PRH | Aux | 4.3 • A Lyapunov Certificate for the Accelerated Collatz Map

A Lyapuno Ce i ica e o he Accele a ed Colla z Map
Backwa d Spec a wi h Blu
Aleksanda Pe išić
Sep embe 2025
Abs ac
We p esen a ini e Lyapuno ce i ica e o he accele a ed Colla z map ha emains
alid unde an explici “blu ” budge . Fix
k≥
1and le
S
=
{
1
,
3
,...,
2
k−
1
}
be he
odd esidues modulo 2
k
(
|S|
= 2
k−1
). De ine
Fk
(
)
≡odd
(3
+ 1) (
mod
2
k
). We exhibi a
unc ion
ϕ
:
S→R
and pa ame e s
δ >
0,
ρ≥
0wi h
ρ
+
δ < log
(4
/
3) such ha he 2
k−1
di e ence cons ain s
log 3 − 2(3 + 1) log 2 + ρ+ϕ
Fk( )≤ϕ( )−δ(∀ ∈S { ∗}),
and
log 3 −klog 2 + ρ+ϕ
Fk( ∗)≤ϕ( ∗)−δ
hold, whe e
∗
is he unique odd class wi h 3
∗
+ 1
≡
0 (
mod
2
k
). Fo he in ege ajec o y
N7→ F♯(N) = odd(3N+ 1), his yields he d i inequali y
log N +1 +ϕodd(N +1) mod 2k≤log N +ϕodd(N ) mod 2k−δ+ρ−ε(N ),
wi h
ε
(
N
) =
log
(1 + 1
/
(3
N
)). Telescoping d i es odd alues below a scale h eshold; in
gene al a ini e bo om- ange check hen su ices o o ce a hi o 1. Fo ou e i ied ins ance,
he pe -s ep d i is al eady posi i e o e e y odd
N≥
3, so no bo om- ange check is needed.
The e i ica ion uses in e al a i hme ic and we explici ly un he checke wi h he s ong
lag
–– o ce-excep ional
, which en o ces he conse a i e weigh
a
(
∗
) =
log
3
−klog
2
on he excep ional esidue.
1
A quick ou : wha we measu e, why we blu , and a 5001
mic o-walk
Th ee lines i s . (1) We wa ch only he odd e ms, using he accele a ed s ep
F♯(N) = 3N+ 1
2 2(3N+1) = odd(3N+ 1) (Nodd),
so each mo e is “3
N+
1 hen di ide ou all 2’s.” (2) The scale we ack is
u
=
log N
, so he
one–s ep change is
∆(u) = logF♯(N)
N= log 3 − 2(3N+ 1) log 2
| {z }
disc e e jump se by esidue
+ log1 + 1
3N
| {z }
iny co ec ion ε(N)
.
(3) Raw s eps can spike up o down, bu a iny a e age o e he las ew bi s (ou “blu ”) cancels
he spikes and e eals a eliable small nega i e d i . The pape hen u ns ha in o a igo ous,
ini e ce i ica e.
1
Why esidues mod 2kma e (and he one excep ional class)
Fix
k≥
1and le
≡N
(
mod
2
k
)be he odd esidue. Excep o a single “excep ional” class
∗( he unique odd solu ion o 3 + 1 ≡0 (mod 2k)), he alue
( ) := 2(3N+ 1)
is comple ely de e mined by
and does no depend on which
N≡
(
mod
2
k
)you picked. Thus,
up o a iny ε(N), he jump
∆(u)≈log 3 − ( ) log 2
is a classwise cons an . The excep ional class only sa is ies
(
∗
)
≥k
, so we ea i conse a i ely
by aking ( ∗)=k.
The “las 4 bi s” blu (wha i is and why i helps)
Pick
k
= 4 as a oy pic u e. E e y odd
N
lies in a 16-block wi h he same uppe bi s and all
odd 4-bi endings:
{M+ 1, M + 3, . . . , M + 15}, M ≡0 (mod 16).
Ou blu simply a e ages he one–s ep scale change ∆(
u
)o e hose eigh add esses. Since he
wild pa o ∆is he disc e e
(
)
∈ {
1
,
2
,
3
,...}
, and hose eigh esidues mix se e al
(
)’s
(some up, some down), hei a e age cancels he coin- lip spikes. Wha ’s le is a small, s eady
nega i e slope. In he ull p oo we use
k
= 13 and a ini e able
ϕ
(
) o gua an ee a pe –s ep
dec ease wi h a ce i ied ma gin.
