Scale-Gauged Cosmological Obse a ion Theo y
— Rigo ous Equi alence o “Expansion ≡Resolu ion Enhancemen ”,
Axioma ic Readou , Rela i is ic Re o mula ion, In o ma ion Bounda y,
and Tu ing Seman ics
Anonymous Au ho
Ve sion: 1.15
Abs ac
Abs ac : We es ablish a cosmological obse a ion heo y cen e ed on he duali y be-
ween he scale gauge (deno ed a( )) and he in e nal obse a ional me ic (c-lock, deno ed
R( )). Se ing a( ) = R( )−1and κ( ) := ˙a/a =−˙
R/R, cosmological edshi and ime
dila ion uni y as Mellin dila ion on he ene gy axis: 1 + z=a( 0)/a( e) = R( e)/R( 0),
ν0=νe/(1 + z), ∆ 0= (1 + z)∆ e. All eadou s a e aligned on he “mo he scale”:
ρ(E) = −ξ′(E) = 1
2π Q(E) = 1
2πi ∂Elog de S(E) = φ′(E)
2π,Q:= −i S†S′,
whe e ρis he ela i e densi y o s a es, Qis he Wigne –Smi h g oup delay ma ix, and
φis he o al sca e ing phase. The mo he scale is equi alen ly cha ac e ized by he
Bi man–K e˘ın o mula and he g oup delay de ini ion, p o iding c oss-de ice compa able
calib a ion uni y ( ixed uni s; dimension E−1). Readou e o s obey “Nyquis –Poisson–
Eule –Maclau in (NPE) ini e-o de closu e”: Nyquis cu o elimina es aliasing, Poisson
summa ion b idges disc e e–con inuous, and ini e-o de Eule –Maclau in (EM) encapsu-
la es endpoin e o s wi h Be noulli laye s and ail bounds. Unde Landau densi y h eshold
and Wexle –Raz dual ame condi ions, window sh inking ↓main ains non-inc easing
singula i y; whe eas he mono onici y and scaling o Fishe in o ma ion wi h espec o
depend on noise model and no maliza ion choice, and no uni e sal mono onici y
o −2lowe bound exis s in gene al. The linea s abilize p ese ing “ligh cone + mo he
scale” uniquely co esponds o he Lo en z g oup; in he FRW backg ound, he uni ied
equency shi law 1 + z= ((kµuµ)e)/((kµuµ)o) na u ally yields E he ing on’s dis ance du-
ali y DL= (1 + z)2DAand Tolman’s su ace b igh ness (1 + z)−4. Via e e sible causal
au oma on (RCA/QCA) seman ics, we p o ide a uni ied o mula ion o “ligh ”, “ edshi ”,
“sha ed mas e equency”, and “c-limi ed alloca ion o a en ion/ac ion esou ces”. This
pape dis inguishes “gauge” om “ ue dynamics” and p oposes alsi iabili y c i e ia and an
enginee ing “ esolu ion alloca ion ma ix” app oach.
1 No a ion, Axioms, and Con en ions
1.1 Obse a ional Objec s and Sca e ing Geome y
1. Backg ound Hilbe space H; ene gy pa ame e E∈R.
2. Sca e ing ma ix S(E) and Wigne –Smi h ma ix Q(E) = −i S†(E)∂ES(E).
3. To al sca e ing phase φ(E) = a g de S(E).
4. T ini y mo he scale:
ρ(E) := −ξ′(E) = 1
2π Q(E) = 1
2πi∂Elog de S(E) = φ′(E)
2π,
1
whe e he equi alence ollows om he Bi man–K e˘ın o mula de S(E) = e−2πi ξ(E)(wi h
ξ(E) he spec al shi unc ion) and he g oup delay de ini ion; hus ρ(E) = −ξ′(E)
holds. This ela ion holds in gene alized sca e ing and geome ic se ings [1].
1.2 Scale Gauge / c-lock
The ex e nal scale ac o a( ) and he in e nal obse a ional me ic R( ) sa is y
a( ) = R( )−1, κ( ) := ˙a
a=−˙
R
R.
Le he in e nal mas e equency be clk( ) := c
ℓclk( )=c
R( )ℓ∗, whe e ℓ∗is a ixed mo he scale
leng h cons an and R( ) is he dimensionless in e nal me ic sa is ying a( ) = R( )−1. Any
“ esolu ion enhancemen ” ope a ion e e s o ↓o sampling densi y dens(Λ) ↑.
