Abs ac
Wi hin abs ac amewo k o de B anges–K e˘ın canonical sys ems and mul i-channel
sca e ing, his pape es ablishes uni ied sys em in pu e heo e ical language o “ope a o –
measu e– unc ion”, independen o expe imen al na a i e: welding “phase de i a i e–
ela i e s a e densi y–Wigne –Smi h g oup delay ace” as uni e sal measu e coo di-
na e o same pa en scale, cha ac e izing ini e esou ces and obse a ional choices ia
windowed eadou s o Toepli z/Be ezin comp ession. On e e sible obse a ional ans-
o ma ion g oup gene a ed by base au omo phisms, phase gauge and e e sible il e -
ing, p o es in a iance o ime scale wi hin blocks cons uc ed om windowed delay
in eg al; gi es non-asymp o ic e o closu e and s able p inciple o “singula i y non-
inc easing/pole = dominan scale” unde “ ini e-o de ” Eule –Maclau in and Poisson
discipline. Thus, in EBOC s a ic block uni e se eplaces ex e nal elapsed ime wi h
in insic Tin ; ob ains uni ied me ic unde RCA e e sible compu a ion’s isomo phic
eno maliza ion. Scale iden i y o his sys em holds almos e e ywhe e on absolu ely
con inuous spec um:
φ′(E)
π=ρ el(E) = 1
2π Q(E),
whe e S(E)∈UN(E)sca e ing ma ix, Q(E) := −i S(E)†∂ES(E) Wigne –Smi h
delay ma ix, φ(E) := 1
2A g de S(E), ρ el ela i e s a e densi y ela i e o e e ence
channel/ ee Hamil onian. Windowed eadou de ined by Toepli z/Be ezin comp ession
map Cwand i s co a ian symbol Kw(E):
Tw(E) := 1
2π Kw(E)Q(E), Tin (I) := ZI
Tw(E) dE,
emaining in a ian unde e e sible obse a ional ans o ma ions, becoming uni-
e sal ime scale wi hin EBOC blocks. Abo e ini y scale connec ed by Wigne –Smi h
delay ma ix and Bi man–K e˘ın spec al shi –de e minan o mula, p o iding uni ied
coo dina e om phase o densi y, om sca e ing o measu e, and s able benchma k o
a ia ional/op imiza ion.
1 No a ion & Axioms / Con en ions
1. Obse a ional iple (H, w, S): HHilbe space; S(E)∈UN(E)sca e ing ma-
ix a ene gy scale E(a.e. on absolu ely con inuous spec um); wwindow, inducing
analysis–syn hesis map Πwand Toepli z/Be ezin comp ession map Cw[X] := ΠwXΠ†
w;
co a ian symbol Kw(E) := Πw(E)†Πw(E)≥0, de e mining eadou unc ional. Win-
dow amilies ake bandlimi ed o exponen ial decay classes, sa is ying egula i y in
ep oducing-ke nel con ex .
2. Window amily no maliza ion (Pa se al/ igh ame, componen -wise): In di-
ec in eg al decomposi ion o absolu ely con inuous spec um and channel ibe s, choose
window amily such ha wi hin each h eshold egula componen Jha e Kw(E)≡
NJ(a.e. E∈J). This no maliza ion compa ible wi h ep oducing-ke nel egula i y,
ensu ing windowed eadou phase gauge e ms only p oduce endpoin cons an s.
3. Th eshold se and egula domains: Deno e h eshold se T:= {E:N(E+)=
N(E−)}. Fo all in eg als in Theo em 4.2, de aul I∩ T =∅, o equi alen ly i s
subdi ide Ialong T hen componen wise in eg a e and agg ega e.
1
4. Phase b anch and di e en iabili y: Fix con inuous b anch o A g de S(E) on each
h eshold egula componen J, hus φ′(E) exis s a.e. on J; ac oss h esholds and
disc e e spec um ea in dis ibu ional sense (Le inson- ype ansi ions).
5. Scale iden i y ca d: Axioma ize
φ′(E)/π =ρ el(E) = (2π)−1 Q(E),Q:= −i S†∂ES.
Equi alence be ween phase de i a i e and Q om Wigne –Smi h de ini ion and de e -
minan di e en ial iden i y; equi alence wi h ρ el gi en by Bi man–K e˘ın o mula and
spec al shi unc ion di e en ial connec ion.
6. Fini e-o de EM+Poisson ca d: Fo sum–in eg al ans o ma ion and ene gy dis-
c e iza ion, uni o mly use ini e-o de Eule –Maclau in and Poisson summa ion o
non-asymp o ic e o closu e; explici bound cons an s depend on ini e no ms o win-
dow and symbol; singula i y non-inc easing and “pole = dominan scale”.