A 5001 mic o-walk ( ou accele a ed s eps) and he las -4-bi s blu
No a ion. W i e he accele a ed odd s ep as
T(n):=F♯(n) = 3n+ 1
2 2(3n+1) (nodd),
and measu e one–s ep change on he log2scale by
∆2(n) := log2T(n)−log2n= log23− 2(3n+ 1)
| {z }
ex disc e ejump
+ log21 + 1
3n
| {z }
iny ε2(n)
.
Fou s eps om
N0
= 5001.We only ack odd- o-odd mo es (e en di isions a e al eady
abso bed in T). Use log23≈1.58496.
•N0= 5001:3N0+ 1 = 15004 = 4 ·3751 ⇒ 2= 2.
Then N1=T(N0) = 3751 and ∆2(N0)≈1.58496 −2 = −0.415 bi s (down).
•N1= 3751:3N1+ 1 = 11254 = 2 ·5627 ⇒ 2= 1.
Then N2= 5627 and ∆2(N1)≈1.58496 −1 = +0.585 bi s (up).
•N2= 5627:3N2+ 1 = 16882 = 2 ·8441 ⇒ 2= 1.
Then N3= 8441 and ∆2(N2)≈+0.585 bi s (up).
•N3= 8441:3N3+ 1 = 25324 = 4 ·6331 ⇒ 2= 2.
Then N4= 6331 and ∆2(N3)≈ −0.415 bi s (down).
Raw pa e n: down, up, up, down. The spikes come om he in ege
2
(3
n+
1)
∈ {
1
,
2
,...}
.
2
The las -4-bi s blu (p ecise ecipe). Fix he uppe bi s ( he 16-block) and a e age o e
he eigh odd 4-bi endings; his cancels he coin- lip spikes.
•Decompose n= 16q+ wi h ∈ {1,3,5,7,9,11,13,15}.
•The blu neighbo hood a scale qis
N(q):={16q+ ′: ′∈ {1,3,...,15} },
i.e. same uppe bi s q, all odd 4-bi endings.
•Fo each neighbo n′∈ N(q), pe o m one accele a ed s ep
T(n′) = 3n′+ 1
2 2(3n′+1) ,∆2(n′) = log2T(n′)−log2n′.
•De ine he blu ed d i used a n( eally a i s block q) by
∆2(q) := 1
8X
′∈{1,3,...,15}
∆216q+ ′.
Because he eigh esidues con ibu e a mix o
2
(3
n′
+ 1) = 1
,
2
, . . .
, he ups and downs
almos neu alize, and
∆2
(
q
) u ns s ably nega i e. The co ec ion
ε2
(
n
) =
log2
(1 +
1
3n
)is
al eady
<
10
−3
bi s a
n
= 5001 and decays o 0wi h
n
. Tha pe sis en nega i e mean is he
engine. In he o mal p oo , we eplace his a e age by a ini ely checked esidue po en ial
ϕ
modulo 2
k
(wi h
k
= 13), which ce i ies he same pe -s ep dec ease wi hou any p obabilis ic
assump ions.
Wha we will p o e (in p ecise o m)
We o malize he sa e y slope wi h a ini ely-checked po en ial
ϕ
on esidues
mod
2
k
. The co e
ini e inequali ies say
log 3 − ( ) log 2+ρ+ϕ
Fk( )≤ϕ( )−δ o e e y odd esidue ,
wi h a iny bu e
ρ
ha accoun s o blu /calib a ion and a ixed ma gin
δ >
0. Summing along
he odd ajec o y gi es he Lyapuno dec ease
log N +1 +ϕ(odd(N +1) mod 2k)≤log N +ϕ(odd(N ) mod 2k)−δ+ρ−ε(N ),
so he odd alues mus descend in o a ini e ange and (in ou e i ied ins ance) in ac d op
e e y s ep o all odd
N≥
3. The es o he pape shows how his ce i ica e is buil and
checked—no global andomness assump ions, jus esidue a i hme ic and a small amoun o blu
encapsula ed in ρ.
2 Se up: accele a ed map, esidues, and blu
Accele a ed odd s ep. W i e he accele a ed Colla z odd s ep as
F♯(N) := 3N+ 1
2 2(3N+1) = odd(3N+ 1) o odd N,
whe e odd(m) := m/2 2(m)deno es he odd pa o mand 2(m)is he 2-adic alua ion.