1.3 NPE Fini e-O de Closu e (Non-Asymp o ic)
Nyquis : Fo band-limi ed a ge s, aliasing is ze o when sampling a e exceeds wice he
bandwid h; o non-band-limi ed cases, aliasing e ms a e explici ly accoun ed o [2].
Poisson: Disc e e–con inuous b idging ia Poisson summa ion, allowing la ice sums o
swi ch o equency-domain comb spec a [3].
Eule –Maclau in ( ini e-o de ): Endpoin laye s and ail bounds gi en by Be noulli
polynomials, wi h unca ion o de p ixed and e o bounds audi able [4].
E o no a ion de ini ion: Deno e by εalias( )≥0 he L1-uppe bound o aliasing e ms
in oduced ia Poisson summa ion; when he Nyquis condi ion is sa is ied, εalias( ) = 0. Deno e
by εEM(δ; ;p)≥0 he “endpoin laye + ail” budge a e unca ing he Eule –Maclau in
o mula a ixed o de p(δis he sampling s ep/equi alen g id spacing). The e exis s a cons an
C2p( ) such ha
εEM(δ; ;p)≤C2p( )δ2p.
In gene al, no uni e sal powe -law ela ion wi h is claimed; i u he egula i y such as
w∈W2p,1,h∗ρ∈L1
loc is gi en, one may de i e C2p( )≲ −2p, whence εEM(δ; ;p) = O(δ2p −2p)
as a model-dependen conclusion. This o de bound equi es assuming h∗ρis piecewise C2p
on he hickening K↑(whe e K↑:= K+ supp(w ∗h)) o he wo king compac domain Kwi h
bounded de i a i es; o unde weake BV assump ions, all jump con ibu ions a e inco po a ed
in o he Be noulli endpoin laye s be o e es ima ion (yielding only a BV - e sion bound, no
equi alen o piecewise C2p). Beyond his egula i y egime, his pape does no claim such
o de .
1.4 F ame and Densi y Th eshold
We adop he Gabo /Weyl–Heisenbe g amewo k: o window wand la ice Λ, we equi e
Landau necessa y densi y and Wexle –Raz duali y o ensu e s able in e ible econs uc ion [5].
2 Expansion ≡Resolu ion Enhancemen : Mellin Dila ion and
Uni ied F equency Shi
De ini ion 2.1 (Scale Gauge).A choice (a, R)sa is ying a( ) = R( )−1is called a scale gauge.
Unde his gauge, ex e nal “expansion” and in e nal “ esolu ion enhancemen ” a e igo ously
equi alen o Mellin dila ion on he ene gy axis.
2
Theo em 2.2 (Redshi –Dila ion Equi alence).Fo he same pho on obse ed a emission e
and obse a ion 0, we ha e
1 + z=a( 0)
a( e)=R( e)
R( 0), ν0=νe
1 + z,∆ 0= (1 + z)∆ e.
P oo ske ch. In he FRW me ic, he equency is ω=−kµuµ, hence 1+z= ((kµuµ)e)/((kµuµ)o);
pa allel anspo o kµalong null geodesics yields ω∝a−1. Subs i u ing a=R−1comple es
he p oo [6].
P oposi ion 2.3 (E he ing on Duali y and Tolman Decay).I pho on numbe is conse ed,
geome y is desc ibed by me ic g a i y, and ligh ollows unique null geodesics, hen
DL= (1 + z)2DA, Iobs =Iem
(1 + z)4.
This conclusion is independen o he choice o scale gauge and is a geome ic–coun ing in a ian
[7].
3 Rela i is ic Windowed Re o mula ion: S abilize o Ligh Cone
+ Mo he Scale
Theo em 3.1 (Lo en z G oup = S abilize ).The g oup o linea ans o ma ions p ese ing
he Minkowski ligh cone s uc u e and he mo he scale is isomo phic o SO+(1,3).
A gumen . A e ixing he o igin ( emo ing ansla ional eedom), he g oup o linea au-
omo phisms p ese ing causal o de is gene a ed by R+×SO+(1,3); equi ing mo he scale
in a iance (excluding global dila ion) lea es only SO+(1,3). This is consis en wi h Alexand o –
Zeeman- ype heo ems [8].
P oposi ion 3.2 (GR Local Co a ian iza ion and Uni ied F equency Shi ).By locally la -
ening “ligh cone + mo he scale” a each poin o he mani old, he uni ied equency shi
law
1 + z=(kµuµ)e
(kµuµ)o
,
is compa ible wi h he s a iona y-phase condi ion o geodesic equa ions and consis en wi h s an-
da d SR/GR kinema ics [9].
