The b-Closu e o Ma hema ics
A Blu –Fi s P og am o Closing Theo ies
Aleksanda Pe iˇsi´c
No embe 2025
Abs ac
Blu means eplacing sha p, b i le objec s (ha d cu o s, poin e alua ions, oscilla o y
ke nels, exac selec o s) by posi i e, no malized app oxima e iden i ies o by conse a i e
closings ha espec wha we can and canno access. I a gue ha blu is he missing o ganizing
p inciple ha le s us close ma hema ics in a p agma ic bu igo ous sense: each heo y can
be comple ed up o an in o ma ion budge , and s a emen s a e s abilized unde blu be o e
we a emp o unsmoo h hem. This yields a p og am— he
b
–closu e o ma hema ics— ha
(i) s a i ies heo ies by hei in o ma ion con en and e godici y, (ii) supplies a plan o
close wha is closable, and (iii) spli s he una ainable in o manageable descendan s. The
same logic uni ies physics and ma h: physical laws s a blu ed; ma hema ics can adop
he same discipline wi hou loss o igo . Along he way I ou line an “in o ma ional speed
limi ” analogy (a no– ee–lunch bound ha includes he p ime/explici – o mula wo ld) and
a p ac ical esea ch/lea ning wo k low whe e us a ion d ops because p og ess is measu ed
a he blu ed, s able le el.
Why blu , and wha does i buy us?
Blu is he ac o eplacing a sha p p obe by a posi i e, no malized ke nel o a conse a i e
closing unde which he s a emen o in e es s abilizes. Con olu ion wi h a molli ie , Abel
ins ead o Ces`a o summa ion, Poisson/Fej´e windows, o passing o an almos –e e ywhe e
selec o a e classical ins ances. In each case, a ha d heo em educes o a one–line because
posi i i y and domina ed con e gence do he hea y li ing; he eal a lies in unsmoo hing back
o he sha p o m when needed.
De ini ion (Blu ope a o s and
b
–s abili y).Fix a amily (
Kσ
)
σ>0
o posi i e, no malized
ke nels on a space
X
(e.g.
Kσ≥
0,
RKσ
= 1,
Kσ→δ0
). Fo
:
X→R
w i e (
Bσ
)(
x
) =
R
(
x−u
)
Kσ
(
u
)
du
. A p ope y P(
) is
b
–s able a
x
i he e exis s a scale egime
σ↓
0 such
ha P(
Bσ
) holds uni o mly o all su icien ly small
σ
; we hen w i e P(
) holds in
b
–closu e a
x.
In logic, we mi o his wi h conse a i e closings: we may wo k inside an en iched bu
conse a i e ex ension so long as he possibili y o comple e he a gumen su i es all such
closings; hen a a ge ed exis ence claim holds al eady in he base heo y (see §).
The b–closu e p inciple
P inciple (b–closu e).To close a heo y Ta an in o ma ion budge I:
C1.
Calib a e a blu . Choose admissible ke nels o closings such ha key claims a e
b
–s able.
C2.
Fac o signal s. en elope. Sepa a e disc e e/a omic ea u es om he smoo h en elope
wi hin he blu ed pic u e.
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C3.
Quan i y in o ma ion. S a e explici ly wha da a leng h/p ecision is needed o a
conclusion a he blu ed le el.
C4.
Unsmoo h judiciously. Only emo e he blu when he in o ma ion budge ac ually
suppo s i ; o he wise, keep he blu ed s a emen as he canonical heo em.
Unde
b
–closu e, “exis ence” means exis ence wi h s able access: you can e i y i in he
blu ed egime and you know wha esou ces a e equi ed o pass o sha pe o ms.
A axonomy by g ow h and e godici y
Many heo ies s a i y na u ally by how in o ma ion agg ega es and by he p esence o e -
godic/mixing elemen s. A ough, delibe a ely coa se axonomy:
•Linea : esponses add and blu dis ibu es pe ec ly (Fou ie /Abel ools i ialize).
•Polynomial: in e ac ions emain locally ame; blu s ill egula izes wi h bounded leakage.
•Exponen ial: small pa ame e shi s ampli y; blu mus be bandwid h–awa e.
•
Supe –exponen ial/chao ic: s ong sensi i i y; only s a is ical/e godic summa ies a e
ealis ically b–s able.
Each bucke occu s wi h/wi hou e godic elemen s. An e godic elemen he e simply means an
in insic a e aging mechanism ha makes blu ed p obes ep esen a i e o he whole. The
p ac ical impo is in o ma ional: in some domains (e.g. p imes) no amoun o cle e ness
subs i u es o da a o size Nwi h p ecision exceeding a h eshold—no because we lack ideas,
bu because he objec does no e eal ce ain s a is ics below ha budge . In ha case,
b
–closu e
eco ds he bes s able s a emen s now and schedules he nex a ge s when da a budge s g ow.
P og am: closing wha can be closed
P oposal (A b–closu e wo k low).Fo any a ge heo y/p oblem:
P1.
Decla e he p obe. Fix admissible blu ke nels (Gaussian, Paley–Wiene , Slepian/DPSS)
and/o conse a i e closings.
P2.
S a e he blu ed heo em. P o e he
b
–s able co e ha posi i i y and app oxima e
iden i y gua an ee; quan i y leakage.
P3.
Sepa a e channels. P ojec a omic (disc e e) s. smoo h sec o s a e blu ; ea each
wi h i s na u al ools.
P4.
Budge he unsmoo hing. A ach explici da a/p ecision h esholds (“how much is
enough”).