E en s eps pe o m hal ing; we analyze only he odd subsequence
N0, N1, . . .
wi h
N +1
=
F♯(N ).
3
Residue dynamics modulo 2
k
.Fix
k≥
1. Le
S
=
{
1
,
3
,...,
2
k−
1
}
be he odd esidues
modulo 2k, and de ine he esidue map
Fk:S→S, Fk( )≡odd(3 + 1) (mod 2k).
I ≡odd(N ) (mod 2k), hen +1 ≡Fk( ) (mod 2k).
Lemma 2.1 (Excep ional class exis s and is unique).The e is a unique
∗∈S
such ha
3 ∗+ 1 ≡0 (mod 2k).
P oo .
Since
gcd
(3
,
2
k
)=1,3has a unique in e se 3
−1
(
mod
2
k
). Se
∗≡ −
3
−1
(
mod
2
k
). As
he in e se o an odd numbe modulo 2kis odd, ∗∈S; uniqueness is immedia e.
Exac one-s ep log change (sa e, classwise). Fo an odd
N
wi h esidue
≡N
(
mod
2
k
),
∆ log N:= log3N+ 1
2 2(3N+1) −log N
=log(3N+1)−log N− 2(3N+ 1) log 2
= log 3 − 2(3N+ 1) log 2 + log1 + 1
3N
| {z }
ε(N)
≤a( ) + log1 + 1
3N≤a( ) + log4
3.(1)
whe e we de ine he classwise bound (using Theo em 2.1)
a( ) := 


log 3 − 2(3 + 1) log 2,i 2k∤(3 + 1),
log 3 −klog 2,i 2k|(3 + 1) (i.e. = ∗).
Fo

=
∗
one in ac has
2
(3
N
+ 1) =
2
(3
+ 1) o all
N≡
(
mod
2
k
); o
=
∗
only
2(3N+ 1) ≥kis gua an eed, hence he piecewise de ini ion abo e.
Lemma 2.2 (Residue de e mines
2
(3
N
+ 1) o he excep ional class).Fix
k≥
1and
∈S
.
I

=
∗
and
:=
2
(3
+ 1)
< k
, hen o e e y odd
N≡
(
mod
2
k
)one has
2
(3
N
+ 1) =
.
I = ∗, hen 2(3N+ 1) ≥k(and may a y wi h N).
P oo .
W i e any odd
N≡
(
mod
2
k
)as
N
=
+ 2
km
. Then 3
N
+ 1 = (3
+ 1) + 3
·
2
km
.
Le
=
2
(3
+ 1). I

=
∗
hen
< k
, and we ac o 3
+ 1 = 2
u
wi h
u
odd and
3
·
2
km
= 2
(3
·
2
k− m
)wi h he b acke e en; hence 3
N
+ 1 = 2
(
u
+
e en
)has odd b acke , so
2(3N+ 1) = . I = ∗ hen 3 ∗+ 1 ≡0 (mod 2k), hence 2(3N+ 1) ≥k.
Blu budge . We model h ee small posi i e e ec s as a single nonnega i e budge
ρ:= blog 2 + κ+ζ, κ := −log(1 −p), b, p, ζ ≥0,
and assume
ρ+δ < log(4/3).(2)
3 Po en ial inequali ies and he ce i ica e
De ini ion 3.1 (One-s ep ce i ica e on esidues).Fix
k≥
1. A unc ion
ϕ
:
S→R
and
pa ame e s δ > 0,ρ≥0 o m a one-s ep ce i ica e i
a( )+ρ+ϕ(Fk( )) ≤ϕ( )−δ(∀ ∈S),(3)
wi h a( )as de ined abo e.
4
Cla i ica ion ( ini e scope o he g aph analysis). Th oughou , all algo i hmic and
po en ial–cons uc ion s eps ake place only on he ini e esidue g aph
S={1,3,...,2k−1}, 7→ Fk( ) = odd(3 + 1) mod 2k.