4 Essence o Resolu ion Enhancemen : In o ma ion Geome y
and Singula i y Conse a ion
Le he no malized window w (x) := 1
w(x/ ), whe e w≥0, w∈W1,1(R), RRw= 1
(op ionally: Rx w(x)dx = 0), con olu ion ke nel h, and obse able
g (E)=(w ∗h∗ρ)(E).
P oposi ion 4.1 (Scale Bound and Con e gence o G adien Response).Le w≥0,w∈
W1,1(R),Rw= 1, and h∗ρ∈L1
loc (o BV ), wi h g =w ∗h∗ρ. Fo compac domain K,
|∂Eg |L1(K)≤|w′|L1
|h∗ρ|L1(K↑).
Con e gence by cases:
3
(i) I h∗ρ∈W1,1(K↑)and w≥0,Rw= 1, hen
lim
↓0|∂Eg −(h∗ρ)′|L1(K)= 0,|∂Eg |L1(K)≤ |(h∗ρ)′|L1(K↑).
(ii) I h∗ρ∈BV (K↑)(no necessa ily in W1,1), hen
g −−→
↓0h∗ρin L1(K),|∂Eg |L1(K)≤TV(h∗ρ;K↑),
and ∂Eg
∗
⇀ D(h∗ρ)in he weak∗sense in measu e space. We do no claim con e gence o
|∂Eg −(h∗ρ)′|L1in his case.
NPE es ima o e sion ( o disc e e implemen a ion bg ):
|∂Ebg |L1(K)≤|w′|L1
|h∗ρ|L1(K↑)+εalias( ) + εEM(δ; ;p).
I he Nyquis condi ion is sa is ied, εalias( )=0, lea ing only he EM endpoin – ail budge .
The abo e shows: educing imp o es edge app oxima ion, bu does no p oduce a uni-
e sal 1/ lowe bound g ow h. He e K↑:= K+ supp(w ∗h) deno es he hickening o he
compac domain Kby he e ec i e suppo o he con olu ion ke nel [4].
P oposi ion 4.2 (Model Dependence o Fishe In o ma ion).Unde he p emise ha Nyquis
and NPE e o budge s (εalias,εEM) a e audi able, he mono onici y and scaling o I (θ)wi h
espec o depend on noise model and no maliza ion choice; in gene al, no uni e -
sal −2lowe bound o mono onici y conclusion exis s. Once noise s a is ics (e.g.,
AWGN/Poisson) and window no maliza ion (e.g., Rw= 1 o |w |2 ixed) a e speci ied, one
may de i e he co esponding -scaling and compa ison esul s [2].
Theo em 4.3 (Non-Inc easing Singula i y).Legi ima e window swi ching (w7→ w wi h ixed-
o de EM budge ) co esponds o smoo hing ha does no in oduce new singula i ies o h∗ρ;
hus unde alias con ol and audi able EM e o , esolu ion enhancemen does no “manu ac u e
spu ious peaks”. The loca ion and o de o singula i ies may be a ec ed by smoo hing; his
pape makes no in a iance claims [3].
5 S able Recons uc ion and F ame Th eshold
Theo em 5.1 (Landau Necessa y Densi y).S able sampling o band-limi ed Paley–Wiene -
ype spaces equi es lowe Beu ling densi y no less han he bandwid h olume cons an ; i
insu icien , econs uc ion condi ion numbe explodes [5].
Theo em 5.2 (Wexle –Raz Duali y and Tigh F ames).Fo Gabo sys ems, he Wexle –Raz
bio hogonali y ela ion cha ac e izes he o hogonali y condi ion o dual windows; he e exis s
a pa ame e egime whe e igh ames hold, making econs uc ion obus . Mul i-window u-
sion educes es ima ion a iance om σ2 o app oxima ely σ2/K unde s a is ical independence
app oxima ion [10].
Rema k 5.3 (Balian–Low Ba ie ).O hono mal bases a c i ical densi y canno simul ane-
ously achie e good ime- equency localiza ion (Balian–Low), sugges ing he need o edundan
ames a he han c i ical bases [11].
4
6 Gauge s. T ue Dynamics: Falsi iabili y Finge p in s
De ine cosmological s a e inge p in s: decele a ion q:= −¨aa/˙a2, je k j:= d3a/d 3
aH3. De ine
η(z) := DL
(1 + z)2DA
.
