coupling hem all.
Rema k 13.1 (Didac ic eading).No hing in Sec ion 13 is needed o he basic cascade pic u e;
i simply illus a es how, a e passing o complex
s
, global esonances (ze os) eme ge as
highe -o de s uc u es ha speak ac oss e e y local heo y a once.
14 Cascades, su p ises, and en ichmen : a galle y
•S a a 1.Only 1is a heo em; he e is no mo ion.
•Tu n on 2.The in ini e chain 2,4,8,16, . . . appea s.
6
•
Add 3.A second chain 3
,
9
,
27
, . . .
appea s and a mixed cascade 6
,
12
,
18
,
24
, . . .
in e lea es
wi h he powe s o 2and 3. Al eady he pa e ning (e.g. densi y, gaps) becomes in e es ing.
•
Add 5.The playg ound becomes h ee-dimensional:
{
2
n
3
m
5
k}
. The eye s a s ca ching
mo i s (e.g. numbe s cong uen o 0o 1modulo small moduli; shapes in loga i hmic plo s).
•
Selec i e powe s. Wi h selec o s
ap,k
, we can keep 2
,
8
,
32
, . . .
bu d op 4
,
16
, . . .
, o o bid
9while keeping 3and 27. This c ea es pa e ned sub heo ies wi h hei own combina o ics.
•
Full closu e. Opening e e y hing e u ns
ζ
(
s
); s ep in o complex
s
and me a-s uc u e
(ze os) becomes isible.
15
P ime-side closu e and he Game o Li e: a oy me a- heo y
This sec ion inishes he p ime-side pic u e by isola ing a iny sel - e e en ial ke nel a each
p ime and hen pi o s o Conway’s Game o Li e as a didac ic labo a o y o how ma hema ical
heo ies g ow, s abilize, and some imes un in o undecidabili y.
15.1 P ime-side comple ion: sel -image, closu e, and a h ee- old cycle
Fix a p ime p. Conside he minimalis “p- heo y” whose only con en is he in ini e ladde
p, p2, p3, . . . ( ead: “pkis a powe o p”).
As a Di ichle a a a , he closu e o his ladde is he Eule ac o
Ip(s) := 1
1−p−s= 1+p−s+p−2s+··· (ℜs>1),(15.1)
and oggling all p imes on gi es he global closu e
Y
p
Ip(s) = ζ(s).
How he closu e closes: a 3-cycle unde a geome ic ans o m. In oduce he o mal
(Möbius) ans o m
T(X) := 1
1−X,
he ope a ion o “closing unde a geome ic se ies”
Pk≥0Xk7→
1
/
(1
−X
). Applying
T
h ee
imes e u ns o he s a :
T3(X)=X, hence XT
−→ 1
1−X
T
−→ 1−1
X
T
−→ X.
Specializing o X=p−sgi es he explici o mal bu i id cycle:
(a om) p−sPk≥0Xk
−−−−−−−→
∞
X
k=0
p−ks =1
1−p−s
Pk≥0Xk
−−−−−−−→
∞
X
k=0
1
1−p−sk=1
1−1
1−p−s
= 1 −ps
Pk≥0Xk
−−−−−−−→
∞
X
k=0
(1 −ps)k=1
1−(1 −ps)=1
ps=p−s.
(15.2)
7
Rema ks. (i) The equali ies a e he i s line a e pu ely o mal ( hey igno e analy ic con e gence
domains); he poin is he algeb aic closu e mechanism and i s idempo ence a e h ee s eps.
(ii) Thus he in insic s uc u e he e is h ee- old: a om
→
Eule ac o
→
complemen a y ac o
→a om.
De ini ion 15.1 (Global aces and he 3-cycle).Fo mally w i e
G0(s) := Y
p
p−s(a omic ace; o mal),
G1(s) := Y
p
1
1−p−s=ζ(s) (ℜs>1),
G2(s) := Y
p
(1 −ps) = 1
ζ(−s)( ia analy ic con inua ion),
Gumb(s):=G(s)(a selec o -d i en umb ella gene a o ).
Unde he ans o m
T
ac ing componen wise on Eule ac o s, he iple
{G0, G1, G2}
cycles
wi h pe iod 3as in (15.2); Gumb in e pola es be ween spa se selec ions and ull closu e.
15.2 The Game o Li e is mo e han a oy
Jus as he p ime-side pic u e shows how i ial local axioms (
pk
is a powe o
p
) close up in o
ζ
and i s me a-s uc u e, he Game o Li e does he same o spa ial- empo al dynamics: ex emely
simple local ules gene a e ich global engines and undecidable on s. Conway’s Game o Li e
(GoL) is an axioma ic sys em wi h wo ules:
De ini ion 15.2 (Game o Li e ules).On he squa e g id wi h Moo e neighbo hood, a each
disc e e ime s ep:
•Su i al: a li e cell wi h 2o 3li e neighbo s s ays ali e;
•Bi h: a dead cell wi h exac ly 3li e neighbo s becomes ali e;
•all o he cells become/s ay dead.