7. Language and objec s: Window/ eadou uni o mly ea ed as “ope a o –measu e–
linea unc ional” objec s; a oid expe imen al p ocedu e na a i e. Toepli z/Be ezin
comp ession and Be ezin ans o m used o map unc ion symbols in ene gy–phase an-
aly ic pla o m (such as de B anges space, Paley–Wiene /Mellin models) o ope a o s.
8. No a ion: de ! deno es egula ized (F edholm) de e minan ; ace; Pac absolu ely
con inuous spec al p ojec ion; “a.e.” all e e o almos e e ywhe e on absolu ely
con inuous spec um.
2 Sca e ing Phase, G oup Delay and Spec al Shi :
T ini y Coo dina e
Le Hand e e ence H0sel -adjoin , sa is ying usual aceable pe u ba ion condi ions mak-
ing S(E) exis and uni a y. De ine Q(E) := −i S(E)†∂ES(E) and φ(E) := 1
2A g de S(E).
Wigne –Smi h gi es He mi ici y o Qand i s ela ion wi h ene gy de i a i e o S; ace sa is-
ies Q(E) = ∂EA g de S(E)=2φ′(E). On o he hand, Bi man–K e˘ın o mula de S(E) =
exp(−2πi ξ(E)) connec s sca e ing de e minan wi h spec al shi unc ion ξ, hus ξ′(E) =
−1
2π Q(E) = −φ′(E)/π. Taking ρ el(E) := −ξ′(E) yields scale iden i y.
Co olla y 2.1. On absolu ely con inuous spec um a.e., measu es induced by h ee objec s
sa is y dµφ=dµρ=dµQ, and dµQ(E) = (2π)−1 Q(E) dE. This p o ides pa en scale o
subsequen windowed eadou and ans o ma ion consis ency.
3 Windowed Readou and Toepli z/Be ezin Comp es-
sion
Take ep oducing-ke nel space H(such as de B anges, Paley–Wiene o Mellin–Ha dy) as
ene gy–phase analy ic pla o m. Window winduces analysis–syn hesis map Πw. De ine com-
p ession map
Cw[X] := ΠwXΠ†
w,
and i s co a ian symbol
Kw(E) := Πw(E)†Πw(E)≥0.
2
De ini ion 3.1 (Channel Fibe Comp ession).Unde di ec in eg al decomposi ion o abso-
lu ely con inuous spec um Hac ≃R⊕CN(E)dE, analysis map Πw(E) : CN(E)→CN(E)gi es
co a ian symbol
Kw(E) := Πw(E)†Πw(E)∈CN(E)×N(E), Kw(E)≥0.
Thus windowed densi y and eadou Tw(E) := (2π)−1 Kw(E)Q(E),Tin (I) := RITw(E) dE,
unde Pa se al no maliza ion, o each h eshold egula componen J, sa is y Kw(E)≡NJ
(a.e. E∈J).
Fo ene gy-local ma ix symbol A(E), de ine windowed ace ⟨A⟩w:= R Kw(E)A(E)dE.
Windowed densi y o g oup delay de ined as Tw(E) := (2π)−1 Kw(E)Q(E), hus Tin (I) :=
RITw(E) dE. Toepli z/Be ezin sys em ensu es posi i i y and egula limi s o Kw, and con-
sis ency on symbol algeb a.
4 Re e sible Obse a ional Equi alence and Gauge In-
a ian s
De ini ion 4.1 (Re e sible Obse a ional T ans o ma ion).Re e sible obse a ional ans-
o ma ions gene a ed by:
(i) Au omo phism Uo H( ixing ene gy scale);
(ii) Phase gauge:S7→ eiθ(E)S, whe e θ∈W1,1(I)∩C0(I) and o each componen
in e al endpoin θ(Ej,±) = 0;
(iii) Re e sible window eno maliza ion (ene gy-independen ): On each h eshold
egula componen J ake ixed channel basis U∈U(NJ). Window eno maliza ion
w7→ ewinduces
Πew= ΠwU†, Kew(E) = UKw(E)U†.
Theo em 4.2 (Gauge In a iance o Windowed Delay–No malized Ve sion).Unde con-
di ions o no maliza ion and De ini ion, o any h eshold egula ini e union in e al
I=FJ
j=1[Ej,−, Ej,+]⊂R(i.e., I∩ T =∅), quan i y
Tin (I) := ZI
1
2π Kw(E)Q(E)dE
in a ian unde e e sible obse a ional ans o ma ions (au omo phism, phase gauge, e-
e sible window eno maliza ion).
