A Scep ic In oduc ion o he P oo o he Colla z Conjec u e
Fo Reade s Who A i e Ready o Re u e
Aleksanda Pe išić
Oc obe 2025
I you came he e o poke holes, you’ e in he igh place. This no e is he as , human- acing
on- amp: minimal o mulas, maximal cla i y, and a conc e e checklis o wha would ac ually
b eak he p oo .
Whe e he au ho s a e coming om (skep icism included)
Fi s , es assu ed: he au ho s sha e you skep icism—ampli ied by ha ing o audi e e y seam.
We did no se ou o “p o e Colla z.” We ca ied a me hod ( he blu iewpoin ) h ough a ew
o he p oblems and hen—hesi an ly—poin ed i a Colla z, knowing se e al ou comes we e
possible:
1.
Fail: blu ing could no be u ned back in o sha p, ce i iable inequali ies (a hin Colla z
migh be alse o a leas esis an o his lens).
2.
Semi-win: a Lyapuno - ype ce i ica e exis s in p inciple, bu wi h no usable bound on
i s size.
3.
Py hic win: a ce i ica e exis s bu li es a an as onomically la ge modulus (“Google-
numbe scale”)—co ec bu useless.
4. Leas likely: a conc e e, small ce i ica e ma e ializes.
We ully expec ed (3). In ha mindse , any hing small would be dismissed—jus as checking
he i s billions o numbe s doesn’ p o e Colla z. Ye he da a kep saying he same hing: a
closed ce i ica e a modulus 213. We ied o b eak i . Repea edly.
The one idea ha looks i ial only a e he ac
Two non-ob ious mo es u ned ou o be decisi e:
1.
Wo k on a ini e esidue g aph. Reduce o he odd esidues modulo 2
k
, wi h one
ou going edge pe class. No in ini e g aphs, no he oic combina o ics. Jus a ini e unc ional
dig aph.
2.
Acknowledge a small, hones blu budge .Mixing +and
×
is like swi ching lenses
( ime/ equency). The e’s an una oidable “ oll.” T ack i as a small nonnega i e numbe
ρ
.
Wi hou i , he key inequali y e uses o close.
Once you do hese wo hings, he es is s anda d max-plus olklo e: cycle means nonposi i e
⇐⇒
he e exis s a po en ial
ϕ⇐⇒
Bellman–Fo d elaxa ion s abilizes, all on he ini e esidue
g aph. Tha po en ial hen yields a one-line d i inequali y o he ac ual in ege s, by algeb aic
elescoping. No algo i hm e e ouches an in ini e g aph.
1
“Bu why 2k? Couldn’ you pick ano he modulus?”
You could, bu 2
k
is canonical he e. The alua ion
2
(3
n
+ 1) becomes esidue-de e mined o a
single excep ional class, gi ing a clean, iny able o edge weigh s and one conse a i e edge.
O he moduli mos ly c ea e mo e excep ions and a la ge ini e check wi hou changing he s o y.
Wha we hough we we e seeing (and wha s ayed when we
s ipped i down)
We kep igh ening he analysis. Pa adoxically, he mo e ca e ully we bounded e e y e m, he
smalle he obse ed slack became un il i ma ched he igh ba o ced by he unique mod-8
edge. In hindsigh his is exac ly wha should happen: p ecision emo es wish ul slack. A he
end o he imming, wha emained was he mos bo ing hing in he wo ld: a ini e able, a se
o linea (max-plus) inequali ies, and he wo d PASS.
Okay, wha would ac ually alsi y he p oo ?
He e is he sho , su gical lis . Any one o hese b eaks i :
1. A di ec ed esidue cycle wi h s ic ly posi i e a e age weigh (con adic s easibili y).
2. A single iola ed esidue inequali y in he ce i ica e able.
3.
A bad in e al bound (e.g. o
log
2,
log
3, o a iny geome ic ail) used in he e i ica ion.
4. An algeb aic slip in de i ing he d i inequali y om he esidue inequali ies.
I ems (1)–(2) a e ini e and ully machine-checkable. I ems (3)–(4) a e local and easy o isola e.
Why Isabelle/HOL?
Because he ce i ica e is ini e and he bounds a e a ional in e als, we could o malize he
check. We po ed he essen ials o Isabelle/HOL and asked i , in e ec : “Please ind a iola ion.”
I did no . We also egene a ed he 2
13
able ia independen code pa hs (C#; a sepa a e
sys em); e e y line ag eed; we su ounded each alue wi h a ce i ied in e al. This wasn’ abou
hea ics; i was abou elimina ing he las “ his eels oo clean” doub .