When we la e in oke Bellman–Fo d, cycle–mean duali y, o elaxa ion a gumen s, hese a e
applied o his ini e unc ional dig aph on
S
. No e sion o Bellman–Fo d is e e applied o he
in ini e Colla z g aph on
N
. The in ini e in ege dynamics en e s only a e wa ds, ia he d i
inequali y de i ed om he ini e ce i ica e. This sepa a ion ensu es ha e e y compu a ional
o logical s ep in he cons uc ion is ini e and e i iable.
Theo em 3.2 (Ce i ica e
⇒
d i o he in ege chain).Assume
(3)
holds. Then o he odd
subsequence N0, N1, . . . o any Colla z ajec o y, wi h ≡odd(N ) (mod 2k), one has
log N +1 +ϕ( +1)≤log N +ϕ( )−δ+ρ−ε(N ),(4)
o all ≥0, whe e ε(N ) = log(1 + 1/(3N )) ∈(0,log(4/3)].
P oo . Combine (1) wi h (3) o ge
∆ log N ≤a( )+ε(N )≤ −ρ−δ+ϕ( )−ϕ( +1)+ε(N ),
and ea ange o ob ain (4).
Co olla y 3.3 (En y in o a ini e se ).Assume
(3)
and
(2)
. Fix any
ζ⋆∈
(0
, δ
+
ρ
)and choose
N⋆:= l1
3exp(ζ⋆)−1mso ha ε(N)≤ζ⋆ o all N≥N⋆.
Le δ′=δ+ρ−ζ⋆>0. Fo any Jsuch ha N ≥N⋆ o all 0≤ < J, summing (4) yields
log NJ+ϕ( J)≤log N0+ϕ( 0)−Jδ′.
Hence he odd subsequence mus en e he ini e se
{
1
,
3
, . . . , N⋆−
2
}
in ini e ime. I , in
addi ion, e e y odd
N < N⋆
e en ually eaches 1(a ini e check), hen he odd subsequence
eaches 1.
Rema k 3.4 (No global s uc u e is needed once
ϕ
is e i ied).The p oo uses only he local
inequali ies
(3)
o all esidues and he bound
(1)
. One does no need o analyze he unc ional
g aph o Fk, no any dis ibu ion beyond 2-adic alua ions.
4 A 2-adic lemma and he igh mod-8 class
Lemma 4.1. Fo odd ,
≡1 (mod 8) ⇒ 2(3 + 1) = 2,
≡3,7 (mod 8) ⇒ 2(3 + 1) = 1,
≡5 (mod 8) ⇒ 2(3 + 1) ≥3.
P oo .
W i e
= 1
,
3
,
5
,
7 (
mod
8) and expand 3
+ 1 explici ly. Sha pening: in ac , i
≡
5
(mod 16) hen 2(3 + 1) ≥4.
5 Cons uc ing ϕ( easible po en ials)
In e p e
(3)
as edge cons ain s on he dig aph wi h e ices
S
and edges
→Fk
(
)ca ying
weigh
w
(
) :=
a
(
)+
ρ
+
δ
. A able
ϕ
:
S→R
is easible i o all
∈S
,
ϕ
(
)
≥ϕ
(
Fk
(
))+
w
(
).
Feasibili y is unchanged by adding a cons an o ϕ.
5

Cycle-mean condi ion. By max–plus duali y (Ka p), easibili y holds i e e y di ec ed cycle
Csa is ies 1
|C|P ∈Cw( )≤0, wi h s ic <0implying a ma gin.
Linea - ime elaxa ion (Bellman–Fo d s yle). Ini ialize ϕ( ) = 0 o all . I e a e
ϕ( )←max{ϕ( ), ϕ(Fk( ))+w( )}(∀ ∈S),
un il no upda e occu s. This e mina es in
O
(
|S|
)passes on ou unc ional dig aph and yields a
easible ϕ. (Any addi i e shi o ϕis also easible.)
LP iewpoin . Equi alen ly, sol e he linea p og am
min P ϕ
(
)subjec o
ϕ
(
)
−ϕ
(
Fk
(
))
≥
w
(
) o all
. The elaxa ion abo e e u ns an op imal solu ion up o an addi i e cons an
whene e all cycle means a e ≤0.
6 Machine-checkable e i ica ion p o ocol
6.1 Inpu s
•An in ege k≥1(we use k= 13 in he ins ance below);
•A able ϕ:S→R o S={1,3,...,2k−1};
•Pa ame e s δ > 0and ρ=blog 2 −log(1 −p)+ζ≥0.