Despi e i s ba ebones axioms, GoL suppo s an as onishing menage ie: s ill li es (s able
pa e ns), oscilla o s (pe iodic pa e ns), and spaceships/glide s (sel -p opaga ing pa e ns).
C ucially, GoL is Tu ing-comple e: wi h app op ia e ci cui y o s able/oscilla o y gadge s, one
can embed a bi a y ini e compu a ions. As a consequence, many na u ally posed ques ions
abou long- e m beha io a e compu a ionally in ac able, and some a e ou igh undecidable.
In sho , he oy hides a ull-scale engine.
A didac ic dic iona y. Li e can se e as a me apho o how ma hema ical heo ies g ow:
Li e Ma hema ics (me apho ) Reading
Cell S a emen /Sen ence “ali e” =cu en ly ac i e/use ul
Neighbo suppo Lemmas/p emises 2o 3suppo s = iable hypo hesis
S ill li e P o ed heo em S able unde in e ence/in e ac ion
Oscilla o Dynamic u h Al e na es oles ac oss con ex s
Spaceship/glide T ans e p inciple Mo es in o ma ion ac oss he heo y
Gadge s Lemmas/ ools Reusable subp oo s in la ge builds
Guns/emi e s Theo em schemes In ini e amilies by con olled ou pu
8
Unde his lens, disco e y looks like inke ing wi h axioms o ind su i able cons uc ions.
Some ial pa e ns die in a ew gene a ions (disca ded heu is ics); o he s s abilize (lemmas);
s ill o he s become mobile and powe la ge a chi ec u es ( ans e p inciples, induc ion de ices).
Because GoL can implemen a bi a y compu a ion, i can, in p inciple, encode p oo sea ch;
con e sely, he undecidabili y o ce ain Li e ques ions mi o s he impossibili y o deciding all
s a emen s om ixed axioms in su icien ly exp essi e ma hema ical sys ems.
Why his ma e s.
•
F om a oms o closu e. Jus as he p ime-side closu e
Qp
(1
−p−s
)
−1
agg ega es i ial
local ladde s in o
ζ
(
s
), Li e agg ega es iny local in e ac ions in o global engines (compu e s,
cons uc o s).
•
S abili y as p oo . A s ill li e is a “p o ed” s a emen : i pe sis s unde he ules. Oscilla o s
model u hs ha a e s able only modulo a con ex shi ; spaceships ep esen heo ems ha
anspo s uc u e.
•
Limi s a e in o ma i e. In Li e, some eachabili y/long- e m ques ions a e undecidable; in
ma hema ics, en i e classes o p oblems a e p o ably ou o each om gi en axioms wi hou
en ichmen . Recognizing hese limi s ea ly is s a egically aluable.
Example 15.3 (Nes ed eme gence).The e exis Li e con igu a ions ha emi glide s (“glide
guns”), which in u n build o he machines, which hen simula e compu a ions. This is a
h ee- ie eme gence: gadge
→
cons uc o
→
compu a ion. On he p ime side, we saw a oms
→
closu es
→
me a-couplings ia ze a’s ze os. In bo h wo lds, simple axioms yield s a i ied
s uc u e.
Rema k 15.4 (Scope and ca ea s).We a e no claiming ha Li e is ma hema ics. The poin
is pedagogical: Li e is an unusually anspa en sandbox in which o wa ch he dynamics
o heo y-building—s abili y, in e ac ion, anspo , and he appea ance o ha d/undecidable
on s—play ou unde axioms so simple ha he en i e “ heo y” i s on a pos ca d.
Takeaway. P ime heo ies and Li e sha e a mo al: i ial-looking axioms can ha bo deep
in e nal o ganiza ion. On he p ime side, he o ganiza ion is mul iplica i e and collapses o ze a;
on he Li e side, i is spa ial- empo al and collapses o uni e sal compu a ion. Bo h p o ide
in ui ion o how local ules scale o global s uc u e—and whe e he limi s (idempo ence o
undecidabili y) begin o bi e.
16 Ou look
The pu pose o his no e is illus a i e: o show how e y simple “local” heo ies—one pe p ime,
asse ing no hing bu “being a powe o ”—o e lap, cascade, and in e wine in o a ich global
pic u e, and how a compac umb ella gene a o in e pola es be ween spa se oy wo lds and ull
closu e. Once complex analysis is admi ed, he same oy becomes a window on o me a-s uc u e:
global modes (ze os) ha whispe ac oss e e y local s and.
Op ional exe cises.
1.
P o e Equa ion (11.2) con e ges absolu ely on
Re s >
1whene e
bp≤
1and
Ppbpp−σ<∞
o some σ > 1.
2.
Fo ixed
S
, coun he numbe o heo ems o T
S
up o
X
and iden i y he leading cons an
in e ms o |S|.
3.
Using selec o s
ap,k
, design a pa e n whe e e e y hi d powe o 2is pe mi ed and o he s
a e o bidden; desc ibe he esul ing se mul iplica i ely.
9