P oo . Use ace and simila i y in a iance ge (UKwU†·UQU†) = (KwQ). Phase gauge
con ibu es e m θ′(E) Kw(E); by componen -wise no maliza ion Kw(E)≡Nj(E∈
[Ej,−, Ej,+]) and endpoin condi ion θ(Ej,±) = 0, ge RIθ′(E) Kw(E) dE=PjNj[θ]Ej,+
Ej,−= 0,
gauge e m anishes, in a iance holds.
Co olla y 4.3 (Uni e sal Time Scale).Tin independen o obse a ional ep esen a ion, con-
s i u es in insic ime scale in EBOC s a ic blocks.
3
5 Uni e sal Measu e Coo dina e and T ans o ma ion
Consis ency
P oposi ion 5.1 (a.c. Th ee-Measu e Consis ency and Dis ibu ional Ex ension).Le S(E)
sa is y usual aceable pe u ba ion and limi ing abso p ion condi ions. Then on absolu ely
con inuous spec um a.e. ha e
dµac
φ=dµac
ρ=dµac
Q,
whe e dµφ(E) = φ′(E)
πdE,dµρ(E) = ρ el(E) dE,dµQ(E) = 1
2π Q(E) dE. I inco po a -
ing disc e e spec um/ h esholds in o ull spec um, h ee consis en in dis ibu ional sense:
dµρcon ains δ-masses a disc e e spec um, φexhibi s phase jumps (Le inson- ype), Q akes
bounda y alues. The e o e any windowed eadou compa able and ans o mable unde same
coo dina e, ans o ma ion e o bounded by uni ied cons an s o Sec ion 5.
6 S able E o Theo y: Fini e-O de Eule –Maclau in
and Poisson
Le wbelong o bandlimi ed class o exponen ial class, a(E) su icien ly smoo h ene gy sym-
bol, {En}ene gy pa i ion (gene a ed by window o spec al ube). Fo ene gy domain
I=FJ
j=1[Ej,−, Ej,+], exis s m∈Nand cons an s Cm, C′
m(depending only on window amily
and ini e-o de de i a i e semino ms) such ha
X
n
a(En)−ZI
a(E) dE−
m
X
k=1
B2k
(2k)!
J
X
j=1 a(2k−1)(E)Ej,+
Ej,−≤CmRm(a, w),
X
k=0 ba(2πk)bw(2πk)≤C′
mPm(a, w).
whe e B2kBe noulli numbe s, Rm,Pme o unc ionals composed o ini e semino ms.
Fo a= Q, φ′, ρ el apply same cons an chain; ob ain uni ied e o budge on windowed
eadou s o h ee objec s; and singula i y non-inc easing and “pole = dominan scale” hold.
7 EBOC In insic Time and RCA Re e sible Compu-
a ion Isomo phic Reno maliza ion
De ini ion 7.1 (In insic Time Scale).Fo ene gy domain Ide ine
Tin (I) = ZI
1
2π Kw(E)Q(E)dE,
as ela ional p og ession ime scale in EBOC s a ic blocks; in a ian unde e e sible
obse a ional ans o ma ions.
Theo em 7.2 (RCA Isomo phic Reno malizabili y).Embed s ep dep h o e e sible cellula
au oma on Uin o Tin me ic: i wo obse a ional iples (Hi, wi, Si) e e sibly equi alen ,
hen RCA “dep h” co esponding o hei bounda y–channel coupling measu ed by same ime
scale. P oo depends on in a iance o Theo em 4.2 and uni ied coo dina e o scale iden i y,
ob aining isomo phic eno maliza ion unde di e en bases/encodings.
4
8 Discussion and Ou look
This wo k es ablishes:
1. T ini y scale uni ica ion φ′/π =ρ el = (2π)−1 Q ia Wigne –Smi h delay and Bi man–
K e˘ın o mula
2. Windowed eadou amewo k ia Toepli z/Be ezin comp ession wi h co a ian symbol
Kw(E)
3. Gauge in a iance o in insic ime scale Tin unde e e sible obse a ional ans o ma-
ions
4. Non-asymp o ic e o closu e ia ini e-o de Eule –Maclau in and Poisson summa ion
5. Connec ion o EBOC s a ic block uni e se and RCA e e sible compu a ion
6. F ame- heo e ic ounda ions ia Wexle –Raz, Balian–Low, Landau densi y
7. de B anges–K e˘ın analy ic pla o m and He glo z–Ne anlinna s uc u e
Key o mulas:
Scale iden i y: φ′/π =ρ el = (2π)−1 Q
Windowed ime: Tin (I) = RI(2π)−1 (KwQ) dE
In a iance: Tin unchanged unde (U, θ, w) ans o ma ions
Fu u e di ec ions:
Nume ical implemen a ion and benchma king
Ex ension o open quan um sys ems and non-He mi ian sca e ing
Connec ions o quan um in o ma ion and complexi y heo y
Applica ions o quan um g a i y and eme gen space ime
5