6.2 Di e ence cons ain s o check
Fo each odd esidue ∈S, compu e
2(3 + 1), Fk( )≡odd(3 + 1) (mod 2k),
and e i y
a( )+ρ+ϕFk( )+δ≤ϕ( ),(5)
wi h
a
(
)as de ined abo e. On he excep ional class
∗
we un he e i ie wi h he explici
lag
–– o ce-excep ional
, i.e. we check wi h he conse a i e weigh
a
(
∗
) =
log
3
−klog
2
(en o cing 2=kwhile keeping he canonical a ge ).
Excep ional-class policy (explici o he a ached a i ac s)
In ou shipped ins ance a
k
= 13 he able lis s he obse ed alua ion
2
(3
∗
+ 1) = 14 and
he canonical esidue a ge
Fk
(
∗
) =
odd
(3
∗
+ 1)
≡
1 (
mod
2
k
). Fo e i ica ion we un wi h
he explici lag
–– o ce-excep ional
, which imposes he s ic e (mo e conse a i e) policy
a
∗
: i ixes
2
:=
k
(i.e. 13) while keeping he canonical a ge
Fk
(
∗
)=1. This makes he
inequali y a
∗
s ic ly ha de han using he obse ed
2
= 14 ( he le -hand side inc eases
by exac ly
log
2). E en unde his s onge policy, he in e al slack is app oxima ely
−
10
−6
(nega i e), so he check passes. I a ully li -agnos ic a ge is desi ed, one may e ine only his
edge o modulus 2k+1 and e i y he esul ing ini e se .
6.3 In e al a i hme ic (no loa ing-poin us )
Pick a ionals
L2< U2
and
L3< U3
wi h
L2<log
2
< U2
,
L3<log
3
< U3
. Also bound
κ
=
−log
(1
−p
)by a a ional
Uκ
(e.g., by di ec sandwiching, o use he inequali y
−log
(1
−p
)
≤
p+p2
2(1−p) o a ional p∈[0,1)). Then (5) is implied by
U3− 2(3 + 1) L2
| {z }
uppe bound o log 3− 2log 2
+b U2+Uκ+ζ
| {z }
uppe bound o ρ
+ϕFk( )+δ≤ϕ( ),
6
in e p e ing
2
(3
+ 1) :=
k
o he class
=
∗
. This elimina es oundo . The e i ica ion is a
linea pass o e S.
6.4 Slack epo
De ine he slack
slack( ) := a( ) + ρ+ϕ(Fk( ))+δ−ϕ( ).
A success ul check has
max slack
(
)
≤
0; he minimum equals app oxima ely
−log
(4
/
3)
−
(
ρ
+
δ
)

when ≡1 (mod 8).
7 Pa ame e oles in ρ=blog 2 −log(1 −p)+ζ
•blog
2co e s ac ional-bi phase misalignmen on he scale ci cle; c . he Lipschi z/ en -
molli ie bound in Appendix B.
•−log
(1
−p
)is he log-momen unca ion penal y o a boxca ex ac ion; c . Appendix A.
•ζabso bs any esidual molli ie /calib a ion ji e .
These e ec s a e subaddi i e in he analysis, so a single budge
ρ
=
blog
2
−log
(1
−p
) +
ζ
is sa e.
8 Ve i ied ins ance a k=13
Fo he a ached able ϕa k= 13 and pa ame e s
(δ, b, p, ζ) = (0.10,0.05,0.05,0.02), ρ =blog 2 −log(1 −p) + ζ≈0.1059506534,
we checked all 2
12
= 4096 inequali ies
(5)
wi h he conse a i e excep ional weigh
a
(
∗
) =
log 3 −klog 2 by unning he e i ie wi h –– o ce-excep ional. Resul s:
•All cons ain s hold: max
slack( )≤0(in e al-checked; nume ically ≈0).
•Tigh ness: min
slack( )≈ −log(4/3) −(ρ+δ), ma ching he mod-8 igh class.
Ve i ica ion no e o
∗
.Fo his ins ance
2
(3
∗
+1) = 14 is obse ed, bu he inequali y a
∗
emains alid when one o ces he conse a i e
2
=
k
= 13 ( he e ec o
–– o ce-excep ional
);
he in e al slack is abou −10−6, i.e. s ill nega i e.