Why i ook so long (and why i s ill eels wei d)
Colla z has a cul u al g a i y: i ’s disc e e, so many expec a pu ely disc e e p oo . Bu addi ion
and mul iplica ion pull you in o di e en lenses ( ime s. equency); he unce ain y p inciple
be ween hem is eal in e e yday algeb a, and igno ing i jus hides he oll. The “blu ” isn’ a
ashion choice; i is he book-keeping needed o c oss ha lens bounda y hones ly. Once you
pay he oll, he max-plus ce i ica e alls in o he mos classical o ini e amewo ks.
Wha his is (and isn’ )
•
Is: a ini e, e i iable Lyapuno ce i ica e on he esidue g aph, yielding a mono one d i
o he in ege dynamics and an explici , small modulus 213.
•
Isn’ : a plea o accep con inuous me hods on ai h. The blu is condensed o a numbe
ρ
.
E e y hing ha ma e s happens in a ini e able and inequali ies you can ead and check.
2
How o engage as a skep ic (quick pa h)
1. Open he a i ac (ce i ica e CSV) and he e i ie .
2. Check he in e al bounds ( hey a e spelled ou ; ails a e ini e geome ic es ima es).
3.
Run he Bellman–Fo d s yle elaxa ion on he ini e esidue g aph and con i m s abiliza ion.
4. Inspec he wo s -slack edge; compa e o he igh mod-8 bound.
5.
I you like, simula e andom odd ajec o ies; wa ch he d i dec ease a e he small
h eshold.
I any o his ails, you ha e a eal bug. I all o i passes, you’ e ep oduced he p oo .
Whe e o go nex (and why we published a lo )
We delibe a ely pu e e y hing on Zenodo: a compac p oo , an Isabelle/HOL check, a s uden -
iendly wo kshee wi h
∼
30 asks, a plain-language essay on he blu idea, and code o ebuild
he ce i ica e. No o o e whelm; o emo e ambigui y and lowe he ba ie o independen
e i ica ion. We will also elease a sho ideo walking h ough he ini e esidue g aph and he
d i pic u e—no ba age o equa ions, jus he mo ing pa s.
Final hough . I you s ill eel ha “some hing his simple can’ be igh ,” y lipping he
ques ion: once you admi a iny, hones oll o swi ching lenses, wha , p ecisely, o bids a ini e
max–plus ce i ica e om exis ing a some modulus? In ou case i appea s a 2
13
. Tha is he
whole mechanism.
Why his no e exis s (and a gen le wa ning). Colla z a ac s a en ion no jus because i is
concise, bu because i si s on a seam be ween addi ion and mul iplica ion— wo ope a ions
we o en ea as i hey sha ed a single “na i e” lens. They do no . The blu budge is no
a s ylis ic lou ish; i is a small, explici p ice o c ossing ha seam. We a e no claiming
exclusi i y o app oach, no asking anyone o like his one on aes he ic g ounds. We a e saying:
i one insis s a p io i on a pu ely disc e e pa h and dismisses any a gumen ha makes he
lens-swi ch oll isible, one may keep missing he same s uc u al poin inde ini ely. Whe he o
no his p oo is o you as e, please do no le as e be he eason o igno e he deepe message:
p oblems ha mix +and
×
eliably su ace an “unce ain y” be ween lenses. Accoun ing o
i —e en in a minimal, ini e way—can u n an allegedly in angible phenomenon in o a checkable
ce i ica e. Ou hope is ha , e en o eade s who emain uncon inced by his ou e, he no e
nudges a en ion owa d ha seam, whe e many o he ha d p oblems seem o li e.
Acknowledgemen s and ep oducibili y
All e i ica ion a i ac s ( ables, bounds, sou ce, and o mal sc ip s) a e a chi ed wi h DOIs and
include command-line ins uc ions o ep oduce a PASS. The o mal componen in Isabelle/HOL
(lemmas, ce i ica es, and checks) is kep as close o he p ose no a ion as he p oo assis an
allows.
Bibliog aphy
•
Pe išić, A. (2025). So -Addi ion and So -Mul iplica ion and he Channel–Swi ch E o .
Zenodo. doi:10.5281/zenodo.17218980.
3
•
Pe išić, A. (2025). Colla z Wi hou he Mys ique o Addi ion and Mul iplica ion. Zenodo.
doi:10.5281/zenodo.17220058.
•Pe išić, A. (2025). Colla z P oo Ve i ica ion: Isabelle/HOL. Zenodo.
doi:10.5281/zenodo.17308278.
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