P5.
I una ainable, spli . Fac o he objec in o sub– heo ies whose
b
–closu es a e sepa a ely
a ainable now.
P6.
Publish bo h he blu ed heo em (s able, now) and he oadmap o i s sha p o m
( u u e).
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Physics and ma hema ics: he same discipline a di e en ole -
ances
E e y physical law is bo n blu ed: New on linea izes Eins ein; Eins ein blu s quan um ields a
classical scales; Ohm and Amp`e e a e i s –o de en elopes o iche elec odynamics; Tesla’s
AC enginee ing exploi s lossless limi s as blu ed ideals. The p ac ical di e ence is ha physics
codi ies i s blu and ole ances; ma hema ics o en p e ends o ha e none. The
b
–closu e s ance
is: keep exac ness whe e i ma e s (logical consequence, de inabili y), bu adop blu as he
p ima y p o ing g ound. Rigo is no diminished: s a emen s a e p o ed a he blu ed le el and
only hen selec i ely sha pened.
A p ime case s udy: explici o mula, bands, and an in o ma ional
speed limi
In he Mellin
u
=
ln | |
pic u e, p ime powe s li e on he p ime band Ω =
{klog p}
and he
explici o mula spli s in o an a omic comb on Ω, a smoo h a chimedean en elope, and endpoin
esidues. Wi h band–limi ed o Gaussian blu , he wo sec o s decouple cleanly; de ec o s
buil om phase–scanned es s ead o disc e e s. smoo h mass wi h con ollable leakage.
Ope a ionally: any o –line ze o would injec an exponen ially g owing signa u e in o he p ime
channel; unde he Riemann hypo hesis (RH), g ow h s ays polynomial. This is an in o ma ional
law: you canno ex ac mo e om he p imes han he calib a ed blu and da a budge allow. In
his sense he e’s a “speed limi ”: no me hod can conju e signal ha is no p esen a he blu ed,
budge ed scale. (The poin is me hodological; he de ails li e in he blu /p ime machine y.)
Ta ge ed choice wi hou selec o s: an axiom o exis ence unde
blu
Blu has a logical a a a : some imes we allow in e nal ools (like agmen s o choice) du ing a
conse a i e closing, and a he end all selec o s anish om he inal s a emen . The Axiom o
Blu ed Choice (ABC) packages his: i a e e e y conse a i e closing i emains possible o
comple e he a gumen using a ixed choice p inciple and ob ain a choice unc ion o a coded
amily, hen al eady in he base heo y he ba e exis ence claim holds—no selec o need be
exhibi ed. This is exac ly he p ac ice o “use i inside, e ase i a he end”, made o mal and
con ined o he ele an exis en ial.
Resea ch and lea ning wi hou ch onic us a ion
A blu – i s wo k low makes daily p og ess isible:
•Igno e wha you canno know ye . P o e he blu ed heo em you can p o e now.
•Collec wha you do know. Calib a e de ec o s and budge s; eco d s able gains.
•Read ahead. Explo e he sha p pic u e o plan he unsmoo hing s eps.
•
Combine. Each pass igh ens he blu o expands da a un il he sha p o m is wi hin each.
F us a ion d ops because he uni o p og ess is he s able blu ed claim, no he all–o –no hing
sha p inish.
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Wha b–closu e p omises (and wha i does no )
This p og am does no magically sol e e e y hing. I does p omise:
1. a uni o m language o in o ma ion budge s and de ec o s,
2. immedia e, publishable blu ed heo ems (o en d ama ically simple ),
3. anspa en oadmaps om blu ed o sha p,
4. p incipled spli s when an objec is oo s i o close a once.
Ambi ious imelines (e.g., “4–5 yea s o a cla i ied landscape”) a e bes ea ed as p og amma ic:
he
b
–closu e makes he e ain legible and he a ge s conc e e; he ac ual pace depends on
budge s (da a, compu a ion, and p oo s).
Appendix A: Minimal blu calculus
De ini ion (Admissible blu ).A ke nel
Kσ
is admissible i
Kσ≥
0,
RKσ
= 1,
Kσ→δ0
, and
ei he (i)
Kσ
is band–limi ed in Fou ie wi h bandwid h Λ(
σ
)
↑ ∞
, o (ii)
Kσ
is Schwa z wi h
apidly decaying ails.
Rema k (Blu
⇒
classical limi s unde mild egula i y).I
has a classical limi
L
a
x0
,
hen
Bσ
(
x0
)
→L
. Con e sely, i
Bσ
(
x0
)
→L
and
has anishing local oscilla ion a
x0
(bounded a ia ion/Lipschi z/mono one su ices), hen he classical limi is
L
. This jus i ies
ea ing b–s able laws as he canonical o m un il unsmoo hing is unded.
Appendix B: A disciplined spli o any heo y
Gi en a a ge T:
1. decla e admissible blu /closings and he obse ables,
2. p o e he blu ed heo em wi h explici leakage,
3. p ojec o disc e e s. smoo h channels,
4. a ach budge s o he unsmoo hing,
5. i necessa y, spli T=T1⊕ · · · ⊕ Tmand i e a e.
Final wo d. The
b
–closu e o ma hema ics is no a es ic ion bu an emancipa ion: i legi imizes
wha we al eady do in o mally (smoo h, linea ize, pass o a.e., igno e null/meag e s uc u e),
makes i i s –class, and u ns i in o a common p og am ac oss numbe heo y, dynamics,
p obabili y, analysis, and beyond.
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