Thus Theo em 3.2 and Theo em 3.3 apply and o ce odd alues below
N⋆
; in his ins ance, he
uni o m d i is posi i e al eady o e e y odd N≥3, so no bo om- ange check is needed.
Rema k 8.1 (Nume ical co olla y o he ins ance).Fo (
δ, b, p, ζ
) = (0
.
10
,
0
.
05
,
0
.
05
,
0
.
02) one
has
δ
+
ρ≈
0
.
2059506534 and
ε
(3) =
log
(10
/
9)
≈
0
.
1053605157. Hence
δ′
=
δ
+
ρ−ε
(3)
≈
0
.
1005901377
>
0. The e o e
(4)
yields a uni o m one-s ep dec ease o all odd
N≥
3, so he
odd subsequence eaches 1wi hou any addi ional bo om- ange e i ica ion.
P oposi ion 8.2 (Uni o m one-s ep dec ease o all odd
N≥
3).Fo he e i ied pa ame e s
(
δ, b, p, ζ
) = (0
.
10
,
0
.
05
,
0
.
05
,
0
.
02) one has
δ
+
ρ−ε
(3)
>
0. Hence
(4)
yields a s ic dec ease
o e e y odd
N≥
3. Since 1is he only odd
<
3and
F♯
(1) = 1, he odd subsequence eaches 1
wi h no sepa a e bo om- ange check.
Rema k 8.3 (Mixed wo-s ep a ian ).Summing
(3)
o
and
Fk
(
)yields a wo-s ep ce i ica e
wi h d i 2δ, i.e. a( )+a(Fk( )) + 2ρ+ϕ(F2
k( )) ≤ϕ( )−2δ,
which can be use ul o diagnos ics. I is implied by he one-s ep sys em and needs no ex a
hypo heses.
7
Rema k 8.4 (Excep ional esidue is handled conse a i ely).Le
∗
be he unique odd class
wi h 3
∗
+ 1
≡
0 (
mod
2
k
). Fo any li
N≡ ∗
(
mod
2
k
)one has
2
(3
N
+ 1)
≥k
, hence
log
3
− 2
(3
N
+1)
log
2
≤log
3
−klog
2. We he e o e check he
∗
-cons ain wi h he conse a i e
weigh
a
(
∗
) =
log
3
−klog
2. Because he nex esidue
odd
(3
N
+ 1)
mod
2
k
may depend on he
li
N
, one may ei he bind his edge o he canonical a ge
Fk
(
∗
) =
odd
(3
∗
+ 1)
≡
1 (
mod
2
k
)
(as done in ou a i ac and s ill alid unde
2
=
k
), o e ine only his edge o modulus 2
k+1
o
make he a ge single- alued and check he esul ing ini e inequali ies.
9 A i ac s and ep oducibili y
We p o ide:
•phi_k13_conse a i e_ce i ica e.cs
: ows (
, ϕ
(
)
, 2
(3
+ 1)
, Fk
(
)) o all
∈S
,
k
= 13. (On he excep ional class
∗
he en y uses he conse a i e weigh
a
(
∗
) =
log 3 −klog 2 in he policy; in he ile, he obse ed alua ion is also lis ed.)
•P og am.cs
: a e i ie ha (i) compu es
Fk
and
2
(3
+ 1) i missing, (ii) checks
(5)
using
a ional in e als
L2< U2
o
log
2,
L3< U3
o
log
3, and a ce i ied ail o
κ
=
−log
(1
−p
),
(iii) epo s slacks and
N⋆
, and (i ) can simula e ajec o ies. Use
–– o ce-excep ional
o
en o ce he conse a i e excep ional alua ion.
•Ce i ica e. x
/ un summa y: shows pa ame e s, ma gin
ln
(4
/
3)
−
(
ρ
+
δ
), and PASS
(bo h loa and in e al) on all 212 cons ain s.
Usage.
do ne un -- ile phi_k13_conse a i e_ce i ica e.cs
-- epo epo .cs --s a N 513 -- o ce-excep ional
A success ul un p in s “PASS: CERTIFICATE VERIFIED” wi h all 2
12
= 4096 cons ain s
sa is ied.
10 Wha is ac ually needed (and wha is no )
•
Needed: A able
ϕ
:
S→R
and numbe s
δ, ρ
sa is ying he 2
k−1
linea inequali ies
(5)
,
e i ied wi h in e al a i hme ic. On he excep ional class
∗
, always use
a
(
∗
) =
log
3
−klog
2.