C i e ion: I η(z)≡1, and unde NPE budge closu e and mo he scale in a iance he e
a e no new singula i ies/spu ious peaks, hen i belongs o gauge laye consis ency; i
η(z)= 1 o new singula i ies/spu ious peaks appea , i poin s o ue dynamics/new
physics (such as op ical dep h, non-me ic e ec s, o pho on non-conse a ion) [12].
7 RCA/QCA Seman ics: Ligh Cone, Redshi , and c-Limi ed
Alloca ion
De ini ion 7.1 (Causal Cone and “Ligh ”).Local e e sible upda e la ice dynamics sa is ying
Lieb–Robinson bounds induce an e ec i e “ligh cone”; he minimal no a ion low sa u a ing
his bound is called “ligh ” [13].
P oposi ion 7.2 (Disc e e Fo mula ion o Redshi ).Timing wi h mas e equency clk( ) =
c
R( )ℓ∗, he disc e e pe iod o he same symbol s eam obse ed sa is ies
P0= (1 + z)Pe, ν0=νe/(1 + z),
i.e., cosmological edshi ’s disc e e ime dila ion, consis en wi h he con inuous o mula ion.
(Di ec disc e iza ion o he uni ied equency shi law om
§
2.2.)
P oposi ion 7.3 (c-Limi ed Alloca ion o A en ion/Ac ion).Le esou ce densi y– lux pai
(ρ, J)sa is y conse a ion ∂ ρ+∇· J=sand cons ain |J| ≤ c ρ; hen in luence can p opaga e
wi hin he causal cone only a speeds no exceeding c; his “scheduling ligh speed” is consis en
wi h he Lieb–Robinson eloci y [14].
8 In o ma ion Bounda y and Veloci y Limi
P oposi ion 8.1 (P ocessing Ra e Uppe Bound: Quan um Speed Limi ).The Mandels am–
Tamm and Ma golus–Le i in bounds gi e he sho es e olu ion ime and maximum s a e change
a e; hus unde gi en ene gy/powe budge , any “ esolu ion enhancemen –p ocessing a e” is
limi ed by hem, no elaxed by scale gauge choice [15].
9 Ope a ional P o ocol o Obse a ion–Recons uc ion–Duali y
Consis ency
P o ocol A (Mo he Scale T iple Closu e): Fo he same objec , compu e simul aneously
φ′(E)/(2π), (2π)−1 Q(E), ρ(E), equi ing cu e and di ec ional pole consis ency o e i y
calib a ion uni y and Bi man–K e˘ın–Wigne –Smi h mu ual e i ica ion [1].
P o ocol B (NPE Budge ): Fo each da a pipeline, epo “alias = 0/= 0, EM o de
p, ail bound”; unde Nyquis sa is ac ion and speci ied noise/no maliza ion, ↓ educes bias
bu a iance ypically inc eases (bandwid h op imiza ion needed); no new singula i ies,
no spu ious peaks gua an eed by
§
4.3 “non-inc easing singula i y” and alias/EM budge [2].
P o ocol C (Geome ic Duali y Check): Cons uc η(z) = DL/[(1 + z)2DA] and pe -
o m Tolman exponen eg ession (expec ing n= 4) as “gauge s. dynamics” consis ency
e idence [7].
5
P o ocol D (Resolu ion Alloca ion Ma ix): In ime/ equency/angle/scale–phase
coo dina es, ake
M⋆= a g max
M⪰0, M=χM, ∇ I ∇ I⊤,
whe e χ > 0 is a ixed esou ce budge cons an (independen o κ( ) = ˙a/a), = ( , ω, ϑ, s)
collec s ime/ equency/angle/scale–phase coo dina es ( ailo able by ask). Repo Fishe in-
o ma ion gain and condi ion numbe imp o emen .
10 Minimal Su iciency o he Theo y
1. Scale gauge a=R−1:Uni ies “ex e nal expansion” and “in e nal esolu ion enhance-
men ” as he same Mellin dila ion, wi hou changing in insic singula i ies.
2. Mo he scale calib a ion: Uni ies eadou ia ρ=−ξ′=1
2π Q=1
2πi ∂Elog de S=φ′
2π,
c oss-de ice compa able [1].
3. NPE ini e-o de closu e: Closes e o budge wi h Poisson–EM ini e-o de discipline;
Nyquis elimina es aliasing [3].
4. F ame h eshold: Landau necessa y densi y and Wexle –Raz duali y ensu e s able in-
e ible econs uc ion [16].