•
No needed: Any global in o ma ion abou he unc ional g aph o
Fk
, equidis ibu ion, o
mo e han 2-adic alua ions (which a e al eady ixed by esidues).
•
Choice o
k
:Any
k≥
1wo ks i
ϕ
passes all cons ain s. In p ac ice,
k
= 13 al eady
su ices o he p o ided able.
11
Conclusion: Whe e he blu li es (and a gene ic de i a i e
p inciple)
A i s glance he inal ce i ica e
(3)
looks “blu – ee.” Tha is misleading. The p oblem is
highly iolen on he scale line
u
=
log N
, and he engine o he a gumen was o allow la ge blu ,
ind a egime whe e such blu can li e, and hen squeeze i un il only a ini e esidue ce i ica e
emains. Conc e ely:
•
We began wi h gene ous smoo hing o he one–s ep change ∆(
u
)(uni o m window / boxca ,
Appendix A) and a bi –phase misalignmen budge (Appendix B). These p oduce he h ee
addends in ρ=blog 2 −log(1 −p) + ζ.
8
•
The goal was no o guess he e minal a ac o ; i was o show he sys em goes down unde
he blu . E en i he e en ual odd cycle we e as onomically la ge, ha would al eady be
success a he “blu esolu ion.”
•
A e loca ing a esolu ion whe e he blu ed d i is nega i e, we igh ened he budge s
un il a simple, ini e ce i ica e appea s. In ou case his happened a
k
= 13, and no hing
la ge was needed.
A gene ic blu -d i en de i a i e. Le
S
be any ansla ion–in a ian , o de –p ese ing
log–momen smoo hing on he scale line (e.g. he boxca Bτ,p). Suppose:
(a) S( +g)≤ S +Sg(Jensen/con exi y in he log–momen sense);
(b)
Fo he
ϕ
-p o ile class,
S
is Lipschi z unde phase shi s:
|
(
S
)(
u
+ ∆
u
)
−
(
S
)(
u
)
|≤b|
∆
u|
;
(c)
The e exis numbe s
ρ≥
0,
δ >
0and a able
ϕ
:
S→R
such ha he esidue cons ain s
(3) hold wi h ha ρ.
Then, w i ing V(N) := log N+ϕ(odd(N) mod 2k), we ha e he gene ic de i a i e
V +1 ≤V −h(δ+ρ)−ε(N )i,
and in pa icula , o e e y h eshold
N⋆
wi h
ε
(
N⋆
)
≤ζ⋆< δ
+
ρ
, he odd subsequence s ic ly
dec eases as long as
N ≥N⋆
. This is exac ly he d i in
(4)
, bu s a ed budge –agnos ically:
i does no ma e how
ρ
is decomposed—only ha he blu penal ies a e abso bed in o some
ini e ρ.
How o euse he me hod.
1. Pick a gene ous blu (la ge τ, small co e 1−p, pe missi e b); compu e an ini ial ρ.
2. Sol e he max–plus sys em (3) o ϕ(Sec ion 5); check d i .
3.
Tigh en he budge s (
τ↓
,
p↓
, sha pen Lipschi z) and educe
k
un il a ini e ce i ica e
appea s.
4.
Repo he pu i ied ce i ica e and keep he budge s as a ep oducible de i a ion pa h. In
ou case, a i ing a 213 was su icien .
This is why he inal p esen a ion looks clean: he blu is no gone—i is encapsula ed by
ρ
. The
ce i ica e is he sha pened bounda y o a ully blu –based cons uc ion.
12 Concluding ema ks
The a gumen is a s anda d max-plus/di e ence-cons ain s ce i ica e: once he esidue-wise
inequali ies a e e i ied, he Lyapuno unc ion
V
(
N
) =
log N
+
ϕ
(
odd
(
N
)
mod
2
k
)dec eases by
a ixed posi i e amoun (a e budge ing blu ) whene e
N
is abo e he scale h eshold, o cing
en y in o a ini e se ; a ini e check he e ules ou non i ial cycles. The e i ica ion is ini e,
uni o m, and can be made ully o mal wi h a ional bounds.
Rep oducibili y no e. The a ached CSV lis s all odd esidues
,
ϕ
(
),
2
(3
+ 1), and
Fk
(
).
A 50-line sc ip su ices o un he in e al check as pe Sec ion 6.
9