5. Rela i is ic consis ency: S abilize o ligh cone + mo he scale yields Lo en z g oup;
in FRW, uni ied equency shi law, E he ing on, and Tolman na u ally hold [8].
11 Appendix: Co espondence o S anda d Resul s wi h This
Pape ’s S uc u e
Wigne –Smi h g oup delay and “densi y–phase de i a i e” iple equi alence:
G oup delay ma ix de ini ion and expe imen al measu abili y, and he Bi man–K e˘ın
ela ion be ween de Sand spec al shi unc ion, suppo mo he scale calib a ion [17].
Co a ian o mula ion o edshi : ω=−kµuµand 1 + z= ((kµuµ)e)/((kµuµ)o);
s anda d composi ion and duali y o cosmological dis ance measu es [9].
Tolman (1 + z)−4and duali y es : Obse a ional calib a ion and me hodological
guidance [18].
Alexand o –Zeeman heo em: Causal s uc u e de e mines (up o global dila ion)
Lo en z–Poinca ´e g oup; emo ing global dila ion yields Lo en z g oup [8].
Landau densi y, Wexle –Raz, Balian–Low: Th ee-poin balance o s able sampling–
duali y–impossibili y o simul aneous localiza ion [5].
Lieb–Robinson and QCA: E ec i e “ligh speed” on la ice and causal cone o e-
e sible upda es [13].
Quan um speed limi : Resolu ion enhancemen and p ocessing a e uni o mly con-
s ained by MT/ML- ype bounds [15].
6
P oo Appendix (Selec ed)
A. Bi man–K e˘ın–G oup Delay–Phase De i a i e T ini y
Le S(E) be a uni a y sca e ing ma ix. By Smi h’s de ini ion Q(E) = −i S†S′,
Q=−i (S†S′) = −i ∂Elog de S ,
using ∂Elog de S= (S−1S′) = (S†S′) (since Sis uni a y). By de S(E) = e−2πiξ(E)and
Q=−i ∂Elog de S, we ge 1
2π Q=−ξ′(E). Also φ(E) = a g de S(E) = −2πξ(E), hence
φ′(E) = −2πξ′(E). Thus
ρ(E) = −ξ′(E) = 1
2π Q(E) = 1
2πi∂Elog de S(E) = φ′(E)
2π.
[1]
B. Geome ic O igin o E he ing on and Tolman
Unde unique null geodesics, pho on numbe conse a ion, and me ic g a i y, ans o ma ion
o angula a ea elemen and in insic luminosi y yields DL= (1 + z)2DA; combining he (1 +
z)−1×(1 + z)−1 ac o o pho on ene gy/a i al a e pe uni ime–uni a ea lux wi h he
(1 + z)−2scaling o isual angle a ea, we ob ain Tolman su ace b igh ness decay (1 + z)−4[7].
C. Alexand o –Zeeman S abilize o Lo en z G oup
A e ixing he o igin, linea maps p ese ing causal o de a e gene a ed by R+×SO+(1,3);
in oking mo he scale in a iance emo es global dila ion, lea ing SO+(1,3) [8].
D. Landau–Wexle –Raz–Balian–Low F ame T iangle
Landau lowe bound gi es necessa y sampling poin densi y; Wexle –Raz cha ac e izes dual
windows making econs uc ion ope a o iden i y; Balian–Low decla es o hono mal bases a
c i ical densi y canno be simul aneously well-localized, hence enginee ing uses edundan igh
ames [16].
Tooling De ini ions and Symbol Index
a( ): scale ac o ; R( ) = a( )−1: in e nal me ic; κ= ˙a/a.
S(E), Q(E), φ(E), ρ(E): ini y mo he scale objec s [1].
NPE: Nyquis (aliasing accoun /cu o )–Poisson (summa ion b idge)–Eule –Maclau in
( ini e-o de Be noulli laye s and ail) [2].
F ame densi y and duali y: Landau necessa y densi y, Wexle –Raz duali y, Balian–Low
es ic ion [5].
Uni ied equency shi : 1 + z= ((kµuµ)e)/((kµuµ)o) [9].
Re e ences
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7
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[13] Lieb–Robinson bounds. h ps://en.wikipedia.o g/wiki/Lieb-Robinson_bounds
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052108_1_online.pd
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//chaosbook.o g/lib a y/Wigne Delay55.pd
[18] The Tolman Su ace B igh ness Tes o he Reali y o he Expansion. h ps://a xi .
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8
